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+[    {        "title": "0704.3218v2.Density_of_bulk_trap_states_in_organic_semiconductor_crystals__discrete_levels_induced_by_oxygen_in_rubrene.pdf",        "content": "arXiv:0704.3218v2  [cond-mat.mtrl-sci]  19 Jun 2007Density of bulk trap states in organic semiconductor crysta ls:\ndiscrete levels induced by oxygen in rubrene\nC. Krellner,∗S. Haas, C. Goldmann,†K. P. Pernstich, D. J. Gundlach,‡and B. Batlogg§\nLaboratory for Solid State Physics, ETH Zurich, CH-8093 Zur ich, Switzerland\n(Dated: October 26, 2018)\nAbstract\nThe density of trap states in the bandgap of semiconducting o rganic single crystals has been\nmeasured quantitatively and with high energy resolution by means of the experimental method of\ntemperature-dependent space-charge-limited-current sp ectroscopy (TDSCLC). This spectroscopy\nhas been applied to study bulk rubrene single crystals, whic h are shown by this technique to be\nof high chemical and structural quality. A density of deep tr ap states as low as ∼1015cm−3is\nmeasured in the purest crystals, and the exponentially vary ing shallow trap density near the band\nedge could be identified (one decade in the density of states p er∼25meV). Furthermore, we have\ninduced and spectroscopically identified an oxygen-relate d sharp hole bulk trap state at 0.27 eV\nabove the valence band.\n1I. INTRODUCTION\nThe performance of organic thin-film transistors (OTFTs) is stead ily improving, and\nthe charge carrier mobility, as a key figure of merit, has reached va lues comparable to\nthat of hydrogenated amorphous silicon.1,2,3,4,5As with most semiconductors, the electrical\nperformance is determined to a high degree by modifications of the id eal crystal, such as\nintentional doping and other chemical or structural effects which create electrically active\nstates in the band gap. For a thorough understanding of the intrin sic capabilities and\nlimitations of organic semiconductors, it is now highly desirable to quan titatively study\nsuch in-gap states, their density of states (DOS) spectrum, the ir origin, and their stability.\nField-effect transistors (FETs) on organic single crystals are well s uited for determining\nthe surface properties, with mobilities near 20cm2/V s routinely achieved (for a review see\nGershenson et al.6). Bulk properties, however, cannot be determined with this surfa ce-\nsensitive method and thus one has to use alternative techniques, s uch as the time-of-flight\nmethod as shown in the pioneering work by Karl and co-workers,7thermally stimulated\ncurrent, or space-charge-limited-current (SCLC) measuremen ts. For instance, SCLC mea-\nsurements with coplanar electrodes were used by Lang et al.8to detect a metastable trap\nstate in pentacene single crystals, where the SCLC changes over s everal orders of magnitude,\nindicating trap filling.\nAdditional experimental techniques are used to detect defect st ates in organic semicon-\nductors: Recently, oxygen-related states at the surface of na turally oxidized rubrene single\ncrystals were detected by photoluminescene measurements.9Kelvin-probe force microscopy\nnot only offers a direct imaging of the potential across the channel of an OTFT, but addi-\ntionally allows one to extract the DOS spectrum at the semiconducto r/insulator interface.10\nInthisstudy we go beyond thebasicconcept of SCLC andusethe sp ectroscopic character\nof temperature-dependent SCLC spectroscopy (TDSCLC) as de scribed in the theory by\nSchaueret al.11to derive the bulk DOS spectrum in ultrapure rubrene single crystals . We\nfound that these crystals of high structural and chemical quality have a broad distribution of\ndeep states with a low total density of trap states, in addition to a s teep band tail. We also\nuse the spectroscopic technique to identify the energetic position of oxygen-induced bulk\ntrap states in rubrene single crystals.\n2II. METHOD\nA central aspect of TDSCLC is to exploit the spectroscopic charac ter inherent in the\ntemperature dependence of the SCLC due to the energy window as sociated with the Fermi-\nDirac statistics.11It is assumed that the SCLC is dominated by the charge carriers tha t are\nthermally excited from a localized trap into delocalized band states. T herefore the valence\nband edge is seen as separating localized and delocalized states, and it is chosen as the\nreference point for the energy scale ( Ev= 0), with positive energy toward the midgap. The\nbasic equations are Ohm’s law in the form j=eµ0nf(x)F(x) and the Poisson equation\ndF/dx=−ens(x)/(ǫǫ0). HereF(x) is the electric field strength along the direction xof\ncurrent flow, jis the current density, ǫǫ0is the product of the dielectric constant and electric\npermittivity (3.5 for rubrene), µ0is the microscopic band mobility, nf(x) is the density of\nfree carriers, ns(x) is the total density of carriers (free and trapped), which is given by\nthe convolution of the density of trap states h(E) in the energy gap with the Fermi-Dirac\nfunction f(E,EF,T), i.e.,ns=/integraltext\nEh(E)f(E,EF,T)dE. The shape of the current-voltage\ncharacteristic j(U) reflects the increment of the space charge with respect to the s hift of the\nFermi energy and thus mirrors the energy dependence of the DOS ,\ndns\ndEF=1\nkBTǫǫ0\neL2(2m−1)\nm2(1+C) (1)\nwith\nC=B(2m−1)+B2(3m−2)+d[ln(1+B)]/dlnU\n1+B(m−1). (2)\nHereListhethickness ofthecrystal withelectrodesonoppositefaces, Utheappliedvoltage,\nkBthe Boltzmann constant, m=dlnj/dlnUthe logarithmic slope of the j(U) curve, and B\ncontains higher-order derivatives of j(U),B=−[dm/dlnU]/[m(m−1)(2m−1)]. The right\nhand side of Eq. (1) can be calculated from the current-voltage ch aracteristic measured at\nonly one temperature. For a complete reconstruction of the DOS, however, it is necessary\nto relate a given voltage Uto the energy of states which are being filled at this value of U.\nA first starting point is to extract an activation energy EAfrom the Arrhenius plot of the\nTDSCLC data for a given U, i.e.EA=−dlnj/d(kBT)−1. Because h(E) in general is not\nsymmetric around EF,12the energy EDof the incremental change of space charge is slightly\nshifted from EA, typically by a fraction of kBT13\nED=EA+(3−4m)n\n(2m−1)(m−1)mkBT. (3)\n3Heren=−d(EA/kBT)/dlnUis the derivative of the activation energy with respect to the\napplied voltage. To extract the DOS from the shape of the j(U) curves, it is necessary to\ndeconvolute Eq. (1) with respect to df/dE F, using a high accuracy deconvolution method\nbased on spline functions.14\nIII. EXPERIMENTAL DETAILS\nTherubrenecrystalsweregrownbyphysicalvaportransport15underastreamofultrapure\nArgon 6.0. The starting material (Aldrich purum) was sublimed three times in vacuum.\nConsiderable effort was made to avoid contamination with any organic substances, e.g.,\nby using glass tubes cleaned in acids. Typical crystals are platelike, w here the direction\nperpendicular to the surface corresponds to the long axis of the o rthorhombic unit cell.16\nWe note that these high-quality rubrene single crystals show in-plan e field-effect mobilities\nat the surface of up to 10 cm2/V s.20\nAn optimized sample preparation method was found in a slightly adapte d “flip-crystal”\napproach,21benefiting from the minimized sample handling. The thin crystals (pref erred\nthickness <2µm) are placed on glass substrates with 20nm Au on 5nm Cr electrode s tripes,\nwhere they stick by electrostatic adhesion. A 20nm gold top electro de is then evaporated\nonto the crystal, which is slightly cooled during the evaporation ( Tmask=−8◦C) to minimize\nthermal damage. The overlap oftheelectrodes results inatypical measurement crosssection\nofA∼2·10−5cm2. Finally, electrical connections to a sample holder were made with silve r\nepoxy and 25- µm-thick gold wires. The thickness of the crystals was measured by a tomic\nforce microscopy, as optical inspection turned out to be unreliable for the ultrathin crystals.\nThe electrical measurements were performed in a closed-cycle cry ostat in inert helium\natmosphere and in darkness, covering a maximal temperature ran ge of 30–350K. By the use\nof helium exchange gas, a constant and reliable sample temperature , measured at the back\nsideofthesampleholder, couldbeadjustedwitharesistiveheaterc oil(drivenbyaLakeshore\n331 controller). A Keithley 6517A electrometer was used for the ele ctrical measurements.\nBy a proper shielding, leakage currents well below 1pA at 100V were a chieved. To inject\nholes from the bottom contact, a negative voltage was applied at th e top contact; V >0\nresults in hole injection from the top contact. The superiority of lam inated vs evaporated\ncontacts was reported previously.22,23\n4The typical measurement procedure for TDSCLC was as follows. Fir st, initial tests at\nroom temperature were performed in order to check the reprodu cibility of repeated mea-\nsurements of the j(U) characteristic. Thereafter the sample was cooled to typically 100 K\nat a rate of 2K/min. Prior to the measurement of the current-volt age characteristic (at\neach chosen temperature), an initial delay of 20min ensured therm al equilibrium. The\nj(U) curves were measured by a stepwise increase of the applied voltag e and measuring the\nquasiequilibrium current. A current measurement delay of 10s turn ed out to be sufficiently\nlong compared to the settling time of the system. Typically 50 points w ere measured per\nvoltage decade. The maximal voltage was limited such that the curre nt density would not\nexceed 0.1A/cm2in order to avoid crystal damage.\nUsuallyj(U) was measured every 10K between 100 and 200K. At lower tempera tures\nit is difficult to get accurate j(U) curves, because the current increases extremely rapidly\nwith voltage. At higher temperatures, the broadening of the Ferm i statistics has an adverse\ninfluence on the spectroscopic character of TDSCLC. In particula r, shallow states can only\nbe reliably measured at low temperature.\nThe issue of possible long-term charge trapping as result of a j(U) sweep needs careful\nattention during the course of a measurement. In most crystals, subsequent sweeps at the\nsame low temperature yield identical curves. This indicates a detrap ping of the charge on\na time scale faster than that of a sweep ( ∼10 min). For a few samples, however, it was\nnecessary to warm the crystal to room temperature after ever y sweep in order to restore the\ninitial condition. Subsequent j(U) sweeps at the same low temperature then yield the same\nresults. If charge trapping or sample deterioration influence the m easurement, the current\nat a given fixed voltage will not be thermally activated (as opposed to the Arrhenius plot in\nFig. 2). Therefore, the measurement itself is a valuable self-consis tency test.\n5/s48/s46/s49 /s49 /s49/s48/s49/s48/s45/s49/s48/s49/s48/s45/s56/s49/s48/s45/s54/s49/s48/s45/s52/s49/s48/s45/s50\n/s48/s46/s48/s49 /s48/s46/s49 /s49/s49/s48/s45/s54/s49/s48/s45/s52/s49/s48/s45/s50\n/s84 /s32/s61/s32/s51/s48/s48/s32/s75/s82/s117/s98/s114/s101/s110/s101\n/s115/s105/s110/s103/s108/s101/s32/s99/s114/s121/s115/s116/s97/s108\n/s84 /s32/s61/s32/s50/s48/s48/s32/s75/s84 /s32/s61/s32/s49/s49/s48/s32/s75\n/s32/s32/s106/s32/s91/s65/s47/s99/s109/s50\n/s93\n/s85 /s32/s91/s86/s93\n/s32/s32\n/s32/s32\nFIG. 1: Space-charge-limited-current (SCLC) density vs ap plied voltage at different temperatures\nfor a rubrene single crystal (Ru65-2, L= 0.6µm). The temperature step is 10 K. The inset\nshowsj(U) at 300 K. The straight line indicates the Ohmic behavior of t hermally generated charge\ncarriers.\nIV. EXPERIMENTAL RESULTS\nA. DOS in rubrene single crystals\nIn Fig. 1, the j(U) curves at different temperatures are shown, measured perpen dicular\nto the molecule layers in a rubrene single crystal (Ru65-2, L= 0.6µm). The “Ohmic”\nregion at low voltages is indicated by a straight line (see inset of Fig. 1) . The onset of SCLC\nat∼0.1V indicates that holes are effectively injected from the laminated go ld contact. At\nlower temperatures the number of thermally excited carriers decr eases exponentially and the\nOhmic region disappears below the sensitivity of the measurement se tup (∼10−14A).\nIn Fig. 2 the Arrhenius plots of the current density are shown for t he same data set as\nin Fig. 1. For clarity we show only data points for selected voltages. T he slope yields the\nactivation energy EA(U) for each applied voltage Uand we note the high quality of these\nplots. The resulting EA(U) is given in the inset and is in agreement with recent FET data24.\nThe effective Fermi energy is moved from ∼0.45eV at the lowest voltage to ∼0.1eV at the\nhighest injection voltage. The distinct change of slope near 0.27eV is worth noting, as it\nobviously reflects amarked increase ofthetrapdensity. Wenotet hat, amongotherevidence,\nthe smooth variation of EAat lowUindicates space-charge-limited-transport rather than\n6/s53 /s54 /s55 /s56 /s57/s49/s48/s45/s56/s49/s48/s45/s54/s49/s48/s45/s52/s49/s48/s45/s50/s49/s48/s48\n/s49 /s49/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s48/s46/s50/s54/s48/s46/s52/s50/s48/s46/s55/s50/s49/s46/s54/s82/s117/s35/s54/s53/s45/s50\n/s32/s32/s32/s106/s32/s91/s65/s47/s99/s109/s50\n/s93\n/s49/s48/s48/s48/s47 /s84 /s32/s91/s75/s45/s49\n/s93/s85 /s91/s86/s93\n/s51/s46/s49/s69\n/s65/s32/s40 /s85 /s41\n/s32/s85 /s32/s91/s86/s93/s32/s69\n/s65/s32/s91/s101/s86/s93\nFIG. 2: Current density for a fixed applied voltage Uvs the inverse temperature for the data\nplotted in Fig. 1. The straight lines are the Arrhenius fits us ed to determine the activation energy\nEA(U). The inset shows the resulting EA(U).\ntransport limited by contacts.25\nThe analysis described above is used to extract the density of trap states from these\nTDSCLC measurements. In order to calculate higher-order deriva tives of the experimental\nj(U) andEA(U) curves according to Eq.(1), a smoothing spline fit was applied to th e\nmeasured data, keeping the fit within 1% of the raw data. The result ing density of trap\nstates after the deconvolution is shown in Fig. 3. The edge of the va lence band is used as\nthe reference level and the positive energy axis points to the cent er of the band gap. Three\nmain features can be discerned. (1) An exponential increase of th e DOS toward the band is\nobserved for all rubrene single crystals. However, the characte ristic energy kBTtover which\nthe DOS is reduced by a factor of evaries from sample to sample. The sample Ru52-3\nhas a broad exponential distribution with kBTt= 210meV and the highest density of trap\nstates. Broad tail states with kBTt= 180meV were recently reported for pentacene single\ncrystals26. The sample Ru65-1 with the lowest deep trap density of order 1015cm−1eV−1in\nthe energy range from 0.45 to 0.1eV has a characteristic exponent ial distribution parameter\nofkBTt= 180meV. (2) Of particular interest in this sample are the shallow tra p states\nbelow∼0.1eV with a steeper slope ( kBTt= 11meV), reminiscent of band tail states. Due\nto the large increase of the trap density the quasi Fermi level is pin ned close to the band\nedge, and the amount of injected charge cannot fill these shallow t ail states. In this energy\n7/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s49/s48/s49/s53/s49/s48/s49/s54/s49/s48/s49/s55/s49/s48/s49/s56\n/s82/s117/s35/s53/s50/s45/s51\n/s82/s117/s35/s54/s53/s45/s49/s82/s117/s35/s54/s53/s45/s50/s82/s117/s35/s53/s50/s45/s50\n/s32/s32/s68/s79/s83 /s32/s91/s99/s109/s45/s51\n/s101/s86/s45/s49\n/s93\n/s69 /s32/s91/s101/s86/s93\nFIG. 3: (Color online) Density of states above the valence ba nd for four rubrene single crystals.\nrange the charge transport is still activated ( EA∼0.05eV). (3) The fine structure of the\nDOS indicates small features for all crystals, due to discrete trap levels in the band gap.\nThe observed fine structure was confirmed using the energy depe ndence of the statistical\nshift.11\nIncomparison, data of organicthin films show a very similar qualitative DOSspectrum,10\nexhibiting tail states and exponentially increasing deep states, albe it at much higher con-\ncentrations than in the purest rubrene single crystals. Again, the FET geometry in Ref. 10\nmight have emphasized defects near the interface, because FET d evices made from the same\ntype of high quality rubrene single crystals show approximately thre e orders of magnitude\nhigher interface trap density than the bulk trap density presente d here.27\nIt is remarkable that the representative selection of rubrene cry stals vary in their DOS,\nalthough grown under basically identical conditions.28This variation may originate from\nindividual micro-conditions during and after crystal growth, i.e., diff erent actual growth\ntemperature, growth rate, (thermo)mechanical strain, and diff erent atmospheric conditions\nin the device fabrication process. On the other hand, measuremen ts of different cross sec-\ntions on the samecrystal result in virtually identical DOS spectra, which is a convincing\nverification of the data evaluation.\n8/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s49/s48/s49/s53/s49/s48/s49/s55/s49/s48/s49/s57/s49/s48/s49/s53/s49/s48/s49/s55/s49/s48/s49/s57/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53\n/s82/s117/s35/s55/s49/s45/s52\n/s110/s111/s116/s32/s105/s108/s108/s117/s109/s105/s110/s97/s116/s101/s100/s82/s117/s35/s54/s53/s45/s50\n/s105/s108/s108/s117/s109/s105/s110/s97/s116/s101/s100\n/s32/s82/s117/s98/s114/s101/s110/s101/s32\n/s32/s82/s117/s98/s114/s101/s110/s101/s32/s43/s49\n/s79\n/s50/s32/s43/s32/s79\n/s51\n/s32/s68/s79/s83 /s32/s91/s99/s109/s45/s51\n/s101/s86/s45/s49\n/s93\n/s69 /s32/s91/s101/s86/s93/s32\n/s32/s82/s117/s98/s114/s101/s110/s101/s32\n/s32/s82/s117/s98/s114/s101/s110/s101/s32/s43/s49\n/s79\n/s50\n/s32/s32\nFIG. 4: (Color online) Density of trap states in rubrene sing le crystals before and after exposure\nto1O2. The oxygen-induced defect acts as a hole trap at an energy of 0.27eV above the valence\nband edge. Sample Ru65-2 was illuminated in an oxygen atmosp here to form the1O2directly at\nthe rubrene crystal surface. Sample Ru71-4 was exposed to ox ygen excited by uv light.\nB. Oxygen-induced bulk trap states\nIn order to understand the role of bulk traps in organic single cryst als and to demonstrate\nthe power of TDSCLC spectroscopy, we investigated the influence of oxygen on the trap\ndensity of rubrene single crystals. It is well known that the reactio n of rubrene with singlet\noxygen1O2formstheendoperoxide.29Rubreneitselfcanactasthesensitizer whenthetriplet\nstate of rubrene is populated by an intersystem crossing from the singlet state excited with\nvisible light. The energy of this triplet state at 1.2 eV above the groun d state is transferred\nto the molecular oxygen resulting in1O2which reacts with the rubrene molecule.\nAfter measuring the full DOS in the as-grown crystals (open symbo ls in Fig. 4), we\nilluminated the sample Ru65-2 with visible light under oxygen atmospher e for four hours.\n9For comparison the sample Ru71-4 was directly exposed for four ho urs to1O2by exciting\nmolecular oxygen with uv light in the vicinity of the sample; the sample its elf was held\nin the dark. The exposure of rubrene to1O2results in a large peak in the density of\ntrap states at 0.27 eV above the valence band (filled symbols in Fig. 4) . The amount of\noxygen-induced trap states in sample Ru65-2 is estimated to be Nox\nt≈2·1017cm−3, which\ncorresponds to ∼100 ppm. Because the sample was illuminated in an oxygen atmosphere\nwithout removing from the cryostat we can exclude origins other th an oxygen for this hole\ntrap. We note that the energetic position of the identified oxygen t rap state is in agreement\nwith recently published photoluminescence measurements on oxydiz ed rubrene.9In sample\nRu71-4the oxygen-induced trapstates arewithin themeasureme nt error at thesame energy,\nbut the density of the trap states is slightly higher than for the illumin ated sample ( Nox\nt≈\n3·1017cm−3). In addition a shoulder appears in the energetic distribution on the side closer\nto the valence band. This difference may be due to the formation of O 3under uv light which\nmight also react with the rubrene single crystal.\nThe trap level at E= 0.27eV already exists in the pristine rubrene single crystals, with\nmuch lower concentrations. This is due to the device fabrication pro cess; the samples are\nhandled in room air under microscope illumination, which results in a similar reaction as\nfor the illuminated sample Ru65-2. The oxygen-related trap level wa s also observed in other\nrubrene samples with various concentrations, probably as a result of a different exposure\ntime to air during handling.\nV. CONCLUSIONS\nIn conclusion, we have implemented temperature-dependent spac e-charge-limited-current\nspectroscopy and demonstrate it to be a powerful tool to quant itatively measure the density\nof bulk trap states with high energy resolution. Applied to high-qualit y rubrene crystals\nthis method reveals the existence of states within ∼0.1 eV of the band edge, reminiscent of\nband tails, and a smooth distribution of states deeper in the gap. Dis crete peaks are also\nobserved, and through a controlled exposure of the crystals to a ctivated oxygen, a distinct\nand stable trap level at 0.27eV has been created. 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Jpn.\n52, 380 (1979).\n12"    },    {        "title": "0705.1738v1.Equation_of_state_of_isospin_asymmetric_nuclear_matter_in_relativistic_mean_field_models_with_chiral_limits.pdf",        "content": "arXiv:0705.1738v1  [nucl-th]  12 May 2007Equation of state of isospin-asymmetric nuclear matter in\nrelativistic mean-field models with chiral limits\nWei-Zhou Jiang1,2, Bao-An Li1, and Lie-Wen Chen1,3\n1Department of Physics, Texas A&M University-Commerce, Com merce, TX 75429, USA\n2Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China\n3Institute of Theoretical Physics, Shanghai Jiao Tong Unive rsity, Shanghai 200240, China\nAbstract\nUsing in-medium hadron properties according to the Brown-Rho sca ling due to the chiral\nsymmetryrestorationat highdensities and consideringnaturalnes softhe couplingconstants,\nwe have newly constructed several relativistic mean-field Lagrang ianswith chiral limits. The\nmodelparametersareadjustedsuchthat thesymmetricpartof theresultingequationofstate\natsupra-normaldensitiesisconsistentwiththatrequiredbythec ollectiveflowdatafromhigh\nenergy heavy-ion reactions, while the resulting density dependenc e of the symmetry energy\nat sub-saturation densities agrees with that extracted from the recent isospin diffusion data\nfrom intermediate energy heavy-ion reactions. The resulting equa tions of state have the\nspecial feature of being soft at intermediate densities but stiff at h igh densities naturally.\nWith these constrained equations of state, it is found that the rad ius of a 1.4M⊙canonical\nneutron star is in the range of 11.9 km ≤R≤13.1 km, and the maximum neutron star mass\nis around 2.0 M⊙close to the recent observations.\nKeywords: Equation of state, nuclear matter, symmetry energy , relativistic mean-field\nmodels, chiral limits. PACS numbers: 21.65.+f, 26.60.+c, 11.30.Rd\n1 Introduction\nThe Equation of State (EOS) of isospin asymmetric nuclear ma tter plays a crucial role in many\nimportant issues in astrophysics, see, e.g., Refs. [1, 2, 3] . It is also important for understanding\nboth the structure of exotic nuclei and the reaction dynamic s of heavy-ion collisions, see, e.g.,\nRef. [4]. Within the parabolic approximation, the energy pe r nucleon in isospin asymmetric\nnuclear matter can be written as E/A=e(ρ)+Esym(ρ)δ2wheree(ρ) is the EOS of symmetric\nnuclearmatter, the Esym(ρ)isthesymmetryenergyand δ= (ρn−ρp)/ρistheisospinasymmetry.\nBoth thee(ρ) and theEsym(ρ) are important in astrophysics although maybe for different i ssues.\n1For instance, themaximum mass of neutronstars is mainly det ermined by theEOSof symmetric\nnuclear matter e(ρ) while the radii and cooling mechanisms of neutron stars are determined\ninstead mainly by the symmetry energy Esym[1, 5]. The nuclear physics community has been\ntrying to constrain the EOS of symmetric nuclear matter usin g terrestrial nuclear experiments\nfor more than three decades, see, e.g.[6], for a review. On th e other hand, a similarly systematic\nandsophisticated studyonthedensitydependenceofthesym metryenergy Esymusingheavy-ion\nreactions only started about ten years ago stimulated mostl y by the progress and availability of\nradioactive beams[7]. Compared to our current knowledge ab out the EOS of symmetric nuclear\nmatter, the symmetry energy Esymis still poorly known especially at supra-normal densities [8]\ngiven the recent progress in constraining it at densities le ss than about 1.2 ρ0using the isospin\ndiffusion data from heavy-ion reactions [9, 10, 11].\nThe aim of this work is to investigate the EOS of isospin asymm etric nuclear matter within\nthe relativistic mean-field (RMF) model with in-medium hadr on properties governed by the BR\nscaling. From the point of view of hadronic field theories, th e symmetry energy is governed\nby the isovector meson exchange. Studying in-medium proper ties of isovector mesons is thus\nof critical importance for understanding the density depen dence of the symmetry energy. We\nfirst construct model Lagrangians respecting the chiral sym metry restoration at high densities.\nThe model parameters are adjusted such that the symmetric pa rt of the resulting EOS at\nsupra-normal densities is consistent with that required by the collective flow data from high\nenergy heavy-ion reactions [6], while the resulting densit y dependence of the symmetry energy\nat sub-saturation densities agrees with that extracted fro m the recent isospin diffusion data\nfrom intermediate energy heavy-ion reactions [9, 10, 11]. T he constrained EOS is then used to\ninvestigate several global properties of neutron stars.\n2 Relativistic mean-field models with chiral limits\nIn-medium properties of the isovector meson ρcan be studied through the special symmetry\nbreaking and restoration. The local isospin symmetry in the Yang-Mills field theory, where the\nρmeson may be introduced as a gauge boson of the strong interac tion, can serve as a possible\ncandidatetostudythein-mediumpropertiesof ρmeson. However, since πNinteractionsactually\ndominate the strong interaction in hadron phase, it was rath er difficult to understand how the\nin-medium properties of the massive ρmeson could be consistent with the restoration of local\n2isospin symmetry. On the other hand, within the microscopic theory for the strong interaction,\nnamely, the QCD which is a color SU(3) gauge theory, the chira l symmetry is approximately\nconserved. The spontaneous chiral symmetry breaking and it s restoration can be manifested\nin effective QCD models. Based on the latter, Brown and Rho (BR) proposed the in-medium\nscaling law [12] implying that hadron masses and meson coupl ing constants in the Welacka\nmodel [13] approach zero at the chiral limit. The scaling was treated in the hadronic phase\nbefore the chiral symmetry restoration.\nAs an effective QCD field theory, the hidden local symmetry theo ry has been developed\nto include the ρmeson in addition to the pion in the framework of the chiral pe rturbative\ntheory by Harada and Yamawaki [14, 15] and it is shown that the ρmeson becomes massless\nat the chiral limit. This supports the mass dropping senario of the BR scaling. There are\nalso experimental indication for the mass dropping, i.e., t he dielectron mass spectra observed\nat the CERN SPS [16, 17], the ωmeson mass shift measured at the KEK [18] and the ELSA-\nBonn [19], as well as the analysis of the STAR data [20, 21]. Ho wever, data from the NA60\nCollaboration for the dimuon spectrum[22] seem to favor the explanation of ρmeson broadening\nbased on a many-body approach [23]. So far, the controversy i s still unsettled [21]. The chiral\nsymmetry and its spontaneous breaking are closely related t o the mass acquisition and dropping\nof hadrons. Since the chiral symmetry is a characteristic of the strong interaction within the\nQCD, it is favorable to include in the RMF models effects of the c hiral symmetry through\nthe BR scaling law. However, this does not mean that the contr ibution of the many-body\ncorrelations [24] is excluded. Actually, the contribution of the many-body correlations can be\nincluded phenomenologically into the RMF models to reprodu ce the saturation properties of\nnuclear matter.\nThe in-medium ρmeson plays an important role in modifying the density depen dence of the\nsymmetry energy. For most RMF models that the ρmeson mass is not modified by the medium,\nthe symmetry energy is almost linear in density. The introdu ction of the isoscalar-isovector\ncoupling in RMF models can soften the symmetry energy at high densities [3]. Meanwhile,\nit reproduces the neutron-skin thickness in208Pb as that given by the non-relativistic models\n(about 0.22fm) [25], consistent with the available data [26 ]. In well-fitted RMF models that\ngive a value of incompressibility κ= 230 MeV, a large coefficient of non-linear self-interacting\nωmeson term is required [25] and thus the naturalness breaks d own. Moreover, the isoscalar-\n3isovector coupling in the RMF models increases the effective ρmeson mass with density, which\nleads the model to be far away from the chiral limit.\nThe Walecka model with the density-dependent parameters is the simplest version to incor-\nporate the effects of chiral symmetry. The Lagrangian is writt en as\nL=ψ[iγµ∂µ−M∗+g∗\nσσ−g∗\nωγµωµ−g∗\nργµτ3bµ\n0]ψ+1\n2(∂µσ∂µσ−m∗2\nσσ2)\n−1\n4FµνFµν+1\n2m∗2\nωωµωµ−1\n4BµνBµν+1\n2m∗2\nρb0µbµ\n0 (1)\nwhereψ,σ,ω, andb0are the fields of the nucleon, scalar, vector, and isovector- vector mesons,\nwith their in-medium scaled masses M∗,m∗\nσ,m∗\nω, andm∗\nρ, respectively. The meson coupling\nconstants and hadron masses with asterisks denote the densi ty dependence, given by the BR\nscaling. The energy density and pressure read, respectivel y,\nE=1\n2C2\nωρ2+1\n2C2\nρρ2δ2+1\n2˜C2\nσ(m∗\nN−M∗)2+/summationdisplay\ni=p,n2\n(2π)3/integraldisplaykFi\n0d3k E∗, (2)\np=1\n2C2\nωρ2+1\n2C2\nρρ2δ2−1\n2˜C2\nσ(m∗\nN−M∗)2−Σ0ρ+1\n3/summationdisplay\ni=p,n2\n(2π)3/integraldisplaykFi\n0d3kk2\nE∗(3)\nwhereCω=g∗\nω/m∗\nω,Cρ=g∗\nρ/m∗\nρ,˜Cσ=m∗\nσ/g∗\nσ,E∗=/radicalBig\nk2+m∗\nN2withm∗\nN=M∗−g∗\nσσthe\neffective mass of nucleon, and kFis the Fermi momentum. The incompressibility of symmetric\nmatter can be expressed explicitly as [27]\nκ= 9ρ∂µ\n∂ρ= 9ρ(C2\nω+2Cωρ∂Cω\n∂ρ+∂EF\n∂ρ−∂Σ0\n∂ρ) (4)\nwhere the chemical potential is given by µ=∂E/∂ρand the Fermi energy is EF=/radicalBig\nk2\nF+m∗\nN2.\nThe rearrangement term is essential for the thermodynamic c onsistency to derive the pressure\nin (3) and its expectation value in the mean field Σ 0is given by\nΣ0=−ρ2Cω∂Cω\n∂ρ−ρ2δ2Cρ∂Cρ\n∂ρ−˜Cσ∂˜Cσ\n∂ρ(m∗\nN−M∗)2−ρs∂M∗\n∂ρ. (5)\nThe density dependence of parameters is usually described b y the scaling functions that are the\nratios of thein-mediumparameters tothoseinthefreespace . Thechoice of scaling functionsand\ntheir coefficients are constrained by thesaturation propert iesof nuclear matter andexperimental\ndataaboutthein-mediummass droppingofvector mesons. Mor eover, wealsouseasaconstraint\nthepressurewithin the density range 2 −4.6ρ0extracted from measurements of nuclear collective\nflows in heavy-ion collisions [6]. The scaling function may t ake the form [28]:\nΦ(ρ) =1\n1+yρ/ρ0(6)\n4withy= 0.28 for the vector meson mass, giving Φ( ρ0) = 0.78 found in QCD sum rules [29].\nRecently, in Ref. [30] where the memory effect in dimuon yield w as studied by considering the\nmass dropping of ρmeson, the authors cited a scaling function [31]\nΦ(ρ) = 1−yρ/ρ0 (7)\nwithy= 0.15. Data extracted from the γ−Areaction by the TAP collaboration indicate a value\nofy≈0.13 [32]. In addition, data from the KEK photon-induced nucle ar reaction indicate that\ntheωmeson mass dropping is about the same order of magnitude at th e saturation density [33].\nSong firstly built effective models based on the BR scaling usin g the scaling functions (6)\nfor both hadron masses and vector coupling constants [28]. A reasonable incompressibility for\nnuclear matter was obtained by introducing the non-linear s elf-interacting meson terms with\ncoefficients satisfying the hypothesis of naturalness that is originated from the chiral symmetry\nand QCD scaling [34, 35, 36, 37]. In Ref. [38], along the line o f [28], the BR scaling function\n(6) was considered only for hadron masses with a small value o fyin the RMF models to study\nnuclear matter properties. In a more recent work [39], the BR scaling function (6) was taken\nfor the scalar and vector meson coupling constants with resp ective values of y, and the scaling\nfunction (7) was taken for hadron masses in the RMF models wit hout the self-interacting meson\ninteractions. Thoughthemodelsbuilt intheseworkscan giv e rather gooddescriptions of nuclear\nsaturation properties, the pressures calculated in the den sity region of ρ= 2−4.6ρ0, however,\nare still far away from that extracted from measurements of n uclear collective flows in heavy-ion\ncollisions [6]. People have already tried to improve the sit uation by including the non-linear\nmeson self-interacting terms. Unfortunately, it is not sat isfactory because even unreasonably\nlarge coefficients of non-linear meson terms that break down t he hypothesis of naturalness can\nnot reduce the pressure lower enough to pass the experimenta l pressure-density region.\nThe effective nucleon mass is not dominated by the BR scaling (a t least at the normal\nnuclear matter density) since it is usually around 0 .65M∗at normal density while the mass\ndropping given by the BR scaling is less than 15% ( y≤0.15 in (7)). This implies that the\nscalar meson coupling constant that plays a crucial role in t he effective nucleon mass can adopt\na different density-dependent scaling from that of the vector meson. In particular, the high\npressure predicted by various models at high densities shou ld be lowered, and this requires the\nscalar coupling constant to decrease more slowly. Furtherm ore, if the scaling function (7) for\ntheωmeson mass is preferred by experiments, one may take the same scaling function (7) for\n5Table 1: Parameter sets fitted at the saturation density ρ0= 0.16fm−3. The vacuum hadron\nmasses are M= 938MeV, mσ= 500MeV, mω= 783MeV and mρ= 770MeV except for mσ=\n600MeV for the parameter set S3. The coupling constants give n here are those at zero density.\nFor parameter sets SL3 and S3, the non-linear σself-interacting coefficients are introduced (see\ntext). The parameter set SL1∗has two more parameters yρ= 0.654 andyω= 0.0365. The\nsymmetry energy is fitted to 31.6MeV at ρ= 0.16fm−3for all models. The critical density ρc\nfor the chiral symmetry restoration is given by the value yin (7) for zero hadron mass.\ngσgωgρy x κ (MeV)M∗/M M∗\nn/M ρ c/ρ0\nSL1 8.6388 10.4634 3.7875 0.126 0.234 230.0 1.0 0.679 7.94\nSL1∗9.7414 12.5535 5.8644 0.126 0.238 230.0 1.0 0.600 7.94\nSL2 6.1664 10.9682 3.9866 0.11 0.381 219.5 0.89 0.763 9.09\nSL3 9.8627 12.4928 3.6128 0.126 - 250.0 1.0 0.620 7.94\nS3[28] 5.3210 15.3134 3.6035 0.28 - 250.0 0.78 0.617 -\nthe coupling constant of ωmeson to avoid the infinity of pressure at the chiral limit whe re the\nscalar density vanishes. In this way, the effect of density dep endence from the vector meson part\nis cancelled out since the energy density (3) only relies on t he ratiosCωandCρ. Generally, we\nmay take the scaling functions for the coupling constants of vector mesons as\nΦρ(ρ) =1−yρ/ρ0\n1+yρρ/ρ0,Φω(ρ) =1−yρ/ρ0\n1+yωρ/ρ0. (8)\nFor hadron masses (including nucleons, if they have), (7) is taken with the same value of yused\nin (8). For the σmeson coupling constant, the same form as (6) is taken but wit h a coefficient\ndenoted by x:\nΦσ(ρ) =1\n1+xρ/ρ0. (9)\n3 Results and discussions\nWe first adjust the parameters to reproduce the saturation pr operties including the binding\nenergy per nucleon E/A−M=−16 MeV, the zero pressure, the incompressibility κand the\neffective nucleon mass m∗\nNat saturation density ρ0= 0.16fm−3. The resulting parameter sets\nSL1 and SL2 and the corresponding saturation properties are tabulated in Table 1. In SL1 the\nnucleon mass scaling is not considered and the effective nucle on mass is just m∗\nN=M−g∗\nσσ. In\nSL2, the nucleon mass scaling is included, and a much larger e ffective nucleon mass at normal\ndensity is obtained. There are just two coefficients xandyused in the scaling functions in SL1\nand SL2. Without the inclusion of the non-linear meson self- interacting terms, the saturation\n6110102103\n2 4 6 8FsuGoldSL1\nSL1*SL2SL3 S3\n ρB/ρ0 p (MeV/fm3)\nFigure 1: The pressure as a function of density for different mo dels. The shaded region is given\nby experimental error bars[6].\npropertiesat ρ0= 0.16fm−3(seeTable1) andthepressurewithinthedensity region ρ= 2−4.6ρ0\nare both nicely reproduced with the SL1 and SL2 (see Fig. 1). I t is worth mentioning that the\nvector potential which is quadratic in density for the const antCωis known to result in a higher\npressure above the experimental region. The softening of th e pressure here is attributed to the\ncontribution of rearrangement terms. For the SL1, only the r earrangement term from the σ\nmeson survives.\nThe parameter set SL3 does not consider the scaling of scalar meson coupling constant but\nincludes in the Lagrangian the non-linear σself-interacting terms U(σ) =g2σ3/3+g3σ4/4 with\ncoefficients g2= 23.095 andg3=−29.678. Though these large coefficients are needed to fit the\nsaturation properties, they seem to be inconsistent with th e hypothesis of naturalness [35, 36].\nIn S3, we introduce a g2=−1.096 to obtain the given κin Table 1. This is a little different\n7from the original one [28]. As a comparison, we can see in Fig. 1 that the SL3 and S3 parameter\nsets are not consistent with the pressure constrained by the collective flow data.\nWith (3), the symmetry energy in the RMF models can be derived as\nEsym=1\n2∂2(E/ρ)\n∂δ2=1\n2C2\nρρ+k2\nF\n6EF. (10)\nIt consists of contribution from the ρmeson (potential part) and nucleons (kinetic part). In\nSL1 and SL2, we adopt the same scaling function for coupling c onstants of ρandωmesons:\nΦρ(ρ) = Φω(ρ) = 1−yρ/ρ0. In this way, the ratio Cρis just a constant that does not rely on\nthe density. An almost linear dependence of the symmetry ene rgy on the density is expected\nfrom Eq.(10). The symmetry energy as a function of density is shown in Fig.2. In Refs. [10, 11],\nthe symmetry energy extracted from the isospin diffusion data is parameterized as Esym=\n31.6(ρ/ρ0)γwith 0.69≤γ≤1.05 . All the parameter sets based on the BR scaling discussed\nabove lead to the symmetry energies between those parameter ized withγ= 0.69 and 1.05 in the\nwhole density region. In Ref. [11], the authors pointed out t hat with the in-medium nucleon-\nnucleon cross sections, a symmetry energy of Esym(ρ) = 31.6(ρ/ρ0)0.69forρ<1.2ρ0was found\nmostacceptable comparedtotheisospindiffusiondata. Howev er, thesymmetryenergy athigher\ndensities is not constrained at all. It is thus interesting t o examine predictions within the RMF\nmodels with the chiral limit at high densities.\nOur strategy is to first fit the symmetry energies constrained at low densities by modifying\nthe coefficient yρin (8), and then predict the symmetry energy at high densitie s. The modified\nsymmetry energy is shown in Fig.2 with the dashed blue curve d enoted as SL1∗. Considering\nthe nucleon effective mass at normal density is a little large i n SL1, we introduce a coefficient\nyωin (8) to reduce it to the value of 0 .6M. The new parameter set is called SL1∗and also listed\nin Table 1. The different parameterizations in SL1 and SL1∗lead to significantly difference in\npressure at high densities as shown in Fig.1. However, the su ppression of the symmetry energy\nin SL1∗at high densities is dominated by the density dependence of t he ratioCρin (10). This\ncan be seen by comparing results shown in Table 1 and Fig.2 tha t the density dependence of\nthe symmetry energy does not change much by the different nucle on effective masses of various\nparameter sets. In the density-dependent RMF model [24], th e dropping of ratios like the Cωor\nCρis induced by the many-body correlation contributions. Tho ugh the mechanism is different\nfrom the present study, the correlation effect beyond the mean -field approximation may play\nsome role in obtaining the coefficients yρandyω.\n80100200300\n0 2 4 6MDI(x=0)\nMDI(x=-1)\nSL2\nSL3\nSL1*\nSL1\n ρB/ρ0 Esym (MeV) 020\n0 1ρ/ρ0Esym\nFigure 2: The symmetry energy as a function of density for diffe rent models. The MDI(x=0)\nand MDI(x=-1) results are taken from [10].\nThe symmetry energy at high densities from the MDI interacti on with x=0 is quite close\nto that given by the SL1∗. This is not surprising since the SL1∗is rendered to have the same\nsymmetry energies as the MDI(x=0) at low densities. On the ot her hand, this justifies the\nconsistency of the symmetry energy given by both models at hi gh densities. One can freely\nadjustyρto fit the symmetry energies given by the MDI(x=-1) model at lo w densities and then\nexamine the behavior at high densities. However, the Cρwill increase with density by doing\nso, which is contrary to the empirical cases [24, 25]. Theref ore, based on the BR scaling the\nsymmetry energy at high densities is expected to be between t hose given by the SL1 and the\nSL1∗.\nWe now turn to the astrophysical implications of the EOS’s co nstrained above. Generally\nspeaking, these EOS’s have the special features of being sof t at moderate densities and stiff at\n9012\n10 12 14SL3SL2\nSL1 SL1*\nSL1′R1.4\n11.9≤R1.4≤13.1\n R (km)M / MO⋅\nFigure 3: The neutron star mass versus the radius for various models. The parameter set SL1′\nis just the same as the SL1 but with yρ= 0.526.\nhigh densities (see Fig. 1). It is thus interesting to compar e predictions using these EOS’s with\ntherecent astrophysical observations. Recent studies on t he millisecond pulsarPSR J0751+1807\nsuggest that it has a mass of 2 .1±0.2(+0.4\n−0.5) with 1σ(2σ) confidence[40]. Since the maximum\nmass of neutron star is more sensitive to the EOS at high densi ties, this requires a rather stiff\nEOS at least at high densities. In Fig.3, we plot the mass-rad ius correlation of neutron stars\nobtained from solving the standard TOV equation. In the mode ls with chiral limits, hadron\nmasses approach zero at certain critical baryon densities ( see Table 1). The maximum central\nenergy density in a neutron star is that calculated at the max imum baryon density. With\nSL1, the maximum neutron star mass is 2.04 M⊙with a radius of R=9.89km. With SL2, these\ntwo observables are 2.17 M⊙and R=10.13km, respectively. As the nucleon effective mass at\nnormal density is reduced by introducing the parameter yω, the maximum neutron star mass\n10also drops. This is owing to that the introduction of yωsoftens the EOS at high densities and\nthe energy density at the critical density is thus lowered. W ith SL1∗, the nucleon effective\nmass isM∗\nn= 0.6Matρ0, and the maximum neutron star mass is 1 .7M⊙. For a moderate\nvalue ofM∗\nn= 0.65M, we can obtain a moderately larger neutron star mass 1 .94M⊙. Recent\nmeasurements on the neutron star EXO 07482-676 gave a mass of M= 2.10±0.28M⊙and a\nradius of 13 .8±1.8km [41]. The author indicated that the existence of such a ma ssive neutron\nstar rules out all the soft EOS’s of neutron-star matter. Amo ng the models studied here,\nonly the parameter set SL3 can give values close to the measur ement for the EXO 07482-676.\nHowever, the parameter set SL3 gives a pressure at ρ≤4.6ρ0that is too strong compared to\nthat constrained by the collective flow data in heavy-ion col lisions [6]. For the other parameter\nsets, the radii of maximum mass neutron stars are just around 10km, which is below that given\nin [41], though the maximum mass can be within the error bars o f the measurement (except for\nSL1∗).\nCompared to the EOS obtained using the FSUGold parameter set [25] the EOS’s in the\npresent study are stiffer at high densities although they have the same incompressibility κ= 230\nMeV at normal density (see Fig.1). The EOS’s obtained here th us also result in larger maximum\nmasses of neutron stars. In Ref. [5], the authors predicted a radius span of 11.5km <R<13.6km\nfor the 1.4M⊙canonical neutron star. Using the EOS’s obtained in the pres ent work, we obtain\nhere a radius span of 11.9km <R<13.1km for the 1.4 M⊙neutron star, which is comparable with\nthat obtained in Ref. [5].\n4 Summary\nIn summary, within the RMF framework we have constructed sev eral new model Lagrangians\nusingin-mediumhadronpropertiesaccordingtotheBrown-R hoscaling duetothechiral symme-\ntry restoration at high densities. The model parameters are determined such that the symmetric\npart of the resulting EOS’s at supra-normal densities are co nsistent with that required by the\ncollective flow data from high energy heavy-ion reactions, w hile the resulting density depen-\ndence of the symmetry energy at sub-saturation densities ag rees with that extracted from the\nrecent isospin diffusion data from intermediate energy heavy -ion reactions. The rearrangement\nterms are found to play an important role in softening the EOS at moderate densities. The\nsymmetry energy depends on the in-medium hadron properties that are characteristic of the\n11chiral symmetry restoration at the critical density accord ing to the BR scaling. The resulting\nEOS’s are then used to examine global properties of neutron s tars. It is found that the radius\nof a 1.4M⊙neutron star is in the range of 11.9km ≤R≤13.1km. Compared to other EOS’s, the\ncurrent EOS’s have the special feature of being soft at inter mediate densities but stiff at high\ndensities naturally. This feature is important to produce a heavier maximum neutron star mass\naround 2.0M⊙consistent with recent observations.\nAcknowledgement\nWe thank P. Krastev and G.C. Yong for useful discussions. The work was supported in part by\nthe US National Science Foundation under Grant No. PHY-0652 548, the Research Corporation,\nthe National Natural Science Foundation of China under Gran t Nos. 10405031, 10575071 and\n10675082, MOE of China under project NCET-05-0392, Shangha i Rising-Star Program under\nGrant No. 06QA14024, the SRF for ROCS, SEM of China, and the Kn owledge Innovation\nProject of the Chinese Academy of Sciences under Grant No. KJ XC3-SYW-N2.\nReferences\n[1]J. M. Lattimer and M. Prakash, Phys. Rep. 333, 121 (2000); Astrophys. J. 550, 426 (2001); Science\n304, 536 (2004).\n[2] A. W. Steiner, M. Prakash, J. M. Lattimer and P. J. Ellis, P hys. Rep. 411, 325 (2005).\n[3] C.J. Horowitz, J. Piekarewicz, Phys. Rev. Lett. 86, 5647 (2001).\n[4] Isospin Physics in Heavy-Ion Collisions at Intermediat e Energies, Eds. Bao-An Li and W.\nUdo Schr¨ oder (Nova Science Publishers, Inc, New York, 2001 ).\n[5] B. A. Li and A. W. Steiner, Phys. Lett. B 642, 436 (2006).\n[6] P. 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Yacoby1,3 \n \n1 Department of Condensed Matte r Physics, Weizmann Instit ute of Science, Rehovot \n76100, Israel. \n2 Max-Planck-Institut für Festkörper forschung, Heisenbergstrasse 1, D-70569 \nStuttgart, Germany.  \n3 Department of Physics, Harvard Un iversity, Cambridge, MA 02138, USA. \n  \nThe electronic density of states of graphe ne is equivalent to that of relativistic \nelectrons [1-3]. In the absence of diso rder or external dopi ng the Fermi energy \nlies at the Dirac point where the density of  states vanishes. Although transport \nmeasurements at high carrier densities indicate rather high mobilities [4-6], many questions pertaining to disorder [7 -14] remain unanswere d. In particular, \nit has been argued theoretically, that wh en the average carrier density is zero, \nthe inescapable presence of disorder will lead to electron and hole puddles with equal probability. In this work, we use a scanning single electron transistor to \nimage the carrier density landscape of gra phene in the vicinity of the neutrality \npoint. Our results clearly show the elec tron-hole puddles expected theoretically \n[13]. In addition, our measurement tec hnique enables to determine locally the \ndensity of states in graphene. In contrast to previously studied massive two dimensional electron systems, the kineti c contribution to the density of states \naccounts quantitatively for the measured  signal. Our results suggests that \nexchange and correlation effects are eith er weak or have canceling contributions . \n The kinetic energy of Dirac particles in gr aphene increases linearly with momentum \n[1,15,16]. The total energy per particle however , also referred to as the chemical \npotential,\nµ, contains additional exchange and co rrelations contribu tions that arise \nfrom the Coulomb interaction K ex c EEEµ=++ [17-22]. HereKE,exE andcE are the \nkinetic, exchange and correlation terms resp ectively. Therefore, the chemical potential \nand its derivative with respect to density, known as the inverse compressibility or \ndensity of states, provide direct insight in to the properties of th e Coulomb interaction \nin such a system. Compressibility meas urements of conventional, massive, two \ndimensional electron  systems made for exampl e of Si or GaAs have been carried out \nby several groups [19,23-26]. It has been sh own that at zero magnetic field the \nchemical potential may be described rath er accurately within the Hartree-Fock \napproximation. Quantitatively it has been found that Coulomb interactions add a substantial contribution to the compressibil ity and become dominant at low carrier \ndensities. In a perfectly uniform a nd clean graphene sample, the inverse \ncompressibility is expected to diverge at the Dirac point in vi ew of the vanishing \ndensity of states.  This divergence, however, is expected to be rounded off by disorder on length scales smaller than our tip size. Long range disorder on the other hand will \ncause local shifts in the Dirac point indicative of a non-ze ro local density. In this work \nwe measure the spatial dependence of the local compressibility versus carrier density across the sample. Fluctuations in the Dirac point across the sample are translated into carrier density maps through which quantit ative information about the degree and \nlength scale of disorder is obtained.   The preparation of graphene monolayers is performed in a similar manner as in Ref. \n[4,27]. We use Nitto tape to peel a large graphite flak e from an HOPG crystal and \npress it onto a Si/SiO\n2 wafer. Smaller and thinner pieces of this flake will then stick to \nthe surface of the substrate. Once a suitable monolayer is  selected in an optical \nmicroscope, two electrical contacts to the layer are patterned w ith optical lithography. \nTypical dimensions of our monolayers are 410× µm2. The conducting Si substrate \nserves as a back-gate to vary the carrier density in the graphene sheet [4]. A back-gate \nvoltage difference of 1V corresp onds to a dens ity change of -2 10cm 107⋅ . This \nconversion factor has been extracted from  magnetotransport data (Fig. 1A). The \nunorthodox Hall quantization in graphene monolay ers allows to distinguish them from \nmultilayer flakes [5,6]. Fig. 1A shows the two terminal conductance pt2G  as a \nfunction of density and ma gnetic field. Maxima in pt2G  can be clearly seen at filling \nfactors 2, 6, 10… for both electrons and hol es. This behavior conc lusively identifies \nthe flake as a monolayer.  The local compressibility measurements of graphene described here are carried out \nusing a scanable single electron transist or. Previous scanning tunneling microscopy \n(STM) studies on bulk graphite surfaces in strong magnetic fields already \ndemonstrated the power of local methods to  probe this class of materials [28-30]. \nHowever, unlike STM that probes the single particle density of states, the inverse \ncompressibility measures the many-body density  of states that includes information \non the exchange and correlation energies as  well. The experiments are performed at \n0.3K. A schematic picture of the experimental setup is depicted in the inset to Fig. 1B. \nDetails of our experimental method are descri bed in Ref. [31,32]. The diameter of the \nSET is about 100 nm and the distance between  SET and the sample is roughly 50 nm.  \nThe SET tip is capable of measuring the local electrostatic potential with microvolt \nsensitivity and a high spatial resolution close to its size. The inverse compressibility \ncan be measured by monitoring the change in the local electrostatic potential,\ntotalΦ , \nwhen modulating the carrier density in the graphene fl ake with the back-gate, \nntotal∂∂Φ . We note that the small size of the graphene flake requires to consider \nposition dependent direct pick-up from the fringing electric fields between the back-\ngate plane and the graphene flake as discu ssed in the methods section. In the absence \nof any transport current, any change in th e local electrostatic potential is equal in \nmagnitude and opposite in sign to the changes in the local chemical potential of the \ngraphene, n netotal\n∂∂−=∂∂ µ Φ. Figure 1B shows th e inverse compressibility for a fixed \nlocation of the SET as a function of the b ack-gate voltage at ze ro magnetic field. \nBased on the linear dispersion of the gra phene bandstructure, the kinetic energy \ncontribution to the inverse compressibility is  expected to exhibit an unusual density \ndependence with a singularity at the neutrality point given by \nsFgnvπ⋅= . Here Fv \nis the Fermi velocity, sg is the band degeneracy, and n is the carrier density \nmeasured from the neutrality point. At zero magnetic field 4sg= due to both spin and band symmetries. The maximum in the experimental n∂∂µ -trace clearly identifies \nthe position of the Dirac point at this par ticular location. The absence of a singularity \nat the Dirac point in experiment is ascribed to disord er broadening. Since the kinetic \nterm in graphene has the same density dependence as the leading exchange and correlations terms it is instructive to fit the data to the kinetic term with a single fit \nparameter that may be thought of as an eff ective Fermi velocity. The red line in Fig. \n1B shows the result of such a fit and yields \n661.1 10 0.1 10eff\nFvm s=⋅±⋅ . The indicated \nuncertainty reflects the accura cy with which the direct pick-up correction can be \ncarried out. Since the Fermi velocity fr om band-structure calculations [1], \ns/m vF6101⋅= , lies within our experimental uncertainty, we conclude that the data \ncan be described quantitatively by the ki netic term alone. It indicates that the \nexchange and correlation contributions are eith er weak or cancel out. Also theory has \nsuggested only weak modifications of the compressibility due to exchange [20-22]. \nThis is in marked contrast to massive two dimensional systems such as GaAs where at \nlow densities the exchange term domin ates [19,23-25]. Moreover, in these \nconventional two-dimensional electron systems quantitative agreement between \nexperiment and theory is only accomplished when taki ng into account the non-zero \nthickness of the 2D layer [19,24] . The finite width is responsible for a softening of the \nCoulomb interaction between the electrons as well as a Stark like shift of the confinement energies in the potential well when the density is tuned. Both effects \nproduce non-generic changes to the comp ressibility terms, which depend on the \ndetails of the heterostructure. These comp lications however do not arise for graphene \nas its thickness is negligible.   The peaked behavior of the inverse compressi bility as a function of back-gate voltage \nor density may be used to map out the dens ity fluctuations in the graphene sheet. At \ndifferent locations across the sample the local density is added to the externally induced density by the back-gate. Therefore,  different back-gate voltages are required \nin order to zero out the dens ity at each location and reach the Dirac point. In practice \nthis means that the entire inverse compressibility curve is shifted along the back-gate voltage axis as one moves from one point across the sample to another. Figure 2A \nshows a measurement of the inverse compressi bility along an arbitrarily chosen line \nacross the sample. The Dirac point is loca ted within the red colored band, which \ncorresponds to reduced compressi bility. As expected, it app ears at different back-gate \nvoltages for different locations. The black  dotted line shows th e dependence of the \nDirac point with position obtained by fitting the kinetic term of the compressibility to \neach back-gate voltage line scan. The sma llest length scale on which density \nvariations are observed is roughly 150 nm. This scale is most likely limited by our \nspatial resolution, i.e. the size of the SET. The underlying  density fluctuations are \ntherefore bound to occur on even smaller le ngth scales and with higher amplitudes. \nDirect evidence for that is given in subsequent paragraphs.    \n The chemical potential variations within the graphene are likely due to charged \nimpurities above and below the layer. In the process of scanning the SET above the layer one picks up not only the spatial variatio ns in the chemical potential but also the \npotential emanating directly from the char ges above the layer al ong with their image \ncharges in the layer. As these potentials ar e much larger than the chemical potential \nvariations it would be desira ble to find means to subtract them out and be left with \nonly the chemical potential variations. This can be readily achieved by subtracting off the measured potential at zero average density  from that at a high carrier density. As \nthe carrier density in the graphene sheet in creases it is expected  to screen better. \nHence, at large carrier density the poten tial landscape in the graphene should be \nnearly constant and the only potential seen by the SET is that due to static charges \nabove the layer. Using this method one can extract the local carri er density from the \nmeasurements of the surface potential only. A formal description of this subtraction \nprocess, which validates this method, is given in the supplementary material. Figure \n2B shows a comparison of th e density variations as ex tracted from the inverse \ncompressibility measurements (black dotted lin e, which is identical to the one in Fig. \n2A) with those extracte d from the surface potential (blue solid line). There is striking \nquantitative agreement between the two met hods. Fig. 3A depict s a two-dimensional \nmap of the density variations of the graphene  sheet when the average carrier density is \nzero. It was obtained from surface potential measurements. The red regions correspond to electrons and the blue re gions correspond to holes. These data \nconstitute direct evidence for the puddle mo del [13]. A statistica l analysis of the \ndensity fluctuations present in this two- dimensional map is s hown in Fig. 3B. The \nhistogram plots the number of patches in which the density lies within a certain \ndensity interval. From a fit with a Gaussian distribution, we extract from its standard deviation that the density fluc tuations are on the order of \n2 10\nT0B,D2 cm 109.3 n−\n= ⋅±= ∆ . \n \nThe fluctuations in density discussed thus  far have been resolution limited by our \ntechnique. However, the intrinsic density fluctuations may be extracted by going into \nthe quantum Hall regime. A large magnetic fi eld applied perpendicular to the sample \nproduces bands of localized and extended st ates that are manifested in universal \ntransport properties [33]. Our previous studies on GaAs two-dimensional electron \nsystems have shown that at sufficiently larg e magnetic fields the width in density of \nthe band of localized states, also referred to as the incompressible band, becomes field \nand also filling factor independent. This width constitutes a di rect measure of the \ndensity fluctuations in th e sample [34]. Fig. 4 show s a color rendering of the n∂∂µ  \nmeasured on our graphene flake as a func tion of the magnetic field and density. A \nsingle density scan at the fixed field of 11 T is incl uded at the top. It is composed of a \nseries of maxima at the integer fillings 2, 6, 10,…, corresponding to regions of low \ncompressibility.  These maxima can be fitted  well by Gaussians of identical variance, \nwhich is in accordance with th e filling independent width of the incompressible bands \nobserved in conventional 2D systems.  Th e variance may serve as a measure for the \ndisorder amplitude. A best fit to the data is obtained assuming density fluctuations of \napproximately 2 11\nT11B,D2 cm 103.2 n−\n= ⋅±= ∆ . A similar analysis at lower magnetic \nfields shows that the varian ce still drops slightly with decreasing magnetic field and \nhence the value T11B,D2n=∆  is a lower bound for the disord er strength as bands of \nlocalized states are not well separated ye t. The disorder amplitude extracted from \nthese measurements in the presence of a magne tic field is about a factor of six larger \nthan the disorder estimate T0B,D2n=∆  from the B = 0 T measurement in Fig. 3. This \ndifference allows us to deduce the intrinsic disorder length scale, ldisorder . For the zero \nfield estimate the density fluctuations are averaged over an area determined by the tip \nsize with a characteristic dimension of a pproximately 150 nm (see Fig. 2B). The ratio \nof these averaged density fluctuations and th e intrinsic ones is simply the square root \nof the ratio of the disorder area to the averaged area. Therefore we end up with an upper-bound for the intrinsic di sorder length scale of a bout 30 nm.  We note as a \ncuriosity that a similar number for the disorder length scale is also obtained from the \nEinstein relation between conductivity and compressibility, 2 neDσµ∂=∂, when \nplugging in the expression for the diffusi on constant valid for conventional 2D \nsystems: 2F Dv l= . Here, l is the mean free path. The c onductivity near the neutrality \npoint is approximately 24e\nhσ≈  and n\nµ∂\n∂ is shown in figure 1. We conclude that the \nintrinsic disorder length scale in graphe ne is approximately 30 nm. At high carrier \ndensity, the high compressibility of graphene  smoothes out the disorder landscape. As \none approaches the neutrality point howeve r, screening becomes poor and the intrinsic \ndisorder length turns relevant.  We acknowledge helpful discussions on the pr eparation of graphene flakes with K. \nNovoselov and A. Geim. We al so would like to acknowledge fruitful discussions with \nF. von Oppen.  References: \n \n1.\n N. B. Brandt, S. M. Chudinov, Y. G.  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Fukuyama, \nScanning tunneling microscopy and spect roscopy of the electronic local \ndensity of states of grap hite surfaces near monoatmic step edges, of  Phys. \nRev. B 73, 085421 (2006). 31. M. J. Yoo, T. A. Fulton, H. F. Hess, R. L. Willett, L. N. Dunkleberger, R. \nJ. Chichester, L. N. Pfeiffer, K.  W. West, Scanning Single-Electron \nTransistor Microscopy: Imaging Individual Charges, Science 276, 579-582 \n(1997).  \n32. A. Yacoby, H. F. Hess, T. A. Fult on, L. N. Pfeiffer and K. W. West, \nElectrical imaging of the quantum  Hall state, Solid State Comm. 111, 1-13 \n(1999).  \n33. R. E. Prange, S. M. Girvin, The quant um Hall effect (Springer, New York, \n1990).  \n34. S. Ilani, J. Martin, E. Teitelbaum, J. H. Smet, D. Mahalu, V. Umansky, A. Yacoby, The microscopic nature of lo calization in the quantum Hall effect, \nNature \n427, 328-332 (2004). \n  \n \n \nFig. 1: A. Color rendering of the two-terminal conductance G 2pt of a graphene monolayer in \nthe density versus magnetic field plane. The conductance maxima follow slopes \ncorresponding to integer fillings 2, 6, 10, …. , an unequivocal signature for a single \nmonolayer. B. The inverse compressibility measur ed at an arbitrary, fixed location on the \ngraphene sample as a function of the back-gate voltage or carrier density (blue line). The red \nline is a best fit to the data using an effective Fermi velocity as a single fit parameter in the \nkinetic energy contribution predicted from the graphene band-structure. The inset depicts the \nexperimental arrangement consisting of an al uminum based SET evaporated on a glass fiber \ntip of a scanning probe microscope. The graphen e monolayer is contacted with two leads on \ntop of an oxidized and heavily doped Si wafer.  A back-gate voltage induces charge carriers \nand modulates the density for the measurement of the local compressibility.          \n \n \nFigure 2: A. Color rendering of  the inverse compressibility measured along a line across the \nsample as a function of the back-gate voltage. The dotted black line marks the location of the \nDirac point obtained from a fit of the kinetic ener gy term to each line scan. B.  Comparison of \nthe spatial variation of the Dirac point extracted using two methods: the inverse \ncompressibility (black dotted line, identical to the dotted line in panel A) and through \nsubtracting surface potential scans at high and zero average carrier density (blue line). The \nbar marks the smallest length scale (approxim ately 150 nm) on which density variations are \nobserved.  \n \n \nFig. 3: A. Color map of the spatial density variations in the graphene flake extracted from \nsurface potential measurements at high density an d when the average carrier density is zero. \nBlue regions correspond to holes and red regions to electrons. The black contour marks the \nzero density contour. B. Histogram of the density distribution in A.   \n \nFig. 4: A. Color rendering of the inverse comp ressibility as a function of density and magnetic \nfield. B. A single line scan from  plot A of the measured inverse compressibility (black line) at a \nmagnetic field of 11 T. The red curve is a fi t to the data composed of Gaussians with equal \nvariance for each maximum. This variance meas ures the width of the incompressible regions \n(maxima) centered around integer fillings and prov ides an estimate for the intrinsic amplitude \nof the density fluctuations. \n \n  \n \nMethod Section \n \n A. Determination of the local carrier  density from the surface potential: \n \nHere we derive that a subtraction of th e surface potential measur ed at high carrier \ndensity and at zero average carrier density yields the local density profile in the \ngraphene flake. The most general expression  for the surface poten tial detected by the \nSET reads   \n()()() r,n rer,n etotal µΦ Φ += ,       ( 1 )  \n \nwhere()rΦ  is the electrostatic potential due to local doping/adsorbates and ()r,nµ  is \nthe local chemical potential of the gra phene sheet. The influence of disorder \nemanating from below and above the graphene  flake on the local chemical potential is \nincluded in ()r,nµ . We assume that µ only depends on the local density \n)r(n n)r(n0δ+= .  Here BGg 0 VC n=  is the average density in graphene which can \nbe tuned with a back-gate and nδis the disorder induced spatial density fluctuation. \nClose to the Dirac point where the average density 0 n0=equation (1) reduces to: \n \n ()()( ) )r(n rer,0 etotal δµΦ Φ +=                    (2) \n \nAt large voltages on the back-gate the aver age density will beco me larger than the \ndensity fluctuations n n0δ>>  and the surface potential may be written as: \n \n() ( ) ( ) () )r(n nn  n rer,n e 0 0 0 total δµµΦ Φ∂∂++≈              (3) \n \nWhen we substract the average value of  the surface potential measured at high \nelectron density (positive back -gate voltage, Eq. (3)) and equal hole density (negative \nback-gate voltage) from the surface potential  recorder close to the Dirac point (Eq. \n(2)), we derive:  \n)r(nn )n( ))r(n(2)r,n( )r,n( e)r,0( e0 0 total 0 totaltotal δµδµΦ ΦΦ∂∂− =+−−  \n               \n  )r(nn )n( \nn ))r(n( 0δµ δµ⋅⎟\n⎠⎞⎜\n⎝⎛\n∂∂−∂∂≈  \n \n \n      )r(nn ))r(n( δδµ⋅∂∂≈   (4) \n \n \nThe density variations in the graphe ne sheet can then be obtained from: \n   ()()()\n()\nn )r(n 2r,n er,n er,0 e\n)r(n0 total 0 totaltotal\n∂∂+−−\n≈δµΦ ΦΦ\nδ    (5) \n \n \nB. Direct pick-up: \n \nThe small size of the graphene flake and the finite distance between the single \nelectron transistor (SET) and th e graphene flake result in a directly measured potential \ncontribution from the highly doped Si-back gate. The inset to Fig. 5 illustrates that some electrical field lines originating from the back-gate reach th e SET. The parasitic \ncharge associated with these fringing fi elds is also detected by the SET.  \n The detected amount of charge depends both on the distance of the SET to the \ngraphene edge as well as on the vertical di stance between tip and flake. Fig. 5 shows \nthe dependence of the direct pickup measur ed as a function of height above a gold \nelectrode. The gold electrode serves as a re ference since it entirely screens the back-\ngate potential underneath and only the fringi ng fields are detected . The distance from \nthe edge is chosen to be equal to the one in the real measurement on top of the \ngraphene flake.   As expected, once the SET comes close to  the surface the dir ect pick-up of the \nfringing fields drops and appr oaches zero. For typical he ights and distances to the \nsample edge we derive a contribution from  the fringing fields of approximately 0.15 \nmeV 10\n-10 cm2. \n \n \n \n \nFig. 5: The direct pick-up cont ribution to the measured inverse compressibility as a function of \nthe distance between the tip and the sample. The ti p is placed above a gold electrode at a  \nfixed distance from the border of the electrode typical for the configuration in the experiment \non a graphene flake. The inset schematically illu strates the origin of this direct pick-up. \n \n    \n "    },    {        "title": "0705.3003v1.Negative_Energy_in_Superposition_and_Entangled_States.pdf",        "content": "arXiv:0705.3003v1  [quant-ph]  21 May 2007Negative Energy in Superposition and Entangled States\nL.H. Ford∗\nInstitute of Cosmology\nDepartment of Physics and Astronomy\nTufts University, Medford, MA 02155\nThomas A. Roman†\nDepartment of Mathematical Sciences\nCentral Connecticut State University\nNew Britain, CT 06050\nAbstract\nWe examine the maximum negative energy density which can be a ttained in various quantum\nstatesofamasslessscalarfield. Weconsiderstatesinwhich eitheroneortwomodesareexcited, and\nshow that the energy density can be given in terms of a small nu mber of parameters. We calculate\nthese parameters for several examples of superposition sta tes for one mode, and entangled states\nfor two modes, and find the maximum magnitude of the negative e nergy density in these states.\nWe consider several states which have been, or potentially w ill be, generated in quantum optics\nexperiments.\nPACS numbers: 04.62.+v,03.65.Ud,42.50.Pq\n∗Email: ford@cosmos.phy.tufts.edu\n†Email: roman@ccsu.edu\n1I. INTRODUCTION\nIt has been proven, beginning in the early 1960’s [1], that there alway s exist states\nwith negative energy density in quantum field theory. Some specific e xamples include the\nCasimir effect [2] and squeezed states [3], both of which have been ex perimentally realized.\n(Although the energy density itself is far too small to be directly mea sured.) Negative\nenergy is also required for black hole evaporation, and hence for th e consistency of the\nlaws of black hole physics with those of thermodynamics. On the othe r hand, unrestricted\namounts of negative energy could produce bizarre effects, for ex ample, violations of the\nsecond law of thermodynamics [4, 5]. However, the same laws of quan tum field theory which\nallow the existence of negative energy also appear to severely rest rict its magnitude and\nduration in such a way as to prevent gross large-scale effects. The se bounds are known\nas quantum inequalities, and quite a large body of work now exists on t he subject. For\nsome recent reviews of quantum inequalities, see Refs. [6, 7, 8]. Qu antum inequality bounds\nhave been proven, for example, for the minimally coupled scalar, elec tromagnetic, and Dirac\nfields. It should be pointed out that the potential macroscopic pro blems arise not because\nof the existence of negative energy per se, but from the arbitrar y separation of negative and\npositive energy. It is this behavior which the quantum inequalities pro hibit. Many possible\nconfigurations of separated negative and positive energy can eas ily be ruled out, and known\npermitted examples involve the subtle intertwining of the two [9]. Whet her the currently\nknown examples are representative of the general case is unknow n. Hence, the study of\nfurther examples could prove useful.\nSince the negative energy densities in these states aretoo small to be directly measurable,\nexperiments in quantum optics may offer the best possibilities for indir ect detection of\nnegative energy. (However, see also Refs. [10, 11].) A first link betw een quantum optics and\nthe work on quantum inequalities has been forged in a recent paper b y Marecki [12]. For\nsqueezed states, he proved quantum inequality-type bounds on t he magnitude and duration\nof the squeezing.\nQuantum optics has seen enormous experimental and theoretical advances in the last\ntwenty years. This marriage of optics with quantum field theory has resulted in experiments\nwhich were formerly purely “gedanken” becoming those which are no w routinely performed\nin the laboratory. Highly non-classical states, such as Schr¨ oding er “cat states” and squeezed\n2states, have been produced and play a part in everything from qua ntum computers to\nnoise reduction in laser interferometer gravitational wave detect ors. The “cat states” of the\nelectromagnetic field are superpositions of coherent states and h ave been created experi-\nmentally [13, 14]. The experiments which have been done so far have p roduced mesoscopic\nsuperpositions, in which the mean photon number is of order 10. This is somewhat short of\na true Schr¨ odinger cat state, which would be a superposition of tw o or more classical config-\nurations, that is, coherent states with very large occupation num bers. More recently, there\nhave been proposals for methods of creating superpositions of sq ueezed vacuum states [15].\nAn interesting question arises: can one start with two quantum sta tes which do not\ninvolve negative energy and by superposing them obtainnegative en ergy? The answer is yes;\nthe classic standard example being the vacuum + two-particle state (for a nice discussion\nsee Ref. [16]). Is this true for the superposition of other states a s well? More generally,\nwhat effects does superposition have on negative energy? Could on e also go the other way,\ni.e., start with two states involving negative energy and by superpos ing them diminish or\neradicate the negative energy? In this paper, we will address such questions for several\nclasses of states. In Sect. II, we develop some formalism for para meterizing the maximum\nmagnitude of negative energy that can occur for states of a minima lly coupled scalar field\nin Minkowski spacetime with either one or two modes excited. We give s everal examples\nof superpositions for a single mode in Sect. III, including superposit ions of two coherent\nstates, two squeezed vacuum states, and a coherent state with a squeezed vacuum state. In\nSect. IV, we move to the two-mode case. This allows us to consider e xamples of entangled\nstates involving either squeezed vacua or coherent states for th e two modes. A summary of\nour conclusions is presented in Section V.\nII. ENERGY DENSITY WITH ONE OR TWO MODES ARE EXCITED\nIn this paper, we will consider a massless scalar field in flat spacetime, for which the\nstress tensor operator is\nTµν=ϕ,µϕ,ν−1\n2gµνϕ,σϕ,σ. (1)\nThe normal-ordered energy density operator is\n:T00:=1\n2[: ˙ϕ2: + : (∇ϕ)2:], (2)\n3where\nϕ=/summationdisplay\nk(akfk+ak†fk∗), (3)\nwith thefk(x,t) being the mode functions.\nA. Two Modes Excited\nWe wish to consider the case where all modes except for two are in th e vacuum state. For\nthe first mode, let f1,a, anda†be the mode function, annihilation operator, and creation\noperator, respectively, and let f2,b, andb†be the corresponding quantities for the second\nmode. The expectation value of the energy density in an arbitrary q uantum state can be\nexpressed as\nρ=/an}b∇acketle{t:T00:/an}b∇acket∇i}ht= Re/braceleftbigg\n/an}b∇acketle{ta†a/an}b∇acket∇i}ht(|˙f1|2+|∇f1|2)+/an}b∇acketle{ta2/an}b∇acket∇i}ht[˙f2\n1+(∇f1)2]+/an}b∇acketle{tb†b/an}b∇acket∇i}ht(|˙f2|2+|∇f2|2)\n+/an}b∇acketle{tb2/an}b∇acket∇i}ht[˙f2\n2+(∇f2)2]+2/an}b∇acketle{ta†b/an}b∇acket∇i}ht(˙f∗\n1˙f2+∇f1∗·∇f2)+2/an}b∇acketle{tab/an}b∇acket∇i}ht(˙f1˙f2+∇f1·∇f2)/bracerightbigg\n.(4)\nLet\nn1=/an}b∇acketle{ta†a/an}b∇acket∇i}ht, n2=/an}b∇acketle{tb†b/an}b∇acket∇i}ht, R1eiγ1=/an}b∇acketle{ta2/an}b∇acket∇i}ht, R2eiγ2=/an}b∇acketle{tb2/an}b∇acket∇i}ht, R3eiγ3=/an}b∇acketle{ta†b/an}b∇acket∇i}ht, R4eiγ4=/an}b∇acketle{tab/an}b∇acket∇i}ht.\n(5)\nAll of the information needed to give the two-mode energy density, Eq. (4), at a given\nquantum state is encoded in the above set of six amplitudes and four phases.\nIn the case of a traveling waves, we may take the mode functions to be\nfj=i/radicalBig\n2ωjVei(kj·x−ωjt), (6)\nwhereωj=|kj|, forj= 1,2 andVis a normalization volume. In this case, the mean energy\ndensity may be expressed as\nρ=1\nV/braceleftbigg\nn1ω1+n2ω2+R1ω1cos[2(k1·x−ω1t)+γ1]+R2ω2cos[2(k2·x−ω2t)+γ2]\n+R3√ω1ω2(1+ˆk1·ˆk2) cos[(k2−k1)·x−(ω2−ω1)t+γ3]\n+R4√ω1ω2(1+ˆk1·ˆk2) cos[(k2+k1)·x−(ω2+ω1)t+γ4]/bracerightbigg\n. (7)\nWe will also consider the case of a standing wave which depends upon o nly one space\ncoordinate, in which case the mode functions can be taken to be\nfj=1/radicalBig\nωjVsin(ωjx)e−iωjt. (8)\n4The energy density now becomes\nρ=1\nV/braceleftbigg\nn1ω1+n2ω2+R1ω1cos(2ω1x) cos(2ω1t−γ1)+R2ω2cos(2ω2x) cos(2ω2t−γ1)\n+ 2R3√ω1ω2cos[(ω2−ω1)x] cos[(ω2−ω1)t−γ3]\n+ 2R4√ω1ω2cos[(ω1+ω2)x] cos[(ω1+ω2)t−γ4]/bracerightbigg\n. (9)\nB. One Mode Excited\nA useful special case is when only one mode is excited. In this case, w e may setn1=n,\nR1=R,γ1=γ, andR2=R3=R4=γ2=γ3=γ4= 0. In this case, we need only the\nthree real numbers n,R, andγto determine the energy density in a given state. For the\ncase of a traveling wave, we have\nρ=ω\nV{n+Rcos([2(k·x−ωt)+γ]} (10)\nWe can see from Eq. (10) that the minimum value of ρis\nρmin=−ω\nV(R−n), (11)\nand hence we can have negative energy density only if R > n. In the case of a standing\nwave, Eq. (9) becomes\nρ=ω\nV[n+Rcos(2ωx) cos(2ωt−γ)]. (12)\nAgain, the minimum value of ρis given by Eq. (11).\nIII. SUPERPOSITIONS FOR ONE MODE\nIn this section, we examine some explicit examples of superpositions in volving a single\nmode. In each case, we need only calculate the quantity R−nto determine the maximum\nmagnitude of the negative energy.\nA. Superposition of Two Coherent States\nFirst we consider a superposition of coherent states. Coherent s tates are eigenstates of\nthe annihilation operator, that is\na|α/an}b∇acket∇i}ht=α|α/an}b∇acket∇i}ht. (13)\n5Let\nψ/an}b∇acket∇i}ht=N[|α/an}b∇acket∇i}ht+η|β/an}b∇acket∇i}ht], (14)\nwhere|α/an}b∇acket∇i}htand|β/an}b∇acket∇i}htare two different coherent states for the same mode, ηis a complex\nnumber, and Nis a normalization factor (see, for example, Sec. 7.6 of Ref. [17]). W e also\nassume that the states are normalized so that /an}b∇acketle{tα|α/an}b∇acket∇i}ht=/an}b∇acketle{tβ|β/an}b∇acket∇i}ht= 1. As a result we have that\n/an}b∇acketle{tψ|ψ/an}b∇acket∇i}ht= 1 =N2[1+|η|2+η/an}b∇acketle{tα|β/an}b∇acket∇i}ht+η∗/an}b∇acketle{tβ|α/an}b∇acket∇i}ht]. (15)\nThe coherent states are not orthonormal; their overlap integral is given by (see for example,\nEq.(3.6.24) of Ref. [18]):\n/an}b∇acketle{tα|β/an}b∇acket∇i}ht=e−1\n2(|α|2+|β|2−2α∗β). (16)\nTherefore the square of the normalization factor is\nN2= [1+|η|2+2e−1\n2(|α|2+|β|2)Re(ηeα∗β)]−1. (17)\nThe mean number of particles is found to be\nn=N2/bracketleftBig\n|α|2+|ηβ|2+2e−1\n2(|α|2+|β|2)Re(ηα∗βeα∗β)/bracketrightBig\n, (18)\nand\n/an}b∇acketle{ta2/an}b∇acket∇i}ht=N2/bracketleftBig\nα2+|η|2β2+e−1\n2(|α|2+|β|2)(ηβ2eα∗β+η∗α2eαβ∗)/bracketrightBig\n. (19)\nLet\nα=|α|eiδ1, β=|β|eiδ2,andη=|η|eiδ. (20)\nThen the quantities n,R, andγare functions of six real parameters, the magnitudes and\nphases ofα,β, andη. However, one finds that only γdepends upon all six. The magnitudes\nnandRdepend only upon the difference δ2−δ1, and are hence functions of five parameters.\nWe are primarily interested in the quantity R−n, which measures the maximum magnitude\nof the negative energy density. Hence set δ1= 0 and write\nF(|α|,|β|,|η|,δ2,δ) =R−n, (21)\nand letGbe a five-dimensional vector given by\nG=/parenleftBigg∂F\n∂|α|,∂F\n∂|β|,∂F\n∂|η|,∂F\n∂δ2,∂F\n∂δ/parenrightBigg\n. (22)\n6One may use Eqs. (21) and (22) as the basis of a numerical algorithm to search for points\nof maximum Fand hence maximally negative energy density. Start at a random poin t in\nthe five-dimensional parameter space, and compute FandG. IfF >0, then this choice of\nparameters is a quantum state with negative energy density. The c omponents of Gindicate\nthe direction in which Fis increasing most rapidly. One then moves along this direction\nuntil a local maximum of Fis located. A preliminary, non-exhaustive, search located two\nsuch local maxima, at ( |α|,|β|,|η|,δ2,δ)≈(0.8,0.8,1,3.14,0) and at ( |α|,|β|,|η|,δ2,δ)≈\n(0,1.61,1,0,0). (One can trivially generate a third maximum by interchange of αandβ\nin the latter case.) The first example corresponds to α=−β= 0.8 and the second to a\nsuperposition of a coherent state and the vacuum. Interestingly , the maximum magnitude\nof the negative energy density is about the same in both examples, w ithF=R−n≈0.278,\nand henceρmin≈ −0.278ω/V. We do not have an explanation as to why these two choices\ngive the same value of R−n. The mean particle number is n≈0.36 in the first example and\nn≈1.0 in the second. This example illustrates that a superposition of two c oherent states\ncan produce negative energy density, and the maximum negative en ergy density arises for\nmean particle number of order one.\nB. Superposed Squeezed Vacuum States\n1. A Single-Mode Squeezed Vacuum State\nWe begin with a review of the features of the expectation value of th e energy density in\na single squeezed vacuum state. Our state is given by:\n|ψ/an}b∇acket∇i}ht=|ξ/an}b∇acket∇i}ht,withξ=reiδ, (23)\nwhereris the squeeze parameter and δis a phase parameter. The squeeze operator S(ξ) is\ngiven by\nS(ξ) =e1\n2[ξa2−ξ∗(a†)2]. (24)\nThis operator is unitary since\nS†(ξ) =S(−ξ) =S−1(ξ). (25)\n7Thesingle-modesqueezed state |ξ/an}b∇acket∇i}htisproducedbythesqueeze operatoractingonthevacuum\nstate\n|ξ/an}b∇acket∇i}ht=S(ξ)|0/an}b∇acket∇i}ht. (26)\nThe state |ξ/an}b∇acket∇i}htcan be written, after some work (see Eq. (3.7.5) of Ref. [18]), in te rms of the\neven Fock states as\n|ξ/an}b∇acket∇i}ht=√\nsechr∞/summationdisplay\nn=0/radicalBig\n(2n)!\nn!/bracketleftbigg\n−1\n2eiδtanhr/bracketrightbiggn\n|2n/an}b∇acket∇i}ht. (27)\nWe also have that\nS†(ξ)aS(ξ) =acoshr−a†eiδsinhr,\nS†(ξ)a†S(ξ) =a†coshr−ae−iδsinhr. (28)\nIn the state |ξ/an}b∇acket∇i}htwe have the expectation values\nn=/an}b∇acketle{ta†a/an}b∇acket∇i}ht=/an}b∇acketle{t0|S†(ξ)a†aS(ξ)|0/an}b∇acket∇i}ht=/an}b∇acketle{t0|S†(ξ)a†S(ξ)S†(ξ)aS(ξ)|0/an}b∇acket∇i}ht= sinh2r,\n/an}b∇acketle{ta2/an}b∇acket∇i}ht=/an}b∇acketle{t0|S†(ξ)a2S(ξ)|0/an}b∇acket∇i}ht=/an}b∇acketle{t0|S†(ξ)aS(ξ)S†(ξ)aS(ξ)|0/an}b∇acket∇i}ht=−eiδsinhrcoshr,(29)\nwhere we have made use of Eqs. (28). Thus R= sinhrcoshrand\nR−n= sinhr(coshr−sinhr) (30)\nattains its maximum value of 0 .5 asr→ ∞.\n2. Superposition of Squeezed Vacuum States\nWe now calculate the energy density in a superposition of two single-m ode squeezed\nvacuum states of the form\n|ψ/an}b∇acket∇i}ht=N[|ξ/an}b∇acket∇i}ht+η|−ξ/an}b∇acket∇i}ht], (31)\nwhere, for simplicity, we will choose\nξ=r, δ= 0, (32)\nand set\nη=|η|eiθ. (33)\n8In this state we have\nn=/an}b∇acketle{ta†a/an}b∇acket∇i}ht=N2[/an}b∇acketle{tξ|a†a|ξ/an}b∇acket∇i}ht+|η|2/an}b∇acketle{t−ξ|a†a|−ξ/an}b∇acket∇i}ht+η/an}b∇acketle{tξ|a†a|−ξ/an}b∇acket∇i}ht+η∗/an}b∇acketle{t−ξ|a†a|ξ/an}b∇acket∇i}ht]\n=N2/bracketleftbigg\nsinh2r(1+|η|2)−2|η|cosθsechrtanh2r\n(1+tanh2r)3/2/bracketrightbigg\n, (34)\nwhere we have made use of Eq. (A4) in the Appendix. A similar calculatio n, using Eq. (A5),\nyields\n/an}b∇acketle{ta2/an}b∇acket∇i}ht=N2/bracketleftbigg\n(|η|2−1)sinhrcoshr+2i|η|sinθsechrtanhr\n(1+tanh2r)3/2/bracketrightbigg\n, (35)\nand\nR=|/an}b∇acketle{ta2/an}b∇acket∇i}ht|=N2/bracketleftbigg\n(|η|2−1)2sinh2rcosh2r+4|η|2sin2θsech2rtanh2r\n(1+tanh2r)3/bracketrightbigg1\n2.(36)\nThe normalization of our state is given by\n/an}b∇acketle{tψ|ψ/an}b∇acket∇i}ht= 1 =N2[1+|η|2+η/an}b∇acketle{tξ|−ξ/an}b∇acket∇i}ht+η∗/an}b∇acketle{t−ξ|ξ/an}b∇acket∇i}ht]. (37)\nThe square of the normalization factor is then\nN2=/bracketleftbigg\n1+|η|2+2|η|cosθ/radicalBig\nsech(2r)/bracketrightbigg−1\n, (38)\nwhere we have used Eq. (A8) of the Appendix.\nFrom Eqs. (34) and (36), we can compute the quantity R−n, which gives the maximum\nmagnitude of the negative energy density, as function of θandr. For fixed θ, one typically\nfinds thatR−nattains a maximum value for some value of r, usually of order one. A\ntypical case of θ= 0 is illustrated in Fig. 1. The case η= 0 is just the single squeezed\nvacuum state discussed in Sect. IIIB1 . This case gives the maximum negative energy\ndensity,R−n= 0.5, for large r. All other values of η, corresponding to superposed\nsqueezed vacua, give somewhat smaller amounts of negative energ y density, and attain their\nmaximum negative energy density at finite values of r.\nC. Superposition of Coherent and Squeezed Vacuum States\nIn this subsection, we consider states of the form\n|ψ/an}b∇acket∇i}ht=N[|ξ/an}b∇acket∇i}ht+η|α/an}b∇acket∇i}ht], (39)\n90.5 11.5 2\n-0.4-0.20.20.4\nrR - n θ = 0\nη = 0\nη = 0.25\nη = 0.5\nFIG. 1: The quantity R−nfor two superposed squeezed vacua, Eq. (31), is plotted for t he case\nθ= 0 for various values of η. The case η= 0 is the single squeezed vacuum, and gives more\nnegative energy density than do any of the superpositions. F or non-zero η, there is a maximum\nvalue for R−nat a finite value of r.\nwhere|ξ/an}b∇acket∇i}htis a squeezed vacuum state, and |α/an}b∇acket∇i}htis a coherent state. We may use Eq. (A10) to\nfind\nN2=/braceleftbigg\n1+|η|2+2√\nsechre−1\n2|α|2Re/bracketleftbigg\nηexp/parenleftbigg\n−1\n2e−iδα2tanhr/parenrightbigg/bracketrightbigg/bracerightbigg−1\n.(40)\nSimilarly, we find\nn=/an}b∇acketle{ta†a/an}b∇acket∇i}ht=N2/braceleftbigg\nsinh2r+|ηα|2\n−2e−1\n2|α|2√\nsechrtanhrRe/bracketleftbigg\nηe−iδα2exp/parenleftbigg\n−1\n2e−iδα2tanhr/parenrightbigg/bracketrightbigg/bracerightbigg\n(41)\nand\n/an}b∇acketle{ta2/an}b∇acket∇i}ht=N2/braceleftbigg\n−sinhrcoshreiδ+|η|2α2+ηα2√\nsechre−1\n2|α|2exp/parenleftbigg\n−1\n2e−iδα2tanhr/parenrightbigg\n+η∗√\nsechre−1\n2|α|2[(α∗)2eiδtanhr−1]eiδtanhrexp[−1\n2eiδ(α∗)2tanhr]/bracerightbigg\n,(42)\nusing Eqs. (A11) and (A13).\nLet us consider the case where δ= 0,η= 1, andαis real. In this case, R−nis plotted\nin Fig. 2 for various values of α. Here the maximum negative energy density, R−n≈0.23\n100.20.40.60.8 1\n-0.4-0.20.2\nα = 1α = 0\nα = 0.8   α= 0.6α = 0.4R - n\nr\nFIG. 2: Here R−nis plotted for the superposition of a coherent and a squeezed vacuum state,\nEq. (39) with η= 1, as a function of rfor various values of α. Note that for α/ne}ationslash= 0,R−ncan\nbe positive, corresponding to negative energy, for both sma ller and larger values of r, but has an\nintermediate region where there is no negative energy.\nis attained for large r. Note that this state has less negative energy than does the sque ezed\nvacuum by itself. [See Eq. (30).] For αnon-zero, we find that R−ninitially decreases,\nreaches a minimum value, and then increases again. For the case α= 0.6, for example, there\nis negative energy for r <0.2 and again for r >0.65, but not for intermediate values of r.\nNote thatr= 0 is a superposition of the vacuum and a coherent state, a special case of the\nstate treated in Sect. IIIA.\nA limit of special interest is when α= 0 and we have the superposition of the vacuum\nwith a squeezed vacuum state. In this case,\nN2=/bracketleftBig\n1+|η|2+2Re(η)√\nsechr/bracketrightBig−1, (43)\nn=/an}b∇acketle{ta†a/an}b∇acket∇i}ht=N2sinh2r, (44)\nand\n/an}b∇acketle{ta2/an}b∇acket∇i}ht=−N2sinhrcoshreiδ/bracketleftBig\n1+η∗(sechr)5\n2/bracketrightBig\n. (45)\nTheα= 0 curve in Fig. 2 is this limit for η= 1. Ifηis real and negative, η=−|η|, we then\n111 2 3 4\n-2.5-2-1.5-1-0.5R - n\nr\nFIG. 3: Here R−nis plotted as a functionof rforα= 0andη=−1, asuperpositionof thevacuum\nand a squeezed vacuum. The maximum negative energy occurs wh enr≈2, where R−n≈0.3.\nhave\nR=|/an}b∇acketle{ta2/an}b∇acket∇i}ht|=N2sinhrcoshr/vextendsingle/vextendsingle/vextendsingle1−|η|(sechr)5\n2/vextendsingle/vextendsingle/vextendsingle, (46)\nand\nR−n=sinhr/bracketleftBig\ncoshr/vextendsingle/vextendsingle/vextendsingle1−|η|(sechr)5\n2/vextendsingle/vextendsingle/vextendsingle−sinhr/bracketrightBig\n1+|η|2−2|η|√\nsechr. (47)\nIn the case that η=−1 the right-hand side of Eq. (47) is plotted as a function of rin Fig. 3.\nHere we find the maximum negative energy, R−n≈0.3 atr≈2. This is slightly less\nnegative energy than can be found in a single squeezed vacuum stat e. Note that as r→0,\nthis state becomes |2/an}b∇acket∇i}ht, a two-particle state with positive energy density everywhere. Th is is\nthe reason that the behavior in Fig. 3 differs from the α= 0 curve in Fig. 2. In the latter\ncase,η= 1, and there is negative energy for all values of r.\nIV. TWO-MODE ENTANGLED STATES\nIn this section, we will consider several examples of entangled stat es involving two modes.\n12A. An Entangled Squeezed State - the Barnett-Radmore State\nOur first example of a two-mode entangled squeezed state was des cribed by Barnett and\nRadmore [18] and is defined by\n|ψ/an}b∇acket∇i}ht=SAB|0/an}b∇acket∇i}ht, (48)\nwhere|0/an}b∇acket∇i}htis the vacuum state for both modes, and\nSAB= e(ξ∗ab−ξa†b†)(49)\nis a two-mode squeeze operator. If one were to expand the state |ψ/an}b∇acket∇i}htin terms of number\neigenstates, the expansion would contain states with an even tota l number of particles, with\nhalf of these particles in each mode. One has the following identities [18 ]\nSAB(−ξ)aSAB(ξ) =acoshr−b†eiδsinhr\nSAB(−ξ)a†SAB(ξ) =a†coshr−be−iδsinhr\nSAB(−ξ)bSAB(ξ) =bcoshr−a†eiδsinhr\nSAB(−ξ)b†SAB(ξ) =b†coshr−ae−iδsinhr. (50)\nNote that\nS†\nAB(ξ) =S−1\nAB(ξ) =SAB(−ξ). (51)\nOne may use these relations to show that\nn1=n2= sinh2r, R 1=R2=R3= 0, R4= sinhrcoshr,andγ4=δ+π.(52)\nThe minimum energy density in this state is\nρmin(BR) =−sinhr\nV[2√ω1ω2coshr−(ω1+ω2)sinhr]. (53)\nThis is never more negative than the minimum energy density that wou ld be obtained if\nthe two modes were individually in squeezed vacuum states. The latte r energy density is\nρmin(2SQ) =−(ω1+ω2)(R−n)/V, whereR−nis given by Eq. (30). Thus we can write\nρmin(BR)−ρmin(2SQ) =sinhrcoshr\nV(√ω1−√ω2)2, (54)\nwhich is always non-negative and approached zero only when the two modes have nearly the\nsame frequency.\n13B. An Second Entangled Squeezed State - the Zhang State\nIn this subsection, we will consider a second possibility for an entang led two-mode\nsqueezed state, which was discussed by Zhang [15]. This state is defi ned by\n|ψ/an}b∇acket∇i}ht=N/parenleftBig\n|¯ξ/an}b∇acket∇i}hta|¯η/an}b∇acket∇i}htb+eiθ|ξ/an}b∇acket∇i}hta|η/an}b∇acket∇i}htb/parenrightBig\n, (55)\nwhere|¯ξ/an}b∇acket∇i}htaand|ξ/an}b∇acket∇i}htaare single-mode squeezed vacuum states for mode a, and|¯η/an}b∇acket∇i}htband|η/an}b∇acket∇i}htbare\nsuch states for mode b. In general, ξ,¯ξ,η, and ¯ηcan be four arbitrary complex parameters.\nHowever, we will restrict our attention to the case where they are real and satisfy\nξ=η=−¯ξ=−¯η. (56)\nIn this case,\nN=/braceleftBig\n2[1+Re(eiθ/an}b∇acketle{t−ξ|ξ/an}b∇acket∇i}hta/an}b∇acketle{t−η|η/an}b∇acket∇i}htb)]/bracerightBig−1\n2= [2(1+cos θsech2r)]−1\n2, (57)\nwhere we have used Eq. (A8) for each of the two modes. Similarly, we find\nn1=n2= 2N2sinh2r/bracketleftBigg\n1−cosθ\n(cosh2r)3\n2/bracketrightBigg\n, (58)\nand\nR1=R2=N2|sinθ|tanh2r√\ncosh2r. (59)\nHere we have use Eqs. (A4) and (A5), as well as the identity sinh2r+cosh2r= cosh(2r).\nIn addition, we find R3=R4= 0 andγ1=γ2=−π/2.\nIn this case, the energy density, Eq. (7), becomes\nρ=1\nV/parenleftbigg\nn1(ω1+ω2)+R1{ω1cos[2(k1·x−ω1t)+γ1]+ω2cos[2(k2·x−ω2t)+γ1]}/parenrightbigg\n(60)\nWe can always choose the spatial position xand timetso as to make both cosine functions\nequal to −1, in which case we achieve the minimum allowed energy density in this sta te of\nρmin=−ω1+ω2\nV(R1−n1). (61)\nFrom Eqs. (57), (58) and (59), we find R1−n1as a function of θandr. In general,\nthe behavior of this entangled state is similar to that of the superpo sed squeezed vacua\nillustrated in Fig. 1. However, there is one limit of particular interest, which is when r≪1\n140.01\r 0.02\r 0.03\r 0.04\r\n-1\r-0.75\r-0.5\r-0.25\r0.25\r0.5\r0.75\r1\r θ = 0.99 π\nθ = 0.95 π\nθ = 0.99 πr\nFIG. 4: The quantities R1−n1(solid lines) and n1(dashed lines) are plotted for two values of θ\nas functions of rfor the entangled squeezed state defined in Eqs. (55) and (56) . In the limit that θ\nis close to π, one can have appreciable negative energy at small values of r. We see that the peak\nnegative energy, R1−n1≈0.25 occurs at r≈0.007 forθ= 0.99πand atr≈0.035 forθ= 0.95π,\nwhereas the mean particle number is about the same for both ca ses,n1≈0.2.\nand 0<|π−θ| ≪1. (Note that if θ=π, thenR1= 0, and there is no negative energy.) If\nwe take the limit r≪1, for fixed θ/ne}ationslash=π, then we find the asymptotic forms\nn1∼1−cosθ\n1+cosθr2, (62)\nand\nR1−n1∼|sinθ|\n1+cosθr. (63)\nIn the case that 0 <|π−θ| ≪1, the coefficient in the expression for n1can be large, so\nwe can have an unusually large particle number in relation to the value o fr. The quantities\nR1−n1andn1are plotted in Fig. 4 for two values of θclose toπ. In this case, we can have\na reasonable amount of negative energy at very small values of the squeeze parameter, r.\n15C. Entangled Coherent States\nIn this subsection, we consider a state of the same form as that in E q. (55), but involving\nentangled coherent states for two modes, which was also discusse d by Zhang [15]. Let\n|ψ/an}b∇acket∇i}ht=N/parenleftBig\n|α/an}b∇acket∇i}hta|β/an}b∇acket∇i}htb+eiθ|α′/an}b∇acket∇i}hta|β′/an}b∇acket∇i}htb/parenrightBig\n, (64)\nwhere|α/an}b∇acket∇i}hta,etcare single-mode coherent states. We will restrict our attention to the case\nwhere the magnitudes of the four complex coherent state parame ters are all equal, and\nα′=−αandβ′=−β. Thus\n|α|=|β|=|α′|=|β′|=σ, (65)\nand\nδ1−δ′\n1=±π, δ 2−δ′\n2=±π. (66)\nHereδ1,δ′\n1,δ2,δ′\n2are the phases of α,α′,β,β′, respectively. In this case, we find\nN=/bracketleftBig\n2(1+cosθe−4σ2)/bracketrightBig−1\n2, (67)\nand\nn1=n2= 2σ2N2(1−cosθe−2σ2),\nR1=R2= 2σ2N2(1+cosθe−2σ2),\nR3=σ2N2(1−cosθe−4σ2),\nR4=σ2, (68)\nas well asγ1= 2δ′\n1,γ2= 2δ′\n2,γ3=δ2−δ1, andγ4=δ1+δ2. Letφ1=k1·x−ω1tand\nφ2=k2·x−ω2t. We then set\nφ1+δ1=φ2+δ2=π\n2, (69)\nwhich can always be done by a suitable choice of xandt. The energy density for a two-mode\ntraveling wave state, Eq. (7) now becomes\nρ=1\nV/bracketleftBig\nn1ω1+n2ω2−R1ω1−R2ω2+(R3−R4)√ω1ω2(1+ˆk1·ˆk2)/bracketrightBig\n.(70)\nIf the two modes are close in wavenumber, so that ω1≈ω2=ωandˆk1≈ˆk2, and we set\nθ= 0 then\nρ=−4ω\nVf(σ), (71)\n160.5 11.5 20.050.10.150.2f (σ) \nσ\nFIG. 5: The function f(σ), given by Eq. (72), is plotted. Its maximum, at σ≈0.7, describes the\ncase of maximal negative energy density for the entangled co herent state defined in Eqs. (64), (65)\nand (66).\nwhere\nf(σ) =σ2e−2σ2(1+e−2σ2)\n1+e−4σ2. (72)\nThe function f(σ) is plotted in Fig. 5, where we see that it attains a maximum value of\nabout 0.22 atσ≈0.7. This corresponds to a negative energy density of ρ≈ −0.88ω/V,\nwhich is about three times as negative as the maximally negative energ y density found in\nthe superposed coherent states discussed in Sect. IIIA.\nV. SUMMARY\nIn this paper we have developed a formalism for parameterizing the e nergy density in\nstates of a massless scalar field in which either one or two modes are e xcited. We found\nexplicit expressions for the energy density for the cases of trave ling waves and of standing\nwaves in one spatial direction. In all cases, the maximum negative en ergy density which can\nbe achieved in a given state can be expressed in terms of our parame ters.\nWe next applied this approach to find the maximum negative energy de nsity in several\nstates, including some states which are of current interest in quan tum optics. For the case\n17of a single mode, we considered three possible superposition states : (1) two coherent states,\n(2) two squeezed vacuum states, and (3) a coherent state and a squeezed vacuum state.\nThe superposition of two coherent states can be described as a Sc hr¨ odinger “cat state” in\nthe sense that it would be a superposition of two classical configura tions in the limit of\nlarge coherent state parameter. Here we find that the maximal ne gative energy density\nis achieved with mean photon numbers slightly less than one. This is an e xample where a\nquantumsuperpositionstatehasnegativeenergydensity, event hougheachcomponentofthe\nsuperposition would have positive energy density by itself. Intheca se of thesuperposition of\ntwo squeezed vacuum states, one finds the opposite effect. Altho ugh here the superposition\nstate does has negative energy density, it is somewhat less negativ e than in the case of a\nsingle squeezed vacuum state. Furthermore, the most negative e nergy density now occurs\nfor small mean photon number, as opposed to large number in the ca se of a single squeezed\nvacuum state. In the case of a superposition of a coherent state and a squeezed vacuum\nstate, we find that for fixed coherent state parameter, there is negative energy density for\nsmall squeeze parameter, and again for larger values, but there is an intermediate range\nwhere the energy density is always positive.\nWe next examined some two-mode states involving entanglement bet ween the two modes,\nincludingtwoexamplesofentangledsqueezedvacuumstates. Thefi rstexample, theBarnett-\nRadmore state [18], exhibits somewhat less negative energy density than would be found if\neach mode were separately in a squeezed vacuum state. In the sec ond example, the Zhang\nstate [15], we find results similar to those in the superposition of sque ezed vacuum states.\nThere is negative energy in the Zhang state, but only for small mean particle numbers.\nFinally, we examined a two-mode entangled coherent state, which als o exhibits negative\nenergy for small mean particle number. It is also similar to the case of a superposition\nof coherent states of a single mode, but the entangled state has s omewhat more negative\nenergy density.\nOneofthemotivationsforthisinvestigation istodrawlinks between t heoretical studies of\nviolations of the weak energy condition, and experimental work in qu antum optics. We hope\nthat this line of work will lead to further experimental studies of sub vacuum phenomena.\n18Acknowledgments\nWe would like to thank Piotr Marecki for useful discussions. This wor k was supported in\npart by the National Science Foundation under Grant PHY-055575 4 to LHF.\nAPPENDIX A\nIn this appendix, we will calculate some of the matrix elements of oper ators such as a†a\nanda2which are needed to find the energy density in the states treated in this paper. We\nbegin with matrix elements between squeezed vacuum states. The d iagonal matrix elements\n/an}b∇acketle{tξ|a†a|ξ/an}b∇acket∇i}htand/an}b∇acketle{tξ|a2|ξ/an}b∇acket∇i}htare given by Eq. (29). We need off-diagonal matrix elements of the\nform/an}b∇acketle{t−ξ|a†a|ξ/an}b∇acket∇i}htand/an}b∇acketle{t−ξ|a2|ξ/an}b∇acket∇i}ht, whereξis real. If we set δ= 0, so that ξ=r, then Eq. (27)\nbecomes\n|ξ/an}b∇acket∇i}ht=√\nsechr∞/summationdisplay\nn=0/radicalBig\n(2n)!\nn!/parenleftbigg\n−1\n2tanhr/parenrightbiggn\n|2n/an}b∇acket∇i}ht. (A1)\nThis leads to the result\n/an}b∇acketle{t−ξ|a†a|ξ/an}b∇acket∇i}ht=/an}b∇acketle{tξ|a†a|−ξ/an}b∇acket∇i}ht= 2sechr∞/summationdisplay\nn=1(2n)!\nn!(n−1)!(−1)n/parenleftbigg1\n2tanhr/parenrightbigg2n\n.(A2)\nUse the fact that\n∞/summationdisplay\nn=1(2n)!\nn!(n−1)!(−1)n/parenleftbigg1\n2x/parenrightbigg(2n−2)\n=−2\n(1+x2)3/2, (A3)\nto find\n/an}b∇acketle{t−ξ|a†a|ξ/an}b∇acket∇i}ht=/an}b∇acketle{tξ|a†a|−ξ/an}b∇acket∇i}ht=−sechrtanh2r\n(1+tanh2r)3/2. (A4)\nSimilarly, we may use Eq. (A1) to show that\n/an}b∇acketle{t−ξ|a2|ξ/an}b∇acket∇i}ht=−/an}b∇acketle{tξ|a2|−ξ/an}b∇acket∇i}ht= sechr∞/summationdisplay\nn=1(2n)!\nn!(n−1)!(−1)n/parenleftbigg1\n2tanhr/parenrightbigg(2n−1)\n=−sechrtanhr\n(1+tanhr)3/2.\n(A5)\nBecause these matrix elements are real, we have that\n/an}b∇acketle{t−ξ|(a†)2|ξ/an}b∇acket∇i}ht=/an}b∇acketle{tξ|(a†)2|−ξ/an}b∇acket∇i}ht=/an}b∇acketle{t−ξ|a2|ξ/an}b∇acket∇i}ht. (A6)\nIn the present case, the squeeze operator is\nS(ξ) =S(r) = e1\n2r[a2−(a†)2]. (A7)\n19From this relation, we see that\n/an}b∇acketle{t−ξ|ξ/an}b∇acket∇i}ht=/an}b∇acketle{tξ|−ξ/an}b∇acket∇i}ht=/an}b∇acketle{t0|S2(r)|0/an}b∇acket∇i}ht=/an}b∇acketle{t0|S(2r)|0/an}b∇acket∇i}ht=/radicalBig\nsech(2r). (A8)\nNext we derive the matrix elements involving both a coherent state a nd a squeezed\nvacuum state that are needed in Sect. IIIC. A coherent state ma y be represented in terms\nof number eigenstates as [18]\n|α/an}b∇acket∇i}ht= e−1\n2|α|2∞/summationdisplay\nℓ=0αℓ\n√\nℓ!|ℓ/an}b∇acket∇i}ht. (A9)\nThis may be combined with Eq. (27) to show that\n/an}b∇acketle{tξ|α/an}b∇acket∇i}ht=/an}b∇acketle{tα|ξ/an}b∇acket∇i}ht∗= e−1\n2|α|2√\nsechr∞/summationdisplay\nn=0α2n\nn![−1\n2e−iδtanhr]n\n= e−1\n2|α|2√\nsechrexp/parenleftbigg\n−1\n2e−iδα2tanhr/parenrightbigg\n, (A10)\nand\n/an}b∇acketle{tξ|a†a|α/an}b∇acket∇i}ht=/an}b∇acketle{tα|a†a|ξ/an}b∇acket∇i}ht∗=−e−1\n2|α|2√\nsechre−iδα2tanhrexp/parenleftbigg\n−1\n2e−iδα2tanhr/parenrightbigg\n.(A11)\nNote that\n/an}b∇acketle{tξ|a2|α/an}b∇acket∇i}ht=/an}b∇acketle{tα|(a†)2|ξ/an}b∇acket∇i}ht∗=α2/an}b∇acketle{tξ|α/an}b∇acket∇i}ht. (A12)\nFinally, we show that\n/an}b∇acketle{tξ|(a†)2|α/an}b∇acket∇i}ht=/an}b∇acketle{tα|a2|ξ/an}b∇acket∇i}ht∗= e−1\n2|α|2√\nsechr∞/summationdisplay\nn=12n(2n−1)\nn!α2n−2/parenleftbigg\n−1\n2e−iδtanhr/parenrightbiggn\n= e−1\n2|α|2√\nsechrd2\ndα2∞/summationdisplay\nn=0α2n\nn!/parenleftbigg\n−1\n2e−iδtanhr/parenrightbiggn\n= e−1\n2|α|2√\nsechrd2\ndα2exp/parenleftbigg\n−1\n2e−iδα2tanhr/parenrightbigg\n= e−1\n2|α|2√\nsechr/parenleftBig\ne−iδα2tanhr−1/parenrightBig\ne−iδtanhr\n×exp/parenleftbigg\n−1\n2e−iδα2tanhr/parenrightbigg\n. (A13)\n[1] H. Epstein, V. Glaser, and A. Jaffe, Nuovo Cim. 36, 1016 (1965).\n[2] H.B.G. Casimir, Proc. K. Ned. Akad. Wet. B51, 793 (1948); L.S. Brown and G.J. Maclay,\nPhys. Rev. 184, 1272 (1969).\n20[3] R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55,\n2409, (1985).\n[4] L. H. Ford, Proc. Roy. Soc. Lond. A364, 227 (1978).\n[5] L. H. Ford, Phys. Rev. D43, 3972 (1991).\n[6] L.H. 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Auffeves, P. Maioli, T. Meunier, S. Gleyzes, G. Nogues, M. Brune, J. M. Raimond, and S.\nHaroche, Phys. Rev. Lett. 91, 230405 (2003).\n[14] P. K. Pathak and G. S. Agarwal, Phys. Rev. A 71, 043823 (2005).\n[15] Zhi-Ming Zhang, Generating superposition and entanglement of squeezed vac uum states ,\nquant-ph/0604128.\n[16] M.J. Pfenning and L.H. Ford, Quantum Inequality Restrictions on Negative Energy Densiti es\nin Curved Spacetimes Doctoral Dissertation, (Dept. of Physics and Astronomy, Tu fts Univer-\nsity), gr-qc/9805037, Sec. 2.1.1.\n[17] C.C. Gerry and P.L. Knight, Introductory Quantum Optics , (Cambridge University Press,\nCambridge, 2005).\n[18] S.M. Barnett and P.M. Radmore, Methods in Theoretical Quantum Optics , (Oxford University\nPress, New York, 1997), Chap. 3.\n21"    },    {        "title": "0705.3574v1.The_geometry_of_density_states__positive_maps_and_tomograms.pdf",        "content": "arXiv:0705.3574v1  [quant-ph]  24 May 2007The geometry of density states, positive maps\nand tomograms\nV.I. Man’ko,1G. Marmo,2E.C.G. Sudarshan3and F. Zaccaria2\n1P.N. Lebedev Physical Institute, Leninskii Prospect, 53, N oscow 119991 Russia\n2Dipartimento di Scienze Fisiche, Universit` a “Federico II ” di Napoli and Istituto\nNazionale di Fisica Nucleare, Sezione di Napoli, Complesso Universitario di Monte\nSant Angelo, Via Cintia, I-80126 Napoli, Italy\n3Physics Department, Center for Particle Physics, Universi ty of Texas, 78712\nAustin, Texas, USA\nEmailsmanko@sci.lebedev.ru marmo@na.infn.it\nsudarshan@physics.utexas.edu zaccaria@na.infn.it\nAbstract. The positive and not completely positive maps of density mat rices,\nwhich are contractive maps, are discussed as elements of a se migroup. A new kind\nof positive map (the purification map), which is nonlinear ma p, is introduced.\nThe density matrices are considered as vectors, linear maps among matrices are\nrepresented by superoperators given in the form of higher di mensional matrices.\nProbability representation of spin states (spin tomograph y) is reviewed and U(N)-\ntomogram of spin states is presented. Properties of the tomo grams as probability\ndistribution functions are studied. Notion of tomographic purity of spin states is\nintroduced. Entanglement and separability of density matr ices are expressed in\nterms of properties of the tomographic joint probability di stributions of random\nspin projections which depend also on unitary group paramet ers. A new positivity\ncriterion for hermitian matrices is formulated. An entangl ement criterion is given\nin terms of a function depending on unitary group parameters and semigroup of\npositive map parameters. The function is constructed as sum of moduli of U(N)-\ntomographic symbols of the hermitian matrix obtained after action on the density\nmatrix of composite system by a positive but not completely p ositive map of the\nsubsystem density matrix. Some two-qubit and two-qutritt s tates are considered as\nexamples of entangled states. The connection with the star- product quantisation is\ndiscussed. The structure of the set of density matrices and t heir relation to unitary\ngroupandLie algebra oftheunitarygroup are studied.Nonli near quantumevolution\nof state vector obtained by means of applying purification ru le of density matrices\nevolving via dynamical maps is considered. Some connection of positive maps and\nentanglement with random matrices is discussed and used.\nKeywords: unitary group, entanglement, adjoint representation, tom ogram, oper-\nator symbol, random matrix.\n1. Introduction\nThestates in quantum mechanics are associated with vectors in Hilbert\nspace [1] (it is better to say with rays) in the case of pure sta tes.\nFor mixed state, one associates the state with density matri x [2, 3].\nc/circlecopyrt2018Kluwer Academic Publishers. Printed in the Netherlands.\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.12 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nIn classical mechanics (statistical mechanics), the state s are associ-\nated with joint probability distributions in phase space. T here is an\nessential difference in the concept of states in classical and quantum\nmechanics. This difference is clearly pointed out by the pheno menon of\nentanglement. Thenotion of entanglement [4] is related to t he quantum\nsuperposition rule of the states of subsystems for a given mu ltipartite\nsystem. For pure states, the notion of entanglement and sepa rability\ncan be given as follows.\nIf the wave function [5] of a state of a bipartite system is rep resented\nas the product of two wave functions depending on coordinate s of the\nsubsystems, the state is simply separable; otherwise, the s tate is sim-\nply entangled. An intrinsic approach to the entanglement me asure was\nsuggested in [6]. The measure was introduced as the distance between\nthe system density matrix and the tensor product of the assoc iated\nstates. For the subsystems, the association being realized via partial\ntraces. There are several other different characteristics an d measures\nof entanglement considered by several authors [7–13]. For e xample,\nthere are measures related to entropy (see, [14–24]). Also l inear entropy\nof entanglement was used in [25–27], “concurrences” in [28, 29] and\n“covariance entanglement measure” in [30]. Each of the enta nglement\nmeasures describes some degree of correlation between the s ubsystems’\nproperties.\nThe notion of entanglement is not an absolute notion for a giv en\nsystem but depends on the decomposition into subsystems. Th e same\nquantum state can be considered as entangled, if one kind of d ivision\nof the system into subsystems is given, or as completely dise ntangled,\nif another decomposition of the system into subsystems is co nsidered.\nFor instance, the state of two continuous quadratures can be entan-\ngled in Cartesian coordinates and disentangled in polar coo rdinates.\nCoordinates are considered as measurable observables labe lling the\nsubsystems of the given system. The choice of different subsys tems\nmathematically implies the existence of two different sets of the sub-\nsystems’ characteristics (we focus on bipartite case). We m ay consider\nthe Hilbert space of states H(1,2) orH(1′,2′). The Hilbert space for\nthe total system is, of course, the same but the index (1 ,2) means\nthat there are two sets of operators P1andP2, which select subsystem\nstates 1 and 2. The index (1′,2′) means that there are two other sets\nof operators P′\n1andP′\n2, which select subsystem states 1′and 2′.The\noperatorsP1,2andP′\n1′,2′have specific properties. They are represented\nas tensor products of operators acting in the space of states of the\nsubsystem 1 (or 2) and unit operators acting in the subsystem 2 (or\n1). In other words, we consider the space H, which can be treated as\nthe tensor product of spaces H(1) andH(2) orH(1′) andH(2′). In\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.2The geometry of density states, positive maps and tomograms 3\nthe subsystems 1 and 2, there are basis vectors |n1/angbracketrightand|m2/angbracketright, and in\nthe subsystems 1′and 2′there are basis vectors |n′\n1/angbracketrightand|m′\n2/angbracketright.The\nvectors|n1/angbracketright |m2/angbracketrightand|n′\n1/angbracketright |m′\n2/angbracketrightform the sets of basis vectors in\nthe composite Hilbert space, respectively. These two sets a re related\nby means of unitary transformation. An example of such a comp osite\nsystem is a bipartite spin system.\nIf one has spin- j1[the space H(1)] and spin- j2[the space H(2)]\nsystems, the combined system can be treated as having basis\n|j1m1/angbracketright |j2m2/angbracketright.\nAnotherbasisinthecomposite-system-state spacecanbeco nsidered\nintheform |jm/angbracketright, wherejisoneofthenumbers |j1−j2|,|j1−j2|+1,...,\nj1+j2andm=m1+m2. The basis |jm/angbracketrightis related to the basis\n|j1m1/angbracketright |j2m2/angbracketrightby means of the unitary transform given by Clebsch–\nGordon coefficients C(j1m1j2m2|jm). From the viewpoint of the given\ndefinition, the states |jm/angbracketrightare entangled states in the original basis.\nAnother example is the separation of the hydrogen atom in ter ms of\nparabolic coordinates used while discussing the Stark effect .\nThe spin states can be described by means of the tomographic\nmap [31–33]. For bipartite spin systems, the states were des cribed\nby the tomographic probabilities in [34, 35]. Some properti es of the\ntomographic spin description were studied in [36]. In the to mographic\napproach, the problems of the quantum state entanglement ca n be\ncast into the form of some relations among the probability di stribution\nfunctions. On the other hand, to have a clear picture of entan glement,\none needs a mathematical formulation of the properties of th e density\nmatrix of the composite system, a description of the linear s pace of\nthe composite system states. Since a density matrix is hermi tian, the\nspace of states may be embedded as a subset of the Lie algebra o f\nthe unitary group, carrying the adjoint representation of U(n2), where\nn2= (2j+ 1)2is the dimension of the spin states of two spinning\nparticles. Thus one may try to characterize the entanglemen t phenom-\nena by using various structures present in the space of the ad joint\nrepresentation of the U(n2) group.\nThe aim of this paper is to give a review of different aspects of\ndensity matrices and positive maps and connect entanglemen t prob-\nlems with the properties of tomographic probability distri butions and\ndiscuss the properties of the convex set of positive states f or composite\nsystembytakingintoaccount thesubsystemstructures.Weu sed[6]the\nHilbert–Schmidt distance to calculate the measure of entan glement as\nthe distance between a given state and the tensor productof t he partial\ntraces of the density matrix of the given state. In [37] anoth er measure\nof entanglement as a characteristic of subsystem correlati ons was intro-\nduced. This measure is determined via the covariance matrix of some\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.34 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nobservables. A review of different approaches to the entangle ment no-\ntion and entanglement measures is given in [38] where the app roach to\ndescribe entanglement and separability of composite syste ms is based,\ne.g., on entropy methods.\nDue to a variety of approaches to the entanglement problem, o ne\nneeds to understand better what in reality the word “entangl ement”\ndescribes. Is it a synonym of the word “correlation” between two sub-\nsystems or does it have to capture some specific correlations attributed\ncompletely and only to the quantum domain?\nThe paper is organized as follows.\nIn section 2 we discuss division of composite systems onto su bsys-\ntems and relation of the density matrix to adjoint represent ation of\nunitary group in generic terms of vector representation of m atrices;\nwe study also completely positive maps of density matrices. In sec-\ntion 3 we consider a vector representation of probability di stribution\nfunctions and notion of distance between the probability di stributions\nand density matrices. In section 4 we present definition of se parable\nquantum state of a composite system and criterion of separab ility. In\nsection 5 the entanglement is considered in terms of operato r symbols.\nParticular tomographic probability representation of qua ntum states\nand tomographic symbols are reviewed in section 6. Symbols o f mul-\ntipartite states are studied in section 7. In section 8 spin t omography\nis reviewed. An example of qubit state is done in section 9. Th e uni-\ntary spin tomogram is introduced in section 10 while in secti on 11\ndynamical map and corresponding quantum evolution equatio ns are\ndiscussed as well as examples of concrete positive maps. Con clusions\nand perspectives are presented in section 12.\n2. Composite system\nIn this section, we review the meaning and notion of composit e system\nin terms of additional structures on the linear space of stat e for the\ncomposite system.\n2.1.Difference of states and observables\nIn quantum mechanics, there are two basic aspects, which are associ-\nated with linear operators acting in a Hilbert space. The firs t one is\nrelated to the concept of quantum state and the second one, to the\nconcept of observable. These two concepts of state and obser vable are\npaired via a map with values in probability measures on the re al line.\nOften states are described by Hermitian nonnegative, trace -class, ma-\ntrices. The observables are described by Hermitian operato rs. Though\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.4The geometry of density states, positive maps and tomograms 5\nbothstates and observables are identified with Hermitian ob jects, there\nis an essential difference between the corresponding objects . The ob-\nservables have an additional product structure. Thus we may consider\nthe product of two linear operators corresponding to observ ables.\nFor the states, the notion of product is redundant. The produ ct of\ntwo states is not a state. For states, one keeps only the linea r structure\nof vector space. For finite n-dimensional system, the Hermitian states\nand the Hermitian observables may be mapped into the Lie alge bra\nof the unitary group U(n). But the states correspond to nonnegative\nHermitian operators. Observables can be associated with bo th types\nof operators, including nonnegative and nonpositive ones. The space\nof states is therefore a convex-linear space which, in princ iple, is not\nequipped with a product structure. Due to this, transformat ions in the\nlinearspaceofstates neednotpreserveanyproductstructu re.Intheset\nofobservables,onehastobeconcernedwithwhatishappenin gwiththe\nproduct of operators when some transformations are perform ed. State\nvectors can be transformed into other state vectors. Densit y operators\nalso can be transformed. We will consider linear transforma tions of the\ndensityoperators. Thedensity operator hasnonnegative ei genvalues. In\nany representation, diagonal elements of density matrix ha ve physical\nmeaning of probability distribution function.\nDensity operator can be decomposed as a sum of eigenprojecto rs\nwith coefficients equal to its eigenvalues. Each one of the pro jectors\ndefines a pure state. There exists a basis in which every eigen projector\nof rank one is represented by a diagonal matrix of rank one wit h only\none matrix element equal to one and all other matrix elements equal\nto zero. Other density matrices with similar properties bel ong to the\norbit of the unitary group on the starting eigenprojector. D ependingon\nthe number of distinct nonzero values determines the class o f the orbit.\nSince density matrices of higher rank belong to an appropria te orbit\nof a convex sum of the different diagonal eigenprojectors (in s pecial\nbasis), we may say that generic density matrices belong to th e orbits of\nthe unitary group acting on the diagonal density matrices wh ich belong\nto the Cartan subalgebra of the Lie algebra of the unitary gro up. Any\nconvex sum of density matrices can be treated as a mean value o f a\nrandom density matrix. The positive coefficients of the conve x sum can\nbe interpreted as a probability distribution function whic h makes the\naveragingprovidingthefinalvalueoftheconvex sum.Theset ofdensity\nmatrices may be identified with the union of the orbits of the u nitary\ngroup acting on diagonal density matrices considered as ele ments of\nthe Cartan subalgebra.\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.56 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\n2.2.Matrices as vectors, density operators and\nsuperoperators\nWhen matrices represent states it may be convenient to ident ify them\nwithvectors. Inthiscase, adensitymatrixcanbeconsidere dasavector\nwith additional properties of its components. If the identi fications are\ndone elegantly, we can see the real Hilbert space of density m atrices in\nterms of vectors with real components. In this case, linear t ransforms\nof the matrix can be interpreted as matrices called superope rators. It\nmeans that density matrices–vectors undergoing real linea r transfor-\nmations are acted on by the matrices representing the action of the\nsuperoperators of the linear map. This construction can be c ontinued.\nThus we can get a chain of vector spaces of higher and higher di men-\nsions. Let us first introduce some extra constructions of the map of\na matrix onto a vector. Given a rectangular matrix Mwith elements\nMid, wherei= 1,2,...,nandd= 1,2,...,m, one can consider the\nmatrix as a vector /vectorMwithN=nmcomponents constructed by the\nfollowing rule:\nM1=M11,M2=M12,Mm=M1m,\n(1)\nMm+1=M21,...MN=Mnm.\nThus we construct the map M→/vectorM=ˆt/vectorMMM.\nWe have introduced thelinear operator ˆt/vectorMMwhich maps the matrix\nMonto avector /vectorM.Now weintroducetheinverseoperator ˆ p/vectorMMwhich\nmapsagivencolumnvectorinthespacewithdimension N=mnontoa\nrectangular matrix. Thismeansthat given avector /vectorM=M1,...,MN,\nwe relabel its components by introducing two indices i= 1,...,nand\nd= 1,...,m. The relabeling is accomplished according to (1). Then\nwe collect the relabeled components into a matrix table. Eve ntually we\nget the map\nˆp/vectorMM/vectorM=M. (2)\nThe composition of these two maps\nˆt/vectorMMˆp/vectorMM/vectorM=1·/vectorM (3)\nacts as the unit operator in the linear space of vectors.\nGiven an×nmatrix the map considered can also be applied. The\nmatrix can be treated as an n2-dimensional vector and, vice versa, the\nvector of dimension n2may be mappedby this procedureonto the n×n\nmatrix.\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.6The geometry of density states, positive maps and tomograms 7\nLet us consider a linear operator acting on the vector /vectorMand related\nto a linear transform of the matrix M. First, we study the correspon-\ndence of the linear transform of the form\nM→gM=Ml\ng (4)\nto the transform of the vector\n/vectorM →/vectorMl\ng=Ll\ng/vectorM. (5)\nOne can show that the n2×n2matrixLl\ngis determined by the tensor\nproduct of the n×nmatrixgandn×nunit matrix, i.e.,\nLl\ng=g⊗1. (6)\nAnalogously, the linear transform of the matrix Mof the form\nM→Mg=Mr\ng (7)\ninduces the linear transform of the vector /vectorMof the form\n/vectorM →/vectorMr\ng=ˆt/vectorMMMr\ng=Lr\ng/vectorM, (8)\nwhere then2×n2matrixLr\ngreads\nLr\ng= 1⊗gtr. (9)\nSimilarity transformation of the matrix Mof the form\nM→gMg−1(10)\ninduces the corresponding linear transform of the vector /vectorMof the form\n/vectorM →/vectorMs=Ls\ng/vectorM, (11)\nwhere then2×n2matrixLs\ngreads\nLs\ng=g⊗(g−1)tr. (12)\nStarting with vectors, one may ask how to identify on them a pr oduct\nstructure which would make ˆ p/vectorMNinto an algebra homomorphism. An\nassociative algebraic structure on the vector space may be d efined by\nimitating the procedure one uses to define star-products on t he space\nof functions on phase space. One can define the associative pr oduct of\ntwoN-vectors/vectorM1and/vectorM2using the rule\n/vectorM=/vectorM1⋆/vectorM2, (13)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.78 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nwhere\n/vectorMk=N/summationdisplay\nl,s=1Kk\nls(/vectorM1)l(/vectorM2)s. (14)\nIf one applies a linear transform to the vectors /vectorM1,/vectorM2,/vectorMof the form\n/vectorM1→/vectorM′\n1=L/vectorM1,/vectorM2→/vectorM′\n2=L/vectorM2,/vectorM →/vectorM′=L/vectorM,\nand requires the invariance of the star-product kernel, one finds\n/vectorM′\n1⋆/vectorM′\n2=/vectorM′,ifL=G⊗G−1tr, G∈GL(n).\nThe kernel Kk\nls(structure constants) which determines the associative\nstar-product satisfies a quadratic equation. Thus if one wan ts to make\nthe correspondence of the vector star-product to the standa rd matrix\nproduct (row by column), the matrix Mmust be constructed appro-\npriately. For example, if the vector star-product is commut ative, the\nmatrixMcorrespondingtothe N-vector/vectorMcanbechosenasadiagonal\nN×Nmatrix.Thisconsiderationshowsthatthemapofmatrices on the\nvectors provides the star-product of the vectors (defining t he structure\nconstantsorthekernelofthestar-product)and,conversel y, ifonestarts\nwith vectors and uses matrices with the standard multiplica tion rule,\nit will be the map to be determined by the structure constants (or by\nthe kernel of the vector star-product).\nThe constructed space of matrices associated with vectors e nables\noneto enlarge thedimensionality of thegroup acting in the l inear space\nof matrices in comparison with the standard one, i.e., we may relax the\nrequirement of invariance of the product structure. In gene ral, given\nan×nmatrixMthe left action, the right action, and the similarity\ntransformation of the matrix are related to the complex grou pGL(n).\nOn the other hand, the linear transformations in the linear s pace ofn2-\nvectors/vectorMobtained by using the introduced map are determined by\nthematrices belongingto thegroup GL(n2). Thereare transformations\non the vectors which cannot be simply represented on matrices. If\nM→Φ(M) is a linear homogeneous function of the matrix M, we\nmay represent it by\nΦab=Baa′,bb′Ma′b′.\nUnder rather clear conditions, Baa′,bb′can be expressed in terms of its\nnonnormalized left and right eigenvectors:\nBaa′,bb′=/summationdisplay\nνxaa′(ν)y†\nbb′(ν),\nbeing an index for eigenvalues, which corresponds to\nΦ(M) =xMy†=n2/summationdisplay\nν=1x(ν)My†(ν).\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.8The geometry of density states, positive maps and tomograms 9\nTherearepossiblelineartransformsonthematrices andcor respond-\ning linear transforms on the induced vector space which dono t give rise\nto a group structure but possess only the structure of algebr a. One can\ndescribe the map of n×nmatricesM(source space) onto vectors /vectorM\n(target space) usingspecificbasisinthespaceofthematric es. Thebasis\nis given by the matrices Ejk(j,k= 1,2,...,n) with all matrix elements\nequal to zero except the element in the jth row and kth column which\nis equal to unity. One has the obvious property\nMjk= Tr(MEjk). (15)\nInourprocedure,thebasismatrix Ejkismappedontothebasiscolumn-\nvector/vectorEjk, which has all components equal to zero except the unity\ncomponent related to the position in the matrix determined b y the\nnumbersjandk. Then one has\n/vectorM=n/summationdisplay\nj,k=1Tr(MEjk)/vectorEjk. (16)\nFor example, for similarity transformation of the finite mat rixM, one\nhas\n/vectorMs\ng=N/summationdisplay\nj,k=1Tr/parenleftig\ngMg−1Ejk/parenrightig/vectorEjk. (17)\nNow we will define the notion of ‘composite’ vector which corr e-\nsponds to dividing a quantum system into subsystems.\nWe will use the following terminology.\nIn general, the given linear space of dimensionality N=mnhas\na structure of a bipartite system, if the space is equipped wi th the\noperator ˆp/vectorMMand the matrix M(obtained by means of the map) has\nmatrix elements in factorizable form\nMid→xiyd. (18)\nThisM=x⊗ycorresponds to the special case of nonentangled states.\nOtherwise, one needs\nM=/summationdisplay\nνx(ν)⊗y(ν).\nIn fact, to consider in detail the entanglement phenomenon, in the\nbipartite system of spin-1/2, one has to introduce a hierarc hy of three\nlinear spaces. The first space of pure spin states is the two-d imensional\nlinear space of complex vectors\n|/vector x/angbracketright=/parenleftbiggx1\nx2/parenrightbigg\n. (19)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.910 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nIn this space, the scalar product is defined as follows:\n/angbracketleft/vector x|/vector y/angbracketright=x∗\n1y1+x∗\n2y2. (20)\nSo it is a two-dimensional Hilbert space. We do not equip this space\nwith a vector star-product structure. In the primary linear space, one\nintroduces linear operators ˆMwhich are described by 2 ×2 matrices\nM. Due to the map discussed in the previous section, the matric es\nare represented by 4-vectors /vectorMbelonging to the second complex 4-\ndimensional space. The star-product of the vectors /vectorMdetermined by\nthe kernel Kk\nlsis defined in such a manner in order to correspond to\nthe standard rule of multiplication of the matrices.\nIn addition to the star-product structure, we introduce the scalar\nproduct of the vectors /vectorM1and/vectorM2, in view of the definition\n/angbracketleft/vectorM1|/vectorM2/angbracketright= Tr(M†\n1M2), (21)\nwhich is the trace formula for the scalar product of matrices .\nThis means introducing the real metric gαβin the standard notation\nfor scalar product\n/angbracketleft/vectorM1|/vectorM2/angbracketright=4/summationdisplay\nα,β=1(M1)∗\nαgαβ(M2)β, (22)\nwhere the matrix gαβis of the form\ngαβ=\n1 0 0 0\n0 0 1 0\n0 1 0 0\n0 0 0 1\n, gαjgjβ=δαβ. (23)\nThe scalar product is invariant under the action of the group of non-\nsingular 4 ×4 matricesℓ, which satisfy the condition\ng=ℓ†gℓ. (24)\nThe product of matrices ℓsatisfies the same condition since g2= 1.\nThus, the space of operators ˆMin the primary two-dimensional\nspace of spin states is mapped onto the linear space which is e quipped\nwith a scalar product (metric Hilbert space structure) and a n associa-\ntive star-product (kernel satisfying the quadratic associ ativity equa-\ntion). In the linear space of the 4-vectors /vectorM, we introduce linear\noperators (superoperators), which can be associated with t he algebra\nof 4×4 complex matrices.\nLet usnow focuson densitymatrices. Thismeans thatourmatr ixM\nis considered as a density matrix ρwhich describes a quantum state.\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.10The geometry of density states, positive maps and tomograms 11\nWe consider here the action of the unitary transformation U(n) of\nthe density matrices and corresponding transformations on the vector\nspace. If one has the structure of a bipartite system, we also consider\nthe action of local gauge transformation both in the “source space” of\ndensitymatrices andinthe“target space”of thecorrespond ingvectors.\nThen×ndensity matrix ρhas matrix elements\nρik=ρ∗\nki,Trρ= 1,/angbracketleftψ|ρ|ψ/angbracketright ≥0. (25)\nSince the density matrix is hermitian, it can always be ident ified as an\nelement of the convex subset of the linear space associated w ith the Lie\nalgebra ofU(n) group, on which the group U(n) acts with the adjoint\nrepresentation\nρ→ρU=UρU†. (26)\nThe system is said to be bipartite if the space of representat ion is\nequipped with an additional structure. This means that for\nn2=n1·n2,\nwhere, for simplicity, n1=n2=n, one can make first the map of\nn×nmatrixρonton2-dimensional vector /vector ρaccording to the previous\nprocedure, i.e., one equips the space by an operator ˆt/vector ρρ. Given this\nvector one makes a relabeling of the vector /vector ρcomponents according to\nthe rule\n/vector ρ→ρid,ke, i,k= 1,2,...,n 1, d,e= 1,2,...,n 2,(27)\ni.e., obtaining again the quadratic matrix\nρq= ˆpρq/vector ρ/vector ρ. (28)\nThe unitary transform (26) of the density matrix induces a li near\ntransform of the vector /vector ρof the form\n/vector ρ→/vector ρU= (U⊗U∗)/vector ρ. (29)\nThere exist linear transforms (called positive maps) of the density ma-\ntrix, which preserve its trace, hermicity, and positivity. In some cases,\nthey have the following form introduced in [39]\nρ0→ρU=LUρ0=/summationdisplay\nkpkUkρ0U†\nk,/summationdisplay\nkpk= 1,(30)\nwhereUkare unitary matrices and pkare positive numbers.\nIf the initial density matrix is diagonal, i.e., it belongs t o the Cartan\nsubalgebra of the Lie algebra of the unitary group, the diago nal ele-\nmentsoftheobtainedmatrixgivea“smoother”probabilityd istribution\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.1112 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nthan the initial one. A generic transformation preserving p reviously\nstated properties may be given in the form (see [39, 40])\nρ0→ρV=LVρ0=/summationdisplay\nkVkρ0V†\nk,/summationdisplay\nkV†\nkVk= 1.(31)\nFor example, if Vk(k= 1,2,...,N) are taken as square roots of or-\nthogonal projectors onto complete set of Nstate, the map provides\nthe map of the density matrix ρ0onto diagonal density matrix ρ0d\nwhich has the same diagonal elements as ρ0has. In this case, the\nmatricesVkhave the only nonzero matrix element which is equal to\none. Such a map may be called “decoherence map” because it rem oves\nall nondiagonal elements of the density matrix ρ0killing any phase\nrelations. In quantum information terminology, one uses al so the name\n“phase damping channel.” More general map may be given if one takes\nVkasNgeneric diagonal density matrices, in which eigenvalues ar e\nobtained by Ncircular permutations from the initial one. Due to this\nmap, one has a new matrix with the same diagonal matrix elemen ts\nbut with changed nondiagonal elements. The purity of this ma trix is\nsmaller then the purity of the initial one. This means that th e map\nis contractive. All matrices with the same diagonal element s up to\npermutations belong to a given orbit of the unitary group.\nFor a large number of terms with randomly chosen matrices Vkin\nthesumin(31), theabovemapgives themost stochasticdensi ty matrix\nρ0→ρs=L1ρ0= (n)−11.\nIts four-dimensional matrix L1for the qubit case has four matrix el-\nements different from zero. These matrix elements are equal to one.\nThey have the labels L11,L14,L41,L44. The map with two nonzero\nmatrix elements L41=L44= 0provides pure-statedensity matrix from\nanyρ0. The transform (30) is the partial case of the transform (31) . We\ndiscuss the transforms separately since they are used in the literature\nin the presented form.\nOne can see that the constructed map of density matrices onto\nvectors provides the corresponding transforms of the vecto rs, i.e.,\n/vector ρ0→/vector ρU=/summationdisplay\nkpk(Uk⊗U∗\nk)/vector ρ0 (32)\nand\n/vector ρ0→/vector ρV=/summationdisplay\nk(Vk⊗V∗\nk)/vector ρ0. (33)\nIt is obvious that the set of linear transforms of vectors, wh ich preserve\ntheir properties of being image of density matrices, is esse ntially larger\nthan the standard unitary transform of the density matrices .\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.12The geometry of density states, positive maps and tomograms 13\nFormulae (32) and (33) mean that the positive map superopera tors\nacting on the density matrix in the vector representation ar e described\nbyn2×n2matrices\nLU=/summationdisplay\nkpk(Uk⊗U∗\nk) (34)\nand\nLV=/summationdisplay\nkVk⊗V∗\nk, (35)\nrespectively.\nThe positive map is called “noncompletely positive” if\nL=/summationdisplay\nkVk⊗V∗\nk−/summationdisplay\nsvs⊗v∗\ns,/summationdisplay\nkV†\nkVk−/summationdisplay\nsv†\nsvs= 1.\nThis map is related to a possible “nonphysical” evolution of a subsys-\ntem.\nFormula(34) can beconsideredinthecontext of randommatri x rep-\nresentation. In fact, the matrix LUcan be interpreted as the weighted\nmean value of the random matrix Uk⊗U∗\nk. The dependence of matrix\nelements and positive numbers pkon indexkmeans that we have a\nprobabilitydistributionfunction pkandaveragingoftherandommatrix\nUk⊗U∗\nkby means of the distribution function. So the matrix LUreads\nLU=/angbracketleftU⊗U∗/angbracketright. (36)\nLet us consider an example of a 2 ×2 unitary matrix. We can consider\na matrix of the SU(2) group of the form\nu=/parenleftbiggα β\n−β∗α∗/parenrightbigg\n,|α|2+|β|2= 1. (37)\nThe 4×4 matrix LUtakes the form\nLU=\nℓ m m∗1−ℓ\n−n s −q n\n−n∗−q∗s∗n∗\n1−ℓ−m−m∗ℓ\n. (38)\nThe matrix elements of the matrix LUare the means\nm=/angbracketleftαβ∗/angbracketright,\nℓ=/angbracketleftαα∗/angbracketright,\nn=/angbracketleftαβ/angbracketright, (39)\ns=/angbracketleftα2/angbracketright,\nq=/angbracketleftβ2/angbracketright.\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.1314 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nThe moduli of these matrix elements are smaller than unity.\nThe determinant of the matrix LUreads\ndetLU= (1−2ℓ)/parenleftig\n|q|2−|s|2/parenrightig\n+4Re/bracketleftig\nq∗m∗n+mns∗/bracketrightig\n.(40)\nIf one represents the matrix LUin block form\nLU=/parenleftbiggA B\nC D/parenrightbigg\n, (41)\nthen\nA=/parenleftbiggℓ m\n−n s/parenrightbigg\n, B=/parenleftbiggm∗1−ℓ\n−q n/parenrightbigg\n, (42)\nand\nD=σ2A∗σ2, C=−σ2B∗σ2, (43)\nwhereσ2is the Pauli matrix.\nOne can check that the product of two different matrices LUcan\nbe cast in the same form. This means that the matrices LUform a\n9-parameter compact semigroup. It means that the product of two\nmatrices from the set (semigroup) belongs to the same set. It means\nthat composition is inner like the one for groups. There is a u nity\nelement in the semigroup, however, there exist elements whi ch have no\ninverse. In our case, these elements are described, e.g., by the matrices\nwith zero determinant. Also the elements, which are matrice s with\nnonzero determinants, have no inverse elements in this set, since the\nmap corresponding to the inverse of these matrices is not pos itive one.\nFor example, in the case ℓ= 1/2 andm= 0, one has the matrices\nA=/parenleftbigg1/2 0\n−n s/parenrightbigg\n, B=/parenleftbigg0 1/2\n−q n/parenrightbigg\n. (44)\nThe determinant of the matrix LUin this case is equal to zero. All the\nmatrices LUhave the eigenvector\n/vector ρ0=\n1/2\n0\n0\n1/2\n, (45)\ni.e.,\nLU/vector ρ0=/vector ρ0. (46)\nThis eigenvector corresponds to the density matrix\nρ1=/parenleftbigg1/2 0\n0 1/2/parenrightbigg\n, (47)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.14The geometry of density states, positive maps and tomograms 15\nwhich is obviously invariant of the positive map.\nFor random matrix, one has correlations of the random matrix ele-\nments, e.g., /angbracketleftαα∗/angbracketright /negationslash=/angbracketleftα/angbracketright/angbracketleftα∗/angbracketright.\nThe matrix Lp\nLp=\n1 0 0 0\n0 0 1 0\n0 1 0 0\n0 0 0 1\n(48)\nmaps the vector\n/vector ρin=\nρ11\nρ12\nρ21\nρ22\n(49)\nonto the vector\n/vector ρt=\nρ11\nρ21\nρ12\nρ22\n. (50)\nThis means that the positive map (48) connects the positive d ensity\nmatrix with its transpose (or complex conjugate). This map c an be\npresented as the connection of the matrix ρwith its transpose of the\nform\nρ→ρtr=ρ∗=1\n2/parenleftig\nρ+σ1ρσ1−σ2ρσ2+σ3ρσ3/parenrightig\n.\nThere is no unitary transform connecting these matrices.\nThere is noncompletely positive map in the N-dimensional case,\nwhich is given by the generalized formula (for some ε)\nρ→ρs=−ερ+1+ε\nN1N.\nIn quantum information terminology, it is called “depolari zing chan-\nnel.”\nFor the qubit case, matrix form of this map reads\nL=\n1−ε\n20 01+ε\n2\n0−ε0 0\n0 0−ε0\n1+ε\n20 01−ε\n2\n. (51)\nThusweconstructedthematrixrepresentationofthepositi vemapof\ndensityoperators ofthespin-1/2system. Thisparticulars et of matrices\nrealize the representation of the semigroup of real numbers −1≤ε≤1.\nIf one considers the product ε1ε2=ε3, the result ε3belongs to the\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.1516 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nsemigroup. Only two elements 1 and −1 of the semigroup have the\ninverse. These two elements form the finite subgroup of the se migroup.\nThe semigroup itself without element ε= 0 can be embedded into the\ngroup of real numbers with natural multiplication rule. Eac h matrix\nLhas an inverse element in this group but all the parameters of the\ninverse elements ηlive out of the segment −1,1. The group of the real\nnumbersiscommutative. Thematricesofthenonunitaryrepr esentation\nof this group commute too. It means that we have nonunitary re ducible\nrepresentation of the semigroup which is also commutative. To con-\nstruct this representation, one needs to use the map of matri ces on the\nvectors discussed in the previous section. Formulae (31) an d (35) can\nbe interpreted also in the context of the random matrix repre sentation,\nbut we use the uniform distribution for averaging in this cas e. So one\nhas equality (35) in the form\nLV=/angbracketleftV⊗V∗/angbracketright (52)\nand the equality\n/angbracketleftV†V/angbracketright= 1, (53)\nwhich provides constraints for the random matrices Vused.\nUsing the random matrix formalism, the positive (but not com -\npletely positive) maps can be presented in the form\nL=/angbracketleftV⊗V∗/angbracketright−/angbracketleftv⊗v∗/angbracketright,/angbracketleftV†V/angbracketright−/angbracketleftv†v/angbracketright= 1.\nOne can characterize the action of positive map on a density m atrixρ\nby the parameter\nκ=Tr(Lρ)2\nTrρ2=µLρ\nµρ≤1.\nAs a remark we note that in [39] the positive maps (30) and (31) were\nused to describe the non-Hamiltonian evolution of quantum s tates for\nopen systems.\nWe haveto pointout that, in general, such evolution isnot de scribed\nby first-order-in-time differential equation. As in the previ ous case, if\nthere are additional structures for the matrix in the form\nρid,ke→xiydzkte, (54)\nwhich means associating with the initial linear space two ex tra lin-\near spaces where xi,zkare considered as vector components in the\nn1-dimensional linear space and yd,teare vector components in n2-\ndimensional vector space, we see that one has bipartite stru cture of\nthe initial space of state [bipartite structure of the space of adjoint\nrepresentations of the group U(n)].\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.16The geometry of density states, positive maps and tomograms 17\nUsually the adjoint representation of any group is defined pe r se\nwithout any reference to possible substructures. Here we in troduce\nthe space with extra structure. In addition to being the spac e of the\nadjointrepresentation of thegroup U(n), ithasthestructureofabipar-\ntite system. The generalization to multipartite ( N-partite) structure is\nstraightforward. One needs only the representation of posi tive integer\nn2in the form\nn2=N/productdisplay\nk=1n2\nk. (55)\nIf one considers the more general map given by superoperator (35)\nrewritten in the form\nLV=/angbracketleftV⊗V∗/angbracketright,/angbracketleftV†V/angbracketright= 1,\nthe number of parameters determining the matrix LVcan be easily\nevaluated. For example, for n= 2,\nV=/parenleftbigga b\nc d/parenrightbigg\n, V∗=/parenleftbigga∗b∗\nc∗d∗/parenrightbigg\n,\nwhere the matrix elements are complex numbers, the normaliz ation\ncondition provides four constraints for the real and imagin ary parts of\nthe matrix elements of the following matrix:\nLV=\n/angbracketleft|a|2/angbracketright /angbracketleftab∗/angbracketright /angbracketleftba∗/angbracketright /angbracketleftbb∗/angbracketright\n/angbracketleftac∗/angbracketright /angbracketleftad∗/angbracketright /angbracketleftbc∗/angbracketright /angbracketleftbd∗/angbracketright\n/angbracketleftca∗/angbracketright /angbracketleftcb∗/angbracketright /angbracketleftda∗/angbracketright /angbracketleftdb∗/angbracketright\n/angbracketleftcc∗/angbracketright /angbracketleftcd∗/angbracketright /angbracketleftdc∗/angbracketright /angbracketleftdd∗/angbracketright\n,\nnamely,\n/angbracketleft|a|2/angbracketright+/angbracketleft|c|2/angbracketright= 1,/angbracketleft|b|2/angbracketright+/angbracketleft|d|2/angbracketright= 1,/angbracketlefta∗b/angbracketright+/angbracketleftc∗d/angbracketright= 0.\nDuetothestructureofthematrix LV,therearesixcomplexparameters\n/angbracketleftab∗/angbracketright,/angbracketleftac∗/angbracketright,/angbracketleftad∗/angbracketright,/angbracketleftbc∗/angbracketright,/angbracketleftbd∗/angbracketright,/angbracketleftcd∗/angbracketright\nor 12 real parameters.\nThe geometrical picture of the positive map can be clarified i f one\nconsiders the transform of the positive density matrix into another\ndensity matrix as the transform of an ellipsoid into another ellipsoid. A\ngeneric positive transform means a generic transform of the ellipsoid,\nwhich changes its orientation, values of semiaxis, and posi tion in the\nspace. But the transform does not change the ellipsoidal sur face into a\nhyperboloidalor paraboloidal surface. For purestates, th epositive den-\nsity matrix defines the quadratic form which is maximally deg enerated.\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.1718 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nIn this sense, the “ellipsoid” includes all its degenerate f orms corre-\nsponding to the density matrix of rank less than n(inn-dimensional\ncase). The number of parameters defining the map /angbracketleftV⊗V∗/angbracketrightin the\nn-dimensional case is equal to n2(n2−1).\nThe linear space of Hermitian matrices is also equipped with the\ncommutator structure defining the Lie algebra of the group U(n). The\nkernel that defines this structure (Lie product structure) i s determined\nby the kernel that determines the star-product.\n3. Distributions as vectors\nIn quantum mechanics, one needs the concept of distance betw een the\nquantum states. In this section, we consider the notion of di stance\nbetween the quantum states in terms of vectors. First, let us discuss\nthe notion of distance between conventional probability di stributions.\nThis notion is well known in the classical probability theor y.\nGiven the probability distribution P(k),k= 1,2,...N, one can\nintroduce the vector /vectorPin the form of a column with components P1=\nP(1), P2=P(2),..., P N=P(N).The vector satisfies the condition\nN/summationdisplay\nk=1Pk= 1. (56)\nThis set of vectors does not form a linear space but only a conv ex sub-\nset. Nevertheless, in this set one can introduce a distance b etween two\ndistributions by using the one suggested by the vector space structure\nof the ambient space:\nD2=/parenleftig/vectorP1−/vectorP2/parenrightig2=/summationdisplay\nkP1kP1k+/summationdisplay\nkP2kP2k−2/summationdisplay\nkP1kP2k.(57)\nOf course, one may use other identifications of distribution s with\nvectors.\nSince allP(k)≥0, one can introduce Pk=/radicalbig\nP(k) as components\nof the vector /vectorP. The/vectorPcan be thought of as a column with nonnegative\ncomponents. Then the distance between the two distribution s takes the\nform\nD2=/parenleftig/vectorP1−/vectorP2/parenrightig2= 2−2/summationdisplay\nk/radicalig\nP1(k)P2(k). (58)\nThe two different definitions (56) and (57) can be used as distan ces\nbetween the distributions.\nLet us discuss now the notion of distance between the quantum\nstates determined by density matrices. In the density-matr ix space (in\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.18The geometry of density states, positive maps and tomograms 19\nthe set of linear space of the adjoint U(n) representation), one can\nintroduce distances analogously. The first case is\nTr(ρ1−ρ2)2=D2(59)\nand the second case is\nTr(√ρ1−√ρ2)2=D2. (60)\nIn fact, the distances introduced can be written naturally a s norms of\nvectors associated to density matrices\nD2=|/vector ρ1−/vector ρ2|2(61)\nand\nD2=/parenleftig/vector(√ρ1)−/vector(√ρ2)/parenrightig2, (62)\nrespectively.\nIn the above expressions, we use scalar product of vectors /vector ρ1and/vector ρ2\nas well as scalar products of vectors/vector(√ρ1) and/vector(√ρ2), respectively.\nBoth definitions immediately follow by identification of eit her matri-\ncesρ1andρ2with vectors according tothe map of theprevious sections\nor matrices√ρ1and√ρ2with vectors. Since the density matrices ρ1\nandρ2have nonnegative eigenvalues, the matrices√ρ1and√ρ2are\ndefined without ambiguity. This means that the vectors/vector(√ρ1) and\n/vector(√ρ2) are also defined without ambiguity. It is obvious that using this\nconstruction and introducing linear map of positive vector s/vector√ρ, one in-\nduces nonlinear map of density matrices. Other analogous fu nctions, in\naddition to square root function, can be used to create other nonlinear\npositive maps.\n4. Separable systems and separability criterion\nAccording to the definition, the system density matrix is cal led sep-\narable (for composite system) but not simply separable, if t here is\ndecomposition of the form\nρAB=/summationdisplay\nkpk/parenleftig\nρ(k)\nA⊗ρ(k)\nB/parenrightig\n,/summationdisplay\nkpk= 1,1≥pk≥0.(63)\nThis is Hilbert’s problem of biquadrates. Is a positive biqu adratic the\npositive sum of products of positive quadratics? In this for mula, one\nmay use also sum over two different indices. Using another labe lling in\nsuch sum over two different indices, this sum can be always repr esented\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.1920 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nas the sum over only one index. The formula does not demand ort hogo-\nnality of the density operators ρ(k)\nAandρ(k)\nBfor different k. Since every\ndensity matrix is a convex sum of pure density matrices, one c ould\ndemand that ρ(k)\nAandρ(k)\nBbe pure. This formula can be interpreted in\nthe context of random matrix representation reading\nρAB=/angbracketleftρA⊗ρB/angbracketright, (64)\nwhereρAandρBare considered as random density matrices of the\nsubsystems AandB, respectively. One can use the clarified structure\nofthedensity matrix set as theunionoforbits obtained by ac tion of the\nunitary group on projectors of rank one with matrix form cont aining\nonly one nonzero matrix element. Then the separable density matrix\nof bipartite composite system can be always written as the su m of\nn1n2tensor products (or corresponding mean tensor product), i. e., in\n(64) the factors are state projectors. Each of tensor produc ts contains\nrandom unitary matrices of local transforms of the fixed loca l projector\nfor one subsystem and for the second subsystem. It means that an\narbitrary projector of rank one of a subsystem can be always p resented\nin the product form ρ(k)\nA=u(k)\nAρAu(k)†\nA, whereu(k)\nAis a unitary local\ntransform and ρAis a fixed projector.\nThere are several criteria for the system to be separable. We suggest\nin the next sections a new approach to the problem of separabi lity\nand entanglement based on the tomographic probability desc ription of\nquantum states. The states which cannot be represented in th e form\n(63) by definition are called entangled states [38]. Thus the states are\nentangled if in formula (63) at least one coefficient (or more) piis\nnegative which means that the positive ones can take values g reater\nthan unity.\nLet us discuss the condition for the system state to be separa ble.\nAccording to the partial transpose criterion [41], the syst em is sepa-\nrable if the partial transpose of the matrix ρAB(63) gives a positive\ndensity matrix. This condition is necessary but not sufficien t. Let us\ndiscuss this condition within the framework of positive-ma p matrix\nrepresentation. For example, for a spin-1/2 bipartite syst em, we have\nshown that the map of a density matrix onto its transpose belo ngs\nto the matrix semigroup of matrices L. One should point out that\nthis map cannot be obtained by means of averaging with all pos itive\nprobability distributions pk. On the other hand, it is obvious that the\ngeneric criterion, which contains the Peres criterion as a p artial case,\ncan be formulated as follows.\nLet us map the density matrix ρABof a bipartite system onto vector\n/vector ρAB. Let the vector /vector ρABbe acted upon by an arbitrary matrix, which\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.20The geometry of density states, positive maps and tomograms 21\nrepresents the positive maps in subsystems AandB. Thus we get a\nnew vector\n/vector ρ(p)\nAB=/parenleftig\nLA⊗LB/parenrightig\n/vector ρAB. (65)\nLet us construct the density matrix ρ(p)\nABusing the inverse map of the\nvectors onto matrices. If the initial density matrix is sepa rable, the new\ndensity matrix ρ(p)\nABmust be positive (and separable).\nIn the case of the bipartite spin-1/2 system, by choosing LA= 1 and\nwithLBbeing the matrix coinciding with the matrix gαβ, we obtain\nthe Peres criterion as a partial case of the criterion of sepa rability\nformulated above. Thus, our criterion means that the separa ble matrix\nkeeps positivity under the action of the tensor product of tw o semi-\ngroups. In the case of the bipartite spin-1/2 system, the 16 ×16 matrix\nof the semigroup tensor product of positive contractive map s (52) is\ndetermined by 24 parameters. Among these parameters, one ca n have\nsome correlations.\nLet us discuss the positive map (52) which is determined by th e\nsemigroup for the n-dimensional system. It can be realized also as\nfollows.\nThen×nHermitian genericmatrix ρcan bemappedontoessentially\nrealn2-vector/vector ρby the map described above. The complex vector /vector ρis\nmapped onto the real vector /vector ρrby multiplying by the unitary matrix\nS, i.e.,\n/vector ρr=S/vector ρ, /vector ρ =S−1/vector ρr. (66)\nThe matrix Sis composed from nunity blocks and the blocks\nS(jk)\nb=1√\n2/parenleftbigg1 1\n−i i/parenrightbigg\n, (67)\nwherejcorresponds to a column and kcorresponds to a row in the\nmatrixρ.\nFor example, in the case n= 2, one has the vector /vector ρrof the form\n/vector ρr=\nρ11√\n2Reρ12√\n2Imρ12\nρ22\n. (68)\nOne has the equalities\n/vector ρ2\nr=/vector ρ2= Trρ2. (69)\nThe semigroup preserves the trace of the density matrix. Als o the\ndiscrete transforms, which are described by the matrix with diagonal\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.2122 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nmatrix blocks of the form\nD=\n1 0 0 0\n0 1 0 0\n0 0−1 0\n0 0 0 1\n, (70)\npreserve positivity of the density matrix.\nFor the spin case, the semigroup contains 12 parameters.\nThus, the direct product of the semigroup (52) and the discre te\ngroup of the transform Ddefines positive map preserving positivity\nof the density operator. One can include also all the matrice s which\ncorrespond to other not completely positive maps. The consi dered rep-\nresentation contains onlyreal vectors andtheirreal posit ivetransforms.\nThis means that one can construct representation of semigro up of\npositive maps by real matrices.\n5. Symbols, star-product and entanglement\nIn this section, we describe how entangled states and separa ble states\ncan be studied using properties of symbols and density opera tors of\ndifferent kinds, e.g., from the viewpoint of the Wigner functi on or\ntomogram. The general scheme of constructing the operator s ymbols is\nas follows [36].\nGiven a Hilbert space Hand an operator ˆAacting on this space,\nlet us suppose that we have a set of operators ˆU(x) acting transitively\nonHparametrized by n-dimensional vectors x= (x1,x2,...,x n). We\nconstruct the c-number function fˆA(x) (we call it the symbol of the\noperator ˆA) using the definition\nfˆA(x) = Tr/bracketleftigˆAˆU(x)/bracketrightig\n. (71)\nLet us suppose that relation (71) has an inverse, i.e., there exists a set\nof operators ˆD(x) acting on the Hilbert space such that\nˆA=/integraldisplay\nfˆA(x)ˆD(x)dx,TrˆA=/integraldisplay\nfˆA(x)TrˆD(x)dx.(72)\nOne needs a measure in xto define the integral in above formulae.\nThen, we will consider relations (71) and (72) as relations d etermin-\ning the invertible map from the operator ˆAonto the function fˆA(x).\nMultiplying both sides of Eq. (2) by the operator ˆU(x′) and taking the\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.22The geometry of density states, positive maps and tomograms 23\ntrace, one can satisfy the consistency condition for the ope ratorsˆU(x′)\nandˆD(x)\nTr/bracketleftigˆU(x′)ˆD(x)/bracketrightig\n=δ/parenleftbigx′−x/parenrightbig. (73)\nThe consistency condition (73) follows from the relation\nfˆA(x) =/integraldisplay\nK(x,x′)fˆA(x′)dx′. (74)\nThe kernel in (74) is equal to the standard Dirac delta-funct ion, if the\nset of functions fˆA(x) is a complete set.\nIn fact, we could consider relations of the form\nˆA→fˆA(x) (75)\nand\nfˆA(x)→ˆA. (76)\nThe most important property of the map is the existence of the asso-\nciative product (star-product) of the functions.\nWe introduce the product (star-product) of two functions fˆA(x) and\nfˆB(x) corresponding to two operators ˆAandˆBby the relationships\nfˆAˆB(x) =fˆA(x)∗fˆB(x) := Tr/bracketleftigˆAˆBˆU(x)/bracketrightig\n. (77)\nSince the standard product of operators on a Hilbert space is an asso-\nciative product, i.e., ˆA(ˆBˆC) = (ˆAˆB)ˆC, it is obvious that formula (77)\ndefines an associative product for the functions fˆA(x), i.e.,\nfˆA(x)∗/parenleftig\nfˆB(x)∗fˆC(x)/parenrightig\n=/parenleftig\nfˆA(x)∗fˆB(x)/parenrightig\n∗fˆC(x).(78)\nUsingformulae(71) and (72), onecan writedown acompositio n rule\nfor two symbols fˆA(x) andfˆB(x), which determines the star-product\nof these symbols. The composition rule is described by the fo rmula\nfˆA(x)∗fˆB(x) =/integraldisplay\nfˆA(x′′)fˆB(x′)K(x′′,x′,x)dx′dx′′.(79)\nThe kernel in the integral of (79) is determined by the trace o f the\nproduct of the basic operators, which we use to construct the map\nK(x′′,x′,x) = Tr/bracketleftigˆD(x′′)ˆD(x′)ˆU(x)/bracketrightig\n. (80)\nThe kernel function satisfies the composition property K∗K=K.\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.2324 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\n6. Tomographic representation\nIn this section, we will consider an example of the probabili ty repre-\nsentation of quantum mechanics [42]. In the probability rep resentation\nof quantum mechanics, the state is described by a family of pr obabil-\nities [43–45]. According to the general scheme, one can intr oduce for\nthe operator ˆAthe function fˆA(x), where\nx= (x1,x2,x3)≡(X,µ,ν),\nwhich we denote here as wˆA(X,µ,ν) depending on the position Xand\nthe parameters µandνof the reference frame\nwˆA(X,µ,ν) = Tr/bracketleftigˆAˆU(x)/bracketrightig\n. (81)\nWe call the function wˆA(X,µ,ν) the tomographic symbol of the oper-\natorˆA. The operator ˆU(x) is given by\nˆU(x)≡ˆU(X,µ,ν) = exp/parenleftbiggiλ\n2(ˆqˆp+ ˆpˆq)/parenrightbigg\nexp/parenleftbiggiθ\n2/parenleftig\nˆq2+ ˆp2/parenrightig/parenrightbigg\n|X/angbracketright/angbracketleftX|\n×exp/parenleftbigg\n−iθ\n2/parenleftig\nˆq2+ ˆp2/parenrightig/parenrightbigg\nexp/parenleftbigg\n−iλ\n2(ˆqˆp+ ˆpˆq)/parenrightbigg\n=ˆUµν|X/angbracketright/angbracketleftX|ˆU†\nµν. (82)\nThe tomographic symbol is the homogeneous version of the Moy al\nphase-space density. The angle θand parameter λin terms of the\nreference phase-space frame parameters are given by\nµ=eλcosθ, ν =e−λsinθ,\nthat is, ˆqand ˆpare position and momentum operators\nˆq|X/angbracketright=X|X/angbracketright (83)\nand|X/angbracketright/angbracketleftX|is the projection density. One has the canonical transform\nof quadratures\nˆX=ˆUµνˆqˆU†\nµν=µˆq+νˆp,\nˆP=ˆUµνˆpˆU†\nµν=1+/radicalbig\n1−4µ2ν2\n2µˆp−1−/radicalbig\n1−4µ2ν2\n2νˆq.\nUsing the approach of [46] one obtains the relationship\nˆU(X,µ,ν) =δ(X−µˆq−νˆp).\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.24The geometry of density states, positive maps and tomograms 25\nIn the case we are considering, the inverse transform determ ining the\noperator in terms of the tomogram [see Eq. (72)] will be of the form\nˆA=/integraldisplay\nwˆA(X,µ,ν)ˆD(X,µ,ν)dXdµdν, (84)\nwhere\nˆD(x)≡ˆD(X,µ,ν) =1\n2πexp(iX−iνˆp−iµˆq).(85)\nThe trace of the above operator, which provides the kernel de ter-\nmining the trace of an arbitrary operator in the tomographic represen-\ntation, reads\nTrˆD(x) =eiXδ(µ)δ(ν).\nThe function wˆA(X,µ,ν) satisfies the relation\nwˆA(λX,λµ,λν ) =1\n|λ|wˆA(X,µ,ν). (86)\nThis means that the tomographic symbols of operators are hom oge-\nneous functions of three variables.\nIf one takes two operators ˆA1andˆA2, which are expressed through\nthe corresponding functions by the formulas\nˆA1=/integraldisplay\nwˆA1(X′,µ′,ν′)ˆD(X′,µ′,ν′)dX′dµ′dν′,\n(87)\nˆA2=/integraldisplay\nwˆA2(X′′,µ′′,ν′′)ˆD(X′′,µ′′,ν′′)dX′′dµ′′dν′′,\nandˆAdenotes theproductof ˆA1andˆA2,then thefunction wˆA(X,µ,ν),\nwhichcorrespondsto ˆA,isthestar-productofthefunctions wˆA1(X,µ,ν)\nandwˆA2(X,µ,ν). Thus this product\nwˆA(X,µ,ν) =wˆA1(X,µ,ν)∗wˆA2(X,µ,ν)\nreads\nwˆA(X,µ,ν) =/integraldisplay\nwˆA1(x′′)wˆA2(x′)K(x′′,x′,x)dx′′dx′,(88)\nwith kernel given by\nK(x′′,x′,x) = Tr/bracketleftigˆD(X′′,µ′′,ν′′)ˆD(X′,µ′,ν′)ˆU(X,µ,ν)/bracketrightig\n.(89)\nThe explicit form of the kernel reads\nK(X1,µ1,ν1,X2,µ2,ν2,Xµ,ν)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.2526 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\n=δ/parenleftig\nµ(ν1+ν2)−ν(µ1+µ2)/parenrightig\n4π2exp/parenleftbiggi\n2/braceleftig\n(ν1µ2−ν2µ1)+2X1+2X2\n−/bracketleftigg\n1−/radicalbig\n1−4µ2ν2\nν(ν1+ν2)+1+/radicalbig\n1−4ν2µ2\nµ(µ1+µ2)/bracketrightigg\nX/bracerightigg/parenrightigg\n.\n(90)\n7. Multipartite systems\nLet us assume that for multimode ( N-mode) system one has\nˆU(/vector y) =N/productdisplay\nk=1⊗ˆU/parenleftig\n/vector x(k)/parenrightig\n, (91)\nˆD(/vector y) =N/productdisplay\nk=1⊗ˆD/parenleftig\n/vector x(k)/parenrightig\n, (92)\nwhere\n/vector y=/parenleftig\nx(1)\n1,x(1)\n2,...,x(1)\nm,x(2)\n1,x(2)\n2,...,x(N)\nm/parenrightig\n. (93)\nThis means that the symbol of the density operator of the comp osite\nsystem reads\nfρ(/vector y) = Tr/bracketleftig\nˆρN/productdisplay\nk=1⊗ˆU(/vector x(k))/bracketrightig\n. (94)\nThe inverse transform reads\nˆρ=/integraldisplay\nd/vector yfρ(/vector y)N/productdisplay\nk=1⊗ˆD(/vector x(k)), d/vector y =N/productdisplay\nk=1m/productdisplay\ns=1dx(k)\ns.(95)\nNow we formulate the properties of the symbols in the case of\nentangled and separable states, respectively.\nGiven a composite m-partite system with density operator ˆ ρ.\nIf the nonnegative operator can be presented in the form of a “ prob-\nabilistic sum”\nˆρ=/summationdisplay\n/vector zP(/vector z)ˆρ(a1)\n/vector z⊗ˆρ(a2)\n/vector z⊗···⊗ˆρ(am)\n/vector z, (96)\nwith positive probability distribution function P(/vector z), where the com-\nponents of /vector zcan be either discrete or continuous, we call the state a\n“separable state.” Without loss of generality, all factors in the tensor\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.26The geometry of density states, positive maps and tomograms 27\nproducts can be considered as projectors of rank one. This me ans that\nthe symbol of the state can be presented in the form\nfρ(/vector y) =/summationdisplay\n/vector zP(/vector z)m/productdisplay\nk=1f(ak)\nρ(/vector xk,/vector z). (97)\nAnalogous formula can be written for the tomogram of separab le state.\nWe point out that in the multipartite case one can introduce r an-\ndom symbols and represent the symbol of separable density ma trix\nof composite system as mean value of pointwise products of sy mbols of\nsubsystem density operators. As in the bipartite case, one c an use sum\nover different indices but this sum can be always reduced to the sum\nover only one indexcommon forall the subsystems.Itis impor tant that\nfor separable state its symbol always can be represented as t he sum\ncontaining number of summants which is equal to dimensional ity of\ncomposite system. Each term in the sum is equal to mean value o f ran-\ndom projector. The random projector is constructed as the pr oduct of\ndiagonal projectorsofrankoneineach subsystemconsidere dinrandom\nlocal basis obtained by means of random unitary local transf orms.\n8. Spin tomography\nBelow we concentrate on bipartite spin systems.\nThe tomographic probability (spin tomogram) completely de ter-\nmines the density matrix of a spin state. It has been introduc ed in\n[31, 32, 36]. The tomographic probability for the spin- jstate is defined\nvia the density matrix by the formula\n/angbracketleftjm|D†(g)ρD(g)|jm/angbracketright=W(j)(m,/vector0), m=−j,−j+1,...,j,\n(98)\nwhereD(g) is the matrix of SU(2)-group representation depending on\nthe group element gdetermined by three Euler angles. It is useful to\ngeneralize the construction of spin tomogram.\nOne can introduce unitary spin tomograms w(m,u) by replacing in\nabove formula (98) the matrix D(g) by generic unitary matrix u. For\nthe case of higher spins j= 1,3/2,2,..., then×nprojector matrix\nρ1=\n1 0···0\n0 0···0\n· · ··· ·\n· · ··· ·\n0 0···0\n, n= 2j+1 (99)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.2728 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nhas the unitary spin tomogram denoted as\nw1(j,u) =|u11|2, w1(j−1,u) =|u12|2, ... w 1(−j,u) =|u1n|2.\n(100)\nOther projectors\nρk=\n0 0··· ···0\n· · ··· · ·\n0···1···0\n· · · ··· ·\n0 0· ···0\n, (101)\nin which unity is located in kth column, have the tomogram wk(m,u)\nof the form\nwk(j,u) =|uk1|2, wk(j−1,u) =|uk2|2, ... w k(−j,u) =|ukn|2.\n(102)\nIn connection with the decomposition of any density matrix i n theform\nρ=/summationdisplay\njkρjkEjk, (103)\nthe unitary spin tomogram can be presented in form of the deco mpo-\nsition\nwρ(m,u) =/summationdisplay\njkρjkwjk(m,u), (104)\nwherewjk(m,u) are basic unitary spin symbols of transition operators\nEjkof the form\nwjk(m,u) =/angbracketleftjm|u†Ejku|jm/angbracketright. (105)\nIf one uses a map\nρ→ρ′, (106)\nthe unitary spin tomogram is transformed as\nwρ(m,u)→w′\nρ(m,u) =/summationdisplay\njkρ′\njkwjk(m,u). (107)\nIf the transform (106) is a linear one\nρjk→ρ′\njk=Ljk,psρps, (108)\nthe transform reads\nw′\nρ(m,u) =/summationdisplay\npsρpsw′\nps(m,u). (109)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.28The geometry of density states, positive maps and tomograms 29\nHere\nw′\nps(m,u) =/summationdisplay\njkLjk,pswjk(m,u) (110)\nis the linear transform of the basic tomographic symbols of t he opera-\ntorsEjk.\nLet us now discuss some properties of usual spin tomograms.\nThe set of the tomogram values for each /vector0 is an overcomplete set.\nWe need only a finite number of independent locations which wi ll give\ninformationonthedensitymatrixofthespinstate. Duetoth estructure\nof the formula, there are only two Euler angles involved. The y are\ncombined into the unit vector\n/vector0 = (cosφsinϑ,sinφsinϑ,cosϑ). (111)\nThis is the Hopf map from S3toS2.\nThe physical meaning of the probability W(m,/vector0) is the following.\nIt is the probability to find, in the state with the density mat rixρ,\nthe spin projection on direction /vector0 equal tom. For a bipartite system,\nthe spin tomogram is defined as follows:\nW(m1m2/vector01/vector02) =/angbracketleftj1m1j2m2|D†(g1)D†(g2)ρD(g1)D(g2)|j1m1j2m2/angbracketright.\n(112)\nItcompletely determinesthedensitymatrix ρ.Ithasthemeaningofthe\njoint probability distribution for spin j1andj2projections m1andm2\non directions /vector01and/vector02.Since the map ρ⇋Wis linear and invertible,\nthedefinitionof separablesystemcan berewritteninthefol lowingform\nfor the decomposition of the joint probability into a sum of p roducts\n(of factorized probabilities):\nW(m1m2/vector01/vector02) =/summationdisplay\nkpkW(k)(m1/vector01)˜W(k)(m2/vector02).(113)\nThis form can be considered to formulate the criterion of sep arability\nof the two-spin state.\nOne can present this formula in the form\nW(m1m2/vector01/vector02) =/angbracketleftW(m1/vector01)˜W(m2/vector02)/angbracketright, (114)\nwherewe interpretthepositive numbers pkas probability distributions.\nThus separability means the possibility to represent joint probability\ndistribution in the form of average product of two random pro bability\ndistributions.\nThe state is separable iff the tomogram can be written in the fo rm\n(113) with/summationtext\nkpk= 1,pk≥0.Itseems that wesimplyusethedefinition\nbut, in fact, we cast the problem of separability into the for m of the\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.2930 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nproperty of the positive joint probability distribution of two random\nvariables. This is an area of probability theory and one can u se the\nresults and theorems on joint probability distributions. I f one does not\nuse any theorem, one has to study the solvability of relation (113)\nconsidered as the equation for unknown probability distrib utionpkand\nunknown probability functions W(k)(m1/vector01) andW(k)(m2/vector02).\n9. Example of spin-1/2 bipartite system\nFor the spin-1/2 state, the generic density matrix can be pre sented in\nthe form\nρ=1\n2(1+/vector σ·/vector n), /vector n= (n1,n2,n3), (115)\nwhere/vector σare Pauli matrices and /vector n2≤1,with the vector /vector nfor a pure\nstate being the unit vector. This decomposition means that w e use as\nbasis in 4-dimensional vector space the vectors correspond ing to the\nPauli matrices and the unit matrix, i.e.,\n/vector σ1=\n0\n1\n1\n0\n, /vector σ 2=\n0\n−i\ni\n0\n, /vector σ 3=\n1\n0\n0\n−1\n,/vector1 =\n1\n0\n0\n1\n.\n(116)\nThe density matrix vector\n/vector ρ=\nρ11\nρ12\nρ21\nρ22\n(117)\nis decomposed in terms of the basis vectors\n/vector ρ=1\n2/parenleftig\n/vector1+n1/vector σ1+n2/vector σ2+n3/vector σ3/parenrightig\n. (118)\nThis means that the spin tomogram of the spin-1/2 state can be given\nin the form\nW/parenleftbigg1\n2,/vector0/parenrightbigg\n=1\n2+/vector n·/vector0\n2, W/parenleftbigg\n−1\n2,/vector0/parenrightbigg\n=1\n2−/vector n·/vector0\n2.(119)\nWe can consider tomograms of specific spin state. If the state is pure\nstate with density matrix\nρ+=/parenleftbigg1 0\n0 0/parenrightbigg\n, (120)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.30The geometry of density states, positive maps and tomograms 31\nthe spin tomogram W(m,/vector0), where\nm=±1\n2,/vector0 = (sinθcosϕ,sinθsinϕ,cosθ)\nhas the values\nW+/parenleftbigg1\n2,/vector0/parenrightbigg\n= cos2θ\n2, W+/parenleftbigg\n−1\n2,/vector0/parenrightbigg\n= sin2θ\n2.(121)\nThe tomogram of the pure state\nρ−=/parenleftbigg0 0\n0 1/parenrightbigg\n, (122)\nhas the values\nW−/parenleftbigg1\n2,/vector0/parenrightbigg\n= sin2θ\n2= cos2π−θ\n2,\n(123)\nW−/parenleftbigg\n−1\n2,/vector0/parenrightbigg\n= cos2θ\n2= sin2π−θ\n2.\nThe spin tomogram of the diagonal density matrix\nρd=/parenleftbiggρ110\n0ρ22/parenrightbigg\n(124)\nequals\nWd(m,/vector0) =ρ11W+(m,/vector0+)+ρ22W−(m,/vector0−). (125)\nThe generic density matrix which has eigenvalues ρ11andρ22can be\npresented in the form\nu0ρdu†\n0, (126)\nwhere the unitary matrix u0has columns containing components of\nnormalized eigenvectors of the density matrix ρ.\nThis means that the tomogram of the state with the matrix ρreads\nWρ(m,/vector0) =/angbracketleftm|u†u0ρdu†\n0u|m/angbracketright. (127)\nThe elements of the group can be combined\nu†u0= ˜u. (128)\nThus the tomogram becomes\nWρ(m,/vector0) =Wd(m,/vector0′), (129)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.3132 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nwhere the angle /vector0′corresponds to the Euler angle calculated from the\nproduct of two unitary matrices u†\n0u.\nOnecanusethepropertyofnumbers ρ11andρ22tointerpretformula\n(125) as averaging\nWd(m,/vector0) =/angbracketleftW(m,/vector0′)/angbracketright, (130)\nwhere one interprets two functions W+(m,/vector0) andW−(m,/vector0′) as the\nrealization of “random” probability distribution functio nW±(m,/vector0).\nThen one has\nWρ(m,/vector0) =/angbracketleftW(m,/vector0′)/angbracketright. (131)\nThe new vector /vector0′has the parameter θ′obtained from the initial pa-\nrameterθby action of the unitary matrix on the initial unitary matrix\nu.\nInserting these probability values into relation (113) for each value\nofk, we get the relationships\nW/parenleftbigg1\n2,1\n2,/vector01,/vector02/parenrightbigg\n=1\n4+1\n2/parenleftigg/summationdisplay\nkpk/vector nk/parenrightigg\n·/vector01+1\n2/parenleftigg/summationdisplay\nkpk/vector n∗\nk/parenrightigg\n·/vector02\n+/summationdisplay\nkpk/parenleftig\n/vector nk·/vector01/parenrightig/parenleftig\n/vector n∗\nk·/vector02/parenrightig\n, (132)\nW/parenleftbigg1\n2,−1\n2,/vector01,/vector02/parenrightbigg\n=1\n4+1\n2/parenleftigg/summationdisplay\nkpk/vector nk/parenrightigg\n·/vector01−1\n2/parenleftigg/summationdisplay\nkpk/vector n∗\nk/parenrightigg\n·/vector02\n−/summationdisplay\nkpk/parenleftig\n/vector nk·/vector01/parenrightig/parenleftig\n/vector n∗\nk·/vector02/parenrightig\n, (133)\nW/parenleftbigg\n−1\n2,1\n2,/vector01,/vector02/parenrightbigg\n=1\n4−1\n2/parenleftigg/summationdisplay\nkpk/vector nk/parenrightigg\n·/vector01+1\n2/parenleftigg/summationdisplay\nkpk/vector n∗\nk/parenrightigg\n·/vector02\n−/summationdisplay\nkpk/parenleftig\n/vector nk·/vector01/parenrightig/parenleftig\n/vector n∗\nk·/vector02/parenrightig\n. (134)\nOne has the normalization property\n1/2/summationdisplay\nm1,m2=−1/2W(m1m2/vector01/vector02) = 1. (135)\nOne easily gets\nW/parenleftbigg1\n2,1\n2,/vector01,/vector02/parenrightbigg\n+W/parenleftbigg1\n2,−1\n2,/vector01,/vector02/parenrightbigg\n=1\n2+/parenleftigg/summationdisplay\nkpk/vector nk/parenrightigg\n·/vector01.(136)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.32The geometry of density states, positive maps and tomograms 33\nThis means that the derivative in /vector01on the left-hand side gives\n∂\n∂/vector01/bracketleftbigg\nW/parenleftbigg1\n2,1\n2,/vector01,/vector02/parenrightbigg\n+W/parenleftbigg1\n2,−1\n2,/vector01,/vector02/parenrightbigg/bracketrightbigg\n=/parenleftigg/summationdisplay\nkpk/vector nk/parenrightigg\n.(137)\nAnalogously\n∂\n∂/vector02/bracketleftbigg\nW/parenleftbigg1\n2,1\n2,/vector01,/vector02/parenrightbigg\n+W/parenleftbigg\n−1\n2,1\n2,/vector01,/vector02/parenrightbigg/bracketrightbigg\n=/parenleftigg/summationdisplay\nkpk/vector n(⋆)\nk/parenrightigg\n.(138)\nTaking the sum of (133) and (134)) one sees that\n1\n2∂\n∂/vector0i∂\n∂/vector0j/bracketleftbigg\nW/parenleftbigg1\n2,−1\n2,/vector01,/vector02/parenrightbigg\n+W/parenleftbigg\n−1\n2,1\n2,/vector01,/vector02/parenrightbigg/bracketrightbigg\n=−/summationdisplay\nkpk(nk)i(n(⋆)\nk)j. (139)\nSince we look for the solution where pk≥0, we can introduce\n/vectorNk=√pk/vector nk,/vectorN(⋆)\nk=√pk/vector n(⋆)\nk. (140)\nThis means that the derivative in (139) can be presented as a t ensor\n−Tij=/summationdisplay\nk(Nk)i(N(⋆)\nk)j. (141)\nOne has/summationdisplay\nkpk/vector nk=/summationdisplay\nk√pk/vectorNk, (142)\n/summationdisplay\nkpk/vector n⋆\nk=/summationdisplay\nk√pk/vectorN(⋆)\nk. (143)\nThe conditions of solvability of the obtained equations is a criterion for\nseparability or entanglement of a bipartite quantum spin st ate. Using\nthe arguments on the representation of the tomogram (tomogr aphic\nsymbol) as sum of random basic projector symbols we get that f or two\nqubits the separable state has the tomogram with following p roperties.\nAll four values of joint probability distribution function are equal to\nmean values of product of two cosine of two different angles squ ared,\nproduct of sine of two different angles squared and product of s ine and\ncosine squared, respectively. The entangled matrix does no t provide\nsuch structure.\nAs an example, we consider the Werner state. For the Werner st ate\n(see, e.g., [47]) with the density matrix\nρAB=\n1+p\n40 0p\n2\n01−p\n40 0\n0 01−p\n40\np\n20 01+p\n4\n,\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.3334 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\n(144)\nρA=ρB=1\n2/parenleftbigg1 0\n0 1/parenrightbigg\n,\nonecan reconstruct theknownresultsthat for p<1/3 thestate is sepa-\nrable and for p>1/3 the state is entangled, since in the decomposition\nof the density operator in the form (113) the state\nρ0=1\n4\n1 0 0 0\n0 1 0 0\n0 0 1 0\n0 0 0 1\n(145)\nhas the weight p0= (1−3p)/4.\nForp>1/3, the coefficient pobecomes negative.\nThere is some extension of the presented consideration.\nLet us consider the state with the density matrix (nonnegati ve and\nHermitian)\nρ=\nR110 0R12\n0ρ11ρ120\n0ρ21ρ220\nR210 0R22\n,Trρ= 1. (146)\nUsing the procedure of mapping the matrix onto vector /vector ρand applying\nto the vector the nonlocal linear transform corresponding t o the Peres\npartial transposeandmakingtheinversemapofthetransfor medvector\nonto the matrix, we obtain\nρm=\nR110 0ρ12\n0ρ11R120\n0R21ρ220\nρ210 0R22\n. (147)\nIn the case of separable matrix ρ, the matrix ρmis a nonnegative\nmatrix. Calculating the eigenvalues of ρmand applying the condition\nof their positivity, we get\nR11R22≥ |ρ12|2, ρ 11ρ22≥ |R12|2. (148)\nViolation of these inequalities gives a signal that ρis entangled. For\nWerner state (144), Eq. (148) means\n1+p>0,1−p>2p, (149)\nwhich recovers the condition of separability p<1/3 mentioned above.\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.34The geometry of density states, positive maps and tomograms 35\nThe joint probability distribution (112) of separable stat e is positive\nafter making the local and nonlocal (partial transpose-lik e) transforms\nconnected with positive map semigroup. But for entangled st ate, func-\ntion (112) can take negative values after making this map in t he func-\ntion and replacing on the right-hand side of this equality th e product of\ntwo matrices D(g) by generic unitary transform u. This is a criterion\nof entanglement in terms of unitary spin tomogram of the stat e of\nmultiparticle system.\nA simpler and more transparent case is the generalized Werne r\nmodel with density matrix\nρ=1\n4(1+µ1σ1⊗τ1+µ2σ2⊗τ2+µ3σ3⊗τ3).(150)\nHere the density matrix is expressed in terms of tensor produ cts of two\nsets of Pauli matrices σkandτk(k= 1,2,3), which are chosen in the\nstandard form.\nIts eigenvalues are\n1−µ1−µ2−µ3,1+µ1+µ2−µ3,1+µ1−µ2+µ3,1−µ1+µ2+µ3.\nThese eigenvalues are related to the vertices of a regular te trahedron.\nThe partially time-reversed density matrix is\n/tildewideρ=1\n4(1−µ1σ1⊗τ1−µ2σ2⊗τ2−µ3σ3⊗τ3),(151)\nwhich may be viewed as\n/tildewideρ=L(1)⊗L(2)ρwithL(1)ρ(1)=ρ(1), L(2)ρ(2)= 1−ρ(2).\n(152)\nThe eigenvalues of this are\n1+µ1+µ2+µ3,1+µ1−µ2−µ3,1−µ1+µ2−µ3,1−µ1−µ2+µ3.\nTheseform an inverted tetrahedron and they have the common d omain\nwhich is a regular octahedron. The unitary spin tomograms ca n be\nwritten down by inspection and we may verify that all the rela tions\nrequired by the separability criterion (see the next sectio n for details)\naresatisfied byany pointinsidetheoctahedron for ρandforL(1)⊗L(2)ρ\nbuttherelations connected withpositivity condition expr essedinterms\nof positivity of unitary spin tomogram fail when it lies outs ide.\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.3536 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\n10. Tomogram of the group U(n)\nIn this section we discuss in more detail the separability cr iterion using\nintroduced notion of unitary spin tomogram.\nIn order to formulate a criterion of separability for a bipar tite spin\nsystem with spin j1andj2, we introduce the tomogram w(/vectorl,/vector m,g(n))\nfor the group U(n), where\nn=n1n2, n 1= 2j1+1, n 2= 2j2+1,\nandg(n)are parameters of the group element. Vectors /vectorland/vector mlabel a\nbasis|/vectorl,/vector m/angbracketrightof the fundamental representation of the group U(n). For\nexample, since this representation is irreducible, being r educed to the\nrepresentation of the U(n1)⊗U(n2) subgroup of the group U(n), the\nbasis can be chosen as the product of basis vectors:\n|j1,m1/angbracketright |j2,m2/angbracketright=|j1,j2,m1,m2/angbracketright. (153)\nDue to the irreducibility of this representation of the grou pU(n) and\nits subgroup, there exists a unitary transform u/vectorl/vector m\nj1j2m1m2|/vectorl,/vector m/angbracketrightsuch\nthat\n|j1,j2,m1,m2/angbracketright=/summationdisplay\n/vectorl/vector mu/vectorl/vector m\nj1j2m1m2|/vectorl,/vector m/angbracketright, (154)\n|/vectorl/vector m/angbracketright=/summationdisplay\nm1m2(u−1)/vectorl/vector m\nj1j2m1m2|jl,j2,m1,m2/angbracketright.(155)\nOne can define the U(n) tomogram for a Hermitian nonnegative n×n\ndensity matrix ρ, which belongs to the Lie algebra of the group U(n),\nby a generic formula\nw(/vectorl,/vector m,g(n)) =/angbracketleft/vectorl,/vector m|U†(g(n))ρU(g(n))|/vectorl,/vector m/angbracketright.(156)\nFormula (156) defines the tomogram in the basis |/vectorl,/vector m/angbracketrightfor arbitrary\nirreducible representation of the unitary group. But below we focus\nonly on tomograms connected with spins.\nLet us define the U(n) tomogram using the basis |j1,j2,m1,m2/angbracketright\nnamely for fundamental representation, i.e.,\nw(j1,j2)(m1,m2,g(n))\n=/angbracketleftj1,j2,m1,m2|U†(g(n))ρU(g(n))|j1,j2,m1,m2/angbracketright.(157)\nThis unitary spin tomogram becomes the spin-tomogram [34] f or the\ng(n)∈U(2)⊗U(2) subgroup of the group U(n). The properties of this\ntomogram follow fromits definition as the joint probability distribution\nof two random spin projections m1,m2depending on g(n)parameters.\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.36The geometry of density states, positive maps and tomograms 37\nOne has the normalization condition\n/summationdisplay\nm1,m2w(j1,j2)(m1,m2,g(n)) = 1. (158)\nAlso all the probabilities are nonnegative, i.e.,\nw(j1,j2)(ml,m2,g(n))≥0. (159)\nDue to this, one has\n/summationdisplay\nm1,m2|w(j1,j2)(ml,m2,g(n))|= 1. (160)\nFor the spin-tomogram,\ng(n)→/parenleftig/vectorO1,/vectorO2/parenrightig\n(161)\nand\nw(j1,j2)(ml,m2,g(n))→w(m1,m2,/vectorO1,/vectorO2). (162)\nThe separability and entanglement condition discussed in t he previ-\nous section for a bipartite spin-tomogram can be considered also from\nthe viewpoint of the properties of a U(n) tomogram. If the two-spin\nn×ndensitymatrix ρisseparable,itremainsseparableundertheaction\nofthegenericpositivemapofthesubsystemdensitymatrice s. Thismap\ncan be described as follows.\nLetρbemappedontovector /vector ρwithn2components.Thecomponents\nare simply ordered rows of the matrix ρ, i.e.,\n/vector ρ=/parenleftig\nρ11,ρ12,...,ρ1n,ρ21,ρ22,...,ρnn,/parenrightig\n. (163)\nLet then2×n2matrixLbe taken in the form\nL=/summationdisplay\nspsL(j1)\ns⊗L(j2)\ns, p s≥0,/summationdisplay\nsps= 1,(164)\nwhere then1×n1matrixL(j1)\nsand then2×n2matrixL(j2)\nsdescribe the\npositive maps of density matrices of spin- j1and spin-j2subsystems,\nrespectively. We map vector /vector ρonto vector /vector ρL\n/vector ρL=L/vector ρ (165)\nand construct the n×nmatrixρL, which corresponds to the vector /vector ρL.\nThen we consider the U(n) tomogram of the matrix ρL, i.e.,\nw(j1,j2)\nL(ml,m2,g(n))\n=/angbracketleftj1,j2,m1,m2|U†(g(n))ρLU(g(n))|j1,j2,ml,m2/angbracketright.(166)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.3738 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nUsing this tomogram we introduce the function\nF(g(n),L) =/summationdisplay\nm1,m2/vextendsingle/vextendsingle/vextendsinglew(j1,j2)\nL(m1,m2,g(n))/vextendsingle/vextendsingle/vextendsingle. (167)\nFor separable states, this function does not depend on the U(n)-group\nparameterg(n)and positive-map matrix elements of the matrix L.\nFor the normalized density matrix ρof the bipartite spin-system,\nthis function reads\nF(g(n),L) = 1. (168)\nFor entangled states, this function depends on g(n)andLand is not\nequalto unity. Thispropertycan bechosen asanecessary and sufficient\ncondition for separability of bipartite spin-states. We in troduce also\ntomographic purity parameter µkofkth order by the formula\nµk(g(n),L) =/summationdisplay\nm1m2/vextendsingle/vextendsingle/vextendsinglew(j1,j2)\nL(m1,m2,g(h))/vextendsingle/vextendsingle/vextendsinglek.\nFor identity semigroup element Land specific g(n)\n0unitary transform\ndiagonalizing the density matrix, the tomographic purity µ2is identical\nto purity parameter of the state ρ. The parameters for k= 2,3,...,\ncorrespond to Tr ρk+1.\nIn fact, the formulated approach can be extended to multipar tite\nsystems too. The generalization is as follows.\nGivenNspin-systems with spins j1,j2,...,jN, let us consider the\ngroupU(n) with\nn=N/productdisplay\nk=1nk, n k= 2jk+1. (169)\nLet us introduce the basis\n|/vector m/angbracketright=N/productdisplay\nk=1|jkmk/angbracketright (170)\ninthelinear spaceofthefundamental representation ofthe groupU(n).\nWe define now the U(n) tomogram of a state with the n×nmatrixρ:\nwρ(/vector m,g(n)) =/angbracketleft/vector m|U†(g(n))ρU(g(n))|/vector m/angbracketright. (171)\nFor a positive Hermitian matrix ρwith Trρ= 1, we formulate the\ncriterion of separability as follows.\nLet the map matrix Lbe of the form\nL=/summationdisplay\nsps/parenleftigN/productdisplay\nk=1⊗L(k)\ns/parenrightig\n, p s≥0,/summationdisplay\nsps= 1,(172)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.38The geometry of density states, positive maps and tomograms 39\nwhereL(k)\nsis the positive-map matrix of the density matrix of the\nkth spin subsystem. We construct the matrix ρLas in the case of the\nbipartite system using the matrix L. The function\nF(g(n),L) =/summationdisplay\n/vector m|wρL(/vector m,g(n))| ≥1 (173)\nis equal to unity for separable state and depends on the matri xLand\nU(n)-parameters g(n)for entangled states.\nThis criterion can beapplied also in the case of continuous v ariables,\ne.g., for Gaussian states of photons. Function (173) can pro vide the\nmeasure of entanglement. Thus one can use the maximum value ( or a\nmean value) of this function as a characteristic of entangle ment. In the\nprevious section, we considered the generalized Werner sta tes. Using\nthe above criterion, one can get the domain of values of the pa rameters\nof the states for which one has separability or entanglement . In fact,\nthe separability criterion is related to the following posi tivity criterion\nof finite or infinite (trace class) matrix A. The matrix Ais positive iff\nthe sum of moduli of diagonal matrix elements of the matrix UAU†\nis equal to a positive trace of the matrix Afor an arbitrary unitary\nmatrixU.\n11. Dynamical map and purification\nIn this section, we consider the connection of positive maps with pu-\nrification procedure. In fact, formula\nρ→ρ′=/summationdisplay\nkpkUkρU†\nk, (174)\nwhereUkare unitary operators, can be considered in the form\nρ→ρ′=/summationdisplay\nkpkρk, p k≥0,/summationdisplay\nkpk= 1.(175)\nHere the density operators ρkread\nρk=UkρU†\nk, (176)\nand the maps which are not sufficiently general keep the most de gener-\nate density matrix fixed.This formis theform of probabilist ic addition.\nThis mixture of density operators can be purified with the hel p of a\nfiducial rank one projector P0\nρ′→ρ′′=N\n/summationdisplay\nkj√pkpjρkP0ρj/radicalbigTrρkP0ρjP0\n, (177)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.3940 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nwhereNis a normalization constant\nN−1= Tr\n/summationdisplay\nkj√pkpjρkP0ρj/radicalbigTrρkP0ρjP0\n. (178)\nThe normalization is unnecessary if all ρkare mutually orthogonal. We\ncall this map a purification map. It maps the density matrix of mixed\nstate on the density matrix of pure state.\nThe map (174) could be interpreted as the evolution in time of the\ninitialmatrix ρ0consideringunitaryoperators Uk(t)dependingontime.\nThus one has\nρ0→ρ(t) =/summationdisplay\nkpkUk(t)ρ0U†\nk(t). (179)\nIn this case, the purification procedure provides the dynami cal map of\na pure state\n|ψ0/angbracketright/angbracketleftψ0|→|ψ(t)/angbracketright/angbracketleftψ(t)|, (180)\nwhere|ψ(t)/angbracketrightobeys a nonlinear equation and, in the general case, this\nmap does not definea one parameter group of transformations n ot even\nlocally.\nFor some specific cases, the evolution (179) can be described by a\nsemigroup.Thedensitymatrix(179)obeysthenafirst-order differential\nequation in time for this case [27–29].\nMore specifically, the reason why there is no differential equa tion in\ntime for the generic case is due to the absence of the property\nρij(t2) =/summationdisplay\nmnKmn\nij(t2,t1)ρmn(t1), (181)\nwhere the kernel of evolution operator satisfies\nKmn\nij(t3,t2)Kpq\nmn(t2,t1) =Kpg\nij(t3,t1). (182)\nThus, via a purification procedure and a dynamical map applie d to\na density matrix we get a pure state (nonlinear dynamical map ). This\nmapcanbeusedinnonlinearmodelsofquantumevolution.Man ylinear\npositive maps both completely positive and not completely p ositive are\ncontractive. We define a positive map Las “contractive” or “dilating”\nifTr(Lρ)2≤Tr(ρ)2orTr(Lρ)2≥Tr(ρ)2, respectively. Thismeans, for\nexample, that purity parameter µ= Trρ2after performing the positive\nmap generically becomes smaller. There are maps for which th e purity\nparameter is preserved, for example,\nρ→ρtr, ρ→ −ρ+2\nN1. (183)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.40The geometry of density states, positive maps and tomograms 41\nThese linear maps include also unitary transform\nρ→uρu†. (184)\nThere are maps which provide dilation. For qubit system, mat rix\nL=\n1 0 0 1\n0 0 0 0\n0 0 0 0\n0 0 0 0\n. (185)\nacting on arbitrary vector /vector ρ0correspondingto a density matrix ρ0gives\nthe pure state\nρf=/parenleftbigg1 0\n0 0/parenrightbigg\n. (186)\nThe matrices Lεthe inverse matrices exist for ε/negationslash= 0. But these inverse\nmatricesdonotprovidepositivetracepreservingmaps.Sin cethepurifi-\ncation procedure provides a positive map, which increases t he purity\nparameter, the composition of linear map with the purificati on map\nprovides the possibility to recover the initial density mat rixρwhich\nwas the object of action of a positive linear map. It means tha t the\npurification map ˆLpcan give\nˆLP0(Lρ) = 1ρ (187)\nfor any density matrix ρbut the choice of fiducial projector depends\nonρ(the initial condition).\nThus one has also for completely positive maps\nρ→ρ′=/summationdisplay\nkρ′\nk, ρ k=VkρV†\nk,/summationdisplay\nkV†\nkVk= 1.(188)\nMaking polar decomposition\nρk=√ρ0kUk, U kU†\nk= 1, ρ 0k≥0\nand introducing the positive numbers pk= Trρ0k, we construct the\nmap\nρ′→ρ′′=\n\n/summationdisplay\nkj√pkpj˜ρkP0˜ρj+ ˜ρjP0˜ρk/radicalbigT˜ρkP0˜ρjP0\n\n,/summationdisplay\nkpk= 1, pk˜ρk=ρk.\n(189)\nThematrix ρ′′isamatrixofrankoneforanyrankonefiducialprojector\nP0. The projector P0is restricted to be not orthogonal to the generic\nmatrixρk. TakingNorthogonal projectors P(s)\n0(s= 1,2,...,N) and\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.4142 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nobtainingNprojectorsρ′′\ns, one can combine them in order to get the\ninitial matrix ρ. It means that one can take convex sum of the N\npure states ρ′′\nsto recover the initial mixed state ρ. Another way to\nmake the state with higher purity was demonstrated usingthe modified\npurification procedure in [48]. For qubit state, one has\nρ=p1ρ1+p2ρ2+κ√p1p2ρ1P0ρ2+ρ2P0ρ1√Trρ1P0ρ2P0, p1+p2= 1,(190)\nwhere the decoherence parameter 0 ≤κ≤1 is used. If κ∼1, we\nincrease purity.\nLet us discuss the map (188) using its matrix form, i.e.,\nραβ→ρ′\nαβ=/summationdisplay\nijLαβ,ijρij. (191)\nThe matrix Lαβ,ijis expressed in terms of the matrices Vkas\nLαβ,ij=/summationdisplay\nk(Vk)αi(V∗\nk)βj. (192)\nOne can construct another positive map [49]\nρ→ρ′=/summationdisplay\nkrkTr(Rkρ), (193)\nwhererkare density matrices and Rkare positive operators satisfying\nthe normalization condition\n/summationdisplay\nkRk= 1. (194)\nThe matrix corresponding to this map (called entanglement b reaking\nmap [50]) reads\nLb\nαβ,ij=/summationdisplay\nk(rk)αβ(R∗\nk)ij. (195)\nThe entanglement breaking map is contractive positive map. There\nexist some special cases of completely positive maps. For ex ample,\nρ→ −ερ+1+ε\nNρ (196)\ndiffers from the depolarizing map by replacing the unity opera tor by\nthe density operator. Another map reads\nρ→1−diagρ\nN. (197)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.42The geometry of density states, positive maps and tomograms 43\nThe decoherence map (phase damping map) of the kind\nρij→/braceleftbiggρij, i=j\nλρij, i/negationslash=j,(198)\nwhere|λ|<1 provides contractive map with uniform change of off-\ndiagonal matrix elements of the density matrix.\nLet us discuss the property of tomogram of bipartite system w ith\ndensity matrix ρ12. If the density matrix is separable, than the depo-\nlarizing map of the second subsystem provides the following density\nmatrix\nρ12→ρε=−ερ12+1+ε\nN2ρ(1)⊗12, (199)\nwhere\nρ(1)= Tr2(ρ12) (200)\nand 12is theN2-dimensional unity matrix. Then one has the property\nof unitary spin tomogram\nwε(m1,m2,g(n)) =−εw12(m1,m2,g(n))+1+ε\nN2w(m1,m2,g(n)),(201)\nwhereg(n)is matrix of U/parenleftig\n(2j1+1)(2j2+1)/parenrightig\nunitary transform;\nwε(m1,m2,g(n)) is the tomogram of transformed density matrix of bi-\npartite system;\nw(m1,m2,g(n)) is the unitary spin tomogram of tensor product of par-\ntial traceρ(1)over the second subsystem’s coordinates of the density\nmatrixρ12and unity operator 1 2;\nw12(m1,m2,g(n))istheunitaryspintomogram ofthestatewithdensity\nmatrixρ12.\nThe criterion of separability means\nj1/summationdisplay\nm1=−j1j2/summationdisplay\nm2=−j2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+ε\n2j2+1w(m1,m2,g(n))−εw12(m1,m2,g(n))/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 1\n(202)\nfor arbitrary g(n)andε.\nFor Werner states ρW, the tomogram of transformed state (in this\ncase, it means that p→ −εp) is related to the initial-state tomogram\nwW\nwε(m1,m2,g(n)) =−εw12(m1,m2,g(n))+1+ε\n4.(203)\nThe criterion of separability yields\n1/2/summationdisplay\nm1,m2=−1/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+ε\n4−εwW(m1,m2,g(n))/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 1.(204)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.4344 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nEquality (204) takes place for arbitrary g(n)andεonly for |p| ≤1/3.\nForp>1/3, the above sum depends on g(n)andεand it is larger than\none.\nIt is obvious if one calculates the tomogram using the elemen t of the\nunitary group of the form\ng(n)\n0=\n0 0 0 1\n0 1 0 0\n0 0 1 0\n1 0 0 0\n. (205)\nAt this point, the sum (204) reads\n1/2/summationdisplay\nm1,m2=−1/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+ε\n4−εwW(m1,m2,g(n))/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 3/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+εp\n4/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−3pε\n4/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\n(206)\nOne can see that this sum equals to one independently on the va lue\nof parameter |ε| ≤1 only for values |p| ≤1/3.Forp= 1, the maxi-\nmum value of the sum equals 2 = (1 + 3 ε)/2 (ε= 1). This value can\ncharacterize the degree of entanglement of Werner state.\nWe have introduced positive nonlinear map of density matrix which\nis purification map. The purification map can be combined with con-\ntractive maps discussed.Thetomographic-probability dis tributions un-\nder discussion can be completely described by their charact eristic func-\ntions. This means that the relation of tomogram property to e ntan-\nglement can be formulated in terms of the properties of chara cteristic\nfunctions.\nOne can also check the criterion using example of two-qutrit e pure\nentangled state with wave function\n|ψ/angbracketright=1√\n31/summationdisplay\nm=−1|um/angbracketright |vm/angbracketright. (207)\nThe sum defining the criterion of separability for specific U(9) trans-\nformg(n)\n0which is diagonalizing the hermitian matrix Lε|ψ/angbracketright/angbracketleftψ|\nreads\nF(ε,g(n)\n0) = 8/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+ε\n9/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−8ε\n9/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (208)\nFor 1/2> ε >1/8, this sum is larger than one, that means that the\nstate is entangled. For ε= 1/2, the function has maximum and it is\nequal to 5/3.\nThe entanglement of the considered state can be detected usi ng\npartial transposition criterion too.\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.44The geometry of density states, positive maps and tomograms 45\nFor the case of pure entangled state of two-qutrite system wi th the\nwave function\n|Φ/angbracketright=1√\n2/parenleftig\n|u1/angbracketright |v1/angbracketright+|u0/angbracketright |v0/angbracketright/parenrightig\n, (209)\ninwhichthestateswithspinprojections m=−1donotparticipate, the\npartial transpose criterion does also detect entanglement . Our criterion\nyields forspecific U(9) transform g(n)\n0, which diagonalizes thehermitian\nmatrixLε|Φ/angbracketright/angbracketleftΦ|the following expression for the function F(ε,g(n)\n0),\nwhich reads\nF(ε,g(n)\n0) = 5|1+ε|\n6+|1−5ε|\n6. (210)\nThe function takes maximum value for ε= 1/2 that equals to 3/2.\nThis value is smaller than 5/3 of the previous case. It corres ponds\nto our intuition that the superposition of three product sta tes of two\nqutrite system is more entangled than the superposition of o nly two\nsuch product states.\nThe criterion can be extended to multipartite spin system.\nWe have to apply for n-partite system the transform of the density\nmatrixρof the form\nL/vector ε=L(1)\nε1⊗L(2)\nε2⊗...⊗L(n)\nεn, (211)\nwhere the transform L(k)\nεkacts as depolarizing map on the kth subsys-\ntem. If the state is separable\nρ=/summationdisplay\nkpkρ(1)\nk⊗ρ(2)\nk⊗...⊗ρ(n)\nk,/summationdisplay\nkpk= 1, pk≥0,(212)\neach of the terms ρ(j)\nk(j= 1,2,...,n) in the tensor product is replaced\nby the term\nρ(j)\nk→ −εjρ(j)\nk+1+εj\nNj1j. (213)\nThis means that the transformed density matrix reads\nρ→L/vector ερ=/summationdisplay\nkpk\nn/productdisplay\nj=1⊗/parenleftigg\n−ερ(j)\nk+1+εj\nNj1j/parenrightigg\n.(214)\nThe unitary spin tomogram of the transformed density matrix takes\nthe form (/vector ε=ε1,ε2,...,εn)\nw/vector ε(m1,m2,...,m n,g(N)) =/summationdisplay\nkpkw(k)\npr(m1,m2,...,m n,g(N),/vector ε),\n(215)\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.4546 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nwhereN=/producttextn\ns=1(2js+ 1) and element g(N)is the unitary matrix in\nN-dimensional space. The tomogram w(k)\npr(m1,m2,...,m n,g(N),/vector ε) is\nthe joint probability distribution of spin projections ms=−js,−js+\n1,...,js, which dependson theunitarytransform g(N)in thestate with\ndensity matrix\nρk=n/productdisplay\ns=1⊗/parenleftbigg\n−εsρ(s)\nk+1+εs\nNs1s/parenrightbigg\n. (216)\nFor the elements\ng(N)\npr=n/productdisplay\ns=1⊗us(2js+1),\nwhereus(2js+1) is unitary matrix, the tomogram (215) takes the form\nof sum of the products\nw/vector ε(m1,m2,...,m n,g(N)\npr) =/summationdisplay\nkpkn/productdisplay\ns=1wk/parenleftig\nms,us(2js+1),εs/parenrightig\n,(217)\nwithwk(ms,us(2js+ 1),εs) being the unitary spin tomograms of the\nsthspinsubsystem with transformeddensity matrix Lεsρ(s)\nk. If oneuses\nas the matrix us(2js+ 1), the matrix of unitary irreducible represen-\ntation of the SU(2) group, the tomogram wkdepends only on the two\nparameters defining the point on the sphere S2.\nFor a separable state of the multipartite system, one has\n/summationdisplay\nm1,...,mn/vextendsingle/vextendsingle/vextendsinglew/vector ε(m1,m2,...,m n,g(N))/vextendsingle/vextendsingle/vextendsingle= 1 (218)\nfor all elements g(N)and all parameters /vector ε.\nFor entangled state, there can be some values of parameters /vector εand\ngroup elements g(N)for which the sum is larger than one.\n12. Conclusions\nWe summarize the results of the paper.\nThenotionofentangledstates (firstdiscussedbySchr¨ odin ger[4, 51])\nhas attracted a lot of efforts to find a criterion and quantitati ve charac-\nteristics of entanglement. A criterion based on partial tra nspose trans-\nform of subsystem density matrix (complex conjugation of th e subsys-\ntem density matrix or its time reverse) provides the necessa ry and\nsufficient condition of separability of the system of two qubi ts and\nqubit-qutritt system [52]. The phase-space representatio n of the quan-\ntum states and time reverse transform (change of the signs of the\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.46The geometry of density states, positive maps and tomograms 47\nsubsystem momenta) of the Wigner function in the case of Gaus sian\nstate was applied to study the separability and entanglemen t of pho-\nton states in [13]. Recently it was pointed out that the tomog raphic\napproach of reconstructing the Wigner function of quantum s tate [43–\n45] can be developed to consider the positive probability di stribution\n(tomogram) as an alternative to density matrix (or wave func tion)\nbecause the complete set of tomograms contains the complete informa-\ntion on the quantum state [42]. This representation (called probability\nrepresentation) was constructed also for spin states inclu ding a bi-\npartite system of two spins. Up to now the problem of entangle ment\nwas not discussed in the tomographic representation. Some r emarks\non tomograms and entanglement of photon states in the proces s of\nRaman scattering were done in [53]. The tomographic approac h has\nthe advantage of dealing with positive probabilities and on e deals with\nstandard probability distributions which are positive and normalized.\nWe studied the properties of separable and entangled state o f mul-\ntipartite system using the tomographic probability distri butions. The\npositive and completely positive maps of density matrices [ 39, 54] in-\nduce specific properties of the tomograms. The properties of the posi-\ntive maps were studied in [55]. We formulated necessary and s ufficient\nconditions of separability and entanglement of multiparti te systems in\nterms of properties of the quantum tomogram. Since the tomog rams\nwere shown [36] to be related to the star-product quantizati on proce-\ndure [56], we discuss entanglement and separability proper ties in terms\nof generic operator symbols. The tomographic symbols of gen eric spin\noperators were studied in [36]. Then we focused on propertie s of entan-\nglement and separability of a bipartite system using spin to mograms\n(SU(2)-tomograms) and tomograms of the U(N)-group.\nThe idea of the approach suggested can be summarized as follo wing.\nThe positive but not completely positive linear maps of a sub system\ndensity matrix do preserve the positivity of separable dens ity matrix\nof the composite system. These maps contain also maps which d o not\npreservethepositivity oftheinitial densitymatrixof ane ntangled state\nfor the composite system. It means that the set of all linear p ositive\nmaps of the subsystem density matrix (this set is semigroup) creates\nfrom the initial entangled positive density matrix of compo site system\na set of hermitian matrices including the matrices with nega tive eigen-\nvalues. To detect the entanglement we use the tomographic sy mbols\nof the obtained hermitian matrices.The tomographic symbol s of state\ndensity matrices (state tomograms)are standard probabili ties.In view\nof this the tomographic symbols of the obtained hermitian ma trices\ncorresponding to initial separable state preserve all the p roperties of\nthe probability representation including positivity and n ormalization.\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.4748 V.I. Man’ko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria\nBut in case of entangled state the tomographic symbols of the obtained\nhermitian matrices can take negative values.The different be haviour\nof tomograms of separable and entangled states of composite systems\nunderaction ofthesemigroupofpositivemapsofthesubsyst emdensity\nmatrix provides the tomographic criterion of the separabil ity.\nTo conclude, we point out the main result of the work.\nWefoundthecriterionofseparabilitywhichisgivenbyequa tion(173).\nThe criterion is valid for multiparticle spin system. The cr iterion can\nbe called “tomographic criterion” of separability. The tom ographic cri-\nterion can be considered also for symplectic tomograms of mu ltimode\nphoton states. The condition of separability is sufficient be cause there\nalways exists a unitary group element by means of which any he rmitian\nmatrix can be diagonalized.It means that tomographic symbo l of non-\npositivehermitianmatrixhasnonpositivevaluesforsomeu nitarygroup\nparameters. The suggested criterion is connected with prop erties of\nthe constructed function (173) which for given density matr ix depends\non unitary group parameters gand the parameters of positive map\nsemigroupL. For separable density matrix the dependence on unitary\ngroup parameters and the semigroup parameters disappears a nd the\nfunction becomes constant equal to unity. For entangled sta tes the\nfunction differs from unity and depends on both group and semig roup\nparameters. The suggested criterion can be considered as so me com-\nplementary test of separability together with other criter ia available in\nthe literature (see, for example, [38, 52]). We point out tha t suggested\ncriterion differs from available usual ones by the kind of the n ecessary\nnumerical calculations. To apply this criterion one needs t o calculate\nthe sum of moduli of diagonal matrix elements of product of th ree\nmatrices. One of the matrices is hermitian and two others are unitary\nones. This proceduredoes not need the calculation of the eig envalues of\na matrix. The structure of positive (including not complete ly positive)\nmap semigroup with elements Lneeds extra investigation (see, for\nexample, [57]). We found also a test of entanglement based on the\nproperty of unitary spin tomogram.\nThe discussed purification map can be applied to find new quant um\nevolution equations in addition to known ones [58–61]. The a pplication\nof different forms of positive maps [55, 62] and supermatrix re presen-\ntation of the maps [63, 64] are useful for better understandi ng of the\ncomputations. Entanglement phenomena can be considered us ing sym-\nbols of density matrix of different kinds, e.g., particular qu asidistribu-\ntions[65] aswell astomographicsymbols[36]. Thedifference ofsymbols\nof entangled and separable density operators for different sc hemes of\nthe star-product quantization needs further investigatio ns as well as\ntest of entanglement of some generalizations of Werner stat e [66, 40]\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.48The geometry of density states, positive maps and tomograms 49\nin multipartite case. A relation of tomographic approach to different\npositive maps [67] should be investigated. The tomographic symbols\nare analytic in group parameters. This can be used to find extr ema of\ntomograms which give information on degree of entanglement .\nAcknowledgments\nV. I. M. and E. C. G. S. thank Dipartimento di Scienze Fisiche, Uni-\nversit´ a “Federico II” di Napoli and Istitito Nazionale di F isica Nu-\ncleare, Sezione di Napoli for kind hospitality. V. I. M. is gr ateful to\nthe Russian Foundation for Basic Research for partial suppo rt under\nProject No. 01-02-17745.\nReferences\n[1] P.A.M. 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Kossakowski 2003 “A class of linear positive maps in m atrix algebras” ArXiv\nquant-ph/0307132 v1\nMMSZ-Bregenz.tex; 10/11/2018; 12:33; p.51MMSZ-Bregenz.tex; 10/11/2018; 12:33; p.52"    },    {        "title": "0706.3116v1.Series_expansion_for_the_density_of_states_of_the_Ising_and_Potts_models.pdf",        "content": "SERIES EXPANSION FOR THE DENSITY OF STATES OF THE\nISING AND POTTS MODELS\nDANIEL ANDR \u0013EN\n1.Introduction\nThe Lentz-Ising model of ferromagnetism has been thoroughly studied since its\nconception in the 1920's[Len20]. It was solved in the 1-dimensional case by Ising\nhimself in 1925[Isi25] and in the 2-dimensional case without an external \feld by\nOnsager in 1944[Ons44]. For an introduction to the model, see Cipra [Cip87].\nThe partition function on a graph Gonnvertices and medges is de\fned as\n(1) Z(G;x;y) =X\ni;jaijxiyj:\nHereaijcounts the number of induced subgraphs of Gwith (n\u0000j)=2 vertices and\n(m\u0000i)=2 edges in the boundary. We refer to the index ias the energy and the\nindexjas the magnetization.\nThe traditional partition function studied in statistical physics is then obtained\nby evaluating it at a certain point\n(2) Z\u0000\nG;eK;eH\u0001\nwhereK=\u0000J=kBT,H=\u0000h=kBTandJandhare parameters describing the\ninteraction through edges and with an external magnetic \feld respectively, Tis the\ntemperature and kBthe Boltzmann constant.\nThe main goal in the study of the Ising model on a graph G, or some family\nof graphs, is usually to study the model in the vicinity of a critical temperature ,\ndenotedTc, where the model undergoes a phase transition and there determine the\nbehaviour of various critical properties.\n2.Definitions and notation\nLetG= (V;E) be a graph with vertex set VwithjVj=nvertices and edge set\nEwithjEj=medges. Let a state \u001bbe a function from the vertices Vto the set\nf\u00061gand let \n be the set of all states. We can then de\fne the energy of the graph\nG= (V;E) in state\u001bto be\n(3) E(G;\u001b) =X\nuv2E\u001b(u)\u001b(v)\nthemagnetization to be\n(4) M(G;\u001b) =X\nv2V\u001b(v)\nand formulate a generating function that counts them all as\n(5) Z(G;x;y) =X\n\u001b2\nxE(G;\u001b)yM(G;\u001b)=X\nijaijxiyj\nwhere the last equality de\fnes the coe\u000ecients aij=aij(G). We often drop Gwhen\nwe can deduce the graph from the context.\nWe will also need these de\fnitions later:\n1arXiv:0706.3116v1  [cond-mat.str-el]  21 Jun 20072 DANIEL ANDR \u0013EN\nDe\fnition 1 (T-join) .A T-join (T;A) in a graph G= (V;E) is a subset T\u0012V\nof vertices and a subset A\u0012Eof edges such that each vertex in Tis incident with\nan odd number of edges in Aand each vertex in VnTis incident with an even\nnumber of edges from A.\nObserve that the cardinality of Thas to be even since we can not have a subgraph\nwith an odd number of vertices of odd degree.\nDe\fnition 2 (Cut) .A cut [S;\u0016S] in a graph G= (V;E) is a subset of edges, induced\nby a partition S[\u0016S=V, that have one endpoint in Sand the other in \u0016S. Let\nj[S;\u0016S]jbe the number of edges in the cut.\nDe\fnition 3 (Locally vertex transitive) .We say that a sequence of graphs fGig1\ni=1\nare locally vertex transitive if for each Rthere exists an Nsuch that all balls of\nradiusR, in the graph metric, around each vertex in each graph in the subsequence\nfGig1\ni=Nare isomorphic.\n3.Series expansion\nWe can try to make our problem simpler by setting y= 1 and get\n(6) Z(x;1) =Z(x) =X\nijaijxi1j=X\niaixi\nwhere once again the last equality is the de\fnition of the aicoe\u000ecients. What do\nthese coe\u000ecients count? Since the state \u001bpartitions the vertex set Vin two parts\nand theE(G;\u001b) counts edges with one vertex in one part and the other vertex in\nthe other part as negative and the other edges as positive we get that aiis twice\nthe number of cuts of sizem\u0000i\n2(we count each cut twice since we can interchange\nthe partitions). This has a natural reformulation using even subgraphs, namely:\nTheorem 4 (van der Waarden) .Letaibe the number of cuts of sizem\u0000i\n2and let\nbibe the number of even subgraphs with iedges, then:\n(7)X\niaieiK= 2 coshmKX\nibitanhiK\nProof. The \frst sum in (7) is the moment generating function for the sequence ai.\nThekthmoment of aican be written as\n(8) \u0016k=X\niaiik=X\n\u001b2\n X\nuv2E\u001b(u)\u001b(v)!k\nWe now expand the multinomial\u0000P\nuv2E\u001b(u)\u001b(v)\u0001kwhere each term can be seen\nas an choice of kout ofmedges (not necessarily distinct). Now observe that if we\nhave chosen an even number of edges incident with a vertex, vsay, we will have an\neven number of \u001b(v)'s in the product so they contribute +1. If we have chosen an\nodd number of vertices we can \fnd a smallest (in some arbitrarily order) such odd\nvertexvand we see that if we change the state \u001bto the state \u001b0with\u001b(v) =\u0000\u001b0(v)\nand all other values equal, we will get a bijection between states witch contribute\n+1 and\u00001 and with at least one vertex of odd degree. Our conclusion is that\nwe only count the choices where we have an even degree at each vertex. We will\nhowever count subgraphs where we have the opportunity to choose each edge a\nmultiple number of times. If we reduce the multiple edges modulo 2 we get a\nsimple subgraph of even degree. The \\surviving\" edges are the ones that where\nchosen an odd number of times so an even number of those \\odd\" edges have to be\nincident at each vertex.SERIES EXPANSION FOR THE DENSITY OF STATES OF THE ISING AND POTTS MODELS 3\nIf we now change our view and instead of adding up the kmoments, change the\norder of summation, and add up along the index of the number of \\surviving\" odd\nedgesiwe get a simple connection between the simple subgraphs and subgraphs\nwith multiple edges. We can construct an even multiedge subgraph by \frst select\na simple subgraph with even degree at each vertex and then multiply each edge an\nodd number of times and then select a number of edges not in the even subgraph\nto multiply an even number of times. So if we \frst choose an even subgraph with i\nedges and multiply each edge an odd number of times we get the generating func-\ntionbisinhiK, and then choose a number of edges outside the even subgraph and\nmultiply these an even number of times, we get the generating function coshm\u0000iK,\nand we end up with an multiedge subgraph with an even number of edges incident\nto each vertex. If we now sum over all iwe get\n(9)X\nibisinhjKcoshm\u0000iK= coshmKX\nibitanhiK\nand we see each of these graphs twice and therefor this gives (7). \u0003\nTo formulate the full two variable connection we need T-joins instead of even\ndegree subgraphs and also consider the size of the sets in the vertex partition\ninduced by the state. Also note that in this theorem we have slightly changed the\nmeaning of aijandbijto make the proof using standard graph theoretic notation.\nThe following theorem can be found in e.g. [Big77]:\nTheorem 5. LetG=G(V;E)be a graph, aijthe number of cuts [S;\u0016S]with\nj[S;\u0016S]j=iandjSj=j. Letbijbe the number of T-joins (T;A)withjAj=iand\njTj=j. Then\nX\nijbijxiyj= 2\u0000jVjX\nijaij(1\u0000x)i(1 +x)jEj\u0000i(1\u0000y)j(1 +y)jVj\u0000j\nProof. Fix a subset T\u0012Vof vertices and a subset A\u0012Eof edges from the graph\nG=G(V;E). LetS\u0012Vbe another subset of vertices and [ S;\u0016S] be the cut de\fned\nby the edges from Sto\u0016S=VnS. Let the weight of the vertices in T\\Sbe\u0000y,\nthe weight of the vertices in T\\\u0016Sbey, the weight of the edges from Athat lies in\nthe cut [S;\u0016S] be\u0000xand the rest of the edges from Ahave weight x. Let the total\nweight of (T;A) with respect to the cut [ S;\u0016S] be the product of the weights of the\nedges and vertices in ( T;A). We say that the weight is positive if the coe\u000ecient in\nfront ofxjAjyjTjis positive and negative otherwise. By magnitude we denote the\nweight without the sign.\n(T;A) can fail to be a T-join in basically three ways. First the cardinality of\nTcan be odd, secondly there can exist a smallest vertex (in an arbitrary order of\nthe vertices) vthat is incident with an odd number of edges from Aand does not\nbelong toT, and \fnally there can exist a smallest vertex vthat is incident with an\neven number of edges from Aand belongs to T.\nIn the \frst case we have two cuts [ S;\u0016S] and [ \u0016S;S] in which the magnitude of the\nweight will be the same but with opposite sign.\nIn the two latter cases we have a bijection between cuts with v2Sandv =2S\n(we simply move the vertex vbetweenSand \u0016S) that once again give the same\nmagnitude and di\u000berent signs of the weight. If we sum over all cuts the total\ncontribution of such a choice of ( T;A) will cancel.\nIf (T;A) indeed is a T-join the weight will always be positive since we either have\nan even number of vertices in T\\Sand an even number of edges crossing the cut\nor an odd number of vertices in T\\Sand an odd number of vertices crossing the\ncut. All in all we end up with an even number of minus signs and thus a positive\nweight.4 DANIEL ANDR \u0013EN\nIf we now sum over all choices ( T;A) andSwe will count each T-join 2jVjtimes.\nIf we rearrange our summation (i.e. we \frst choose a cut and then go through all\nchoices ofTandA) we get the theorem. \u0003\nTo get Theorem 4 we have to shift the indices and substitute xfore\u00002Kandy\nfor 0 and \fnally double all values since we count all cuts twice in (7).\n(10) 2emKX\ni;jaije\u00002iK0j= [since 00= 1] =X\ni2aie(m\u00002i)K=\n2emK2\u0000mX\ni;jbij(1\u0000e\u00002K)i(1 +e\u00002K)m\u0000i1n=\n2\u0012eK+e\u0000K\n2\u0013mX\nibi\u00121\u0000e\u00002K\n1 +e\u00002K\u0013i\n=\n2 coshmKX\nibitanhiK\nwherebidenotes the number of T-joins with iedges. Now, since aicounts the\nnumber of cuts with iedges in Theorem 5, 2 aie(m\u00002i)Kcorresponds to the aieiKin\nTheorem 4 via reindexing and the fact that each cut is counted twice in Theorem\n4.\nSince we have a symmetry between T-joins and cuts we have the following corol-\nlary:\nCorollary 6. With the same notation as in theorem 5 we have\nX\ni;jaijxiyj= 2\u0000jEjX\ni;jbij(1\u0000x)i(1 +x)jVj\u0000i(1\u0000y)j(1 +y)jEj\u0000j\nProof. If we choose a T-join instead of a cut the weight of ( T;A) will always be\npositive if and only if ( T;A) is a cut. In other cases the contributions once again\ncancel out. \u0003\n3.1.The thermodynamic limit. In Physics we are interested in the so called\nthermodynamic limit of a sequence of graphs fGjg1\nj=1. This is de\fned as\n(11) f(x) = lim\nj!11\njV(Gj)jlogZ(Gj;x;1)\nwhen it exists. An example of such a family is fCn\u0002Cng1\nn=3. If the graph family\nis locally vertex transitive its easy to see that the number of connected T-joins\nof \fxed size will grow proportionally to jV(G)j. From that follows that the total\nnumber of T-joins of a \fxed size will grow as a polynomial with degree equal to the\nmaximal number of connected components in the T-joins of that size and thus will\nthe thermodynamic limit exist.\nIf we change notation so that our index set instead is the number of vertices n\nin our graph sequence and use bj(n) to denote the number of T-joins with jedges\nwe see that the thermodynamic limit is\n(12) lim\nn!11\nnlog 2 coshmxX\njbj(n) tanhjx= log 2 +dcoshx+X\njbjtanhjx\nwhich de\fnes a new set of bj:s that happens to be rational numbers and d=m=n,\nthe average degree.SERIES EXPANSION FOR THE DENSITY OF STATES OF THE ISING AND POTTS MODELS 5\n3.2.Taylor series. It can be of interest to plot these functions, or at least some\napproximation of them. Since we have a phase transition in all interesting cases its\nhard to \fnd one function that works for the entire interval. In this section we will\ninstead develop two Taylor approximations, one for the high-temperature case and\none for the low-temperature case. The \frst one is simple, take\n(13)u(K) =X\niaiKi= log 2 +dlog coshK+ lim\nn!11\nnlogX\nibi(n) tanhiK\nand Taylor expand around K= 0. If you want the function K(u) you can invert\nu(K) as a Taylor series. Its often better to plot a Pad\u0013 e-approximation of u(K).\nThe second one needs a little more work:\n(14)d\ndKu(K) =d\ndK\"\nlog 2 +dlog coshK+ lim\nn!11\nnlogX\nibi(n) tanhiK#\n=\n\u0014x= tanhK\nd\ndK= (1\u0000x2)d\ndx\u0015\n=dx+ (1\u0000x2)d\ndxlim\nn!11\nnlogX\nibi(n)xi=\ndx+ (1\u0000x2)d\ndxlim\nn!11\nnlog1\n2nX\niam\u00002i(n)(1\u0000x)i(1 +x)m\u0000i=\ndx+ (1\u0000x2)d\ndxlim\nn!1\"\n\u0000log 2 +dlog(1 +x) +1\nnlog exp \nnX\niai\u00121\u0000x\n1 +x\u0013i!#\n=\ndx+d(1\u0000x2)\n1 +x+ (1\u0000x2)d\ndxX\nai\u00121\u0000x\n1 +x\u0013i\n=d+ (1\u0000x2)d\ndxX\nai\u00121\u0000x\n1 +x\u0013i\n=\nd+2(x\u00001)\n(x+ 1)X\nai\u00121\u0000x\n1 +x\u0013i\u00001\n=d\u00002X\nai\u00121\u0000x\n1 +x\u0013i\n=\nd\u00002X\nai\u00121\u0000tanhK\n1 + tanhK\u0013i\nBy not doing the last substitution x= tanhKits easy to invert this function and\nplot it in the low temperature (high energy) portion of the scale.\n4.Two other series expansions\nFrom the basic thermodynamic limit one can construct two other sequences that\nare of a more combinatorial \ravour. These are sequences with integer coe\u000ecients.\nThis is because they, instead of weighting the counts, make an explicit order in\nwhich you have to choose things and thus avoid dividing with large factorials. The\ncorrespondence with the thermodynamic limit series are:\n(15)X\ni\u00150bi(n)xi= exp0\n@nX\ni\u00151bixi1\nA=Y\ni\u00151\u00121\n1\u0000xi\u0013n\fi\n=Y\ni\u00151\u00121\n1\u0000b\fixi\u0013n\nThe coe\u000ecients are connected trough these simple triangular linear equations:\nibi=X\ntjit\ft (16)\nibi=X\nrs=irb\fs\nr (17)\ni\fi=X\nrst=i\u0016(r)tb\fs\nt (18)6 DANIEL ANDR \u0013EN\n4.1.What they are counting. The new series can be seen as placing connected\nsubgraphs at each vertex in some order. First we place all subgraphs (including the\nempty one) rooted at vertex one, then the subgraphs rooted at vertex two and so\non in such a way that we never form a new connected component. In this way we\navoid symmetries and we get a integer sequence. We can still calculate, in principle,\nthe di\u000berent \fiandb\fifor locally vertex transitive graphs.\n5.Plots\nWe shall now compare these Taylor expansions of the series with some sampled\ndata to compare how well behaved the series are around the critical energy. As we\nshall see, the series are rather far from what can be expected to be the truth.\n0.2 0.4 0.6 0.8 10.10.20.30.40.5\nFigure 1. K(u) for the three-dimensional simple cubic lattice to-\ngether with sampled data for the cubes of linear order 32, 64 and\n128.\n5.1.Simple cubic lattice. We start of with the simple cubic lattice in three,\nfour and \fve dimensions. As can be expected, the longest series expansion is for\nthe three-dimensional simple cubic lattice. The four- and \fve-dimensional simple\ncubic lattices have much shorter high- and low-temperature expansions. Figure\n1 to 3 shows K(u) curves for the three lattices respectively. The pictures are of\ndiagonal Pad\u0013 e-approximants of the Taylor expansions. The Pad\u0013 e-approximant for\nthe function of the three-dimensional simple cubic lattice is fairly accurate to about\nK\u0019:22. For the four- and \fve-dimensional lattices the accuracy is a lot lower as\ncan be seen from the pictures.\n6.Guessing the radius of convergence\nThe interesting part is of course to try to \fnd the radius of convergence for\nthe series of the function u(K) since that would give the critical temperature Kc.\nA fairly good guess for Kcfor the simple cubic lattice is that it is approximately\n0:2216546 (see e.g. [HRL+]). A simple observation is that the radius of convergence\nmust be smaller than any 1 =b\fi1=isince otherwise the in\fnite product (15) will notSERIES EXPANSION FOR THE DENSITY OF STATES OF THE ISING AND POTTS MODELS 7\n0.2 0.4 0.6 0.8 10.10.20.30.40.5\nFigure 2. K(u) for the four-dimensional simple cubic lattice to-\ngether with sampled data for the cube of linear order 4, 6, 8, 12,\n16, 32, 48 and 64.\n0.2 0.4 0.6 0.8 10.10.20.30.40.5\nFigure 3. K(u) for the \fve-dimensional simple cubic lattice to-\ngether with sampled data for the cube of linear order 4, 6, 8, 12,\n16 and 32.\nconverge. In Table 1 the row labelled min is the minimum over all known values of\nb\fifor the simple cubic lattice. In trying to extrapolate this value we have used a\npolynomial of degree 3 and a min square approximation to the data set (1 =i;1=b\fi1=i)\nand looked at the constant term in the resulting polynomial. This gives the result in\nthe row labelled f(0). Since the coe\u000ecients is some form of combinatorial quantity8 DANIEL ANDR \u0013EN\nb\fibib\u000bi+b\u000bi\u0000a+\nia\u0000\ni\nmin 0.291989 0.291989 0.618531 0.616299 0.618531 0.616307\nf(0) 0.227727 0.227839 0.54278 0.543852 0.542846 0.539196\n\u000bii\f\r0.221418 0.221451 0.523726 0.520324 0.523747 0.542904\n\u000bi\r 0.258385 0.258429 0.578131 0.565115 0.578137 0.56453\nTable 1. Di\u000berent ways to try to guess the radius of convergence\nfor the simple cubic lattice.\nit is not unthinkable that they grow exponentially. We have thus also tried to\n\ft the data ( i;b\fi) to the functions \u000bii\f\rand\u000bi\rand calculated 1 =\u000bthat gives\nus the guessed radius of convergence. This is the last two lines of Table 1. The\nsecond column is the same thing for the coe\u000ecients of bi. When we come to the low\ntemperature part we get into some trouble since the coe\u000ecients of the sequences\naiandb\u000bihave both positive and negative signs. If we try to use the same analysis\nas for the (all positive) biandb\fi, we get very erratic numbers, so instead we split\nthe sequences in a positive and a negative part and do the analysis separately. As\nobserved by others, the low temperature series seems to have a complex root that\nis closer to the origin than the physically important real root. Fortunately does\nthe extrapolation with \u000bii\f\rfor thebiandb\fiseries give a decent idea of what the\ncritical temperature may be.\nReferences\n[Big77] Norman Biggs. Interaction models . Cambridge University Press, Cambridge, 1977. Course\ngiven at Royal Holloway College, University of London, October{December 1976, London\nMathematical Society Lecture Note Series, No. 30.\n[Cip87] Barry A. Cipra. An introduction to the Ising model. Amer. Math. Monthly , 94(10):937{\n959, 1987.\n[HRL+] Roland H aggkvist, Andres Rosengren, Per H\u0017 akan Lundow, Klas Markstr om, Daniel\nAndr\u0013 en, and Petras Kundrotas. On the Ising model for the simple cubic lattice. Manu-\nscript.\n[Isi25] Ernst Ising. Beitrag zur Theorie des Ferromagnetismus. Z.Physik , 31:253{258, 1925.\n[Len20] Wilhelm Lenz. Beitrag zum Verst andnis der magnetishen Erscheinungen in festen\nK orpern. Z. Physik , 21:613{615, 1920.\n[Ons44] Lars Onsager. Crystal statistics. I. A two-dimensional model with an order-disorder tran-\nsition. Phys. Rev. (2) , 65:117{149, 1944.SERIES EXPANSION FOR THE DENSITY OF STATES OF THE ISING AND POTTS MODELS 9\nAppendix A.Tables\nWe have collected the various series we have found in this appendix. Some of\nthese are old and thus not especially long and some are from newer calculations\nand longer.\nn \u000bn b\u000bn an\n6 1 1 1\n10 3 3 3\n12 -4 -4 -7/2\n14 15 15 15\n16 -33 -33 -33\n18 104 104 313/3\n20 -282 -285 -561/2\n22 849 849 849\n24 -2460 -2470 -9847/4\n26 7485 7485 7485\n28 -22542 -22647 -45069/2\n30 69392 69384 346966/5\n32 -213738 -214299 -427509/2\n34 666750 666750 666750\n36 -2086785 -2092121 -12520405/6\n38 6583341 6583341 6583341\n40 -20852223 -20892996 -83409453/4\n42 66425750 66424630 464980286/7\n44 -212410377 -212770353 -424819905/2\n46 682202205 682202205 682202205\n48 -2198562644 -2201602421 -17588511087/8\n50 7110521070 7110521022 35552605353/5\n52 -23065955826 -23093964696 -46131904167/2\n54 75045653088 75045278168 675410878105/9\n56 -244806881325 -245063348553 -979227570369/4\n58 800606679471 800606679471 800606679471\n60 -2624325216574 -2626724535242 -13121625909861/5\n62 8621219166681 8621219166681 8621219166681\n64-28379404026078 -28402366460136 -113517616531821/4\nTable 2. Simple cubic lattice10 DANIEL ANDR \u0013ENn \fnb\fn bn\n4 3 3 3\n6 22 22 22\n8 186 183 375/2\n10 1980 1980 1980\n12 24032 23793 24044\n14 319170 319170 319170\n16 4514664 4497993 18059031/4\n18 67003462 66999920 201010408/3\n20 1032455736 1030496478 5162283633/5\n22 16397040750 16397040750 16397040750\n24 266958785298 266673642443 266958797382\n26 4437596650548 4437596650548 4437596650548\n28 75078511535604 75027576950427 525549581866326/7\n30 1289656872697576 1289654284203514 6448284363491202/5\n32 22447149807206352 22437033558570606 179577198475709847/8\n34 395251648062268272 395251648062268272 395251648062268272\n36 7031220729573330428 7028971745154518717 21093662188820520521/3\n38 126225408651399082182 126225408651399082182 126225408651399082182\n40 2284608766597864770492 2284077801219364370517 4569217533196761997785/2\n42 41655898158709803301884 41655887320814523000436 291591287110968623857940/7\n44 764611114761740442269316 764476683289070060492337 8410722262379235048686604/11\n4614120314204713719766888210 14120314204713719766888210 14120314204713719766888210\nTable 3. Simple cubic latticeSERIES EXPANSION FOR THE DENSITY OF STATES OF THE ISING AND POTTS MODELS 11\nn a4\nnb4\nna5\nnb5\nn\n4 0 6 0 10\n6 0 76 0 180\n8 1 1371 0 5025\n10 0 30152 1 178696\n18 0 5\n20 28 -11/2\n22 -64 45\n24 127/3 -95\n26 228 50\n28 -834 5\n30 1116 1471/3\n32 5911/4 -1685\n34 -10404 1885\n36 21460 -775/2\n38 -1956 5445\n40 -595179/5 -112951/4\n42 1076092/3 49545\n44 -344316 -62795/2\n46 -1132588 71442\n48 10842287/2\n50 -9187444\n52 -5820150\n54 73867260\n56-1294335811/7\n58 95069292\n60 2609680726/3\n62 -3217644924\nTable 4. Simple cubic lattice in dimension 4 ( x4\nn) and 5 (x5\nn)12 DANIEL ANDR \u0013EN\nn an bn\n4 0 6\n6 2 44\n7 0 36\n8 0 384\n9 0 688\n10 6 4572\n11 6 11148\n12 -14 66158\n13 0 190662\n14 30 1051668\n15 60 3471452\n16 -108 17917704\n17 -144 65438160\n18 1118/3 971627006/3\n19 498 1265584728\n20 -714 30625029636/5\n21 -2366 25078631014\n22 3270 -3816568476\n23 7704\n24 -8106\n25 -27372\n26 19842\n27 114342\n28 -68892\n29 -377376\n30 643952/5\n31 1431726\n32 -137718\n33 -5365756\n34 -33330\n35 18644574\n36 -970922/3\n37 -69012330\n38 -32516754\n39 249820316\n40 162829320\n41 -869879742\n42-5660822830/7\n43 3155460756\nTable 5. Simple cubic lattice, 3-states Potts model"    },    {        "title": "0707.3099v3.Existence_of_a_Density_Functional_for_an_Intrinsic_State.pdf",        "content": "arXiv:0707.3099v3  [nucl-th]  19 Aug 2008Existence of a Density Functional for an Intrinsic State\nB. G. Giraud\nbertrand.giraud@cea.fr, Institut de Physique Th´ eorique ,\nDSM, CE Saclay, F-91191 Gif/Yvette, France\nB. K. Jennings\njennings@triumf.ca, TRIUMF, Vancouver BC, V6T2A3, Canada\nB. R. Barrett\nbbarrett@physics.arizona.edu, Department of Physics,\nUniversity of Arizona, Tucson, AZ 85721, USA\n(Dated: November 1, 2018)\nA generalization of the Hohenberg-Kohn theorem for finite sy stems proves the existence of a\ndensity functional (DF) for a symmetry violating intrinsic state, out of which a physical state with\ngood quantum numbers can be projected.\nI. INTRODUCTION\nDensity functional [1] theory (DFT) was initially defined for ground s tates. These have good quantum numbers.\nEverynuclearphysicistknowsthat, forinstance, thegroundsta teof20Ne isa0+andthat its densityis, thus, isotropic,\nnot an ellipsoid. Every molecular physicist knows that, for instance, the ground state of the ammonia molecule is a\ngood parity state, not just the pyramid described by the Born-Op penheimer approximation. In particular, the nuclear\nDF must generate spherical solutions for the some thousand 0+nuclear ground states, whether nuclei are intrinsically\ndeformed or not. The same need for isotropic solutions extends to the non-local generalization of the DFT [2] - [3].\nBut the theory of rotational bands and/or parity vibrations, whe ther in nuclear, atomic or molecular physics, most\noften relates ground states to wave packets, often named “intr insic states”, which are symmetry breaking, namely do\nnot transform in an irreducible representation (irrep) of the symm etry group Sof the Hamiltonian. Therefore, one\nmay raise the question of DFT for intrinsic states rather than eigen states.\nGiven the physical Hamiltonian Hwith its symmetry group S,calculations providing a “non S-irrep” state as a\nsolution for a minimum energy cannot be labelled as the result of “the” DF. Such a state, labelled intrinsic, is actually\njust a convenient wave packet, to be subsequently projected on to good quantum numbers to account for physical\nlevels. Such intrinsic calculations should rather exhibit a special Hamilt onian, which might be called an intrinsic\nHamiltonian, distinct from the physical one, if such calculations are t o be legitimized. Or they should be interpreted\nas one variety of the Hartree-Fock, Hartree-Bogoliubov, etc. v ariational approaches. This is implicit or even explicit\nin calculations with an energy density functional, implying non-localities, see for instance [4] - [7]. Energy d ensity and\nparticle density are different concepts.\nIt turns out that the particle density which has been used for the f oundation of DF theory mainly concerns eigen-\nstates of the physical Hamiltonian, in principle at least, while the ener gy density, used for Skyrme force calculations\nin nuclear physics for instance, mainly provides intrinsic states. This note presents a particleDF theory for intrinsic\nstates, not for eigenstates of the Hamiltonian. We show how the ph ysical Hamiltonian can be reconciled with the\nproper definition of a DF for an intrinsic state and how the resulting in trinsic state can be accepted as a useful wave\npacket, out of which states with good quantum numbers can be pro jected. In particular, in the case of molecules,\nour approach will consider both the electrons and the nuclei. Our int rinsic state can take into account both kinds of\ndegrees of freedom. Section II describes a functional out of whic h a variational principle derives for an intrinsic state,\nand out of which a DF for the intrinsic density is obtained. Section III gives an example of variational equations to\nbe solved in practice. Section IV rewrites the formalism into a slightly s impler form. Section V contains a discussion\nof our result and suggests an ansatz for intrinsic Hamiltonians.\nII. BASIC FORMALISM\nFor a first argument, dealing with one kind of identical particles only, letHbe their physical Hamiltonian and\nφ,∝angbracketleftφ|φ∝angbracketright= 1,be a trial wave packet, most often not transforming under an irre p of the symmetry group SofH.\nFor instance, for fermions, φmay be an arbitrary Slater determinant, but we let φbe also a more general wave\nfunction, including some amount of correlations. States ψ∝Pφwith good quantum numbers can then be projected\nout ofφby a projector P,a fixed operator. In the following, we shall systematically use the pr operties,P2=Pand2\n[P,H] = 0.It may happen that ∝angbracketleftφ|P|φ∝angbracketrightvanishes, but such cases usually make a domain of zero measure in th e usual\nvariational domains, where φevolves. In any case, since His an operator bounded from below, the functional of φ,\n∝angbracketleftφ|PH|φ∝angbracketright/∝angbracketleftφ|P|φ∝angbracketright,is bounded from below. Embed now the system in an external, local fie ld,U=/summationtext\niu(ri).The\nlocal, real potential uis taken bounded from below, but is otherwise arbitrary. In particu lar, it may usually have\nnone of the symmetries of H.Then, given u,the following functional of φ,\nF[φ] =∝angbracketleftφ|PH|φ∝angbracketright\n∝angbracketleftφ|P|φ∝angbracketright+∝angbracketleftφ|U|φ∝angbracketright, (1)\nis bounded from below. To find the lowest energy with the quantum nu mbers specified by Pone can use a constrained\nsearch [8], in which one first considers only states that show a given d ensity profile τ(r),then one lets τvary,\nInfφ/bracketleftbigg∝angbracketleftφ|PH|φ∝angbracketright\n∝angbracketleftφ|P|φ∝angbracketright+∝angbracketleftφ|U|φ∝angbracketright/bracketrightbigg\n= Infτ/bracketleftbigg/parenleftbigg\nInfφ→τ∝angbracketleftφ|PH|φ∝angbracketright\n∝angbracketleftφ|P|φ∝angbracketright/parenrightbigg\n+/integraldisplay\ndrτ(r)u(r)/bracketrightbigg\n. (2)\nThe process goes in two steps, namely, i) a minimization within a given pa rticle density profile, τ(r)≡ ∝angbracketleftφ|c†\nrcr|φ∝angbracketright,for\nNparticles, with c†\nrandcrthe usual creation and annihilation operators at position r,then, ii) a minimization with\nrespect to the profile. The inner minimization clearly defines a DF, F[τ]≡Infφ→τ(∝angbracketleftφ|PH|φ∝angbracketright/∝angbracketleftφ|P|φ∝angbracketright).\nActually, it is more general [9] - [13] to use many-body density matric esBinN-body space, meaning mixed as well\nas pure states, and yielding a density τ(r) in one-body space,\nInfτ/bracketleftbigg\nInfB→τ/parenleftbiggTrBPH\nTrBP+TrBU/parenrightbigg/bracketrightbigg\n, (3)\nbut we shall use wave-functions in the following, namely B=|φ∝angbracketright∝angbracketleftφ|,for obvious pedagogical reasons. We shall assume\nthat thisInf φactuallydefinesanabsoluteminimum, Min φ,reachedatsomesolutionΦ ofthecorrespondingvariational\nprinciple. Moreover, we shall assume, temporarily at least, that th e solution Φ is unique. Uniqueness is not obvious,\nhowever, if only because many φ’s can give the same P|φ∝angbracketright,and, whenuvanishes, this variational principle, Eq. (2),\nreduces to the well-known “variation after projection” [14] metho d for Hartree-Fock calculations for instance.\nAnyhow,τanduare clearly conjugate in a functional Legendre transform, with δF/δτ=−u.Finally, ifρ(r)\ndenotes the profile of Φ when u→0,then the lowest energy with good quantum numbers is nothing but F[ρ].\nThe minimization, with respect to τ,of the functional, F[τ],provides simultaneously the density of the intrinsic\n(unprojected!) state and the projected energy. Notice, inciden tally, thatF[τ] depends on the choice of the variational\nset of trial functions φwhere the “inner minimization” is performed. Furthermore, it obviou sly depends on P.\nAmore generalargumentis possible, with morethan one kind ofident ical particles. Trialstatescan be, forinstance,\nproducts of determinants, one for each kind of fermions, and per manents, one for each kind of bosons. Consider for\ninstance the ammonia molecule, with i) its active electrons, ii) its three protons and iii) its nitrogen ion. It is trivial\nto include a center-of-mass trap into Hto factorize into a spherical wave packet the center of mass motio n of this\nself-bound system and avoid translational degeneracy problems. The complete Hamiltonian H, trial states φand\ndensity operators Bdepend on and describe simultaneously the electron, proton and nit rogen ion coordinates and\nmomenta. A DF in just the electronic density space, however, resu lts from the definition,\nF[τ] = Inf B→τTrBPH\nTrBP, (4)\nwherePprojectsgood quantumnumbers forthe wholesystem and traces aretakenoverall degreesoffreedom, while τ\nis set as only an electronic density. Interactions between heavy de grees of freedom, between heavy and electronic ones,\nand between electrons, are taken into account by the trace in the numerator. No Born-Oppenheimer approximation\nis needed for this “global” definition. For the sake of simplicity, howev er, we return in the following to the case of one\nkind of particles only. Most considerations which follow have obvious g eneralization for multicomponent systems.\nIII. VARIATIONAL EQUATIONS\nLetδφbe an infinitesimal variation of the trial function in its allowed domain. T hen, at first order, one obtains,\nδF=∝angbracketleftδφ|PH|φ∝angbracketright\n∝angbracketleftφ|P|φ∝angbracketright+∝angbracketleftδφ|U|φ∝angbracketright−∝angbracketleftδφ|P|φ∝angbracketright∝angbracketleftφ|PH|φ∝angbracketright\n(∝angbracketleftφ|P|φ∝angbracketright)2+\n∝angbracketleftφ|PH|δφ∝angbracketright\n∝angbracketleftφ|P|φ∝angbracketright+∝angbracketleftφ|U|δφ∝angbracketright−∝angbracketleftφ|P|δφ∝angbracketright∝angbracketleftφ|PH|φ∝angbracketright\n(∝angbracketleftφ|P|φ∝angbracketright)2. (5)3\nIf one defines the “gradient operator”,\nG=PH\n∝angbracketleftφ|P|φ∝angbracketright+U−P∝angbracketleftφ|PH|φ∝angbracketright\n(∝angbracketleftφ|P|φ∝angbracketright)2, (6)\nthen, obviously, δF=∝angbracketleftδφ|G|φ∝angbracketright+∝angbracketleftφ|G|δφ∝angbracketright.Note, incidentally, that Gis Hermitian.\nAt the minimum position Φ, the variation δFvanishes for any δφ.Replaceδφbyiδφto see that the difference,\n−∝angbracketleftδφ|G|Φ∝angbracketright+∝angbracketleftΦ|G|δφ∝angbracketright,vanishes as well. Then, trivially, at Φ ,both∝angbracketleftδφ|G|Φ∝angbracketrightand∝angbracketleftΦ|G|δφ∝angbracketrightvanish simultaneously,\n∝angbracketleftδφ|G|Φ∝angbracketright=∝angbracketleftΦ|G|δφ∝angbracketright= 0,∀δφ. (7)\nIn the special case of Slater determinants, let |ph∝angbracketright ≡c†\npch|Φ∝angbracketrightdenote any particle-hole state built upon |Φ∝angbracketrightas the\n“reference vacuum” for quasi-particles. Here c†andcare the familiar fermionic creation and annihilation operators,\nrespectively. Then the particle-hole matrix elements of Gvanish,\n∝angbracketleftph|G|Φ∝angbracketright= 0,∀ph. (8)\nAs long as a solution of this stationarity condition, Eq. (8), is not rea ched, the matrix elements, ∝angbracketleftph|G|φ∝angbracketright,define\nthe direction of the gradient of Fin the hyperplane tangent to the manifold of Slater determinants. A gradient\ndescent algorithm, |δφ∝angbracketright=−η/summationtext\nph|ph∝angbracketright∝angbracketleftph|G|φ∝angbracketright,whereηis a small step parameter, then leads to the solution. Notice,\nhowever, that the phrepresentation is covariant with φ.Thephbasis has to be recalculated at each step. Being state\ndependent, Gmust also be recalculated at each step.\nIV. SIMILAR THEORY, WITH A LAGRANGE MULTIPLIER\nThe slightly complicated gradient operator, Eq. (6), leads to a varia tional condition, Eq. (7), which combines\nthe matrix elements of three operators, namely PH, PandU.Define the number λ=∝angbracketleftΦ|PH|Φ∝angbracketright/∝angbracketleftΦ|P|Φ∝angbracketrightas a yet\nunknown Lagrange multiplier; it can be considered as an arbitrary pa rameter and shall be adjusted self-consistently\nlater, when Φ is reached. Then Eq. (7) also reads,\n∝angbracketleftδφ|(PH−λP+∝angbracketleftΦ|P|Φ∝angbracketrightU)|Φ∝angbracketright= 0. (9)\nIfφwere completely unrestricted, namely if δφwere completely general, this equation, Eq. (9), would mean that Φ is\nan eigenstate of the operator G.Since intrinsic states are understood to belong to restricted sets of states, the result\nΦ is only an approximate eigenstate of G.\nTo avoid the cumbersome coefficient, ∝angbracketleftΦ|P|Φ∝angbracketright,which multiplies U,it is convenient to define the auxiliary operator,\nH=PH−λP+W, (10)\nwhereW=/summationtext\niw(ri) is, likeU,an arbitrary, local, real, external field, bounded from below. It is o bvious that Gand\nHdefine a common solution Φ if uandware suitably proportional to each other, w=∝angbracketleftΦ|P|Φ∝angbracketrightu.In the following,\nhowever, we set Hab initio. It is an operator bounded from below. We are interested in its “almo st ground state”\nΞ and assume that this state is unique. A connection between a solut ion Ξ in this section and a solution Φ in the\nprevious section can easily be tested later.\nDefine again a constrained search for the lowest energy,\nInfφ∝angbracketleftφ|H|φ∝angbracketright= Infτ(Infφ→τ∝angbracketleftφ|H|φ∝angbracketright) = Inf τ/parenleftbigg\nFλ[τ]+/integraldisplay\ndrτ(r)w(r)/parenrightbigg\n, (11)\nwhere theλ-dependent DF, Fλ,is defined as,\nFλ[τ]≡Infφ→τ∝angbracketleftφ|(PH−λP)|φ∝angbracketright. (12)\nIt is again convenient, for pedagogy at least, to assume that this I nfφinduces an absolute minimum, reached at a\nposition Ξ in the variational space. The same assumption states tha t, givenλ,the absolute minimum of Fλ[τ] is\nFλ[σ] =∝angbracketleftΞ|(PH−λP)|Ξ∝angbracketright, (13)\nwhereσis the density of Ξ .LetEdenote this energy, E(λ)≡ ∝angbracketleftΞ|(PH−λP)|Ξ∝angbracketright.A simple manipulation then gives,\ndE\ndλ=−∝angbracketleftΞ|P|Ξ∝angbracketright. (14)\nA Legendre transform, using λand∝angbracketleftΞ|P|Ξ∝angbracketrightas conjugate variables, is thus available to return the matrix elemen t\n∝angbracketleftΞ|PH|Ξ∝angbracketrightas a function of the matrix element ∝angbracketleftΞ|P|Ξ∝angbracketright.Then one has just to locate the minimum of their ratio.\nNote again that the theory depends on the variational space wher eφevolves. But, in any case, one obtains\nsimultaneously the density of Ξ,the best intrinsic state, and the energy of its projected state P|Ξ∝angbracketright.4\nV. SUMMARY, DISCUSSION AND CONCLUSION\nIf only because of the need for spin densities [15] in the description of polarizable systems, the problem of symmetry\nconservation, or restoration, in DF theory has already received m uch attention in atomic and molecular physics [16]\n- [19]. It has been revisited here, in the spirit of the projected Hart ree-Fock method with variation afterprojection\n[14]: a variational principle for the density of an intrinsic state, witho ut symmetry, optimizes the energy of a state\nwith good quantum numbers. The idea was already introduced in the c ontext of particle number projection [20]. We\nhave shown in Secs. II and IV that our approach allows generalizatio ns of the Hohenberg-Kohn existence theorem.\nIt can be stressed that the present approach is concerned with t he density of an intrinsic state, not that of an\neigenstate. This is a major difference with all the other DF theories t hat we are aware of. Note, in particular, how\nour functional differs from a functional of a symmetrized [16] dens ity .\nWe showed in Sec. IV that a way to define the intrinsic Hamiltonian amou nts to a linear combination, H=\n−λP+PH,of the projector Pon the desired quantum numbers, and the laboratory Hamiltonian mu ltiplied by that\nsameP.Here, a subtle question must be raised, that of the nature of the in trinsic state. The more flexible the trial\nfunctions for this state, the better the projected state and th e lower the projected energy. However, full flexibility\ncontradicts simplicity, and, moreover, uniqueness of the intrinsic s tate; many different packets |φ∝angbracketrightcan give the same\nP|φ∝angbracketright.Symmetry projection brings correlations which, therefore, mustbe absent from the intrinsic state. This is why\nvariational domains for intrinsic states must necessarily be much narrower than the full Hilbert space.\nIn practice, fortunately, intrinsic states are confined to non-line ar, curved [21] manifolds, such as coherent states,\nSlater determinants, etc., which do not make linear subspaces. The intrinsic state, therefore, is not an exact eigenstate\nofH.It just minimizes a related quantity, the projected energy. It mus t be concluded that DF theory for an intrinsic\nstate necessarily depends on two factors, namely, i) obviously the quantum numbers to be projected out, but also ii)\nthe variational space retained for this intrinsic state.\nAcknowledgements : It is a pleasure for B.R.B. and B.G.G. to thank the TRIUMF Laborator y, Vancouver, B.C.,\nCanada, for its hospitality, where part of this work was done. The N atural Science and Engineering Research Council\nof Canada is thanked for financial support. TRIUMF receives fede ral funding via a contribution agreement through\nthe National Research Council of Canada. B.R.B. also thanks Servic e de Physique Th´ eorique, Saclay, France, and\nthe Gesellschaft f¨ ur Schwerionenforschung mbh Darmstadt, Ge rmany, for their hospitality, where parts of this work\nwere carried out, and acknowledges partial support from the Alex ander von Humboldt Stiftung and NSF Grant\nPHY0555396. Thecontributionoftwoanonymousrefereesinmakin gthispaperclearerisalsogratefullyacknowledged.\n[1] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)\n[2] T.L. Gilbert, Phys. Rev. B 12, 2111 (1975)\n[3] R. M. Dreizler and E. K. U. Gross, Density Functional Theory , Springer, Berlin/Heidelberg (1990); see also the referen ces\nin their review\n[4] J. Dobaczewski, H. Flocard and J. Treiner, Nucl. Phys. A 422, 103 (1984)\n[5] J. Meyer, J. Bartel, M. Brack, P. Quentin and S. Aicher, Ph ys. Lett. B 172, 122 (1986)\n[6] T. Duguet, Phys. Rev. C 67, 044311 (2003); T. Duguet and P. Bonche, Phys. Rev. C 67, 054308 (2003)\n[7] G.F. Bertsch, B. Sabbey and M. Uusn¨ akki, Phys. Rev. C 71, 054311 (2005)\n[8] M. Levy, Proc. Natl. Acad. Sci. USA 766062 (1979)\n[9] E.H. Lieb, Int. J. Quant. Chem. 24, 243 (1983)\n[10] H. Englisch and R. Englisch, Phys. Stat. Solidi B123, 711 (1984); B124, 373 (1984)\n[11]´A. Nagy and M. Levy, Phys. Rev. A 63, 052502 (2001)\n[12] J.P. Perdew and S. Kurth, in A Primer in Density Functional Theory , C. Fiolhais, F. Nogueira and M. Marques, eds.,\nLecture Notes in Physics, 620, Springer, Berlin (2003); see also the references in their r eview\n[13] R. van Leeuwen, Advances Quant. Chem. 43, 25 (2003)\n[14] R. Dreizler, P. Federman, B.G. Giraud and E. Osnes, Nucl . Phys.A 113145 (1968); S. Das Gupta, J.C. Hocquenghem\nand B.G. Giraud, Nucl. Phys. A 168625 (1971)\n[15] O. Gunnarson and B. J. Lundquist, Phys. Rev. B 13, 4274 (1976)\n[16] A. G¨ orling, Phys. Rev. A 47, 2783 (1993)\n[17] A. G¨ orling, Phys. Rev. A 59, 3359 (1999)\n[18] M. Levy and ´A. Nagy, Phys. Rev. Lett. 83, 4361 (1999)\n[19] A. G¨ orling, Phys. Rev. Lett. 85, 4229 (2000)\n[20] M.V. Stoitsov, J. Dobaczewski. R. Kirchner, W. Nazarew icz and J. Terazaki, Phys. Rev. C 76014308 (2007); J.A. Sheik\nand P. Ring, Nucl. Phys. A 66571 (2000)\n[21] B.G. Giraud and D.J. Rowe, J. Physique Lett. 40, L177-L180 (1979); B.G. Giraud and D.J. Rowe, Nucl. Phys. A 330,\n352 (1979)"    },    {        "title": "0707.3759v1.The_space_of_density_states_in_geometrical_quantum_mechanics.pdf",        "content": "arXiv:0707.3759v1  [quant-ph]  25 Jul 2007THE SPACE OF DENSITY STATES IN GEOMETRICAL\nQUANTUM MECHANICS\nJES´US CLEMENTE-GALLARDO AND GIUSEPPE MARMO\nAbstract. We present a geometrical description of the space of density states\nof a quantum system of finite dimension. After presenting a br ief summary\nof the geometrical formulation of Quantum Mechanics, we pro ceed to describe\nthe space of density states D(H) from a geometrical perspective identifying\nthe stratification associated to the natural GL(H)–action on D(H) and some\nof its properties. We apply this construction to the cases of quantum systems\nof two and three levels.\nKeywords :Density states, projective space, geometric quantum mechanics\nPACS: 03.65.-w, 03.65.Ta\n1.Introduction\nA comparison of the frameworks underlying classical and quantum m echanics\nshows that the two descriptions have several common mathematic al structures.\nHowever, a striking difference emerges: the classical setting is geo metrical and non-\nlinear while the quantum is algebraic and linear. The emphasis on the und erlying\nlinearity in quantum mechanics is usually attributed to the description of the inter-\nferencephenomena[10]. Therefore,thecarrierspaceofquantu msystemsisrequired\nto be a Hilbert space Hfrom the beginning. The Hermitian structure is required\nto describe the probabilistic interpretation of Quantum Mechanics. However, it\nis exactly this probabilistic interpretation which forces on us the iden tification of\nphysical states not with the Hilbert space but rather with the spac e of rays, i.e.\nthe complex projective space of H, sayRH. Of course RHis a genuine nonlinear\nmanifold and on it the Hermitian structure gives rise to a K¨ ahler stru cture.\nThe appearance of this manifold in the quantum setting calls for a geo metrical\nformulation of Quantum Mechanics. It is clear that in the manifold view point we\nhave to give up the usual “superposition of states” and the notion of operators,\neigenvectors and eigenstates as usually presented. Nevertheles s, due to their phys-\nical relevance and interpretation we must be able to recover these “attributes” for\nquantum systems also at the manifold level. The overall formulation m ust allow\nfor nonlinear transformations and therefore only tensorial obje cts should be iden-\ntified with physically relevant quantities. To fully exploit the geometric al picture,\none prefers to work with real differential manifolds, i.e. one replace s the complex\nvector space Hwith its realification HR. The Hermitian structure then splits into a\ncomplex structure, a symplectic structure and a Riemannian struc ture (compatible\namongthem to define a K¨ ahlerstructure) Hermitian operatorsar etransformed into\nfunctions by replacing them by their expectation values. These fun ctions project\nontoRH. With the help of the Poissontensor associated with the symplectic s truc-\nture it is possible to give rise to a flow by integrating the Hamiltonian vec tor field\n12 JES ´US CLEMENTE-GALLARDO AND GIUSEPPE MARMO\nassociated with the expectation value function corresponding to a given Hermitian\noperator [1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 23].\nThe symplectic structure appearing in Quantum Mechanics makes als o possible\nto consider it as a “classical field theory” associated with a Lagrang ian description\nwith relativity group the Galilei group [21], since we deal with non-rela tivistic\nQuantum Mechanics. Thus, classical and quantum descriptions hav e in common a\nsymplectic structure. However, this is the only quantum feature t hat has a direct\nclassical analogue. Some characteristic features like the quantum uncertainties\nand state vector reduction in a measurement process are strictly related to the\nadditional complex structure, available in Quantum Mechanics but no t present in\nClassicalMechanics [20]. This additional structure lies at the heart o f the difference\nbetween the mathematical structures underlying the two theorie s, much more than\nthe linear structure.\nGoing back to the manifold view point introduced in Quantum Mechanics , i.e.\nthe identification of RHas the true manifold of physical states, we have to recover\nthe notion of superposition of physical states [19] . This has been d one and creates\na deep relation with Pancharatnam connection, Bargmann invariant s and geomet-\nric phases [22]. In addition we have to recover the notion of “eigenve ctor” and\n“eigenvalue”. As a matter of fact by considering the expectation v alue function\nassociated with any operator we find that their critical points will co rrespond to\neigenstates and their values at those critical points correspond t o the eigenvalues.\nIf we consider the expectation value-functions of generic operat ors we get com-\nplex valued functions on RHand they provide us with a C∗–algebra, thus paving\nthe way for the geometrization of the C∗–algebraic approach to quantum theories.\nIn this latter approach usually the space of states is identified with t he space of\nnormalized positive linear functionals on the C∗–algebra describing the quantum\nsystem. As a provisional geometrization of this approach we shall c onstruct these\nspaces with the help of the momentum map associated with the symple ctic action\nof the unitary group on the K¨ ahler manifold RH.\n2.A brief exposition of geometrical quantum mechanics\nThe aim ofthis section is to present a brief summary ofthe set of the geometrical\ntools which characterize the description of Quantum Mechanics [13 , 14, 18].\n2.1.The states. The first step consists in replacing the usual complex vector\nspace structure of the Hilbert space Hof a quantum system by the corresponding\nrealification of the vector space. We shall denote as HRthe resulting vector space.\nIn this realification process the complex structure on Hwill be represented by a\ntensorJonHR.\nThe natural identification is then provided by\nψR+iψI=ψ∈ H /ma√sto→(ψR,ψI)∈ HR.\nUnder this transformation, the Hermitian product becomes, for ψ1,ψ2∈ H\n/an}bracketle{t(ψ1\nR,ψ1\nI),(ψ2\nR,ψ2\nI)/an}bracketri}ht= (/an}bracketle{tψ1\nR,ψ2\nR/an}bracketri}ht+/an}bracketle{tψ1\nI,ψ2\nI/an}bracketri}ht)+i(/an}bracketle{tψ1\nR,ψ2\nI/an}bracketri}ht−/an}bracketle{tψ1\nI,ψ2\nR/an}bracketri}ht).\nTo consider HRjust as a real differential manifold, the algebraic structures avail-\nable onHmust be converted into tensor fields on HR. To this end we have to\nintroduce the tangent bundle THRand its dual the cotangent bundle T∗HR. TheTHE SPACE OF DENSITY STATES IN GEOMETRICAL QUANTUM MECHANIC S 3\nlinear structure available in HRis encoded in the vector field ∆\n∆ :HR→THRψ/ma√sto→(ψ,ψ)\nWe can consider the Hermitian structure on HRas an Hermitian tensor on THR.\nWith every vector we can associate a vector field\nXψ:φ→(φ,ψ)\nTherefore, the Hermitian tensor, denoted in the same way as the s calar product\nwill be\n/an}bracketle{tXψ1,Xψ2/an}bracketri}ht=/an}bracketle{tψ1,ψ2/an}bracketri}ht\nThe scalar product above is written as /an}bracketle{tψ1,ψ2/an}bracketri}ht=g(Xψ1,Xψ2) +iω(Xψ1,Xψ2),\nwheregis now a symmetric tensor and ωa skew-symmetric one. It is also possible\nto write them as a pull-back by means of the dilation vector field ∆ as:\n(∆∗(g+iω))(ψ,φ) =/an}bracketle{tψ,φ/an}bracketri}htH\nThe properties of the Hermitian product ensure that:\n•the symmetric tensor is positive definite and non-degenerate, and hence\ndefines a Riemannian structure on the real vector space.\n•the skew-symmetrictensorisalsonondegenerate, andisclosedwit h respect\nto the natural differential structure of the vector space. Henc e, the tensor\nis a symplectic form.\nAs the inner product is sesquilinear, it satisfies\n/an}bracketle{tψ1,iψ2/an}bracketri}ht=i/an}bracketle{tψ1,ψ2/an}bracketri}ht,/an}bracketle{tiψ1,ψ2/an}bracketri}ht=−i/an}bracketle{tψ1,ψ2/an}bracketri}ht.\nThis implies\ng(Xψ1,Xψ2) =ω(JXψ1,Xψ2).\nWe also have that J2=−I, and hence that the triple ( J,g,ω) defines a K¨ ahler\nstructure. This implies, among other things, that the tensor Jgenerates both\nfinite and infinitesimal transformations which are orthogonal and s ymplectic.\nLinear transformations are converted into (1 ,1)–tensor fields by setting A→TA\nwhere\nTA:THR→THR(ψ,φ)/ma√sto→(ψ,Aφ).\nThe association A→TAis an associative algebra isomorphism. It is possible to\nrecover the Lie algebra of vector fields by setting XA=TA(∆). Complex linear\ntransformations will be represented by (1 ,1)–tensor fields commuting with J.\nFor finite dimensional Hilbert spaces it may be convenient to introduc e adapted\ncoordinates on HandHR. Fixing an orthonormal basis {|ek/an}bracketri}ht}of the Hilbert space\nallows us to identify this product with the canonical Hermitian produc t ofCn:\n/an}bracketle{tψ1,ψ2/an}bracketri}ht=/summationdisplay\nk/an}bracketle{tψ1,ek/an}bracketri}ht/an}bracketle{tek,ψ2/an}bracketri}ht\nThe group of unitary transformations on Hbecomes identified with the group\nU(n,C), its Lie algebra u(H) withu(n,C) and so on.\nThe choice of the basis also allows us to introduce coordinates for th e realified\nstructure:\n/an}bracketle{tek,ψ/an}bracketri}ht= (qk+ipk)(ψ),\nand write the geometrical objects introduced above as:\nJ=∂pk⊗dqk−∂qk⊗dpkg=dqk⊗dqk+dpk⊗dpkω=dqk∧dpk4 JES ´US CLEMENTE-GALLARDO AND GIUSEPPE MARMO\nIf we combine them in complex coordinates we can write the Hermitian s tructure\nin a simple way zn=qn+ipn:\nh=/summationdisplay\nkd¯zk⊗dzk\nIn an analogous way we can consider a contravariant version of the se tensors. It\nis also possible to build it by using the isomorphism THR↔T∗HRassociated to\nthe Riemannian tensor g. The result in both cases is a K¨ ahler structure for the\ndual vector space H∗\nRwith the dual complex structure J∗, a Riemannian tensor G\nand a (symplectic) Poisson tensor Ω: The coordinate expressions w ith respect to\nthe natural base are:\n•the Riemannian structure G=/summationtextn\nk=1/parenleftBig\n∂\n∂qk⊗∂\n∂qk+∂\n∂pk⊗∂\n∂pk/parenrightBig\n,\n•the Poisson tensor Ω =/summationtextn\nk=1/parenleftBig\n∂\n∂qk∧∂\n∂pk/parenrightBig\n•while the complex structure has the form\nJ=n/summationdisplay\nk=1/parenleftbigg∂\n∂pk⊗dqk−∂\n∂qk⊗dpk/parenrightbigg\n2.2.The observables. The space of observables (i.e. of self-adjoint operators\nacting on H) may be identified with the dual u∗(H) of the real Lie algebra u(H),\naccording to the pairing between the unitary Lie algebra and its dual given by\nA(T) =i\n2TrAT\nUnder the previous isomorphism, u∗(H) becomes a Lie algebra with product\ndefined by\ni[A,B] = [A,B]−= (AB−BA)\nWe can also transfer the Jordan product:\n[A,B]+= 2A◦B=AB+BA\nBoth structures are compatible. As a result, u∗(H) becomes a Jordan-Lie algebra\n(see [11, 16]).\nWe can also define a suitable scalar product, given by:\n/an}bracketle{tA,B/an}bracketri}ht=1\n2TrAB\nwhichturnsthespaceintoarealHilbertspace. Thisscalarproduct istherestriction\nof the one on gl(H) defined as /an}bracketle{tM,N/an}bracketri}ht=1\n2TrM†N.\nBesides this scalar product is compatible with the Lie-Jordan struct ure in the\nfollowing sense:\n/an}bracketle{t[A,ξ],B/an}bracketri}htu∗(H)=/an}bracketle{tA,[ξ,B]/an}bracketri}htu∗(H)/an}bracketle{t[A,ξ]+,B/an}bracketri}htu∗(H)=/an}bracketle{tA,[ξ,B]+/an}bracketri}htu∗(H)\nThese algebraic structures may be given a tensorial translation in t erms of the\nassociation A/ma√sto→TA. However we can also associate complex valued functions with\nlinear operators A∈gl(H) by means of the scalar product\ngl(H)∋A/ma√sto→fA=1\n2/an}bracketle{tψ,Aψ/an}bracketri}htH.\nIn more intrinsic terms we may write\nfA=1\n2(g(∆,XA)+iω(∆,XA)).THE SPACE OF DENSITY STATES IN GEOMETRICAL QUANTUM MECHANIC S 5\nHermitian operators give rise thus to quadratic real valued functio ns.\nThe association of operators with quadratic functions allows also to recover the\nalgebraic structures on u(H) andu∗(H) by means of approapriate (0 ,2)–tensors on\nHR. By using the contravariant form of the Hermitian tensor G+iΩ given by:\nG+iΩ = 4∂\n∂zk⊗∂\n∂¯zk=∂\n∂qk⊗∂\n∂qk+∂\n∂pk⊗∂\n∂pk+i∂\n∂qk∧∂\n∂pk,\nit is possible to define a bracket\n{f,h}H={f,h}g+i{f,h}ω\nIn particular, for quadratic real valued functions we have\n{fA,fB}g=fAB+BA= 2fA◦B{fA,fB}ω=−ifAB−BA\nThe imaginary part, i.e. {·,·}ω, defines a Poisson bracket on the space of func-\ntions. Both brackets allow us to define a tensorial version of the Lie -Jordan algebra\nof the set of operators.\nFor Hermitian operators we recover previously defined vector field s:\ngradfA=/tildewideA; HamfA=/tildewideriA\nwhere the vector fields associated with operators, we recall, are d efined by:\n/tildewideA:HR→THRψ/ma√sto→(ψ,Aψ)\n/tildewideriA:HR→THRψ/ma√sto→(ψ,JAψ)\nWe can alsoconsider the algebraicstructure associatedto the full bracket{·,·}H,\nas we associated above the Jordan product and the commutator o f operators to the\nbrackets {·,·}gand{·,·}ωrespectively. It is simple to see that it corresponds to the\nassociative product of the set of operators, i.e.\n{fA,fB}H={fA,fB}g+i{fA,fB}ω=fAB+BA+ifAB−BA= 2fAB\nThis particular bilinear product on quadratic functions may be writte n also as\na star product\n{fA,fB}H= 2fAB=/an}bracketle{tdfA,dfB/an}bracketri}htH∗=fA⋆fB\nThe set of quadratic functions endowed with such a structure tur ns out to be a\nC∗–algebra.\nWe see then that we can reconstruct all the information of the alge bra of oper-\nators starting only with real-valued functions defined on HR. We have thus\nProposition 1. The Hamiltonian vector field Xf(defined as Xf=ˆΩ(df)) is a\nKilling vector field for the Riemannian tensor Gif and only if fis a quadratic\nfunction associated with an Hermitian operator A, i.e. there exists A=A†such\nthatf=fA.\nFinally, we can consider the problem of how to recover the eigenvalue s and\neigenvectors of the operators at the level of the functions of HR. It is simple to see\nthat\n•eigenvectors correspond to the critical points of functions fA, i.e.\ndfA(ψ∗) = 0 iffψ∗is an eigenvector of A6 JES ´US CLEMENTE-GALLARDO AND GIUSEPPE MARMO\n•the corresponding eigenvalue is recovered by the value\nfA(ψ∗)\n/an}bracketle{tψ∗,ψ∗/an}bracketri}ht\nThus we can conclude that the K¨ ahler manifold ( HR,J,ω,g) contains all the\ninformation of the usual formulation of Quantum Mechanics on a com plex Hilbert\nspace.\nUp to now we have concentrated our attention on states and obse rvables. If we\nconsider observables as generators of transformations, i.e. we c onsider the Hamil-\ntonian flow associated to the corresponding functions, the invaria nce of the tensor\nGimplies that the evolution is actually unitary. It is, therefore, natur al, to consider\nthe action of the unitary group on the realification of the complex ve ctor space.\n3.The momentum map: geometrical structures on g∗\nThe unitary action of U(H) onHinduces a symplectic action on the symplectic\nmanifold ( HR,ω). By using the association\nF:HR×u(H)→R(ψ,A)/ma√sto→1\n2/an}bracketle{tψ,Aψ/an}bracketri}ht=fiA(ψ),\nwe find, with FA=fiA:HR→R, that\n{F(A),F(B)}ω=iF([A,B]).\nThus if we fix ψ, we have a mapping F(ψ) :u(H)→R. Thus with any element\nψ∈ Hwe have an element in u∗(H). Hence it defines a momentum map\nµ:H →u∗(H),\nwhich provides us with a symplectic realization of the natural Poisson manifold\nstructure available in u∗(H). We can write the momentum map from HRtou∗(H)\nas\nµ(ψ) =|ψ/an}bracketri}ht/an}bracketle{tψ|\nIf we make the convention that the dual u∗(H) of the (real) Lie algebra u(H)\nis identified with Hermitian operators by means of a scalar product, t he product\npairing between Hermitian operators A∈u∗(H) and the anti-Hermitian element\nT∈u(H) will be given by\nA(T) =i\n2Tr(AT)\nIf we denote the linear function on u∗(H) associated with the element iA∈u(H)\nbyˆA, we have\nµ∗(ˆA) =fA\nThe pullback of linear functions on u∗(H) is given by the quadratic functions on\nHRassociated with the corresponding Hermitian operators.\nIt is possible to show that the contravariant tensor fields on HRassociated with\nthe Hermitian structure are µ–related with a complex tensor on u∗(H):\nµ∗(G+iΩ) =R+iΛ;\nwhere the two new tensors Rand Λ are defined by\nR(ξ)(ˆA,ˆB) =/an}bracketle{tξ,[A,B]+/an}bracketri}htu∗=1\n2Tr(ξ(AB+BA))THE SPACE OF DENSITY STATES IN GEOMETRICAL QUANTUM MECHANIC S 7\nand\nΛ(ξ)(ˆA,ˆB) =/an}bracketle{tξ,[A,B]−/an}bracketri}htu∗=1\n2iTr(ξ(AB−BA))\nClearly,\nG(µ∗ˆA,µ∗ˆB)+iΩ(µ∗ˆA,µ∗ˆB) =µ∗(R(ˆA,ˆB)+iΛ(ˆA,ˆB)).\nAs we know that u∗(H) is foliated by symplectic manifolds, we wish to consider\nmore closely the map from HRto the minimal symplectic orbit on u∗(H).\n4.The complex projective space\nAs we have already remarked, the association of states of the qua ntum system\nwith vectors in the Hilbert space needs further qualifications becau se of the prob-\nabilistic interpretation required in Quantum Mechanics. More specific ally, states\nshould be identified with rays in the Hilbert space, i.e. equivalence class es of vec-\ntors, orbits of non-null vectors under the action of C0=C−{0}. The equivalence\nclass of the vector ψ∈ Hwill be denoted then as\n[ψ] ={λψ,λ∈C0}\nAs the infinitesimal generatorsofthe realand imaginarypartsoft he actionof C0\nonHRaregivenby the dilationvectorfield ∆ and the vectorfield J(∆) respectively,\nit is clear that we have to undertake the projection of the relevant tensors on HR\nto the complex projective space or ray space RH.\nWithout entering in too many details, we find that we have to modify Gand Ω\nby a conformal factor to turn them into projectable tensors. Sp ecifically we have\n˜G=g(∆,∆)G,˜Λ =g(∆,∆)Λ.\nThese tensors are projectable onto non-degenerate contrava riant tensors on RH\nand givesrise to aLie-Jordanalgebrastructureon the spaceofre alvalued functions\nwhose Hamiltonian vector fields are also Killing vector fields for the pro jection˜G.\nAs a matter of fact a theorem by Wigner allows us to state that thes e functions\nare necessarily projections of expectation values of Hermitian ope rators\neA(ψ) =/an}bracketle{tψ,Aψ/an}bracketri}ht\n/an}bracketle{tψ,ψ/an}bracketri}ht.\nThe action of the unitary group may also be projected and gives rise to a sym-\nplectic action on RH. The momentum map from HRprojects onto the momentum\nmap from RHbecause it is equivariant with respect to the action of ∆ HonHR\nand the action of ∆ u∗onu∗(H).\nFromµ∗(ˆA) =fAwe find\n∆Hµ∗(ˆA) = 2µ∗(∆u∗ˆA) = 2µ∗(ˆA)\nThe momentum map for the projected action may be written in the fo rm\nµ([ψ]) =|ψ/an}bracketri}ht/an}bracketle{tψ|\n/an}bracketle{tψ,ψ/an}bracketri}ht=ρψ.\nThis map identifies RHwith the Hermitian operatorsin u∗(H) which are of rank\none and are projectors, i.e.\nρψρψ=ρψ,Trρψ= 1.\nAs the ray space is a principal bundle with base manifold RHand structure\ngroupC0, we may look for a connection one form.8 JES ´US CLEMENTE-GALLARDO AND GIUSEPPE MARMO\nThe connection one-form θis given by\nθ(ψ) =/an}bracketle{tψ,dψ/an}bracketri}ht\n/an}bracketle{tψ,ψ/an}bracketri}ht,\nwith associated curvature form ω=dθ, because the structure group is Abelian.\nThis curvature form coincides with the symplectic structure on RHarising from\nthe projection of ˜Λ (conformally related to Λ).\nIt is also possible to write the Hermitian tensor which coincides with the Her-\nmitian tensor on RHwhen evaluated on horizontal vector fields. We thus have\n/an}bracketle{tdψ,dψ/an}bracketri}ht\n/an}bracketle{tψ,ψ/an}bracketri}ht−/an}bracketle{tψ,dψ/an}bracketri}ht/an}bracketle{tdψ,ψ/an}bracketri}ht\n/an}bracketle{tψ,ψ/an}bracketri}ht2.\nIt is not difficult to see that both ∆ and J(∆) are annihilated by this tensor.\nThe embedding of RHintou∗(H) by means of the momentum map allows us\nto consider convex combinations of the image µ(RH)⊂u∗(H). The convex com-\nbinations will generate the space of density states, i.e. normalized p ositive linear\nfunctionals on the Lie-Jordan algebra of observables. This convex body inherits\nsome structures from those existing on u∗(H), which are particularly important\nand useful when we are interested in describing evolutions of state s which are not\nunitary. In particular they inherits a Poisson structure and a Jord an structure. In\nthe next section we shall study more closely the space of density st ates.\n5.The space of density states\nAs we have already remarked the space of density states is the spa ce of positive\nnormalized linear functionals on the real linear space of observables . A theorem by\nGleason[12] assertsthat these functionals may be represented b y suitable operators\nwhen the trace is used as a bilinear pairing. By using this theorem we ca n start by\nconsidering states directly as appropriate operators.\nLet us introduce first the space of all non-negatively defined oper ators, i.e. the\nspace of all those ρ∈gl(H) which can be written in the form\nρ=T†T T∈gl(H).\nWe shall denote by PHthis space of operators, which is a convex cone in u∗(H).\nBy imposing the condition Tr ρ= 1 we select in PHthe convex body of density\nstates which we denote by D(H). We shall also consider non-negative Hermitian\noperatorsanddensitystatesofrank k, anddenoteas Pk(H)andDk(H)respectively\nthe corresponding spaces.\nThe complex projective space is in one-to-one correspondence wit hD1(H). In-\ndeed, any state in D(H) can be written as a convex combination of distinct states\nρ=p1ρ1+(1−p1)ρ2, with 0≤p1≤1. We shall call extremal states those which\ncan not be written in this form (i.e. as convex combination of two ρ1andρ2). The\nextremal states are thus given by D1(H).\nAsΛandRarenotinvertiblein u∗(H), itisconvenienttousethepairingbetween\nu(H) andu∗(H) defined by the trace, to introduce two tensor fields on u∗(H). We\nset then\n˜J,R:Tu∗(H)→Tu∗(H)THE SPACE OF DENSITY STATES IN GEOMETRICAL QUANTUM MECHANIC S 9\ndefined as\n˜Jξ(XA) = (ξ,[A,ξ]−) = Λ(ξ)(dˆA)\nRξ(XA) = (ξ,[A,ξ]+) =R(ξ)(dˆA)\nThe image of ˜Jis the Hamiltonian involutive distribution associated with linear\nHamiltonian functions, and we shall denote it as DΛ. The image of Ris also a\ndistribution, which we shall denote as DR, but in this case it is not involutive.\nIt is possible to see that combining DRandDΛwe can define two distributions\nD0=DR∩DAandD1=DR+DΛwhich are indeed involutive.\nWe noticethat the tensors ˜JandRcommute, i.e. ˜J◦R=R◦˜J. Morespecifically\nwe have\n˜J(ξ)◦R(ξ)(XA) =R(ξ)◦˜J(ξ)(XA) = [A,ξ2]−.\nAs a result, we find that the distribution D0becomes:\nD0(ξ) ={[A,ξ2];A∈u∗(H)}.\nOnRH,D0coincides with DΛ.\nThe distribution D1is involutive and the leaves are related to orbits of the\nfollowingGL(H)–action:\nGL(H)×u∗(H)→u∗(H) (T,ξ)/ma√sto→TξT†.\nWe obtain some interesting results [13, 14]:\n(1) The Hermitian operators ρandρ′belong to the same GL–orbit if and only\nif they have the same number K+of positive eigenvalues and the same\nnumberK−of negative eigenvalues (counted with multiplicities).\n(2) AnyGL–orbitintersectingthe positive cone PHlies entirely in PH; sothat\nPHis stratified by the GL–orbits. These GL–orbits in PHare determined\nby the rank of the operator, i.e. they are exactly Pk(H).\nWhen we restrict to the space of density states by imposing the con dition Trρ=\n1, thisGL–action will not preserve the states. It is however possible to defin e a\nnew action that maps D(H) into itself by setting\nGL(H)×D(H)→ D(H) (T,ρ)/ma√sto→TρT†\nTr(TρT†).\nThis action does preserve the rank of ρand then the following proposition holds\ntrue:\nProposition 2. The decomposition of the convex body of density states D(H)into\norbits of the GL(H)–actionρ/ma√sto→TρT†\nTr(TρT†)is exactly the stratification\nD(H) =n/uniondisplay\nk=1Dk(H),\ninto states of a given rank.\nTheboundaryofthe convexbody ofdensitystatesconsistsofst atesofranklower\nthann, i.e.∂D(H) =/uniontextn−1\nk=1Dk(H), and each stratum is a smooth submanifold in\nu∗(H). However, the boundary ∂D(H) is not smooth (for n >2). We have the\nfollowing theorem:10 JES ´US CLEMENTE-GALLARDO AND GIUSEPPE MARMO\nTheorem 1. Every smooth curve γ:R→u∗(H)through the convex body of density\nstates is tangent, at every point, to the stratum to which it b elongs, i.e.\nγ(t)∈ Dk(H)⇒Tγ(t)∈Tγ(t)Dk(H).\nOne may gain further insight on the “location” of the boundary by us ing the\nnotion of “face”\nDefinition 1. A non-empty closed convex subset K0of a closed convex set Kis\ncalled a faceofKif any closed segment in Kwith an interior point in K0lies\nentirely inK0.\nThus, for any ρ∈ D(H) we may consider the decomposition H= Imρ+ Kerρ\ninto the kernel and the image of ρ. We have:\nProposition 3. The face of D(H)throughρ∈ Dk(H)consists of states Awhich\nare “projectable” with respect to the projection defined by Kerρ, i.e.KerA⊂Kerρ.\nThe face through ρis then equivalent to D(Imρ).\nThe inner product defined by the trace allows to define a probability t ransition\nfunction\np(ρ1,ρ2) = Trρ1ρ2,\nwhenρ1andρ2belong to the boundary ∂eD(H), the space of extremal states.\nThis function satisfies\n0≤p(ρ1,ρ2)≤1p(ρ1,ρ2) =p(ρ2,ρ1).\nMoreover,p(ρ1,ρ2) = 1 if and only if ρ1=ρ2.\nIt is not difficult to show that the Hamiltonian vector fields which leave Rin-\nvariant will preserve also the probability transition functions. This r esult is related\nto a theorem by Wigner and may be used to recover the space of den sity states\nstarting with a Poisson space carrying a compatible probability trans ition function.\nFurther details and a full treatement of Poisson spaces with a tran sition probability\nfunction have been considered by Landsman (see [17]).\n6.Two examples: gu(2)andgu(3)\n6.1.States of a two level system. Weshallconsiderinsomedetailtwoexamples.\nThe first one is the two level system with carrier space H=C2. We consider u(2)\nandu∗(2) and make a specific choice of basis\nσ0=/parenleftbigg1 0\n0 1/parenrightbigg\nσ1=/parenleftbigg0i\n−i0/parenrightbigg\nσ2=/parenleftbigg0 1\n1 0/parenrightbigg\nσ3=/parenleftbigg1 0\n0−1/parenrightbigg\nWe recall that\nσ1σ2=iσ3σ2σ3=iσ1σ3σ1=iσ2,\nalong with\nσ2σ1=−iσ3σ3σ2=−iσ1σ1σ3=−iσ2;\nwhich may be obtained considering the conjugate-transpose of an y product.\nWe define coordinate functions by writing\ny0(A) =1\n2Tr(σ0A), ya(A) =1\n2Tr(σaA).THE SPACE OF DENSITY STATES IN GEOMETRICAL QUANTUM MECHANIC S 11\nIn these coordinates the corresponding Poisson brackets for th e canonical Lie-\nPoisson structure on the dual of the Lie algebra read:\n{y0,ya}= 0{ya,yb}= 2ǫabcyc.\nThe expression of the Poisson tensor thus becomes:\nΛ = 2/parenleftbigg\ny1∂\n∂y2∧∂\n∂y3+y2∂\n∂y3∧∂\n∂y1+y3∂\n∂y1∧∂\n∂y2/parenrightbigg\nIt is also possible to construct the Riemann-Jordan tensor in the fo rm:\nR=∂\n∂y0⊗s/parenleftbigg\ny1∂\n∂y1+y2∂\n∂y2+y3∂\n∂y3/parenrightbigg\n+\ny0/parenleftbigg∂\n∂y0⊗∂\n∂y0+∂\n∂y1⊗∂\n∂y1+∂\n∂y2⊗∂\n∂y2+∂\n∂y3⊗∂\n∂y3/parenrightbigg\nwhere⊗smeans the symmetrized tensor product.\n6.2.Distributions associated with ΛandR.It is easy to see that the Hamil-\ntonian distribution is generated by\nH1=y3∂\n∂y2−y2∂\n∂y3, H2=y1∂\n∂y3−y3∂\n∂y1, H3=y2∂\n∂y1−y1∂\n∂y2,\nwhile the distribution associated with the Riemann-Jordan tensor is\nX0=ya∂\n∂ya+y0∂\n∂y0Xa=ya∂\n∂y0+y0∂\n∂ya\nIt is clear that X0is central and {Xa}are boosts of a four dimensional Lorentz\ngroup, therefore their commutator will provide us with the Lie algeb ra of the rota-\ntion group:\n[Xa,Xb] =ya∂\n∂yb−yb∂\n∂ya.\nThe intersection of the distribution associated with Rand the Hamiltonian dis-\ntribution associated will indeed be generated by the Hamiltonian vect or fields and\nis involutive with leaves which are symplectic two dimensional spheres. The dis-\ntribution generated by the union of the two distributions is the full L orentz group\ncentrally extended with the dilations. As the Lorentz group admits a s a covering\nSL(2,C)the central extension is isomorphic to GL(2,C). This is a generalproperty\nholding true in any dimension (see [14]).\nWe find that\nLemma 1. The rank of Λis zero ify2\n1+y2\n2+y2\n3= 0and the rank is equal to 2 if\ny2\n1+y2\n2+y2\n3>0.\nThe situation is richer with R:\nLemma 2. The rank of Ris\n•zero ify2\n0+y2\n1+y2\n2+y2\n3= 0\n•two ify0= 0andy2\n1+y2\n2+y2\n3>0.\n•three fory2\n0=y2\n1+y2\n2+y2\n3\n•four ify2\n0/ne}ationslash=y2\n1+y2\n2+y2\n312 JES ´US CLEMENTE-GALLARDO AND GIUSEPPE MARMO\n6.3.Density states. As we have already seen in the previous sections the set\nof states is identified with a subset of u∗(H) satisfying a positivity condition and\na normalization condition. In the specific situation we are considering , a generic\nHermitian matrix A=y0σ0+yaσawill define a state if\nTrA= 1 0≤y0+y3≤1,0≤y0−y3≤1,detA≥0.\nExplicitly we have\ny0=1\n2,/parenleftbigg1\n2+y3/parenrightbigg/parenleftbigg1\n2−y3/parenrightbigg\n−((y1)2+(y2)2)≥0,\nor\n(y3)2+(y2)2+(y1)2≤1\n4.\nThus in our parametrization states are determined by points in R4on the hy-\nperplaney0=1\n2, and on this three dimensional space are identified by the points\nin the ball of radius1\n2. When referring to states we replace Awithρand write:\nρ=/parenleftbigg1\n2+y3y2+iy1\ny2−iy11\n2−y3/parenrightbigg\n. (1)\nThe pure states corresponding to the vector ( z1,z2)∈C2with unit norm z1¯z1+\nz2¯z2= 1 has a density state\nρ=/parenleftbigg¯z1\n¯z2/parenrightbigg\n⊗(z1,z2) =/parenleftbiggz1¯z1¯z1z2\n¯z2z1z2¯z2/parenrightbigg\n.\nWithin the previous parametrization we find\ny3=1\n2(z1¯z1−z2¯z2), y1= Im(¯z1z2), y2= Re(¯z1z2),\nand for these points the inequality is saturated thus implying that th ey lie on the\nsurface of the ball of radius1\n2. We shall denote the set of density states by D. This\nset is the convex hull of the sphere of pure states. For any ρ∈ Dthere exist pure\nstatesρ1andρ2and a positive number psuch thatρ=pρ1+(1−p)ρ2.\nThese states, points on the surface, are in one-to-one corresp ondence with the\nunit rays in C2and the map is given by the momentum map associated with the\nsymplectic action of U(2) onRH ∼CP1. The ball of the density states is foliated\nby symplectic leaves associated with the coadjoint action of U(2), which coincide\nalso with the orbits of the SU(2) group.\nThe analysis of these orbits may also be done by considering the orbit s passing\nthrough diagonal matrices, in other terms\nρ=S/parenleftbigga0\n0b/parenrightbigg\nS†a+b= 1a≥0, b≥0.\nWe visualize the situation with the help of the following diagram: the seg ment\nconnecting (1\n2,1\n2) with (1,0) parametrizes the family of two dimensional spheres.\nThe point (1\n2,1\n2) coincides with the center of the ball and (1 ,0) belongs to the\noutmost sphere of pure states.THE SPACE OF DENSITY STATES IN GEOMETRICAL QUANTUM MECHANIC S 13\n(1,0)(0,1)\n(0.5,0.5)\nPSfrag replacementsab\nWhat we have described is usually known as the Bloch sphere represe ntation of\none qubit. The decomposition of a density states ρ, a point in the ball, as a convex\nsum of two pure states ρ1=|ψ1/angbracketright/angbracketleftψ1|\n/angbracketleftψ1,ψ1/angbracketrightandρ2=|ψ2/angbracketright/angbracketleftψ2|\n/angbracketleftψ2,ψ2/angbracketright, is given geometrically by\ndrawing a straight line through ρ: the states ρ1andρ2are the intersections of the\nline with the sphere. Evidently this decomposition may be done in a two p arameter\nfamily of ways.\nPSfrag replacements\ny1y2y3\nρρ1\nρ2ρ′\n1\nρ′\n2\n6.4.States of a three level system. NowH=C3. The states are normalized\npositive3 ×3matricesinside u∗(3). Wefirstconsiderthegeometricaltensorsdefined\nby means of the momentum map construction. We choose a basis for u(3) given by\nthe Gell-Mann matrices\nλ1=\n0 1 0\n1 0 0\n0 0 0\nλ2=\n0−i0\ni0 0\n0 0 0\nλ3=\n1 0 0\n0−1 0\n0 0 0\n\nλ4=\n0 0 1\n0 0 0\n1 0 0\nλ5=\n0 0−i\n0 0 0\ni0 0\nλ6=\n0 0 0\n0 0 1\n0 1 0\n14 JES ´US CLEMENTE-GALLARDO AND GIUSEPPE MARMO\nλ7=\n0 0 0\n0 0−i\n0i0\nλ8=1√\n3\n1 0 0\n0 1 0\n0 0−2\nλ0=/radicalbigg\n2\n3\n1 0 0\n0 1 0\n0 0 1\n\nThey satisfy the scalar product relation\nTrλµλν= 2δµν\nTheir commutation and anti-commutation relations are written in ter ms of the\nantisymmetric structure constants and symmetric d–symbols dµνρ. We find\n[λµ,λν] = 2iCµνρλρ[λµ,λρ]+= 2/radicalbigg\n2\n3λ0δµν+2dµνρλρ.\nThe numerical values turn out to be\nC123= 1, C 458=C678=√\n3\n2\nC147=−C156=C246=C257=C345=−C367=1\n2\nThe values of these symbols show the different embeddings of SU(2) intoSU(3)⊂\nU(3). For the other coefficients we have\ndjj0=−d0jj=−dj0j=/radicalbigg\n2\n3j= 1,···,8\n−d888=d8jj=djj8=dj8j=1√\n3j= 1,2,3\nd8jj=djj8=dj8j=−1\n2√\n3j= 4,5,6,7\nd3jj=djj3=dj3j=1\n2j= 4,5d3jj=djj3=dj3j=−1\n2j= 6,7\nd146=d157=d164=d175=−d247=d256=d265=−d274=1\n2\nd416=−d427=d461=−d472=d517=d526=d562=d571=1\n2\nd614=d625=d641=d652=d715=−d724=d751=−d742=1\n2\nThe indices appearing in the non-null structure constants are iden tifying the\ncorresponding λ–matrices whose pairwise commutators define SU(2)–subgroups.\nIt is now possible to introduce coordinate functions\nyµ(A) =1\n2TrλµA.\nIn these coordinates, a generic Hermitian matrix Acan be written as\nA=y0λ0+yrλr\nThe vector ( y0,/vector y)∈R9plays a similar role to the one we saw on U(2). Under\nconjugation with S∈SU(3), any matrix Acan be written as\nA=S\na0 0\n0b0\n0 0c\nS†.THE SPACE OF DENSITY STATES IN GEOMETRICAL QUANTUM MECHANIC S 15\nThe scalar product induced on vectors on R8will be invariant under the action\nofSO(8). It is now possible to write the Poisson tensor\nΛ = 2Cµνρyρ∂\n∂yµ∧∂\n∂yν\nand the Riemann-Jordan tensor\nR=∂\n∂y0⊗syµ∂\n∂yµ+y0∂\n∂yr⊗∂\n∂yr+dµνρyµ∂\n∂yν⊗s∂\n∂yρ.\nNow the analysis of the various distributions is more cumbersome, ho wever it is\neasy to identify a few elements:\nR(dy0) =yµ∂\n∂yµ,\nwhich is the dilation vector field on R9; whileR(dyr) =yr∂\n∂y0+y0∂\n∂yr+dµνryµ∂\n∂yν,\nwhere it is possible to identify a boost structure plus a correction du e to the d–\nsymbols. InanycasetheunionoftheHamiltoniandistributionandthe Riemannian-\nJordan distribution generates GL(3,C).\nThe set of states will again be identified as the subset of the Hermitia n matrices\nwhich are normalized and satisfy the positivity condition. If we set\nρ=\na¯h g\nh b¯f\n¯g f c\na,b,c∈Rf,g,h∈C.\nThe conditions for ρto be a state are:\n•a+b+c= 1\n•a≥0,b≥0,c≥0.\n• |f|2≤bc,|g|2≤ca,|h|2≤ab.\n•detρ=abc+2Re(fgh)−(a|f|2+b|g|2+c|h|2)≥0\nThese matrices form a convex set of R8. The trace condition allows to identify\nthis subset as a subset of the vector space of R8corresponding to the dual space of\nthe Lie algebra of SU(3).\nExtremal states are in one-to-one correspondence with the minim al symplectic\norbit of the unitary group according to the coadjoint action and co rresponds to\nCP2, the complex projective space of C3.\nPure states, rank one projectors, are given by vectors ( z1,z2,z3)∈C3with the\nnormalization condition z1¯z1+z2¯z2+z3¯z3= 1 as\n\nz1¯z1¯z1z2¯z1z3\n¯z2z1z2¯z2¯z2z3\n¯z3z1z3¯z2¯z3z3\n\nPrevious inequalities are saturated by these states.\nThese extremal states may be written in terms of the λ–matrices\nρψ=|ψ/an}bracketri}ht/an}bracketle{tψ|\n/an}bracketle{tψ,ψ/an}bracketri}ht=1\n3(I+√\n3naλa),\nwithnana= 1 andn⋆n=n, the star product being\n(a⋆b)l=√\n3dljkajbk16 JES ´US CLEMENTE-GALLARDO AND GIUSEPPE MARMO\nBy using the “radial-angular” parametrization of states, say\nρ=s\na0 0\n0b0\n0 0c\ns†a≥0,b≥0,c≥0, s∈SU(3),\nwe may study the structure of this union of symplectic orbits by con sidering the\nfamily of diagonal matrices with the positivity condition (elements of a positive\nWeyl chamber in the Abelian Cartan subalgebra). The hyperplane Tr ρ= 1 iden-\ntifies a triangle with the intersection with positive axes ( Oa,Ob,Oc ); i.e. in the\npositive octant.\nPSfrag replacements\nabc\n0\nEach internal point of the triangle correspondsto a 6–dimensional symplectic orbit,\nout of which we may consider convex combinations. Due to the action ofSU(3)\ncontaining the action of the discrete Weyl group, the symplectic or bits are actually\nparametrized by the following smaller triangle.THE SPACE OF DENSITY STATES IN GEOMETRICAL QUANTUM MECHANIC S 17\na=1c=0b=1a=0b=0c=1\nPSfrag replacements\na\nb\nc\n0\nWhena=b=c=1\n3we have the “maximally mixed state” which play a crucial\nrole when we consider composite systems and entangled states (th e orbit passing\nthrough this point degenerates to a zero dimensional orbit). On th e boundary\nof the bigger triangle the rank of ρis either 1 or 2. However the orbits passing\nthrough these points are diffeomorphic to CP2. For a generic point, the orbits are\ndiffeomorphic to SU(3)/U(1)×U(1). It appears quite clearly that the set of states\nis a stratified manifold characterized by the rank of the state. We s hall not indulge\nfurther on the geometrical analysis and refer to the literature fo r further details\nand applications.\nReferences\n[1] M.C. Abbati, R. Cirelli, P. Lanzavecchia and A. Mani´ a, Pure states of general quantum\nmechanical systems as K¨ ahler bundle , Nuovo Cimento B 83, pp 43–60, 1984\n[2] J.S. Anandan A Geometric approach to Quantum Mechanics Found.Phys. 21, pp 1265-1284,\n1991\n[3] A. Ashtekar and T.A. Schilling, Geometrical formulation of Quantum Mechanics , onEin-\nstein’s path , pp 23-65, New York: Springer, 1999\n[4] A.Benvegn´ u, N. Sansonetto and M. Spera Remarks on geometric quantum mechanics\nJ.Geom.Phys. 51, pp 229-243, 2004\n[5] A. Bloch, An infinite-dimensional Hamiltonian system on a projective Hilbert space , Trans.\nAMS302, pp 787–796, 1987\n[6] D. Brody and L.P. Hughston, Geometric quantum mechanics , J. Geom. Phys. 38, pp 19–53,\n2001\n[7] R. Cirelli and P. Lanzavecchia, Hamiltonian vector fields in Quantum Mechanics , Nuovo\nCimento B, 79, pp 271–283, 1984\n[8] R. Cirelli, A. Mani´ a and L. Pizzocchero, J.Math.Phys. 3 1, pp 2891-2903, 1990 (part I and II)\n[9] D. Cruscinski and A. Jamiolkowski Geometric phases in classical and Quantum Me-\nchanics Birkhauser,Boston,2004\n[10] P.A.M. Dirac, The Principles of Quantum Mechanics , Clarendon Press, Oxford, 2nd edition,\n1936.\n[11] G.G. Emch, Foundations of 20th century Physics , North Holland, Amsterdam, 198418 JES ´US CLEMENTE-GALLARDO AND GIUSEPPE MARMO\n[12] A.M.Gleason Measures on the closed subspaces of a Hilbert space J.Math.Mech 6, pp 885-893,\n1957\n[13] J. Grabowski, M. Ku´ s and G. Marmo, Geometry of quantum systems: density states and\nentanglement , J. Phys. A:Math. Gen 38, pp 10217-10244, 2005\n[14] J. Grabowski, M. Ku´ s and G. Marmo, Symmetry, group actions and entanglement , Open\nsys. & Information dyn.13, pp 343-362, 2006\n[15] A. Heslot, Quantum mechanics as a classical theory , Phys Rev D 31, pp 1341–1348, 1985\n[16] N.P. Landsman, Mathematical topic between Classical and Quantum Mechanic s,\nSpringer-Verlag, 1998\n[17] N.P. Landsman, Poisson spaces with a transition probability , Rev Math Phys 9, pp 29-57,\n1997\n[18] V. I. Manko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria, The geometry of density states ,\nRep. Math. Phys 55, pp 405-422, 2005\n[19] V. I. Manko, G. Marmo, E.C.G. Sudarshan and F. Zaccaria, Interference and entanglement:\nan intrinsic approach , J. Phys A: Math Gen 35, pp 7137-7157, 2002\n[20] G.Marmo, G. Scolarici, A.Simoni, F. Ventriglia, The Quantum-Classical Transition:The Fate\nof the Complex Structure Int. J.Mod.Geom.Meth.Phys. 2, pp 127-145, 2005\n[21] G.Marmo and G.Vilasi, Symplectic Structures and Quantum Mechanics Mod.Phys.Letters\nB10,pp 545, 1996\n[22] N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, R. Simon, Bargmann invariants,\nnull phase curves and a theory of geometric phase , Phys Rev A 67, 042114, 2003\n[23] D.J. Rowe, A. Ryman and G. Rosensteel, Many body quantum mechanics as a symplectic\ndynamical system , Phys Rev A 22, pp 2362-2372, 1980\nBIFI-Universidad de Zaragoza, Corona de Arag ´on 42, 50009 Zaragoza-SPAIN\nDipartamento di Scienze Fisiche, Universit ´a Federico II and INFN- Sezione Napoli,\nVia Cintia I-80126 Napoli-ITALY"    },    {        "title": "0707.4428v2.Only_n_Qubit_Greenberger_Horne_Zeilinger_States_are_Undetermined_by_their_Reduced_Density_Matrices.pdf",        "content": "arXiv:0707.4428v2  [quant-ph]  12 Feb 2008Onlyn-Qubit Greenberger-Horne-Zeilinger States are Undetermi ned by their\nReduced Density Matrices\nScott N. Walck∗and David W. Lyons†\nLebanon Valley College, Annville, PA 17003\n(Dated: December 4, 2007)\nThe generalized n-qubit Greenberger-Horne-Zeilinger (GHZ) states and thei r local unitary equiv-\nalents are the only states of nqubits that are not uniquely determined among pure states by their\nreduced density matrices of n−1 qubits. Thus, among pure states, the generalized GHZ state s are\nthe only ones containing information at the n-party level. We point out a connection between local\nunitary stabilizer subgroups and the property of being dete rmined by reduced density matrices.\nPACS numbers: 03.67.Mn,03.65.Ta,03.65.Ud\nQuantifying and characterizing multi-party quantum\nentanglement is a fundamental problem in the field of\nquantum information. Roughly speaking, one expects\nthe states that are “most entangled”to be the most valu-\nable resourcesfor carryingout quantum information pro-\ncessing tasks such as quantum communication and quan-\ntum teleportation, and to give the most striking philo-\nsophical implications in terms of the rejection of local\nhidden variable theories [1].\nAlthough no single definition of “most entangled”\nseems possible, since we know that multi-party entangle-\nment occurs in many types that admit at best a partial\norder [2], it is still worthwhile to consider properties that\ncarry some of the spirit of “most entangled.”\nOne such property is the failure of a state to be de-\ntermined by its reduced density matrices. As reduced\ndensity matrices contain correlation information pertain-\ning to fewer than the full number of parties in the sys-\ntem, states exhibiting entanglement involving allparties\nmust possess information beyond that contained in their\nreduced density matrices. Linden, Popescu, and Woot-\nters put forward this suggestion in [3, 4] and proved the\nsurprising result that almost all n-party pure states are\ndetermined by their reduced density matrices. In other\nwords, the set of n-party pure states undetermined by\ntheir reduced density matrices is a set of measure zero.\nIn [5], Di´ osi gave a constructive method that succeeds in\nalmost all casesfor determining a 3-qubit pure state from\nits reduced density matrices. Nevertheless, the question\nof precisely which states are determined by their reduced\ndensity matrices remained open.\nIn this Letter, we show that the only n-qubit states\nthat are undetermined among pure states by their\nreduced density matrices are the generalized n-qubit\nGreenberger-Horne-Zeilinger (GHZ) states,\nα|00···0/an}bracketri}ht+β|11···1/an}bracketri}ht, α,β /ne}ationslash= 0\nand their local unitary (LU) equivalents. This means\nthat, among pure states, the generalized GHZ states are\nthe only ones containinginformation at the n-partylevel.\nFor the case n= 3, this result was reported previously in[3]. Part of our argument employs the methods of [5] in\nan essential way.\nLetDnbe the set of n-qubit density matrices. If ρ∈\nDnis ann-qubit density matrix, and j∈ {1,...,n}is a\nqubit label, wemayforman( n−1)-qubitreduceddensity\nmatrixρ(j)= trjρby taking the partial trace of ρover\nqubitj. Let\nPTr :Dn→Dn\nn−1\nbe the map ρ/ma√sto→(ρ(1),...,ρ (n)) that associates to ρits\nn-tuple of (n−1)-qubit reduced density matrices. The\nmap PTr is neither injective (one-to-one) nor surjective\n(onto). Its failure to be surjective means that there are\nn-tuples of (n−1)-qubit density matrices that cannot be\nproduced from any n-qubit density matrix by the partial\ntrace. The question of whether a collection of ( n−1)-\nqubit reduced density matrices could have come from an\nn-qubit density matrix by the partial trace is the subject\nof recent and ongoing investigations [6, 7]. The failure\nof PTr to be injective means that multiple n-qubit states\ncan have the same reduced density matrices. States ρ1/ne}ationslash=\nρ2with PTr(ρ1) = PTr(ρ2) require more information for\ntheir determination than is contained in their ( n−1)-\nqubit reduced density matrices.\nLet\nPn={ρ∈Dn|ρ2=ρ}\nbe the set of pure n-qubit states. If we are interested\nprimarily in pure states, we can restrict the partial trace\nmap to pure n-qubit states.\nptr = PTr |Pn:Pn→Dn\nn−1\nGiven a pure state ψwith|ψ/an}bracketri}ht/an}bracketle{tψ| ∈Pn, the set\nptr−1(ptr(ψ)) contains all pure states with the same re-\nduced density matrices as ψ. (We abbreviate ptr( ψ) =\nptr(|ψ/an}bracketri}ht/an}bracketle{tψ|).)\nWedefineastate ψtobedetermined among pure states\nif ptr−1(ptr(ψ)) contains only |ψ/an}bracketri}ht/an}bracketle{tψ|, andundetermined\namong pure states if ptr−1(ptr(ψ)) contains more than2\none state. Similarly, we define a state ρ∈Dnto bede-\ntermined among arbitrary states if PTr−1(PTr(ρ)) con-\ntains onlyρ, andundetermined among arbitrary states if\nPTr−1(PTr(ρ)) contains more than one state.\nThe surprising result of Linden and Wootters [4] is\nthat almostall n-qubit pure statesaredeterminedamong\narbitrary states.\nNevertheless, there are pure states that are undeter-\nmined among pure states (and consequently undeter-\nmined among arbitrary states). For example, consider\nthe one-parameter family of n-qubit states\n|η/an}bracketri}ht=1√\n2|00···0/an}bracketri}ht+η√\n2|11···1/an}bracketri}ht,\nwhereηisacomplexnumberwithmagnitudeone. If η1/ne}ationslash=\nη2, then|η1/an}bracketri}htand|η2/an}bracketri}htare different states with different\ndensity matrices |η1/an}bracketri}ht/an}bracketle{tη1| /ne}ationslash=|η2/an}bracketri}ht/an}bracketle{tη2|, yet they share the\nsame reduced density matrices, that is, ptr( |η1/an}bracketri}ht/an}bracketle{tη1|) =\nptr(|η2/an}bracketri}ht/an}bracketle{tη2|).\nWe see that almost all pure n-qubit states are deter-\nmined among pure states, yet n-qubit GHZ states are\nundetermined among pure states. The question then be-\ncomes, precisely which states ψare undetermined among\npure states?\nMain Result. Ann-qubit state ψis undetermined\namong pure states if and only if ψis LU-equivalent to\na generalized n-qubit GHZ state.\nProof.Letψbe ann-qubit pure state.\nSuppose that ψis LU-equivalent to a generalized n-\nqubit GHZ state, so we have\nU|ψ/an}bracketri}ht=α|00···0/an}bracketri}ht+β|11···1/an}bracketri}ht,\nwhereUis a local unitary transformation. Define\nU|ψ′/an}bracketri}ht=α|00···0/an}bracketri}ht −β|11···1/an}bracketri}ht. Then |ψ/an}bracketri}ht/an}bracketle{tψ| /ne}ationslash=\n|ψ′/an}bracketri}ht/an}bracketle{tψ′|, sinceαβ/ne}ationslash= 0, but ptr( ψ) = ptr(ψ′). Hence,\nψis undetermined among pure states.\nConversely, suppose that |ψ/an}bracketri}htis undetermined among\npure states. Then there is an n-qubit state vector |ψ′/an}bracketri}ht /ne}ationslash=\neiα|ψ/an}bracketri}htthat hasthe samereduceddensitymatricesas |ψ/an}bracketri}ht.\nClaim: If |ψ/an}bracketri}htand|ψ′/an}bracketri}hthave the same reduced density\nmatrices, then for each qubit j∈ {1,...,n}, there is\na one-qubit local unitary transformation Ljsuch that\n|ψ′/an}bracketri}ht=Lj|ψ/an}bracketri}ht.\nTo prove this, let j∈ {1,...,n}be a qubit label. Let\nρjdenote the one-qubit reduced density matrix of |ψ/an}bracketri}htfor\nqubitj. We write ρjas a spectral decomposition,\nρj=1/summationdisplay\nij=0pij\nj|ij/an}bracketri}ht/an}bracketle{tij|,\nfor some orthonormal basis |ij/an}bracketri}ht, wherep0\njandp1\njare the\neigenvalues of ρj. Ifp0\nj/ne}ationslash=p1\nj, then the orthonormal basis\n|ij/an}bracketri}htis uniquely determined up to a phase. If p0\nj=p1\nj,\nthen any one-qubit orthonormal basis can be used.The (n−1)-qubit reduced density matrix\nρ(j)= trj|ψ/an}bracketri}ht/an}bracketle{tψ|,\nobtained by taking the partial trace of |ψ/an}bracketri}ht/an}bracketle{tψ|over qubit\nj, has the same nonzero eigenvalues as ρj,\nρ(j)=1/summationdisplay\nij=0pij\nj|ij;(j)/an}bracketri}ht/an}bracketle{tij;(j)|.\nIfp0\nj/ne}ationslash=p1\nj, then the (n−1)-qubit eigenvectors |0;(j)/an}bracketri}htand\n|1;(j)/an}bracketri}htare unique up to a phase. If p0\nj=p1\nj, then the\neigenvectorsof ρ(j)with eigenvalue p0\nj=p1\nj= 1/2 consti-\ntute a two-dimensional subspace of the 2n−1-dimensional\nvector space of ( n−1)-qubit vectors, and any orthonor-\nmal pair of vectors in this subspace may be chosen as a\nbasis.\nWe choose the one-qubit orthonormal basis |ij/an}bracketri}htand\nthe (n−1)-qubit orthonormal basis |ij;(j)/an}bracketri}htso that|ψ/an}bracketri}ht\ncan be written\n|ψ/an}bracketri}ht=/radicalBig\np0\nj|0/an}bracketri}ht⊗j|0;(j)/an}bracketri}ht+/radicalBig\np1\nj|1/an}bracketri}ht⊗j|1;(j)/an}bracketri}ht,\nwhere⊗jis the tensor product that inserts a one-qubit\nket just before the jth factor in the ( n−1)-qubit ket\n|ij;(j)/an}bracketri}ht.\nNow|ψ′/an}bracketri}htcan be regarded as the state of a bipartite\nsystem composed of qubit jand all qubits but j, and has\na Schmidt decomposition with respect to those subsys-\ntems,\n|ψ′/an}bracketri}ht=/radicalBig\nq0\nj|0′/an}bracketri}ht⊗j|0′;(j)/an}bracketri}ht+/radicalBig\nq1\nj|1′/an}bracketri}ht⊗j|1′;(j)/an}bracketri}ht,\nwhere|0′/an}bracketri}ht,|1′/an}bracketri}htare orthonormal one-qubit vectors and\n|0′;(j)/an}bracketri}ht,|1′;(j)/an}bracketri}htare orthonormal ( n−1)-qubit vectors.\nTaking the partial trace over qubit j, we have\ntrj|ψ′/an}bracketri}ht/an}bracketle{tψ′|=q0\nj|0′;(j)/an}bracketri}ht/an}bracketle{t0′;(j)|+q1\nj|1′;(j)/an}bracketri}ht/an}bracketle{t1′;(j)|.\nSince this must be equal to ρ(j), it must have the eigen-\nvalues ofρ(j),q0\nj=p0\njandq1\nj=p1\nj. We consider two\ncases, depending on whether ρ(j)has distinct eigenvalues\nor not. Let us treat first the case of distinct eigenvalues,\np0\nj/ne}ationslash=p1\nj. In this case, the eigenvector |0′;(j)/an}bracketri}htcan be\noff by at most a phase from the eigenvector |0;(j)/an}bracketri}ht, and\nsimilarly for |1′;(j)/an}bracketri}ht. The same argument applied to the\none-qubit reduced density matrix ρjshows that |0′/an}bracketri}htcan\nbe off by at most a phase from |0/an}bracketri}ht, and similarly for |1′/an}bracketri}ht.\nIn this case, then, we can write\n|ψ′/an}bracketri}ht=/radicalBig\np0\nj(Lj|0/an}bracketri}ht)⊗j|0;(j)/an}bracketri}ht+/radicalBig\np1\nj(Lj|1/an}bracketri}ht)⊗j|1;(j)/an}bracketri}ht,\n(1)\nwithLja 2×2 diagonal unitary matrix.\nLet us treat next the case of repeated eigenvalues,\np0\nj=p1\nj. In this case, the eigenvectors |0′;(j)/an}bracketri}htand\n|1′;(j)/an}bracketri}htmustmerelyspanthesametwo-dimensionalcom-\nplex space that is spanned by |0;(j)/an}bracketri}htand|1;(j)/an}bracketri}ht. In this3\ncase, the primed eigenvectors must be related to the un-\nprimed eigenvectors by a two-dimensional unitary trans-\nformation,\n|0′;(j)/an}bracketri}ht=u00|0;(j)/an}bracketri}ht+u01|1;(j)/an}bracketri}ht\n|1′;(j)/an}bracketri}ht=u10|0;(j)/an}bracketri}ht+u11|1;(j)/an}bracketri}ht\nwith\n/bracketleftbiggu00u01\nu10u11/bracketrightbigg\n∈U(2).\nThe same argument applied to the one-qubit reduced\ndensity matrix ρjshows that there must be some 2 ×2\nunitary matrix vlmwith\n|0′/an}bracketri}ht=v00|0/an}bracketri}ht+v01|1/an}bracketri}ht\n|1′/an}bracketri}ht=v10|0/an}bracketri}ht+v11|1/an}bracketri}ht.\nIn this case, then, we can write equation (1) with Ljthe\n2×2unitary matrixequaltothe product ofthe transpose\nofvlmwithulm. (We have abused notation by using the\nsymbolLjto represent both the 2 ×2 unitary matrix,\nandalsothelocalunitarytransformationon n-qubitstate\nvectors\nI⊗···⊗I⊗Lj⊗I⊗···⊗I,\nwith the 2 ×2 matrixLjin thejth slot of this tensor\nproduct, and one-qubit (2 ×2) identity operators in all\nother slots.) This completes the proof of the Claim.\nFor each pair of qubit labels j,k, we have\n|ψ/an}bracketri}ht=L−1\nkLj|ψ/an}bracketri}ht.\nNext, spectrally decompose each Ljwith unitary matri-\ncesUjso that\nDj=UjLjU−1\nj\nare diagonal. We have\nD−1\nkDjU1···Un|ψ/an}bracketri}ht=U1···UnU−1\nkD−1\nkUkU−1\njDjUj|ψ/an}bracketri}ht\n=U1···Un|ψ/an}bracketri}ht.\nUsing the multi-index I= (i1i2···in), where each ijis\nzero or one, and the basis\n|I/an}bracketri}ht=|i1i2···in/an}bracketri}ht=|i1/an}bracketri}ht⊗|i2/an}bracketri}ht⊗···⊗|in/an}bracketri}ht,\nexpand\nU1···Un|ψ/an}bracketri}ht=/summationdisplay\nIcI|I/an}bracketri}ht\nand write\nDj=eiαj/bracketleftbiggeiβj0\n0e−iβj/bracketrightbigg\n,whereeiβj/ne}ationslash=e−iβj, since|ψ′/an}bracketri}htis a different state from\n|ψ/an}bracketri}ht. Now we have\ncI=cIexp{i[αj−αk+(−1)ijβj−(−1)ikβk]}\nfor all multi-indices Iand allj,k∈ {1,...,n}. We see\nthat for each multi-index I, eithercI= 0 or\nexp{i[αj−αk+(−1)ijβj−(−1)ikβk]}= 1\nfor allj,k∈ {1,...,n}. LetJbe a multi-index with\ncJ/ne}ationslash= 0. IfIis any multi-index that agrees with Jin at\nleast one entry (say the jth qubit entry), and disagrees\nwithJin at least one entry (say the kth qubit entry),\nthencI= 0. We conclude that cJandcJ, whereJis the\nmulti-index consisting of the complements of each of the\nnbits in multi-index J, are the only nonzero coefficients\ninU1···Un|ψ/an}bracketri}ht. Consequently, |ψ/an}bracketri}htis LU-equivalent to a\ngeneralized n-qubit GHZ state.\nMuch ofthis argumentcarries overto the more general\nsituation of a system of nparties in which party jhas\ndimensiondj(partyjis a qubit if and only if dj= 2). In\nparticular, the Claim carries over. Suppose that |ψ/an}bracketri}htand\n|ψ′/an}bracketri}htare states of a system of nparties in which party j\nhas dimension dj. If|ψ/an}bracketri}htand|ψ′/an}bracketri}hthave the same reduced\ndensitymatrices,thenforeachparty j∈ {1,...,n},there\nis a one-party local unitary transformation Ljsuch that\n|ψ′/an}bracketri}ht=Lj|ψ/an}bracketri}ht. The second part of the argument is com-\nplicated by the possibility of repeated eigenvalues in the\ntransformations Dj. We leave this as a question for fu-\nture work.\nIt is worthwhile to point out the significance of stabi-\nlizer subgroups of the local unitary group in this work.\nThe local unitary group for n-qubit density matrices is\nthe groupG=SU(2)n, consisting of a special unitary\ntransformationoneachqubit. Each(pureormixed)state\nρhas a stabilizer subgroup Iρconsisting of elements of G\nthat leaveρfixed under the action gρg−1forg∈G. We\nhave seen that a state that is undetermined among pure\nstates has a special type of enlarged stabilizer subgroup.\nThere are two alternative formulations of the main re-\nsult that may suggest promising avenues for the classi-\nfication of entanglement types. One can precisely char-\nacterize the pure n-qubit states that are undetermined\namong pure states in terms of the stabilizer subalgebra\nof the state, that is the Lie algebra of the stabilizer sub-\ngroup of the state. It is shown in [8] that the generalized\nn-qubit GHZ state has (for n≥3) stabilizer subalgebra\nKρ=\n\nn/summationdisplay\nj=1itjZj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay\nj=1tj= 0\n\n,\nwhereZjisthe Paulimatrix σzapplied toqubit j. States\nofnqubits undetermined amongpure states areprecisely\nthose which are LU-equivalent to states with this subal-\ngebra.4\nA second alternative formulation of the main result is\nin terms of the dimension of the stabilizer subgroup. For\nn= 3andn≥5,ann-qubitstateisundeterminedamong\npure states if and only if it is not a product state and its\nstabilizer subgroup has dimension n−1 [9].\nWe have shown that all n-qubit states other than gen-\neralizedn-qubit GHZ states and their LU-equivalents are\ncompletelydeterminedbytheirreduceddensitymatrices.\nIs it necessaryto specify allofthe reduced density matri-\nces? Which states are undetermined by specifying only\nn−1 (rather than all n) of the (n−1)-qubit reduced\ndensity matrices? Is it a larger set than the generalized\nGHZs? The answer is yes, and stabilizers can help us\nunderstand this. For example, the state\n|χ/an}bracketri}ht=1√\n3(|0000/an}bracketri}ht+|0001/an}bracketri}ht+|1111/an}bracketri}ht)\nis undetermined by its 3-qubit reduced density matrices\nobtained by taking the partial trace over qubit 1, the\npartialtraceoverqubit 2, andthepartialtraceoverqubit\n3. It is not LU-equivalent to a generalized 4-qubit GHZ\nstate (it has a different stabilizer subalgebra structure,\nand stabilizer subalgebra structure is an LU-invariant).\nNote thatZ1|χ/an}bracketri}ht=Z2|χ/an}bracketri}ht=Z3|χ/an}bracketri}ht /ne}ationslash=eiα|χ/an}bracketri}ht. The state\nZ1|χ/an}bracketri}hthas the same 3-qubit reduced density matrices as\n|χ/an}bracketri}htwhen taking the partial trace over qubit 1, 2, or 3.\nThese two states have a different 3-qubit reduced density\nmatrix when taking the partial trace over qubit 4.\nA pure state’s LU-equivalence class is often considered\nto contain all of the information about the entanglement\nofthestate. Aninterestingquestionis: Arethere n-qubit\npure states with entanglement information that is not\ncontained in their reduced density matrices? We might\ninterpret this question as equivalent to the question: Are\ntheren-qubit pure states for which the LU-equivalence\nclass of the state is undetermined by its reduced densitymatrices? The answer to this question is no. Every n-\nqubit pure state can be determined (among pure states)\nup to a local unitary transformation by its reduced den-\nsity matrices. This can be seen directly from the Claim\nat the beginning of our proof. Two pure states that have\nthe same reduced density matrices must be LU equiva-\nlent.\nWe have not answered the question of which n-qubit\npure states are undetermined among arbitrary states by\ntheir reduced density matrices. The set of n-qubit states\nundetermined among arbitrary states must contain the\ngeneralized GHZ states, bit it could be strictly larger\nthan the set that is undetermined among pure states.\nThis remains an open question.\nThe authors thank the National Science Foundation\nfor their support of this work through NSF Award No.\nPHY-0555506.\n∗Electronic address: walck@lvc.edu\n†Electronic address: lyons@lvc.edu\n[1] D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter,\nand A. Zeilinger, Phys. Rev. Lett. 82, 1345 (1999).\n[2] W.D¨ ur, G. Vidal, andJ.I.Cirac, Phys.Rev.A 62, 062314\n(2000), arXiv:quant-ph/0005115.\n[3] N. Linden, S. Popescu, and W. K. Wootters, Phys. Rev.\nLett.89, 207901 (2002).\n[4] N. Linden and W. K. Wootters, Phys. Rev. Lett. 89,\n277906 (2002).\n[5] L. Di´ osi, Phys. Rev. A 70, 010302(R) (2004).\n[6] Y.-J. Han, Y.-S. Zhang, and G.-C. Guo, Phys. Rev. A 70,\n042309 (2004).\n[7] J.-M. Cai, Z.-W. Zhou, S. Zhang, and G.-C. Guo, Phys.\nRev. A75, 052324 (2007).\n[8] S. N. Walck and D. W. Lyons, Phys. Rev. A 76, 022303\n(2007), arXiv:0706.1785 [quant-ph].\n[9] D. W. Lyons, S. N. Walck, and S. A. Blanda,\narXiv:0709.1105 [quant-ph]."    },    {        "title": "0709.1973v1.Density_of_states_of_helium_droplets.pdf",        "content": "arXiv:0709.1973v1  [physics.atm-clus]  13 Sep 2007Density of states of helium droplets\nKlavs Hansen1\nDepartment of Physics, G¨ oteborg University, SE-412 96 G¨ o teborg, Sweden\nMichael D. Johnson, Vitaly V. Kresin\nDepartment of Physics and Astronomy,\nUniversity of Southern California, Los Angeles, Californi a 90089-0484,\nUSA\nNovember 1, 2018\nAbstract\nAccurate analytical expressions for the state densities of liquid4He droplets are\nderived, incorporatingthe ripplonand phonondegrees of fr eedom. Themicrocanon-\nical temperature and the ripplon angular momentum level den sity are also evalu-\nated. The approach is based on inversions and systematic exp ansions of canonical\nthermodynamic properties.\n1 Introduction\nImportant dynamical processes in finite systems such as nuclei, po lyatomic molecules,\nnanoclusters, atomic clouds, droplets frequently turn out to be s tatistical in nature:\nevaporation/fragmentation, radiation, emission of electrons, eq uilibration between inter-\nnal degrees of freedom or between host and solvent molecules. Wh en such a system\nis thermally isolated, e.g. when flying in a beam or suspended in a trap, t he proper\nstatistical-mechanics treatment is that of the microcanonical ens emble where the energy\nEis fixed and not the temperature of an external heat bath. The de nsity of states, or\nlevel density, ρ(E)dErepresents the number of quantum states between energy Eand\nE+dE. For separable degrees of freedom this is number of normal mode c ombination\nsuch that their energies add up to a total internal energy lying in th is interval. The func-\ntion plays a crucial role in the thermal description of microcanonical systems. For low\nexcitation energies ρ(E) can be represented by a sum of delta functions, corresponding\nto excitations of a only a few of the individual modes, but for even mo derate excitation\n1email: klavs@physics.gu.se\nphone:+46 (0)31 772 3432\nFAX: +46 (0)31 772 3496\n1energies the density of these delta functions becomes so large tha t is is well described as\na continuous function of energy. In this situation it is most convenie nt to use a density\nsmoothed over the discreteness of the energy levels. In addition t o energy systems de-\nscribed with the microcanonical ensemble have a conserved total a ngular momentum, so\nthe correspondingly resolved density of states, ρ(E,J), is often of relevance.\nFree liquid helium nanodroplets [1, 2] represent an interesting syste m for a statistical\ntreatment. One reason is that helium is the only element which cannot be described in\nterms of classical dynamics for any internal degrees of freedom u nder the experimental\nconditions used to study the droplets. This makes the system inter esting in its own right.\nFor example, ”magic number” maxima in the size distributions of small4He clusters have\nbeen shown to correlate with the ability of the cluster to accommoda te elementary ex-\ncitation modes [3]. A second reason is the use of the droplets as micro -cryostats used\nto investigate other clusters and molecules. Evaporative cooling ge nerates internal ener-\ngies corresponding to temperatures of ≈0.4 K and is used to thermalize impurities to\nthis otherwise unreachable temperature for gas phase molecule an d cluster beams. Under-\nstanding these processes requires accurate density-of-state s expressions for the elementary\nexcitation spectrum.\nThe calculation of level densities requires that the excitation spect rum is known. At\nlowtemperatures, therelevantnormalmodesof4HeNclusterswithintheliquiddropmodel\nareripplonswhich arequantized capillarysurfacewaves, andphono nswhicharequantized\nbulk compression waves. For large droplets these modes are separ able to a good approxi-\nmation [4], a fact that greatly facilitates a statistical analysis of the excitation spectrum.\nFor a spherical droplet, both ripplon and phonon modes possess we ll-defined eigenvalue\nspectra characterized by angular momentum for ripplons, and ang ular momentum and\nmode index for phonons. A calculation of the total density of state s requires enumeration\nof all possible normal mode combinations, with individual energies and angular momenta\nadding up to a given total EandJ.\nA leading-order evaluation of ρ(E) for ripplons was carried out by Brink and Stringari\n[5]. Subsequently, Lehmann [6] presented a comprehensive discuss ion of the densities of\nstates for ripplons and phonons computed by direct numerical cou nting, and showed that\nthe resultant plots of the logarithm of the level densities could be we ll parameterized by\npolynomial fits. These fits were then used to calculate other therm odynamic functions\nand to analyze droplet cooling with angular momentum conservation c onstraints [7, 8].\nIn this paper, we show that accurate density of states functions can be obtained by\nanalytic evaluation. This is appealing in its own right, as the calculations take advantage\nof several elegant and generally useful tools from the literature. In addition, having ana-\nlytic expressions for various types of elementary excitations prov ides a systematic method\nfor treating situations where several types of normal modes are excited simultaneously, or\nwhen the spectrum of elementary excitations is modified.\nThe plan of the paper is as follows. In Section 2 we calculate the ripplon density of\nstates as a function of energy, Section 3 considers its angular mom entum dependence, Sec-\ntion 4 is devoted to phonon excitations, Section 5 to the angular mom entum of phonons,\nand Section 6 to the total ρ(E) function. Section 7 comments on the similarity between\n2the spectra considered here and those of multielectron bubbles in b ulk liquid helium, and\npresents a summary.\n2 Ripplon density of states\nAs mentioned above, ripplons are quantized waves on the droplet su rface. For a spherical\nliquid drop, the elementary excitation spectrum is given by [9]\nεℓ= ¯hω0/radicalBig\nℓ(ℓ−1)(ℓ+2). (1)\nHereℓ≥2 is the angular momentum quantum number of the wave and\nω0=/radicalbiggσt\nDR3=/radicalBigg\n4πσt\n3maN, (2)\nwhereσtis the coefficient of surface tension, Dthe mass density, Rthe droplet radius, ma\nthe atomic mass, and Nthe number of atoms in the droplet. If the parameters of bulk\nliquid helium are used, we have ¯ hω0≈/parenleftBig\n3.8/√\nN/parenrightBig\nK in temperature units [6]. Below, the\nripplon energy will be expressed in dimensionless units scaled to the qu antity ¯hω0. Each\nmode has a degeneracy of (2 ℓ+1).\nCanonical approximation\nAfirstapproximationtothelevel densitycanbederivedinthecanon icalensemble picture,\nwhere it is assumed that the system possesses a definite temperat ureT, and the system’s\ninternal energy is associated with its most probable value. The ener gy density of states\nof a finite system is then given by [5, 10, 11, 12]\nρ(E) =eS\n/radicalBig\n−2π(∂E/∂β), (3)\nwhereβ≡(kBT)−1and\nS=βE+lnZ (4)\nis the entropy; Zis the canonical partition function. In the following we will use units\nwherekB= 1. The square root appearing in the equation involves the heat cap acity\nand appears because the canonical entropy includes an approxima tely gaussian integral\nover the thermally populated states with a width given by the heat ca pacity and the\ntemperature, see, e.g., Ref. [12].\nSince the ripplon elementary excitations are bosons we have\nlnZ=−ℓmax/summationdisplay\nℓ=2(2ℓ+1)ln/parenleftBig\n1−e−βεℓ/parenrightBig\n. (5)\n3The canonical thermal energy of the ripplon ensemble is\nE=−∂(lnZ)/∂β. (6)\nTo leading order, we can replace the sum in Eq. (5) by an integral fro m zero to\ninfinity, and approximate the energy eigenvalues (1) by εℓ≈ℓ3/2. The integral then\nstraightforwardly evaluates to\nlnZ= Γ/parenleftbigg7\n3/parenrightbigg\nζ/parenleftbigg7\n3/parenrightbigg\nβ−4/3= 1.685β−4/3, (7)\nand from Eq. (6) the (dimensionless) energy is\nE= 2.247β−7/3. (8)\nAssembling everything into Eq.(3) and expressing the answer in term s of the energy,\nwe find\nρrip(E)≈0.311E−5/7exp(2.476E4/7), (9)\nwhich is the same answer as in Ref. [5].\nMicrocanonical ensemble\nThe above calculation can be improved in two places. One obvious refin ement is to\nevaluate the sum (5) with greater care and to use more precise eige nvalues. A deeper\nconceptual question is how to compute thermodynamic quantities f or a finite isolated\nsystem for which the total internal energy is a conserved quantit y and not an expectation\nvalueandtheuseofa”temperature”mustbecarefullydefined. At horoughdiscussion was\ngiven by Andersen et al. in Ref. [12] with the conclusion that the conv enient canonical\nformalism may be retained, but the canonical expression for the en ergy (6) must be\ncorrected as follows:\nE=−∂(lnZ)/∂β−β−1. (10)\nHereEis the fixed excitation energy of the system and βis understood as the ”micro-\ncanonical temperature” defined as\nβ≡∂[lnρ(E)]/∂E. (11)\nThe procedure taken is as follows. First, the sum in Eq. (5) is calculat ed using the\nfirst three terms of the Euler-Maclaurin summation formula [13]. With the form of the\nspectrum given in Eq.(1), this formula becomes\n−lnZ=∞/integraldisplay\n2(2ℓ+1)ln/parenleftBig\n1−e−βεℓ/parenrightBig\ndℓ+5\n2ln/parenleftBig\n1−e−βε2/parenrightBig\n(12)\n−1\n12d\ndℓ/bracketleftBig\n(2ℓ+1)ln/parenleftBig\n1−e−βεℓ/parenrightBig/bracketrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nℓ=2+...\n4Theupper limit, ℓmax, hasbeenset toinfinityasbefore. Theactualvalueisontheorder of\nℓmax≈2πR/λmin≈2πR/(2d),whereλisthewavelength and distheinteratomicdistance\n[4]. In the liquid drop approximation ( R=N1/3d/2) one then has ℓmax≈πN1/3/2. In\nview of Eqs. (1, 2) this yields a size-independent ripplon Debye tempe rature of εmax≈7.5\nK. Using this value to estimate the error in ln Z, the leftover terms are found to be on the\norder of ( βεmax/4−7ℓmax/6)exp(−βεmax).ForT= 1Kthis is a relative contribution to\nln(Z) of less than 10−2/N1/3which will be ignored.\nA tedious calculation of Eq.(12) involving expansions of exponentials in powers of β\nresults in [14]\nlnZ= 1.685β−4/3+0.639β−2/3−349\n96+7\n3ln(2√\n2β)+... (13)\nThe first term coincides with Eq. (7), and the rest are finite-size an d spectral correc-\ntions. Note that all the numerical coefficients derive from explicit ex pressions involving\nspecial functions. The expansion Eq.(13) has been checked agains t a numerical sum. The\ncomparison is shown in Fig. 1 for Helmholtz’ free energy, F=−Tln(Z). Already at tem-\nperatures where Tis equal to the lowest excitation energy ε2= ¯hω0√\n6, the free energy\nis well represented by the above expression. At higher energies th e agreement improves\nmonotonically.\nKnowing the partition function, we can now use Eq. (10) to determin e the relation\nbetween the microcanonical energy and temperature:\nE= 2.247β−7/3+0.426β−5/3−10\n3β−1. (14)\nAgain, the first term reproduces Eq. (8).\nIn order to proceed with the calculation of the entropy, the heat c apacity and the\nlevel density in Eq.(3), we need to invert the relation (14) which expr essesE(β) to get\nβ(E). This is done by the iterative method of successive approximations . The result is\nan expansion for β−1in powers of E−2/7,\nβ−1≡0.7069E3/7−0.07239E1/7+0.7212E−1/7+..., (15)\nwhere the coefficients are calculated from those entering Eq.(14). Now the prefactor and\nthe exponent in Eq.(3) can be evaluated, using Eqs. (4),(13),(14), andβ(E), finally\nyielding\nρrip(E) = 0.205E−12/7exp/parenleftBig\n2.476E4/7+0.507E2/7/parenrightBig\n. (16)\nLet us emphasize again that all the numerical coefficients encode an alytical expressions.\nEq.(16), which is the main result of this section, may be compared with the canonical\napproximation (9), an exact numerical count carried out with the h elp of the Beyer-\nSwinehart algorithm [15], and the form written down in Ref. [6] as an e mpirical fit to\nthe numerical count in the interval E=50-2500. Fig. 2 shows such a comparison, and\ndemonstrates that the analytical expression gives an excellent re presentation of the exact\nresult [16].\n53 Ripplon angular momentum density\nThe next step is to generalize the ripplon state density to a function which is not only\nenergy- but also angular momentum-resolved. This problem has bee n comprehensively\nstudied in nuclear physics [10, 11]. One way of visualizing the net angula r momentum of\na large distribution of excitations with varying ( ℓ,ℓz) is as the result of random angular\nmomentum coupling, in which case the central limit theorem applies and one expects to\nfind a normal distribution. Indeed, the above references show th atρ(E,J) is essentially\na product of ρ(E) and a Gaussian factor:\nρ(E,J) =ρ(E)2J+1\n2(2π)1/2σ3e−J(J+1)\n2σ2. (17)\nIt is permissible here to replace J(J+1) by ( J+1\n2)2.\nIt is still necessary to establish the variance σ2. An elegant way to do this to leading\norder by means of an extended grand canonical distribution is desc ribed in Bethe’s review\n[10], where the method is applied to a system of non-interacting ferm ions in a spheri-\ncal potential box. Here we follow the same procedure for a system of bosonic ripplon\nexcitations.\nThe idea is first to evaluate the projection Mof the net angular momentum /vectorJof the\ndroplet in terms of the contributions of individual normal modes at t emperature T. The\nfact that Mis a conserved quantity is accounted for by a separate Lagrange m ultiplier,\nor ”chemical potential” γ, such that\nγ=−∂S/∂M (18)\n(Sis the entropy). We can calculate Mdirectly by summing over all modes:\nM=∞/summationdisplay\nℓ=2ℓ/summationdisplay\nm=−ℓm\neβεℓ−γm−1≈∞/integraldisplay\n0dℓℓ/integraldisplay\n−ℓm·dm\neβℓ3/2−γm−1(19)\n(in reduced energy units). Expanding the integrand to first order inγ[10], we find\nM=20\n27Γ/parenleftbigg5\n3/parenrightbigg\nζ/parenleftbigg5\n3/parenrightbigg\nγβ−4/3, (20)\nand Eq. (8) allows us to express the result in terms of the droplet en ergy. To leading\norder we have:\nγ= 1.776ME−8/7. (21)\nNow we can use Eq. (18) with Eq. (21) to obtain the entropy variatio n:\nS(E,M) =S(E,0)−M2/(2σ2) (22)\nwith /parenleftBig\n2σ2/parenrightBig−1= 0.888E−8/7. (23)\n6The second term in Eq. (22) leads to a normal distribution in M. The distribution in\nJcan be shown to have the same variance [10, 11]. Therefore Eqs. (1 7) and (23) define\nρrip(E,J).\nThenumerical evaluationoftherotationaldensity ofstatesinRef . [6]ledtoessentially\nthesameformofthestatedensity function, withthefactorcorr esponding to(2 σ2)−1fitted\nas 0.868E−8/7+0.964E−13/7, which affirms the analytical result (23): the factors deviate\nby less than 2% for E=100-2500.\nA shorter estimate of the variance is illustrative. The number of qua nta in one mode\n(ℓ,m) is on the order of T/εℓfor levels up to εℓ≃Tand zero for higher quantum energies.\nThe total number of excited quanta is then\nn≈T2/3/summationdisplay\nℓ=2(2ℓ+1)T/ℓ3/2≈4T4/3, (24)\nwhere the sum was approximated by an integral and Tis written in terms of the ω0\nunit. With the energy-temperature relation (14) we get the leading order value for en-\nergy per quantum ∝angbracketlefte∝angbracketright=E/n= 2.247T/4,and from this an average of ∝angbracketleftℓ∝angbracketright=∝angbracketlefte∝angbracketright2/3=\nT2/3(2.247/4)2/3.The standard deviation σofℓis then, according to the ’random walk’\nargument used above, σ=√n∝angbracketleftℓ∝angbracketright. Inserting the calculated ∝angbracketleftℓ∝angbracketrightand expressing the result\nin terms of the total energy, one has\n/parenleftBig\n2σ2/parenrightBig−1= 2−1/3(2.247)−4/21E−8/7= 0.68E−8/7, (25)\nin surprisingly sensible agreement with the above result.\nOne may seek to describe the angular momentum distribution in the lan guage of\na rotational energy and a moment of inertia I, associating [11] the exponential in Eq.\n(17) with a Boltzmann factor involving β¯h2J(J+ 1)/(2I), i.e.,I= ¯h2βσ2. Using the\ncanonical-ensemble results, Eqs.(8) and (23), we can express the ”ripplon moment of\ninertia” in terms of the ripplon excitation energy (in reduced units):\nI= 0.797E5/7. (26)\n4 Phonon density of states\nSurface ripplons are the lowest-temperature droplet excitations ; bulk phonons appear\nnext. These are compression sound waves which arise as solutions o f the wave equation\nwithin the volume of the spherical drop. As such, their energies are given by\nεn,ℓ= ¯hukn,ℓ (27)\nwhereuis the speed of sound and the wave number kn,ℓis determined by the boundary\ncondition at the surface. If the Dirichlet boundary condition is adop ted [4, 6], then kn,ℓ=\nan,ℓ/R, wherean,ℓis thenth root of the jℓspherical Bessel function. For a free surface, a\nmore appropriate boundary condition is the Neumann one, in which ca sekn.ℓ=a′\nn,ℓ/R,\n7witha′\nn,ℓthe root of the Bessel function derivative, j′\nℓ. The energy scale is set by the\nlongest wave length, i.e., k∼π/R, so we can express phonon energies in units of\n˜ε= ¯huπ/R, (28)\nwhich works out to ˜ ε=/parenleftBig\n25.5N−1/3/parenrightBig\nK in temperature units if the speed of sound in bulk\n4He is used [6]. The leading-order behavior of the phonon state densit y can be determined\nin a straightforward way by invoking the standard expression for t he Debye heat capacity\n(per unit volume) of bulk phonons:\nCbulk=2π2\n15k4\nB\n¯h3u3T3. (29)\nMultiplying this by 4 πR3/3 and using the fact that (in the canonical framework)\nC=∂E/∂Tand (kBT)−1=∂S/∂E, we can use integrations to express Sin terms of E.\nThen, from Eq.(3) we find that to first order, ln ρph(E)≈S(E) = 3.41E3/4. This matches\nthe leading term of the fit to a direct numerical count in Ref. [6] which is 3.331E3/4.\nThe Debye temperature for phonons in liquid4He is≈25 K [21], corresponding to a\ntotal phonon thermal energy (from Eq.31) of ≈1000NK. We can therefore use the low\ntemperature approximation throughout.\nTheprospectofrefiningthecalculationbyanalyticallyevaluatingast atisticalsumover\nthe precise spectrum (27) may seem bleak, as the Bessel function roots which ”contribute\nin an essential manner are just the zeros for which the usual form ulae (like McMahon’s\nexpansion) are bad approximations” [17]. However, rescue comes f rom an elegant math-\nematical framework known as the Weyl expansion [17, 18, 19]. It pr ovides a systematic\nexpression for the smoothed density of eigenmodes in a finite cavity in terms of volume,\nsurface, and curvature terms. As described in the above refere nces, this is a very general\ntheory, valid for both scalar and vector wave equations, and applic able to a wide variety\nof physical phenomena.\nRef. [20] applied this formalism to the specific heat of metal nanopar ticles. The finite-\nsize correction to the specific heat (29) derived there is immediately usable for our droplet\nproblem:\nC=Cbulk+\n(−)9ζ(3)\n4πk3\nB\n¯h2u2T2\nR+1\n6k2\nB\n¯huT\nR2. (30)\nThe + sign applies to the Neumann and the - sign to the Dirichlet bounda ry conditions.\nAlthough we focus on the Neumann condition, the Dirichlet case will be included for\ncompleteness.\nWe now follow almost the same sequence as in the bulk limit described abo ve: Eq.\n(30) is multiplied by the droplet volume and integrated once to obtain ( withEandTin\nunits of ˜ε)\nE(T) =2π6\n45T4+\n(−)ζ(3)π2T3+π2\n9T2, (31)\n8and a second time to obtain S(T) asS=/integraltextT\n0(C/T′)dT′. The first function is inverted by\niteration to yield\nT(E) = 0.391E1/4−\n(+)0.069−0.006E−1/4. (32)\n[Calculating T(E) instead of β(E) is more convenient in this case.] Eq. (3) is then\nused to obtain the density of states. The calculation is done to the fi rst three orders\ninE, in correspondence to the three terms in the heat capacity expan sion (30). The\nmicrocanonical correction (10) in the present case turns out to c ontribute only in the\nnext order of smallness. The result of the calculation is as follows:\nρph(E) =AE−5/8exp/parenleftbigg\n3.409E3/4+\n(−)0.908E1/2+0.482E1/4/parenrightbigg\n(33)\nOnce again, the + sign is for the Neumann boundary condition on the p honon wave at the\ndroplet surface and (-) for the Dirichlet condition. Using the bulk ca nonical heat capacity\nin Eq.(30) gives a pre-exponential factor of A= 0.26.\nFig.3 compares the exact Beyer-Swinehart count for the phonon s pectrum with the\nfull Eq.(33) and with the level density based on the bulk Debye heat c apacity, Eq. (29),\ni.e., where only the first term in the exponent is present. Fig.(4) show s a more detailed\ncomparison of Eq.(33) and the exact-count phonon level density. We find good agreement\nbetween analytical expression and the exact computation, althou gh not as good as for the\nripplon case.\nTheestimate oftheprefactor AinEq. (33)cannot beexpected tobecorrectbecause it\ndoesnotincludehigher-orderexpansiontermsintheexponenttha twouldyieldcorrections\nof the same order. A comparison with the numerical count suggest s a correction in the\nform of a factor exp( −0.62E0.2). Although this correction is larger than the error found\nfor the ripplon level density, it is nevertheless still relatively small. An effective value of\nA≈0.05 can be used for energies below 400.\n5 Phonon angular momentum density\nA computation of the angular momentum resolved phonon level dens ity suffers from the\ndifficulties with expressing the lowest Bessel function eigenvalues wit h a simple functional\nform. In contrast to ρph(E) there is, to our knowledge, no solution in the literature\nfor this problem. As will be clear from the results presented in sectio n 6 below, the\ncontribution to the level density from the phonons is minor compare d to that of the\nripplons, and the required precision in the calculation of the angular s pecified phonon\ncontribution is therefore correspondingly smaller. In this section w e will make an order\nof magnitude estimate, based on the leading order term of McMahon ’s expansion of the\nroots of the Bessel functions [13]. For the Neumann boundary con dition the roots are\n(n+ℓ/2−3/4)π≈(n+ℓ/2)π. With the phonon energy scale used, Eq.(28), the quantum\nenergies are thus n+ℓ/2. When states with energies up to Tare averaged, the linear\ndependence of the quantum energy on ℓgives an average value of ∝angbracketleftℓ∝angbracketright ∼T. Since also the\nn-dependence is linear the constant of proportionality is on the orde r of unity. The total\n9number of states below energy Tis on the order of T3. Combining these estimates give,\nusing the same type of ’random walk’ estimate as Eq. (25) for the rip plons, that\n(2σ2\nph)−1∼1\nT3·T2=/parenleftBigg2π6\n45/parenrightBigg5/4\nE−5/4≈100E−5/4. (34)\nThe ratio of the σ’s for the phonons and ripplons (here denoted σrip) with the leading\norder terms in the caloric curves Eqs.(14,31) and the proper energ y scaling is\nσph\nσrip∼0.002(T[K])7/6N1/6. (35)\nThis is small compared to unity up to extremely large droplet sizes. Th e conclusion\nthat the width of the phonon angular momentum distribution can be ig nored holds very\nwell, even if the estimate of the width should be incorrect by as much a s an order of\nmagnitude.\n6 Combined level density\nA helium droplet may have both ripplon and phonon oscillations excited a t the same time\n(and, at higher temperatures, rotons as well [1]). The coupling bet ween these normal\nmodes is weak at bulk liquid surfaces [22] and in large droplets [4], thus their energy\ncontents may remain independently defined for some length of time, and the individual\nstate densities will then come from Eqs. (16) and (33). The questio n of equilibration\ndynamics of excitations in superfluid droplets and the relevant time s cales is a very in-\nteresting one, and has not yet been addressed in detail. Below, we d iscuss an estimate\nof state densities in circumstances when the ripplon and phonon exc itations do achieve\nstatistical equilibrium.\nIn principle, the level density of combined excitations can be calculat ed by direct\nsummation, as described in Section 2. This would be a very involved pro cedure, because\nthe ripplon and phonon quantum energies have different dispersion r elations and scale\ndifferently with size. Alternatively, one can calculate the level densit y as a convolution.\nAlso in this task does one benefit from formulating the general prob lem in terms of the\nmicrocanonical temperature. The convolution to be performed is\nρ(E) =/integraldisplayE\n0ρrip(E−ε)ρph(ε)dε. (36)\nFor not extremely large droplets the largest part of the excitation energy resides in the\nripplons. Indeed, the ratio between the energies of the ripplon and phonon subsystems is,\ncanonically:\nEph\nErip≈6.8×10−3N1/3(T[K])5/3. (37)\n(Temperature expressed in Kelvins.) It is clear that for temperatu res under 1 K (i.e.,\nthose which lie safely below the Debye cut-off values specified above a nd below the onset\n10ofrotonmodes)anddropletsofuptoseveraltensofthousands ofatomsinsize, thephonon\nenergy contents is a fraction of the ripplon energy. Under these c onditions one can treat\nthe ripplon degrees of freedom as a heat bath and calculate the pho non contribution with\nan expansion of the integrand of Eq. (36) around some small phono n energy. We will use\nthe simplest choice of zero phonon energy, and to increase the pre cision we expand the\nlogarithm of the level density. Thus\nρ(E) =/integraldisplayE\n0ρrip(E)exp/bracketleftBigg\n−εdln[ρrip(E)]\ndE+1\n2ε2d2ln[ρrip(E)]\ndE2−.../bracketrightBigg\nρph(ε)dε.(38)\nThe upper limit of the integral can be replaced by infinity without serio us loss of precision\nbecause the integrand peaks well below this value. We recognize the first derivative in\nthe exponential as the microcanonical temperature 1 /Tof the ripplon system at energy\nE, see Eq. (11), and therefore have\nρ(E) =ρrip(E)/integraldisplay∞\n0e−ε/Tρph(ε)exp/bracketleftBigg1\n2ε2d2ln[ρrip(E)]\ndE2+.../bracketrightBigg\ndε. (39)\nThe second exponential in the integrand can be expanded, with the integral of the first\nterm yielding the phonon canonical partition function at T,Zph(T):\nρ(E) =ρrip(E)/braceleftBigg\nZph(T)+/integraldisplay∞\n0e−ε/Tρph(ε)/bracketleftBigg1\n2ε2d2ln[ρrip(E)]\ndE2+.../bracketrightBigg\ndε/bracerightBigg\n.(40)\nTo leading order and ignoring the difference between the canonical a nd microcanonical\ntemperatures, this simplifies to\nρ(E) =ρrip(E)Zph(T)/braceleftBigg\n1−Cph\n2Crip−E2\nph\n2CripT2+.../bracerightBigg\n. (41)\nHence the ratio of the term which is second order in εto the zero order term in Eq.(40)\nis approximately\nCph\n2Crip+E2\nph\n2CripT2= 6×10−3(T[K])5/3N1/3+4×10−6(T[K])14/3N4/3.(42)\nFor not excessively large or warm droplets we can leave out the corr ection terms and thus\nhave\nρ(E) =ρrip(E)Zph(T), (43)\nwhereZph(T) as stated above is the phonon canonical partition function at the micro-\ncanonical ripplon temperature corresponding to the ripplon energ yE.\nThe exponential part of the phonon canonical partition function c an be calculated,\ne.g., by integration of the standard relation Eq.(6) with the caloric cu rve in Eq.(31). This\nprocedure does not determine the integration constant which tra nslates into a multiplica-\ntive constant on the total level density, Eq. (43). This constant ,c, is approximately the\n11product of the pre-exponential from Eq.(33) and the prefactor that appears in Eq.(3) (i.e.,\nthe value given by a saddlepoint expansion of the phonon level densit y in the calculation\nof the canonical particion function). The result is\nc≈/radicalBig\n2πT2CphAE−5/8, (44)\nwhereCphisagainthephononheatcapacity. Theleadingorderexpressions fo rthephonon\nparameters Cph(T),Eph(T) give, taking into account the different scaling of energies for\nphonons and ripplons, the total level density\nρ(E) =ρrip(E)·0.526N1/6exp/parenleftBig\n0.04713N−1/2T3+0.01317N1/3T2+0.1634N1/6T/parenrightBig\n(45)\nwith the equation given in ripplon energy units and T=T(E) given by Eq.(15). The\nconstant of 0 .05 for the phonon level density pre-exponential, mentioned at the end of\nSec. 4, was used here also.\nThis result is compared with numerical convolutions for N= 103in Fig.5. The\nnumerical convolution was also calculated for N= 104with a similar result.\nOne remark about Eqs.(43,45) is in place: These equations should only be used for\ncalculations of microcanonical properties. For the calculation of ca nonical properties one\nshould use the product partition function, Zripp,ph=ZripZph. A naive application of\nEq.(45) in a calculation of the partition function of the combined ripplo n-phonon system\nwill give a divergent result at all temperatures. The origin of this dive rgence is the\nbreakdown at high excitation energies of the approximations leading to the equation.\n7 Conclusions\nWe have presented an analytical evaluation of the statistical dens ity of states functions\nof the elementary excitations (surface ripplons and volume phonon s) of isolated liquid-\ndrop helium nanoclusters. These functions are expressed in terms of microcanonically\nconserved quantities: energy and angular momentum. The obtaine d formulas accurately\nmatch numerically computed curves as the energy level densities va ry over∼150��300\norders of magnitude.\nOther interesting helium systems to which the results may be applicab le include\nmicron-sized superfluid fog [23] and multielectron bubbles in liquid heliu m. The lat-\nter are spherical voids inside bulk He, with a thin shell of electrons linin g the inner wall\n(see, e.g., Refs. [24, 25] and references therein). They can unde rgo small-amplitude shape\noscillations, i.e., ripplons, whose frequency under zero external ap plied pressure has the\nformω2\nℓ∝(ℓ2−1)(ℓ−2) , which for large ℓapproaches the same form as the droplet\nripplon dispersion, Eq. (1). This implies that the statistical mechanic s of these bub-\nbles should be similar to that of nanodroplets. One distinction is that t he bubble are\nsubmerged into a bulk helium thermal bath, therefore for them the canonical ensemble\ntreatment is rigorously correct and not just a convenient approx imation.\nFinally, it should be pointed out that the results obtained in the prese nt paper have a\nuniversal form and are expressed in terms of dimensionless scaled e nergies, therefore they\nare generally applicable to the statistics of droplets of various subs tances besides helium.\n128 Acknowledgments\nThis work was supported by the Swedish National Research Council (VR), the U.S. Na-\ntional Science Foundation under Grant No. PHY-0245102, and a Lic k fellowship to M.J.\n13References\n[1] J.P. Toennies and A.F. Vilesov, Angew. Chem. Intern. Ed. 43(2004) 2622\n[2] F.Stienkemeier and K.K.Lehmann, J. Phys. B 39(2006) R127\n[3] R. Guardiola, O. Kornilov, J. Navarro, andJ.P. Toennies, J. Chem. Phys. 124(2006)\n084307\n[4] A. Tamura, Phys. Rev. B 53(1996) 14475\n[5] D.M.Brink and S.Stringari, Z. Phys. D 15(1990) 257\n[6] K.K.Lehmann, J. Chem. Phys. 119(2003) 3336\n[7] K.K.Lehmann, J. Chem. Phys. 120(2003) 513\n[8] K.K.Lehmann and A.M.Doktor, Phys. Rev. Lett. 92(2004) 173401\n[9] L.D.Landau and E.M.Lifshitz, Fluid Mechanics , 2nd ed. (Butterworth-Heinemann,\nOxford, 1987), §62\n[10] H.A.Bethe, Rev. Mod. Phys. 9(1937) 69, §53\n[11] T.Ericson, Adv. Phys. 9(1960) 425\n[12] J.U.Andersen, E.Bonderup, and K.Hansen, J. Chem. Phys. 114(2001) 6518\n[13]Handbook of Mathematical Functions , ed. by M.Abramowitz and I.A.Stegun (Dover,\nNew York, 1972)\n[14] K.Hansen et al., to be published\n[15] T.Beyer and D.F.Swinehart, Comm. Assoc. Comput. Machines 16(1973) 379\n[16] To resolve the numerical comparison, we empoyed two more digit s on the leading\norder term in the exponential than written out in the text.\n[17] H. P. Baltes and E. R. Hilf, Spectra of Finite Systems (Bibliographisches Institut,\nMannheim, 1976)\n[18] H. P. Baltes and E. R. Hilf, Comput. Phys. Commun. 4(1972) 208\n[19] M.BrackandR.K.Bhaduri, SemiclassicalPhysics (Addison-Wesley, NewYork1997)\n[20] H. P. Baltes and E. R. Hilf, Solid State Commun. 12(1973) 369\n[21] A.D.B. Woods and R.A. Cowley, Rep. Prog. Phys. 36(1973) 1135\n[22] M.W. Reynolds, I.D. Setija, and G.V. Shlyapnikov, Phys. Rev. B 46(2001) 575\n14[23] H. Kim, K. Seo, B. Tabbert, and G. A. Williams, Europhys. Lett. 58(2002) 395\n[24] M. M. Salomaa and G. A. Williams, Phys. Rev. Lett. 47(1981) 1730\n[25] J. Tempere, I. F. Silvera, and J. T. Devreese, Phys. Rev. Lett. 87(2001) 275301\n151 10 100 10 -1 10 010 110 210 310 410 5\nε2\n  -F [h ω0]\nT [h ω0]\nFigure1: The (negative of) the ripplon free energies, calculated wit h Eq.(13) (dotted line)\nand the summation in Eq.(5) (full line) which is exact apart from settin g the upper limit\nto infinity. The temperature corresponding to the energy of the lo west excitation, ε2, is\nindicated.\n160 2000 4000 6000 8000 10000 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 \n  relative error \nE [h ω0]\nFigure 2: Comparison of the ripplon level densities calculated accord ing to Eq.(16) (full\nline) and Ref.[6] (dashed for E <2500, dotted line for E >2500). The fit in Ref.[6] was\nlimited to energies between 50 and 2500, in the reduced units used he re and is calculated\nas the derivative of the numerical fit to the integrated level densit y. The expressions have\nbeen divided by the exact Beyer-Swinehart result, causing the osc illatory behavior at low\nenergy, and the curves plotted are the logarithms of these ratios . The curve of Ref.[5]\n(not shown) is around 3.\n170 200 400 600 800 1000 1200 0100 200 300 400 500 600 700 800 \nExcitation energy [ ε]\n  ln ρ(E)\n~\nFigure 3: Phonon level densities calculated according to the exact B eyer-Swinehart count\n(open circles), Eq.(33) (full line), and the level density derived fro m the bulk Debye heat\ncapacity, i.e. corresponding to Eq.(33) but including only the first te rm in the exponential\n(dashed line).\n180 50 100 150 200 250 300 350 400 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 \n  ln( ρ) / ln( ρBS ) -1 \nE [ ε]\nFigure 4: A comparison of Eq.(33) and the exact-count phonon leve l density, showing\nessentially the relative difference in the entropy of the phonon syst em in the two calcula-\ntions.\n190 200 400 600 800 1000 020 40 60 80 100 120 \n  \nT = 1K \n(if ripplons \nalone excited) ln( ρ(E) [(h ω0)-1 ]) \nE [h ω0]\nFigure 5: The convoluted level densities for phonons and ripplons fo r droplet size 103.\nThe numerical convolution is the full line, and the approximate result in Eq.(45) the,\nhardly discernible, dashed line. The level densities for ripplons alone ( dotted line) is\ngiven for reference. The arrow indicates the energy content of t he ripplon excitations at\na temperature of 1 K.\n20"    },    {        "title": "0711.2258v1.Density_operators_and_selective_measurements.pdf",        "content": "arXiv:0711.2258v1  [math-ph]  14 Nov 2007Density operators and selective measurements.\nW/suppress lodzimierz M. Tulczyjew\nValle San Benedetto, 2\n62030 Monte Cavallo, Italy\nAssociated with\nDivision of Mathematical Methods in Physics\nUniversity of Warsaw\nHo˙ za 74, 00-682 Warszawa\nand\nIstituto Nazionale di Fisica Nucleare,\nSezione di Napoli\nComplesso Universitario di Monte Sant’Angelo\nVia Cinthia, 80126 Napoli, Italy\ntulczy@libero.it\n1. Introduction.\nIt is widely believed that statistical interpretation of quantum mech anics requires that density op-\nerators representing quantum states be normalized. We present a description of selective measurements\nin terms of density operators. The description is inspired by Schwing er’s Algebra of Microscopic Mea-\nsurements [1], (see also [2]). Density operators used are not norma lized. We do not know applications\nof density operators requiring normalization.\n2. Beams of particles.\nThe physical space is an affine space Mof dimension 3 modelled on a vector space V. There is\nan Euclidean metric tensor\ng:V→V∗. (1)\nWe consider beams of particles of mass mand constant energy Ein the direction of a unit vector\nz∈V. The internal states of the particles are elements of a unitary vec tor spaceUof dimension r\nover the field Cof complex numbers. The elements of Uarekets|u/an}bracketri}htand elements of the dual space\nU∗arebras/an}bracketle{ta|. The unitary structure establishes an antilinear isomorphism of UwithU∗assigning\nto each ket |u/an}bracketri}hta unique bra /an}bracketle{tu†|. The number\n/an}bracketle{tu1†|u2/an}bracketri}ht (2)\nis thescalar product of vectors |u1/an}bracketri}htand|u2/an}bracketri}ht\nWe introduce a sequence of points\nx0,x1,x2,...,x n (3)\nsatifying inequalities\n/an}bracketle{tg(z),xi−xi−1/an}bracketri}ht>0, (4)\nand a corresponding sequence of planes\nXi={x∈M;/an}bracketle{tg(z),x−xi/an}bracketri}ht= 0}. (5)\nIn the immediate neighbourhood of each plane Xithe beam is not subject to external interaction and\nis represented by a plane wave\n|ψi(x)/an}bracketri}ht=|Ai/an}bracketri}htexp (ik/an}bracketle{tg(z),x−x0/an}bracketri}ht+δi),|Ai/an}bracketri}ht ∈U, k =√\n2mE\n¯h(6)\nwith time dependence separated. The probability flux through a unit surface element of the plane Xi\nis expressed by\n¯hk\nm/an}bracketle{tAi†|Ai/an}bracketri}ht. (7)\n1Between the plane Xi−1and the plane Xithe beam passes through a selective device. Its action\non the wave function is described as the action of a linear transition o perator\nMi:U→U:|ψi−1(x)/an}bracketri}ht /ma√sto→ |ψi(x)/an}bracketri}ht=Mi|ψi−1(x)/an}bracketri}ht. (8)\nIf\nM1,M2,M3,...,M n (9)\nis the sequence of transition operators and\n|ψ0(x)/an}bracketri}ht=|A0/an}bracketri}htexp (ik/an}bracketle{tg(z),x−x0/an}bracketri}ht) (10)\nis the initial state, then\n|ψi(x)/an}bracketri}ht=|Ai/an}bracketri}htexp (ik/an}bracketle{tg(z),x−x0/an}bracketri}ht+δi)\n=Mi···M2M1|A0/an}bracketri}htexp (ik/an}bracketle{tg(z),x−x0/an}bracketri}ht)(11)\nand\n/an}bracketle{tψi†(x)|ψi(x)/an}bracketri}ht=/an}bracketle{tA0†|M1†M2†···Mi†Mi···M2M1|A0/an}bracketri}ht (12)\nThe flux of particles through unit surface element of Xiis given by\n¯hk\nm/an}bracketle{tA0†|M1†M2†···Mi†Mi···M2M1|A0/an}bracketri}ht. (13)\n3. Mixed states and density operators.\nThe expression (13) can be presented in the form\n¯hk\nmtr/parenleftbig\nMi···M2M1|A0/an}bracketri}ht/an}bracketle{tA0†|M1†M2†···Mi†/parenrightbig\n. (14)\nIn this new expression the pure initial state is represented by the d ensity operator |A0/an}bracketri}ht/an}bracketle{tA0†|and can\nbe replaced by a mixed state represented by a positive Hermitian den sity operator T. The expression\n¯hk\nmtr/parenleftbig\nMi···M2M1TM1†M2†···Mi†/parenrightbig\n(15)\nis the result.\n4. Selective measurements.\nWe set the density operator Tin the expression (15) equal to\nT=m\n¯hkrI, (16)\nwhereIis the identity operator. The operator Tis normalized in the sense that\n¯hk\nmtrT= 1 (17)\nalthough this normalization is of no importance. The expression\nPi=¯hk\nmtr/parenleftbig\nMi···M2M1TM1†M2†···Mi†/parenrightbig\n=1\nrtr/parenleftbig\nMi···M2M1M1†M2†···Mi†/parenrightbig\n(18)\nrepresents the probability of detecting a particle crossing a unit su rface element of the plane Xiin unit\ntime. The state of the initial beam emitted by a source at X0is totally mixed.\n2We want to describe the following experimental arrangement. The in itial beam emitted at X0\nundergoes a preliminary selection by a sequence of devices represe nted by the sequence of operators\nM1,M2,...,M j. (19)\nThe beam undergoes further selection passing through a sequenc e of devices represented by operators\nN1=Mj+1,N2=Mj+2,...,N n−j=Mn. (20)\nThe particles are detected at Xnby a non selective detector. After the preliminary selection the sta te\nof the beam is represented by the density operator\nM=MjMj−1···M1TM1†···Mj−1†Mj†=m\n¯hkrMjMj−1···M1M1†···Mj−1†Mj†(21)\nwith the probability of non selective detection\nPin=¯hk\nmtr/parenleftbig\nMjMj−1···M1TM1†···Mj−1†Mj†/parenrightbig\n=¯hk\nmtrM (22)\nThe beam arrives at Xnin a state represented by the density operator\nNn−jNn−j−1···N1MjMj−1···M1TM1†···Mj−1†Mj†N1†···Nn−j−1†Nn−j†(23)\nIt is detected with the probability\nPout=¯hk\nmtr/parenleftbig\nNn−jNn−j−1···N1MjMj−1···M1TM1†···Mj−1†Mj†N1†···Nn−j−1†Nn−j†/parenrightbig\n=¯hk\nmtr/parenleftbig\nN1†···Nn−j−1†Nn−j†Nn−jNn−j−1···N1MjMj−1···M1TM1†···Mj−1†Mj†/parenrightbig\n=¯hk\nmtr(NM )(24)\nwith\nN=N1†···Nn−j−1†Nn−j†Nn−jNn−j−1···N1. (25)\nThe density operator Ncharacterizes the selective detector. The probability Poutis measured at Xjby\nthe selective detector. This measurement is performed on the mixe d state represented by the operator\nM. The relative probability\nPout/Pin= tr(NM )/trM (26)\nshould be considered the result of the selectve measurement desc ribed. Arbitrary normalization can\nbe imposed on M. Normalization of Nwould distort the result of the measurement.\n5. An example.\nIn addition to the metric tensor\ng:V→V∗(27)\nwe introduce in the model space Vof the physical space Man orientation odefined as an equivalence\nclass of bases.\nWe analyse the internal states of a beam of particles of spin 1 /2. States of particles are represented\nby wave functions with values in an unitary space of complex dimension 2. The set of hermitian traceless\noperators in Uis a real vector space Sof dimension 3.\nThe trace tr ( ab) of a product is a non negative real number and the mapping\nS×S→R: (a,b)/ma√sto→tr (ab) (28)\n3is bilinear and symmetric. The spectrum of an operator a∈Sis a pair {α,−α}of real numbers and\nthe spectrum of the operator aais the set {α2,α2}. It follows that tr ( aa) = 0 if and only if a= 0. In\nconclusion we have a Euclidean scalar product\n(|) :S×S→R: (a,b)/ma√sto→(a|b) =1\n2tr (ab). (29)\nWe introduce the Pauli morphism\nσ:V→S. (30)\nThis morphism is an isometry such that the operator\n1\n2σ(w) :U→U (31)\nassociated with each unit vector w∈Vis thespin operator in the direction w. Its spectrum is the\nset{1/2,−1/2}and its eigenvectors represent states of the particle with spin 1 /2 and−1/2 in the\ndirection of w.\nWe introduce a number of operators in the space U:\n1) The projection operator\nK(w) =1\n2(I+σ(w)) (32)\nassociated with a unit vector w∈V. This operator projects onto the space of eigenstates of the\nspin operator 1 /2σ(w) corresponding to the eigenvalue 1 /2.\n2) Aphase shift operator\nD(δ) = exp(iδ)I. (33)\n3) Anattenuation operator\nR(ρ) = exp(−ρ/2)I. (34)\n4) A unitary unimodular operator\nG:U→U. (35)\nThis operator represents a rotation\nE:V→V (36)\nin the sense that\nGσ(w)G−1=σ(Ew). (37)\n5) The operator\nQ(w) =1\n2(I+σ(w)) (38)\nassociated with a vector w∈Vof norm/bardblw/bardbl /ne}ationslash= 1. This operator is not a projection operator.\nConsider a beam undergo a preliminary selection by devices represen ted byM1=K(w) and\nM2=D(δ). The vector wis orthogonal to the direction of the beam and the first of the devic es is\na Stern-Gerlach filter. It is accompanied by an unavoidable phase sh ift. The state prepared by these\ndevices is a pure state represented by the density operator\nM=M2M1TM1†M2†=m\n2¯hkK(w). (39)\nThe selective detector is composed of devices represented by ope ratorsN1=R(ρ),N2=D(δ′), and\nN3=K(w′). The attenuation R(ρ) my be due to the beam passing through a potential barrier. The\ndensity operator\nN=N1†N2†N3†N3N2N1= exp(−ρ)K(w′) (40)\n4represents the selective detector. The result of the selective me asurement is the relative probability\nPout/Pin= tr(NM )/trM=1\n2(1 + (w′|w)) (41)\nsince\ntrM= 1 (42)\nand\ntr(NM ) = tr (exp( −ρ)K(w′)K(w))\n=1\n4tr (exp(−ρ)(I+σ(w′))(I+σ(w)))\n=1\n4tr (exp(−ρ)(I+σ(w′) +σ(w) +σ(w′)σ(w)))\n=1\n2(1 + (w′|w))(43)\n6. References.\n[1] J. Schwinger, The Algebra of Microscopic Measurements , Proc. Natl. Acad. Sc. US, 45(1959)\n[2] F. A. Kaempffer, Concepts in Quantum Mechanics , Academic Press, New York and London (1965)\n5"    },    {        "title": "0711.3748v1.On_the_effect_of_weak_disorder_on_the_density_of_states_in_graphene.pdf",        "content": "arXiv:0711.3748v1  [cond-mat.mes-hall]  23 Nov 2007On the effect of weak disorder on the density of states in graph ene\nBal´ azs D´ ora∗\nMax-Planck-Institut f¨ ur Physik Komplexer Systeme, N¨ oth nitzer Str. 38, 01187 Dresden, Germany\nKlaus Ziegler\nInstitut f¨ ur Physik, Universit¨ at Augsburg, D-86135 Augs burg, Germany\nPeter Thalmeier\nMax-Planck-Institut f¨ ur Chemische Physik fester Stoffe, 01 187 Dresden, Germany\n(Dated: October 31, 2018)\nThe effect of weak potential and bond disorder on the density o f states of graphene is studied. By\ncomparing the self-consistent non-crossing approximatio n on the honeycomb lattice with perturba-\ntion theory on the Dirac fermions, we conclude, that the line ar density of states of pure graphene\nchanges to a non-universal power-law, whose exponent depen ds on the strength of disorder like\n1-4g/√\n3πt2, withgthe variance of the Gaussian disorder, tthe hopping integral. This can result\nin a significant suppression of the exponent of the density of states in the weak-disorder limit. We\nargue, that even a non-linear density of states can result in a conductivity being proportional to the\nnumber of charge carriers, in accordance with experimental findings.\nPACS numbers: 81.05.Uw,71.10.-w,72.15.-v\nI. INTRODUCTION\nGraphene is a single sheet of carbon atoms with a honeycomb lattice, exhibiting interesting transport\nproperties1,2,3,4,5. These are ultimately connected to the low-energy quasiparticles o f graphene, i.e. two-dimensional\nDirac fermions. Its conductivity depends linearly on the carrier den sity, and reaches a universal value in the limit of\nvanishingcarrierdensity1,2. Theformerhas been explainedby the presenceofchargedimpurit ies, while the latter does\nnot allow a charged disorder6,7. Moreover, in the presence of a magnetic field, the half-integer qu antum Hall-effect is\nexplained in terms of the unusual Landau quantization and by the ex istence of zero energy Landau level2,3.\nThe density of states in pure graphene is linear around the particle- hole symmetric filling (called the Dirac point),\nand vanishes at the Dirac point. This is a common feature both in the la ttice description and in the continuum.\nIn addition, the lattice model also shows a logarithmic singularity at th e hopping energy, which is absent in the\ncontinuum or Dirac description.\nWhen disorder is present, the emerging picture is blurred. Field-the oretical approaches to related models (quasi-\nparticles in a d-wave superconductor) predict a power-law vanishin g with non-universal8or universal9exponent or a\ndiverging8density of states, depending on the type and strength of disorde r. Away from the Dirac point a power-law\nwith positive non-universal exponent is also supported by numerica l diagonalization of finite size systems10,11. At\nand near the Dirac point the behavior of the density of states is less clear. Some approaches favour a finite density of\nstates at the Dirac point12, whereas others predict a vanishing DOS8,9or an infinite DOS13. There is some agreement\nthat away from the Dirac point and for weak disorder the DOS behav es like a power law with positive exponent\nρ(E)∼ρ0|E|γ(γ >0). (1)\nThe purpose of the present paper is to investigate, how a non-univ ersal (therefore disorder dependent) power-law\nexponent (found numerically in Refs. 10,11) can emerge for weak dis order (compared to the bandwidth), and what\nits physical consequences are. We determine the exponent based on the comparison of the self-consistent non-crossing\napproximation on the honeycomb lattice and of the perturbative tr eatment of the Dirac Hamiltonian. The exponent\ndecreases linearly with disorder. Then, using this generally non-linea r density of states, we evaluate the conductivity\naway from the Dirac point by using the Einstein relation. We show, tha t based on the specific form of the diffusion\ncoefficient, this can result in a conductivity, depending linearly on the carrier concentration1, and in a mobility,\ndecreasing with increasing disorder. These are in accord with recen t experiment on K adsorbed graphene14. By\nvarying the K doping time, the impurity strength was controlled. The conductivity away from the Dirac point still\ndepends linearly on the charge carrier concentration, but its slope , the mobility decreases steadily with doping time.\nOur results apply to other systems with Dirac fermions such as the o rganic conductor15α-(BEDT-TTF) 2I3.2\nII. HONEYCOMB DISPERSION\nWe start with the Hamiltonian describing quasiparticles on the honeyc omb lattice, given by16,17:\nH0=h1σ1+h2σ2, (2)\nwhereσj’s are the Pauli matrices, representing the two sublattices. Here,\nh1=−t3/summationdisplay\nj=1cos(ajk), h 2=−t3/summationdisplay\nj=1sin(ajk), (3)\nwitha1=a(−√\n3/2,1/2),a2=a(0,−1) anda3=a(√\n3/2,1/2) pointing towards nearest neighbours on the hon-\neycomb lattice, athe lattice constant, tthe hopping integral. The resulting honeycomb dispersion is given by\n±/radicalbig\nh2\n1+h2\n2, which vanishes at six points in the Brillouin zone. To take scattering in to account, we consider the\nmutual coexistence of both Gaussian potential (on-site) disorde r (with matrix element Vo,r, satisfying ∝angbracketleftVo,r∝angbracketright= 0 and\nvariance ∝angbracketleftVo,rVo,r′∝angbracketrightg0=goδrr′) and bond disorder in only one direction (in addition to the uniform hop ping with\nmatrix element Vb,r, satisfying ∝angbracketleftVb,r∝angbracketright= 0 and variance ∝angbracketleftVb,rVb,r′∝angbracketright=gbδrr′), which is thought to describe reliably the\nmore complicated case of disorder on all bonds18. In graphene, ripples can represent the main source of disorder, and\nare approximated by random nearest-neighbour hopping rates, w hile potential disorder might only be relevant close\nto the Dirac point19. The corresponding term in the Hamiltonian is\nV=Vo,rσ0+Vb,rσ1, (4)\nwhich results in H=H0+V.\nFIG. 1: (Color online) A small fragment of the honeycomb latt ice is shown. The thick red lines denote the uni-directional bond\ndisorder, on-site disorder acts on the lattice points.\nWithout magnetic field, the self-energy for the Green’s function, w hich takes all non-crossing diagrams to every\norder into account (non-crossing approximation, NCA), can be fo und self-consistently from20\nΣ(iωn) =1\n1−(go+gb)G2\n[(go+gb)−(go−gb)2G2]G−G, (5)\nwhereiωnis the fermionic Matsubara frequency and\nG=G0[iωn−Σ(iωn)]. (6)\nHere,G0is the unperturbed local Green’s function on the honeycomb lattice given by\nG0(z) =Ac\n(2π)2/integraldisplayzd2k\nz2−t2[4cos(√\n3kx/2)cos(3ky/2)+2cos(√\n3kx)+3], (7)\nwhereAc= 3√\n3a2/2 is the area of the unit cell, and the integral runs over the hexagon al Brillouin zone with corners\ngiven by the condition h2\n1+h2\n2= 0. This can further be brought to a closed form using the results o f Ref. 21. On the\nother hand, in the continuum representation, using the Dirac Hamilt onian, the above Green’s function simplifies to\nG0(z) =2Acσ0\n(2π)2/integraldisplayd2k\nz+v(kxσ1+kyσ2)=−Aczσ0\n2πv2ln/parenleftbigg\n1−λ2\nz2/parenrightbigg\n, (8)3\nandv= 3ta/2. The cutoff λcan be found by requiringthe number ofstates in the Brillouin zoneto be preservedin the\nDirac case as well. This leads to λ=/radicalbig\nπ√\n3t. Another possible choice relies on the comparison of the low frequen cy\nparts of the Green’s function in the lattice and in the continuum limit, w hich reveals the presence of ln( ω/3t) terms.\nThis leads to λ= 3t, which coincides with the real bandwidth on the lattice. We are going t o use this form in the\nfollowing. The difference of the variances becomes important when c alculating the 2nd order correction (in variance)\nto the self-energy. It is clear from Eq. (5), that the same self-en ergy is found for pure potential or unidirectional bond\ndisorder. The effect of their coexistence is the strongest, when t hey possess the same variance. From this, the density\nof states follows as\nρ(ω) =−1\nπImG(ω+iǫ) (9)\nwithǫ→0+. Without disorder, we have the linear density of states ρ(ω≪t) =Ac|ω|/2πv2. At zero frequency, in\nthe limit of weak disorder, the self-energy is obtained as\nΣ(0) =−iλexp/parenleftbigg\n−πv2\nAc(go+gb)/parenrightbigg\n, (10)\nwhich translates into a residual density of states as\nρ(0) =λ\nπ(g0+gb)exp/parenleftbigg\n−πv2\nAc(go+gb)/parenrightbigg\n. (11)\nFrom this expression, weak disorder is defined by the condition go+gb≪t2. The exponential term indicates the\nhighly non-perturbative nature of density of states at the Dirac p oint: all orders of perturbation expansion vanish\nidentically at ω= 0.\nFrom Eq. (5) it is evident, that the interference of the mutual coe xistence of both on-site and bond disorder should\nbe the most pronounced when go=gb. The frequency dependence of the density of states on the hone ycomb lattice\ncan be obtained by the numerical solution of the self-consistency e quation, Eq. (5), and is shown in Fig. 2. For small\nfrequency and disorder, there is hardly any difference between pu re on-site or unidirectional bond disorder and their\ncoexistence. However, at higher energies and disorder strength , they start to deviate from each other. At ω=t,\nthe weak logarithmic divergence is washed out with increasing disorde r strength. Such features are absent from the\nDirac description, which concentrates on the low energy excitation s. Interestingly, for weak disorder, the residual\nDOS remains suppressed as suggested by Eq. (11), but the initial s lope in frequency changes. In order to determine,\nwhether the exponent or its coefficient or both change with disorde r, we perform a perturbation expansion in disorder\nstrength using the Dirac Hamiltonian to quantify the resulting densit y of states, and compare it to the numerical\nsolution of the self-consistent non-crossing approximation using t he honeycomb dispersion.\nIII. POWER-LAW EXPONENT\nThe expansion of the one-particle Green’s function in disorder at z=E+iǫleads to\nG(z) =G0+G0VG0+G0VG0VG0+..., (12)\nwhereV=Vo,rσ0+Vb,rσ1describes both Gaussian potential and unidirectional bond disorde r,G= (z−H0−V)−1\nandG0= (z−H0)−1, andHo=v(kxσ1+kyσ2) is the Dirac Hamiltonian. After averaging over disorder, we get\n∝angbracketleftGrr∝angbracketright=G0;rr+g/summationdisplay\nr′G0;rr′G0;r′r′G0;r′r+... (13)\nwithg=go+gb. The Green’s function G0is translational invariant and reads from Eq. (8) at real frequenc ies as\nG0;rr=Ac|E|\n2πv2/bracketleftbigg\nln/parenleftbiggλ2\nE2−1/parenrightbigg\n−iπ/bracketrightbigg\n(14)\nThis implies\n∝angbracketleftGrr∝angbracketright=G0;rr+g(G2\n0)rrG0;rr+o(g2) =G0;rr[1+g(G2\n0)rr]+o(g2) (15)4\n0 0.5 1 1.5 200.050.10.150.20.250.30.350.4\nPSfrag replacements\nω/ttρ(ω)\n00.511.522.533.5400.020.040.060.080.10.120.14\nPSfrag replacements\n(go+gb)/t2tρ(0)\nFIG. 2: (Color online) The density of states is shown in the le ft panel for pure on-site ( gb= 0) or unidirectional bond ( go= 0)\ndisorder (solid line) for ( go+gb)/t2= 0.1, 0.4, 0.8 and 1.6 with decreasing DOS at ω=t. The red dashed line represents\nthe coexisting bond and unidirectional bond disorder with g0=gb. The black dashed-dotted line denotes the free case with a\nlinear density of states at low energies, exhibiting a logar ithmic divergence at ω=tin the pure limit. The right panel shows\nthe residual density of states for on-site or unidirectiona l bond disorder (blue solid line) and their coexistence with g0=gb(red\ndashed line). The black dashed-dotted line denotes the appr oximate expression, Eq. (11), for weak disorder. For go+gb≤0.4t2,\nthe residual density of states is negligible.\n−0.5 −0.25 0.25 0 0.500.020.040.060.080.10.12\nPSfrag replacements\nω/ttρ(ω)\ngoorgb go=gb\n00.1 0.2 0.3 0.4 0.50.550.60.650.70.750.80.850.90.951\nPSfrag replacements\n(go+gb)/t2exponent ( γ)\nFIG. 3: (Color online) The low energy density of states is sho wn in the left panel for ( go+gb)/t2= 0.004, 0.1, 0.2, 0.3, 0.4 and\n0.5 from bottom to top, for pure on-site ( gb= 0) or unidirectional bond ( go= 0)disorder at positive energies (blue solid line),\nand for their coexistence at negative energies at go=gb(red dashed line). The black dashed-dotted line denotes the power-law\nfit asρ(ω) =ρ0+2ρ1(ω/t)γ. The green vertical dotted line separates the two parts. The right panel visualizes the exponents as\na function of the variance of the disorder, go+gb, for on-site or bond disorder (blue solid line) and their coe xistence ( go=gb).\nThe black dashed-dotted line denotes the result of perturba tion theory: γ= 1−4(go+gb)/√\n3πt2. As is seen, the agreement\nis excellent in the limit of weak disorder. Note, that gdenotes the variance of the disorder.\nMoreover, we have with Eq. (14)\n(G2\n0)rr=Ac\n(2πv2)2/integraldisplayd2k\n[z+v(kxσ1+kyσ2)]2=−∂G0;rr\n∂z=Ac\n2πv2/bracketleftbigg\nln(1−λ\nz2)−2λ2\nz2−λ2/bracketrightbigg\nσ0≈Ac\n2πv2/bracketleftbigg\n−2ln/parenleftbigg|E|\nλ/parenrightbigg\n−iπ/bracketrightbigg\n(16)5\nforλ≫ |E| ≫ǫ. Therefore, we obtain\n∝angbracketleftGrr∝angbracketright=G0;rr/bracketleftbigg\n1−Acg\nπv2ln/parenleftbigg|E|\nλ/parenrightbigg\n−igAc\n2v2/bracketrightbigg\n+o(g2). (17)\nFrom this, the density of states follows as\nρ(E) =−1\nπIm∝angbracketleftGrr∝angbracketright=AcE\n2πv2/bracketleftbigg\n1−2gAc\nπv2ln/parenleftbiggE\nλ/parenrightbigg/bracketrightbigg\n. (18)\nIf we further assume that (I) the density of states as a function ofEsatisfies a power law (inspired by Refs. 9,10,11)\nand (II) the disorder strength gis small, we can formally consider Eq. (18) as the lowest order expans ion in disorder,\nand sum it up to a scaling form as\nρ(E) =Acλ\n2πv2/parenleftbiggE\nλ/parenrightbigg1−(2gAc/πv2)\n. (19)\nHence, thissuggeststhatthelineardensityofstatesofpuregra phenechangesintoanon-universalpower-lawdepending\non the strength of the disorder as γ= 1−(4g/π√\n3t2). Note, that the exponent does not depend on the ambiguous\ncutoffλ. We mention that Eq. (18) might suggest other closed forms than E q. (19). By using the renormalization\ngroupproceduretoselectthemostdivergentdiagramsatagiveno rderg(similarlytoparquetsummationintheKondo\nproblem), one can sum it up as a geometrical series19,22. However, the resulting expression contains a singularity\naround|Σ(0)|(Eq. (10), playing the role of the Kondo temperature here), and is valid at high energies compared\nto|Σ(0)|as Eq. (18). To avoid such problems, we use a different scaling funct ion, suggested by the results of Refs.\n9,10,11.\nWe compare this expression to the numerical solution of the self-co nsistent non-crossing approximation in Fig. 3.\nTo extract the exponent, we fit the data with ρ(E) =ρ0+2ρ1(|E|/t)γ, and extract ρ0,1and the exponent γ. As can be\nseen in the left panel, the power-law fits are excellent in an extended frequency window up to t/4. This suggests that\nthis effect should also be observable experimentally as well. The obtain ed value of ρ0is negligibly small, as follows\nfrom Eq. (11). From the fits, we deduce the exponent and its coeffi cient, which is shown in the right panel. It agrees\nwell with the result of perturbation theory, Eq. (19) in the limit of we ak disorder. The suppression of the exponent is\nsignificant, and can be as big as 30-35% around ( go+gb)/t2∼0.4. Similar phenomenon has been observed for Dirac\nfermions on a square lattice in the presence of random hopping10,11, where disordered systems were studied by exact\ndiagonalization. The decreasing exponent with disorder agrees with our results.\nIV. CONDUCTIVITY FOR NON-LINEAR DENSITY OF STATES\nNow we turn to the discussion of the conductivity in graphene. A pos sible starting point is the Einstein relation23,\nwhich states for the conductivity\nσ=e2ρD, (20)\nwhereρis the density of states and Dis the diffusion coefficient, both at the Fermi energy EF. Assuming a general\npower-law density of states, as found above, we have ρ(E) =ρ1(E/λ)γ. In the weak-disorder limit and away from the\nDirac point E= 0, we can safely neglect any tiny residual value. Moreover, the diff usion coefficient in this case is of\nthe form D=D1E, which is validated from the Boltzmann approach in the presence of c harged impurities6. On the\nother hand, at the Dirac point there is a exponentially small density o f states and a finite non-zero diffusion coefficient\nD∝g/ρin the presence of uncorrelated bond disorder24such that the conductivity is of order 1, in units of e2/h. In\nthe following, however, we will concentrate on the regime away from the Dirac point. Putting these results together,\nwe find\nσ=e2ρ1D1Eγ+1\nF\nλγ. (21)\nThis can be simplified further by noticing that the total number of ch arge carriers, participating in electric transport,\ncan be expressed as\nn=/integraldisplayEF\n0ρ(E)dE=ρ1Eγ+1\nF\n(γ+1)λγ. (22)6\nBy inserting this back to Eq. (21), we can read off the conductivity a s\nσ=e2D1(γ+1)n. (23)\nFrom this we can draw several conclusions. First, it predicts that a way from the Dirac point, where our approach\npredicts a general power-law density of states, the conductivity varies linearly with the density of charge carriers, in\nagreement with experiments1. Second, the mobility of the carriers, which is the coefficient of the nlinear term in the\nconductivity, behaves as\nµ=e/parenleftbigg\n2−4g\nπ√\n3t2/parenrightbigg\nD1, (24)\nwhere we used our approximate expression for the exponent in the density of states, Eq. (19). This means, that\nwith increasing disorder ( g), the mobility decreases steadily, in agreement with recent experim ents on K adsorbed\ngraphene14. There, the graphene sample was doped by K, representing a sour ce of charged impurities. Nevertheless,\nthese centers also distort the local electronic environment, and a ct as bond and potential disorder as well. The\nobserved conductivity varied linearly with the carrier concentratio nn, similarly to Eq. (23). Moreover, the mobility\n(the slope of the nlinear term) decreased steadily with the doping time (and hence the im purity concentration),\nwhich, in our picture, corresponds to a reduction of the exponent γas well as the mobility, Eq. (24).\nTo study the properties close to the Dirac point we have to go beyon d the perturbative regime. Then we realize\nthat the density of states does not vanish at E= 0. As an approximation we add a small contribution near the Dirac\npoint\nρ(E) =ρ0δη(E)+ρ1/parenleftbiggE\nλ/parenrightbiggγ\n,\nin form of a soft Dirac Delta function\nδη(E) =1\nπη\nE2+η2(η >0).\nThis implies a particle density nwhich does not vanish at the Dirac point:\nn(EF) =/integraldisplayEF\n0ρ(E)dE≈ρ0+ρ1\n(γ+1)λγEγ+1\nF. (25)\nMoreover, the diffusion coefficient does neither diverge nor vanish a t the Dirac point18,24such that we can assume\nD(E) =D0δη(E)+D1E .\nFrom the Einstein relation we get the conductivity which provides an in terpolation between a behavior linear in n\naway from the Dirac point and a minimal conductivity at the Dirac point :\nσ∼e2\nh/braceleftBigg\nD0ρ0δ2\nη(EF) for EF∼0\nD1ρ1Eγ+1\nF/λγ∼(1+γ)nforEF≫0. (26)\nThis, together with Eq. (25), implies for EF≫0 the same behavior as in Eq. (23) with the mobility of Eq. (24).\nThe value of the minimal conductivity can be adjusted by choosing th e parameter ηproperly. Therefore Eq. (26)\nprovides us with a qualitative understanding of the conductivity in gr aphene for arbitrary carrier density.\nV. CONCLUSIONS\nWe have studied the effect of weak on-site and bond disorder on the density of states and conductivity of graphene.\nBy using the honeycomb dispersion, we determine the self-energy d ue to disorder in the self-consistent non-crossing\napproximation. The density of states at the Dirac point is filled in for a rbitrarily weak disorder. We investigate\nthe possibility of observing non-linear density of states away from t he Dirac point, motivated by numerical studies\non disordered Dirac fermionic systems. By comparing the results of non-crossing approximation on the honeycomb\nlattice to perturbation theory in the Dirac case, we conclude, that a disorder dependent exponent can account for\nthe evaluated density of states. The exponent decreases linearly with the variance for weak impurities. Then, by\nusing the obtained power-law DOS, we evaluate the conductivity awa y from the Dirac point through the Einstein\nrelation. We find, that this causes the conductivity to depend linear ly on the carries concentration by assuming that\nthe diffusion coefficient is linear in energy6, and the mobility decreases steadily with increasing disorder. These can\nalso be relevant for other systems with Dirac fermions15.7\nAcknowledgments\nWe acknowledge enlighting discussions with A. V´ anyolos. This work wa s supported by the Hungarian Scientific\nResearch Fund under grant number OTKA TS049881 and in part by t he Swedish Research Council.\n∗Electronic address: dora@pks.mpg.de\n1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Kats nelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov,\nNature438, 197 (2005).\n2A. K. Geim and K. S. Novoselov, Nature Materials 6, 183 (2007).\n3V. P. Gusynin and S. G. Sharapov, Phys. Rev. B 73, 245411 (2006).\n4E. McCann, K. Kechedzhi, V. I. Fal’ko, H. Suzuura, T. Ando, an d B. L. Altshuler, Phys. Rev. Lett. 97, 146805 (2006).\n5V. V. Cheianov and V. I. Fal’ko, Phys. Rev. Lett. 97, 226801 (2006).\n6K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 96, 256602 (2006).\n7A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, arXiv:cond-mat/0709.1163v1.\n8A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, Phys. Rev. B 50, 7526 (1994).\n9A. A. Nersesyan, A. M. Tsvelik, and F. Wenger, Phys. Rev. Lett .72, 2628 (1994).\n10Y. Morita and Y. Hatsugai, Phys. Rev. Lett. 79, 3728 (1997).\n11S. Ryu and Y. Hatsugai, Phys. Rev. B 65, 033301 (2001).\n12K. Ziegler, M. H. Hettler, and P. J. Hirschfeld, Phys. Rev. Le tt.77, 3013 (1996).\n13W. A. Atkinson, P. J. Hirschfeld, A. H. MacDonald, and K. Zieg ler, Phys. Rev. Lett. 85, 3926 (2000).\n14J. H. Chen, C. Jang, M. S. Fuhrer, E. D. Williams, and M. Ishiga mi, arXiv:0708.2408.\n15N. Tajima, S. Sugawara, M. Tamura, R. Kato, Y. Nishio, and K. K ajita, Europhys. Lett. 80, 47002 (2007).\n16G. W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984).\n17N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys. Rev. B 73, 125411 (2006).\n18K. Ziegler, arXiv:cond-mat/0703628.\n19P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. B 74, 235443 (2006).\n20A. V´ anyolos, B. D´ ora, K. Maki, and A. Virosztek, New J. Phys .9, 216 (2007).\n21T. Horiguchi, J. Math. Phys. 13, 1411 (1972).\n22I. L. Aleiner and K. B. Efetov, Phys. Rev. Lett. p. 236801 (200 6).\n23A. A. Abrikosov, Fundamentals of the Theory of Metals (North-Holland, Amsterdam, 1998).\n24K. Ziegler, Phys. Rev. Lett. 97, 266802 (2006)."    },    {        "title": "0712.1194v1.The_Hartree_Fock_ground_state_of_the_three_dimensional_electron_gas.pdf",        "content": "arXiv:0712.1194v1  [cond-mat.str-el]  7 Dec 2007The Hartree-Fock ground state of the three-dimensional ele ctron gas\nShiwei Zhang1and D. M. Ceperley2\n1Department of Physics, College of William and Mary, William sburg, VA 23187, USA\n2NCSA and Department of Physics, University of Illinois at Ur bana-Champaign, Urbana, IL 61801, USA\nIn 1962, Overhauser showed that within Hartree-Fock (HF) th e electron gas is unstable to a spin\ndensity wave (SDW) instability. Determining the true HF gro und state has remained a challenge.\nUsing numerical calculations for finite systems and analyti c techniques, we study the HF ground\nstate of the 3D electron gas. At high density, we find broken sp in symmetry states with a nearly\nconstant charge density. Unlike previously discussed spin wave states, the observed wave vector of\nthe SDW is smaller than 2 kF. The broken-symmetry state originates from pairing instab ilities at\nthe Fermi surface, a model for which is proposed.\nPACS numbers: 71.10.Ca,71.10.-w,71.15.-m,75.30.Fv\nThe three-dimensional electron gas is one of the basic\nmodels of many-body physics, and has been investigated\nfor over 70 years [1, 2, 3, 4, 5, 6, 7]. As the simplest\nmodel system representing an itinerant metal, it consists\nof interacting electrons in a uniform neutralizing charge\nbackground, described by the Hamiltonian:\nH=−¯h2\n2m/summationdisplay\ni∇2\ni+1\n2/summationdisplay\ni/negationslash=je2\n|ri−rj|+constant ,(1)\nwhere the sums are over particle indices. Its properties\nare routinely used in density functional theory, e.g., in\nlocal density approximations, as a reference state in cal-\nculations of electronic structure of real materials [6].\nThe simplest approach to an interacting many-fermion\nsystem suchasthe electrongas(jellium) is the mean-field\nHartree-Fock (HF) method, which finds the Slater deter-\nminant wave function minimizing the variational energy.\nIn unpolarized jellium, the “conventional” solution is a\nparamagnetic state with spin symmetry, the restricted\nHF (rHF) solution in quantum chemistry.\nThe rHF solution, however,is notthe exactHF ground\nstate of jellium. In 1962, Overhauser [3] proved that the\nrHF solution is unstable with respect to spin and charge\nfluctuations at any density. The global minimum en-\nergy state within HF is a spontaneously broken symme-\ntry state. The properties of this global ground state have\nremained unknown [7]. This is surprising, given the fun-\ndamentalimportanceofboth theelectrongasandthe HF\napproach. The correlation energy of the homogeneous\nelectron gas is a commonly used fundamental concept,\nbut its definition is in terms of the HF energy of the\nelectron gas.\nIn this paper, we numerically find the HF ground state\nfor finite systems. Our motivation, aside from solving\nthis mathematical puzzle, was to understand the mech-\nanism for the broken symmetry state. Further, it was\nhoped that the solution would suggest candidate ground\nstates, for jellium or for other systems, that can then\nbe explored by accurate many-body approaches such as\nquantum Monte Carlo [5, 8]. We focus on high andmedium densities. An analytic approach is used to aug-\nment and extend the results to the thermodynamic limit.\nWe find that a different pairing instability characterizes\nthe high-density ground state.\nWe consider N(N↑=N↓=N/2) electrons in a cu-\nbic supercell of volume Ω = L3. The density is spec-\nified by the average distance between electrons: rs≡\n(3Ω/4πN)1/3/aB. We write a Slater determinant as\n|Φ∝an}b∇acket∇i}ht=|φ↑\n1,φ↑\n2,···,φ↑\nN↑∝an}b∇acket∇i}ht⊗|φ↓\n1,φ↓\n2,···,φ↓\nN↓∝an}b∇acket∇i}ht,(2)\nwith|φσ\nj∝an}b∇acket∇i}ht=/summationtext\nkcσ\nj(k)|k∝an}b∇acket∇i}htwhere|k∝an}b∇acket∇i}htis a plane-wave basis\nfunction. In the rHF solution, any |k∝an}b∇acket∇i}htwithk≡ |k| ≤kF\nis fully occupied, while all others are empty. Our basis\ncontains all plane-waves with k < kcut.\nTo find the global ground state, i.e., the unrestricted\nHF (uHF) solution, we use an iterative projection\n|Φ(m+1)∝an}b∇acket∇i}ht=e−τHHF(Φ(m))|Φ(m)∝an}b∇acket∇i}ht, (3)\nwhereHHF(Φ(m)) is the HF Hamiltonian, i.e., the mean-\nfield approximation of Eq. ( 1). The wave function re-\nmains a single Slater determinant in the projection [8].\nIfτis sufficiently small, the energy will decrease in each\nstep and the projection will converge to a HF solution as\nm→ ∞. To ensure that the solution is not a local mini-\nmum, we often start from multiple randominitial states\n|Φ(0)∝an}b∇acket∇i}htand verify that the same final state is reached.\nThe smallness of the energy scale relevant to the sym-\nmetry breaking at high density presents a difficulty;\nfinite-size effects can easily be larger than differences in\nenergy of different phases. As a consequence, the stable\nstructures vary wildly with N. There are subtle com-\nmensuration effects in both r-space (Wigner crystal) and\nk-space (spin waves). For example, in open-shell systems\nthere is always a uHF solution, since a broken symmetry\nstate can be formed in a partially filled shell to lower the\nexchange energy, with no cost to the kinetic or Hartree\nenergy (see below). In closed-shell systems, on the other\nhand, there seems to be a critical value of rc\ns(N), below\nwhich no uHF state exists for a given value of Nunder\nperiodic boundary condition (PBC).2\nTo breakthe shell structure, we impose twisted bound-\nary conditions [9] on the orbitals (and hence on the wave-\nfunction): φ(r+Lˆα) =ei2πθαφ(r) where α=x,y,z.\nThis applies a shift of kθ= 2π/vectorθ/Lto the plane-wave ba-\nsis./vectorθ= 0 corresponds to PBC, the Γ-point for solids.\nFor generic twist angles /vectorθ, the rHF solution is non-\ndegenerate.\nLetuswritetheHamiltonianinsecondquantizedform,\nomitting an overall constant:\nˆH=¯h2\n2m/summationdisplay\nσ,kk2c†\nk,σck,σ+1\n2Ω/summationdisplay\nΛ′4πe2\nQ2ˆV(Λ),(4)\nwherec†\nk,σandck,σare creation and annihilation op-\nerators. In the second term, Λ denotes the variables\n{k,k′,Q,σ,σ′},Qis a reciprocal lattice vector, the′on\nthe summation indicates Q∝ne}ationslash= 0, and\nˆV(Λ)≡c†\nk−Q,σc†\nk′+Q,σ′ck′,σ′ck,σ. (5)\nThe HF Hamiltonian ˆHHF(Φ(m)) needed in Eq. ( 3) is\nthe same as ˆH, but with ˆV(Λ) replaced by the linearized\nformˆVHF(Λ) = ˆv(Λ)−∝an}b∇acketle{tˆv(Λ)∝an}b∇acket∇i}ht/2, where\nˆv(Λ)≡2/bracketleftbig\n∝an}b∇acketle{t/summationtext\nk′′c†\nk′′,σ′ck′′−Q,σ′∝an}b∇acket∇i}htδk′,k+Q(6)\n−∝an}b∇acketle{tc†\nk′+Q,σ′ck+Q,σ′∝an}b∇acket∇i}htδσ,σ′/bracketrightbig\nc†\nk,σck′,σ.\nThe expectation ∝an}b∇acketle{t..∝an}b∇acket∇i}htis with respect to |Φ(m)∝an}b∇acket∇i}ht. The vari-\national energy is Ev(Φ)≡ ∝an}b∇acketle{tΦ|ˆH|Φ∝an}b∇acket∇i}ht/∝an}b∇acketle{tΦ|Φ∝an}b∇acket∇i}ht. We also\ncompute a “growth estimator” of the energy, Eg≡\n−ln[∝an}b∇acketle{tΦ(m+1)|Φ(m+1)∝an}b∇acket∇i}ht/∝an}b∇acketle{tΦ(m)|Φ(m)∝an}b∇acket∇i}ht]/2τ, in the projec-\ntion. At convergence, Ev(Φ(m)) =Ev(Φ(m+1)) =Eg,\nwhich means that |Φ(m)∝an}b∇acket∇i}htis an eigenfunction of ˆHHF.\nHence, the projection gives a true solution of the HF\nHamiltonian, not just a Slater determinant with a varia-\ntional energy lower than the rHF value.\nCalculations were carried out with a set of random /vectorθ\nand the results averaged and errors estimated. For each\nN, a fixed set of kθ-points were used at different values\nofrsto correlate the runs. At larger rs, less statistical\naccuracy is needed and fewer kθ-points were used. Typi-\ncally the plane-wave basis cutoff was set to kcut∼2-3kF,\ni.e., a kinetic energy cutoff of 5-10 EF. The resulting ba-\nsis set error is negligible for all but the largest rs(= 7).\nFast Fourier transform (FFT) techniques were used to\nspeed up each step in the projection [8], and the orbitals\nre-orthonormalized as necessary.\nThe top panel of Fig. 1shows the energy difference δE\nbetweentheuHFgroundstateandtherHFstate. As rsis\nreduced, the magnitude of δEdecreases before becoming\nnearly flat at high densities ( rs<3) for each value of N.\nThe energy lowering remains finite across rs, showing\nthat the broken symmetry uHF solution exists for all\ndensities, consistent with Overhauser’s proof [3]. At low\nrs,δEis roughly −10−4Ry in the large-sized systems.1 1.5 2 2.5 3 3.5 4-0.001-0.0008-0.0006-0.0004-0.00020δE\nN=54\nN=64\nN=66\nN=128\nN=246\nN=512\nN=528 (a)\n1 2 3 4 5 6 7\nrs (a.u.)-0.04-0.03-0.02-0.0100.010.02δE\n-0.002-0.00100.0010.002K\nVH\nVex\n(b)\nFIG. 1: (color online) Energy differences (in Ry) between the\nHF ground state and the rHF state. (a) The energy lowering\nper electron, δE= (E−ErHF)/N, vs.rsfor different values of\nN [10]. (b) The three components of δE, kinetic ( K), Hartree\n(VH), and exchange ( Vex), forN= 54 (solid lines) and N=\n66 (dashed lines). The inset is an enlargement of the low\nrs. Error bars are estimated from the results for various kθ-\npoints.\nCalculations for larger systems are needed to clarify its\nbehavior at small rs. The energy differences are very\nsmall compared to the rHF energy which, at N→ ∞, is\nErHF/N= (2.21/r2\ns−0.916/rs)Ry: the relativeenergy\nreduction vanishes as rs→0. From the bottom panel,\nwe see that the lower total energy of the uHF ground\nstate is achieved by reducing the exchange energy at the\ncost of increased kinetic energy (exciting electrons into\n|k∝an}b∇acket∇i}htstates with k > k F). The Hartree energy remains\nunchanged up to rs∼3.\nWe find that the momentum distribution in the uHF\nsolutionis spin-independent, i.e., n↑(k) =n↓(k) is anun-\nbroken symmetry in the broken symmetry ground state.\nThis is illustrated in Fig. 2forrs= 3, but holds in all\nour calculations when fully converged, at all rsandN.\nFigure2also shows that, as rsdecreases, the occu-\npancy of k > k Fstates becomes less pronounced, and\nsignificant modifications to the Fermi sphere become in-\ncreasinglyconfinedtotheimmediatevicinityoftheFermi\nsurface (FS). At small rs, such modifications tend to be\nprincipally single pairing states. An example is seen at\nrs= 1 inN= 54: two plane-wave vectors are involved,\nsuch that a pair of rHF orbitals |φ↑∝an}b∇acket∇i}ht=|φ↓∝an}b∇acket∇i}ht=|k∝an}b∇acket∇i}htbecome\n|φ/arrowbothv∝an}b∇acket∇i}ht=ck|k∝an}b∇acket∇i}ht±ck′|k′∝an}b∇acket∇i}ht (7)\nwherek≤kFandk′> kF, and|ck|2+|ck′|2= 1. Such a\npairing state by itself forms a linear spin-density wave3\n0.5 1 1.5 2 2.5 3\nk/kF00.20.40.60.81n(k)\nrs=1\nrs=2\nrs=3\nrs=3, dn\nrs=4\nrs=70.9 1 1.1 1.2\nk/kF-10-4010-4n(k) - nrHF(k)\nN=246\nN=54\nFIG. 2: (color online) Momentum distribution n(k) at dif-\nferentrsvs.k/kF. The main graph has N= 54, with\n/vectorθ= (−0.368,0.172,−0.364). The arrow indicates kF. The\ninset shows [ n(k)−nrHF(k)] atrs= 1 for N= 246, with\n/vectorθ= (−0.494,−0.425,0.144). Note the primary spike at kF,\nand the paired smaller spikes with one on each side of kF.\n(SDW), with constant charge density and unchanged\nHartree energy. As the inset in Fig. 2shows, additional\npairing states form in the uHF solution involving wave\nvectors further from the FS. Although their amplitudes\nbecome very small as rsis reduced, these are important\nto the true uHF ground state, as we discuss later.\nReal-space properties are examined in Fig. 3. The\nchargedensityis ρ(r) =n↑(r)+n↓(r)andthespindensity\nisσ(r) =n↑(r)−n↓(r). We measure their Fourier trans-\nforms, e.g., Sρ(q) =|/integraltext\nρ(r)eiq·rdr|2/N. Atrs= 7, the\nN= 54 system is an antiferromagnetic bcc Wigner crys-\ntal. Asrsdecreases, electrons become less localized and\nless particle-like. Fluctuation in the charge density be-\ncomes much smaller, as indicated by the rapid reduction\ninSρ(q). This is consistent with the vanishing Hartree\nenergy in Fig. 1. Although Sσ(q) also decreases with\nrs, it is much larger and spin symmetry remains bro-\nken. Atrs= 4, the SDW no longer has a bcc structure,\nand its symmetry between x,y, andzis broken. At\nsmallrs, the electrons are highly delocalized and wave-\nlike. Charge variations are effectively compensated for\nby spatial “double occupancy” of ↑and↓electrons, as in\nthe pairing state discussed in Fig. 2.\nWe measure the characteristic wave vector of the spin\norchargedensitywaveby q=|q|, whereqisthepeakpo-\nsition of S(q). A wave vector of qσ∼2kFseems to have\nalways been assumed in previous investigations of the\nSDW states [3, 7]. However, we find the maximum spin\nordering is at smaller wave vectors, as shown in Fig. 3\nforN= 54. Consistent results are seen for larger N, e.g.\natrs= 2,qσ/kF= 1.2(2), 1.4(2), 1.5(3), and 1 .5(2) for\nN= 66, 128, 246, and 528, respectively.\nWe next prove analytically that an SDW instability\nwhose wave vector is qσ<2kFindeed exists. Let us con-10-510-310-2100S(q)spin\ncharge\n0 1 2 3 4 567\nrs  (a.u.)123q/kF5 10 15 20\n5101520\nFIG. 3: (color online) Left: Peak values and locations of the\nFourier transforms of spin and charge densities for differen t\nvalues of rsinN= 54. Errors are estimated from the values\nat different kθ-points. Note the logarithmic scale in S(q).\nRight: Contour plot of the spin density σ(x,y,z) for a slice\nparallel tothe x-yplane. The systemhas N= 512and rs= 1.\nTheq-vector [2 ¯6¯3] has the leading Sσ(q) value, followed by\n[33¯3], and then [4 ¯15] with Sσfour times smaller.\nsider a system of large but finite N, withkθ= 0. From\nthe rHF reference state, we create a broken symmetry\nstate with two pairing orbitals as in Eq. ( 7), using two\npoints at the FS, k(a highest occupied state) and k′(a\nlowest unoccupied state), as illustrated in Fig. 4. The\nenergy cost consists of kinetic and exchange terms [7]:\n∆K∼2|ck′|2¯h2\nmkF∆k (8)\n∆Vex∼ |ck′|2e2\nπ∆kln2kF\n∆k, (9)\nwherekF= 1/(αrs), withα= (4/9π)1/3. Our choice\nofkandk′gives: ∆ k=|k′| − |k| ∼(1/2lF)(2π/L) =\n1/(2l2\nFαrs), where lFis defined by kF≡lF(2π/L), i.e.,\nlF= (3/8π)1/3N1/3. For fixed rs, the exchange term\ndominates if Nis sufficiently large. Choosing ckandck′\nto be real and of O(1), we can write Eq. ( 9) as:\n∆Vkk′\nex∼e2\nπαrslnlF\nl2\nF. (10)\nWe now createa “satellite”pairingstate in the vicinity\nofkandk′,i.e., with k′−k=s′−sand|s−k| ∼ O(2π/L).\nWe choose the excitation amplitude to be of the particu-\nlar form: cs′∼1/(lnlF)1+δ(δ >0), and thereby cs∼1.\nUsing Eq. ( 9) and noting that the difference in the mag-\nnitude of the wave vectors, ∆ s≡ |s′| − |s|, is less than\nthe size of circles in Fig. 4, we obtain an upper bound to\nthe energy cost for creating the {s,s′}-pairing state\n∆Vss′\nex∼2e2\nπαrs1\nlF(lnlF)1+2δ. (11)\nThedecreaseinexchangeenergybecauseof“constructive\ninterference” between the two parallel pairs is:\n∆Vks\nex∼ −2ckck′cscs′4πe2\nL31\n|k−s|24\nk\nk's\ns'FS\nFS\nFIG. 4: (color online) Cartoon of pairing state with qσ<\n2kF. The primary pairing state {k,k′}is at the FS. The\nsatellite pairing state {s,s′}can be in the shaded areas,\nwithinO(2π/L) ofkandk′. Dashed arrow lines illustrate\nOverhauser[3, 7] pairing at 2 kF.\n∼ −e2\n2π2αrs1\nlF(lnlF)1+δ. (12)\nFor sufficiently large lF, i.e., a large enough system size,\n|∆Vks\nex|can always be made larger than the energy costs\nin Eqs. ( 10) and (11). Hence this is an SDW state with\nlower energy than the rHF state.\nThewavevectoroftheconstructedSDWis qσ=k′−k.\nAs Fig.4shows,qσdoes not need to be 2 kF. The angle\nbetween k′andqσ,θ, can range from 0 ( qσ= 2kF) to\nπ/2 (qσ= 0). As θincreases, more {k,k′}pairing states\nbecome available on the FS, while the number of possi-\nble satellite pairs, i.e., the volume of the shaded areas,\ndecreases. The optimal choice would be in between. In\nfact, as a crude estimate, the number of {k,k′}pairs is\n∝2πk2\nFsinθ, and the number of {s,s′}pairs for each is\n∝(π−θ). Maximizing their product gives qσ∼1.52kF,\nwhich is consistent with our data.\nIn previous approaches [3, 7], pairing is constructed\nfrom orbitals directly across the FS. The energy lowering\nis driven by interference between such pairs (dashed ar-\nrows in Fig. 4), which in our model belong to primary\npairing states. Our approach differs by including the\nsatellite pairing states. The interference between the pri-\nmary and satellite states is what makes a general qσpos-\nsible. This model is supported by the exact numerical\ndata in Fig. 2. Clearly, the true ground state goes be-\nyond this model: the energy will be further lowered by\nhaving more {s,s′}pairs and multiple {k,k′}states, etc.\nWe have shown that the uHF states at intermediate\nand high densities have nearly constant charge density.\nThey arewave-like,and arisefrom pairingbetween states\non the FS separated by distance qσ. The momentum\ndistribution is spin-independent. Its deviation from the\nFermi sea is increasingly confined to the vicinity of the\nFS asrs→0. The SDW wave vector is qσ∼1.5(2)kF,\nand its structures are determined primarily by the short-\nrange exchange potential [11]. In the rHF solution, the\nsizeofthe exchangeholeis rx≡2.34rs[6]. AlinearSDWwith a wave length (i.e., characteristic like-spin separa-\ntion) ofrxhas wave vector q= 1.40kF. Asrs→0, the\nnumerical results are sensitive to the detailed topology\nof the FS in the finite-size systems. The outcome of the\nSDWstructurecanvarygreatly,asitisadelicatebalance\nto optimize pairing among a small number of plane-wave\nstates at the FS that can participate. Our results indi-\ncate that, at high densities, the HF ground state tends\nto further break spatial symmetry and favor one or two\ndimensions. As illustrated in the right panel of Fig. 3,\nfor example, multiple ’pockets’ can coalesce into locally\norevenglobally(e.g., stripe-like)connectedstructures,in\ncontrastwiththeWignercrystalstatewhereeachpocket,\ncorresponding to one electron, is fully localized.\nTo conclude, we have determined the true HF ground\nstate for finite electron gas. Combining numerical and\nanalytic results, we have described the origin and char-\nacteristics of the broken symmetry state at high density,\nand the novel pairing mechanism that drives it.\nThis work was supported by NSF (DMR-0535529 and\nDMR-0404853) and ARO (48752PH). We are grateful\nto H. Krakauer for help with the plane-wave machinery.\nWe acknowledgeuseful discussions with H. Krakauerand\nR. M. Martin.\n[1]E. P. Wigner, Phys. Rev. 46, 1002, (1934); Trans. Fara-\nday Soc. 34678 (1938).\n[2]F. Bloch, Z. Phys. 57, 549 (1929).\n[3]A. W. Overhauser, Phys. Rev. Lett. 3, 414 (1959); Phys.\nRev.1281427 (1962).\n[4]J. R. Trail et. al.Phys. Rev. B 68, 045107 (2003).\n[5]D. Ceperley, Phys. Rev. B 18, 3126 (1978); D. M. Ceper-\nley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980);\nF. H. Zong et. al., Phys. Rev. E 66, 036703 (2002).\n[6]R. M. Martin, “Electronic Structure: basic theory and\npractical methods”, Cambridge University Press, 2004.\n[7]G. F. Giuliani and G. Vignale, “Quantum Theory of the\nElecton Liquid”, Cambridge University Press, 2005.\n[8]S. Zhang and H. Krakauer, Phys. Rev. Lett. 90, 136401\n(2003); M. Suewattana et. al., Phys. Rev. B 75, 245123\n(2007).\n[9]C. Linet. al., Phys. Rev. E 64, 016702 (2001).\n[10]Systems with up to N= 66 were projected to conver-\ngence following this approach. Some larger systems were\nquenched from a “frozen core” state, where Nfc(< N)\nelectrons are frozen in the rHF state while the remaining\nN−Nfcelectrons are active in the projection. Nfcis then\ngradually reduced in subsequent projections. This proce-\ndure can sometimes get “stuck” in a local minimum. It is\nthus possible that the energy gain in the larger systems\nis underestimated and is a lower bound.\n[11]The structure which maximizes the near-neighbor dis-\ntance (NND) between like-spin electrons is the close-\npacking fcc, which would lead to an NaCl-like SDW, but\nwith a characteristic NND of 2 .28rs, less than rx."    },    {        "title": "0801.0852v1.Statistics_of_local_density_of_states_in_the_Falicov_Kimball_model_with_local_disorder.pdf",        "content": "arXiv:0801.0852v1  [cond-mat.str-el]  6 Jan 2008Statistics of local density of states in\nthe Falicov-Kimball model with local disorder\nMinh-Tien Tran\nAsia Pacific Center for Theoretical Physics, Pohang, Republ ic of Korea, and\nInstitute of Physics and Electronics, Vietnamese Academy o f Science and Technology, Hanoi, Vietnam.\nStatistics of the local density of states in the two-dimensi onal Falicov-Kimball model with local\ndisorder is studied by employing the statistical dynamical mean-field theory. Within the theory\nthe local density of states and its distributions are calcul ated through stochastic self-consistent\nequations. The most probable value of the local density of st ates is used to monitor the metal-\ninsulator transition driven by correlation and disorder. N onvanishing of the most probable value\nof the local density of states at the Fermi energy indicates t he existence of extended states in the\ntwo-dimensional disordered interacting system. It is also found that the most probable value of the\nlocal density of states exhibits a discontinuity when the sy stem crosses from extended states to the\nAnderson localization. A phase diagram is also presented.\nPACS numbers: 71.27.+a, 71.23.An, 71.30.+h, 71.10.Fd\nI. INTRODUCTION\nElectron interaction and disorder strongly influence\nthe properties of materials. In particular, the motion\nof charge carrier particles can be suppressed by Coulomb\ninteraction and disorder, and the suppression leads to\na metal-insulator transition (MIT). In the pure system\nwithout disorder the MIT can occur and is purely driven\nby electron correlations.1The transition is commonly re-\nferred to the Mott-Hubbard MIT. In the presence of dis-\norder, the MIT can occur even without electron interac-\ntions. The state of system changes from extended phase\nto the Anderson localization due to coherent backscat-\ntering from randomly distributed impurities.2The Mott-\nHubbard MIT is characterized by opening a gap of the\ndensity of states (DOS) at the Fermi energy, while the\nAnderson localization is a gapless insulator. At the An-\nderson localization the spectra change from continuous\nto dense discrete points. It is characterized by vanishing\nof the most probable value of the local density of states\n(LDOS).2,3,4The most probable value of the LDOS dis-\ncriminates between the metal and insulator phases. The\nimportance of the distribution ofthe LDOS in the proper\ndescription of the Anderson localization was stressed by\nAnderson.2,3,4The very distribution of the LDOS deter-\nmines the Anderson localization but not average quan-\ntities of the LDOS. From the distribution of the LDOS\none can detect both the vanishing of the most probable\nvalue of LDOS as well as the gap opening of the total\nDOS, thus the distribution of the LDOS is a valuable\ntool for determining the MIT driven by both disorder\nand correlation.\nIn recent years, a generalized Curie-Weiss mean-field\ntheory, the dynamical mean-field theory (DMFT) was\ndeveloped.5The DMFT essentially captures local tem-\nporal fluctuations. It has been widely applied to study\ncorrelated electron systems. The DMFT describes the\nMott-Hubbard MIT well.5However, it works with the\narithmetic average of the LDOS, and cannot determinethe Anderson localization in disordered systems. A sta-\ntistical variant of the DMFT, which is usually referred\nto the statistical DMFT, has been introduced to study\nsystems with both disorder and interaction.6,7It can\nbe viewed as the DMFT formulated in real space with\ngeneral inhomogeneous solutions.8Within the statistical\nDMFT, the self energy is a local function of frequency,\nbut it also depends on the site index. In the presence of\ndiagonaldisorder, the self energyis also diagonalrandom\nvariables, and gives additional dynamical random contri-\nbutions to disorder. It generates a set of self-consistent\nstochastic equations. The statistical DMFT essentially\ndeals with the LDOS, hence it is capable of studying\nthe Anderson-Mott-Hubbard MIT. In parallel with the\nstatistical DMFT, a typical medium theory (TMT) was\nalso introduced to study the Anderson localization.9The\nTMT is based on the DMFT too. However, instead of\nthe arithmetic average DOS, it works with the geomet-\nric average DOS. The geometric average DOS is incor-\nporated into the self-consistent stochastic DMFT equa-\ntions, which result into self-consistent equations of the\nDMFT fashion. The TMT is essentially a mean-field\ntheory of both disorder and correlation, while in the sta-\ntistical DMFT disorder is treated exactly, and only the\ncorrelation effects are treated in a mean-field manner.\nThe TMT was employed to study the Anderson-Mott-\nHubbard MIT in correlated electron systems with local\ndisorder.10,11\nIn the presence of disorder the LDOS forms a stochas-\ntic ensemble. The stochastic ensemble of the LDOS\nmusthavecharacteristicswhichdiscriminatebetween the\nmetallic and insulator phases, and one has to search the\ncharacteristics for determining the MIT. Examples for\nthe characteristics are the most probable value of the\nLDOS in the original Anderson theory of localization2,3,4\northegeometricaverageoftheLDOSintheTMT.9How-\never, nearby the critical point of the MIT, quantum fluc-\ntuations may induce outliers of the statistical description\nof the stochastic ensemble of the LDOS. For a random\nsample an estimator is called robust if it is insensitive to2\noutliers.12The most probable value, arithmetic average,\nmedian are the examples of robust estimator. The geo-\nmetric average is not a robust estimator because it does\nnot fulfil the linear property of the robust estimator.12\nFrom the point ofview ofrobust statistics the most prob-\nable value is a better estimator of a random sample than\nthe geometric average. Moreover, in general, the geo-\nmetric average DOS is not the most probable value of\nthe LDOS, hence it does not truly represent a typical\nDOS, although it is closer to the most probable value of\nthe LDOS than the arithmetic averageof the LDOS. The\ngeometric average DOS is sensitive to small values of the\nLDOS at individual sites, even when these values do not\nrepresentthemostprobablevalueoftheLDOS.Adecline\nof the geometric average DOS with increasing the disor-\nder strength does not necessarily imply the approach to\nthe Anderson localization.13Nevertheless, when the geo-\nmetric average is embedded into the self-consistent cycle\nof the DMFT, the whole method, i.e the TMT, can de-\nscribe the Anderson localization.9,10,11\nThe interplay between disorder and correlation in the\nMIT theory is a long standing problem. In the clean\nor noninteracting limits the MIT was well studied.5,14\nHowever, the correlation effects in disordered systems\nstill remain unclear. In particular, despite experiments\nfound evidences of the metallic behavior in the two di-\nmensional electron systems, in theory it is not clear how\nelectron correlations induce metallic phase in low dimen-\nsional disordered systems.15In the present paper we con-\nsider the interplay of disorder and short range interac-\ntion in the MIT. Usually, the short range interaction\nis modelled by the Hubbard interaction.16Here we take\nan alternative point of view. Rather than try to study\nthe Hubbard model we take a simpler model, the spin-\nless Falicov-Kimball model (FKM).17The relation of the\nFKM to the Hubbard model is analogous to the relation\nbetween the Ising and Heisenberg models of magnetism.\nThe FKM describes itinerant electrons interacting via a\nrepulsive contact potential with localized electrons (or\nions). It can also be viewed as a simplified Hubbard\nmodel where electrons with down spin are frozen and do\nnot hop. Certainly, within the TMT the phase diagram\nof the Anderson-Mott-Hubbard MIT in the FKM and\nthat in the Hubbard model share common features.10,11\nMoreover, the FKM exhibits a rich phase diagram. In\nhomogeneous phase it exhibits the Mott-Hubbard type\nof MIT, although the model does not describe the Fermi\nliquid picture. At low temperature different phases with\nlong-range order may exist depending on the doping and\ninteraction strength.18,19,20The FKM can also be incor-\nporated into different models to study various aspects\nof electron correlations, for instance the charge ordered\nferromagnetism in manganites.21,22,23In the disordered\nFKM we can study different realizations of the inter-\nplay of disorder and electron correlations. With local\ndisorder the FKM exhibits the Anderson-Mott-Hubbard\nMIT.11It will be studied in the present paper by em-\nploying the statistical DMFT. After solving of the self-consistent equations of the statistical DMFT we obtain\nthe distributions of the LDOS. We determine the most\nprobable value of the LDOS and use it to monitor the\nAnderson localization. We find extended states which\noccur in the region from weak to intermediate strengths\nof interaction and disorder. For intermediate values of\ndisorder or interaction there is a reentrance of the An-\nderson localization. At strong disorder the system is a\ngapless insulator, while at strong interaction the system\nis a Mott-Hubbard insulator. We also find at the cross-\ning point from extended states to localization the most\nprobable value of the LDOS exhibits a discontinuity.\nThe plan of the present paper is as follows. In Sec. II\nwe present the statistical DMFT through its application\nto the FKM with local disorder. Numerical results are\npresented in Sec. III. In Sec. IV conclusion and remarks\nare presented.\nII. STATISTICAL DYNAMICAL MEAN-FIELD\nTHEORY\nIn this section we describe the statistical DMFT\nthrough its application to the FKM with local disorder.\nThe Hamiltonian of the system reads\nH=/summationdisplay\n<i,j>tijc†\nicj+/summationdisplay\niεic†\nici−µ/summationdisplay\nic†\nici\n+Ef/summationdisplay\nif†\nifi+U/summationdisplay\nic†\nicif†\nifi, (1)\nwherec†\ni(ci) andf†\ni(fi) is the creation (annihilation)\noperator of itinerant and localized electrons at site i, re-\nspectively. tijis the hopping integral of itinerant elec-\ntrons between site iandj. In the following we take into\naccount only nearest neighbor hopping, i.e., tij=−tfor\nnearest neighborsites, and tij= 0 otherwise. We will use\ntas the energy unit. Uis the local interaction of itiner-\nantandlocalizedelectrons. µisthechemicalpotentialfor\nitinerant electrons. It controls the electron density. Ef\nis the energy level of localized electrons. It also serves as\nthe chemical potential of localized electrons and controls\nthe density oflocalized electrons. In the following we will\nconsider only the symmetric half filling case. It turns out\nthat it is equivalent to µ=U/2 andEf=−U/2.18εi\nare independent random variables. They represent lo-\ncal disorder in the model. We will consider the random\nvariables with uniform distribution\nP(εi) =1\nWΘ/parenleftbiggW\n2−|εi|/parenrightbigg\n, (2)\nwhere Θ( x) is the step function, and Wrepresents the\ndisorder strength. When the disorder is absent, model\n(1) is the pure FKM. In homogeneous phase which oc-\ncurs at high temperature it exhibits a Mott-Hubbard\nMIT.18,19,20When the interactionisabsent( U= 0), itin-\nerant and localized electrons are decoupled, and the itin-\nerant electron part of model (1) represents the Anderson3\nmodel, and it would exhibit the Anderson localization.2\nThus when both disorder and interaction are present,\nmodel (1) would exhibit the complex Anderson-Mott-\nHubbard MIT transition at high temperature.11The\nFKM may be realized by loading two kinds of fermion\natomswith light and heavymassesin optical lattice. The\nlight atoms play the role of itinerant electrons, while the\nheavyatoms are kept immobile as the localized electrons.\nThe main idea of the statistical DMFT is to formulate\nthe DMFT of the system with a realization of disorder in\nrealspace.6Whenthedisorderisrealized,thelocalGreen\nfunction is not homogeneous anymore. In this case we\ncan adopt the inhomogeneous DMFT8for treating the\ninteraction part. Within the approach the disorder is\ntreated exactly, while the effects of electron interaction\nresult into the self energy which is self-consistently calcu-\nlated by the DMFT equations. However, the self energy\ndepends on the site index. The electron Green function\ncan be written in real space\nG(ω) = [G−1\n0(ω)−Σ(ω)]−1, (3)\nwhereΣ ij(ω)istheselfenergyoftheelectronGreenfunc-\ntionGij(ω) =/angbracketleft/angbracketleftci|c†\nj/angbracketright/angbracketrightω.G0(ω) is the noninteracting\nGreen function. For the FKM with a realization of dis-\norderG0(ω) = [ωδij−tij−εi+µδij]−1.εiis random\nvariables realized accordingly to the probability distribu-\ntion (2). Equation (3) is just the Dyson equation written\ninthematrixform. Within thestatisticalDMFT,theself\nenergy is approximated by a local function of frequency.\nHowever, this local function can vary from site to site,\ni.e.,\nΣij(ω) =δijΣi(ω). (4)\nThe approximation is strictly local. In infinite dimen-\nsions the self energy is purely local. For finite dimensions\nthe approximation neglects nonlocal correlations. The\nnonlocal correlations can be systematically incorporated\nby cluster extensions of the DMFT.24The site depen-\ndence of the self energy is generated via random vari-\nablesεi. As a consequence the self energy Σ i(ω) is also\nstochastic variables. With this feature the effective mean\nfield and the local Green function also are local stochas-\ntic variables. The Dyson equation (3) shows that the\nself energy gives additional local random contributions\nto disorder of the system. These contributions are due\nto both interaction and the interplay between interac-\ntion and disorder. However, in difference to the random\nvariables εithe contributions are dynamical. They take\ninto account temporal local quantum fluctuations gen-\nerated by interaction and disorder. They also broaden\nthe random energy levels generated by disorder. The self\nenergy Σ i(ω) is determined from an effective single site.\nOnce the effective single site is solved the self energy is\ncalculated by the Dyson equation\nΣi(ω) =G−1\ni(ω)−G−1\ni(ω), (5)\nwhereGi(ω) is the bare Green function of the effective\nsingle site and represents the effective mean field actingon sitei.Gi(ω) is the electron Green function of the\neffectivesinglesite. Theself-consistentconditionrequires\nthat the Green function Gi(ω) of the effective single site\nmust concisewith the local Green function of the original\nlattice. i.e.,\nGi(ω) =Gii(ω). (6)\nIn the Appendix we show the exact derivation of the self\nconsistentequationforinhomogeneoussystemsininfinite\ndimensions. Equations (3)-(6) form the self-consistent\nsystemofequationsforthelatticeGreenfunctionandthe\nselfenergy. They areprincipalequationsofthe statistical\nDMFT. Since the local Green function is stochastic vari-\nables, the self-consistent equations are naturally stochas-\ntic too. In Eq. (6) the right hand side is a functional of\nstochastic variables Gi(ω), thus the self-consistent con-\ndition (6) generates a stochastic chain of Gi(ω) via the\niteration process. The LDOS is defined as usually\nρi(ω) =−1\nπImGii(ω+iη),\nwhereη= 0+is an infinitely small positive number.\nFor the FKM the effective single site problem has the\nfollowing action\nSi[c†\ni,ci] =−/integraldisplayβ\n0dτdτ′c†\ni(τ)G−1\ni(τ−τ′)ci(τ′)\n+/integraldisplayβ\n0dτU(c†\nicif†\nifi)(τ)+βEff†\nifi,(7)\nwhereβ= 1/Tis the inverse of temperature. The parti-\ntion function corresponding to the action is\nZi= Trfi/integraldisplay\nDc†\niDcie−Si[c†\ni,ci]. (8)\nThispartitionfunctioncanbecalculatedexactly, because\nthetraceoverthelocalizedelectronsisindependentofthe\ndynamics of itinerant electrons. We obtain20\nZi= 2exp/bracketleftBig/summationdisplay\nnln/parenleftBigG−1\ni(iωn)\niωn/parenrightBig\neiωnη/bracketrightBig\n+\n2exp/bracketleftBig\n−βEf+/summationdisplay\nnln/parenleftBigG−1\ni(iωn)−U\niωn/parenrightBig\neiωnη/bracketrightBig\n,(9)\nwhereωn= (2n+1)πTis the Matsubara frequency. The\nGreen function can directly be calculated from the par-\ntition function. Without difficulty one obtains\nGi(iωn) =W0i\nG−1\ni(iωn)+W1i\nG−1\ni(iωn)−U,(10)\nwhereW1i=f(/tildewideEi),W0i= 1−W1i. Here f(x) =\n1/(exp(βx)+1) is the Fermi-Dirac distribution function,\nand\n/tildewideEi=Ef+T/summationdisplay\nnln/parenleftBig1\n1−UGi(iωn)/parenrightBig\neiωnη.(11)4\nNote that the weight factors W0i,W1iare not simply a\nnumber. They are functionals of the local Green func-\ntion. One can show that W1i=/angbracketleftf†\nifi/angbracketrightis the density\nof localized electrons at site i. So far we have obtained\nthe complete solution of the effective single site. This to-\ngether with the statistical DMFT equations (3)-(6) fully\ndeterminethedynamicsofitinerantelectronswithafixed\ndisorder realization. Within the statistical DMFT, the\ndisorder is treated exactly, while the correlation effects\nare taken into account through the mean field contribu-\ntions of the DMFT. Once the self-consistent equations\nof the statistical DMFT are solved we obtain the LDOS.\nWith many different realizations of disorder a data en-\nsembleofthe LDOS isobtained. The sizeofthe ensemble\ndepends on the lattice size and the number of disorder\nrealizations. From the ensemble of the LDOS we can de-\ntermine its probability distributions as well as the most\nprobable value of the LDOS. The most probable value of\nthe LDOS is used to monitor the MIT driven by disorder\nand correlation.\nThe present approach proposes a statistical local de-\nscription of the MIT in disordered interacting systems.\nThe statistical aspect is a proper description since it\nworks directly with the ensemble of LDOS and use\nthe most probable value of the LDOS to monitor the\nMIT.2,3,4The local treatment in the spirit of DMFT ne-\nglects nonlocal correlations. This weakness may be seri-\nous in low dimensional systems. However, for the two di-\nmensional FKM nonlocal correlations give nonsignificant\ncontributions to the DOS in the homogeneous phase.25\nThe DMFT calculations for the two-dimensional FKM\nalso show reasonable results.26Nevertheless, the statis-\ntical DMFT can be considered as a simplest approach\nincorporating both disorder and correlation to the MIT.\nIII. NUMERICAL RESULTS\nIn this section we present the numerical results of the\nstatistical DMFT equations. In general, the matrix in-\nversion in Eq. (3) can be performed only for a finite size\nlattice. We consider a two dimensional square lattice\nwith the linear size L. Thus, we perform numerical cal-\nculations for the lattice of size Land finite number Nd\nof disorder realizations. In particular numerical calcu-\nlations were performed for L= 12 and Nd= 100. We\ntake temperature T= 1 which is high enough for homo-\ngeneous phase and avoiding any long-range order. Cer-\ntainly, the homogeneous phase is insensitive to temper-\nature, however at low temperature it is unstable against\nthe long-range ordered phase.20The Mott-Hubbard like\nMIT in the FKM occursonly for the homogeneousphase.\nStrictly speaking, the Anderson localization is well de-\nfined only at zero temperature. Here we study the in-\nterplay between the Anderson localization and the Mott-\nHubbard MIT by investigating the single particle Green\nfunction at finite temperature. However, in the absence\nof long range orders the single particle Green function/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53\n/s48/s53/s49/s48/s49/s53\n/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s48/s53/s49/s48\n/s45/s49/s53 /s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48 /s49/s53/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54\n/s48/s53/s49/s48/s49/s53/s97/s40 /s41/s85/s61/s50\n/s87/s61/s52\n/s32/s124/s100\n/s97/s40 /s41/s47/s100 /s124/s32/s32\n/s97/s40 /s41\n/s32/s85/s61/s56\n/s87/s61/s52\n/s32/s124/s100\n/s97/s40 /s41/s47/s100 /s124\n/s85/s61/s56\n/s87/s61/s49/s48/s32\n/s97/s40 /s41\n/s32\n/s32/s124/s100\n/s97/s40 /s41/s47/s100 /s124\nFIG. 1: The total DOS ρa(ω) and its derivative |dρa(ω)/dω|\nfor various interactions and disorders.\nis insensitive to temperature. Hence, we can consider\nthe single particle Green function at finite temperature\nas it would be at zero temperature. The symmetric half\nfilling case with fixed µ=U/2 and localized electron\ndensitynf= 1/2 is considered. The energy level Efof\nlocalized electrons is determined accordingly by condi-\ntionnf=/summationtext\ni/angbracketleftf†\nifi/angbracketright/L2for each disorder realization. We\nsolvethe statisticalDMFT equationsinrealfrequencyby\niterations. The small positive number η= 0.01 is used.\nWithout disorder the FKM exhibits the Mott-Hubbard\nMIT with critical value Uc≈4. Thus when system is\nclean the system state is metallic for U < U c, and is\ninsulator for U > U c.\nA. Total DOS and band edge\nFirst we determine the band edge of the system. Usu-\nally, the band edge is determined from vanishing condi-\ntion of the total DOS. The total DOS is defined as the\narithmetic average of the LDOS\nρa(ω) =1\nNdL2/summationdisplay\ndisorder/summationdisplay\niρi(ω). (12)\nHowever, strictly speaking, in the numerical calcula-\ntions the total DOS never vanishes since η= 0.01 was\nused. In Fig. 1 we plot the total DOS and its deriva-\ntive|dρa(ω)/dω|. One can see that at the band edge the\ntotal DOS sharply changes and its derivative exhibits a5\n/s48/s49/s48/s50/s48/s51/s48\n/s48/s53/s49/s48\n/s48/s53/s49/s48\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50/s48/s53/s49/s48/s49/s53\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s48/s49/s48/s50/s48/s51/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48\n/s32/s32\n/s87/s61/s50\n/s32/s32\n/s87/s61/s52/s32/s80/s40\n/s105/s41\n/s32/s87/s61/s54\n/s32 /s32/s87/s61/s56\n/s32\n/s105/s32/s87/s61/s49/s48\n/s32/s32\n/s32/s32\n/s105/s87/s61/s49/s50\nFIG. 2: (Color online) Probability distribution P(ρi) of the\nLDOSρiat the Fermi energy for various disorder Win the\nweak interaction case ( U= 2).\npronounced peak. We use the position of the peak to\ndetermine the band edge. The value of the total DOS\nat the band edge, ρbe, also serves as the cutoff of the\nDOS. Below this value ρbethe DOS approximately van-\nishes. For strong interaction and weak disorder, the total\nDOS opens a gap at the Fermi energy. It is similar to\nthe Mott-Hubbard insulator in the clean case. For such\ncases we also use the peaks of |dρa(ω)/dω|to determine\nthe band gap. For strong interaction and strong disor-\nder, the gap opened at the Fermi energy closes. As we\nwill see later, the system is still localized, but gapless. It\nis a crossover from the Mott-Hubbard to the Anderson\ninsulator by disorder.\nB. Anderson-Mott-Hubbard MIT\nThe probability distribution of the LDOS is con-\nstructed from statistical data of the LDOS which is ob-\ntained after solving the statistical DMFT equations. In\nFig. 2 we present the histogram of the probability dis-\ntribution of the LDOS at the Fermi energy for U= 2\nand various disorders. This value of interaction ( U= 2)\ncorresponds to the metallic phase in the clean limit.\nFor weak disorders the probability distribution has a\nmonomodal structure with a shaped peak at its most\nprobable value. In this regime the most probable value\nof the LDOS has a finite value, thus the system is in\nan extended state. This is an unambiguous evidence of/s48 /s53 /s49/s48 /s49/s53/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50/s48/s46/s49/s54\n/s97\n/s103\n/s109/s112/s118\n/s32/s32\n/s87\nFIG. 3: The most probable value ρmpv, arithmetic average ρa\nand geometric average ρgof the LDOS at the Fermi energy\nvia disorder strength Win the weak interaction case ( U= 2).\nthe existence of extended states in the two-dimensional\ndisordered interacting system. As the disorder strength\nincreases the peak broadens, and then a second peak is\ndeveloped. Thus the probability distribution has a bi-\nmodal structure. The high value mode is nearly fixed\nalmost independently on the disorder strength. The low\nvalue mode moves towardsto zerovalue. Actually, as the\ndisorder strength increases the low value mode becomes\nthe most probable value of the LDOS, and it approxi-\nmately vanishes at some value of disorder. The vanishing\nofthe most probablevalueofthe LDOS is detected in the\nsense that its value is below the density cutoff ρbe. The\nvanishing of the most probable value of the LDOS mani-\nfeststhe Andersonlocalization. Thusthe systemexhibits\na MIT from extended state to the Anderson localization\nas the disorder strength increases. Perhaps, the bimodal\nstructure of the probability distribution of the LDOS is\nduetospecialfeaturesoftheFKM.Thelowvaluemodeis\ndueto the Andersonlocalizationwhen disorderincreases.\nThe high value mode reflectsthe non-Fermi liquid behav-\nior of the FKM in the weak interaction regime. For weak\ninteractions the chemical potential remains pinned at the\neffectivelevelof flocalizedelectrons.27In the presenceof\ndisorder most of the LDOS still persist with the pinning\nproperty, thus most of the LDOS at the chemical poten-\ntial keep the same value. The bimodal structure may be\nabsent in the systems where the Fermi liquid properties\nare maintained.\nIn principle, we can determine the most probablevalue\nas the value at which the probability distribution reaches\nits global maximum. However, the probability distribu-\ntion is constructed by histogram, its most probable value\nis sensitive to the width of histogram bars. To avoid the\ninaccuracy, we determine the most probable value by the\nhalf sample mode algorithm.12This algorithm is a fast\nroutine for locating the most probable value of a finite\nstatistical sample. The half sample mode algorithm is\nbased on finding the smallest interval that contains half\nnumber of the sample points. The most probable value6\n/s48/s52/s48/s56/s48/s49/s50/s48\n/s48/s49/s48/s50/s48/s51/s48\n/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s48/s46/s48/s48 /s48/s46/s48/s53/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s32\n/s87/s61/s52\n/s32/s32/s32\n/s32/s87/s61/s53\n/s32/s32/s80/s40\n/s105/s41/s87/s61/s54\n/s32/s32 /s32/s87/s61/s49/s48\n/s32\n/s105/s32/s87/s61/s49/s49\n/s32\n/s105/s32 /s32/s87/s61/s49/s50\nFIG. 4: (Color online) Probability distribution P(ρi) of the\nLDOSρiat the Fermi energy for various disorder Win the\nintermediate interaction case ( U= 6).\nmust lie in the obtained half sample. Repeat this half\nsample procedure until obtain the half sample with two\nor three sample points. Then one can easily locate the\nmost probable value of the sample. In Fig. 3 we plot the\nmost probable value as well as the arithmetic and geo-\nmetric averageof the LDOS at the Fermi energy for com-\nparison. It shows as the disorder strength increases the\nmost probable value shifts from the higher value mode to\nlowervalueone. Afteracriticalvalueofdisorderstrength\n(Wc≈11.1) the most probable value approximately van-\nishes, thus the system changes to the Anderson localized\nphase. At the crossing point the most probable value ex-\nhibits a discontinuity. The most probable value of the\nLDOS is not an order parameter of MIT, since it does\nnot associate with any symmetry breaking, but the dis-\ncontinuity of the most probable value of the LDOS may\nbe considered as a sign of the first order phase transi-\ntion. Figure 3 also shows that both the arithmetic and\ngeometric average of the LDOS never coincide with the\nmostprobablevalueoftheLDOS.Thearithmeticandge-\nometric averages of the LDOS monotonously decrease as\nthe disorder strength increases. However, the decreases\ndo not necessarily imply the approach to the Anderson\nlocalization.13\nIn Fig. 4 we present the histogram for the probability\ndistribution of the LDOS at the Fermi energy for various\ndisorders and U= 6. This value of interaction ( U= 6)\ncorresponds to the insulator phase in the clean limit. In\ndifference to the weak interaction case, at weak disorders\nthe probabilitydistribution ofthe LDOS at the Fermi en-ergy has almost a delta function like structure. It means\nthat almost all LDOS at the Fermi energy vanish. The\ntotal DOS opens a gap at the Fermi energy. It is anal-\nogous to the Mott-Hubbard insulator in the clean limit.\nAs the disorder strength increases, the delta peak broad-\nens, and the probability distribution of the LDOS shows\na long tail. However, the most probable value of the\nLDOS still approximately vanishes, while the arithmetic\nand geometric average are finite. The total DOS now\ncloses the opened gap at the Fermi energy. The system\nstate is still localized, however gapless. It is analogous to\nthe Anderson localized phase in the noninteracting limit.\nWe still refer it to the Anderson localization, although\nthe physics nature may be different. In general, at finite\ninteraction and finite disorder there are no precise def-\ninitions of the Mott-Hubbard and Anderson insulating\nphases.10These phases rigorously exist only in the clean\nor noninteracting limits. In disordered interacting sys-\ntems, both phases are characterized by vanishing of the\nmost probable value of the LDOS at the Fermi energy.\nHowever, the total DOS of the Mott-Hubbard insulator\nopens a gap at the Fermi energy, while the one of the\nAnderson localization is gapless. The scenario of clos-\ning the opened gap by disorder in the strong interaction\ncase can be understood from the atomic limit.28In the\nclean atomic limit each site has two energy levels, one\nsingle occupancy and one double occupancy, separated\nby interaction strength U. At the half filling, each site\nis occupied either by one itinerant electron or by one\nlocalized electron. The double occupancy levels remain\nempty, thus the charge gap is equal to U. When disorder\nis added, each of these two energy levels is shifted by ran-\ndomly fluctuating site energy −W/2< εi< W/2. For\nW < U the situation remains unchanged, thus there is\nstill a charge gap for electron excitations. When W > U,\nthe double occupancy levels at some sites may be shifted\nlower than the single occupancy levels. Thus the double\noccupancy levels of a fraction of sites are either occu-\npied or empty. As a result the charge gap is closed. The\nphase may be interpreted as a mixture of Anderson and\nMott-Hubbard insulators.\nIn contrast to the weak interaction case, where the\nprobability distribution of the LDOS has the bimodal\nstructure, in the intermediate and strong interaction\ncases the probability distribution of the LDOS keeps its\nmonomodal structure. As the disorder strength increases\nfurther, the most probable value of the LDOS first in-\ncreases from zero value and then decreases back to zero\nvalue. In the first stage the system changes from the lo-\ncalized state to extended state, while in the second stage\nthe system changes back from the extended state to lo-\ncalized state. It is a reentry effect of the Anderson local-\nization. In Fig. 5 we plot the most probable value, the\narithmetic and geometric average of the LDOS at the\nFermi energy as a function of the disorder strength for\nU= 6. In difference to the weak interaction case, as the\ndisorder strength increases, the most probable value and\nthe averages of the LDOS increase from zero value, reach7\n/s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52\n/s97\n/s103\n/s109/s112/s118\n/s32/s32\n/s87\nFIG. 5: The most probable value ρmpv, arithmetic average ρa\nandgeometric average ρgof theLDOSat theFermi energyvia\ndisorder strength Win the strong interaction case ( U= 6).\ntheirmaximum, andthendecrease. Italsoshowsthatthe\ngeometric average of the LDOS approximately vanishes\nonly in the Mott-Hubbard insulator phase. However in\nthis phase the arithmetic average and the most proba-\nble value of the LDOS approximately vanish too. In the\nAnderson localized phase only the most probable value\nof the LDOS vanishes. In the region of intermediate val-\nues of the disorder strength, the most probable value of\nthe LDOS is finite. This evidence unambiguously indi-\ncates the existence of extended states for intermediate\ndisorders. It also shows that disorder can drive the sys-\ntem from the insulating state to extended one. However,\nthe insulating state should be the gapless localized state.\nDisorder cannot drive a Mott-Hubbard insulator directly\nto extended state. One may speculate the MIT scenario\nof the intermediate interaction case as a screening of dis-\norder which leads to close the gap at the Fermi energy,\nand then the standard scenario of the Anderson localiza-\ntion as in the weak interaction case. At the transition\npoint the most probable value also shows a discontinuity\nlike in the weak interaction case. We can use the dis-\ncontinuity to detect the crossing point from extended to\nlocalized states.\nIn Fig. 6 we plot the phase diagram. It clearly dis-\ntinguishes three phase regions. Extended phase is those\nstates that the most probable value of the LDOS at the\nFermi energy is finite. The insulator phase is charac-\nterized by the vanishing of the the most probable value\nof the LDOS at the Fermi energy. This phase is sep-\narated into the Anderson localization where the total\nDOS is gapless and the Mott-Hubbard insulator where\nthe total DOS opens a gap at the Fermi energy. The\nAnderson localization occurs for strong disorder, while\nthe Mott-Hubbard insulator occurs for strong correla-\ntion. The extended phase appears only for weak and\nintermediate disorder and correlation. For a weak dis-\norder, as the interaction increases, the system changes\nfrom the extended phase to the Anderson localization,\nand finally to the Mott-Hubbard insulator. For an in-/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s32/s32/s87\n/s85/s101/s120/s116/s101/s110/s100/s101/s100/s32/s112/s104/s97/s115/s101\n/s77/s111/s116/s116/s45/s72/s117/s98/s98/s97/s114/s100\n/s32/s32/s32/s32/s32/s32/s32/s112/s104/s97/s115/s101/s65/s110/s100/s101/s114/s115/s111/s110/s32/s108/s111/s99/s97/s108/s105/s122/s97/s116/s105/s111/s110\nFIG. 6: Phase diagram for the Mott-Hubbard-AndersonMIT.\nThe dotted line separates the phase region with finite gap in\nthe total DOS.\ntermediate value of disorder, as the interaction increases,\nthe system changesfrom the Anderson localizationto the\nextended phase, and then back again to the Anderson lo-\ncalization. This is a reentry effect of the Anderson local-\nization. For a weak interaction as the disorder strength\nincreases the system changes from extended to localized\nphase. For intermediate and strong interactions there is\na crossover from the Mott-Hubbard insulator to the An-\nderson insulator by closing the gap at the Fermi energy\nby disorder. For a fixed intermediate interaction there is\nalso the reentry effect of the Anderson localization when\nthe disorder strength is varied. The phase diagram is\nqualitatively analogous to the one calculated within the\nTMT.11Although the geometric mean alone does not in-\ndicate the Anderson localization, its fully embedding in\nthe self consistent cycle of the DMFT may describe the\nAnderson localization.9,10,11However, within the statis-\ntical DMFT there is a discontinuity of the most proba-\nble value of the LDOS at the phase boundary, whereas\nwithin the TMT the geometric average is continuous at\nthe phase boundary. The two limiting cases W= 0 and\nU= 0 are special. In the clean limit W= 0, there is\nthe Mott-Hubbard MIT, although the metallic phase is\nnot a Fermi liquid. The noninteracting limit U= 0 is\ncontroversial in two dimension. Our result agrees well\nwith the real space renormalization group calculations.29\nHowever,scalingtheorydid notfind anytruemetallicbe-\nhaviorin the system.30There is only a crossoverfrom ex-\nponentially to logarithmicallylocalized states. Certainly,\nthe present phase diagram is constructed from statistics\nof the LDOS, and the actual transport properties still\nremain unclear. Nevertheless, it was demonstrated that\nthe two dimensional disordered Hubbard model can have\na delocalizing effect.31,32,33,348\n/s48/s46/s48/s55/s48/s48/s46/s48/s55/s53\n/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48\n/s32/s32/s97\n/s32/s87/s61/s49/s48/s46/s53\n/s32/s87/s61/s49/s49/s46/s53\n/s32\n/s32/s78\n/s100/s32\n/s109/s112/s118\nFIG. 7: The statistical average of the most probable value\n(mpv) and the arithmetic average (a) of the LDOS at the\nFermi energy via the number of disorder realizations. The\nerror bars are their standard deviations. ( U= 2,L= 12,\nNb= 5).\nC. Finite size effects\nThe results of the previous subsections basically per-\nmit finite size effects. There are two sources of the finite\nsize effects. One is the finite size Lof the lattice, and\nthe second is the finite number Ndof disorder realiza-\ntions. First, we study the finite size effects of Nd. We\nfix the lattice size L= 12, and consider different num-\nbers of disorder realizations. For each Ndwe generate\nNbdifferent bins of size L2Ndfor disorder realizations.\nFor each bin we calculate the most probable value as well\nas the arithmetic average of the LDOS at the Fermi en-\nergy. Then we calculate their statistical average ραand\nstandard deviation σα, i.e.,\nρα=1\nNbNb/summationdisplay\nn=1ρ(n)\nα,\nσ2\nα=1\nNb−1Nb/summationdisplay\nn=1/parenleftbig\nρ(n)\nα−ρα/parenrightbig2,\nwhereαdenotes the most probable value (mpv) or the\narithmetic average (a). ρ(n)\nαis the most probable value\nor the arithmetic average of the LDOS, obtained from\nthe nth bin calculations. In Fig. 7 we plot the statistical\naverage and the standard deviation of the most proba-\nL810121416\nNd2401501108060\nN1568015000158401568015360\nTABLE I: The lattice size L, the number of disorder realiza-\ntionsNd, and the total number N, which are used for study\nof the finite size effects./s48/s46/s48/s54/s53/s48/s46/s48/s55/s48/s48/s46/s48/s55/s53\n/s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48/s32/s87/s61/s49/s48/s46/s53\n/s32/s87/s61/s49/s49/s46/s53\n/s32/s32/s97\n/s32\n/s32/s76/s32\n/s109/s112/s118\nFIG. 8: The most probable value ρmpvand the arithmetic\naverageρaof the LDOS at the Fermi energy via the lattice\nsizeL. The number of disorder realizations for each lattice\nsize is given in Table I. ( U= 2).\nble value or the arithmetic average of the LDOS at the\nFermi energy for various NdandNb= 5. We choose\ntwo values of disorder nearby the transition point. One\nis in the extended phase, and the other is in Anderson\nlocalized phase. Figure 7 shows the arithmetic average\nof the LDOS (the total DOS) has little fluctuations even\nforNd= 50. It almost independent on Ndalready from\nNd= 50. The most probable value of the LDOS fluctu-\nates in the extended phase more strongly than in the lo-\ncalized phase. Certainly, in the localized phase the most\nprobablevalueoftheLDOSapproximatelyvanishes, that\nthe fluctuations of the vanishing value is negligible when\nNdvaries. The most probable value of the LDOS seems\nto be reasonable from Nd= 100. Thus at Nd= 100 the\nfinite size effects of Ndare small and do not significantly\nchange the results.\nNext, we study the finite size effects of the lattice size.\nThe DMFT calculations for clean systems show the fi-\nnite site effects are small and controllable.8In disordered\nsystems, one can notice that the finite size effects of the\nensemble of the LDOS depend mostly on the size of the\nensemble, i.e., on the number N=L2Nd.Nis the total\nnumber of LDOS which are obtained in numerical cal-\nculations. Therefore to study the finite size effects of\nLalone, when the lattice size Lis varied, we have to\nchangeNdaccordingly, that the total number Nkeeps\nmore or less the same value. In Table I we present sev-\neral values of L, and corresponding values of Ndthat the\ntotal number Nis around 15000. We use the parame-\nter values in Table I for study of the finite size effects of\nL. For all cases the whole bin of disorder realizations is\nkept more or less the same. We also choose two values\nof disorder nearby the transition point. One is in the\nextended phase, and the other is in the localized phase.\nThe most probable value and the arithmetic average of\nthe LDOS at the Fermi energy via the lattice size are9\nplotted in Fig. 8. It shows that the arithmetic average\nof the LDOS is almost independent on the lattice size,\nat least from L= 8. The most probable value of the\nLDOS also slightly fluctuates as the lattice size varies.\nThese fluctuations mainly are due to the statistical fluc-\ntuations of finite size of disorder realizations. As we al-\nready showed in Fig. 7, the statistical fluctuations of the\nmost probable value of the LDOS in the extended phase\nis larger than in the localized phase. This feature is con-\nsistent with Fig. 8, where the most probable value of the\nLDOS fluctuates in the extended phase stronger than in\nthe localized phase. Both results show that the finite size\neffects in our study are small and do not change signifi-\ncantly the picture of MIT. It is interesting to note that\nthe DMFT can be performed for finite size lattices, and\nthe obtained results are not significantly different from\nthe ones of the thermodynamical limit.8\nIV. CONCLUSIONS\nIn this paper we have studied the Mott-Hubbard-\nAnderson MIT in the two-dimensional FKM with local\ndisorder by the statistical DMFT. Within the statisti-\ncal DMFT the correlation effects are resulted into addi-\ntional dynamical local random variables, which are self-\nconsistently determined from the local single site dynam-\nics. The probability distribution and the most probable\nvalue of the LDOS are calculated. The localized phase\nis detected by vanishing condition of the most probable\nvalue of the LDOS at the Fermi energy. The scenario\nof the MIT in the system depends on the interaction.\nFor weak interactions, which correspond to the metallic\nphase in the clean limit, the system changes from ex-\ntended to localized states as the disorder strength in-\ncreases. For intermediate interactions, which correspond\nto the insulating phase in the clean limit, as the disor-\nder strength increases the system crosses from the Mott-\nHubbard to the Anderson insulator, and then it transits\nto extended states and goes back again to the Anderson\nlocalized phase. Thus there is a reentrance of the An-\nderson localization. At the crossing point from extended\nto localized states the most probable value of the LDOS\nexhibits a discontinuity. For strong interactions only lo-\ncalized states exist. There is only a crossover from the\nMott-Hubbard to the Anderson insulator by closing the\nopened gap at the Fermi energy by disorder. The results\nalsoconfirmthedelocalizingeffect inthetwodimensional\ndisordered interacting system. However, the phase dia-\ngramwasdetermined only by statistics ofthe LDOS, and\nthe actual transport properties of the system still remain\nunclear. We leave the problem for further study.\nAcknowledgments\nThe author would like to thank the Asia Pacific Cen-\nter forTheoreticalPhysicsfor the hospitality. He alsoac-knowledgesuseful discussionswithHanyonChoiandJae-\njun Yu. The author is grateful to thank the Max Planck\nInstitute for the Physics of Complex Systems at Dresden\nfor sharing computer facilities where the numerical cal-\nculations were performed. This work was supported by\nthe Asia Pacific Center for Theoretical Physics, and by\nthe Vietnam National Program on Basic Research.\nAPPENDIX A: SELF CONSISTENT EQUATION\nOF THE INHOMOGENEOUS DMFT IN\nINFINITE DIMENSIONS\nIn this Appendix we present the derivation of the self\nconsistent equation of the inhomogeneous DMFT in infi-\nnite dimensions. Formally, we can derive the self consis-\ntentequationforinhomogeneoussystemsinthesameway\nas for homogeneous systems.5Certainly, the derivation\nfor homogeneous systems is based on the cavity method\nand the Hilbert transform.5Since the cavity method for\nhomogeneoussystems is also formulated in real space, we\ncan follow it closely. First a disorder realization is fixed,\nand then all fermions are traced out except for a single\nsitel. In the infinite dimensions we obtain the Green\nfunction which represents the effective mean field5\nG−1\nl(iωn) =iωn+µ−εl−∆l(iωn),(A1)\n∆l(iωn) =/summationdisplay\nijtliG(l)\nij(iωn)tjl, (A2)\nwhereG(l)\nij(iωn) is the Green function of the model with\nsitelremoved. ∆ l(iωn) can be considered as a hybridiza-\ntion function. The cavity Green function can be ex-\npressed through the original lattice Green function5\nG(l)\nij(iωn) =Gij(iωn)−Gil(iωn)Glj(iωn)\nGll(iωn).(A3)\nInserting Eq. (A3) into Eq. (A2) one obtains\n∆l(iωn) =/bracketleftbig\nt·G(iωn)·t/bracketrightbig\nll\n−/bracketleftbig\nt·G(iωn)/bracketrightbig\nll/bracketleftbig\nG(iωn)·t/bracketrightbig\nll\nGll(iωn),(A4)\nwheretisthehoppingmatrix. ThelatticeGreenfunction\ncan be rewritten as\nG(iωn) =/bracketleftbig\nξ(iωn)−t−Σ(iωn)/bracketrightbig−1,(A5)\nwhereξij(iωn) = (iωn+µ−εi)δij. In infinite dimensions\nthe self energy is purely local,5hence the self energy ma-\ntrixΣ(iωn) is diagonal. For homogeneous systems the\nterms in the right hand side of Eq. (A4) are calculated\nby using the Fourier and Hilbert transforms.5For inho-\nmogeneous systems the Fourier and Hilbert transforms\nare replaced by the matrix multiplication and inversion\nin real space. One can notice that\nt·G·t=t·G·/bracketleftbig\nξ−Σ−G−1/bracketrightbig10\n=t·G·/parenleftbig\nξ−Σ/parenrightbig\n−t, (A6)\nt·G=/bracketleftbig\nξ−Σ−G−1/bracketrightbig\n·G\n=/parenleftbig\nξ−Σ/parenrightbig\n·G−1. (A7)\nUsing relations (A6)-(A7), from Eq. (A4) we obtain\n∆l(iωn) =−/bracketleftbig\nξ(iωn)−Σ(iωn)/bracketrightbig\nll\n+/bracketleftBig/parenleftbig\nξ(iωn)−Σ(iωn)/parenrightbig\n·G(iωn)·/parenleftbig\nξ(iωn)−Σ(iωn)/parenrightbig/bracketrightBig\nll\n−/bracketleftbig/parenleftbig\nξ(iωn)−Σ(iωn)/parenrightbig\n·G(iωn)−1/bracketrightbig\nll/bracketleftbig\nG(iωn)·/parenleftbig\nξ(iωn)−Σ(iωn)/parenrightbig\n−1/bracketrightbig\nll/Gll(iωn).(A8)\nHere we have used tll= 0. Since both matrices ξ(iωn)\nandΣ(iωn) are diagonal, from Eqs. (A1) and (A8) we\nobtain\nG−1\nl(iωn) = Σl(iωn)+1/Gll(iωn).(A9)Equation(A9)isjusttheselfconsistentequations(5)-(6).\nIt is exact in infinite dimensions for any inhomogeneous\nsystem. The above derivation of the self consistent equa-\ntion can be considered as a generalization of the homo-\ngeneous one. In infinite dimensions, the sum in Eq. (A2)\nruns over infinitely many neighbouring sites, that the hy-\nbridization function is replaced by its average value. As\na consequence, all spatial fluctuations ofthe environment\nsurrounding a given site are suppressed, thus the Ander-\nson localization is prohibited.7At infinite dimensions the\nAnderson localization is absent in any disordered inter-\nacting system.\n1N. F. Mott, Metal-Insulator Transition , 2nd ed. (Taylor\nand Francis, London, 1990).\n2P. W. Anderson, Phys. Rev. 109, 1492 (1958).\n3P. W. Anderson, Rev. Mod. Phys. 50, 191 (1978).\n4R. Abou-Chacra, P. W. 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Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).\n31P. J. H. Denteneer, R. T. Scalettar, and N. Trivedi, Phys.\nRev. Lett. 83, 4610 (1999).\n32I. F. Herbut, Phys. Rev. B 63, 113102 (2001).\n33B. Srinivasan, G. Benenti, and D. L. Shepelyansky, Phys.\nRev. B67, 205112 (2003).\n34D. Heidarian and N. Trivedi, Phys. Rev. Lett. 93, 126401\n(2004)."    },    {        "title": "0801.3131v1.Density_induced_suppression_of_the_alpha_particle_condensate_in_nuclear_matter_and_the_structure_of_alpha_cluster_states_in_nuclei.pdf",        "content": "arXiv:0801.3131v1  [nucl-th]  21 Jan 2008Density-induced suppression of the α-particle condensate in nuclear matter and the\nstructure of αcluster states in nuclei\nY. Funaki1, H. Horiuchi2, G. R¨ opke3, P. Schuck4,5, A. Tohsaki2, T. Yamada6\n1Nishina Center for Accelerator-Based Science, The Institu te of Physical\nand Chemical Research (RIKEN), Wako, Saitama 351-0198, Jap an\n2Research Center for Nuclear Physics (RCNP), Osaka Universi ty, Osaka 567-0047, Japan\n3Institut f¨ ur Physik, Universit¨ at Rostock, D-18051 Rosto ck, Germany\n4Institut de Physique Nucl´ eaire, 91406 Orsay Cedex, France\n5Universit´ e Paris-Sud, F-91406 Orsay-C´ edex, France and\n6Laboratory of Physics, Kanto Gakuin University, Yokohama 2 36-8501, Japan\nAt low densities, with decreasing temperatures, in symmetr ic nuclear matter α-particles are\nformed, which eventually give raise to a quantum condensate with four-nucleon α-like correlations\n(quartetting). Starting with a model of α-matter, where undistorted αparticles interact via an\neffective interaction such as the Ali-Bodmer potential, the suppression of the condensate fraction at\nzero temperature with increasing density is considered. Us ing a Jastrow-Feenberg approach, it is\nfound that the condensate fraction vanishes near saturatio n density. Additionally, the modification\nof the internal state of the αparticle due to medium effects will further reduce the conden sate.\nIn finite systems, an enhancement of the Sstate wave function of the c.o.m. orbital of αparticle\nmotion is consideredas thecorrespondence tothecondensat e. Wavefunctions havebeenconstructed\nfor self-conjugate 4 nnuclei which describe the condensate state, but are fully an tisymmetrized on\nthe nucleonic level. These condensate-like cluster wave fu nctions have been successfully applied to\ndescribe properties of low-density states near the nαthreshold. Comparison with OCM calculations\nin12C and16O shows strong enhancement of the occupation of the S-state c.o.m. orbital of the\nα-particles. This enhancement is decreasing if the baryon de nsity increases, similar to the density-\ninduced suppression of the condensate fraction in αmatter. The ground states of12C and16O show\nno enhancement at all, thus a quartetting condensate cannot be formed at saturation densities.\nI. INTRODUCTION\nThe properties of nuclear matter at very low densities\nand low temperatures are dominated by the formation\nof clusters, in particular αparticles. As a well-known\nconcept,αmatter has been introduced where symmetric\nnuclear matter is described by a system of αparticles,\nweakly interacting via effective α-αpotentials fitted\nto the scattering phase shifts, such as the Ali-Bodmer\ninteraction potential [1, 2, 3].\nThis concept becomes less valid with increasing den-\nsity. First, at finite temperatures other correlations and\nalso single nucleon states appear so that we have a mix-\nture of different constituents, described in chemical equi-\nlibrium by a mass action law. Secondly, at higher densi-\nties the internal fermionic structure of the αparticles be-\ncomes of relevance so that the four-nucleon bound state\nwill be modified by medium effects. A consistent ap-\nproach can be given by quantum statistical methods [4].\nUsingthermodynamic Greenfunctions, the effects ofself-\nenergyand Pauli blockingare included so that the bound\nstates are dissolved when the density exceeds a critical\nvalue. For αparticles this critical density, which is also\ndependent on temperature, is about ρ0/5, withρ0= 0.17\nfm−3as the saturation density [5].\nAn important phenomenon is the formation of a quan-\ntum condensate with strong four nucleon correlations at\nlow temperatures [6]. At low densities where αparticles\nare well defined weakly interacting constituents of sym-\nmetric nuclear matter, we have Bose-Einstein condensa-tion ofαparticles. With increasing density, quartetting\noccurs with medium-modified αparticles and disappears\nat a density of about ρ0/3. Note that quartet condensa-\ntion has recently also been considered in the context of\ncold atom physics [7].\nThe Bose-Einstein condensation for ideal quantum\ngases is a well-known phenomenon. The occupation of\nsingle-particle states is given by the Bose distribution\nfunction. Below a critical temperature Tc, to obey nor-\nmalization, the state of lowest energy is macroscopically\noccupied. This macroscopically enhanced coherent occu-\npation of the lowest quantum state is denoted as quan-\ntum condensate. As well known, the fraction of bosons\nfound in the condensate results for the ideal Bose gas as\nncond/n= 1−(T/Tc)3/2.\nHowever, this simple picture is no longer valid, if inter-\naction is taken into account. For a recent determination\nofTcin the interacting case, see Ref. [8]. Here, we want\nto concentrate on interaction effects at zero temperature.\nIn general, the condensate fraction is given by the prop-\nerties of the density matrix which contains a part which\nfactorizes. According to Penrose and Onsager [9], the\nquantum condensate in a homogeneousinteracting boson\nsystem at zero temperature is given by the off-diagonal\nlong-rangeorderinthe densitymatrix. Thenon-diagonal\ndensity matrix in coordinate representation can be fac-\ntorized so that in the limit |r−r′| → ∞follows\nlim\n|r−r′|→∞ρ(r,r′) =ψ∗\n0(r)ψ0(r′)+γ(r−r′).(1)\nThe last contribution γ(r) disappears at large distances,2\nwhereas the first contribution determines the condensate\nfraction in infinite matter as\nn0=∝an}bracketle{tΨ|a†\n0a0|Ψ∝an}bracketri}ht\n∝an}bracketle{tΨ|Ψ∝an}bracketri}ht. (2)\nExploratory calculation of the condensate fraction of α\nmatterwillbegiveninthefollowingSec.II.Incontrastto\nRef. [6, 8] where the transition temperature Tcfor quar-\ntetting was considered, we considerhere the zerotemper-\naturecaseand analyzethe groundstatewavefunction. It\nwill be shown that due to the interaction, the condensate\nfraction is suppressed with increasing density.\nAn important question is whether such properties of\ninfinite nuclear matter are of relevance for finite nuclei.\nAs well known, e.g., pairing obtained in nuclear matter\nwithin the BCS approach is also clearly seen in finite nu-\nclei. Nuclei with densities near the saturation density are\nwell described by the quasiparticle picture which leads to\nthe shell model for finite nuclei. At low densities, a fully\ndevelopedαcluster structure similar to αmatter is ex-\npected. Cluster structures in finite nuclei have been well\nestablished. A density functional approach is able to in-\nclude correlations and to bridge between infinite matter\nand finite nuclei.\nAn interesting aspect of finite nuclei is the enhance-\nment of the occupation of single α-particle states sim-\nilar to Bose-Einstein condensation in α-particle matter\nor condensation of bosonic atoms in traps. Recently,\ngas-like states have been investigated in self-conjugate\n4nnuclei [10], and a special ansatz for the wave function\n(THSR ansatz), which is similar to the condensate state\nin infinite matter, has been shown to be appropriate in\ndescribing low-density isomers. In particular,8Be and\nthe Hoyle state of12C are well described with this THSR\nwave function. Investigation of states near the four α\nthreshold in16O is in progress [11, 12]. Predictions for\n20Ne have been given in [13].\nIn Sect. III, we will explain how the suppression of\nthe condensate fraction, calculated for infinite nuclear\nmatter, is also seen in the low-density isomers of self-\nconjugate 4 nnuclei, in particular for n= 3 (12C). First\nresults forn= 4 (16O) are also given. General conclu-\nsions are drawn in Sec. IV.\nII. SUPPRESSION OF CONDENSATE\nFRACTION IN αMATTER AT ZERO\nTEMPERATURE\nThetheoryofPenroseandOnsager[9]wasfirstapplied\nto a system with hard core repulsion. Depending on the\nfilling factor, the suppression of the condensate was cal-\nculated. In particular, for liquid4He with a filling factor\nof 28% at normal conditions, the condensate fraction is\nreduced to ≈8% in good agreement with experimental\nobservations. To give an estimation for αmatter, with\nan “excluded volume” of about 20 fm3[14], such a fillingfactor of 28 % would arise at ≈ρ0/3 so that a substan-\ntial reduction of the condensate fraction already below\nsaturation densities is expected for αmatter.\nWithinamoresystematicapproach,wefollowthework\nofClarketal. [15]. Wecalculatethereductionofthecon-\ndensate fraction as function of the baryon density within\nperturbation theory. A uniform Bose gas of αparticles,\ninteracting via the potential Vα(r), is considered, disre-\ngardingany change of the internal structure of the αpar-\nticles at increasing density. In particular, the dissolution\nof theαparticle as a four-nucleon bound state because\nof the Pauli blocking is not taken into account.\nThe simplest form of a trial wave function incorpo-\nrating the strong spatial correlations implied by the\ninteraction potential is the familiar Jastrow choice,\nψ(r1,...,rA) =/producttext\ni<jf(|ri−rj|). Within our ex-\nploratory calculation we consider the lowest approxima-\ntion with respect to the density in order to show the ten-\ndency of condensate suppression due to the interaction.\nNormalization gives for the variational function the con-\nstraint\n4πρα/integraldisplay∞\n0[f2(r)−1]r2dr=−1, (3)\nρα=ρ/4 being the density of αparticles .\nIn the low density limit, the binding energy per α-\nparticle is given by\nE[f] = 2πρα/integraldisplay∞\n0/braceleftBigg\n/planckover2pi12\n4M/parenleftbigg∂f(r)\n∂r/parenrightbigg2\n+Vα(r)f2(r)/bracerightBigg\nr2dr,\n(4)\nMbeing the nucleon mass. The condensate fraction is\ncalculated according to\nn0= exp/braceleftbigg\n−4πρα/integraldisplay∞\n0[f(r)−1]2r2dr/bracerightbigg\n.(5)\nNote that these approximations[15] only hold in the low-\ndensity limit. At higher densities, the pair correlation\nfunction has to be evaluated. A more advanced approach\nbased on a HNC calculation has been given by Clark,\nRistig and others, see [15, 16].\nFor the evaluation of the condensate fraction (5) we\nuse the Ali-Bodmer α-αinteraction potential [2]\nVα(r) = 457e−(0.7r/fm)2MeV−130e−(0.475r/fm)2MeV.\n(6)\nAccording to Johnson and Clark [15] we choose the vari-\national function as\nf(r) = (1−e−ar)(1+be−ar+ce−2ar).(7)\nAfter determining the parameters a,b,cfrom the mini-\nmum of energy [17], the condensate fraction can be eval-\nuated, see Fig. 1.\nIn Fig. 1, the full line represents the result for the\ncondensate fraction as function of the baryonic density\naccording to the perturbative treatment. In the zero3\n㪇㪅㪇 㪇㪅㪉 㪇㪅㪋 㪇㪅㪍 㪇㪅㪏 㪈㪅㪇 㪈㪅㪉㪇㪅㪇㪇㪅㪉㪇㪅㪋㪇㪅㪍㪇㪅㪏㪈㪅㪇\n㪁\n㩷㩷㪺㫆㫅㪻㪼㫅㫊㪸㫋㪼㩷㪽㫉㪸㪺㫋㫀㫆㫅\nU㪆U\u0013㪁\n\nFIG. 1: Reduction of condensate fraction in αmatter with\nincreasing baryon density ( ρ0denotes the saturation density).\nFull line - perturbation approach, crosses - HNC calculatio ns\nby Johnson and Clark [15], stars - Hoyle state (see Sec. III).\ndensity limit this fraction is expected to go to 1. Cal-\nculations performed by Johnson and Clark [15] using a\nHNC calculation for the pair distribution function are\ngiven by crosses, showing a stronger suppression of the\ncondensate fraction near the saturation density.\nAs found from the calculation of the critical tempera-\nture for the formation of a quartetting condensate [6], we\nexpect that near the saturation density the condensate\nfraction will disappear. For this, we have not only to\ntake into account the HNC type improvement of the pair\ndistribution function, but also the Pauli blocking effects\nwhich modify the internal structure of the αparticle so\nthat the use of the Ali-Bodmer interaction potential is\nno longer justified. Recently, improved versions of the\nα-αinteraction have been proposed [18]. Three- αforces\nhave been considered in order to give a better estimation\nfor the critical point of αmatter which should be posi-\ntioned below saturation density [19]. Thus, the repulsive\npart of the α-αinteraction (which also is a consequence\nof the Pauli blocking with respect to the internal nucle-\nonic structure) is only a part of the suppression of the\ncondensate, which is described here.\nAnother effect is the medium modification of the inter-\nnal structure ofthe αparticle aswell as ofthe interaction\nwhich can be elaborated within a cluster-mean field ap-\nproximation [4]. The dissolution of α-like bound states\ndue to Pauli blocking has been evaluated for an uncorre-\nlated medium solving the Faddeev-Yakubowsky equation\n[5]. It has been shown [6] that the four-particle correla-\ntions in the condensate disappear due to Pauli block-\ning at around ρ0/3 within a variational approach, ap-\nproximating the 4-nucleon wave function by the solution\nof the two-particle problem and describing the relative\nc.o.m. motion by a Gaussian wave function. Therefore,a medium dependent α-αinteraction of the Ali-Bodmer\ntype may be expected to account for the features of this\neffect in an exploratory way. In principle, an ab ini-\ntiocalculation based on interacting nucleons should be\nperformed, with Green functions, variational, or AMD\ntechniques.\n  !\" !# !$ !% !& !' !( !)\n*+,-./012324 5,6740123248\"0\n9\"\n*\"\n8#\n9#\n*#\n8$\n9$\n*$\nFIG. 2: Occupation of the single- αorbitals of the ground\nstate of12C compared with the Hoyle state [28]. For expla-\nnation see the text.\n㪇㪅㪇 㪇㪅㪉 㪇㪅㪋 㪇㪅㪍 㪇㪅㪏 㪈㪅㪇 㪈㪅㪉㪇㪅㪇㪇㪅㪉㪇㪅㪋㪇㪅㪍㪇㪅㪏㪈㪅㪇\n㩷㩷\n㪪㪄㫆㫉㪹㫀㫋㩷㫇㫉㫆㪹㪸㪹㫃㫀㫋㫐\nU㪆U\u0013\nFIG. 3: Occupation of the S1 orbital as function of density\nusing the 3 αOCM [28].\nIII. ENHANCEMENT OF CLUSTER C.O.M. S\nORBITAL OCCUPATION IN 4 nNUCLEI\nSignatures akin to Bose-Einstein condensation should\narise already in finite nuclei. Low-density states of self-\nconjugate 4 nnuclei clearly show an αcluster structure,4\nin particular for n= 2 andn= 3 (Hoyle state). The\ncounterpart of a condensate in infinite αmatter, where\nthe occupation of the ground state is enhanced and be-\ncomes of the same order as the total particle number,\nwill be the enhancement of the occupation number of a\nsingle-αorbital of the α-clusters in a low density state of\nthe nucleus.\nTheαclustering nature of the nucleus12C has been\nstudied by many authors using various approaches [20].\nAmong these studies, solving the fully microscopic three-\nbody problem of αclusters gives us the most important\nand reliabletheoreticalinformation of αclustering in12C\nwithin the assumption that no αcluster is distorted or\nbroken except for the change of the size parameter of the\nαcluster’s internal wave function. First solutions of the\nmicroscopic 3 αproblem where the antisymmetrization of\nnucleons is exactly treated, have been given by Uegaki et\nal. [21] and by Kamimura et al. [22]. In those works, the\n12C levels are described by the wave function of the form\nA{χ(s,t)φ3\nα}withAstanding for the antisymmetrizer,\nφ3\nα≡φ(α1)φ(α2)φ(α3) for the product of the internal\nwave functions of three αclusters, and sandtfor the\nJacobi coordinates of the center-of-mass motion of three\nαclusters. Here φ(αi) (i= 1,2,3) is the internal wave\nfunction of the α-clusterαihaving the form φ(αi)∝\nexp[−(1/8b2)/summationtext4\nm>n(ri,m−ri,n)2]. The wave function\nχ(s,t) of the relative motion of 3 αclusters is obtained\nbysolvingtheenergyeigenvalueproblemofthefullthree-\nbody equation of motion; ∝an}bracketle{tφ3\nα|(H−E)|A{χ(s,t)φ3\nα}∝an}bracketri}ht=\n0, whereHis the microscopic Hamiltonian consisting of\nthe kinetic energy, effective two-nucleon potential, and\nthe Coulomb potential between protons. The difference\nbetween the works by Uegaki et al. and Kamimura et\nal. lies in the adopted effective two-nucleon force, besides\nthe differing techniques of solution.\nBoth calculations by Uegaki et al. and Kamimura et\nal. reproduced reasonably well the observed binding en-\nergy and r.m.s. radius of the ground 0+\n1state which is\nthe state with normal density, while they both predicted\na very large r.m.s. radius for the second 0+\n2state which\nis larger than the r.m.s. radius of the ground 0+\n1state\nby about 1 fm, i.e. by over 30%. The observed 0+\n2state\nlies slightly above the 3 αbreakup threshold and the en-\nergies of the calculated 0+\n2state reproduced reasonably\nwell the observed value although the value by Uegaki et\nal. is slightly higher than the 3 αbreakup threshold by\nabout 1 MeV. The second 0+state of12C is well known\nas the key state for the synthesis of12C in stars (Hoyle\nstate) and also as one of the typical mysterious 0+states\nin light nuclei which are very difficult to understand from\nthe point of view of the shell model [23].\nAlternatively, the 0+\n2state with dilute density can be\ndescribed by a gas-likestructure of 3 α-particles which in-\nteract weakly among one another, predominantly in rel-\nativeSwaves. The S-wave dominancy in the 0+\n2state\nstructure had been already suggested by Horiuchi on the\nbasisofthe3 αOCM(orthogonalityconditionmodel)cal-\nculation [24]. It should be mentioned that not only thebinding energy, but also other properties of the 0+\n2state\nsuchaselectronscatteringformfactorsarewelldescribed\nwithin the calculations given in Refs. [21, 22, 24].\nRecently, based on the investigations of the possibility\nofα-particle condensation in low-density nuclear mat-\nter [6], the present authors proposed a conjecture that\nnear thenαthreshold in self-conjugate 4 nnuclei there\nexist excited states of dilute density which are composed\nof a weekly interacting gas of self-bound αparticles and\nwhich can be considered as an nαcondensed state [10].\nThis conjecture was backed by examining the structure\nof12Cand16Ousinganew α-cluster wavefunction ofthe\nα-cluster condensate type. The new α-cluster wave func-\ntion, which will be denoted as THSR wave function, ac-\ntually succeeded to place a level of dilute density (about\none third of saturation density) in each system of12C\nand16O in the vicinity of the 3 respectively 4 αbreakup\nthreshold, without using any adjustable parameter. In\nthe case of12C, this success of the new α-cluster wave\nfunction mayseemrathernatural, asweexplained above.\nThemicroscopic3 αclustermodelshadpredictedthatthe\n0+\n2in the vicinity of the 3 αbreakup threshold has a gas-\nlike structure of 3 α-particles which interact weakly with\neach other predominantly in relative Swaves. Having\nput forward that Hoyle like states in 4 nself-conjugate\nnuclei may be a general and common phenomenon is the\nmerit of the work in [10].\nThe THSR wave function of the α-cluster condensate\ntype used in Ref. [10] represents a condensation of α-\nclusters in a spherically symmetric state. This is clearly\nseen by the following expression\n|Ψ∝an}bracketri}ht=P(C†\nα)n|vac∝an}bracketri}ht, (8)\nwith\n∝an}bracketle{t1234|C†\nα|vac∝an}bracketri}ht= Φ(P)δP,p1+p2+p3+p4φα(1234)a†\n1a†\n2a†\n3a†\n4,\n(9)\nΦ(P) describing the c.o.m. motion of the αcluster, and\nφthe internal wave function of the four-nucleon cluster.\nThe operator Pis projecting out the total c.o.m. motion\nof the 4nnucleus. In the limit of infinite nuclear matter,\nthe Φ orbitals are plane waves, and the projection op-\neratorPcan be neglected. In the case considered here,\nthe use of Gaussians allows the explicit separation of the\nc.o.m. motion of the four-nucleon cluster as well as of\nthe whole 4 nnucleus. It should also be noted that Eq.\n(8) contains two limits exactly: the one of a pure Slater\ndeterminant relevant at higher densities and the one of\na 100 percent ideal α-particle condensate in the dilute\nlimit [10]. All intermediate scenarios are also correctly\ncovered.\nThe present authors extended the wave function so\nthat it can describe the α-cluster condensate with spatial\ndeformation [25]. They applied this new wave function\nto8Be and succeeded to reproduce not only the bind-\ning energy of the ground state but also the energy of\nthe excited 2+state. In addition, they found that al-\nthough the effect of the spatial deformation is not large,5\nthe introduction of the spatial deformation brought forth\na 100 % overlap of the THSR wave function with the\n“exact” wave function given by the microscopic 2 αclus-\nter model which solves the 2 α-cluster equation of mo-\ntion,∝an}bracketle{tφ2\nα|(H−E)|A{χ(r)φ2\nα}∝an}bracketri}ht= 0. This fact forces us\nto modify our understanding of the8Be structure from\nthe 2α“dumb-bell” structure to the 2 αdilute (gas-like)\nstructure. It was shown that the 0+\n2wave function of\n12C which was obtained long time ago by solving the full\nthree-body problem of the microscopic 3 αcluster model\nis almost completely equivalent to the wave function of\nthe3αTHSRstate. Thisresultgivesusstrongsupportto\nour opinion that the 0+\n2state of12C has a gas-like struc-\nture of 3αclusters with “Bose-condensation”. The rms\nradius for this THSR state was calculated as R(0+\n2)THSR\n= 4.3 fm which fits well with experimental data for the\nform factor of the Hoyle state, see Ref. [26]. It confirms\ntheassumptionoflowdensityasaprerequisiteforthefor-\nmation of an α-cluster structure for which the Bose-like\nenhancement of the occupation of the Sorbit is possible.\nRecently, a fermionic AMD calculationbased on nucle-\nons with effective interactions has been performed [26]\nwhich supports the applicability of the THSR state to\ndescribe the Hoyle state. It is found hat the form fac-\ntor calculated for the 0+\n2state of12C coincides with the\nform factor obtained from the THSR wave function. In\nparticular, the low density of nucleons, the formation of\nfour-nucleon clusters and the dominant contribution of\nthe gas-like distribution has been confirmed.\nA very interesting analysis of the applicability of the\nTHSR wave function can be performed by comparing\nwith stochastic variational calculations [27] and OCM\ncalculations [28]. The αdensity matrix ρ(r,r′) defined\nby integrating out of the total density matrix all intrin-\nsicα-particle coordinates, is diagonalized to study the\nsingle-αorbitsand occupationprobabilitiesin12Cstates.\nFig. 2 shows the occupation probabilities of the L-orbits\nwithS,DandGwaves belonging to the k-th largest oc-\ncupation number (denoted by Lk), for the ground and\nHoyle state of12C obtained by diagonalizing the density\nmatrixρ(r,r′). We found that in the Hoyle state the\nα-particleSorbit with zero node ( S1 in Fig. 2) is occu-\npied to more than 70 % by the three α-particles (see also\nRef.[27]andFig.1). Takingintoaccountthefinitesizeof\nthe nucleus, a reduction of the condensate fraction from\n100 % to about 70 % is not surprising, and the remaining\nfraction (about 30 %) is due to higher orbits originat-\ning from antisymmetrization among nucleons. This huge\npercentagemeansthatanalmostideal α-particleconden-\nsate is realized in the Hoyle state. One should remember\nthat superfluid4He has only 8 percent of the particles in\nthe condensate, what represents a macroscopic amount\nof particlesnonetheless. Pleasealsonote that the S-wave\noccupancy of the Hoyle state is at least by a factor ten\nlargerthan the occupancyofany otherstate (Fig. 2). In-\ndependent ofthe absoluteoccupancy ofthe S-wavestate,\nthisis aclearsignatureofquantumcoherence,i.e. ofcon-\ndensation.On the other hand, in the ground state of12C, the\nα-particle occupations are equally shared among S1,D1\nandG1 orbits, where they have two, one, and zero nodes,\nrespectively, reflecting the SU(3)( λµ) = (04) character\nof the ground state [28]. This fact thus invalidates a\ncondensate picture for the ground state.\nTo get a more extended analysis, OCM calculations\nhave been performed [28] for studying the density de-\npendence of the S-orbit occupancy in the 0+state of\n12C on the different densities ρ/ρ0∼(R(0+\n1)exp/R)3, in\nwhich the rms radius ( R) of12C is taken as a parame-\nter andR(0+\n1)exp=2.56 fm. A Pauli-principle respected\nOCM basis ΨOCM\n0+(ν) with a size parameter νis used,\nin which the value of νis chosen to reproduce a given\nrms radius Rof12C, and the αdensity matrix ρ(r,r′)\nwith respect to ΨOCM\n0+(ν) is diagonalized to obtain the S-\norbit occupancy in the 0+wave function. The results are\nshown in Fig. 3. The S-orbit occupancy is 70 ∼80 %\naroundρ/ρ0∼(R(0+\n1)exp/R(0+\n2)THSR)3= 0.21, while\nit decreases with increasing ρ/ρ0and amounts to about\n30∼40 % in the saturation density region. A smooth\ntransition of the S-orbit is observed from the zero-node\nS-wave nature ( ρ/ρ0≃0.2) to a two-node S-wave one\n(ρ/ρ0∼1) with increasing ρ/ρ0[28]. The feature of the\ndecrease of the enhanced occupation of the Sorbit is in\nstriking correspondence with the density dependence of\nthecondensatefractioncalculatedfornuclearmatter(see\nFig. 1).\nAn interesting item is whether there exist other nuclei\nshowing the Bose condensate-like enhancement of the S-\norbit occupation number. Then, the suppression of the\ncondensate with increasingdensity is alsoof relevance for\nthose nuclei. After we discussed the case of12C corre-\nsponding to n= 3 we will now shortly discuss the sit-\nuation in the next nucleus16O corresponding to n= 4,\nwhere great efforts are performed recently to investigate\nlow-density excitations in the 0+spectrum in theory as\nwell as in experiments.\nIn analogy to the aforementioned OCM calculation for\n12C [28], we recently performed a quite complete OCM\ncalculation also for16O, including many of the cluster\nconfigurations, just mentioned (a full account will be\ngiven in a separate publication [12]). We were able to\nreproduce the full spectrum of 0+states with 0+\n2at 6.4\nMeV, 0+\n3at 9.4 MeV, 0+\n4at 12.6 MeV, 0+\n5at 14.1 MeV,\nand 0+\n6at 16.5 MeV. Also the rms radii are obtained.\nThe largest values are found as R(0+\n6)OCM= 5.6 fm, fol-\nlowed byR(0+\n4)OCM= 4.0 fm. We tentatively make a\none to one correspondence of those states with the six\nlowest 0+states of the experimental spectrum. In view\nof the complexity of the situation, the agreement can be\nconsidered as very satisfactory. The analysis of the diag-\nonalization of the α-particle density matrix ρ(r,r′) (as\nwas done in Ref. [28]) showed that the newly discovered\n0+state at 13.6 MeV [31], as well as the well known 0+\nstate at 14.01 MeV, corresponding to our states at 12 .6\nMeV and 14 .1 MeV, respectively, have, contrary to what6\nwe assumed previously [32], very little condensate occu-\npancy of the zero-node S-orbit (about 20 percent). On\nthe other hand, the sixth 0+state at 16.5 MeV calcu-\nlated energy, to be identified with the experimental state\nat 15.1 MeV, has 61 percent of the αparticles being in\nthe zero-node S-orbit.\nTheseresultsconfirmourstatementthatthe α-particle\ncondensate in nuclear matter is suppressed with increas-\ning density and, consequently, a well developed con-\ndensate state in nuclei can be expected only at very\nlow densities. For16O, the relative densities ρ/ρ0\nare estimated as ( R(0+\n1)exp/R(0+\n4)OCM)3= 0.32 and\n(R(0+\n1)exp/R(0+\n6)OCM)3= 0.12. Therefore we expect a\nsignificant enhancement of the Sorbit occupation num-\nber only for the 0+\n6state, in full agreement with the\nOCM calculation cited above. The very large radius of\nthat state is again a clear indication of an α-particle gas\n(Hoyle)-like state, and the THSR wave function is ex-\npected to describe this state in a sufficient approxima-\ntion. Work in determining the complete spectrum of\nTHSR states in16O showing the relevance of a Bose-\ncondensate like state is in progress [11].\nIV. CONCLUSIONS\nMultiple successful theoretical investigations, concern-\ning the Hoyle state in12C, have established, beyond any\ndoubt, thatitisadilutegas-likestateofthree α-particles,\nheld together only by the Coulomb barrier, and describ-\nabletofirstapproximationbyawavefunctionoftheform\n(C†\nα)3|vac∝an}bracketri}htwhere the three bosons ( C†\nα) are condensed\ninto theS-orbital. There is no objective reason, why in\n16O,20Ne,···there should not exist similar ’Hoyle’-like\nstates. At least the calculations with THSR and OCM\napproaches show this to be the case, systematically. In\nthis work,wegivepreliminaryresultsofacomplete OCM\ncalculation which reproduces the six first 0+states of\n16O to rather good accuracy. In that calculation the 0+\n6\nstate at 16 .5 MeV, which might be identified with the\nexperimental 0+state at 15.1 MeV, shows the character-\nistics typical for a Hoyle-like state, that is high α-particle\nS-wave occupancy combined with an unusually large ra-\ndius.\nTherefore, the main quantity for the formation of an\nαcluster state is the density which should be low. Then,\nthe occurrence of a THSR state where all αparticles oc-\ncupy the same orbit with respect to the c.o.m. motion\nis an interesting effect which corresponds to the forma-\ntion of an α-particle condensate in symmetric nuclear\nmatter. The condensate fraction is decreasing with in-\ncreasing density because of correlations, as known from\ninteracting Bose systems. In addition, the internal struc-\nture of the four nucleon cluster is changed due to Pauli\nblocking if density is increasing.\nOnly in the very low density limit, the αparticles maybe considered as independent bosons moving relatively\nfree like quasi particles. A mean field approach of the\ninteraction which is assumed to be week would give a\nGross-Pitaevskii equation [33]. Then we can apply the\napproach of a non-interacting Bose gas where the αpar-\nticles may occupy the samec.o.m. orbital. The enhanced\noccupation of the ground state (plane wave) in infinite\nmatter is the standard description of Bose-Einstein con-\ndensation. This corresponds, in finite nuclei, to the en-\nhanced occupation of the same orbital for the c.o.m. mo-\ntionsothattheTHSRstatewillbeagoodapproximation\nfor the many-nucleon wave function. We stress the simi-\nlarity to two-particle pairing where the concept of a BCS\nstate was successfully applied to finite nuclei. The ques-\ntion of finite number of Cooper pairs in the nuclear BCS\nstate is also to be considered in analogy with the finite\nnumber ofα-particles in the THSR state.\nWith increasing contribution of the interaction, e.g.\nwith increasing density, the condensate state becomes\nmore complex. Calculations in infinite matter ( T= 0)\nshowthatthecondensatestatebecomesincreasinglynon-\nideal (the condensate fraction is smaller than one). The\nsame is also observed in OCM calculations for finite nu-\nclei where with increasing density the condensate state\nbecomes gradually depleted. We conclude that there are\nsimilarities between the structure of the ground state\nwave function of αmatter and the αgas like states in\nfinite nuclei.\nIn addition to the effect of interaction, mixing higher\nstates of c.o.m. orbits to the ground state wave function,\nthere is also the dissolution of the internal wave func-\ntion of the αparticle due to medium effects. The tran-\nsition from the cluster picture with well defined αstates\nto a shell model where nucleons move independently in\na mean field is also reproduced in harmonic oscillator\napproximation, but needs a first principle approach to\ncalculate the many-nucleon wave function.\nThese results are also of relevance for other phenom-\nena which ariseif the local density approachis used. Low\ndensity matter arises in the halo of heavy nuclei so that\npreformation of α-clusters is an interesting issue there,\nbut also in heavy ion reactions or during supernova ex-\nplosions. Cluster condensation very likely will soon also\nbecome an important subject in cold atom physics. The-\noretical investigations already have appeared [7]. So far\nnuclear physics is at the forefront of this subject.\nAcknowledgements\nThepresentauthors(G.R. andT.Y.) expressthanksto\nM.L.RistigandJ.W.Clark,andalsototheorganizersof\nthe International Workshop on Condensed Matter The-\nories (CMT31) at Bangkok, Thailand, (V. Sa-yakanit:\nchairperson), where this paper was completed.7\n[1] D.M. Brink, J.J. Castro, Nucl. Phys. A 216, 109 (1973).\n[2] S. Ali, A.R. Bodmer, Nucl. Phys. A 80, 99 (1966).\n[3] Y.C. Tang, K. Wildermut, A Unified Theory of the Nu-\ncleus(Academic, New York, 1977).\n[4] G. R¨ opke, L. M¨ unchow, H. Schulz, Nucl. Phys. A 379,\n536 (1982); A 399, 587 (1983).\n[5] M. Beyer, S. A. Sofianos, C. Kuhrts, G. R¨ opke, and P.\nSchuck, Phys. Lett. B 488, 247 (2000).\n[6] G. R¨ opke, A. Schnell, P. Schuck, and P. Nozieres, Phys.\nRev. Lett. 80, 3177 (1998).\n[7] S. Capponi, G. Roux, P. Lecheminant, P. Azaria, E.\nBoulat, S.R. White, arXiv: 0706.0609.; B. Doucot, J.\nVidal, Phys. Rev. Lett. 88, 227005 (2002).\n[8] A.Sedrakian, H.Muether, P. Schuck, Nucl.Phys. A 766,\n97 (2006).\n[9] O. Penrose, L. Onsager, Phys. Rev. 104, 576 (1956).\n[10] A. Tohsaki, H. Horiuchi, P. Schuck, and G. R¨ opke, Phys.\nRev. Lett. 87, 192501 (2001).\n[11] Y. Funaki, H. Horiuchi, G. R¨ opke, P. Schuck, A. Tohsaki ,\nT. Yamada, in preparation.\n[12] Y. Funaki, T. Yamada, P. Schuck, H. Horiuchi,\nA. Tohsaki, and G. R¨ opke, in preparation.\n[13] A. Tohsaki, H. Horuichi, P. Schuck and G. R¨ opke, Nucl.\nPhys.A 738, 259 (2004).\n[14] J. M. Lattimer, F. D. Swesty, Nucl. Phys. A 535, 331\n(1991).\n[15] M. T. Johnson and J. W. Clark, Kinam2, 3 (1980)\n(PDF available at Faculty web page of J. W. Clark\nat http://wuphys.wustl.edu); see also J. W. Clark and\nT. P. Wang, Ann. Phys. (N.Y.) 40, 127 (1966) and\nG. P. Mueller and J. W. Clark, Nucl. Phys. A 155, 561\n(1970).\n[16] K.A. Gernoth, M.L. Ristig, and T. Lindenau, Int. J.\nMod. Phys. B 21, 2157 (2007); G. Senger, M.L. Ristig,\nC.E.Campbell, and J.W. Clark, Ann. Phys. (N.Y.) 218,\n160 (1992); R. Pantfoerder, T. Lindenau, and M.L.\nRistig, J. Low Temp. Phys. 108, 245 (1997).[17] G. R¨ opke and P. Schuck, Mod. Phys. Lett. A 21, 2513\n(2006).\n[18] Z.F. Shehadeh et al., Int. J. Mod. Phys. B 21, 2429\n(2007); M.N.A. Abdullah et al., Nucl. Phys. A 775, 1\n(2006).\n[19] J. W. Clark, private communication.\n[20] For example, Y. Fujiwara, H. Horiuchi, K. Ikeda, M.\nKamimura, K. Kato, Y. Suzuki, and E. Uegaki, Prog.\nTheor. Phys. Suppl. No.68, 29 (1980).\n[21] E. Uegaki, S. Okabe, Y. Abe, and H. Tanaka, Prog.\nTheor. Phys. 57, 1262 (1977); E. Uegaki, Y. Abe, S. Ok-\nabe, and H. Tanaka, Prog. Theor. Phys. 59, 1031 (1978);\n62, 1621 (1979).\n[22] Y. Fukushima and M. Kamimura, Proc. Int. Conf.\non Nuclear Structure , Tokyo, 1977, ed. T. Marumori\n(Suppl. of J. Phys. Soc. Japan, Vol.44, 1978), p.225; M.\nKamimura, Nucl. Phys. A 351, 456 (1981).\n[23] B. R. Barrett, B. Mihaila, S. C. Pieper, and R. B.\nWiringa, Nucl. Phys. News, 13, 17 (2003).\n[24] H. Horiuchi, Prog. Theor. Phys. 51, 1266 (1974); 53, 447\n(1975).\n[25] Y. Funaki, H. Horiuchi, A. Tohsaki, P. Schuck, and G.\nR¨ opke, Prog. Theor. Phys. 108, 297 (2002).\n[26] M. Chernykh, H. Feldmeier, T. Neff, P. von Neumann-\nCosel, A. Richter, Phys. Rev. Lett. 98, 032501 (2007).\n[27] H. Matsumura, Y. Suzuki, Nucl. Phys. A 739, 238\n(2004).\n[28] T. Yamada, P. Schuck, Eur. Phys. J. A 26, 185 (2005).\n[29] H. Horiuchi and K. Ikeda, Prog. Theor. Phys. 40, 277\n(1968); Y. Suzuki, Prog. Theor. Phys. 55, 1751 (1976);\n56, 111 (1976).\n[30] F. Ajzenberg-Selove, Nucl. Phys. A 460, 1 (1986).\n[31] T. Wakasa et al., Phys Lett B 653, 173 (2007).\n[32] Y. Funaki, H. Horiuchi, G. R¨ opke, P. Schuck, A. Tohsaki ,\nT. Yamada, Nucl. Phys. News, 17(04), 11 (2007).\n[33] T. Yamada, P. Schuck, Phys. Rev. C 69, 024309 (2004)."    },    {        "title": "0803.0779v1.Ultimate_Energy_Densities_for_Electromagnetic_Pulses.pdf",        "content": "arXiv:0803.0779v1  [quant-ph]  6 Mar 2008Ultimate Energy Densities for Electromagnetic Pulses\nMankei Tsang∗\nResearch Laboratory of Electronics, Massachusetts Institu te of Technology,\nCambridge, Massachusetts 02139, USA\n(Dated: November 14, 2018)\nThe ultimate electric and magnetic energy densities that ca n be attained by ban-\ndlimited electromagnetic pulses in free space are calculat ed using an ab initio quan-\ntized treatment, and the quantum states of electromagnetic fields that achieve the\nultimate energy densities are derived. The ultimate energy densities also provide\nan experimentally accessible metric for the degree of local ization of polychromatic\nphotons.\nPACS numbers:\nUltrafastlasershavebecome anindispensible toolinawide spectrum ofscience, including\nnonlinear optics[1, 2], metrology[3], laserfusion[4], biologicalimaging[5], biologicalsurgery\n[6], and chemistry [7]. A key to the success of such lasers is the extre mely high peak energy\ndensity that they can achieve, as the moderate energy of each las er pulse can be confined\nwithin femtoseconds or even attoseconds and focused to a micron -sized area. The high\nenergy density strongly enhances light-matter interactions, and is especially crucial to the\nstudy ofrelativistic nonlinear optics [2]. Given theimportance of anult rahigh optical energy\ndensity inabroadrangeofapplications, thelimit towhichonecanfocu sabroadbandoptical\npulse in three spatiotemporal dimensions and maximize the energy de nsity is a fundamental\nproblem.\nLocalization of electromagnetic pulses has been investigated both in the classical regime\n[8] and the quantum single-photon regime [9, 10]. While these seminal studies have con-\ntributed to our fundamental understanding of electromagnetic e nergy localization, all of\ntheir predictions have not yet been experimentally verified because of the difficulty in im-\nplementing their proposed electromagnetic pulse solutions. Most re quire a spectrum that\ncontains arbitrarily high frequency components [8, 9] and can be ex ceedingly difficult to\nrealize due to the finite laser gain bandwidth or finite transparency r ange of optical compo-2\nnents. The use of a spontaneously emitting atomproposedin Ref. [1 0] couples theproperties\nof the emitted photon to those of the atom and does not seem to be generalizable to other\nsituations. Moreover, the quantum studies [9, 10] are mainly conce rned with the decay of\nthe energy density far away from the center of localization at an ins tant of time for one\nphoton, and thus are not immediately relevant to the more practica l problem of maximizing\nthe energy density at the center of localization for a large number o f bandlimited photons.\nIn this Letter, the ultimate electric and magnetic energy densities t hat can be attained by\nbandlimited electromagnetic pulses in free space are calculated using anab initio quantized\ntreatment, and the quantum states that achieve the ultimate den sities are derived. By\ntaking into account all degrees of freedom of electromagnetic field s and explicitly limiting\nthebandwidth of thepulses, our result overcomes all theshortco mings of theaforementioned\nstudies and is more applicable to experimental situations. Measuring the energy densities\nat a fixed point is also considerably easier experimentally than measur ing the decay of the\nenergydensity ataninstant oftime, sothemaximum achievableener gydensities canbeused\nas an alternative and more accessible metric for the degree of localiz ation of polychromatic\nphotons. Most importantly, the ultimate energy densities impose a f undamental limit to\nwhich a bandlimited optical pulse can be focused spatially and tempora lly, so the presented\nresult should prove useful for designing ultrafast optics experime nts.\nThe procedure of calculating the maximum energy densities is similar to the one used to\ncalculate the multiphoton absorption rate limit for monochromatic ligh t in Ref. [11], except\nthatherewegeneralizetheproceduretopolychromaticlight, such thatalldegreesoffreedom\nare taken into account and the treatment can be considered ab initio. We also calculate the\nmaximum magnetic energy density and the corresponding quantum s tate, as the magnetic\nfield can also play a significant role in relativistic nonlinear optics [2].\nWe first derive the ultimate electric energy density, since it is more imp ortant for most\napplications. Consider the quantized electric field operator in free s pace [12]:\nˆE(r,t) =i\n(2π)3/2/summationdisplay\nσ/integraldisplay\nd3k/parenleftbigg¯hω\n2ǫ0/parenrightbigg1/2\nε(k,σ)ˆa(k,σ)eik·r−iωt+H.c., (1)\nwhereσdenotesthetwotransversepolarizations, ε(k,σ)istheunitelectric-fieldpolarization\nvector,ω=ck=c(k2\nx+k2\ny+k2\nz)1/2is the frequency, ˆ a(k,σ) is the annihilation operator\nsatisfying the commutation relation [ˆ a(k,σ),ˆa†(k′,σ′)] =δ3(k−k′)δσσ′, and H.c. is the\nHermitian conjugate. To impose a limit on the bandwidth, it is necessar y to describe3\nthe optical modes in terms of the frequency variable. This can be do ne by changing the\nmomentum-space coordinates ( kx,ky,kz) to normalized spherical coordinates (Ω ,θ,φ):\nω=ω0Ω, k x=k0Ωsinθcosφ,\nky=k0Ωsinθsinφ, k z=k0Ωcosθ,\ndkxdkydkz=dΩdθdφ/parenleftbig\nk3\n0Ω2sinθ/parenrightbig\n,ˆa(k,σ) = ˆa(Ω,θ,φ,σ)/parenleftbig\nk3\n0Ω2sinθ/parenrightbig−1/2.(2)\nwhereω0is a normalizationfrequency, k0≡ω0/c≡2π/λ0, andtheannihilation operatorhas\nbeen renormalized so that its commutator is [ˆ a(Ω,θ,φ,σ),ˆa†(Ω′,θ′,φ′,σ′)] =δ(Ω−Ω′)δ(θ−\nθ′)δ(φ−φ′)δσσ′. The positive-frequency electric field becomes\nˆE(+)(r,t) =i/parenleftbigg¯hω0\n2ǫ0λ3\n0/parenrightbigg1/2/integraldisplay∞\n0dΩ/integraldisplayπ\n0dθ/integraldisplay2π\n0dφ/parenleftbig\nΩ3sinθ/parenrightbig1/2ε(θ,φ,σ)ˆa(Ω,θ,φ,σ)eik·r−iωt.\n(3)\nIn the continuous Fock space representation [12], the N-photon momentum eigenstate is\ngiven by\n|Ω1,θ1,φ1,σ1,...,ΩN,θN,φN,σN/angbracketright ≡1√\nN!N/productdisplay\nn=1ˆa†(Ωn,θn,φn,σn)|0/angbracketright, (4)\nand the identity operator is\nˆ1 =∞/summationdisplay\nN=0|N/angbracketright/angbracketleftN|, (5)\n|N/angbracketright/angbracketleftN|=/summationdisplay\nσ1,...,σN/integraldisplay\ndΩ1dθ1dφ1...dΩNdθNdφN\n×|Ω1,θ1,φ1,σ1,...,ΩN,θN,φN,σN/angbracketright/angbracketleftΩ1,θ1,φ1,σ1,...,ΩN,θN,φN,σN|.(6)\nAn arbitrary quantum state of electromagnetic fields can thus be e xpressed as\n|Ψ/angbracketright=∞/summationdisplay\nN=0CN|N/angbracketright, CN≡ /angbracketleftN|Ψ/angbracketright, (7)\nand a Fock state as\n|N/angbracketright=/summationdisplay\nσ1,...,σN/integraldisplay\ndΩ1dθ1dφ1...dΩNdθNdφNΦN(Ω1,θ1,φ1,σ1,...,ΩN,θN,φN,σN)\n×|Ω1,θ1,φ1,σ1,...,ΩN,θN,φN,σN/angbracketright, (8)4\nwhere\nΦN(Ω1,θ1,φ1,σ1,...,ΩN,θN,φN,σN)≡ /angbracketleftΩ1,θ1,φ1,σ1,...,ΩN,θN,φN,σN|N/angbracketright(9)\nis theN-photon momentum-space probability amplitude, which must satisfy the normaliza-\ntion condition:\n/summationdisplay\nσ1,...,σN/integraldisplay\ndΩ1dθ1dφ1...dΩNdθNdφN|ΦN(Ω1,θ1,φ1,σ1,...,ΩN,θN,φN,σN)|2= 1,(10)\nand the symmetrization condition:\nΦN(...,Ωn,θn,φn,σn,...,Ωm,θm,φm,σm,...)\n= ΦN(...,Ωm,θm,φm,σm,...,Ωn,θn,φn,σn,...) for any nandm. (11)\nTo impose a limited bandwidth ( α≤Ω≤β) on the electromagnetic fields, we require the\nprobability of photons existing outside the bandwidth to vanish:\nΦN(Ω1,θ1,φ1,σ1,...,ΩN,θN,φN,σN) = 0 for any Ω n< αor Ωn> β. (12)\nWith the theoretical framework put forth, we now proceed to calc ulate a bound on the\nelectric energy density (minus the zero-point energy density) give n by\nUe≡/angbracketleftBig\n:ǫ0\n2ˆE·ˆE:/angbracketrightBig\n=/angbracketleftBig\nǫ0ˆE(−)·ˆE(+)/angbracketrightBig\n. (13)\nA bound on the electric energy density is equivalent to a bound on the energy density for\none component of the electric field:\nU′\ne≡/angbracketleftbigg\n:ǫ0\n2/parenleftBig\np·ˆE/parenrightBig2\n:/angbracketrightbigg\n=ǫ0/angbracketleftBig/parenleftBig\np·ˆE(−)/parenrightBig/parenleftBig\np·ˆE(+)/parenrightBig/angbracketrightBig\n, (14)\nwherepis an arbitrary real unit vector, because UeandU′\neare equivalent if we choose pto\nbe parallel to the electric field. In terms of the momentum-space re presentation,\nU′\ne=¯hω0\n2λ3\n0∞/summationdisplay\nN=0|CN|2N/summationdisplay\nσ2,...,σN/integraldisplay\ndΩ2dθ2dφ2...dΩNdθNdφN\n×/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nσ/integraldisplayβ\nαdΩ/integraldisplayπ\n0dθ/integraldisplay2π\n0dφ/bracketleftBig\ni/parenleftbig\nΩ3sinθ/parenrightbig1/2p·ε(θ,φ,σ)eik·r−iωt/bracketrightBig\n×ΦN(Ω,θ,φ,σ,Ω2,θ2,φ2,σ2,...,ΩN,θN,φN,σN)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (15)5\nwhere the symmetric property of Φ Ngiven by Eq. (11) is used. By virtue of the Schwarz’s\ninequality and the normalization condition given by Eq. (10),\nU′\ne≤¯hω0\n2λ3\n0∞/summationdisplay\nN=0|CN|2N/summationdisplay\nσ1,...,σN/integraldisplay\ndΩ1dθ1dφ1...dΩNdθNdφN\n×|ΦN(Ω1,θ1,φ1,σ1,...,ΩN,θN,φN,σN)|2\n×/summationdisplay\nσ/integraldisplayβ\nαdΩ/integraldisplayπ\n0dθ/integraldisplay2π\n0dφ/parenleftbig\nΩ3sinθ/parenrightbig\n|p·ε(θ,φ,σ)|2\n=π\n3/angbracketleftN/angbracketright¯hω0\nλ3\n0/parenleftbig\nβ4−α4/parenrightbig\n, (16)\nwhere/angbracketleftN/angbracketright ≡/summationtext\nN|CN|2Nis the average photon number. As expected, the bound on U′\ne\ndoes not depend on p, and is therefore also applicable to the total electric energy densit y.\nDefining the actual lower and upper frequencies as ω1=αω0andω2=βω0, respectively,\nand the corresponding wavelengths as λ1,2= 2πc/ω1,2, we obtain the central result of this\nLetter:\n/angbracketleftBig\n:ǫ0\n2ˆE·ˆE:/angbracketrightBig\n≤π\n3/angbracketleftN/angbracketright/parenleftbigg¯hω2\nλ3\n2−¯hω1\nλ3\n1/parenrightbigg\n. (17)\nThis simple expression agrees with the intuition that the ultimate ener gy density is limited\nby the maximum energy of photons ( /angbracketleftN/angbracketright¯hω2) divided by the smallest volume that the\nphotons can occupy ( λ3\n2).\nIn the limit of a small bandwidth compared to the center frequency, we can let ∆ ω≡\nω2−ω1,ω0= (ω1+ω2)/2, ∆ω≪ω0, and obtain\n/angbracketleftBig\n:ǫ0c\n2ˆE·ˆE:/angbracketrightBig\n<∼2\n3/angbracketleftN/angbracketright¯hω0∆ω\nλ2\n0, (18)\nwhich is a bound on the peak intensity in the slowly-varying envelope re gime, and again\nagrees with the intuition that the highest intensity is achieved when t he mean energy of\nthe photons is focused to their minimum pulse width (2 π/∆ω) and beam size ( λ2\n0). This\napproximate bound also agrees with that derived in Ref. [11], where t he monochromatic\napproximation is made at the beginning. Beyond the slowly-varying en velope regime, the\nexact bound given by Eq. (17) depends on the upper frequency to the fourth power, un-\nderlying the importance of high-frequency components in maximizing the energy density, as\nthey have a higher energy as well as a smaller localization volume.\nThe use of the Schwarz’s inequality is reminiscent of the matched filte r concept in com-\nmunication theory [13]. In Eq. (15), the N-photon amplitude can be regarded as the input6\nsignal, and the expression in square brackets can be regarded as a filter transfer function in\nthe measurement of the electric field. An N-photon amplitude that achieves the Schwarz\nupper bound is one that is linearly dependent on the square-bracke ted expression, or in\nother words, when the input signal matches the filter. Assuming a f actorizable Φ N, the\nfollowing N-photon amplitude that achieves the ultimate electric energy densit y can then\nbe obtained:\nΦN=N/productdisplay\nn=1fe(Ωn,θn,φn,σn),\nfe=/braceleftbigg−iC−1/2(Ω3sinθ)1/2p·ε∗(θ,φ,σ)e−ik·r0+iωt0forα≤Ωn≤β,\n0 otherwise,\nC=2π\n3/parenleftbig\nβ4−α4/parenrightbig\n. (19)\nThis state produces the maximum electric energy density at ( r0,t0), with the electric field\nat (r0,t0) polarized along p.\nTo apply the above result to the classical regime, let\nCN=e−/angbracketleftN/angbracketright/2/angbracketleftN/angbracketrightN/2\n√\nN!. (20)\nTogether with a factorizable Φ Nin Eq. (19), the quantum state becomes a coherent state in\nthe continuous mode representation [12], and the mean electric fie ld is then given by\nE(r,t) =i/parenleftbigg/angbracketleftN/angbracketright¯hω0\n2ǫ0λ3\n0/parenrightbigg1/2/integraldisplayβ\nαdΩ/integraldisplayπ\n0dθ/integraldisplay2π\n0dφ\n×/parenleftbig\nΩ3sinθ/parenrightbig1/2ε(θ,φ,σ)fe(Ω,θ,φ,σ)eik·r−iωt+H.c., (21)\nwhich, incidentally, must be an exact solution of the Maxwell equation s. The Fourier trans-\nform ofE(r,t) is proportional to ω3, and the classical power spectrum is then proportional\ntoω6within the allowed frequency band.\nConsider now the magnetic field operator:\nˆB(r,t) =i\n(2π)3/2/summationdisplay\nσ/integraldisplay\nd3k/parenleftbiggµ0¯hω\n2/parenrightbigg1/2\nκ×ε(k,σ)ˆa(k,σ)eik·r−iωt+H.c.,(22)\nwhereκ≡k/k. While the total magnetic energy must be the same as the total elec tric\nenergyforphotonsinfreespace, itisnotdifficulttoshowthatthem agneticenergydensity at\n(r0,t0) is zero where the electric energy density is maximum for the state g iven by Eqs. (19).7\nTo maximize the magnetic energy density instead, we can simply apply t he same procedure\nas above to the magnetic energy density, which turns out to obey t he same bound as the\nelectric one:\n/angbracketleftbigg\n:1\n2µ0ˆB·ˆB:/angbracketrightbigg\n≤π\n3/angbracketleftN/angbracketright/parenleftbigg¯hω2\nλ3\n2−¯hω1\nλ3\n1/parenrightbigg\n. (23)\nThe quantum state with the ultimate magnetic energy density is also s imilar to the electric\ncase,\nΦN=N/productdisplay\nn=1fb(Ωn,θn,φn,σn),\nfb=/braceleftbigg−iC−1/2(Ω3sinθ)1/2m·[κ×ε∗(θ,φ,σ)]e−ik·r0+iωt0forα≤Ωn≤β,\n0 otherwise,(24)\nwheremis the unit vector of the magnetic field at ( r0,t0).\nThe ultimate electric and magnetic energy densities may be challenging to achieve ex-\nperimentally, as they require a power spectrum proportional to ω6within the allowed band,\nspatial focusing in all directions with a specific angular spectrum, an d polarization control.\nThat said, the results set forth impose fundamental limits to which t he energy densities\ncan reach regardless of technological advances in the control of electromagnetic fields, and\ntherefore should prove useful for designing ultrafast optics exp eriments.\nThe author would like to acknowledge financial support from the William M. Keck Foun-\ndation Center for Extreme Quantum Information Theory.\n∗Electronic address: mankei@mit.edu\n[1] T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000).\n[2] G. A. Mourou, T. Tajima, and S. V. Bulanov, Rev. Mod. Phys. 78, 309 (2006).\n[3] S. T. Cundiff and J. Ye, Rev. Mod. Phys. 75, 325 (2003).\n[4] M. Tabak et al., Phys. Plasmas 1, 1626 (1994).\n[5] W. Denk, J. H. Strickler, and W. W. Webb, Science 248, 73 (1990); W. R. Zipfel, R. M.\nWilliams, W. W. Webb, Nat. Biotechnol. 11, 1369 (2003).\n[6] T. Juhasz, F. H. Loesel, R. M. Kurtz, C. Horvath, J. F. Bill e, and G. A. Mourou, IEEE J.\nSel. Topics Quantum Electron. 5, 902 (1999).8\n[7] A. H. Zewail, J. Phys. Chem. A 104, 5660 (2000).\n[8] R. W. Ziolkowski, Phys. Rev. A 39, 2005 (1989); R. W. Hellwarth and P. Nouchi, Phys. Rev.\nE54, (1996).\n[9] W. O. Amrein, Helv. Phys. Acta 8, 2684 (1969); C. Adlard, E. R. Pike, and S. Sarkar, Phys.\nRev. Lett. 79, 1585 (1997); I. Bialynicki-Birula, Phys. Rev. Lett. 80, 5247 (1998).\n[10] K. W. Chan, C. K. Law, and J. H. Eberly, Phys. Rev. Lett. 88, 100402 (2002).\n[11] M. Tsang, e-print arXiv:0802.0516v1.\n[12] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press,\nCambridge, UK, 1995).\n[13] L. W. Couch, Digital and Analog Communication Systems (Prentice Hall, New Jersey, 2001)."    },    {        "title": "0803.0821v1.Finite_doping_of_a_one_dimensional_charge_density_wave__solitons_vs__Luttinger_liquid_charge_density.pdf",        "content": "arXiv:0803.0821v1  [cond-mat.str-el]  6 Mar 2008Finite doping of a one-dimensional charge density wave: sol itons vs. Luttinger liquid\ncharge density\nYuval Weiss, Moshe Goldstein and Richard Berkovits\nThe Minerva Center, Department of Physics, Bar-Ilan Univer sity, Ramat-Gan 52900, Israel\nThe effects of doping on a one-dimensional wire in a charge den sity wave state are studied using\nthe density-matrix renormalization group method. We show t hat for a finite number of extra\nelectrons the ground state becomes conducting but the parti cle density along the wire corresponds\nto a charge density wave with an incommensurate wave number d etermined by the filling. We find\nthat the absence of the translational invariance can be disc erned even in the thermodynamic limit,\nas long as the number of doping electrons is finite. Luttinger liquid behavior is reached only for a\nfinite change in the electron filling factor, which for an infin ite wire corresponds to the addition of an\ninfinite number of electrons. In addition to the half filled in sulating Mott state and the conducting\nstates, we find evidence for subgap states at fillings differen t from half filling by a single electron or\nhole. Finally, we show that by coupling our system to a quantu m dot, one can have a discontinuous\ndependence of its population on the applied gate voltage in t he thermodynamic limit, similarly to\nthe one predicted for a Luttinger liquid without umklapp pro cesses.\nPACS numbers: 73.21.Hb,71.55.-i,71.45.Lr\nI. INTRODUCTION\nThe transition between a Mott insulator and a metal-\nlic phase in one-dimensional (1D) wires has attracted no-\ntable interest in recent years1. Different types of transi-\ntions are possible, controlled by various physical param-\neters, such as the electron-electron interaction range and\nitsstrength. SincetheMottstateexistsonlyforcommen-\nsurate fillings, the electron filling factor inside the wire is\nan important parameter, and it can be varied either by\ncontrolling the chemical potential, or by doping with a\nfinite number of electrons. However, these two methods\nare different: doping with an infinitesimally small num-\nber of electrons breaks immediately the Mott insulator\nstate, while due to the Mott gap, a finite change in the\nchemical potential is required in order to insert the first\nelectron and cause the transition2.\nThe metallic phase resulting from doping is usu-\nally described by the Tomonaga-Luttinger liquid (TLL)\ntheory2,3, although for a small finite doping, a large cur-\nvatureofthe elementaryexcitationspectrumis expected,\nand deviations from the TLL theory are possible. In the\nTLL, manyphysicalpropertiesaredetermined by the pa-\nrameterKwhich describes the interactions. Recently, a\ndiscontinuity in the occupation of a resonant level which\nis coupled to a TLL with K <1/2 was predicted by\nFurusaki and Matveev4. However, since the K <1/2\nregime corresponds to very strong repulsive interactions,\nneither experimental nor numerical evidence for such a\njump has been obtained so far.\nAs a generic model for the wire it is convenient to use\na tight-binding description of a 1D lattice with spinless\nelectrons. When repulsive interactions between nearest-\nneighbor electrons are considered, the Mott state, taking\nthe form of a charge density wave (CDW), occurs for\nstrong enough interactions. The Hamiltonian of such asystem can be written as\nˆHwire=−tL−1/summationdisplay\nj=1(ˆc†\njˆcj+1+H.c.) (1)\n+IL−1/summationdisplay\nj=1(ˆc†\njˆcj−1\n2)(ˆc†\nj+1ˆcj+1−1\n2),\nwhereIdenotes the nearest-neighbor interaction\nstrength, and the hopping matrix element between near-\nest neighbors, t, sets the energy scale. ˆ c†\nj(ˆcj) is the\ncreation (annihilation) operator of a spinless electron at\nsitejin theL-site wire, and a positive background is\nincluded in the interaction term.\nThe model of Eq. (1) is equivalent to that of the XXZ\nspin 1/2 chain by the Jordan-Wigner transformation5.\nThe XXZ model is exactly solvable using the Bethe\nansatz6,7. From this equivalence it is known that for\na half filled system with periodic boundary conditions, a\nphase transition between a TLL phase and a CDW one\noccurs at I= 2t. Of course, for sufficiently long wires\nthe type of the boundary conditions does not change this\nresult8,9. The interactionparameter Kis then givenby10\nK=π\n2cos−1(−I/2t), (2)\nso that in the TLL phase, I <2tandK >1/2, while for\nI >2t, the half filled wire is no longer in the TLL phase,\nbut is rather in a CDW state.\nThe competition between the TLL and the CDW\nphases is attributed to the presence ofumklapp processes\nin this model. For half filling the CDW phase wins over\nthe TLL phase once the interactions are strong enough.\nHowever, when the wire is not exactly half filled, a CDW\nstate cannot emerge since it demands a commensurate\nfilling, and thus the TLL description is valid even for in-\nteraction values which are greater than 2 t. As a result,2\nfor strong enough interactions, and sufficiently close to\nhalf filling, one can then get values of Kwhich are less\nthan 1/2.11\nAnother approach to treat the Hamiltonian of Eq. (1)\nis to map it, using bosonization2, into that of the sine-\nGordon model12. In this model, the elementary excita-\ntions are solitons, which carry half an electron charge2.\nIn the vicinity of the Mott state the value of the TLL\ninteraction parameter is K→1\n4+. The value K= 1/4,\nwhich cannot be obtained using this model, corresponds\nto the Luther-Emeryline, for which the solitonsare effec-\ntively non interacting13. For small deviations from half\nfilling the solitons are only weakly interacting.\nIn this paper we investigate the effects of a finite dop-\ning on a 1D Mott state. We show that although a small\nnumber of electrons results in an insulator-metal transi-\ntion, thechargedistributioninthe metallicsystemwith a\nfinite number of electrons is not uniform, but rather cor-\nresponds to an incommensurate CDW. The added charge\ncan also be considered as delocalized solitons in the orig-\ninal commensurate CDW, whose spatial distribution is\ndetermined by boundary conditions and symmetry con-\nsiderations. Nevertheless, the system is conducting, and\nonceabiasvoltageis appliedthe solitonsarefreeto move\nand transport charge across the wire.\nAn exceptional behavior is exhibited when a lattice\nwith an even number of sites L= 2pis occupied by p±1\nelectrons, i.e., there is a single additional electron (or\nhole) relative to half filling. As will be shown, in this\ncase the insulating behavior of the original Mott state is\nretained even though the filling is incommensurate. This\nunusualcasewill be explained asaresultofsubgapstates\nin the soliton energy spectrum.\nIn addition, we show that since the Mott state with a\nsmall doping allows for the interaction parameter in the\nrange 1/4< K < 1/2, it might exhibit the Furusaki-\nMatveev discontinuity. The appearance of that jump is\nnot a foreclosed conclusion, since the model used by Fu-\nrusaki and Matveev does not contain umklapp processes.\nNevertheless, we find that the jump indeed occurs, and\nshow its implications on the solitons present in the wire\nnear half filling.\nSince the CDW state is obtained for exactly half fill-\ning, it is convenient to denote the extra charge by Q=\nne−L/2, where neis the number of electrons in the sys-\ntem. As will be shown, the parity of Lplays a significant\nrole in the behavior when adding the first few electrons.\nUsing the finite-size version of the density-matrix renor-\nmalization group (DMRG) method14,15the Hamiltonian\nˆHwirewas diagonalized and the ground state was calcu-\nlated for different values of the charge Qand the lattice\nsizeL. In the current paper we present results obtained\nforI= 3t, which is deep inside the CDW regime (for\nhalf filling). Nevertheless, other interaction strengths in\nthe regime of I >2thave been checked as well, and were\nfound to lead to qualitatively similar results.\nThe outline of the paper is as follows: in the next sec-\ntion, Sec. II, we present the charge occupation alongnearly half filled wires, which demonstrates the presence\nof incommensurate CDW or solitons. In Sec. III we dis-\ncuss the excitation spectrum of the solitons, and the ad-\ndition spectrum of the system. For systems which are ex-\nactly half filled, or doped with only one electron or hole,\nwe show that the ground state is insulating; otherwise, it\nis conducting. In Sec. IV we discuss the existence of the\nFurusaki-Matveev jump when the doped wire is coupled\nto a resonant level. Finally we conclude in Sec. V.\nII. CHARGE DISTRIBUTION\nWe begin by presenting the occupation of electrons\nalong the wire for different numbers of extra electrons Q\nfor wires with an odd or an even number of sites (Fig. 1).\nTheQ= 0 case (for an even L) results in a flat distri-\nbution of nj= 1/2 for each lattice site j(dashed line).\nThisstateisalinearcombinationoftwodegenerateCDW\nstates, which are eigenstates of the Hamiltonian in the\nthermodynamic limit. These CDW states, which are also\nshown in Fig. 1, are numerically obtained by applying\nan infinitesimally small potential on the first site, which\nbreaks the degeneracy between them8. The population\nof such a CDW state along the wire (near the center) can\nbe written as\nnj=n+Acos(2πnj+φ), (3)\nwhere the amplitude Adepends on the interaction\nstrength, n= 1/2 is the filling, and φis a phase.\nWhenQ/negationslash= 0, the ground state is no longer degenerate\nand the electron occupation throughout the lead is not\nuniform. Let us start with the Q= 1/2 case (when Lis\nodd), which results in a true CDW state, i.e., for each\neven (odd) site the occupation is low (high). Unlike the\ncase of the degenerate ground states for Q= 0, the Q=\n1/2 ground state is not coupled by the Hamiltonian to\nthe similar CDW ground state of Q=−1/2 due to the\ndifference in their total population.\nIn order to explore the addition of electrons onto the\nCDW states, one can take the uniform distribution of the\nQ= 0 state as a reference, and investigate the difference\nfromitbycalculatingtheaccumulatedpopulation∆ nj=/summationtext\ni≤jni−j/2. As can be seen in Fig. 2, ∆ njfor each\nof the two clean CDW states of Q= 0 (presented in\nsolid lines), has an extra charge of e/4 localized at one of\nthe wire edges, and a compensating charge (i.e., −e/4)\nlocalized at the other. The difference between these two\nstates is thus a soliton (of charge e/2) located at one\nedge of the wire and an antisoliton (of charge −e/2) at\nthe other edge.\nBy taking one of the Q= 0 CDW states, and locating\na soliton at its negatively charged edge, while leaving\nthe other edge as it is, one gets a new CDW state in\nwhich both edges have a positive charge, i.e., a CDW\nstate having Q= 1/2. Similarly, placing an antisoliton\nat the edge with the positive charge of the Q= 0 state3\n0 100 200 300 400 500j0.20.30.40.50.60.70.8nj0\n1\n2\n3\n0 100 200 300 400 500j0.20.30.40.50.60.70.8nj1/2\n3/2\n5/2\nFIG. 1: (Color online) The electrons occupation in a lattice of\n500sites (upperpanel)and499sites (lower panel)for differ ent\nnumber of electrons. The values near the curves denote the\nextra charge Qwhich varies between Q= 0 for half filling and\nQ= 3.\nresults in the Q=−1/2 CDW state. These states are\nalso shown in Fig. 2 (dashed lines).\nFor theQ= 1 case (when Lis even), adding a local-\nized soliton near one of the edges is not sufficient, and it\nrequiresthe addition ofanotherchargeof e/2. This extra\ncharge is obtained by the formation of a delocalized soli-\nton, centered at the middle of the wire. Further increase\nof the filling above Q= 1/2 (Q= 1) for odd (even) wire\nlengths, results in the addition of two delocalized solitons\nfor everyelectron, sothat the total number ofdelocalized\nsolitons is 2 Q−1.\nThe delocalized solitons, which carry a charge of e/2\neach, are free to move along the wire. However, when\nno bias voltage is applied, the charge distribution across\nthe wire is fixed by the boundaries and by symmetry\nconsiderations. Practically these constraints lead to a\ndensitywavewith acosineterm similarto that ofEq.(3),\nin which the filling factor nis modified. This statement\nis true for much larger values of Qas well. For instance,\nFig. 3 shows the electron distribution for a 300-site wire\nwithQ= 14, i.e., with 27 delocalized solitons. It is easy\nto see that although the amplitude of the oscillations is50 100 150 200 250 300\nj-0.5 -0.5-0.25 -0.250 00.25 0.250.5 0.5∆nj\nFIG. 2: (Color online) The extra charge distribution along a\nwire of length L= 300 sites with Q= 0 (solid lines) and of\nL= 299 sites with Q=±1/2 (dashed lines).\n0 50 100 150 200 250 300\nj00.20.40.60.8nj\n100 120 140 160 180 2000.450.50.550.60.65\nFIG. 3: (Color online) The electrons occupation in a lattice of\n300 sites with additional 14 electrons. A fit to Eq. (3) (done\nover the middle region) is shown in the inset (symbols).\nreduced, it doesn’t vanish and it is rather constant near\nthe middle of the wire. Moreover, the oscillations can\nstill be fitted to Eq. (3) using nas a fitting parameter.\nAs can be expected, this results in a value of nwhich\nmatches the filling of the wire.\nThe non-uniform charge distribution for a fixed num-\nber of electrons is not a finite-size effect, and survives in\nthe thermodynamic limit as well. In Fig. 4 one can see\na comparison between wires of 300 and 600 sites, with\nQ= 14 and Q= 28. One can see weaker oscillations\nin theQ= 28 case, attributed to a smaller wave length.\nOn the other hand, increasing the size of the lead from\nL= 300 to L= 600 while maintaining the number of\nextra electrons Qconstant, results in an increase of the\noscillations.\nOne can thus conclude that on the one hand the oscil-4\n240 260 280 300 320 340 360\nj0.50.550.6njL=600   Q=28\nL=600   Q=14\nL=300   Q=14\nFIG. 4: (Color online) The electrons occupation near the mid -\ndle of the system in lattices of 300 or 600 sites with addition al\n14 or 28 electrons. Results for the 300-site wire are shown as\na function of 2 j.\nlations are expected to vanish if a finite filling fraction is\nconsidered, i.e., in the limit of Q→ ∞,L→ ∞, keeping\nQ/Lconstant. On the other hand, when a finite number\nof electrons is added, the oscillations are expected to be\nnoticed even in the limit of L→ ∞.\nIII. EXCITATION SPECTRUM\nUp to now we have shown that the density distribution\nof the electrons along the wire, when it is doped by a\nfinite number of electrons, is not flat, and preserves some\nfeatures of the Mott state. One can still wonder whether\nthese doped states retain the insulating behavior of the\nMott state or not. To clarify that point we study the size\ndependence of the addition spectrum, defined through\n∆2(Q) =E0(Q+1)−2E0(Q)+E0(Q−1),(4)\nwhereE0(Q) is the ground-state energy of the wire with\nQelectrons above half filling. For a Mott state the limit\nof ∆2, asL→ ∞, should be a finite value, corresponding\nto the gap size, while for a conducting state one expects\n∆2→0.\nThe dependence of ∆ 2(Q) on the wire length Lis pre-\nsented in Fig. 5 for even wire sizes (lines) and for odd\nsizes (filled symbols), for different number of electrons\nQ. As can be seen, for Q= 0 and 1 /2, the values of\n∆2converge to the value ∆ 2(∞) (presented by the as-\nterisk symbol at the left margin of the figure), given by\nthe Bethe ansatz6,7. On the other hand, for Q≥3/2\none gets ∆ 2→0, which indicates that these states are\nconducting.\nA deviation from this intuitive picture appears for\nQ= 1, in which an unexpected gap of ∆ 2(∞)/2 occurs.\nNevertheless, this result can be explained by the spec-\ntrum of soliton states near half filling. It is known that\nfor solitons in the sine-Gordon model with open bound-\nary conditions there are two degenerate subgap states at0.001 0.01\n1/L00.10.20.3∆2Q=0\nQ=1\nQ=2\nQ=3\n0.001 0.01\n1/L00.10.20.3∆2Q=1/2\nQ=3/2\n0.001 0.01\n1/L00.10.20.3∆2∆2(∞)\n∆2(∞)/2Even Odd Bethe ansatz\nFIG. 5: (Color online) The dependence of the addition spec-\ntrum on the lattice size L, for different fillings, comparing the\ncases of evenand oddsizes and theexact Bethe ansatz results .\nNote the semi-logarithmic scale.\nzero energy, positioned exactly between the conduction\nand the valence bands, each of them localized near one\nof the edges of the system16. In our case these subgap\nstates are thus the localized solitons, which we identified\nas the difference between the two CDW states of Q= 0\npresented in Fig. 1.\nThusthe energyspectrumofthe solitonsisrepresented\nby valence and conduction bands separated by a gap ∆,\nwhile the subgap states exist at ∆ /2 above the valence\nband (see schematic picture in Fig. 6). The filling of the\nsubgap states with the localized solitons does not change\nthe total energy. However, every additional soliton in-\ncreases the energy in ∆ /2 (in the limit L→ ∞).\n-∆/2∆/2\n0E\n∆\nFIG. 6: (Color online) A schematic picture of the energy\nbands of the solitons according to Ref. 16. The bands are\nseparated by energy ∆, and two discrete states exist at zero\nenergy.\nFor an odd L, the states with Q=±1/2, which have\nthe same energy due to the particle-hole symmetry of\nthe Hamiltonian, differ only in the occupation of the two\nlocalized solitons. The addition of any other electron is5\nequivalent to the addition of two delocalized solitons, so\nthat the energy difference between successive fillings is\n∆. Therefore, E0(1/2)−E0(−1/2) = 0 and E0(3/2)−\nE0(1/2) = ∆, resulting in ∆ 2(1/2) = ∆. For higher\nvalues of Q, one gets E0(Q+ 1)−E0(Q) = ∆, so that\n∆2(Q≥3/2) = 0. These results are summarized in\nTable I.\nQ−1\n21\n23\n25\n2Q′>5\n2\nE000∆2∆(Q′−1\n2)∆\n∆2∆∆000\nTABLE I: Addition spectrum for wires with an odd number\nof sites in the limit L→ ∞.\nOn the other hand, when Lis even, the Q= 0 state\ncorresponds to the occupation of only one localized soli-\nton (or, more precisely, to a linear combination of two\nstates, in each of them one edge soliton state is filled and\nthe other is empty). The addition of the first extra elec-\ntronfillstheadditionallocalizedsolitonstateandasingle\ndelocalized soliton state, so that E0(1)−E0(0) = ∆/2.\nEvery additional electron adds two delocalized solitons,\nthus forQ >1 one gets E0(Q+1)−E0(Q) = ∆. There-\nfore, asonecanseein TableII, ∆ 2(0) = ∆, ∆ 2(1) = ∆/2,\nand ∆ 2(Q≥2) = 0.\nQ−10123Q′>3\nE0∆\n20∆\n23∆\n25∆\n2(Q′−1\n2)∆\n∆2∆\n2∆∆\n2000\nTABLE II: Addition spectrum for wires with an even number\nof sites in the limit L→ ∞.\nBefore commencing with the observation of Furusaki-\nMatveev jump we briefly summarize the results so far.\nWe have demonstrated the existence of CDW states for\nQ= 0 and Q=±1/2, and shown that these states are\ninsulating. The ground states with Q=±1 were found\nto be insulating as well, with an excitation gap which is\nhalf of the Mott gap. States with Q >1 are conduct-\ning and thus should be generally described by the TLL\ntheory. Nevertheless, we have found that the spatial dis-\ntribution of the electron density is not uniform as in the\nregular TLL picture. Furthermore, this distribution can\nbe fitted to an incommensurate CDW form. It is there-\nfore interesting to explore some other predictions of the\nTLL theory for such states.\nIV. FURUSAKI-MATVEEV JUMP\nThe unconventionalbehavior of the gaplessstates with\nthe non-uniform density can be illustrated by studying\nthe discontinuity in the occupation of a resonant level\ncoupledtothewire4. Asmentionedabove,once K <1/2,the level is expected to show a jump in its occupation as\na function of its energy. Nevertheless, one should note\nthat umklapp processes, which are an essential ingredi-\nent in explaining the behavior of our system, were not\nconsidered in the theoretical framework of Ref. 4.\nFurthermore, since here the particle distribution in the\nuncoupledwireisnotflat, theoccurrenceoftheFurusaki-\nMatveev discontinuity raises an additional question. If\nthe electrons occupation profile along the uncoupled wire\nwas uniform, one should have observed Friedel oscilla-\ntions in the wire as soon as it is coupled to the resonant\nlevel. Thepotentialofthe resonantlevelcanthen be con-\ntinuously changed until the predicted jump in the level\noccupation occurs. At that point, an inversion of the\nFriedel oscillations along the wire is expected. On the\nother hand, once the wire is a Mott state doped by a\nfinite number of electrons and the electron density is not\nuniform, the situation is less clear.\nWe first calculate the value of Kfor the doped Mott\nstate, and show that it is below 1 /2. In order to have\nthe same value of Kfor different lengths of wires, one\nmust consider an equal density of additional electrons,\nand we choose to work with doping of Q=L/50, with\nwires of sizes 100sites and above. Kmay be obtained2,17\nby the ratio between ∆ E, the energy difference between\nthe groundstate and the first excited state, and the addi-\ntion spectrum ∆ 2, since for spinless electrons with open\nboundary conditions ∆ E=πvc\nLand ∆ 2=πvc\nKL. More ac-\ncurate result may be obtained by fitting both ∆ 2(L) and\n∆E(L) to a polynom in 1 /L, and then obtaining Kfrom\nthe ratio of the linear coefficients18. Using both methods\nwe find that the value of Kfor our wire is 0 .42.\nIn order to represent the coupling of a resonant level\nor an impurity of energy ǫ0to the left edge of the wire\nthe following term is added to the Hamiltonian:\nˆHimp=ǫ0ˆa†ˆa−V0(ˆa†ˆc1+H.c.), (5)\nwhere ˆa†(ˆa) is a creation (annihilation) operator of an\nelectron in the resonant level, and V0is the hopping ma-\ntrix element between the level and the first site of the\nwire. Interaction between the impurity and the wire is\nnot considered here since it does not change the result\nqualitatively, but only causes a renormalization of the\nhopping amplitude V0towards larger values4,8.\nThe level occupation n0as a function of the level en-\nergyǫ0is presented in Fig. 7 for different wire lengths\nusingV0= 0.2t. As mentioned above, wires in different\nlengths contain different number of additional electrons\nusingQ=L/50, andK= 0.42. When ǫ0is much larger\nthan the chemical potential in the wire µ, the impurity is\nalmost empty, and the wire contains most of the Qextra\nelectrons. On the other hand, for ǫ0≪µthe impurity\nis almost entirely occupied, and then only Q−1 extra\nelectrons are in the wire.\nAlthough the population of the impurity is found to\nbe continuous for all finite wire lengths studied, it is ex-\npected to display an abrupt jump in the thermodynamic\nlimit,L→ ∞. In order to demonstrate this we study the6\n0.35 0.4 0.45 0.5 0.55ε000.20.40.60.81n0\n100150200250300\nL-150-100-50slope\nFIG. 7: (Color online) The occupation of an impurity which\nis coupled to one end of 1D lattices of different sizes. The\nresults shown are for lattice sizes between 100 and 300 (from\nleft to right) in steps of 50. In order to compare cases with\nidentical values of K, the number of additional electrons is\ntaken as Q=L/50, so that K= 0.42. Inset: the slope near\nthe point where n0= 1/2, which shows a linear dependence\non the system size.\ndependence of the occupation slope near n0= 1/2 on the\nwire length L. As is clearly seen in the inset of Fig. 7,\nthe slope scales linearly with L, which is a clear sign of\na first order transition in the thermodynamic limit19.\n50 100 150jnj\nFIG. 8: (Color online) The transition of an electron from the\nresonant level into the wire, which leads to the formation of 2\nadditional solitons. The results shown are for a level coupl ed\nto a 150-site wire, with Q= 3. Different curves correspond\nto values of ǫ0between 0 .498 (top) to 0 .510 (bottom) with\noffsets in the vertical axis for clarity.\nThe finite size transition region, in which the electrontransfersfromthe resonantlevel into the wire, is interest-\ning by itself, since the addition of a single electron to the\nwire is related to the appearance of two additional delo-\ncalized solitons in the electron population inside it. As\ncan be seen in Fig. 8, as ǫ0increases the electron bound\nto the impurity tunnels into the wire and additional two\nsolitons are created in the wire. As mentioned above,\nthis is in sharp contrast to the case of a resonant level\ncoupled to a TLL having a uniform charge distribution,\nin which only a local change of the Friedel oscillations in\nthe charge distribution of the wire is expected.\nV. CONCLUSIONS\nIn this paper we have shown that doping a Mott state\nwith a finite number of electrons yields states which pre-\nserve the charge modulations of the density wave even\nfor an infinite system size. The charge carriers are soli-\ntons, whose spectrum fits the known predictions of the\nsine-Gordon model.\nWe have demonstrated how the solitons are added to\nthe wire. The first charge of e/2 above half filling results\nin the appearance of a localized soliton at one of the\nwire edges. Each additional charge of e/2 results in an\nadditional delocalized soliton, so that for charge Qone\ngets a wire with 2 Q−1 delocalized solitons and a single\nlocalized one.\nThe ground state of the pure Mott state, which is ex-\nactly half filled, is obviously insulating. For wires of odd\nsizes, the ground states for Q=±1/2 are insulating as\nwell, and the excitation spectrum in these three cases\nhas the regular Mott gap. Additional insulating states\nwith a non-trivial half gap were found for Q=±1. This\nunusual gap was explained according to the spectrum of\nthe soliton states in a 1D wire with open boundaries. For\nQ >1, the excitation spectrum was found to be gapless.\nThe unconventional behavior of wires with Q >1,\nwhich are gapless but do not preserve translational in-\nvariance of charge, is checked against a prediction for a\nTLL wire with an interaction parameter K <1/2 cou-\npled to a resonant level. We find that the TLL prediction\n(i.e., the Furusaki-Matveevjump) is indeed obtained also\nfor our system, although it involves the creation of two\nadditional delocalized solitons in the charge distribution\nof the wire, as opposed to the inversion of Friedel oscil-\nlations in the wire characterizing the conventional TLL\nscenario. As a final remark we note that it may be inter-\nesting to investigate, for this regime of parameters, some\nother TLL predictions as well.\nAcknowledgments\nWe thank M. Pepper and M. Kaveh for useful discus-\nsions, and the Israel Science Foundation (Grant 569/07)\nfor financial support.7\n1N. F. Mott, Metal Insulator Transitions (Taylor and Fran-\ncis, London, 1990).\n2T. Giamarchi, Quantum Physics in One Dimension (Ox-\nford University Press, New York, 2003).\n3J. Voit, Rep. Prog. Phys. 57, 977 (1994).\n4A. Furusaki and K. A. Matveev, Phys. Rev. Lett. 88,\n226404 (2002).\n5P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928).\n6H. A. Bethe, Z. Phys. 71, 205 (1931).\n7R. J. Baxter, Exactly Solved Models In Statistical Mechan-\nics, (Academic Press, London, 1989).\n8Y. Weiss, M. Goldstein and R. Berkovits, J. Phys.: Con-\ndens. Matter 19, 086215 (2007).\n9Y. Weiss, M. Goldstein and R. Berkovits, Phys. Rev. B 76,\n024204 (2007).\n10F.WoynarovichandH.P.Eckle, J.Phys.A 20, L97(1987);\nC. J. Hamer, G. R. W. Quispel, and M. T. Batchelor, ibid.20, 5677 (1987).\n11F. D. M. Haldane, Phys. Rev. Lett. 45, 1358 (1980).\n12R. Rajaraman, Solitons and Instantons: An Introduc-\ntion to Solitons and Instantons in Quantum Field Theory\n(North Holland, Amsterdam, 1982).\n13A. Luther and V. J. Emery, Phys. Rev. Lett. 33, 589\n(1974).\n14S. R. White, Phys. Rev. B 48, 10345 (1993).\n15U. Schollw¨ ock, Rev. Mod. Phys. 77, 259 (2005).\n16M. Fabrizio and A. O. Gogolin, Phys. Rev. B 51, 17827\n(1995).\n17H. Schulz, Phys. Rev. Lett. 64, 2831 (1990).\n18S. Ejima, F. Gebhard, S. Nishimoto and Y. Ohta, Phys.\nRev. B72, 033101 (2005).\n19K.Binder andD.P.Landau, Phys.Rev.B 30, 1477(1984)."    },    {        "title": "0805.0542v1.Crossover_in_the_local_density_of_states_of_mesoscopic_SNS_junctions.pdf",        "content": "arXiv:0805.0542v1  [cond-mat.supr-con]  5 May 2008Crossoverinthelocaldensity ofstatesofmesoscopic\nsuperconductor/normal-metal/superconductor junctions\nAlex Levchenko\nSchool of Physics and Astronomy, University of Minnesota, M inneapolis, MN, 55455, USA\n(Dated: May 5, 2008)\nAndreev levels deplete energy states above the superconduc tive gap, which leads to the peculiar non-\nmonotonouscrossoverinthelocaldensityofstatesofmesos copicsuperconductor /normal-metal/superconductor\njunctions. Thiseffect isespeciallypronounced inthe case whenthenormal meta l bridge length Lissmallcom-\npared to the superconductive coherence length ξ. Remarkable property of the crossover function is that it\nvanishes not onlyatthe proximityinduced gap ǫgbutalsoatthe superconductive gap ∆. Analytical expressions\nforthe density of states atthe both gapedges, as well as gene ral structure of the crossover are discussed.\nPACS numbers: 74.45. +c\nExperimental advances in probing systems at the meso-\nscopic scale1,2,3,4,5revived interest to the proximity related\nproblemsinsuperconductor–normalmetal(SN)heterostruc -\ntures.6The most simple physical quantity reflecting proxim-\nity effect is the local density of states (LDOS) ρ(ǫ,r), which\ncan be measured in any spatial point rat given energyǫus-\ning scanning tunneling microscopy. The e ffects of supercon-\nductive correlations on the spectrum of a normal metal are\nespecially dramatic in restricted geometries. For example ,\nin the case of superconductor–normal metal–superconducto r\n(SNS) junction, proximity e ffect induces an energy gap ǫg\nin excitation spectrum of a normal metal with the square\nroot singularityρ(ǫ,r)∝/radicalbigǫ/ǫg−1 in the density of states\njust above the threshold ǫ−ǫg≪ǫgRef. 7,8,9,10,11,12\n(here and in what follows, ρwill be measured in units of the\nbare normal metal density of states νat Fermi energy). The\nmost recent theoretical interest was devoted either to meso-\nscopic13,14,16,17orquantum18,19,20,21fluctuation effects on top\nof mean–field results7,8,9,10,11,12that smear hard gap below ǫg\nand lead to the so called subgaptail states with nonvanishing\nρ∝exp/bracketleftbig−g(1−ǫ/ǫg)(6−d)/4/bracketrightbigatǫg−ǫ/lessorsimilarǫg, where g is the\ndimensionlessnormalwire conductanceand dis the effective\nsystem dimensionality. The latter is essentially a nonpert ur-\nbative result that requires instantonlike approach within σ–\nmodel19,20or relies on methodsof random matrix theory.16,18\nSurprisingly, after all of these advances, there is somethi ng\ninteresting to discuss about proximity induced properties of\nthe SNS junctionseven at the level of quasiclassical approx i-\nmationbyemployingUsadelequations.22Thepurposeofthis\nworkistopointoutasubtlefeatureofthecrossoverinthelo -\ncal density of states of mesoscopic SNS junctions. The latte r\nwasseeninsomeearlyandrecentstudies,8,11,12,14,15however,\nneitheremphasizednortheoreticallyaddressed.\nTo this end, consider normal wire (N) of length Land\nwidthWlocated between two superconductive electrodes\n(S). In what follows, we concentrate on di ffusive quasi–one–\ndimensional geometry and the short wire limit L≪ξ, where\nξ=√DS/∆is superconductivecoherencelength, with DSas\nthediffusioncoefficientinthesuperconductorand ∆astheen-\nergygap(hereafter, /planckover2pi1=1). Thecenterofthewireischosento\nbeatx=0 andboundarieswith superconductorsat x=±L/2\ncorrespondingly,where xisthe coordinatealongthewire.\nUnder the condition L≪ξ, the proximity effect is es-/s40 /s44/s32 /s120 /s41\n/s84/s104/s47 \n/s49\n/s103 /s52 \n/s61\n/s103/s61/s66 /s67/s83 /s40 /s41\nFIG. 1: (Color online) Schematic plot for the local density o f states\ncrossover inthe short L≪ξdiffusive SNSjunction.\npecially strong — superconductive correlations penetrate the\nentire volume of the normal region. As the result, the in-\nduced energy gap in normal wire ǫgis large and turns out to\nbe of the same order as the gap in the superconductor itself\nǫg=∆−δ∆,withthefinitesize correction δ∆∼∆3/ǫ2\nTh≪∆,\nwhereǫTh=DN/L2is the Thoulessenergyand DNis the dif-\nfusion coefficient in the normal bridge. The latter should be\ncontrasted to the long wire limit L≫ξ, where the proxim-\nity effect is weak and the induced gap is ǫg∼ǫTh≪∆. Just\nabove the gapǫ−ǫg≪ǫg, density of states has square root\nsingularity, which is similar to the L≫ξcase,7,8,9,10,11,12see\nFig. 1, which is a robust property of quasiclassical approx-\nimation. However, the prefactor for L≪ξis significantly\nenhancedρ(ǫ,x)∝(ǫTh/∆)2/radicalbigǫ/ǫg−1(noteherethat propor-\ntionalitysignimpliescharacteristicenergydependence, while\nthe exact numerical coe fficientis differentfor a givencoordi-\nnatexalong the wire). The value of ǫgis a property of the\nspectrum,thusit is xindependent.\nOne would naturally expect that above the proximity in-\nduced gapǫg, local density of states goes through the max-\nimumρmax=ρ(ǫ=∆,x) and then crosses over to the BCS2\nlike DOSρBCS=ǫ/√\nǫ2−∆2atǫ >∆, which finally satu-\nrates to unityρ→1 atǫ→∞. Surprisingly, however, the\ncrossover scenario is di fferent. The density of states indeed\nreachesthemaximum,whichoccursat ǫ∼∆−δ∆/2,butthen\ndecreasesand vanishesto zeroat superconductivegap ∆with\nquarctic power law behavior ρ(ǫ,x)∝(ǫTh/∆)3/24√|ǫ/∆−1|\nfor|ǫ−∆|/lessorsimilarδ∆. Finally,ρ(ǫ,x) grows back forǫ>∆and at\nthe energyscaleǫ∗∼∆+δ∆crosses overto the ρBCS. Indeed,\nobserve thatρBCS(ǫ∗)∼ǫTh/∆∼ρ(ǫ∗,x), see Fig. 1. It is\nimportant to mention here that the discussed feature appear s\nonlyatthelevelof localdensityofstates. Energydependence\nof theglobaldensity of states, which is integrated over the\nentire volume, does not show dip at ǫ=∆(this point will be\ndiscussedlaterin thetext).\nIt seems that crossover picture presented in the Fig. 1 was\nalready numerically seen in previous studies,11,12,14,15how-\never the origin of the soft gap at ǫ=∆was never addressed.\nOneshouldalsonotethatnumericswasperformedalwaysfor\nthe not too short junctions L/greaterorsimilarξ. It is certainly unusual to\nfindzerointhedensityofstatesatthegapedge ∆. Inorderto\nget someinsightintothisbizarrefeature,let usconsidera toy\nmodel, which crudelymimicsthe system underconsideration\nand gives some hints for qualitative understanding. Quanti ta-\ntivetheoryoftheoutlinedcrossoverwillbedevelopedfoll ow-\ningthisdiscussion.\nImagine chaotic quantum dot (QD) sandwiched between\ntwo superconductors. A quantum dot is really a zero dimen-\nsional system, in the sense that L/ξ→0 (or equivalently,\n∆/ǫTh→0). However, one will momentarily see that finite\ntunnelingrateintothedot ΓplaysaneffectiveroleofThouless\nenergyǫTh. By following Ref. 23 one may construct scatter-\ning states and determine quasiparticles excitation spectr um.\nIf in addition we assume that QD supports only one trans-\nverse propagating mode, then the discrete spectrum obtaine d\nfromthepolesofthescatteringmatrix S(ǫ)consistsofasingle\nnondegenerateAndreev state at energy ǫo∈[0,∆], satisfying\nΩ(ǫ)+Γǫ2√\n∆2−ǫ2=0,where the function Ω(ǫ) is defined\nbyΩ(ǫ)=(∆2−ǫ2)(ǫ2−Γ2/4). The density of states in the\nsuperconductor–quantumdot–superconductorsystemisgiv en\nbyρ(ǫ)=ρBCS(ǫ)+δρ(ǫ),where\nδρ(ǫ)=1\n2πid\ndǫlnΩ(ǫ)+iΓǫ2√\nǫ2−∆2\nΩ(ǫ)−iΓǫ2√\nǫ2−∆2,(1)\nwhich follows from the relation ρ(ǫ)=ρBCS+\n(1/2πi)(d/dǫ)lnDetS(ǫ)betweenDOSandthescatteringma-\ntrix.24It is easy to check that in the limit Γ≫∆Andreev\nlevel is positioned at ǫo≈∆−8∆3/Γ2, which resembles the\nexpression forǫgin the case of the di ffusive wire discussed\nabove,whereΓindeedplaystheroleofThoulessenergy. The\npresence of an Andreev level changes density of states ρ(ǫ)\nin two ways. The first contribution δρ1(ǫ) originating from\ndlnDetS/dǫtermofEq.(1)belongstothesubgappartofthe\nspectrumǫ∈[0,∆]andhasstructureoftheform\nδρ1(ǫ)=1\n2πid\ndǫln/parenleftiggǫ−ǫo−i0\nǫ−ǫo+i0/parenrightigg\n=δ(ǫ−ǫo),(2)\nwhichis nothingelse but DOS associated with the singleAn-\ndreev state. Most interestingly, Andreev level changes ρ(ǫ)above the superconducting gap as well. Indeed, it follows\nfrom the Eq. (1) that at energies ǫ>∆the density of states\nρ(ǫ)getsthecorrection\nδρ2(ǫ)=−ρBCS(ǫ)Γ\nπ/parenleftbig2∆2−ǫ2/parenrightbigΓ2/4−ǫ2/parenleftbigǫ2+∆2/parenrightbig\n/parenleftbigǫ2−∆2/parenrightbig/parenleftbigǫ2−Γ2/4/parenrightbig2+Γ2ǫ4.(3)\nObservethatintheimmediatevicinityofthesuperconducti ve\ngapǫ−∆/lessorsimilar∆3/Γ2thecorrectionδρ2(ǫ)≈−(Γ/4π∆2)ρBCS(ǫ)\nisnegative, implying that Andreev level suppresses bulk su-\nperconductivedensity of states ρBCS(ǫ) abovethe gap. At en-\nergyǫ∗∼∆+∆3/Γ2the correction given by Eq. (3) reaches\nitsmaximum,while ρBCSisrecoveredatlargeenergies ǫ/greaterorsimilarΓ,\nwhereδρ2decays asδρ2(ǫ)≈Γ/πǫ2. Based on this example,\nit appears that Andreev level tends to depletebulk BCS den-\nsity of states at energies ǫ−∆/lessorsimilar∆3/Γ2. One shouldnote that\nthe Andreev level leads to pure redistribution of the energy\nstates — there are no additional states abovethe gap; indeed , /integraltext∞\n∆δρ2(ǫ)dǫ≡0. If QD supports not only one but a large\nnumberof the transversepropagatingchannels, then cummu-\nlativenegativeδρ2(ǫ) may compensate ρBCS(ǫ), which leads\nto the vanishing density of states ρ(ǫ) at the gap edgeǫ=∆,\nsee Fig.1.\nHaving discussed the qualitative picture of the crossover,\nlet us now turn to the quantitative description. The quasi-\nclassical approachto di ffusive SNS structuresis based on the\nUsadel equation22for the retarded Green’s function GR(r,ǫ).\nForthelatter,wewill employangularparametrization25GR=\nτzcosθ+τxsinθcosφ+τysinθsinφ, whereτiis the set of\nPaulimatrices. Intheabsenceofthephasedi fferencebetween\nthe S terminals, one can set φ≡0 and Usadel equations for\nquasi–one–dimensionSNS geometryacquiresthe form\n/braceleftigg\nDN/parenleftbigd2θN/dx2/parenrightbig+2iǫsinθN=0|x|/lessorequalslantL/2\nDS/parenleftbigd2θS/dx2/parenrightbig+2iǫsinθS+2∆cosθS=0|x|>L/2(4)\nwhereθN(S)(ǫ,x)aretheGreen’sfunctionanglesinN(S)parts\nof the junction, correspondingly. In writing Eq. (4), we as-\nsumedstepfunctionpair–potential ∆(x)=∆η(|x|−L/2),with\nη(x)=1 ifx>0andη(x)=0 otherwise. Theapplicabilityof\nthisapproximationreliesontheconditionthatthewidth Wof\nthejunctionissmallcomparedtothecoherencelength. Inth is\ncase,nonuniformitiesin ∆(r)extendonlyoverthedistanceof\norderWfrom the junction, which is due to the geometrically\nconstrained influence of the narrow junction on the bulk su-\nperconductor.26\nIn the absence of additional tunnel barriers at SN–\ninterfaces, Eqs. (4a) and (4b) are supplemented by the fol-\nlowingboundaryconditions:27\nθN(ǫ,x)|x=±L/2=θS(ǫ,x)|x=±L/2, (5a)\nσNdθN(ǫ,x)\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex=±L/2=σSdθS(ǫ,x)\ndx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglex=±L/2,(5b)\nwhereσN(S)aretheconductivitiesofnormalmetalandsuper-\nconductor. Knowing solutionsof Usadel equations, one finds\nlocaldensityofstates fromthegeneralexpression\nρ(ǫ,x)=Re[cosθ(ǫ,x)]. (6)3\nFor future convenience,we rotate the Green’s function an-\ngles asθN=π/2+iϑNandθS=θBCS+iϑS, whereθBCS=\nπ/2+iarsinhγwithγ=ε/√\n1−ε2,andintroducedimension-\nless variablesε=ǫ/∆,λ=x/L. After the rotation, Usadel\nEq. (4b) for superconducting sides of the junction becomes\nrealandcanbeeasilyintegrated,providing\nϑS(ε,λ)=4artanh/braceleftbigexp[−(|λ|−1/2)/ξS]/bracerightbig,(7)\nwhere we have introduced the energy depending supercon-\nductive coherence length ξ−1\nS=/radicalig\n2∆DN\nǫThDS4√\n1−ε2. Equation\n(4a) mayalso be exactly solvedin terms of elliptic function s;\nhowever, this exact solution is not needed for our purposes.\nIndeed, observe that it follows from the boundary condition\n[Eq. (5a)] that in the energy range 1 −ε≪1, the normal\nmetal phase Re/bracketleftbigϑN(ε,λ)/bracketrightbig∝arcsinhγ≈ln/radicalig\n2\n1−ε≫1 is large\neverywhere in the N part of the junction. Thus, one may ap-\nproximate sinθN≈exp(ϑN)/2 and solve Eq. (4a) in closed\nform\nϑN(ε,λ)=ϑ0(ε)−ln/bracketleftbigcosh2(λ/ξN)/bracketrightbig, (8)\nwith normal side coherence length being defined as ξ−1\nN=/radicalig\n2ε∆\nǫThandϑ0=ϑN(ε,0). We now introduce u0=expϑ0and\nuS=exp(arcsinhγ+ϑB\nS), whereϑB\nS=ϑS(ε,±1/2),to rewrite\ntheboundarycondition[Eq.(5b)]as\nuS/γ−2=κγ(u0−uS), (9)\nwhere the interface parameter κ=σ2\nNDS/σ2\nSDNmeasures\nthe mismatch of conductivities and di ffusion coefficients at\nthe SN boundaries. By using solutions (7) and (8), together\nwithEq.(9),oneeliminatestheunknown u0andarrivestothe\nalgebraicequationfor z=uS/γ−2,\nF(z,κ)=/radicaligg\nγε∆\n8ǫTh, (10)\nwherethesingleparameter κ=κγ2scalingfunction F(z,κ)is\ngivenby\nF(z,κ)=/radicalbiggκ\nz+(z+2)κarctanh/radicalbiggz\nz+(z+2)κ.(11)\nKnowing the solution of Eq. (10), one finds the density of\nstatesat theSN interfacesas\nρ(ǫ,±1/2)=γ\n2Im[z(ǫ)]. (12)\nAt the same time, uS(ǫ) togetherwith Eqs.(6)–(8)providean\nexplicitinformationaboutthe localdensityof states ρ(ǫ,x) at\nanyposition xalongthewire.\nBy looking at Eq. (10), one sees that its right hand side\ngrows to infinity when ε→1, while its left hand side has\nan absolute maximum for certain value of z. This implies\nthatforallenergiesbelowsomethreshold εg,Eq.(10)hasthe\nonly real solution for zprovidingρ(ǫ)≡0 as it follows fromEq. (12). The condition ε=εgwhen Eq. (10) has a complex\nsolution for zfor the first time defines the proximity induced\nenergygapεg. Forenergiesabovethe gap ε>εg, the density\nof states is nonzerosince Im[ z]/nequal0. It turnsout that Eq. (10)\npossesses two qualitatively di fferent solutions depending on\nthevalueofthe interfaceparameter κ.\nThe limit of strong superconductor κ≪∆2/ǫ2\nTh. In this\ncase,κ≪1 andFfunction determined by Eq. (11) has\nthe following asymptotes: F(z,k)≈√z/4κforz≪1 and\nF(z,k)≈√κ/4zln(2/κ) forz≫1. It means that the only\nrelevantz, which determine the maximum of Fare those\nz∼κ. In this region, F(z,κ) may be approximated as\nF(z,κ)≈/radicalig\nκ\nz+2κarctanh/radicalig\nz\nz+2κ.As a result, the absolute max-\nimumFm=F(z=zm) occurs at point zm≈4.5κ, which cor-\nresponds to Fm≈0.5. Then, the gap determining condition\ngives\nFm=/radicaligg\nγgεg∆\n8ǫTh⇒εg=1−1\n8/parenleftigg∆\nǫTh/parenrightigg2\n,(13)\nwhere we have used the notation γg=γ(εg). Just above the\ngapε−εg≪εg,onecanexpand FinTaylorseriesaroundthe\nmaximum F≈Fm+b(z−zm)2, withb=(1/2)(d2F/dz2)z=zm\nto findz≈zm+i/radicalig\n1\nb/radicalbigg/radicalig\nγε∆\n8ǫTh−/radicalig\nγgεg∆\n8ǫTh.By using now def-\ninition (12), one finds for the density of states just above th e\nproximityinducedgapat SN–interface,\nρ(ǫ,±1/2)∝/parenleftbiggǫTh\n∆/parenrightbigg2/radicaligg\nǫ−ǫg\nǫg, ǫ−ǫg≪ǫg,(14)\nwherethenumericalcoe fficientoftheorderofunitywasomit-\nted. For the other limiting case, in the vicinity of the super -\nconductivegapε∼1,Eq.(10)issolvedby z≈−π2p2(1−4ip)\nwithp=/radicalig\nǫTh\n∆4√\n1−ε,whichgivesforthe densityofstates,\nρ(ǫ,±1/2)∝/parenleftbiggǫTh\n∆/parenrightbigg3/24/radicalbigg\n|ǫ−∆|\n∆,|ǫ−∆|/lessorsimilarδ∆.(15)\nThisasymptoticresult holdsabove ∆aswell. Observethat at\nǫ∼ǫg+δ∆/2,Eqs.(14)and(15)crossovertoeachother,while\natǫ∼∆+δ∆, Eq. (15) crossoversto the BCS like density of\nstates.\nThe limit of weak superconductor κ≫∆2/ǫ2\nTh. This\nlimiting case corresponds to the situation when κ≫1 and\nthe expression for Fgreatly simplifies F(z,κ)≈/radicalig\n1\nκ√z\nz+2.At\nzm=2, the function Fhasthe maximum Fm=/radicalbig1/8κg, were\nκg=κγ2\ng, so that the gap determining condition is di fferent\nfromEq.(13) andreads\n/radicaligg\n1\n8κg=/radicaligg\nγgεg∆\n8ǫTh⇒εg=1−1\n2/parenleftiggκ∆\nǫTh/parenrightigg2/3\n.(16)\nTo calculate the asymptotes for density of states at both gap\nedges, one follows the same steps as in the previouscase and4\nfinds\nρ(ǫ,±1/2)∝/parenleftbiggǫTh\nκ∆/parenrightbigg2/3/radicaligg\nǫ−ǫg\nǫg, ǫ−ǫg≪ǫg,(17a)\nρ(ǫ,±1/2)∝/radicalbigg\nǫTh\nκ∆4/radicalbigg\n|ǫ−∆|\n∆,|ǫ−∆|/lessorsimilarδ∆∗,(17b)\nwhereδ∆∗∼∆(κ∆/ǫTh)2/3. Equations (14), (15) and (17)\ncomplement our qualitative considerations presented at th e\nbeginningofthispaper.\nAt this point, let us discuss the obtained results and limits\nof their applicability. (i) We have studied the energy depen -\ndence of the local density of states for short ( L≪ξ) dif-\nfusive SNS junctions. Although ρ(ǫ,x) was analytically cal-\nculated at SN interfaces only it turns out that its energy de-\npendence is generic for any x∈[−L/2,L/2] and given by\nEqs. (14), (15) and(17). The exactnumericalprefactor,how -\never, isxdependent and should be determined numerically.\n(ii) Let us stress that the discussed feature in the ρ(ǫ,x) at\nǫ∼∆disappears at the level of global ∝angbracketleftρ(ǫ)∝angbracketright, which is inte-\ngrated over the volume, density of states. Indeed, the spati alintegration brings an additional factor of ξd\nS, wheredis an\neffective dimensionality of superconductor, which is due to\nthelongspatialtailsofAndreevstatespenetratingdeepin side\nthe superconductor [note that because of these tails, it is n ot\ncorrect to integrate ρ(ǫ,x) overx∈[−L/2,L/2] only]. For\nquasi–one–dimensional geometry discussed here, a factor o f\nξS∝/parenleftbig1\n1−ε/parenrightbig1/4gained after xintegration exactly compensates\nthe dip inρ(ǫ,x) atǫ∼∆[see Eqs. (15) and (17b)], leading\nto the finite value∝angbracketleftρ(ǫ=∆)∝angbracketright∼ǫTh/∆. Ford>1, the density\nofstatesat superconductivegapedge ǫ∼∆shoulddivergeas\na certain power–law ∝angbracketleftρ(ǫ)∝angbracketright∼(ǫ−∆)−pfor p>0. (iii) It fol-\nlowsfromthenumericalanalysis12thatpresenceofadditional\ntunnelbarriersatSNinterfacesalterssoftgapandleadsto the\nnonzerodensity of states at the gap edge ∆. (iv) Finite super-\nconductivephaseφimposedacrossthejunctionshiftsposition\nof the proximityinducedgap9ǫg(φ), such thatǫg(φ=π)=0,\nand also changes shape of the crossover function. However,\nzero in the LDOS at ǫ=∆and asymptotes at both gap edges\npersist.\nNumerous useful discussions with A. Kamenev and\nL. Glazman, which initiated and stimulated this work are\nkindly acknowledged. This research is supported by DOE\nGrantNo. 08ER46482.\n1S.Gu´ eron,H.Pothier,NormanO.Birge,D.EsteveandM.H.De -\nvoret, Phys.Rev. Lett. 77, 3025 (1996).\n2M.F.Goffman,R.Cron,A.LevyYeyati,P.Joyez, M.H.Devoret,\nD.Esteve, and C.Urbina, Phys.Rev. Lett. 85, 170 (2000).\n3E. Scheer, W. Belzig, Y. Naveh, M. H. Devoret, D. Esteve, and\nC.Urbina, Phys. Rev. Lett. 86, 284 (2001).\n4A.Anthore,H.Pothier,andD.Esteve,Phys.Rev.Lett. 90,127001\n(2003).\n5Yong-Joo Doh, Jorden A. van Dam, Aarnoud L. Roest, Erik\nP. A. M. Bakkers, Leo P. Kouwenhoven, Silvano De Franceschi,\nScience309, 272(2005).\n6See, e.g., Mesoscopic Superconductivity , edited by P. F. Bagwell\nspecial issue of Superlattices Microstruct. 25, 5–6 (1999).\n7A.A.Golubov, E.P.Houwman, J.G.Gijsbertsen,V.M.Krasnov ,\nJ. Flokstra, H. Rogalla, and M. Yu. Kupriyanov, Phys. Rev. B 51,\n1073 (1995).\n8W.Belzig,C.Bruder,andG.Sch¨ on,Phys.Rev.B 54,9443(1996).\n9F. Zhou, P. Charlat, B. Spivak, and B. Pannetier, J. 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Usadel,Phys. Rev. Lett. 25, 507(1970).\n23C. W. J. Beenakker and H. van Houten, Single-Electron Tunnel-\ning and Mesoscopic Devices , edited by H. Koch and H. L¨ ubbig,\nSpringer, Berlin,(1992), pp. 175-179.\n24E. Akkermans, A. Auerbach, J. E. Avron, and B. Shapiro, Phys.\nRev. Lett. 66, 76(1991).\n25W.Belzig,F.K.Wilhelm,C.Bruder,G.Sch¨ on, andA.D.Zaiki n,\nSuperlattices Microstruct. 25, 1251 (1999).\n26K.K. Likharev, Rev. Mod. Phys. 51, 101 (1979).\n27M. Yu. Kuprianov and V. F. Lukichev, Sov. Phys. JETP 67, 1163\n(1988)."    },    {        "title": "0805.2148v2.Density_of_states_of_disordered_graphene.pdf",        "content": "arXiv:0805.2148v2  [cond-mat.mes-hall]  18 Oct 2008Density of states of disordered graphene\nBen Yu-Kuang Hu1,2, E. H. Hwang1, and S. Das Sarma1\n1Condensed Matter Theory Center, Department of Physics,\nUniversity of Maryland, College Park, MD 20742-4111 and\n2Department of Physics, The University of Akron, Akron, OH 44 325-4001\n(Dated: October 18, 2008)\nWe calculate the average single particle density of states i n graphene with disorder due to impurity\npotentials. For unscreened short-ranged impurities, we us e the non-self-consistent and self-consistent\nBorn and T-matrix approximations to obtain the self-energy. Among th ese, only the self-consistent\nT-matrix approximation gives a non-zero density of states at the Dirac point. The density of states\nat the Dirac point is non-analytic in the impurity potential . For screened short-ranged and charged\nlong-range impurity potentials, the density of states near the Dirac point typically increases in the\npresence of impurities, compared to that of the pure system.\nPACS numbers: 81.05.Uw; 72.10.-d, 72.15.Lh, 72.20.Dp\nI. INTRODUCTION\nThe recent experimental realization of a single layer\nof carbon atoms arranged in a honey-comb lattice has\nprompted much excitement and activity in both the ex-\nperimental and theoretical physics communities1,2. Car-\nriers in graphene (both electrons and holes) have a lin-\near bare kinetic energy dispersion spectra around the\nKandK′points (the “Dirac points”) of the Brillouin\nzone. The ability of experimentalists to tune the chemi-\ncal potential to lie above or below the Dirac point energy\n(by application of voltages to gates in close proximity to\nthe graphene sheets) allows the carriers to be changed\nfrom electrons to holes in the same sample. This sets\ngraphene apart from other two dimensional (2D) car-\nrier systems that have a parabolic dispersion relation,\nand typically have only one set of carriers, i.e. either\nelectrons or holes. Another unique electronic property,\nthe absence of back-scattering, has led to the specula-\ntion that carrier mobilities of 2D graphene monolayers\n(certainly at room temperature, but also at low temper-\nature) could be made to be much higher than any other\nfield-effect type device, suggesting great potential both\nfor graphene to be the successor to Si-MOSFET (metal-\noxide-semicondcutor field effect transistor) devices and\nfor the discovery of new phenomena that normally ac-\ncompanies any significant increase in carrier mobility3–6.\nIt is therefore of considerable fundamental and techno-\nlogical interest to understand the electronic properties o f\ngraphene2.\nGraphene samples that are currently being fabricated\nare far from pure, based on the relatively low electronic\nmobilities compared to epitaxially grown modulation-\ndoped two-dimensional electron gases (2DEGs) such as\nGaAs-AlGaAs quantum wells. It is therefore important\nto understand the effects of disorder on the properties of\ngraphene. Disorder manifests itself in the finite lifetimes\nof electronic eigenstates of the pure system. In the pres-\nence of scattering from an impurity potential that is not\ndiagonal in the these eigenstates, the lifetime scattering\nrate of the eigenstate, γ, is non-zero and can be measuredexperimentally by fitting the line-shape of the low-field\nShubnikov-de Haas (SdH) oscillations.7,8The effect of\ndisorder scattering on the SdH line shape is equivalent\nto increasing the sample temperature and one can there-\nfore measure this Dingle temperature ( TD) and relate it\nto the single-particle lifetime through /planckover2pi1γ= 2πkBTD. To\navoid potential confusion, we mention that the lifetime\ndamping rate γdiscussed in this paper is notequal to\nthetransport scattering rate which governs the electrical\nconductivity. The lifetime damping rate is the measure\nof the rate at which particles scatters out of an eigen-\nstate, whereas the transport scattering rate is a mea-\nsure of the rate of current decay due to scattering out\nof an eigenstate. In normal 2DEGs, the transport scat-\ntering time can be much larger than the impurity in-\nduced lifetime /planckover2pi1/2γ, particularly in high mobility mod-\nulation doped 2D systems where the charged impurities\nare placed very far from the 2DEG8,9. Recently, the issue\nof transport scattering time versus impurity scattering\nlifetime in graphene has been discussed.10\nThe single-particle level broadening due to the impu-\nrity potential changes many of the physical properties of\nthe system including the electronic density of states.11,12\nThe electronic density of states is an important property\nwhich directly affects many experimentally measurable\nquantities such as the electrical conductivity, thermoele c-\ntric effects, and differential conductivity in tunneling ex-\nperiments between graphene and scanning-tunneling mi-\ncroscope tips or other electron gases. Changes in the\ndensity of states also modify the electron screening,13\nwhich is an important factor in the determination of var-\nious properties of graphene. It is therefore imperative\nto take into account the effects of disorder on the den-\nsity of states, particularly since disorder is quite strong\nin currently available graphene samples.\nIn the present work, we present calculations of the av-\nerage density of states of disordered graphene. This prob-\nlem has been investigated using various models and tech-\nniques, both analytical and numerical.14?–20We take\ninto account scattering effects from long-range and short\nrange impurity potentials. We consider both unscreened2\nand screened short-ranged and screened charged impuri-\nties, using the Born approximation. In addition, for un-\nscreened short-ranged (USR) impurities, we go beyond\nthe Born approximation and include self-consistent ef-\nfects.\nThere is another class of disorder in graphene called\n“off-diagonal” or “random gauge potential” disorder, in\nwhich the hopping matrix elements of the electrons in the\nunderlying honeycomb lattice are random. In this paper\nwe do not consider in this type of disorder, which can\nresult from height fluctuations (ripples) in the graphene\nsheet and lead to qualitatively different results from the\nones presented in this paper.14,19,22,23\nThe rest of the paper is organized as follows: In Sec. II,\nwe describe the approximation schemes that we use.\nSecs. III and IV deal with unscreened short range impuri-\nties and screened short-range/charged impurities, respec -\ntively. In Sec. V, we compare our results to those from\nother workers, and we conclude in Sec. VI.\nII. APPROXIMATIONS FOR THE\nSELF-ENERGY\nThe single-particle density of states for a translation-\nally invariant 2DEG is given by24\nD(E) =−g\nπ/summationdisplay\nλ/integraldisplaydk\n(2π)2Im[Gλ(k, E)] (1)where gis the degeneracy factor (for graphene g= 4 due\nto valley and spin degeneracies) λis the band index, G\nis the retarded Green’s function, and the k-integration\nis over a single valley, which we assume to be a circle of\nradius |k|=kc.Gexpressed in terms of the retarded\nself-energy Σ λ(k, ω) is\nGλ(k, E) = [E−Ek,λ−Σλ(k, E) +iη]−1.(2)\nwhere Ek,λis the bare band energy of the state |kλ∝angbracketright\nandηis an infinitesimally small positive number. (In\nthis paper, the Green’s functions and self-energies are all\nassumed to be retarded.) Eqs. (1) and (2) show that if\nEk,λand Σ λ(k, E) are known, the density of states can\nbe obtained in principle from\nD(E) =g\nπ/summationdisplay\nλ/integraldisplaydk\n(2π)2−Im[Σ λ(k, E)] +η\n(E−Ek,λ−Re[Σ λ(k, E)])2+ (Im[Σ λ(k, E)]−η)2. (3)\nFor pure graphene systems, Σ( k, E) = 0 (ex-\ncluding electron–electron and electron–phonon inter-\nactions, which are not considered here), and hence\nIm[Gλ(k, E)] =−πδ(E−Ekλ). Close to the Dirac points\n(which we choose to the the zero of energy), the disper-\nsion for graphene is (we use /planckover2pi1= 1 throughout this paper)\nEk,λ=λvFk, (4)\nwhere λ= +1 and −1 for the conduction and valence\nbands, respectively, k=|k|is the wavevector with re-\nspect to the Dirac point, and vFis the Fermi velocity\nof graphene. Performing the k-integration in Eq. (1) for\nthe pure graphene case gives\nD0(E) =g\n2π|E|\nv2\nFθ(|E| −Ec), (5)\nwhere Ec=vFkcis the band energy cut-off.\nThe average density of states for a disordered 2DEG\ncan be obtained by averaging the Green’s function overimpurity configurations. The averaging procedure gives\na non-zero Σ which, in general, cannot be evaluated ex-\nactly. Various approximation schemes for Σ have there-\nfore been developed, four of which are described below.\nBorn Approximation — In the Born approximation,\nthe self-energy is given by the Feynman diagram shown\nin Fig. 1(a)24, and the expression for the self-energy is\nΣB,λ(k, E) =ni/integraldisplaydk′\n(2π)2|U(k−k′)|2\n×/summationdisplay\nλ′G0,λ′(k′, E)Fλλ′(k,k′) (6)\nwhere niis the impurity density, U(q) is the Fourier\ntransform of the impurity potential, G0is the bare\nGreen’s function and Fλλ′(k,k′) is square of the the over-\nlap function between the part of the wavefunctions of\n|kλ∝angbracketrightand|k′λ′∝angbracketrightthat are periodic with the lattice (here\nλ, λ′are band indices). For graphene states near the3\n(a)\n(b)\n+ + +...X\nX X X\nFIG. 1: Feynman diagrams for the (a) Born and (b) T-matrix\napproximations for the self-energy. The “x”, dotted line an d\nline with arrow signify the impurity, impurity potential, a nd\nGreen’s function, respectively. The Green’s functions are ei-\nther bare or self-consistent.\nDirac point,\nFλλ′(k,k′) =1\n2(1 +λλ′cosθkk′), (7)\nwhere θkk′is the angle between kandk′.\nSelf-Consistent Born Approximation — The Feynman\ndiagram for this self-energy is the same as Fig. 1(a),\nexcept that the bare Green’s function is replaced by the\nfull one. Consequently, the expression for Σ SBis the same\nas in Eq. (6), except with G0replaced by G.\nT-matrix Approximation — TheT-matrix approxima-\ntion is equivalent to the summation of Feynman diagrams\nshown Fig. 1(b). The expression for Σ Tis the same as\nin Eq. (6), except with Ureplaced by T, theT-matrix\nfor an individual impurity.\nSelf-consistent T-matrix Approximation — The Feyn-\nman diagram for this approximation is the same as in the\nT-matrix approximation, except that the bare Green’s\nfunctions are replaced by full ones. The expression for\nΣSTis the same as in Eq. (6), except with Ureplaced by\nT, and G0replaced by G.\nWhen the potentials for the impurities are not all iden-\ntical (for example, in the case where there is a distribu-\ntion of distances of charged impurities from the graphene\nsheet), one averages |U(k−k′)|2or|T(k−k′)|2over the\nimpurities.\nIII. UNSCREENED SHORT-RANGED (USR)\nDISORDER\nIn the present context, short-ranged impurities are im-\npurities which result from localized structural defects in\nthe honeycomb lattice, which are roughly on the length\nscale of the lattice constant. In this case, it is accept-\nable to approximate U(q) =U0, a real constant, for in-\ntravalley scattering processes. In this paper, we ignore\nthe intervalley processes. (However, we note that if thematrix element joining intervalley states is constant, in-\nclusion of intervalley scattering in our calculations is no t\ndifficult.) This simplification allows us to obtain some\nanalytic expressions for the self-energies in the approxi-\nmation schemes mentioned above.\nA. Self-energy for USR disorder\nFor USR disorder, the four approximation schemes we\nuse give self-energies that are independent of λandk.\n1. Born approximation\nThe self-energy for graphene with USR scatterers in\nthe Born approximation, using U(q) =U0, Eqs. (4) and\n(7) in Eq. (6), is\nΣ(usr)\nB(E) = ˜γBH0(E+iη); (8a)\n˜γB=niU2\n0\n2v2\nF; (8b)\nH0(ζ) = 2v2\nF/summationdisplay\nλ′/integraldisplaydk′\n(2π)2G0,λ′(k′, ζ)Fλλ′(k,k′)\n=/integraldisplayEc\n0dE′\n2π/parenleftbiggE′\nζ−E′+E′\n��+E′/parenrightbigg\n=−ζ\n2πln/parenleftbigg\n1−E2\nc\nζ2/parenrightbigg\n. (8c)\nAs a function of complex ζ,H0(ζ) is real and positive\n(negative) along the real axis from Ecto∞(−Ecto−∞).\nFurthermore, it has a branch cut in on the real axis of ζin\nbetween −EcandEc, so that for −Ec< E(real) < E c,\nH0(E±iη) =−(2π)−1/parenleftbigg\nEln/vextendsingle/vextendsingle/vextendsingle/vextendsingleE2\nc\nE2−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle±iπ|E|/parenrightbigg\n.(9)\nFig. 2 shows H0(E+iη).\nFor real Eand|E| ≪Ec,\nΣ(usr)\nB(E)≈ −˜γB\n2/bracketleftbigg2E\nπln/vextendsingle/vextendsingle/vextendsingle/vextendsingleE\nEc/vextendsingle/vextendsingle/vextendsingle/vextendsingle+i|E|/bracketrightbigg\n.(10)\nThe Born approximation damping rate for state |kλ∝angbracketrightis\nγB(k) =−2Im[Σ(usr)\nB(Ek,λ)] = ˜γBvFk.\n2. Self-consistent Born approximation\nThe self-energy Σ(usr)\nSBfor unscreened short-ranged\nscatterers in the self-consistent Born approximation is\ngiven by the self-consistent equation\nΣ(usr)\nSB(E) = ˜γBHSB(E+iη); (11a)\nHSB(ζ) =−1\n2π/bracketleftBig\nζ−Σ(usr)\nSB(ζ)/bracketrightBig\nln/bracketleftBigg\n1−E2\nc\n[ζ−Σ(usr)\nSB(ζ)]2/bracketrightBigg\n=H0/bracketleftBig\nζ−Σ(usr)\nSB(ζ)/bracketrightBig\n, (11b)4\n-2-1012\nE/E-1.0-0.50.00.51.0H  /E\ncc0\nIm[H  ]0Re[H  ]0\nFIG. 2: Real and imaginary parts of H0(E+iη) =\nΣ(usr)\nB(E)/˜γB[see Eq. (9)], where Σ(usr)\nB(E) is the self-energy\nfor USR impurities in the Born approximation.This shows that if |Σ(usr)\nSB(E)| ≪ |E|, then Σ(usr)\nSB(E)≈\nΣ(usr)\nB(E) (except possibly around E=±Ec, which is\nusually not experimentally relevant).\n3.T-matrix approximation\nIn general the impurity averaged T-matrix for a poten-\ntialU(q) is\nTk0λ0,kλ(E) =ni∞/summationdisplay\nn=1/bracketleftBigg/parenleftBiggn/productdisplay\ni=1/summationdisplay\nλi=±1/integraldisplaydki\n(2π)2U(ki−1−ki)G0,λi(ki, E)Fλi−1,λi(ki−1,ki)/parenrightBigg\nU(ki−k)Fλn,λ(kn,k)/bracketrightBigg\n.\n(12)\nIn this approximation, the self energy is\nΣT,λ(k, E) =Tkλ,kλ(E). (13)\nIfU(q) =U0, a constant, the term in the square paren-\ntheses in Eq. (12) is Un+1\n0Hn\n0(E+iη)/(2v2\nF)n. The sum\nthen gives\nΣ(usr)\nT(E) =˜γBH0(E+iη)\n1−U0\n2v2\nFH0(E+iη). (14)\n4. Self-consistent T-matrix approximation\nAs in self-consistent Born approximation, the H0(E) in\nEq. (14) is replaced by the self-consistent HST(E), giving\nΣ(usr)\nST(E) =˜γBHST(E+iη)\n1−U0\n2v2\nFHST(E+iη); (15a)\nHST(ζ) =H0/parenleftBig\nζ−Σ(usr)\nST(ζ)/parenrightBig\n. (15b)As in the self-consistent Born approximation, this shows\nthat if |Σ(usr)\nST(E)| ≪ | E|, then Σ(usr)\nST(E)≈Σ(usr)\nT(E)\n(except possibly around E=±Ec).\nB. Density of states for k-independent Σ\nWhen the self-energy is k- andλ-independent, the den-\nsity of states for graphene can be calculated analytically\nfrom Eq. (3). The result is\nD(E) =gsgv\n2π2v2\nF/braceleftBigg\nΓ\n2ln/parenleftbigg(E2\nc+ Ω2+ Γ2)2−4E2\ncΩ2\n(Ω2+ Γ2)2/parenrightbigg\n+ Ω/bracketleftbigg\ntan−1/parenleftbiggEc−Ω\nΓ/parenrightbigg\n−tan−1/parenleftbiggEc+ Ω\nΓ/parenrightbigg\n+ 2 tan−1/parenleftbiggΩ\nΓ/parenrightbigg/bracketrightbigg/bracerightBigg\n,\n(16)5\nwhere Γ( E) =−Im[Σ( E)] and Ω( E) =E−Re [Σ(E)].\nC. Density of states at the Dirac point\nThere has been considerable interest in the minimum\nDC electrical conductivity of disordered graphene as the\nFermi energy moves through the Dirac point.25,26There\nis still no consensus on whether the minimum conductiv-\nity is a universal value or not. Since the electrical con-\nductivity is directly proportional to the density of states\nat the Fermi energy, it is important to be able to deter-\nmine the density of states at the Dirac point of disordered\ngraphene. Because the minimum conductivity is non-zero\nas the Fermi energy passes through the Dirac point, the\ndensity of states should be nonzero.\nEq. (3) shows that if Σ( E)→0 when E→0, then\nthe density of states at the Dirac point D(0) = 0. (Note\nthat we have not included the term that is first order in\nthe impurity potential in our self-energy. Since this first-\norder term merely rigidly shifts the band by an amount\nniU0, ignoring this term is equivalent to shifting the zero\nof the energy by −niU0, and hence the Dirac point is still\natE= 0.) A non-zero density of states at the Dirac point\ndepends on a non-zero Im[Σ( E= 0)]. Since H0(E→\n0) = 0, it is clear from Eqs. (8a) and (14) that the Born\nandT-matrix approximations give zero density of states\nat the Dirac point.\nFor the self-consistent Born approximation, Eq. (11b)\ncan be rewritten as\nΣ(usr)\nSB,λ(E) =E\n1 +2π\n˜γBln/parenleftbigg\n1−E2c\n(E−Σ(usr)\nSB,λ(E))2/parenrightbigg\n−1\n,\n(17)\nwhich shows that Σ(usr)\nSB,λ(E→0) = 0, and therefore the\nself-consistent Born approximation also gives D(0) = 0.\nIn the case of the self-consistent T-matrix approxima-\ntion, re-writing Eq (15a) as\nΣ(usr)\nST(E) +niU0=niU0\n1−H0(E−Σ(usr)\nST(E))U0/(2v2\nF),\n(18)\nsetting E= 0, using Eq. (8c) and taking the imaginary\nparts of both sides of this equation gives\nIm[Σ(0)] ≡Γ(0) = Im\nniU0\n1−iU0Γ(0)\n2v2\nFln/parenleftbigg\n1 +E2\nc\nΓ(0)2/parenrightbigg\n.\n(19)\nIn the weak scattering limit, when ˜ γB≪1, the imag-\ninary term in the denominator of Eq. (19) is much less\nthan one (we check for self-consistency later), and thisgives\nΓw(0)≈Im/bracketleftbigg\niniΓw(0)U2\n0\nv2\nFln/parenleftbiggEc\nΓw(0)/parenrightbigg/bracketrightbigg\n= 2˜γBΓw(0)ln/parenleftbiggEc\nΓw(0)/parenrightbigg\n,\n(20)\nwhich implies that\nΓw(0) = Ecexp/parenleftbigg\n−1\n2˜γB/parenrightbigg\n. (21)\nNote that the result is non-analytic in U0. Inserting\nEq. (21) into the imaginary part of the denominator of\nEq. (19) (which we had assumed to be much smaller\nthan 1 in magnitude) gives the self-consistent criterion\nexp(−1/2˜γB)≪niU0/Ecfor the validity of Eq. (21).\nSubstituting this into Eq. (16) gives an average density\nof states at the Dirac point for weak scattering of approx-\nimately\nρw(0) =gsgvEc\nπ2niU2\n0exp/parenleftbigg\n−1\n2˜γB/parenrightbigg\n. (22)\nSimilar results to Eq. (22) have been reported15,19,27in\nstudies of disordered systems of fermions with linear dis-\npersions using other methods.\nWe mention that our calculation of the graphene den-\nsity of states at the Dirac point should only be consid-\nered as demonstrative since electron-electron interactio n\neffects are crucial28at the Dirac point, and the undoped\ngraphene system is not a simple Fermi liquid at the Dirac\npoint.\nIV. SCREENED SHORT-RANGED AND\nCHARGED IMPURITIES\nFree carriers will move to screen a bare impurity po-\ntential Vei(q), resulting in a screened interaction U(q) =\nVei(q)/ε(q), where ε(q) is the static dielectric function.\nTheε(q) results in an q-dependent effective electron-\nimpurity potential U, even in the case of short-ranged\n(q-independent) bare impurity potentials. This makes\nthe calculations much more involved than in the USR\ncase. Therefore, in this paper, we limit our investiga-\ntion of q-dependent screened potentials to the level of\nthe Born approximation.\nFor the dielectric function ε(q), we use the random\nphase approximation (RPA) for dielectric function appro-\npriate for graphene, given by ε(q) = 1−Vc(q)Π0(q) where\nVc(q) = 2πe2/(κq) is the two-dimensional Fourier trans-\nform of the Coulomb potential ( κis the dielectric con-\nstant of the surrounding material), and Π 0(q) is the static\nirreducible RPA polarizability for graphene.29We use\nVei(q) =Vc(q) for charged impurities, and Vei(q) =U0, a\nconstant, for short-range point defect scatterers.\nWe first look at the density of states at the Fermi sur-\nface;i.e., at energy E=λkFvF. To obtain this, we6\ncalculate the single-particle lifetime damping rate γin\nthe Born approximation, which is given by\nγλ(k) =−2Im[Σ λ(k, Ekλ)]\n=ni\n2πk\nvF/integraldisplayπ\n0dθ∝angbracketleft|Vei(q)|2∝angbracketright\nε(q)2(1 + cos θ),(23)\nwhere q= 2ksin(θ/2). Then, assuming that\nIm[Σ λ(k, E)] is relatively constant for kclose to kF,\nwe substitute1\n2γλ(kF) for−Im[Σ λ(k, EF)] into Eq. (3),\nwhich gives Eq. (16) with Γ =1\n2γλ(kF).\nWe assume that the charged or neutral impurities are\ndistributed completely at random on the surface of the\ninsulating substrate on which the graphene layer lies, the\nareal density for the charged and neutral impurities is\nnicandniδ, respectively, and the density of carriers in\nthe graphene layer is n=k2\nF/π, where kFis the Fermi\nwavevector relative to the Dirac point. (This relationship\nbetween nandkFtakes into account the spin and valley\ndegeneracy gs= 2 and gv= 2.) We use the RPA screen-\ning function at T= 0,29to obtain the effective impurity\npotential. The key dimensionless parameter that quan-\ntifies the screening strength is rs=e2/(κvF), which is\ncorresponding to the interaction strength parameter of a\nnormal 2D system ( i.e.the ratio of potential energy to ki-\nnetic energy). The Born approximation lifetime damping\nrates for screened charged-impurities γcandδ-correlated\nneutral impurities γδatkFare\nγc(kF) =nicEF\n4nIc(2rs); (24a)\nγδ(kF) =2EF˜γB\nπIδ(2rs). (24b)\nIn these equations,\nIc(x) =x−πx2\n2+x3f(x); (25a)\nIδ(x) =π\n4+ 3x/parenleftBig\n1−πx\n2/parenrightBig\n+x(3x2−2)f(x),(25b)\nwhere\nf(x) =\n\n1√\n1−x2ln/bracketleftBig\n1+√\n1−x2\nx/bracketrightBig\nforx <1;\n1 for x= 1;\n1√\nx2−1cos−11\nxforx >1.(26)\nIn Fig. 3 we show the calculated damping rates scaled\nby|EF|/=kFvFas a function the interaction param-\neterrs. For rs≪1 we have γc/EF≈nicrs/2nand\nγδ/EF≈˜γB/2. For rs≫1 we have γc/EF≈πnic/16n\nandγδ/EF≈˜γB/(2πrs). Thus, for small (large) rsthe\ndamping rate due to the short-ranged impurity domi-\nnates over that due to the long-ranged charged impu-\nrity. On the other hand, since γc(kF)∝k−1\nF∝n−1\n2and\nγδ(kF)∝kF∝n1\n2[Eq. (24)], in the low (high) carrier\ndensity limit the lifetime damping of single particle state s\nat the Fermi surface is dominated by charged impurity0.01 0.1 1r10-410-310-210-1100γ /E\nsF\nFIG. 3: Calculated damping rates scaled by Fermi energy\nγ/E Fas a function of rs. Graphene on a SiO 2(air) sub-\nstrate has an rs≈0.7 (2). Solid lines indicate damping\nrates ( γc) due to charged impurities with an impurity den-\nsitynic= 1011cm−2for different electron densities n= 1, 10,\n50×1011cm−2(from top to bottom), respectively. Dashed\nline indicate the damping rate ( γδ) due to short-ranged im-\npurity with impurity density niδ= 1011cm−2and potential\nstrength U0=1 KeV ˚A2, which correspond to ˜ γB= 0.11.\nNoteγδ/EFis independent on the electron density.\n(short-ranged impurity) scattering. The crossover takes\nplace around a density\nncross=nic\nniδπv2\nF\n4U2\n0Ic(2rs)\nIδ(2rs). (27)\nUsing Γ = γλ(kF)/2 in Eq. (16) gives (assuming EF,\nΓ≪Ec)\nD(EF)≈D0(EF)/bracketleftbigg1\n2+1\nπtan−1/parenleftbigg|EF|\nΓ/parenrightbigg\n+Γ\n2π|EF|ln/parenleftbiggE2\nc\nE2\nF+ Γ2/parenrightbigg/bracketrightbigg\n.(28)\nFor Γ/|EF| ≪1, this gives\nD(EF)≈D0(EF)/braceleftbigg\n1 +1\nπΓ\n|EF|/bracketleftbigg\nln/parenleftbiggEc\n|EF|/parenrightbigg\n−1/bracketrightbigg/bracerightbigg\n,\n(29)\nand for |EF|/Γ≪1\nD(EF)≈D0(EF)/bracketleftbigg1\n2+|EF|\nπΓ/bracketrightbigg\n+gsgv\n2π2Γ\nv2\nFln/parenleftbiggEc\nΓ/parenrightbigg\n.\n(30)\nWe can apply Eq. (29) for short-ranged impurity scat-\ntering and for charged impurity scattering in high carrier\ndensity limits, and Eq. (30) for charged impurity scat-\ntering in low density limits. Taking the limit |EF| →0\nin Eq. (30), it appears that for the case of screened\nCharged impurities, we obtain a finite density of states7\n-6 -4 -2 0 2 4 6\nω/Ε00.10.2-ImΣ  (k  ,ω)/E(a)\nFFF+\n-6 -4 -2 0 2 4 6\nω/E-0.100.1Re Σ  (k  ,ω)/E(b)\nFFF+\nFIG. 4: (a) Imaginary and (b) real parts of self energy of a dis -\nordered graphene at k=kFfor screened Coulomb scattering\npotential (solid lines) and for screened neutral short-ran ged\nscattering potential (dashed lines).\nat the Dirac point with the Born approximation (since\nΓ∝γc(kF)∝k−1\nF). However, recall that in deriving\nEqs. (29) and (30), we have assumed that Σ( k, EF) is\nconstant with respect to k, which is not necessarily the\ncase at the Dirac point.\nIn general, the damping rate (or the imaginary part of\nthe self energy) is a function of energy and wave vector\nrather than a constant. From Eq. (6), we calculated the\nself-energy of disordered graphene. Fig. 4 we show the\nself-energy of a conduction band electron ( λ= +1) for\nboth screened Coulomb scattering potential and screened\nneutral short-ranged potential. For Coulomb scatter-\ners we use the impurity density nic= 1012cm−2, and\nfor neutral short-ranged scatterers the impurity density\nniδ= 1011cm−2and potential strength U0=1 KeV\n˚A2. The self-energies in the valence band ( λ=−1)\nare related to the self-energy in the conduction band\nby ReΣ +(k, ω) =−ReΣ−(k,−ω) and ImΣ +(k, ω) =\nImΣ−(k,−ω). As ω→0,−ImΣ λ(kF, ω)→ |ω|, and\nReΣ λ(kF, ω)→ωln|ω|for both scattering potentials.\nHowever, for large value of |ω|the asymptotic behaviors\nare different, that is, as |ω| → ∞ − ImΣ(kF, ω)∝ |ω|−1\nfor Coulomb scattering potential and −ImΣ(kF, ω)∝ |ω|\nfor short-ranged potential. Note that by using only\nthe (non-self-consistent) Born approximation in this sec-\ntion, we assume weak scattering and ignoring multiple-\nscattering events in calculated Σ( k, ω). Therefore, the-3 -2 -1 0 1 2 3\nω/E0123D(ω)/D  (E  )\nFF(a)0\n-3 -2 -1 0 1 2 3\nω/E01234D(ω)/D  (E  )\nFF0(b)\nFIG. 5: The density of states in the presence of impurity (a)\nfor screened Coulomb potential and (b) for screened short-\nranged potential. In (a) we use the charged impurity densiti es\nnic= 0, 1, 5 ×1012cm−2(from bottom to top), and in (b)\nthe short-ranged impurity density nid= 0, 0.5, 1 ×1012cm−2\n(from bottom to top) and potential strength U0=1 KeV ˚A2.\nresults are unreliable in the strong disorder limit ( i.e.,\nwhen Σ( k, ω) is modified significantly from its lowest-\norder form).\nIn Fig. 5 the density of states in the presence of im-\npurity is shown for different impurity densities. In Fig.\n5(a) we show the density of states for Coulomb impurity\npotential with impurity densities, nic= 0, 1012cm−2,\n5×1012cm−2(from bottom to top), and in Fig. 5(b) we\nshow the density of states for neutral short-ranged impu-\nrity potential with densities, nid= 0, 5×1011cm−2, 1012\ncm−2(from bottom to top) and potential strength U0=1\nKeV˚A2. The calculated density of states is normalized\nbyD0(EF) = (gsgv/2π)EF/γ2. The density of states is\nenhanced near Dirac point ( E= 0), but as |E| →0 it\ngoes zero as D(E)→ |E|ln|E|for both types of screened\nimpurity scattering. Based on the results of Section III,\nwe expect that this result is an artifact of the Born ap-\nproximation, and that the density of states should in fact\nbe non-zero. The enhancement of the density of states\ncan be explained as follows. In normal 2D system with\nfinite disorder the band edge Eedge,0of the pure conduc-\ntion (valence) band is shifted to Eedge,imp<0 (>0) and8\na band tail forms below (above) the band edge of a pure\nsystem. Thus, the density of states in the presence of\nimpurities is reduced for E > 0 (E < 0) because the\nstates have been shifted by the impurity potential into\nthe band tail. However, for graphene since the conduc-\ntion band and the valence band meet at the Dirac point\nthe band tail (or shift of band edge) cannot be formed,\nwhich gives rise to enhancement of density of states near\nDirac point.\nBefore concluding we point out that our perturbative\ncalculation of the graphene density of states assumes that\nthe system remains homogeneous in the presence of impu-\nrities. It is, however, believed3,30that graphene carriers\ndevelop strong density inhomogeneous (i.e. electron-hole\npuddle) at low enough carrier densities in the presence of\ncharged impurities due to the breakdown of linear scat-\ntering. In such an inhomogeneous low-density regime\nclose to the Dirac point, our homogeneous perturbative\ncalculation does not apply.\nV. COMPARISONS TO OTHER WORKS\nIn this section, we compare and contrast our model of\ndisorder and results to other works in the field.\nPereset al.16studied the effect of disorder in graphene\nby considering the effect of vacancies on the honeycomb\nlattice. For a finite density of vacancies, they found that\nthe density of states at the Dirac point is zero for the\n“full Born approximation” (equivalent to our T-matrix\napproximation) and non-zero for the “full self-consistent\nBorn approximation” (equivalent to our self-consistent T-\nmatrix approximation). Our results are consistent with\ntheirs, even though the regimes that are studied are dif-\nferent. Vacancies correspond to the limit where the im-\npurity potential U0→ ∞ , whereas this work is more\nconcerned with the weak impurity-scattering limit.\nPereira at al.17considered, among several different\nmodels of disorder, both vacancies and randomness in the\non-site energy of the honeycomb lattice. They numeri-\ncally calculated the density of states for these models of\ndisorder. For compensated vacancies (same density of va-\ncancies in both sub-lattices of the honeycomb structure)\nthey found that the density of states increased around\nthe Dirac point. (Ref. 17 also studied the case of un-\ncompensated vacancies, but that has no analogue in our\nmodel of disorder.) For the case of random on-site impu-\nrity potential, they find that “there is a marked increase\nin the DOS (density of states) at ED(the Dirac point)”\nand “the DOS becomes finite at EDwith increasing con-\ncentration” of impurities. Our self-consistent T-matrix\napproximation result is consistent with their numerical\nresults, although it should be mentioned again that it\nstrictly does not apply to the case of vacancies.Wuet al.20numerically investigated the average den-\nsity of states of graphene for the case of on-site disorder.\nThey found that for weak disorder, the density of states\nat the Dirac point increased with both increasing den-\nsity of impurities and strength of the disorder. (In their\nwork, they did not absorb the shift in the band due to\nthe impurity potential in their definition of the energy,\nso the Dirac point had a shift ED=xvwhere xis the\nconcentration of impurities and vis the on-site impu-\nrity energy.) Their numerical results for the minimum in\nthe average density of states (Fig. 3(b) in Ref. 20) seem\nto indicate a non-linear dependence of the value of the\nminimum as a function of the strength of the disorder\npotential, and is at least not inconsistent with Eq. (22).\nThe issue of the effect of screening of the impurity in-\nteractions on the density of states discussed in Section IV,\nto the best of our knowledge has not yet been treated in\nthe literature. Qualitatively, the effect of screened im-\npurities away from the Dirac point is the increase the\ndensity of states, and is consistent with numerical results\nfor random on-site disorder.17,20(Since our treatment of\nscreened impurities is at the level of the Born approx-\nimation, we do not obtain either a non-zero density of\nstates at the Dirac point nor resonances in the density of\nstates.17,18,20)\nVI. CONCLUSION\nWe have calculated the density of states for disordered\ngraphene. In the case of unscreened short-ranged impuri-\nties, we utilized the non-self-consistent and self-consis tent\nBorn and T-matrix approximations to calculate the self-\nenergy. Among these, only the self-consistent T-matrix\napproximation gave a non-zero density of states at the\nDirac point, and the density of states is a non-analytic\nfunction of the impurity potential. We investigated the\ndensity of states in the case of screened short-ranged and\ncharged impurity potentials at the level of the Born ap-\nproximation. We find that, unlike the case of parabolic\nband 2DEGs, in graphene near the band-edge ( i.e., the\nDirac point) the density of states is enhanced by impu-\nrities instead of being suppressed. 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B 75, 245401 (2007).\n19Bal´ aza D´ ora, Klauss Ziegler, and Peter Thalmeier, Phys.\nRev. B 77, 115422 (2008).\n20Shangduan Wu, Lei Jing, Quanxiang Li, Q. W. Shi, Jie\nChen, Haibin Su, Xiaoping Wang, and Jinlong Yang, Phys.\nRev. B 77, 195411 (2008).\n21Aur´ elein Lherbier, X. Blase, Yann-Michel Niquet, Fran coi s\nTriozon, and Stephan Roche, Phys. Rev. Lett. 101, 036808\n(2008).\n22Y. Morita and Y. Hatsugai, Phys. Rev. Lett. 79, 3728\n(1997); Shinsei Ryu and Yasuhiro Hatsugai, Phys. Rev. B\n65, 033301 (2001).\n23F. Guinea, Baruch Horovitz, and P. Le Doussal, Phys. Rev.\nB77, 205421 (2008).\n24Seee.g., G. Rickaysen, Green’s Functions and Condensed\nMatter (Academic Press, New York, 1980); S. Doniach\nand E. H. Sondheimer, Green’s Functions for Solid State\nPhysicists (Imperial College, London, 1998); G. D. Mahan,\nMany Particle Physics , 3rd ed. (Plenum, New York, 2000).\n25Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. H.\nHwang, S. Das Sarma, H. L. Stormer, and P. Kim, Phys.\nRev. Lett. 99, 246803 (2007).\n26K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nY. Zhang, M. I. Katsnelson, I. V. Grigorieva, S. V.\nDubonos, and A. A. Firsov, Nature 438, 197 (2005).\n27P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, Phys.\nRev. B 74, 235443 (2006).\n28S. Das Sarma, E. H. Hwang, and W. K. Tse, Phys. Rev. B\n75, 121406(R) (2007).\n29E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418\n(2007).\n30E. Rossi and S. Das Sarma, arXiv:0803.0963."    },    {        "title": "0805.3870v1.Stability_of_the_density_wave_state_of_a_dipolar_condensate_in_a_pancake_trap.pdf",        "content": "arXiv:0805.3870v1  [cond-mat.other]  26 May 2008Stability of the density-wave state of a dipolar condensate in a pancake trap\nO. Dutta∗, R. Kanamoto†, and P. Meystre\nB2 Institute, Department of Physics and College of Optical S ciences,\nThe University of Arizona, Tucson, AZ 85721, USA\n(Dated: October 31, 2018)\nWe study a dipolar boson-fermion mixture in a pancake geomet ry at absolute zero temperature,\ngeneralizing our previous work on the stability of polar con densates and the formation of a density-\nwave state in cylindrical traps. After examining the depend ence of the polar condensate stability on\nthe strength of the fermion-induced interaction, we determ ine the transition point from a ground-\nstate Gaussian to a hexagonal density-wave state. We use a va riational principle to analyze the\nstability properties of those density-wave state.\nPACS numbers: 03.75.-b, 42.50.-p, 33.80.-b, 03.65.-w\nI. INTRODUCTION\nThe recent realization of a Bose-Einstein conden-\nsate of chromium atoms [1, 2] opens up the study of\nquantum-degenerate gases that interact via the long\nrange, anisotropic magnetic dipole interaction. This is\nan anisotropic and long-range interaction that leads to\nthe appearance of a wealth of new properties past those\ncharacteristic of systems with isotropic interactions [3].\nInthe caseof52Crs-waveinteractionsarenormallymuch\nstronger than the dipolar part, but it has been shown ex-\nperimentallythatbyapplyingamagneticfieldthe s-wave\nscattering length can be lowered to values comparable to\nthe Bohr radius [4], so that the dipole interaction domi-\nnates the system. In the regime where the dipole-dipole\ninteraction is dominant the condensate is characterized\nby the existence of a metastable state whose properties\ndepends on the trapping geometry [5, 6, 7, 9, 10, 11],\nas was experimentally demonstrated in Ref. [8]. A roton\nfeature has also been predicted to exist in these systems\nfor appropriate parameters. There would considerable\ninterest indeed in accessing the associated roton instabil-\nity [12, 13, 14], as it is characterized by the spontaneous\ngeneration of a periodic density modulation in the con-\ndensate [15]. Unfortunately that state is unstable against\ncollapse, but we found recently that this difficulty can\nbe circumvented by the addition of a small fraction of\nnon-interacting fermions, resulting in a significant stabi-\nlization of the bosonic system in cylindrical traps [16].\nThe simultaneous trapping of bosonic and fermionic iso-\ntopes of chromium [17] also points to an interesting di-\nrections, with the possibility to observenovel ground and\nmetastable phases in quantum degenerate polar boson-\nfermion mixtures.\nThis paper extends our previous study of dipolar\nboson-fermionmixturesfromcylindricaltrapstopancake\ngeometries,showingthat dipolarbosonscanbe stabilized\n∗Corresponding author. Electronic address:\ndutta@physics.arizona.edu\n†Present Address: Ochanomizu University, Tokyo 112-8610, J apanconsiderably by increasing the boson-fermion s-wave\nscattering length, and more importantly the density-\nwave states of dipolar bosons can be stable for typical,\nzero-field boson-fermion interaction strengths. Section II\ndescribes the induced potential created by fermions on\nbosons, and Sec. III discusses the general properties of\nthe energy functional as a function of boson-fermion in-\nteractionusing a variationalGaussianansatz. Section IV\npresentsananalysisofthetransitionpointtothedensity-\nwave state for various combinations of trap aspect ratios\nand boson-boson contact interaction strengths. Section\nV introducesa density-waveansatzin the form ofa series\nof shifted Gaussians. The stability of these density-wave\nstates is determined by varying the width of each Gaus-\nsian to seek minima in the interaction energy for various\nboson-boson s-wave interaction. We find a condition for\nthe stability of these types of density modulated states.\nFinally, Section VI is a summary and outlook.\nII. FERMION-INDUCED INTERACTION IN\nPOLAR CONDENSATES\nWe consider a mixture of Nbdipolar bosons of mass\nmbandNfsingle-component fermions of mass mfcon-\nfined in a pancake-shaped trap characterized by a tight\nharmonic potential of frequency ωzalong the z-axis and\na softer harmonic potential of frequency ω⊥in the trans-\nversedirection. Apolarizingexternalelectricormagnetic\nfield, is taken to be along the z-axis . The dipole-dipole\ninteraction between two bosonic particles separated by a\ndistance ris then\nVdd(r) =gdd/parenleftbigg\n1−3z2\nr2/parenrightbigg1\nr3, (1)\nwheregddis the dipole-dipole interaction strength. In\nthe mean-field approximation, the energy functional for\nthe order parameter φ(r) of the dipolar condensate can\nbe expressed as\nE=/integraldisplay\nφ∗(r)H0φ(r)d3r+gNb\n2/integraldisplay\n|φ(r)|4d3r(2)\n+Nb\n2/integraldisplay/integraldisplay\n|φ(r)|2Vdd(r−r′)|φ(r′)|2d3rd3r′+Eind,2\nwhere\nH0=−/planckover2pi12\n2mb∇2+mbω2\nz\n2/bracketleftbig\nλ2/parenleftbig\nx2+y2/parenrightbig\n+z2/bracketrightbig\n(3)\nis the sum of the kinetic energy and the trapping po-\ntential and λ=ω⊥/ωz. The second term in the energy\nfunctional (2) denotes the contact interaction between\nbosons, characterized by the strength g= 4π/planckover2pi12abb/mb\nwithabbbeing the s-wavescatteringlength, andthe third\nterm describes the nonlocal dipole-dipole interaction be-\ntween bosons. Finally, the last term Eindaccounts for\nthe fermion-induced interaction Vind(k) between bosons,\ngiven in linear response theory [18] as follows.\nWe assume a contact boson-fermion interaction of\nstrength gbf= 2π/planckover2pi12abf/mr, where abfis the boson-\nfermions-wavescattering length and mr=mbmf/(mb+\nmf) is the reduced mass. The boson-fermion inter-\naction energy has the form gbf/integraltext\nnf(k)n(−k)d3k, with\nn(k) andnf(k) being the bosonic and fermion densi-\nties in the momentum space. The linear response of the\nfermions to a bosonic density n(k) can be expressed as\nnf(k) =Vind(k)n(k),sothatfortheeffectofthefermions\non the bosonic energy functional is\nEind=1\n2gbfNb\n(2π)3/integraldisplay\nVind(k)n(k)n(−k)d3k.(4)\nThe explicit form of the induced potential is\nVind(k) =gbfχf(k), (5)\nwhereχfis the density response function [18]. It\nis related to the dynamical structure factor S(k,ω),\nwhich is the probability of exciting particle-hole pairs\nwith momentum kout of the Fermi sea, by χf(k) =\n−2/integraltext∞\n0dω′[S(k,ω′)/ω′]. For a non-interacting single-\ncomponent Fermi system we have\nS(k,ω) =∞/summationdisplay\np<kf\n|p+k|>kfδ(ω−ω0\npk), (6)\nwherep=|p|,kfis the Fermi momentum, the excitation\nenergy is ω0\npk=pkcosθ/mf+k2/(2mf), andθis the\nrelative angle between pandk. This expression assumes\nthat the fermions are locally free, so that there is a localFermi sphere in momentum space. In this local density\napproximation, kfis given by [19]\nkfℓz= 1.9N1/6\nfλ1/3/radicalbiggmf\nmb, (7)\nwhereℓz=/radicalbig\n/planckover2pi1/(mbωz) is the oscillator length in the z\ndirection.\nFollowing Ref. [16] the induced potential Vind(k) is\ngiven by\nVind(k)=\n\ngbfν/bracketleftBigg\n−1+∞/summationdisplay\nn=1/parenleftbiggk\n2kf/parenrightbigg2n1\n4n2−1/bracketrightBigg\n, k <2kf,\n−gbfν∞/summationdisplay\nn=1/parenleftbigg2kf\nk/parenrightbigg2n1\n4n2−1, k > 2kf,(8)\nwhereν=kfmf/(π/planckover2pi1)2is the three-dimensional\nfermionic density of states, and k2=k2\nx+k2\ny+k2\nzis\nthe square of the fermionic momentum. The induced po-\ntential is attractive for very low momenta and goes to\nzero with increasing momenta.\nThroughout this paper, we consider the parameters\ncorresponding to a52Cr-53Cr mixture, and fix the num-\nber of fermions to be Nf= 103unless otherwise stated.\nThezero-fieldbosonic s-waveinteractionfor52Cristaken\nto beabb= 103a0, andabf= 70a0for the boson-fermion\nscattering length [20].\nIII. STABILITY OF THE GAUSSIAN DIPOLAR\nCONDENSATE\nWe now proceed to determine the stability of the dipo-\nlar condensate, using a variational ansatz in the parame-\nter space of the fermion-induced interaction and dipolar\nstrength. The variational wave function is taken as the\nGaussian\nφ(r) =1/radicalbig\nπ3/2d2dzexp/bracketleftbigg\n−x2+y2\n2d2−z2\n2d2z/bracketrightbigg\n,(9)\nwithd,dzbeing the variational parameters and/integraltext\nd3r|φ(r)|2= 1. Substituting this Gaussian ansatz and\nits Fourier transform into Eqs. (2) and (4) yields the en-\nergyEgof the condensate as\nmbℓ2\nz\n/planckover2pi12Eg=1\n2/parenleftbigg1\n2+η2/parenrightbigg/parenleftbiggℓz\ndz/parenrightbigg2\n+1\n2/parenleftbigg1\n2+λ2\nη2/parenrightbigg/parenleftbiggdz\nℓz/parenrightbigg2\n+g3d/parenleftbiggℓz\ndz/parenrightbigg3/bracketleftBigg\n2\n3η2−F(η−1)+/parenleftbiggdz\nℓz/parenrightbigg3/braceleftbig\ng<E<\n1+g>E>\n1+gind(E<\n2−E>\n2+E<\n3)/bracerightbig/bracketrightBigg\n, (10)\nwhereη=dz/d, and˜kf=kfℓz. The deriva-\ntion of algebraic forms of the interaction-energy termsE<\n1,E>\n1,E<\n2,E>\n2,E<\n3and of the function Fare straight-3\nforward but lengthy, and their derivations are relegated\nto an Appendix. In Eq. (10) we have also introduced the\neffective three-dimensional dipole-dipole interaction\ng3d=mbNbgdd√\n2π/planckover2pi12ℓz.\nThe coefficients g<andg>find their origin in the\nmomentum-independent contact interaction, given by\nthe summation of the s-wave boson-boson scattering and\nthe constant terms in the induced interaction (8). They\nare given explicitly by\ng<=g−g2\nbfν\n4πgdd,\ng>=g\n4πgdd. (11)\nFinally\ngind=g2\nbfν\n48πgdd˜k2\nf\nis due to the nonlocal induced interaction. We re-\nmark that the effective contact interaction consists of\nthe boson-boson contact interaction as well as the\nmomentum-independent part of the dipole-dipole inter-\naction and of the induced interaction. From Eqs. (10)\nand (11) we have that for low momenta\ngs=2\n3+g<. (12)\nIt is known that in boson-fermion mixtures without\ndipolar interaction, phase separation occurs when gsbe-\ncomes negative [22, 23, 24]. In this paper, in contrast, we\nrestrict our considerations to the case gs>0 by chang-\ning the boson-boson contact interaction g, so that phase\nseparation does not take place.\nThe energyfunctional (10) wasminimized with respect\ntoηanddz/ℓzfor various parameter values, with our re-\nsults summarized in Fig. 1. Without the fermion induced\ninteraction, gind= 0, the ground-state energy of the sys-\ntem is not bounded from below, and the strong trap-\nping inzdirection creates a local minimum in the energy\nlandscape as a function of dzandη[Fig.1 (a)]. For fi-\nnite induced interactions gind>0, we find in contrast\nthat the energy landscape is characterized by two min-\nima, as shown in Fig. 1 (b), (c), and (d). In Fig. 1(b) the\nglobal minimum, which occurs for ( η,dz/ℓz)≃(3,2.3),\nis a Gaussian state with narrow width in the trans-\nversex-yplane. The additional local minimum close to\n(η,dz/ℓz)≃(0.2,1.4) is a metastable state.\nWith increasing boson-fermion interaction gind,\nthough, the ground-state energy corresponding to the\nglobal minimum at η >1 approaches that of the local\nminimum [Fig. 1 (c)], and these minima eventually reach\nequal energies at a critical value of gind. The new ground\nstate past that point is characterized by the parameters\nFIG. 1: Energy landscape Egobtained by the variational\nGaussian ansatz Eq. (9) as a function of η=dz/danddz/ℓz\nat a constant dipolar interaction strength g3d= 30. In these\nfigures the energy is cut off at Eg= 1.5 from above and\nEg=−0.5 from below for viewability. (a) Behavior of Egin\nthe absence of the fermion-induced interaction gind= 0 (b)\nEnergyEgforgind= 0.02. The plot is characterized by pres-\nence of two minima: (i) the true, narrow Gaussian ground\nstate located at ( η,dz/ℓz)≃(3,2.3); and (ii) a pancake-\nshaped metastable state located at ( η,dz/ℓz)≃(0.2,1.4). (c)\nEnergy landscape for gind= 0.06. The energies of the ground-\nand metastable states approach each other. (d) The broad\nGaussian metastable state at ( η,dz/ℓz)≃(0.2,1.4) eventually\nbecomes the true ground state for gind= 0.07. In the figures\nthe global (ground state) and local minima (metastable stat e)\nare indicated by a circle and cross, respectively.\nη <1 anddz∼ℓz[Fig. 1 (d)], that is, it is a wide\nGaussian in x-yplane.\nExperimentally one may either vary the boson-boson\ns-wavescattering length while keeping the boson-fermion\nscattering length constant, or vary gbfwith constant\ns-wave interaction g. In a boson-fermion mixture of\nchromium isotopes, typically gind∼10−3is small, which\ncorresponds to an energy landscape that resembles that\nof Fig. 1(b) and the broad Gaussian corresponds to a\nmetastable state. Such a metastable state has been\nachieved experimentally in experiments by the Stuttgart\ngroup [8]. A system that offers the potential to reach the\nregime of Fig. 1(d) is provided by a mixture of bosonic\n87Rb and fermionic40K with the scattering length of\nrubidium atoms tuned close to zero. In that mixture,\na zero-field scattering length abf≈250a0[21] gives\ngind∼0.2.4\nIV. TRANSITION TO A DENSITY WAVE\nThe excitation spectrum of condensates dominated by\na dipolar interaction is predicted to exhibit a roton min-\nimum [12, 13]. As a consequence, a bosonic density-\nmodulatedstateinthe x-yplanemayariseasalocalmin-\nimum, and it may actually have a lower energy than themetastable Gaussian state and for high enough bosonic\ndensities [15]. This section discusses the transition to the\nappearance of such a state as a function of number of\nbosonic particles for various combinations of trap ratio.\nWe proceed by introducing the new variational wave\nfunction that describes a density-wave structure with tri-\nangular symmetry,\nφt\ndw(x,y,z) =φ(x,y,z)/bracketleftBigg\na0+∞/summationdisplay\nn=1an/braceleftBigg\ncos/parenleftBigg\nn˜k0x\ndz/parenrightBigg\n+2cos/parenleftBigg\nn˜k0x\n2dz/parenrightBigg\ncos/parenleftBigg\nn√\n3˜k0y\n2dz/parenrightBigg/bracerightBigg/bracketrightBigg\n, (13)\nwhereφ(x,y,z) is defined in Eq. (9), nis an integer, and ˜k0=k0dz≪η=dz/d. Substituting this trial wave\nfunction (13) into Eq. (1), we find that the scaled excess energy o f the density-modulated state relative to the\nGaussian state\nǫ(˜k0,a1,a2,a3) =2md2\nz\n/planckover2pi12(Edw−Eg)\nis approximately given by\nǫ(˜k0,a1,a2,a3)\n3≈˜k2\n0\n4(a2\n1+4a2\n2+9a2\n3)+g3dnd\n4/bracketleftBig\n(a2\n1+2a0a1+a1a2+a2a3)2Veff(˜k0)\n+ (a2\n1+2a1a2)2Veff(√\n3˜k0)+1\n4(a2\n1+2a2\n2+4a0a2+2a1a3)2Veff(2˜k0)\n+ 2(a2\n1a2\n2+a2\n1a2\n3+a2\n2a2\n3)Veff(√\n7˜k0)+(2a0a3+a2\n3)2Veff(3˜k0)/bracketrightBig\n, (14)\nwherend≡η2ℓz/dz[25]. The normalization condition reads a2\n0+3(a2\n1+a2\n2+a2\n3)/2 = 1, and the effective transverse\npotential is\nVeff(˜k0)≈\n\n2\n3+g<−/radicalbiggπ\n2˜k0erfcx/parenleftBigg˜k0√\n2/parenrightBigg\n+gind/parenleftbiggℓz\ndz/parenrightbigg2\n˜k2\n0, ˜k0<2kfdz,\n2\n3+g\n4πgdd−/radicalbiggπ\n2˜k0erfcx/parenleftBigg˜k0√\n2/parenrightBigg\n−gind/parenleftbiggℓz\ndz/parenrightbigg2(2kfdz)4\n˜k2\n0,˜k0>2kfdz,(15)\nIn evaluating Eq. (14) we kept only the first four terms\nn= 0,...,3 of Eq. (13) as the energy converges at the\ntransition point. The magnitude of the error in that ap-\nproximate expression is estimated to be of the order of\ne= exp(−˜k2\n0η2/4). In the subsequent calculations we\nconsider values of ˜k0such that this error is less than or\non the order of 10−6.\nWe numerically determine the variational parameters\nthatminimizethe excessenergy ǫ(˜k0,a1,a2,a3)asafunc-\ntion ofg<for fixed ndandg3d. These results are summa-\nrizedinFig.2, whichshowsthecriticalnumberofbosonic\nparticles Nbsuch that ǫhas a minimum for ˜k0/ne}ationslash= 0. By\ntuning the trap aspect ratio λ=ω⊥/ωzto higher val-\nues, the critical number of bosons Nbrequired to achieve\nǫ <0, that is, a transition from a Gaussian to a density-\nwave metastable state with lower energy, is lowered as a\nresult of an increased nd. In addition, we note that thevalue of˜k0determined by the present variationalcalcula-\ntion matches closely the position of roton minima at the\nexcitation spectrum of the condensate as in Ref. [15]. We\nalso checked the energy with density wave having square\nsymmetry and find that the triangular one always has\nlower energy.\nV. DENSITY WAVE STABILITY\nIn the previous section we determined the transition\nto the density-wave instability, assuming the ansatz (13).\nAs alreadymentioned, this transitionis related to the ex-\nistence of a roton minimum in the excitation spectrum.\nHowever, assuming such a density wave does not pro-\nvide an answer to the question of its stability against\ncollapse. This is the issue that we address now, restrict-5\n0.16 0.18 0.2 0.22 0.24 0.260246810Nb(x 104) \ng< \nFIG. 2: Critical number of bosonic52Cr atoms needed for the\ntransition from the Gaussian state to a hexagonal density-\nwave state for the trap aspect ratios λ= 0.25 (solid curve)\nandλ= 0.22 (dashed curve). Here ℓz= 0.15µm.\ning our considerations to the regime where the boson-\nfermion interaction strength is close to its zero-field value\nabf= 70a0. This corresponds in the case of52Cr-53Cr\nisotopes to gind∼10−3. This is the parameter regimecharacterized by a Gaussian metastable state, see Sec. II\nand Fig. 1(b).\nOnce the transition to the density-wave state has oc-\ncurred the condensate can also be described as a super-\nposition of shifted Gaussian wave functions within the\ntwo-dimensional Z≡x+iyplane,\nφdw(Z,z) =1/radicalbig\nMπ3/2ξ2dzM/summationdisplay\nj=1exp/bracketleftbigg\n−(Z−Zj)2\n2ξ2−z2\n2d2z/bracketrightbigg\n(16)\nwherejis the index of the lattice site, Zjthe lattice\nvectors generating periodic density modulations, ξthe\nwidth of each density peak, Mthe total number of den-\nsity peaks, and lthe distance between neighboring den-\nsity peaks. It is given by l= 2π/k0wherek0is the\nposition of roton minimum in the excitation spectrum of\nthe condensate.\nUsing the results of Sec. III to investigate the stabil-\nity of each Gaussian in Eq. (16), we now show that the\npresence of fermions substantially stabilizes the density-\nwave state, noting that a stable density wave should be\ncharacterized by non-zero values of ξ/dzanddz/ℓz.\nFrom Eq. (16), the total interaction energy is found to\nbe [26]\nEint(ξ,dz) =mbℓ2\nzEint\n/planckover2pi12M\ng3d\n=/parenleftbiggℓz\ndz/parenrightbigg3/bracketleftBigg\n2\n3/parenleftbiggdz\nξ/parenrightbigg2\n−F(dz/ξ)+/parenleftbiggdz\nℓz/parenrightbigg3/braceleftbig\ng<E<\n1+g>E>\n1+gind(E<\n2−E>\n2+E<\n3)/bracerightbig\n+f(l,ξ,dz)/bracketrightBigg\n,\n(17)\nwhere we have assumed that ξ≪l, i.e., that neigh-\nboring Gaussian peaks in Eq. (16) have little overlap.\nHereE<\n1,E>\n1,E<\n2,E>\n2,E<\n3andFhave the same form as\nin the Appendix, but σis now a function of ( ξ,dz,θ),\nσ=/radicalBig\nd2zcos2θ+ξ2sin2θ. The function f(l,ξ,dz) is\nresponsible for the particular geometry of the density-\nmodulated state. Other terms, on the other hand, just\narise from the energy of each individual Gaussian and\nwe can thus apply the results of Sec. III to the present\nanalysis.\nAssuming a triangular crystal and including only the\ninteraction between nearest neighbors, we have\nf(l,ξ,dz) = 6/integraldisplay∞\n0Veff(˜k)exp/parenleftBigg\n−˜k2ξ2\n2d2z/parenrightBigg\nJ0/parenleftBigg˜kl\ndz/parenrightBigg\n˜kd˜k,\nwith˜k=kdz, the effective two-dimensional potential\nVeff(˜k) is defined in Eq. (15), and J0is the zeroth-order\nBessel function of the first kind.\nWe now discuss the minimum of the energy functionalEint(ξ,dz) as a function of the effective strength of the\ncontact interaction gs, which is varied by changing g<,\nsee Eq. (12). Without boson-fermion interaction, the in-\nteractionenergyis amonotonically increasingfunction of\nξwithEint(ξ→0,dz)→ −∞. As a result each gaussian\nof the density wave Eq. (16) collapses. A typical exam-\nple of the dependence of the interaction energy on ξis\nshown as the dotted curve in Fig. 3(a). With non-zero\nboson-fermion interaction energy, in contrast, the inter-\naction energy (17) exhibits a minimum at a finite ξas\nillustrated by the solid curve in Fig. 3(a) f or dz= 1.8ℓz\nandg<= 0.17. The existence of that minimum implies\nthe stability of each Gaussian, that is, the stability of the\nwave function (16).\nWe can determine the dependence of the gaussian\nwidthsξon thes-wave boson-boson scattering length by\nchanging g<. To do this we first minimize Eq. (17) as a\nfunction of g<, treating dzandξas variational parame-\nters. Figure3(b)showstheresultingvalue ξ/ℓzasafunc-\ntion ofg<forλ= 0.25,Nf= 1000 and ℓz= 0.15µm. We6\n0.2 0.4 0.6 0.8 1−3−2−10\n0.14 0.16 0.18 0.2 0.22 0.2410−210−1100 int  \ng< ξ/l\nz log10(ξ/lz) (a) \n(b) ε \nFIG. 3: (a) Interaction energy Eintof Eq. (17) as a function\nofξ/ℓzatλ= 0.25,ℓz= 0.15µm,dz= 1.5ℓz, andg<= 0.18.\nForgbf= 0 the energy is unbounded from below, but for\ngbf= 70a0, a minimum appears at ξ/ℓz≈0.1. (b) Log-scale\nplot of the normalized width ξ/ℓzthat minimizes Eintas a\nfunction of g<.\n0 5 10 15 20 25−0.1−0.0500.05 V eff(k⊥dz)  l3z/d3z\nk⊥dz \nFIG. 4: Effective potential Veffas a function of k⊥dzfor the\nparameters of Fig. 3(b) for g<= 0.2 (solid curve), and the\ncritical value of g<≈0.14 (dashed curve).\nobserve that ξdecreases for smaller values of the contact\ninteraction g<, and for g</lessorsimilar0.14, the minimum-energy\nstate corresponds to the collapsed state, ξ= 0. Below\nthat point the density wave is unstable.\nThis behaviorcan be understood from the effective po-\ntential by first integrating out the zdirection to obtain\nthe effective potential ( ℓz/dz)3Veff(k⊥dz), where Veffis\ngiven by Eq. (15) and k2\n⊥=k2\nx+k2\ny. The width dz\nof the Gaussians is determined by the position of the\nminimum in Eq. (17). For g</lessorsimilar0.14,Veff(k⊥dz) is at-tractive for allhigh momenta, but at the critical value\ng<≈0.14, it approaches zero as k⊥→ ∞, as shown by\nthe dashed curve in Fig. 4. Hence the energy is always\nminimized when the wave function has zero width in the\nx-yplane, i.e., ξ→0. Forg<>0.14 the effective po-\ntential is repulsive both at low and high momenta, and\nattractive in-between as illustrated by the solid curve in\nFig. 4. Subsequently the energy is minimized for a finite\nvalue ofξ.\nTo find the range of parameter space characterized by\nthe appearance of density-wave states, we need to con-\nsider, in addition to stability arguments, the transition\npoint discussed in Section III. Table 1 summarizes the\nrange of s-wave scattering lengths and critical numbers\nof52Cr atoms necessary to be inside stable density-wave\nregime.\nTABLE 1\nλℓz(µ)abb(a0)Nb(×104)\n0.15 16-21 ≥0.65\n0.25 0.2 16-19 ≥2.3\n0.25 15-18 ≥5.0\nTABLE I: Tabulation of the scattering length of boson-boson\ncontact interaction abband critical number of bosons Nbfor\ndifferentvalues of aspect ratio λand oscillator length ℓzinside\nthe density-wave regime with Nf= 103andgbf= 70a0for a\nmixture of chromium isotopes.\nVI. CONCLUSION\nInsummary, wehaveanalyzedthe stabilityofadipolar\nbosonic condensate mixed with non-interacting fermions\nin a pancake trap at T= 0. We found that the fermions\nhelp stabilize the condensate for a significant range of\nboson-boson and boson-fermion interaction strengths.\nWe then investigated the transition of the system from\na Gaussian-like to the density-wave ground state as a\nfunction of number of bosons, strength of the contact in-\nteraction, and trap aspect ratio. Our central result is the\nuse of a variational ansatz to show that while in a purely\nbosonic system the density-wave state is always unsta-\nble it can be stabilized by the admixture of even a small\nboson-fermion interaction.\nIn particular, this study leads us to the conclusion that\na pancake-shaped87Rb-40K mixture, which has a large\nboson-fermion s-wave scattering length, should be abso-\nlutely stable in the dipole dominated regime. By tuning\nthes-wave scattering length it is possible to reach a sit-\nuation characterized by the appearance of a roton insta-\nbility in the excitation spectrum, leading to the existence\nof a stable density-wave state. However, due to the small7\ndipole moment of rubidium atoms the transition to this\nstable density-wave regime needs a substantial number\nof atoms, of the order of 106.\nFuture work will discuss the effect of the dipolar na-\nture of the fermionic isotopes on the condensate – with\nchromium atoms in mind –, the existence and stability of\ndensity waves, as well as possible extensions to rotating\nsystems.\nAcknowledgments\nWe thank Prof. Tilman Pfau for several interesting\ndiscussions and his deep insight on dipolar condensates.This work is supported in part by the US Office of Naval\nResearch, by the National Science Foundation, and by\nthe US Army Research Office.\nAPPENDIX A: DERIVATION OF THE ENERGY\nFUNCTIONAL OF EQ. (10)\nIn this appendix we derive the energy functional\nEq. (10) by using the Gaussian ansatz of Eq. (9). First\nwe consider the dipolar interaction energy,\nEdd=gddNb\n3(2π)2/integraldisplay\ndk/parenleftbigg\n3k2\nz\nk2−1/parenrightbigg\nexp/bracketleftbigg\n−1\n2{k2\nzd2\nz+(k2\nx+k2\ny)d2}/bracketrightbigg\n, (A1)\nwherek2=k2\nx+k2\ny+k2\nz. Evaluating this integral gives\nthe form of dipolar energy\nEdd=gddNb\n3(2π)2d3z/bracketleftbigg2\n3η2−F(η−1)/bracketrightbigg\n,(A2)\nwithη=dz/dand\nF(y) =tan−1(/radicalbig\ny2−1)\n(y2−1)3/2−1\ny2(y2−1).\nNext we calculate the energy due to s-wave boson-boson\ninteraction and fermion-induced interaction. To achievethis goal we break the integral of the interaction energies\nexpressed in spherical coordinates into two parts, Es≡\nE<\ns+E>\nsandEdd≡E<\ndd+E>\ndd, according to\n/integraldisplay∞\n0dk=/integraldisplay2kf\n0dk+/integraldisplay∞\n2kfdk. (A3)\nFork <2kfand in spherical coordinate, the interac-\ntion energy including the s-wave and induced interaction\nis given by\nE<\ns+E<\nind=gddNb\n2π/integraldisplay2kf\n0/integraldisplayπ\n0/bracketleftBigg\ng<+g2\nbfν\n4πgdd∞/summationdisplay\nn=1/parenleftbiggk\n2kf/parenrightbigg2n1\n4n2−1/bracketrightBigg\nexp/bracketleftbigg\n−k2\n2(d2\nzcos2θ+d2sin2θ)/bracketrightbigg\nk2dksinθdθ\nwhereg<is defined by Eq. (11). This integral contain\ncontribution from both the local and nonlocal part of\nthe induced interaction. After integrating over kand\nrearranging the terms we obtain the expression\nE<\ns+E<\nind=g<E<\n1+gind(E<\n2+E<\n3),(A4)\nwith\nE<\n1(σ) =/integraldisplayπ/2\n0erf(√\n2kfσ)\nσ3sinθdθ,\nE<\n2(σ) =48k3\nf√\n2π/integraldisplayπ/2\n0exp(−2k2\nfσ2)\nσ2sinθdθ,E<\n3(σ) =3√\n2π∞/summationdisplay\nn=12n+1/2\n4n2−11\n2k2(n−1)\nf×\n/integraldisplayπ/2\n0/bracketleftBigg\nΓ(n+3/2)−γ(n+3/2,2k2\nfσ2)\nσ2n+3/bracketrightBigg\nsinθdθ,\n(A5)\nHereσ=/radicalBig\nd2zcos2θ+d2sin2θis a function of ( θ,dz,d),\nerf(y), erfc(y), Γ, and γdenote the error function, com-\nplementary error function, gamma function, and incom-\nplete gamma function, respectively.8\nSimilarly, the contribution for k >2kfto the interac-\ntion energy that contains both the s-wave boson-bosoninteraction and the nonlocal induced interaction is\nE>\ns+E>\nind=gddNn\n2π/integraldisplay∞\n2kf/integraldisplayπ\n0/bracketleftBigg\ng>−g2\nbfν\n4πgdd∞/summationdisplay\nn=1/parenleftbigg2kf\nk/parenrightbigg2n1\n4n2−1/bracketrightBigg\nexp/bracketleftbigg\n−k2\n2(d2\nzcos2θ+d2sin2θ)/bracketrightbigg\nk2dksinθdθ,\nwhereg>defined in Eq. (11). In this equation the first\nterm in the bracket stems from the s-wave boson-boson\ninteraction and the last term from attractive nonlocal\npart of the induced interaction. Again after evaluating\nthe integral over kwe get\nE>\ns+E>\ndd=g>E>\n1−gindE>\n2 (A6)where\nE>\n1(σ) =/integraldisplayπ/2\n0erfc(√\n2kfσ)\nσ3sinθdθ,\nand\nE>\n2(σ) =3√\n2π∞/summationdisplay\nn=12kf2(n+1)\n4n2−1/integraldisplayπ/2\n0/bracketleftBigg\n−(−1)n/radicalbig\nπ/2\n(2n−1)!!σ2n−3erfc(√\n2kfσ)\n+n−2/summationdisplay\nm=0(−1)m2m+1(2kf)2m\n2kf2n−2m−3(2n−3)(2n−5)....(2n−2m−3)exp(−2k2\nfσ2)σ2m/bracketrightBigg\nsinθdθ, (A7)\nThe total interaction energy is the sum of the contri-\nbutions in Eq. (A2), (A4), (A6) and is given in Eq. (10).\nIn this present paper we took n= 1,...,3 in the series inEqs. (A5), (A7) as the energy Egin Eq. (10) converges\nwith this inclusion.\n[1] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and\nT. Pfau, Phys. Rev. Lett. 94, 160401 (2005).\n[2] Q. Beaufils, R. Chicireanu, T. 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A 68, 033605 (2003).[25] For a fixed number of dipolar bosons, the quantity nd\ncan be changed by tuning the aspect ratio λ.\n[26] The qualitative behavior of the total-energy landscap e,\nhence the stability of the density-wave state, remains un-\nchanged by the inclusion of the kinetic energy and trap-\nping potential."    },    {        "title": "0806.1988v1.Statistical_Characterization_of_a_1D_Random_Potential_Problem___with_applications_in_score_statistics_of_MS_based_peptide_sequencing.pdf",        "content": "arXiv:0806.1988v1  [q-bio.QM]  12 Jun 2008Statistical Characterization of a 1D Random\nPotential Problem – with applications in score\nstatistics of MS-based peptide sequencing\nGelio Alves and Yi-Kuo Yu1\nNational Center for Biotechnology Information, National Libr ary of Medicine,\nNational Institutes of Health, Bethesda, MD 20894\nAbstract\nWe provide a complete thermodynamic solution of a 1D hopping model in the pres-\nence of a random potential by obtaining the density of states . Since the partition\nfunction is related to the density of states by a Laplace tran sform, the density\nof states determines completely the thermodynamic behavio r of the system. We\nhave also shown that the transfer matrix technique, or the so -called dynamic pro-\ngramming, used to obtain the density of states in the 1D hoppi ng model may be\ngeneralized to tackle a long-standing problem in statistic al significance assessment\nfor one of the most important proteomic tasks – peptide sequencing using tandem\nmass spectrometry data.\nKey words: Statistical Significance, Dynamic Programming, Mass Spect rometry,\nDirected Paths in Random Media, Peptide Identification\n1 Introduction\nImportant in both fundamental science and numerous applications , optimiza-\ntion problems of various degrees of complexity are challenging (see [1 ] for\nan excellent introduction). Optimization conditioned by constraints that may\nvary fromevent to event is ofespecial theoretical andpractical importance. As\na first example, when dealing with a system under a random potential, each\nrealization of the random potential demands a separate optimizatio n result-\ning in a different ground state. The thermodynamic behavior of such a system\nin a quenched random potential crucially depends on the random pot ential\nrealized. A similar but practical problem may arise in routing passenge rs at\n1To whom correspondence should be addressed. E-mail address : yyu@ncbi.nlm.nih.gov\nPreprint submitted to Elsevier Science June 6th 2008various cities to reach their destinations. In the latter case, the o ptimal rout-\ning depends on the number of passengers at various locations, the costs from\none location to the others, which likely to vary from time to time. This t ype of\nconditional optimization also occurs in modern proteomics problem, t hat is,\nin the mass spectrometry (MS) based peptide sequencing. In this c ase, each\ntandem MS (MS2) spectrum constitute a different condition for optimization\nwhich aims to find a database peptide or a de novopeptide to best explain\nthe given MS2spectrum.\nWhen the cost function of an optimization problem can be expressed as a sum\nof independent local contributions, the problem usually can be solve d using\nthe transfer matrix method that is commonly employed in statistical physics.\nA well-studied example of this sort in statistical physics is the directe d poly-\nmer/path in a random medium (DPRM) [2,3,4]. Even when a small non-loca l\nenergetics is involved, the transfer matrix approach still proves u seful [5]. As\nan example, the close relationship between the DPRM problem and MS- based\npeptide sequencing, where a small nonlocal energetics is necessar y to enhance\nthe peptide identifications, was sketched in an earlier publication [5] a nd the\ncost value distribution from many possible solutions other than the o ptimal\none is explored. Indeed, obtaining the cost value distribution from allpossible\nsolutions in many cases is harder than finding the optimal solution alon e. In\nthis paper, we will provide the solution to a generic problem that enab les a full\ncharacterization of the peptide sequencing score statistics, inst ead of just the\noptimal peptide. The 1D problem considered is essentially a hopping mo del in\nthe presence of a random potential. The solution to this problem may also be\nuseful in other applications such as in routing of passengers and ev en internet\ntraffic.\nIn what follows, we will first introduce the generic 1D hopping model in a\nrandom potential, followed by its transfer matrix (or dynamic progr amming)\nsolution. Wethendiscuss theutilityofthissolutioninthecontext ofM S-based\npeptide sequencing, and demonstrate with real example from mass spectrum\nin real MS-based proteomics experiments. In the discussion sectio n, we will\nsketch the utility of the transfer matrix solution in other context a nd then\nconclude with a few relevant remarks.\n2 1D hopping in random potential\nAlongthe x-axis,letusconsideraparticlethatcanhopwithasetofprescribed\ndistances {mi}K\ni=1towards the positive ˆ xdirection. That is, if the particle is\ncurrently at location x0, it can move to location x0+m1,x0+m2, ...x0+mK\nin the next time step. At each hopping step, the particle will accumula te an\nenergy−s(x) from location xthat it just visited. The score s(x) (negative\n2of the on-site potential energy) is assumed positive and may only ex ist at a\nlimited number of locations. For locations that s(x) do not exist, we simply\nsets(x) = 0 there. The energy of a path starting from the origin specified\nby the sequential hopping events p≡ {mh1,mh2,...,m hL}would have visited\nlocations {x1,x2,...,x L}withxi≡/summationtexti\nj=1mhjand has energy\nEp(x=xL)≡ −L−1/summationdisplay\ni=1s(xi)≡ −Sp(x).\nIn general, there can be more than one path terminated at the sam e point.\nTreating each path as a state with energy given by Ep, one ends up having\nthe following recursion relation for the partition function Z(x)≡/summationtext\npe−βEp(x)\nZ(x) =K/summationdisplay\ni=1eβs(x−mi)Z(x−mi), (1)\nwhereβ= 1/Tplays the role of inverse temperature (with kB= 1 chosen).\nIf one were only interested in the best score terminated at point x, it will be\ngiven by the zero temperature limit β→ ∞and the recursion relation may be\nobtained by taking the logarithm on both sides of (1) and divided by βthen\ntakingβ→ ∞limit to reach\nSbest(x) = max\n1≤i≤K{s(x−mi)+Sbest(x−mi)}, (2)\nwhereSbest(x) records the best path score among all paths reaching position\nx. This update method, also termed dynamic programming, records t he lowest\nenergy and lowest energy path reaching a given point x. The lowest energy\namong all possible at position xis simply −Sbest(x) and the associated path\ncan be obtained by tracing backwards the incoming steps. It is inter esting to\nobserve thatonecanalsoobtaintheworst scoreateachpositionv iadynamical\nprogramming\nSworst(x) = min\n1≤i≤K{s(x−mi)+Sworst(x−mi)}. (3)\nThe full thermodynamic characterization demands more informatio n than the\nground state energy. In principle, one may obtain the full partition function\nusing eq. (1) evaluated at various temperatures. This procedure , however,\nhinders analytical property such as determination of the average energy\n/an}bracketle{tE/an}bracketri}ht ≡ −∂lnZ\n∂β.\n3A better starting point may be achieved if one can obtain the density of states\nD(E). In this case, we have\nZ≡/integraldisplay\ndEe−βED(E)\n/an}bracketle{tE/an}bracketri}ht=/integraltextdEe−βEED(E)/integraltextdEe−βED(E).\nNote that if the ground energy Egrdof the system is bounded from below,\nthe partition function is simply a Laplace transform of a modified dens ity of\nstates given by\nZ=e−βEgrd∞/integraldisplay\n0dEe−βE˜D(E)\nwhere˜D(E)≡D(E−Egrd) and\n/an}bracketle{tE/an}bracketri}ht=Egrd+/integraltext∞\n0dEe−βEE˜D(E)\n/integraltext∞\n0dEe−βE˜D(E)\nThis implies that the density of states D(E) together with the ground state\nenergyEgrddetermine all the thermodynamic behavior of the system. In the\nnext section, we will explain how to obtain the density of states using the\ndynamical programming technique as well as how to extend this appr oach\nto more complicated situations that will be useful in characterizing t he score\nstatistics in MS-based peptide sequencing.\n3 Obtaining the Density of States\nThe density of states is related to the energy histogram in a simple wa y. The\nnumber of states between energies EandE+η(withη≪1) is given by\nD(E)η. If we happen to use ηas the energy bin size for energy histogram,\nthe count C(E) in the bin with energy Eis simply D(E)ηand the density\nof states D(E) =C(E)/η. For simplicity, we will assume that the all the\non-site energies −s(x) are integral multiple of η. This implies that each path\nenergy/score is also an integral multiple of η. In the following subsections, we\nwill use score density of states instead of energy density of state s.\n43.1 The Simplest Case and its Application\nWe denote by C(x,N) the number of paths reaching position xwith score\nNη. With this notation, we can easily write down the recursion relation fo r\nC(x,N) as follows\nC(x,N) =K/summationdisplay\ni=1C(x−mi,N−s(x−mi)\nη). (4)\nThis recursion relation allows us to compute the density of states in t he same\nmanner as computing the partition function (1) except that we nee d to have\nan additional dimension for score at each position x. As an even simpler ap-\nplication of this recursion relation, suppose that one is only interest ed in the\nnumber of paths reaching position x, one may sum over the energy part on\nboth side of (4) and arrives at\nC(x) =/summationdisplay\niC(x−mi), (5)\nwhich enables a very speedy way to compute the total number of pa ths reach-\ning position x. In the context of de novopeptide sequencing [6], this number\ncorresponds to the total number of ⁀all possible de novopeptides within a given\nsmall mass range. Although simply obtained, this number may be usef ul for\nproviding rough statistical assessment in de novopeptide sequencing.\n3.2 The More Realistic Case\nIn general, one may wish to associate with each hop an energy hor one may\nwish to introduce some kind of score normalization based on the numb er of\nhopping steps. This is indeed the case when applying this framework t o MS-\nbased peptide sequencing where a peptide length factor adding or m ultiplying\nto the overall raw score is a common practice. In this case, it becom es impor-\ntant to keep track the number of hops made in each path. We may fu rther\ncategorize the counter C(x,N) into/summationtext\nLC(x,N,L). That is, we may separate\nthe paths with different number of steps from one another and arr ive at a\nfiner counter C(x,N,L) which records the number of paths reaching position\nxwith score Nηand with Lhopping steps.\nIt is rather easy to write down the recursion relation obeyed by this fine\n5counter\nC(x,N,L) =K/summationdisplay\ni=1C(x−mi,N−s(x−mi)\nη,L−1). (6)\nThis recursion relation allows us to renormalize the raw score based o n the\nnumber of steps taken. For example, for RAId DbS [7], a database search\nmethod we developed, we divide the raw score obtained by 2( L−1) for any\npeptide (path) of Lamino acids (hopping steps) to get better sensitivity in\npeptide identification.\nIn principle, the recursion relations given by (4-6) are all one-dimen sional up-\ndates. The only difference is the internal structure of counters a t each position\nx. For (5), the counter is just an integer and has no further struc ture. For (4),\nthe counter at each position has a 1D structure indexed by the sco re. For\n(6), the counter at each position xhas a 2D structure indexed by both the\nscore and the number of hopping steps. This means that in terms of solving\nthe problem using dynamical programming, it is always a 1D dynamical p ro-\ngramming with different degrees of internal structure that may len gthen the\nexecution time when shifting from the simplest case (5) to the more c ompli-\ncated case (6). Obviously at each position x, there is an upper bound and\na lower bound for score and also for the number of hopping steps ac cumu-\nlated. We shall call them Sbest(x),Sworst(x),Lmax(x) andLmin(x) respectively.\nThe first two quantities may be obtained by eqs. (2) and (3) respec tively. We\nprovide the recursions for the two latter quantities below\nLmax(x)= max\n1≤i≤K{Lmax(x−mi)}+1, (7)\nLmin(x)= min\n1≤i≤K{Lmin(x−mi)}+1. (8)\nEqs. (2-3) and (7-8) provides the ranges for both the scores an d the number\nof cumulative hopping steps at each position xvia simple dynamic program-\nming. As we will discuss later, this information enables a memory-efficie nt\ncomputations of score histograms.\n4 Application in MS-based Peptide Sequencing\nIn this section, we focus on an important subject in modern biology – using\nMS data to identify the numerous peptides/proteins involved in any g iven\nbiological process. Because of the peptide mass degeneracies and the limited\nmeasurement accuracyforthepeptidemass-to-chargeratio,u singMS2spectra\n6is more effective in peptide identifications. In a MS2setup, a selected peptide\nwith its mass identified by the first spectrometer is fragmented by n oble gas,\nand the resulting fragments are analyzed by a second mass spectr ometer.\nAlthough such MS2-based proteomics approaches promise high throughput\nanalysis, the confidence level assignment for any peptide/protein identified is\nchallenging.\nThe majority of peptide identification methods are so-called databa se search\napproaches. The main idea is to theoretically fragment each peptide in a\ndatabase to obtain the corresponding theoretical spectra. One then decides the\ndegree of similarity between each theoretical spectrum and the inp ut query\nspectrum using a scoring function. The candidate peptides from th e database\nare ranked/chosen according to their similarity scores to the quer y spectrum.\nAlthough one may assign relative confidence levels among the candida te pep-\ntides via various (empirical) means, an objective, standardized calib ration\nexists only recently [8]. In our earlier publications [5,9], we proposed to tackle\nthis difficulty by using a de novosequencing method to provide an objective\nconfidence measure that is both database-independent and take s into account\nspectrum-specific noise. In this paper, we will provide concrete alg orithms for\nsuch purpose.\nTo begin, consider a spectrum σwith parent ion mass range [ w−δ,w+δ],\nwe denote by Π( w,δ) the set of all “possible” peptides with masses in this\nrange. Given a peptide πfrom Π(w,δ), the associated quality score S(π,σ)\nis defined by a prescribed scoring system. The score distribution of S(π,σ)\nwithin Π( w,δ) provides naturally a likelihood measure for any given peptide\nπto the the correct one.\nHowever, as described earlier [5], this seemingly straightforward ide a faces two\ndifficulties in terms of implementations. First, unlike the DPRM problem f or\nwhich the function to be optimized is defined without ambiguity, the ch oice\nof the scoring function is somewhat empirical because the paramet ers used\nin the scoring must be trained using a training data set. Further, be cause of\ndifferent instruments and experimental setups, it seems impossible to design a\nscoring system such that the correct peptide for each spectrum has the highest\nscore among all possible peptides; the application of a given scoring function\nto general cases may require a leap of faith. Second, even after t he scoring\nfunction is chosen, it is not known how to find the peptide πothat maximizes\nS(π,σ) as well as the score distribution pdf( S) within Π( w,δ) other than by\nthe generally impractical procedure of examining all members of Π( w,δ).\nThe first difficulty can be alleviated by validating high scoring de novopep-\ntides via database searches [9] and is not the main focus of the curr ent paper.\nNote that a partial solution to the second problem via iterative mapp ing when\nnonlocal score contributions exist is provided earlier [5]. Here we tac kle the\n7second problem head on when the scoring function used does not co ntain\nnonlocal contribution other than a final renormalization with respe ct to the\npeptide length. Our algorithms contains two parts: computer memo ry alloca-\ntions anddynamical programming update. Prior to discussing these two parts,\nhowever, we first address the important issue of choosing a good m ass unit.\n4.1 Choosing a Good Mass Unit\nThe goal here is to choose a mass unit ∆ and expresses the molecular mass\nof each amino acid as an integral multiple of this unit. For example, one may\nchoose ∆ to be 0 .1 Dalton (Da), and round the molecular mass of each amino\nacid to be an integral multiple of 0 .1 Da. Once a mass unit is chosen, all the\nmasses under consideration are integral multiples of this unit. It tu rns out\nthat different choices of the mass unit leads to different maximum cum ula-\ntive mass error. As a specific example, consider using ∆ = 0 .1 Da as the mass\nunit. The mass of Alanine, with true mass 71 .03711538 Da, is now represented\nas 710∆. This molecular mass expression is 0 .03711538 Da smaller than the\ntrue molecular mass of Alanine. When this happens, the integral mas s rep-\nresentation has a mass smaller than the true mass, and we call such type\nof mass error a down-error. Now the amino acid Tryptophan with molecular\nmass 186 .07931613 Da will be assigned an integral mass of 1861∆, which has\nan extra of (0 .1−0.07931613) Da compared to the true mass. We call this\ntype of mass error the up-error.\nThe ratio of the mass error to the real molecular mass when multiplied by\n3000 Da provides the cumulative maximum error that can be induced b y a\nsingle amino acid at 3 ,000 Da mass. For a fixed mass unit, we went over this\nmass error analysis for each of the twenty amino acids and documen ted the\nlargest up-error and down-error. The larger one between the ma ximum up-\nerror and the maximum down-error is called the max-error. To sear ch for best\nmass units that minimize the max-error at 3 ,000 Da, we went over all possible\nmass unit ranging from 0 .005 Da to 1 .005 Da in step of 10−6Da. Interestingly\nenough, we found a discrete list of mass units that have smaller max- error\ncompared to their nearby mass units. These numerically found magicmass\nunits are summarized in table 1.\nOnce a mass unit is chosen, all the amino acid masses are effectively int egers.\nTo obtain the score histogram of all de novo peptides when queried by a\nspectrum σwith parent molecular mass w(with N- and C- terminal groups\nof the peptide stripped away), we first construct a mass array wh ere index\nkcorresponds a molecular mass k∆. To encode all possible peptides with\nmolecular mass up to w, we need to have an array of size w/∆+1.Apparently,\nwhen a larger mass unit is used, the size of the mass array is smaller an d thus\n8Table 1\nA list of best mass units in Da. The abbreviation “m.u.e.” sta nds for “maximum\nup-error,” while “m.d.e.” stands for “maximum down-error. ” The maximum up-\nerror, maximum down-error, and max-error are evaluated in e xtrapolation to 3 ,000\nDa as described in the text. The abbreviation “a.a.w.m.u.” s tands for “amino acid\nwith maximum up-error,”while “a.a.w.m.d.” stands for “ami no acid with maximum\ndown-error.”\nmass unit m.u.e. a.a.w.m.u. m.d.e. a.a.w.m.d. max-error\n0.006070 0.041980 Tryptophan 0.037455 Cysteine 0.041980\n0.007300 0.041495 Methionine 0.061276 Asparagine 0.06127 6\n0.017540 0.094183 Cysteine 0.121977 Proline 0.121977\n0.021500 0.199585 Arginine 0.182283 Asparagine 0.199585\n0.054470 0.453793 Asparagine 0.347792 Alanine 0.453793\n0.065400 0.553492 Lysine 0.536989 Alanine 0.553492\n0.109450 0.908287 Proline 0.900898 Lysine 0.908287\n0.110300 0.962781 Histidine 0.858742 Lysine 0.962781\n0.110320 0.960176 Aspartate 0.907801 Histidine 0.960176\n0.500208 0.980357 Cysteine 0.983149 (Iso)Leucine 0.98314 9\n1.000416 0.980357 Cysteine 0.983149 (Iso)Leucine 0.98314 9\nreduces computation time. However, as one may see from table 1, t he larger\nmassunitisalsoaccompaniedbyalargermax-errorandmightnotbep referred\nwhen high mass accuracy is the first priority.\n4.2 Efficient Memory Allocation\nThe basic idea of our algorithm is to encode all possible peptides in the m ass\narray by linking pointers, analogous to the consecutive hopping ste ps in the\n1D hopping model. For an amino acid a, letn(a) represents its corresponding\ninteger mass in unit of ∆. For a peptide made of [ a1,a2,...,a M], it will have\na hopping trajectory in the molecular array given by [0 ,x1,x2,...,x M] with\nxi≥1≡/summationtexti\nj=1n(aj). Let us also denote xMbyxFto indicate that it is the\nterminatingpointofthepath.Apparently, allpossiblepeptideswith molecular\nmasses equal to xF∆ will all have corresponding hopping paths starting at the\norigin and terminating at xF. Through appropriate pointer linking, one may\ntherefore encode allpossible peptides with molecular mass xF∆ in a one-\ndimensional mass array.\n9For a given spectrum σ, depending on the score function used, one may cal-\nculate local score contributions at each mass index. This step is don e once\nonly for the whole mass array, and need not be repeated for each c andidate\npeptide. In a typical MS2experimental spectrum, there always exists some\nlevel of parent ion mass uncertainty. Once the size of the mass unc ertainty\nis specified, we only need to examine de novopeptides whose corresponding\nhopping paths terminating at a few consecutive mass indices. This ind icates\nthat some of the mass indices of the aforementioned mass array ma y not even\nbe used in this context. Below we describe how to efficiently obtain relevant\nmass indices and only allocate computer memories for those masses.\nAssume that the possible terminating points are F1,F2,...,F kwithFj+1=\nFj+ 1. The update rules described in Eqs. (2-3), (5), and (7-8) will als o be\nused at this stage. The following pseudocode describes our algorith m.\nInitialize the mass index = 0 entry\nSbest=Sworst=Lmax=Lmin= 0;C=1;\nREMARK: Max aa is the maximum number of amino acids considered\nfor (aa index = 0; aa index<Maxaa; aaindex ++) {\nlabel occupancy of n(aa index);\nat n(aa index) attach a pointer back to 0;\nupdateSbest,Sworst,Lmax,Lmin,Cat n(aa index);\n}\nfor (mass index = 1; mass index<=Fk; massindex ++) {\nif (mass index occupied ?) {\nfor (aa index = 0; aa index<Maxaa; aaindex ++) {\nlabel occupancy of (mass index + n(aa index));\nat mass index+n(aa index) attach a pointer to mass index;\nupdateSbest,Sworst,Lmax,Lmin,Cat (mass index + n(aa index));\n}\n}\n}\nfor (mass index = Fk; massindex>=F1; massindex --) {\nbacktrack all possible paths →final occupied entries;\n}\nThelaststepinthealgorithmaboveidentifies relevantmassindices ,massindices\nthat will be traversed by the hopping paths of all peptides with molec ular\nmasses in the range [ F1∆,Fk∆]. We only need to allocate computer memory\nassociated with those sites. For each of these relevant sites, we a lso know the\nvalues of Sbest,Sworst,Lmax,Lmin, and the total number of peptides reaching\nthat site through the algorithm above. One may therefore allocate a 2D array\nof size (Sbest(i)−Sworst(i))/η×(Lmax(i)−Lmin(i)) for each relevant mass index\nifor later use.\n104.3 Main Algorithm and some Results\nOnce memory allocation for relevant mass indices is done, we can efficiently\ngo through those relevant sites to obtain the 2D score histogram t hat we\nmentioned. In the pseudocode below, update is performed using eq . (6). We\nnow demonstrate the very simple main algorithm\nInitialize all the fine counters C(x,N,L) = 0\nexceptC(x= 0,N= 0,L= 0)=1;\nfor (aa index = 0; aa index<Maxaa; aaindex ++) {\nupdateC(x,N,L)atx=n(aaindex);\n}\nfor (mass index in ascendingly ordered relevant mass indices){\nfor (aa index = 0; aa index<Maxaa; aaindex ++) {\nupdateC(x,N,L)atx=(massindex + n(aa index));\n}\n}\nWe now define the final 2D counter\nY(N,L)≡k/summationdisplay\ni=1C(Fi,N,L). (9)\nApparently, in the 1D hopping model when allowing kconsecutive termi-\nnating points, the resulting density of states D(E) can now be expressed as\nD(E=−Nη) =/summationtext\nLY(N,L)/η. If one were interested in normalizing the final\nscore in a path-length dependent manner, one will has the following g eneric\ntransformation\nH(E) =/summationdisplay\nL/integraldisplay\ndE′Y(E′=−Nη,L)\nηδ(E−f(E′,L)) (10)\nwheref(E′,L) is a generic length-normalized energetic function that takes\nthe raw energy E′withLhopping steps and turn them into a new energy\nf(E′,L), and/integraltextdE′→η/summationtext\nNis understood.\nUsing a real experimental MS2spectrum of parent ion mass 2254 .7±3.0\nDa and a raw scoring function (RAId DbS [7] raw score without divided by\n2(L−1) withLbeing the peptide length), we obtained a 2D score histogram.\nFrom this 2D score histogram, we can compute the average peptide length\n/an}bracketle{tL/an}bracketri}htas well. We then transform the 2D score histogram using two differen tf\nfunctions. In the first case, f(E′,L) =E′/2(/an}bracketle{tL/an}bracketri}ht−1), meaning that one just\ndivides the score by a constant given by 2( /an}bracketle{tL/an}bracketri}ht−1). In the second case, we use\n110 1 2 3 4 5 6\nScore−19−14−9−41log10(PDF(Score))RAId_score histogram\nRAId_score theoretical fit\nRaw score histogram\nRaw score theoretical fit\nFigure 1. Score histograms for raw score and RAId DbS score. Note that the two\nhistogram cross each other at large score regime, indicatin g that the raw score\nfunction might not be as effective as the RAId DbS score, see text for details.\nthe RAId DbS scoring function where f(E′,L) =E′/2(L−1). In Figure 1,\nwe show the two resulting score histograms along with the fits to the oretical\ndistribution function [7]. As one may see from the figure, both histog rams are\nwell fitted by the theoretical distribution function over at least 15 order of\nmagnitudes. There is difference, however, in the histograms obtain ed. In the\nfirst case, where the score is merely divided by the average length, we have a\nwider score distribution than that of the second case. This implies th at a high\nscoring hit out of the first type of scoring function will have a larger P-value\nthan that of the second type. This is perfectly reasonable becaus e when using\nthe first type of raw scoring, very long peptides which by random ch ances are\nmore likely to hit on fragment peaks in the mass spectrum are less pen alized\nthan the shorter peptides. As a consequence, one anticipates mo re false long\npeptides out of thefirst type of scoring method thanthat of the s econd scoring\nmethod. Therefore, one should assign a larger P-value to the former case and\na smaller P-value to the latter case. It is apparently important to be able to\nobtain score histograms of the second scoring method. However, this can only\nbeachieved if one keeps thelength informationin thedynamical prog ramming\nupdate, see eq. (6).\n5 Discussion, Summary, and Outlook\nOur method may also be extended to other applications. In the case of passen-\ngerroutings,the x-axisactuallyrepresentstime.Thelocalscoremaybeviewed\n12as the additional cost that may vary for different stops. Once the problem is\nlaid out, the 2D histogram obtained from our solution indicates the nu mber of\nequivalent routes in terms of additional costs and the total numbe r of stops.\nThis problem should be interesting in its own right.\nIn this paper, we developed a new approach to obtain the density of states of\na 1D hopping problem in random potential. We have extended the simple st\ncase scenario and have shown that we can apply this method to prov ide a\ncomplete score histogram for MS-based peptide sequencing problem. This im-\nportant information may be used for a more objective statistical s ignificance\nassignment in peptide identification. Our algorithmmay also serve as a speedy\nde novoalgorithm. If one is only interested in getting the best scoring pep-\ntide with length normalized score, one only needs to keep track of Sbest(x,L).\nFurthermore, it is straightforward to include in our de novoalgorithm post-\ntranslationally modified amino acids. The effect is simply an enlargement of\nthe alphabet. That is, instead of having 20 amino acids, we will simply ha ve\nmore allowed masses but without needing to change any part of the a lgorithm.\nIn the near future, we would like to build a web application that allows th e\nusers toobtaininformationofinterest. Forexample, auser might b einterested\nin knowing: given a parent ion molecular mass and a mass error toleran ce,\nhow many de novopeptides can there be? Furthermore, we plan to provide\nusers with the full score histogram when a query spectrum is provid ed and a\nscoring method is chosen. Our approach, founded on statistical p hysics, can\neasilyaddressthistypeofquestionstoprovideuseful information forbiological\nresearches.\nAcknowledgement\nThis work was supported by the Intramural Research Program of the National\nLibrary of Medicine at the National Institutes of Health.\nReferences\n[1] M.R. Garey and D.S. Johnson, Computers and intractability , W.H. Freeman\nand company, New York (1979).\n[2] D.A. Huse and C.L. Henley, Phys. Rev. Lett. 54, 2708 (1985).\n[3] M. Kardar, Nucl. Phys. B290, 582 (1987).\n[4] D.S. Fisher and D.A. Huse, Phys. Rev. B. 43, 10728 (1991).\n13[5] T.P. Doerr, G. Alves and Y.-K. Yu, Physica A 354, 558-570 (2005).\n[6] J.A. Taylor and R.S. Johnson, Raid Commu. Mass Spect. 11, 1067 (1997) .\n[7] G. Alves, A.Y. Ogurtsov and Y.-K. Yu, Biology Direct 2, art. no. 25 (2007).\n[8] G. Alves, A.Y. Ogurtsov, W.W. Wu, G. Wang, R.-F. Shen and Y .-K. Yu,\nBiology Direct 2, art. no. 26 (2007).\n[9] G. Alves and Y.-K. Yu, Bioinformatics 21, 3726-3732 (2005).\n14"    },    {        "title": "0807.1875v1.The_cluster_glass_state_in_the_two_dimensional_extended_t_J_model.pdf",        "content": "arXiv:0807.1875v1  [cond-mat.supr-con]  11 Jul 2008The cluster glass state in the two-dimensional extended t-J model\nChung-Pin Chou, Noboru Fukushima, and Ting Kuo Lee\nInstitute of Physics, Academia Sinica, NanKang, Taipei 115 29, Taiwan\nThe recent observation of an electronic cluster glass state composed of random domains with\nunidirectional modulation of charge density and/or spin de nsity on Bi2Sr2CaCu 2O8+δreinvigo-\nrates the debate of existence of competing interactions and their importance in high temperature\nsuperconductivity. By using a variational approach, here w e show that the presence of the cluster\nglass state is actually an inherent nature of the model based on the antiferromagnetic interaction\n(J) only,i.e.the well known t−Jmodel. There is no need yet to introduce a competing interact ion\nto understand the existence of the cluster glass state. The l ong-range pairing correlation is not\nmuch influenced by the disorder in the glass state which also h as nodes and linear density of states.\nIn the antinodal region, the spectral weight is almost compl etely suppressed. The modulation also\nproduces subgap structures inside the ”coherent” peaks of t he local density of states.\nPACS numbers: 71.10.-w, 71.27.+a, 74.20.-z, 74.72.-h\nI. INTRODUCTION\nSince the discovery of high temperature superconduc-\ntors (HTS) two decades ago, many anomalous proper-\nties have been reported. One of the most interesting\nproperties is the possible existence of the stripe state\nconsisting of one dimensional charge-density modulation\ncoupled with spin ordering1,2,3. The first direct experi-\nmental evidence4of this stripe state as the ground state\nis for the non-superconducting La1.48Nd0.4Sr0.12CuO4\nwith the hole density about 1 /8 per unit cell. Since then\nmore evidences of the presence of these have been re-\nported in other cuprate samples5,6. However it seems\nthat the stripe is much more prominent in the LaSr-\nCuO (LSCO) family near 1 /8 doping8,9and in partic-\nular forLa1.875Ba0.125CuO4(LBCO−1/8)6where the\ncharge density or spin density modulation could be con-\nsidered as an long-ranged order parameter of a phase\nwith broken symmetries. Very recently the high resolu-\ntion scanning tunneling microscopy (STM) has observed\nunidirectional domains with periodic density of states\nmodulation in two families of Ca2−xNaxCuO2Cl2and\nBi2Sr2CaCu 2O8+δ(BSCCO)10,11. These bond-centered\nelectronic patterns with a width of four lattice constants\nformthesocalledelectronicclusterglassstatewithshort-\nranged modulations. Although the modulation is weak,\nit is still surprising to have the rather high supercon-\nducting (SC) transition temperature in this glass state.\nSTM spectra also provided the new puzzling result that\nthere are two different types of spectral gaps in those\nsystems. One is a larger gap with a broader distribu-\ntion and it seems to be related to the pseudogap. Inside\nthis gap, there is a smaller gap or the so called sub-gap\nkink with clear d-wave like spectra12,13. These results\nand the most recent conflicting results reported by angle-\nresolved-photo-emission spectra (ARPES)14,15raise the\nquestion whether there aretwo energyscalesrelated with\ntwo different underlying mechanisms separately respon-\nsible for the larger pseudogap and the lower-temperature\nSC transition. But one feature that most experimental\nresults agree is the presence of the linear density of state(DOS) near the nodal point. It is interesting to note that\nthis has also been reported in the non-superconducting\nstripe state of LBCO−1/816.\nThe observations of these strongly-modulated inhomo-\ngeneous states like LBCO−1/8 with almost zero SC\ntransition temperature ( Tc) or weakly inhomogeneous\ncluster glass states in BSCCO with quite high Tchave\nfueled the idea about the presence of competing inter-\nactions and underdoped HTS is near the boundary of\ntwo distinct phases with different order parameters. The\nfluctuations3of the order parameters in these adjacent\nphases could be the mechanism of high temperature su-\nperconductivity. Thus it seems like a d´ej`a vuthat after\ntwo decades we are still faced with a daunting question\nabout the appropriatefundamental interactions to model\nand to understand the HTS. Many people are still not\nconvinced by all those successes17,18,19claimed by study-\ning the strong coupling Hubbard model or its equivalent\nt−Jmodel. In this model the nearest-neighbor anti-\nferromagnetic (AF) spin coupling Jis the only relevant\ninteraction besides the usual kinetic energy of electrons\n(represented by the tterm). There are no other compet-\ning interactions and certainly no phase boundaries be-\ntween overdoped and underdoped regimes with different\norder parameters to be worried.\nIn this paper we will show that interactions repre-\nsentedbytandJareactuallycompetingwith eachother.\nThiscompetition, greatlyenhancedbythestrongcorrela-\ntion between electrons, hasmany spatially heterogeneous\nstates almost exactly the same energy as the uniform\nground state. The ground state could easily tolerate lo-\ncal spatialmodulation of chargedensity, spin density and\neven pairing amplitude without much an effect on its SC\norder parameter. The presence of these very low-energy\nclusterglassstateswitharandompatternofshort-ranged\nmodulation is an inherent nature of the t−Jmodel. The\nselection of a particular local electronic pattern as ob-\nservedin scanningtunneling spectroscopy(STS) for HTS\nis most likely determined by the effects of impurities, de-\nfects and electron-lattice interaction, etc. Random dis-\ntribution of impurities and defects will not produce long-2\nranged modulation in the sense of charge-density-wave\norder or spin-density-wave order unless these is a very\nstrong lattice distortion as demonstrated6by the struc-\ntural transition observed in LBCO−1/8. Hence our re-\nsult for the extended t−Jmodel shows that for weakly\ninhomogeneous cuprates like BSCCO there is no need to\nintroduce other strong competing interactions with new\nbroken symmetry phases and long-ranged order parame-\nters to produce the observed glass states. The extended\nt−Jmodel is adequate to explain many of the experi-\nmentalobservations. Depending ontheparticulartypeof\nmodulations, thelocalDOScouldeitherhaveanodewith\nlinear DOS or without a node. These local modulations\nalso produce the sub-gap structures. In addition, these\ncluster glass states have almost completely suppressed\nthe quasi-particle spectral weight near the antinodal re-\ngion as observed by ARPES for BSCCO compounds20.\nBefore we start with the discussion about the clus-\nter glass states, we should point out that the competi-\ntion between the kinetic energy represented by tand the\nmagnetic energy represented by Jis not a new idea. It\nis the main reason for the uniform-state phase diagram\nobtained by the theory of the resonating-valence-bond\n(RVB) state17,18,19. TheJterm prefers the formation of\nspinpairing,thesocalledRVBsinglet,orthelong-ranged\nantiferromagnetism. Because of the strong correlation a\nspin can only hop by exchanging position with a hole,\nhence the kinetic energy is proportional to the density of\nholes. But the more hole the system has, the less number\nof spin there is. Consequently the pairing is also reduced.\nThus the pairing amplitude reduces when hole doping in-\ncreases. Magnetic energy is reduced while kinetic-energy\ngain increases. It is shown below this competition also\nhappens in individual unit cell.\nOn the theoretical side the stripe state also has a long\nhistory. It was first founded almost two decades ago\nin the mean-field treatment of the Hubbard model1,2\nalthough the validity of the method in treating strong\nHubbard interaction is questionable. Then the extended\nt−Jmodels were studied by several numerical methods\nsuch as the exact diagonalization method21, the density\nmatrix renormalization group method22,23and the varia-\ntional Monte Carlo (VMC) method23,24. But the results\nare inconsistent. There are indications that stripe is un-\nstable when the second neighbor hopping, t′, is included\nin thet−Jmodel. However, a later VMC study of the\nt−t′−Jmodel25indicates that the stripe has about\n1% lower energy than the uniform RVB based d-wave SC\nstate for most of the negative values of t′except when\n−t′/tis less than 0 .1. Similar results were also reported\nfortheHubbardmodel26. Theseresultsposeanewdirect\ncontradiction with the experimental findings. Many ex-\nperimental and theoretical studies have found that −t′/t\nis smallest for LSCO family27,28,29, thus the stripe state\nshould not be favorable. Yet as mentioned above, among\nallthecupratesLSCOfamilyhasthemostsolidevidences\nfor the existence of stripe. There are also other prob-\nlems with the proposed stripe states, such as suppressedpairing correlation and absence of the V-shape DOS at\nlow energy, and we will discuss them below. All of the\nprevious works concentrated on studying periodic stripes\nwith a long-ranged order, it is unclear if the result will\nhold for the cluster glass state. There are also differ-\nent kinds of stripes to be considered. It is possible to\nhave only charge density modulation, or only spin den-\nsity modulation, or only pairing amplitude modulation\nor with the linear combination of any two or all three of\nthem30,31. Furthermore the relation between these mod-\nulations could be correlated or anti-correlated. All these\nissuesareaddressedbelowandtheir resultsarecompared\nwith the experiments.\nII. THE STRIPE-LIKE STATES BY THE\nVARIATIONAL MONTE CARLO METHOD\nWe consider the extended t−JHamiltonian,\nH=−/summationdisplay\ni,j,σtij/parenleftBig\n˜c†\niσ˜cjσ+h.c./parenrightBig\n+J/summationdisplay\n<i,j>Si·Sj(1)\nwhich has been used to describe many physical systems\nin the high-temperature superconductors32. The hop-\nping amplitude tij=t,t′, andt′′for sitesiandjbeing\nthe nearest-, the second-nearest, and the third-nearest-\nneighbors, respectively. We restrict the electron creation\noperators ˜c†\niσto the subspace with no-doubly-occupied\nsites.Siis the spin operator at site iand< i,j >\nmeans that the interaction occurs only for the nearest-\nneighboring sites. In the following, we mainly focus on\nthe caset′′=−t′/2 andJ/t= 0.3 at hole doping 1 /8.\nWe shall follow the work by Himeda et al.25to con-\nstruct the variational wave functions. In the mean-field\ntheory we assume a local AF order parameter, the stag-\ngeredmagnetization mi, and nearestneighborpairingor-\nder parameter ∆ ij. Thus the effective mean-field Hamil-\ntonian is reduced to\nHMF=/summationdisplay\ni,j/parenleftBig\nc†\ni↑ci↓/parenrightBig/parenleftbiggHij↑Dij\nD∗\nji−Hji↓/parenrightbigg/parenleftbiggcj↑\nc†\nj↓/parenrightbigg\n,(2)\nwhere the matrix elements\nHijσ=−\ntv/summationdisplay\nβ=N+t′\nv/summationdisplay\nβ=NN+t′′\nv/summationdisplay\nβ=NNN\nδj,i+β\n+/parenleftbig\nρi−µv+σ(−1)xi+yimi/parenrightbig\nδj,i, (3)\nDij=/summationdisplay\nβ=N∆ijδj,i+β. (4)\nHereβ=N,NN, andNNNcorrespond to the nearest-\n, the next-nearest, and the third-nearest-neighbors, re-\nspectively, and σ=↑(1) or↓(-1). The local charge den-\nsity is controlled by ρiandµvis the variational param-\neter for the chemical potential. For periodic stripes we3\nassume charge density ρiand staggered magnetization\nmiare anti-correlated, i.e.there are more holes at sites\nwith minimum staggered magnetization. For simplicity,\nwe assume these spatially varying functions with simple\nforms:\nρi=ρvcos[4πδ·(yi−y0)], (5)\nmi=mvsin[2πδ·(yi−y0)], (6)\nwhereδis the doping density and is 1 /8 in this paper.\nWe have also taken the lattice constant to be our unit.\nHere we assume the stripe extends uniformly along the x\ndirection.y0= 0 (1/2) corresponds to the site- (bond-)\ncentered stripe. In this paper, we will only focus on the\nbond-centered stripe since the variational energy differ-\nence between the site- and bond-centered stripes is very\nsmall (not shown) within the finite cluster33,34.\nBesides staggered magnetization and charge density,\nthe nearest neighbor pairing ∆ ijcan also have a spa-\ntial modulation. There are several different stripes we\ncan choose. If ∆ ijhas the same period 1 /δas the stag-\ngered magnetization but it is πphase shifted, this is\nthe so called ”antiphase” stripe studied by a number of\ngroups25,33,34,35. In this stripe state, the bond-average\n∆ijis zero and there is no net pairing. Hole density is\nmaximum at the sites with maximum pairing amplitude\n|∆ij|and minimum staggered magnetization |mi|.\nAnother moregeneralchoiceis to havethe spatialvari-\nation of ∆ ijof the form,\n∆i,i+ˆx= ∆M\nvcos[4πδ·(yi−y0)]−∆C\nv,\n∆i,i+ˆy=−∆M\nvcos[4πδ·(yi−y0)+2πδ]+∆C\nv.(7)\nThe bond-average∆ ijis determined by the constant∆C\nv.\nIf both ∆M\nvand ∆C\nvare positive, then the hole density is\nmaximum at sites with smallest pairing amplitude |∆ij|\nand smallest magnetization |mi|. This is similar to the\nphase diagram36,37predicted by the uniform RVB and\nAF states, when hole density is small both staggered\nmagnetizationandpairingamplitude arelarger. Thus we\nwilldenotethisstateastheAF-RVBstripe. ForAF-RVB\nstripe the period of ∆ ijis the same as the charge-density\nmodulation ρi. This period, 1 /2δ, is only half of the\nperiod for the antiphase or πphase stripe. Besides the\nantiphase stripe and AF-RVB stripes, we could also have\nthe AF stripe without both ∆M\nvand ∆C\nvor the charge-\ndensity stripe without any staggered magnetization but\nwith pairing amplitude modulation.\nIn general we have total seven variational parameters\nµv,t′\nv,t′′\nv,ρv,mv, ∆M\nv, and ∆C\nvwithtvset to be 1. Once\nthese parametersaregiven, wediagonalizethe mean-field\nHamiltonian in equation (2). By solving the Bogoliubov\nde Gennes (BdG) equations\n/summationdisplay\nj/parenleftbiggHij↑Dij\nD∗\nji−Hji↓/parenrightbigg/parenleftbiggun\nj\nvn\nj/parenrightbigg\n=En/parenleftbigg\nun\ni\nvn\ni/parenrightbigg\n,(8)and then obtain Npositive eigenvalues En(n= 1−\nN) andNnegative eigenvalues ¯Enwith corresponding\neigenvectors ( un\ni,vn\ni) and (¯un\ni,¯vn\ni). The eigenvectors are\nused to construct the mean-field wave function25|ψ∝angbracketrightby\nusing Bogoliubov transformation\n/parenleftbigg\nγn\n¯γn/parenrightbigg\n=/parenleftbigg\nun\nivn\ni\n¯un\ni¯vn\ni/parenrightbigg/parenleftbiggci↑\nc†\ni↓/parenrightbigg\n. (9)\nThe trial wave function P|ψ∝angbracketrightwith a Gutzwiller projec-\ntorPcan be constructed by creating all negative energy\nstates and annihilating all positive energy states on a\nvacuum of electrons |0∝angbracketright. Then we formulate the wave\nfunction in the Hilbert space with the fixed number of\nelectronsNe,\n|Φ∝angbracketright=PPNe|ψ∝angbracketright=PPNe/productdisplay\nnγn¯γ†\nn|0∝angbracketright (10)\n∝P\n/summationdisplay\ni,j(ˆU−1ˆV)ijc†\ni↑c†\nj↓\nNe/2\n|0∝angbracketright,\nwhereˆUij=ui\njandˆVij=vi\nj. We optimize the varia-\ntional energy E=∝angbracketleftΦ|H|Φ∝angbracketright/∝angbracketleftΦ|Φ∝angbracketrightby using the stochas-\ntic reconfiguration algorithm38. Additionally, to reduce\nthe boundary-condition effect in numerical studies45, we\naveragetheenergiesoverthe fourdifferent boundarycon-\nditions which is periodic or antiperiodic in either xory\ndirection.\nFig.1 shows the t′/tdependence of the variational en-\nergies for the hole density δ= 1/8. Firstly, we have\nshown that the AF-RVB stripe state (the empty trian-\ngles) becomes more stable than uniform d-wave RVB\nstate (the filled triangles) as decreasing t′/tfurther from\n−0.05. It is worth noting that the AF-RVB stripe state\nfort′/t <−0.05 has the vanishing pairing parameters\n∆C\nvand ∆M\nvbut largemvand finiteρv. Therefore, we\ncan consider this stripe state without SC order as the an-\ntiphase AF stripe state1. The hole density and the stag-\ngeredmagnetizationareplotted asafunction ofpositions\nfor a typical AF stripe (filled circles) in Fig.2(a) and (b),\nrespectively. The AF stripe pattern with a very large\nhole-density variation is essentially a nano-scale phase\nseparation with hole-rich and hole-poor regions alternat-\ning. This is consistent with what has been obtained by\nHimedaet al.25although they did not include t′′which\nis set to be −t′/2 here.\nIn our previous calculations to study the possibility\nof phase separation in the t−Jmodel46, we found the\ntendency to overestimate the strength of the pairing of\nholes. If we reduce this strength by going beyond the\nsimple trial wave functions used in our discussion above\nwe could push the phase separation boundary to a much\nhigher value of J/t. A simpler way to make this ad-\njustment is to introduce the hole-hole repulsion Jastrow\nfactor39,40,41:\nPJ=/productdisplay\ni<j/parenleftBig\n1−(1−rα\nij·vδj,i+β\nβ)·nh\ninh\nj/parenrightBig\n(11)4\n-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00-0.464-0.460-0.456-0.452-0.448-0.444-0.440-0.436E/t\nt'/t4 8 12 16481216\n0.1190\n0.1250\n0.1310\nFIG. 1: Variational energies of the bond-centered AF-RVB\nstripe state and uniform d-wave RVB state as function of t′/t.\nThe data have been optimized for 1 /8 doping on the 16 ×16\nlattice system. The filled (empty)symbols represent the ene r-\ngies of uniform d-waveRVBstate (AF-RVBstripe state). The\ntriangles (circles) indicate the wave functions without (w ith)\nthe hole-hole correlation of equation (11). The asterisks c or-\nrespond to the random stripe states with the Jastrow factor o f\nequation (11). Inset: The hole density of the random stripe\nstate optimized in the case of ( t′,t′′,J)/t= (−0.3,0.15,0.3).\nwith\nrij=/radicalBigg\nsin2/parenleftbiggπ\nLx(xi−xj)/parenrightbigg\n+sin2/parenleftbiggπ\nLy(yi−yj)/parenrightbigg\n,\nwherenh\ni= 1−Σσc†\niσciσ. The three parameters vβof\nβ=N,NN, andNNNare for short-ranged hole-hole\nrepulsion (attraction) if these values are less (greater)\nthan 1. The factor rα\nijis for long-ranged correlations39\nand it is repulsive if αis positive. LxandLyare the\nnumber of sites in the xandydirection, respectively.\nWe have found that the variational energy of the peri-\nodic stripe state without including the Jastrow factors is\nvery sensitive to boundary conditions. However the Jas-\ntrow factor is capable of reducing the dependence of the\nboundary conditions.\nAs shown in Fig.1, when the hole-hole repulsion of\nequation (11) is included in our VMC calculation, both\nthe uniform d-waveRVB state (filled circles) and the AF-\nRVB stripe state (empty circles) have gained significant\nenergies. In particular, the uniform d-wave RVB state\nhasgainedabout2 ∼3%intotalenergy. Itindicatesthat\nfor the extended t−Jmodel the holesarenot bounded as\ntightly as estimated by the simple RVB trial wave func-\ntions. Italsoimpliesthatthenano-scalephaseseparation\nof the AF stripe will not be preferable. Indeed, the low-\nest energy stripe state does not favor the AF stripe but0.060.080.100.120.140.160.180.20<nh(y)>\n-0.50-0.250.000.250.50<M(y)>\n2 4 6 8 10 12 14 160.0000.0010.0020.0030.0040.005Pxx(y,Ry)\ny(a)\n(b)\n(c)\nFIG. 2: The profiles of (a) hole density, (b) staggered mag-\nnetization and (c) pair-pair correlation function with Ry= 8\nfor the optimized states at 1 /8 doping in the extended t−J−\nmodel with t′/t=−0.3 andJ/t= 0.3. The filled (empty)\ncircles correspond to the AF-RVB stripe state without (with )\nhole-hole repulsion. In (c), the dashed (dashed-dotted) li ne\nindicates the uniform d-wave RVB state without (with) hole-\nhole repulsion. The red solid line corresponds to the random\nstripe state with hole-hole repulsive Jastrow correlation . All\nquantities are calculated on a 16 ×16 lattice system with the\nperiodic and antiperiodic boundary conditions along the x\nand y directions, respectively.\nthe AF-RVB stripe with a much more reduced staggered\nmagnetization as shown in Fig.2(b) and a larger pairing\nparameter ∆C\nv. Now the hole density has a much smaller\nvariation as shown by the empty circles in Fig.2(a).\nIt is surprising to find out that for all t′/t, the AF-\nRVB stripe state is almost degenerate in variational en-\nergy with the uniform d-wave RVB state. This is quite\nremarkable as the two trial wave functions are very dif-\nferent. As an illustration we show in Fig.2(a) and (b)\nthe variation of the hole density and the staggered mag-5\nnetization along the modulation direction, respectively,\nfor an AF-RVB stripe state (empty circles). It should\nbe noticed that not only the hole density is completely\nuniform for d-RVB state but there is also no staggered\nmagnetization at all. Instead of periodic stripes, we have\nalso examined the stripe state with 4 ×4 patches in the\n16×16 lattice system. For each patch, we choose a di-\nrection of the stripe, xory, randomly. We consider this\nstate as random stripe state. For simplicity, we still use\nthe same equations (5-7,11). The hole-density modula-\ntion of a random stripe state has been shown in the inset\nofFig.1. AsshowninFig.1,therandomstripestateswith\nthe Jastrow factors of equation (11) also have the opti-\nmized energies almost identical to uniform d-wave RVB\nand AF-RVB stripe states even though we have not op-\ntimized parameters on very bond or site independently.\nThese optimized random stripe states have finite mvand\n∆C\nvbut smaller ∆M\nv(less than one third of ∆C\nv).\nWe believe this energy degeneracy is caused mainly by\ntwo reasons. The first reason is that the terms in the\nt−JHamiltonian are all local within nearest neighbors\nor next nearest neighbors. The second reason is that the\nhopping terms and spin interaction not only are of the\nsame order of magnitude but they are competing against\neach other. We have found the energy competition be-\ntween the kinetic energy and spin interaction is very ro-\nbust among different states (not shown). Their competi-\ntion is enhanced by the no-doubly occupancy constraint\nas the presence of holes will suppress the spin interaction\nto zero. Thus it is possible to have locally different spin-\nhole configurationswith different emphasis on the kinetic\nenergy or the spin energy. Some of these patterns have\nlowerkinetic energy but higher magnetic energy than the\nuniformd-RVB state and some with opposite energetics.\nThere are at least two important implications of this\nenergy degeneracy. The first one is that the inhomoge-\nneousstatesarequite robustin the extended t−Jmodels\nwithout the need for introducing any other large inter-\nactions. States with different local arrangement of spin\nand holes may have very similar energies. The second\nimplication is that any small additional interaction could\nbreak the degeneracy. If we consider the ”realistic” situ-\nation of cuprates with large numbers of impurities, disor-\nders and electron-latticeinteractions, the inhomogeneous\nstates could be much more numerous and complex than\nwe have expected. Materials made with different pro-\ncessing conditions could also produce different inhomo-\ngeneous states. Since all these are presumably secondary\ninteractionssmallerthan the dominant tandJ, the mod-\nulations of charge density, staggered magnetization and\npairing are expected to be small. If one of the inter-\nactions, like electron-lattice interaction, becomes quite\nstrong as observed in LBCO−1/816, then the modula-\ntion will also become larger and longer ranged.\nWe have also investigated the pair-pair correlation\nfunction for the optimized states with/without hole-hole\nrepulsive Jastrow factors in the case of ( t′,t′′,J)/t=\n(−0.3,0.15,0.3). The singlet pair-pair correlation func-tion along the modulated direction (y direction) is de-\nfined as\nPxx(y,Ry) =1\nLx/summationdisplay\nx|∝angbracketleftΦ|∆†\nx(r)∆x(r+Ry)|Φ∝angbracketright|\n∝angbracketleftΦ|Φ∝angbracketright,(12)\nwherer= (x,y) and ∆†\nx(r) =c†\nr↑c†\nr+ˆx↓−c†\nr↓c†\nr+ˆx↑creates\na singlet pair of electrons among the nearest neighbors\nalongxdirection for each site r. Here, we focus on the\nlong range correlation, and thus set Ry= 8 to be the\nlargest distance on 16 ×16 lattice system. In Fig.2(c), it\nis shown that the hole-holerepulsion suppressesthe long-\nrange pair-pair correlation about three times in the uni-\nformd-waveRVBstate. Wehavealsofound that without\nhole-hole repulsive correlation, the long-range pair-pair\ncorrelation is much more reduced in the AF-RVB stripe\nstate than uniform d-wave RVB state, because the AF-\nRVB stripe state has almost vanishing ∆C\nv. Due to large\nmv, this AF-RVB stripe state shows much larger ampli-\ntude of the Pxxmodulation. However, after considering\nthe hole-hole repulsion, the AF-RVB stripe and uniform\nd-wave RVB states have almost the similar magnitude\nof pair-pair correlation as shown by the empty circles\nand dashed-dotted line in Fig.2(c), respectively. For the\nrandom stripe state shown in the inset of Fig.1 the aver-\nage valuePxxfor the whole system is shown as the red\nsolid line in Fig.2(c). It is essentially the same as the\nvalue of uniform d-wave RVB state and the periodic AF-\nRVB stripe state. Although the staggered magnetization\nfor the random stripe state has larger variation than the\nperiodic stripe state shown in Fig.2(b), the pairing cor-\nrelation is unchanged. These results indicate that the\nlong-range pair-pair correlation is mostly determined by\nthe value of ∆C\nvand is rather insensitive to the modu-\nlation of the hole density and staggered magnetization.\nThus we also expect to have a robust d-wave node ob-\nserved by recent experiments10,13. This will be shown\nbelow.\nIII. DENSITY OF STATES BY THE\nGUTZWILLER APPROXIMATION\nAccording to the VMC calculation for the extended\nt−Jmodel, it is likely that there are a number of inho-\nmogeneous states close in energy to the uniform ground\nstate. Then, some sort of small perturbation may choose\na particular stripe state as the ground state. Assum-\ning such a situation, here we regard a stripe state as the\nground state, and consider the projected quasi-particle\nexcitation spectra. However, calculation of the excited\nstates by the VMC method is computationally very ex-\npensive, and it takes too much time to investigate wide\nparameterrangetoobtaingeneralpropertiesofthestripe\nstates. Furthermore, one can take only a limited system\nsize and it is difficult to obtain dense spectra.\nTherefore, as a first step, we use a Gutzwiller mean-\nfield approximation for this purpose. The minimization6\n/Minus1012\nE/Slash1tv0.050.10.15LDOS/LParen1a/RParen1\n/Minus1012\nE/Slash1tv0.050.10.15LDOS/LParen1b/RParen1\nFIG. 3: Spatially averaged local DOS for the random stripe\nstates calculated bythe non-self-consistent BdG equation . (a)\nFor ∆C\nv/negationslash= 0, we use parameters optimized by the VMC, t′\nv=\n−0.35,t′′\nv= 0.16,m= 0.15,ρ= 0.03, ∆C\nv= 0.28 , ∆M\nv=\n0.02, butµ=−0.875tvis adjusted to realize 1 /8 filling, in\nunits oftv. (b) The same parameters except for ∆C\nv= 0.\nof the total energy yields a BdG equation49. Usually\nthe parameters in the BdG equations are solved self-\nconsistently to find an optimal solution. However, since\nwe already have the assumed inhomogeneous ground\nstate here, we shall use parameter sets obtained from\nVMC results and diagonalize the BdG Hamiltonian only\nonce, instead of solving self-consistently. Furthermore,\nfor convenience, we slightly modify the Gutzwiller pro-\njection by attaching fugacity factors. Namely, we assume\nthatPλ|ψ∝angbracketrightis the ground state and that Pλγ†\nn|ψ∝angbracketrightare the\nexcited states, where Pλ≡P/producttext\niσλniσ\niσ, andλiσis a fu-\ngacity factor to impose the local electron density conser-\nvationforeachspin, namely,/angbracketleftψ|PλˆniσPλ|ψ/angbracketright\n/angbracketleftψ|P2\nλ|ψ/angbracketright=/angbracketleftψ|ˆniσ|ψ/angbracketright\n/angbracketleftψ|ψ/angbracketrightfor\nanyiandσ. Thequasi-particleoperators γ†\nnareobtained\nby solvingthe BdG Hamiltonian. In addition, here we do\nnot take into account the Jastrow factor as it only affects\nthe hole-hole correlation slightly but not the local DOS\nstudied below. Then, under the assumption of the non-\nself-consistency, the BdG Hamiltonian is represented by\nequation (2). With this formulation, Pλγ†\nn|ψ∝angbracketrightof differ-\nentnare approximately orthogonal to each other49, and\nthus we expect that it is suitable to use Pλinstead ofP\nfor our purpose here. Since the result presented below\nis qualitatively not very sensitive to small change of the\nparameters, we expect that such a modification of the\nprojection should not affect the results qualitatively.\nThen, by taking the most dominant terms, the local\nDOS is represented by\nN↑(R,ω) =gt\nR↑/summationdisplay\nn|un\nR|2δ(ω−En),(13)\nN↓(R,ω) =gt\nR↓/summationdisplay\nn|vn\nR|2δ(ω+En),(14)/Minus1.5 /Minus1 /Minus0.500.5 11.5 2\nE/Slash1tv0.050.10.150.2LDOS\nFIG. 4: Local DOS at eight different positions for the ran-\ndom stripe states with ∆C\nv/negationslash= 0 calculated by the non-self-\nconsistent BdG equation.\nwhere index nruns for both positive and negative eigen-\nvalues. Note that only the position dependent constant\ngt\nRσ≡(1−nR↑−nR↓)/(1−nRσ) is multiplied in front\nof the local DOS by the standard BdG formalism. Since\nthe result of the site-centered stripe is very similar to\nthat of the bond-centered stripe, we show only the lat-\nter. Here we shall only discuss our results for the ran-\ndom stripe state. For the random stripe state consid-\nered above, each randomly oriented domain is assumed\nto have the same parameters so that the VMC calcula-\ntion is possible. Ideally we should have optimized these\nvariationalparameters on everybond or every site. Then\nwe expect to have a much broader distribution of these\nparameters. To simulate this effect, we simply replace\neach ∆ijby (1 +ξij)∆ij, whereξijis a random vari-\nable which has the Gaussian distribution around 0 with\nthe standard deviation of 1. We use a supercell of size\n32×32 sites, and the same configuration is repeated as\n20×20 supercells to obtain the local DOS. The Fourier\ntransformwith respect to the supercell index is similarto\na system of small clusters with many twisted boundary\nconditions.\nAs shown in Fig.3(a), the spatially averaged/summationtext\nσNσ(R,ω) for the AF-RVB stripe (∆C\nv∝negationslash= 0) has a V-\nshape at low energy. On the other hand for an antiphase\nstripe which has ∆C\nv= 0 there is no V-shape as shown in\nFig.3(b). The energy level is broadened with a width of\n0.05tv. In Fig.3(b), there is a very small dip at E= 0, it\nwould be bigger if ∆M\nvincreases. Our results are consis-\ntent with the very recent report by Baruch and Orgad31.\nIn general, there is no V-shape DOS for the antiphase\nstripe. For the same system as Fig.3(a), in Fig.4 we plot\nthe position dependence of local DOS at randomly cho-\nsen 8 sites. The low energy spectra seem less influenced\nby the disorder than high energy. This result shows that\nthe node and the low energy V-shape DOS are robust\nagainst this kind of inhomogeneity. This is possible be-\ncausenodal k-pointsdonothavemanystatestomix with\nand also the suppression of impurity scattering50. The\nsub-gap structures12,13also seem to be quite apparent.7\nIf we switch off the random variables ξij, the gap vari-\nations from site to site are small. These gap variations\ngrow larger as distributions of ξijbecome wider.\nThe key to understand the absence of V-shape in the\nLocal DOS for the antiphase stripe is the Fourier trans-\nform of the modulated ∆ ijterm written as,\n∆M\nv/summationdisplay\nkcoskx(e−iθc†\nk+q,↑c†\n−k,↓+eiθc†\nk−q,↑c†\n−k,↓+h.c.)\n−∆M\nv\n2/summationdisplay\nk/bracketleftbigg/parenleftBig\neiky+e−i(ky+qy)/parenrightBig\nei(qy\n2−θ)c†\nk+q,↑c†\n−k,↓\n+/parenleftBig\neiky+e−i(ky−qy)/parenrightBig\ne−i(qy\n2−θ)c†\nk−q,↑c†\n−k,↓+h.c./bracketrightbigg\n,\n(15)\nwhereθ= 0 for site-centered stripes and θ=qy/2 for\nbond-centered stripes; q= (0,π/4) for the antiphase\nstripe, and q= (0,π/2) for the AF-RVB stripe. What\nis important here is that it contains onlypairing with\nnonzerocenter-of-mass momentum as the FFLO state51.\nIn the case of zero-momentum pairing as the conven-\ntional BCS theory, the spin-up electron band couples\nwith the spin-down hole band (the upside-down down\nelectron band). These bands intersect at the Fermi level,\nand a gap opens if ∆ k∝negationslash= 0. In the case of finite- qpair-\ning, however, the spin-up electron band couples with ±q\nshifted spin-down hole bands, and thus the band inter-\nsections occur not at the Fermi level. Therefore, a gap\ndoes not open at the Fermi level. The constant ∆C\nvterm\nwhichformsthe usualCooperpairisnecessaryforhaving\nthe node and the V-shape DOS.\nTo compare with ARPES experiments, A(k,ω) is also\ncalculated. Since A(k,ω) is regarded as the local DOS in\nthek-space, Let us take the Fourier transform of renor-\nmalizedun\nR,vn\nR, namely,\n(˜un\nk,˜vn\nk)≡1√Nsite/summationdisplay\nRe−ikR/parenleftbig\ngt\nR↑un\nR,gt\nR↓vn\nR/parenrightbig\n.(16)\nThen,Aσ(k,ω) is written as\nA↑(k,ω) =/summationdisplay\nn|˜un\nk|2δ(ω−En), (17)\nA↓(k,ω) =/summationdisplay\nn|˜vn\nk|2δ(ω+En). (18)\nFig.5 shows/summationtext\nσAσ(k,ω) of the random stripe state\nwith ∆C\nv∝negationslash= 0, where eachof the δ-function spectraarere-\nplaced with a Lorentzian distribution with δE= 0.05tv.\nIn comparison with the uniform d-wave RVB state, the\nspectral weight around the antinodal region is much\nstrongly reduced than near the node.\nIV. CONCLUSIONS\nIn summary, we have used a variational approach to\nexamine the possibility of having inhomogeneous ground/Minus1.5 /Minus1/Minus0.50\nE/Slash1tv24681012A/LParen1k,E/RParen1/LParen1a/RParen1\nk/Equal/LParen13/SΠaceΠ\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt\n16,3/SΠaceΠ\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt\n16/RParen1k/Equal/LParen14/SΠaceΠ\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt\n16,4/SΠaceΠ\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt\n16/RParen1k/Equal/LParen15/SΠaceΠ\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt\n16,5/SΠaceΠ\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt\n16/RParen1k/Equal/LParen16/SΠaceΠ\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt\n16,6/SΠaceΠ\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt\n16/RParen1\n/Minus1.5 /Minus1/Minus0.50\nE/Slash1tv24681012A/LParen1k,E/RParen1/LParen1b/RParen1\nk/Equal/LParen1Π,2/SΠaceΠ\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt\n16/RParen1k/Equal/LParen1Π,3/SΠaceΠ\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt\n16/RParen1k/Equal/LParen1Π,4/SΠaceΠ\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt\n16/RParen1k/Equal/LParen1Π,5/SΠaceΠ\n/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt\n16/RParen1\nFIG. 5:P\nσAσ(k,E) for the random stripe state with\n∆C/negationslash= 0. Dotted lines are for uniform d-wave RVB state\n(ρv= 0,mv= 0,∆M\nv= 0). (a) Nodal and (b) near antinodal\nregion.\nstates within the extended t−Jmodel with 1 /8 doping.\nWe considered states with spatial modulation of charge\ndensity, staggered magnetization and pairing amplitude.\nBesides the antiphase or inphase stripes considered by\nmany groups we have proposed a new AF-RVB stripe. In\nthis stripe state, we assume there is a constant pairing\namplitude besides the various modulation. In addition\nto considering states with periodic stripes we also con-\nsider random stripe states to simulate the cluster glass\nstate observed by experiments10. By improving the trial\nwave functions with the introduction of hole-hole repul-\nsive correlation,we have greatlyimprovedthe variational\nenergiesbyseveralpercentsforbothuniformRVB d-wave\nSC state and states with periodic AF-RVB stripes for re-\nalistic values of t′/t. Most surprisinglythe random stripe\nstate essentially also has the same energy as the uniform\nstate in spite of our oversimplified assumption that all\nthe stripe domain has the same patterns of modulation\ninstead of each site or bond with different values. This\nrandom stripe state also has about the same long-range\npair-pair correlation as the uniform or periodic stripe\nstate even though there are significant staggered mag-\nnetization and charge variation from site to site. Then\nwe also examined the local DOS and the spectral weight\nof the random stripe state by using Gutzwiller approx-\nimation. We found the V-shape DOS and the node are\nstillpresentateverysite. ThelocalDOSmeasuredatdif-\nferent positions shows a broad variation of the gaps and\nalso it has sub-gap structures seen in experiments12,13.\nThe spectral weight at the antinodal direction is negli-\ngibly small but finite around the node. All these results\narequite consistentwith experimentsreportedfor cluster\nglass state in BSCCO10.\nOur result also resolves an inconsistency with ex-8\nperiments derived from previous theoretical calculations\nwithoutincludingthehole-holerepulsioninthetrialwave\nfunctions. Stripe is neither stabilized nor destabilized by\nthe long range hopping. In fact, due to the competition\nbetween the kinetic energy gain and magnetic interac-\ntion, it is very natural to have the spatial modulation, in\nperiodicorrandomconfiguration,ofchargedensity, mag-\nnetization and even pairing amplitude. The constraint of\ndisallowing doubly occupation of electrons at each lattice\nsite has significantly enhanced the competition. Many\nlocal arrangements of spin and hole configurations could\ngivealmostidenticaltotalenergyasthe uniformsolution.\nRecently, Capello et al.34have also found that the en-\nergy of the periodic RVB stripe state is very close to\nthat of the uniform RVB state by using a variational cal-\nculation. They have considered several possibilities for\nthe stability of the RVB stripe state, such as lattice dis-\ntortion, t’-effect, and long-range Coulomb repulsion with\nthe conclusion that uniform RVB state is still the lowest\nenergy state. This is very consistent with our conclusion\nalthough we haveincluded the antiferromagneticorderin\nthe stripe state. The issue about whether AF is present\nin the stripe will be discussed in the future. They have\nnot considered the cluster glass state with random AF-\nRVB stripe domains which is also a good candidate for\nthe ground state.\nThe presence of inhomogeneous or cluster glass states\nis apparently a very natural consequence of the t−Jmodel. There is no need for introducing additional in-\nteractions to generate such states. In fact we showed\nthat the RVB state with a finite constant pairing is quite\ncompatible with the local variations of charge density,\nmagnetization and even pairing amplitude. As long as\nthis modulation is not overlystrong, the superconductiv-\nity still survives as the node and V-shape DOS are still\npresent. In a realistic material, other interactionssuch as\nimpurity, disorder, and electron-lattice interactions, etc.,\nno doubt will help to determine the most suitable local\nconfigurationofspinsandholesbut they willnot produce\na globally ordered state unless there is a very strong and\ndominant interaction like the electron-lattice interaction\nseen inLa2−xBaxCuO4at 1/8 doping. The verification\nof this is left for future work.\nV. ACKNOWLEDGMENTS\nThisworkissupportedbytheNationalScienceCouncil\nin Taiwan with Grant no.95-2112-M-001-061-MY3. 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Lett. 96, 117006 (2006)."    },    {        "title": "0807.4962v2.Negative_density_of_states__screening__Einstein_relation__and_negative_diffusion.pdf",        "content": "arXiv:0807.4962v2  [cond-mat.str-el]  27 Aug 2008Negative density of states: screening, Einstein relation, and\nnegative diffusion.\nA. L. Efros∗\nDepartment of Physics, University of Utah, Salt Lake City UT , 84112 USA\nAbstract\nIn strongly interacting electron systems with low density a nd at low temperature the thermo-\ndynamic density of states is negative. It creates difficultie s with understanding of the Einstein\nrelation between conductivity and diffusion coefficient. Usin g the expression for electrochemical\npotential that takes into account the long range part of the C oulomb interaction it is shown that at\nnegative density of states Einstein relation gives a negati ve sign of the diffusion coefficient D, but\nunderthis condition thereis nothermodynamiclimitation o n the sign of D. It happensbecausethe\nunipolar relaxation of inhomogeneous electron density is n ot described by the diffusion equation.\nThe relaxation goes much faster due to electric forces cause d by electron density and by neutral-\nizing background. Diffusion coefficient is irrelevant in this c ase and it is not necessarily positive\nbecause process of diffusion does not contribute to the positi ve production of entropy. In the case\nof bipolar diffusion negative Dresults in a global absolute instability that leads to forma tion of\nneutral excitons. Graphene is considered as an example of a s ystem, where the density relaxation\nis expected to be due to electric force rather than diffusion. I t may also have a negative density of\nstates.\nPACS numbers: 71.27. +a,73.50.-h\n1I. INTRODUCTION\nThe idea of the Einstein relation was put forward by Einstein1and Smoluchowski2in\n1905-1906. Both scientists considered the Brownian motion in the p resence of gravitational\nforce. The result is the relation between mobility uin the field and diffusion coefficient D.\nIn case of electric field and particles with the charge eit has a form\neD=Tu, (1)\nwhereTis the temperature in energy units. The main idea was equivalence of a n exter-\nnal force and the density gradient. Of course, both Einstein and S moluchowski did not\ncare about negligible mutual gravitational or any other small intera ctions of the Brownian\nparticles.\nThe formulation of the Einstein relation for electrons is based upon e lectrochemical po-\ntential, thethermodynamic functionthat, like temperature andpr essure, shouldbethe same\nat all points of the system in the equilibrium state. The usual argume nts are asfollows. If an\nexternal potential ψis applied to the system, the condition of thermodynamic equilibrium\nreads\nEec=µ(n)+eψ=Const, (2)\nwhereµ(n) is the chemical potential as a function of inhomogeneous electron densityn.\nIn the equilibrium both nandψare function of coordinates while Eecis constant. The\ntemperature Tshouldalsobeconstant. Therefore, theelectrical currentdens ityjatconstant\nTcan be written in a form3\nj=−σ\ne∇Eec=σE−D∇en, (3)\nwhereσis conductivity and E=−∇ψ. Then one gets relation connecting σandD\nσ\ne2dµ\ndn=D, (4)\nwhich is also called Einstein relation. For the Boltzman gas dµ/dn=T/nand one gets\nEq.(1) ifσ=enu. It looks like derivation of Eq. (4) is independent of the properties o f the\nsystem and this equation can be consider as general thermodynam ic law.\nAsimple observationshows however thatinthe caseof non-ideal ele ctron gasthe Einstein\nrelation needs some comments. We discuss an electron gas on the po sitive background at\n2low temperatures and low densities when dimensionless parameter rsis not very small. Here\nr3\ns= 3/(4πna3\nB) for 3-d case and r2\ns= 1/πn2a2\nB, wherenandn2are 3- and 2-dimensional\nelectron densities respectively and aB=/planckover2pi12κ/me2is the Bohr radius, mis an effective\nelectronic mass, κis an effective permittivity.\nThe problems of dynamic screening and diffusion in slightly non-ideal ele ctron gas(rs<<\n1) with electron-electron interaction were considered in details abo ut 20 years ago (See\nRef.[4,5,6])Inthiscasethethermodynamicdensityofstatesislargeandposit ive. Iconcentrate\nhere on the strongly non-ideal case rs≥1.\nAn electron gas on the positive background at low temperatures an d low densities has\nenergyEof the order of −e2n1/dN/κ, whered= 2,3 is the space dimensionality and nis the\ndensity per area or volume respectively, Nis total number of electrons. Then µ∼ −e2n1/d/κ\nandE,µ, anddµ/dnare negative7,8. The first experimental confirmation of this idea was\ndone by Kravchenko et al9,10, but direct quantitative study of this effect was performed by\nEisenstein et al11,12.\nThe derivative dµ/dnis proportional to the reciprocal compressibility of the electron\ngas. Note that compressibility has to be positive due to the thermod ynamical condition of\nstability. However, this principle cannot be applied to the charged sy stems, like electron gas,\nbecause part of their energy is outside the system in a form of the e nergy of electric field.\nOn the other hand, in the case of a neutral electron-hole plasma, t he situation of negative\ncompressibility can arise leading to collapse of the system. Such a situ ation is considered at\nthe end of Sec. III.\nIt follows from Eq. 4 that if dµ/dnis negative, diffusion coefficient Dand conductivity σ\nhaveoppositesigns. Thisobservationneeds anexplanationbecaus enearthethermodynamic\nequilibrium both of them have to be positive to provide positive entrop y production due to\nthe Joule heat and due to the relaxation of inhomogeneous density.\nII. ELECTROCHEMICAL POTENTIAL AND STATIC SCREENING\nTo resolve this contradiction one should include the long-range part of the Coulomb\npotential created by inhomogeneous electron gas into the functio nEecin Eq. (2). This\ncontribution is a functional of n(r).\nTo findEectaking into account electron-electron interaction one should minimiz e the\n3Helmholtz energy Fwith respect to electron density n(r) at a given value of TandN. For\nlowTone gets\nF=e2\n2κ/integraldisplay /integraldisplayn′(r)n′(r′)d3rd3r′\n|r−r′|+/integraldisplay\nf(n+n′)d3r+\n/integraldisplay\nen′(r)ψd3r−Eec/integraldisplay\nn′(r)d3r, (5)\nwherefis the Helmholtz energy density of a homogeneous electron system t hat results from\nthe interaction in a neutral system, like the Wigner crystal or ”Wign er liquid”. Since this\ninteraction comes mainly from the nearest neighbors and n(r) is a smooth function, one\nmay assume that both fand chemical potential µ=df/dnare local functions of n(r) We\nassume also that n(r) =n+n′(r), wherenis average density and n′≪n.\nMinimization of this expression with respect to n′gives the equation\nEec=µ(n)+eψ+dµ\ndnn′+e2\nκ/integraldisplayn′(r′)d3r′\n|r−r′|. (6)\nIt differs from Eq. (2) by the potential of electrons in the right han d side. Note that\nthis potential is due to the violation of neutrality in a scale much larger than the average\ndistance between electrons. To check this equation we consider th ermodynamic equilibrium\nand find equations for the Thomas-Fermi static screening in 3- and 2-dimensional cases.\nSinceEecis independent of rin thermodynamic equilibrium one may take Eec−µ(n) as a\nreference point for the total potential ϕdefined as\nϕ=ψ+e\nκ/integraldisplayn′(r′)d3r′\n|r−r′|. (7)\nIt follows from Eq. (6) that\neϕ=−dµ\ndnn′. (8)\nThe Poisson equation has a form\n∇2ϕ=−4π(en′−ρext)\nκ, (9)\nwhereρextis density of external charge. Using Eq. (8) one gets final equatio n for the 3-d\nlinear screening\n∇2ϕ=−q2\n3ϕ−4πρext\nκ. (10)\nHere\nq2\n3=4πe2\nκdn\ndµ(11)\n4is the reciprocal 3-dimensional screening radius.\nConsider now a thin layer (x-y plane) with 2d electron gas separating two media with\ndielectric constants κ1andκ2. In this case one should substitute n⇒n2δ(z) andκ⇒¯κ=\n(κ1+κ2)/2. The results is13\n∇2ϕ=−q2ϕδ(z)−4πρext\n¯κ, (12)\nwhere\nq2=2πe2\n¯κdn2\ndµ. (13)\nIt is important that Eqs. (10), (12) are applicable only if the screen ing is linear ( n′≪\nn)14. There is another serious problem of applicability the Thomas-Fermi approximation\nin the case of the negative density of states. Indeed, the dielectr ic permittivity in this\napproximation has a form\nǫ(q) =κ(1−|q2\n3|\nq2) (14)\nin 3-d case and\nǫ(q) =κ(1−|q2|\nq) (15)\nin 2-d case. In both cases it has roots at q=|q3|,|q2|. The expression for the screened\npotentialϕhas a form\nϕ(r) =/integraldisplayϕ0(q)exp(iq·r)dq\nǫ(q), (16)\nwhereϕ0is a bare potential. Thus, the roots of ǫtransform into the first order poles without\nany reasonable way of the detour. Such a detour follows from the c asuality for the ω-plane\nbut not for the q-plane. Moreover, the electrostatic potential s hould be real and one cannot\nadd a small imaginary part in the denominator. Therefore I think tha t the poles do not\nhave any physical sense.\nThereasonisthatnegativesignofthedensityofstatesappearsw henq3,q2areoftheorder\nof average distance between electrons ¯ r. At such distances the very concept of macroscopic\nfield does not have sense. However, if the bare potential has only h armonics with q≪\n|q3|,|q2|, the Eqs.(10,12) have a sense. Consider, for example, the screen ing of the positive\nchargeZat a distance z0from the plane with 2-d gas(plane z= 0. The solution of Eq.(12)\nhas a form13\nϕ(ρ) =/integraldisplay∞\n0Zexp(−qz0)\nκ(q+q2)qJ0(qρ)dq, (17)\n5whereρis a polar radius in the plane z= 0. Suppose that |q2|z0≫1. Now the contri-\nbution to integral Eq.(17) from q≃ |q2|is exponentially small and one can ignore qin the\ndenominator. Then\nϕ(ρ) =Zz0\nκq2(z2\n0+ρ2)3/2. (18)\nNote that at q2<0 a positive charge creates a small negative potential in the plane with\nelectrons. That is what I call ”overscreening”.\nExtra electron density, as calculated from Eq. (8) is\nen′=−Zz0\n2π(z2\n0+ρ2)3/2(19)\nIt is negative and independent of the sign of q2. One can see that the total charge\n/integraldisplay∞\n0en′2πρdρ=−Z (20)\nDue to geometry of the problem electric field is zero below the plane wit h electrons. As\nfollows from Eq. (8), the signs of charge density and potential are opposite if the density of\nstates is negative.\nFor the case of two such planes (double quantum well structure) L uryi15has predicted\na small penetration of electric field through the first plane. He has c onsidered the case of\npositive density of states. Then the small penetrating field betwee n two planes has the same\ndirection as the incident field.\nEisenstein atal.11studiedthiseffectexperimentallyandfoundoutthatatnegativede nsity\nof states the propagating field is opposite to the incident field and th is is also a result of the\noverscreening (see the quantitative theory in Ref.12,16,17).\nNegative density of states was also used18for theexplanation ofmagnetocapacitance data\nby Smith at al.19.\nIII. CONDUCTIVITY VERSUS DIFFUSION\nNow I come back to the problem of the negative diffusion. If the syst em is not in\nequilibrium the electric current can be written in the same form as Eq. (3)\nj=−σ\ne∇Eec. (21)\n6Using Eq. (6) one gets\nj=σE−D∇en′−σe\nκ∇/integraldisplayn′(r′)d3r′\n|r−r′|. (22)\nHere D is connected to σby the Einstein relation Eq. (4). Considering relaxation of the\ncharge density one can ignore external field E. The relaxation is described by the continuity\nequation\n∂(en)\n∂t=−∇·j (23)\nor\n∂(en)\n∂t=σ/parenleftbigg1\ne2dµ\ndn∇2(en′)−4πen′\nκ/parenrightbigg\n. (24)\nThe ratioRof the first (diffusion) term in the right hand side to the second (field ) term is\nR= (q2\n3L2)−1, whereL−2=∇2n′/n′is the characteristic size of the extra charge and q2\n3is\ngiven by Eq. (11). If electron gas is non-ideal, q3∼1/¯r, where ¯ris the average distance\nbetween electrons. However, the very concept of diffusion equat ion is valid at L≫¯r. This\nmeans that for the non-ideal gas |R| ≪1 and the diffusion term in Eq. (24) should be\nignored. Then the equation has a simple solution\nn′(r,t) =n′(r,0)exp−(t/τM), (25)\nwhereτM=κ/(4πσ) is well-known Maxwell’s time. Coefficient Ddoes not enter in this\ncase in the entropy production and it does not have a physical sens e. Thus in 3-dimensional\nnon-ideal electron gas negative dµ/dndoes not create any contradiction with the Einstein\nrelation.\nIn the 3d gas of high density µ∼n2/3andR∼(¯r/L)2/rswithrs<1. In this case R\nmight be large and diffusion is possible. However dµ/dn> 0, andD>0.\nNow we consider the relaxation of the charge density in 2-dimensiona l case. Instead of\nEq.(24) one gets\n∂(en2)\n∂t=σ2(1\ne2dµ\ndn2∇2(en′\n2)\n−e\n¯κ∇2/integraldisplayn′\n2(r′)d2r′\n|r−r′|). (26)\nHeren2,σ2and∇are 2-dimensional density, conductivity, and 2-dimensional gradie nt re-\nspectively. To consider the ratio R2of the first (diffusion) term to the second (field) term it\n7is convenient to make the Fourier transformation. Then one gets\n∂(nq)\n∂t=−σ2(1\ne2dµ\ndn2q2nq+2πq\n¯κnq), (27)\nwherenqis the Fourier transformation of n′\n2.\nNow we find that the ratio of the first ( diffusion) term in the right han d side of Eq.\n(27) to the second (field) term R2=q/q2, whereq2is given by Eq. (13). Similar to the\n3d case in the non-ideal gas |q2| ∼1/¯rand diffusion should be ignored. Then we get the\nDyakonov-Furman equation20\n∂(nq)\n∂t=−vqnq, (28)\nwhere velocity v= 2πσ2/¯κ. The physical meaning of this equation is that extra density of\nelectrons localized initially at some spot propagates in all directions wit h velocityvconserv-\ning the total amount of extra electrons. Of course, this way of re laxation is more efficient\nthan diffusion (random walk), because r∼vtwhiler∼√\nDtin the case of diffusion. Thus,\ndiffusion coefficient Dis irrelevant and negative dµ/dndoes not create any contradiction\nwith the Einstein relationIn a high density electron gas R2=q¯r/rsand diffusion mechanism\nis possible. In this case dµ/dn> 0 andD>0.\nOne can consider this problem from a different point of view. In both 3 d and 2d cases the\nnegative diffusion coefficient Dappears in the term with the highest derivative that leads\nto the absolute instability even if Dis small21. Consider, for example Eq. (24) for 3d case.\nAfter the Fourier transformation the solution for the charge den sityρ=en′can be written\nin a form\nρq=ρ0\nqexp/parenleftbigg\n−4πσt\nκ−Dq2t/parenrightbigg\n, (29)\nwhereDis given by the Einstein relation Eq. (4). One can see that at D <0 solution\nincreases with time exponentially for harmonics with q¯r≥1.\nThe physical explanation is as follows. The Eqs.(24,26) contain avera ge distance between\nelectrons ¯r. So they contain information that the charged liquid has a discreet e lectronic\nstructure. This information comes from the negative density of st ates which originates\nfrom the interaction of the separate electrons. That is why macro scopic equations become\nunstable at small spacial harmonics. The message is that n(r) is rather a set of δ-functions\nthan a continuous function. The instability is absent if Dis positive.\nThe instability of small spatial harmonics at small negative Ddoes not affect larger\nharmonics because Eqs.(24,26) are linear. Due to the linearity differe nt harmonics are in-\n8dependent and transformation of energy from small spacial harm onics to large harmonics is\nforbidden (cp. phenomenon of turbulence in non-linear hydrodyna mics where the transfor-\nmation of energy is not forbidden, but the instability is initiated by larg e harmonics).\nTherefore, I think that at small Dapproximation D= 0 that gives Eqs.(25,28) is correct.\nOne should note that the problem of the non-physical roots of elec tric permittivity dis-\ncussed in the previous section is of the same nature.\nBefore we discussed the unipolar diffusion. Consider the simplest cas e of the ambipolar\ndiffusion assuming that at t= 0 the densities of electrons and holes are equal in some finite\nregion of space and are zero otherwise. Moreover we assume that the local macroscopic\ncharge density ρ(r,t) = 0 and a recombination of carriers is very slow. In this case Eq. (6)\ndescribes the electron-hole system in quasi-equilibrium. At large rsone getsE,µ,dµ/dn< 0\nbut the last term in Eq. (6) is absent. So the smearing of the density of particles is described\nby the equation of diffusion at all rs, but at small density ( rs≥1) coefficient D<0. Then\nthe absolute instability takes place for all harmonics that means a co llapse of the system.\nThus the electron-hole ”Wigner liquid” and crystal are unstable.\nThis result is very transparent. It happens because negative µjust means that the energy\nof the system decreases with increasing density. In bipolar case ne utrality is provided by\nthe particles and we do not consider any background. Thus the inst ability is a result of the\nnegative compressibility in a neutral system. At large enough rsthese particles are classical,\nand the absence of the mechanical equilibrium follows also from the Ea rnshaw theorem. In\nreality quantum mechanics becomes more important with increasing d ensity. As a result\nthe excitons are formed. These neutral particles have a positive d iffusion coefficient Daand\ntheir density smears with time through all available space. This proce ss is described by a\nregular diffusion equation. In the case of optical excitation the car riers may appear in the\nform of the excitons from the very beginning\nFor the coefficient of the ambipolar diffusion Daa textbook equation22\nDa=2DeDh\nDe+Dh(30)\nis often used, where De,hare diffusion coefficients of electron and holes in unipolar case.\nAs follows from the previous discussion, one should be careful with t his equation because\nfor the non-ideal electron (or hole) gas these unipolar coefficients might be negative and\nmeaningless. It happens because in unipolar case there is a deviation from neutrality that\n9creates electric field, while in bipolar case the system is neutral. In th is case Eq. (30) does\nnot work and one should calculate Dain a different way as a diffusion of the exciton.\nIn the recent paper by Zhao23the experimental results for the ambipolar diffusion in\nsilicon-on-insulator system are compared with Eq. (30). At high tem peratures a good\nagreement is found while at low temperatures the observed values o fDaare 6-7 times less.\nThe previously reported values24show similar temperature dependence.\nThe author’s explanation is that coefficients De,hare taken for the bulk silicon using\nEinstein relation and they might be larger than in the film at low tempera tures. However,\nthe reason discussed above cannot be excluded.\nIV. GRAPHENE AS A POSSIBLE EXAMPLE OF A NON-IDEAL ELECTRON\nSYSTEM\nIt is interesting to discuss the single layer graphene as an example of the system with\nnon-ideal electron gas. Graphene is a gapless material with the linea r spectrum of electrons\nand holes near the Dirac point. Due to some reasons, that are not q uite clear now, the\nvelocityvof electrons and holes in equation ǫ=±pvis of the order of e2//planckover2pi1. It follows\nthat at any Fermi energy inside this linear spectrum electron gas in g raphene is non-ideal\nin a sense mentioned above: the absolute value of the chemical pote ntial is of the order of\ninteraction energy e2n1/2. It means that unipolar density relaxation in this system should\nbe described by the Dyakonov-Furman equation rather than by diff usion equation.\nHowever,without magnetic field the electron gas in graphene is margin ally non-ideal. It\ncannot be classical, like an electron gas of a low density with quadratic spectrum. The\nmarginal situation makes theoretical calculations very difficult. Nev ertheless, it is accepted\nthat the Wigner crystal in single layer graphene is absent without ma gnetic field25,26. The\nsign ofdµ/dnis also an interesting question but very difficult for theoretical stud y. Recently\ntunneling microscopy experiment has been done by Martin et al.27. They claim that their\nmeasurement give the thermodynamic density of states and that it is positive. The last\nstatement might be a result of disorder.\n10V. CONCLUSION\nFinallyIarguethatthe negative signofdiffusion coefficient thatfollow s fromthe Einstein\nrelation at negative density of states does not lead to any contrad iction because diffusion\ncoefficient is irrelevant for the unipolar transport under this condit ion. The sign of the\ndiffusion coefficient in this case should not be definitely positive becaus e the diffusion is not\nthe main source of the entropy production. In bipolar situation neg ative diffusion means\nthe collapse of the system and formation of neutral excitons.\nI am grateful to Boris Shklovskii and Yoseph Imry for important dis cussion. I am espe-\ncially indebted to David Khmelnitskii and Emmanuel Rashba for multiple d iscussions and\ncriticism.\n∗Electronic address: efros@physics.utah.edu\n1A. Einstein, Annalen der Physik 17, 549 (1905).\n2M. von Smoluchowsky, Annalen der Physik 21, 756 (1906).\n3L. D. Landau and E. Lifshitz, Electrodynamics of Continuous Media (Butterworth-Heinenann,\n1984), chapter III.\n4B. L. Altshuler, A. G. Aronov, and P. A. Lee, Phys. Rev. Lett 44, 1288 (1980).\n5A. Y. Zyuzin, JETP Lett. 33, 360 (1981).\n6B. L. Altshuler and A. G. Aronov, Electron-Electron Interaction in Disordered Systems (ed. b y\nA. L. Efros and M. Pollak) (North-Holland, Amsterdam, 1985), p. 37.\n7B. I. Shklovskii and A. L. Efros, JETP Lett. 44, 669 (1987).\n8M. S. Bello, E. I. Levin, B. I. Shklovskii, and A. L. Efros, Sov . Phys JETP 53, 822 (1981).\n9S. V. Kravchenko, V. M. Pudalov, and S. G. Semenchinsky, Phys . Lett. A 141, 71 (1989).\n10S. V. Kravchenko, D. A. Rinberg, S. G. Semenchinsky, and V. M. Pudalov, Phys. Rev. B 42,\n3741 (1990).\n11J. P. Eisenstein, L. N. Pfeiffer, and K. West, Phys. Rev. Lett 68, 674 (1992).\n12J. P. Eisenstein, L. N. Pfeiffer, and K. West, Phys. Rev. B 50, 1760 (1994).\n13T. Ando, A. B.Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982).\n14A. L. Efros, Phys. Rev B 45, 11354 (1992).\n1115S. Luryi, Appl. Phys. Lett. 52, 501 (1988).\n16A. L. Efros, F. G. Pikus, and V. G. Burnett, Solid State Comm. 84, 91 (1992).\n17F. G. Pikus and A. L. Efros, Phys. Rev. B 47, 16395 (1993).\n18A. L. Efros, Phys. Rev. B 45, 11354 (1992).\n19T. P. Smith, W. I. Wang, and P. J. Stiles, Phys. Rev. B 34, 2995 (1986).\n20M. I. Dyakonov and A. S. Furman, Sov. Phys. JETP 65, 574 (1987).\n21I am grateful to E. I. Rashba for this comment.\n22K. Seeger, Semiconductor Physics. An Introduction (Springer, 1999), p. 124.\n23H. Zhao, Appl. Phys. Lett. 92, 112104 (2008).\n24M. Rosling, H. Bleichner, P. Jonsson, and E. Nordlander, App l. Phys. Lett. 76, 2855 (1994).\n25H. P. Dahal, T. O. Wehling, K. S. Bedell, J.-X. Zhu, and A. V.Ba latsky, arXiv:cond-\nmat/0706.1689.\n26R. Cote, J.-F. Jobidon, and H. A. Fertig, arXiv:cond-mat/08 06.0573.\n27J. Martin, N. Akrman, G. Ulbricht, T. Lohmann, J. H. Smet, K. v on Klitzing, and A. Yacoby,\nNature Physics 4, 144 (2008).\n12"    },    {        "title": "0809.0795v1.Impurity_states_in_antiferromagnetic_Iron_Arsenides.pdf",        "content": "arXiv:0809.0795v1  [cond-mat.supr-con]  4 Sep 2008Impurity states in antiferromagnetic Iron Arsenides\nQiang Han\nDepartment of Physics, Renmin University, Beijing, China\nZ. D. Wang\nDepartment of Physics and Center of Theoretical and Computa tional Physics,\nThe University of Hong Kong, Pokfulam Road, Hong Kong, China\n(Dated: September 4, 2008)\nWe explore theoretically impurity states in the antiferrom agnetic spin-density wave state of the\niron arsenide. Two types of impurity models are employed: on e has only the intraband scattering\nwhile the other has both the intraband and interband scatter ing with the equal strength. Inter-\nestingly, the impurity bound state is revealed around the im purity site in the energy gap for both\nmodels. However, the impurity state is doubly degenerate wi th respect to spin for the first case;\nwhile the single impurity state is observed in either the spi n-up or spin-down channel for the second\none. The impurity-induced variations of the local density o f states are also examined.\nPACS numbers: 71.55.-i,75.30.Fv,75.10.Lp\nThe recent discovery of iron-based superconductors [1]\nhas triggered intensive efforts to unveil the nature of and\ninterplay between magnetism and superconductivity in\nthis family of materials. Series of iron arsenide have\nbeen synthesized, which possess many similar features\nof the normal and superconducting states. Experimental\nmeasurements have reported that the undoped ReFeAsO\n(where Re= rare-earth metals) and AFe 2As2(where\nA=divalent metals such as Ba, Ca, Sr) compounds ex-\nhibit a long-range antiferromagnetic spin-density-wave\n(SDW) order [2, 3, 4, 5, 6, 7]. Upon electron/hole dop-\ning the SDW phase is suppressed and superconductivity\nemerges with Tcup to above 50 K [8, 9, 10, 11, 12].\nAt present, there is likely certain controversy on\nthe understanding of the SDW state of the undoped\nFeAs-based parent compounds. Two kinds of theories\nhave been put forward: 1) the itinerant antiferromag-\nnetism, which takes advantage of proper Fermi surface\n(FS) nesting (or strong scattering) between different FS\nsheets [13, 14, 15, 16]; and 2) the frustrated Heisen-\nberg exchange model of coupled magnetic moments of\nthe localized d-orbital electrons around the Fe atoms\n[17, 18, 19, 20]. As for the itinerant electronic behav-\nior, first principle band structure calculations [21] based\non the density functional theory (DFT) indicate up to\nfive small Fermi pockets with three hole-like pockets cen-\ntered around the Γ point and two electron-like ones cen-\ntered around the Mpoint of the folded Brillouin zone\nof the FeAs layers, which have partially supported by\nthe angle-resolved photoemission spectroscopy (ARPES)\nfrom different groups [22, 23, 24, 25, 26]. Motivated by\nthe DFT calculation and experimental measurements, in\nRefs.[15, 16], the excitonic mechanism [27] of itinerant\ncarriers are employed taking account of the FS nesting\nbetween electron and hole pockets and the SDW phase\nare associated with triplet excitonic state, which can be\nunderstood as condensate of triplet electron-hole pairs[27].\nIn this paper, we explore theoretically the effect of a\nsingle impurity on the local electronic structure of an Fe-\nbased antiferromagnet in the triplet excitonic phase. It\nis shown that impurity bound states are formed inside\nthe SDW gap, which may be observed experimentally by\nlocal probes. Before introducing the impurity, we first\npropose an effective model Hamiltonian to address the\ntriplet excitonic state,\nˆHMF=/summationdisplay\ni,k,σεΓi(k)d†\nikσdikσ\n+/summationdisplay\nk,σεX(k+X)c†\nk+Xσck+Xσ\n+/summationdisplay\nik,σ,σ′/bracketleftBig\n∆∗\niσσ′(k)d†\nikσck+Xσ′+H.c./bracketrightBig\n,(1)\nwhereΓ= (0,0),X= (π,0). We use the index ito\nlabel different valence bands around Γ point. Around\nXandYpoints, there are two conduction bands. dikσ\nandck+Xσare the annihilation operators of electrons in\nthe ΓiandXbands. Theoretically XandYare two\nequivalent nesting directions. Note that, the structural\nphase transition occurred just above/onthe SDW transi-\ntion breaks this equivalency. Without loss of generality,\nit is assumed that only conduction band around the X\npoint couples with the valence bands around the Γ point,\nwhich is characterized by the mean-field order parame-\nters ∆ iσσ′. For the triplet excitonic phase (SDW), we\nhave real order parameters satisfying ∆ i↑↑=−∆i↓↓and\n∆i↑↓= ∆i↓↑= 0 [27].\nεΓi(k) andεX(k) are used to denote the band disper-\nsionsofthenonmagneticnormalstate. For kinthevicin-\nity of the Γ point (therefore, k+Xin the vicinity of the\nXpoint), the normal-state energy dispersions have ap-2\n-0.03 -0.02 -0.01 0.00 0.01 0.02 \n(a) \nε(k)  (eV) \nX\nεX\n0εΓ2 \n0\nεΓ1 \n0\nΓY\nX Γ12\n(b) \nFIG. 1: Schematic plot of (a) the Fermi surfaces; and (b)\nthe band dispersions of the valence (hole) band and conduc-\ntance (electron) bands in the unfolded Brillouin Zone for th e\nundoped parent compound. See text for detail.\nproximately the 2D parabolic forms\nεΓi(k) =−¯h2(k2\nx+k2\ny)\n2mΓi+ǫΓi\n0, (2)\nεX(k+X) =¯h2(k2\nx+k2\ny)\n2mX−ǫX\n0,(3)\nas schematically shown in Fig. 1. Here mΓiandmX\nare the corresponding effective masses. In describing the\nXband, the elliptic FS is approximated by the circu-\nlar one for simplicity. ǫΓi\n0(ǫX\n0) denotes the top (bot-\ntom) of the hole (electron) bands. According to the\nARPES measurement[22], two hole-like Fermi pockets\nare revealed around the Γ point for undoped BaFe 2As2.\nThe band parameters extracted from the experimental\ndata are as follows. mΓ1≈2.8me,mΓ2≈7.4me,\nandmX≈6.5me, wheremeis the mass of bare elec-\ntron.εΓ1\n0≈4 meV,εΓ2\n0≈16 meV, and εX\n0≈24\nmeV. These parameters indicate that the nesting be-\ntween the Γ2 band and Xband is much better than\nthat of the Γ1 band. Therefore it is natural to assume a\nlarger order parameter ∆ 2and a vanishingly small ∆ 1.\nLetEg= (ǫΓ2\n0+ǫX\n0)/2 andµ0= (ǫΓ2\n0−ǫX\n0)/2. Here\nEG=−2Egdenotes the indirect gap between the top of\nthe Γ band and the bottom of the Xbands. Therefore,\nEg>0 describes a semimetal and Eg<0 a semiconduc-\ntor. Withthehelpof Egandµ0andafurtherassumption\nofmΓ2=mX=m≈7me. we can re-express the energy\ndispersions as\nεΓ2(k) =−ε(k)−µ0, (4)\nεX(k+X) =ε(k)−µ0, (5)\nε(k) =¯h2\n2mk2−Eg, (6)\nNotethat for µ0= 0, the holeandelectronbandsareper-\nfectly nested since εΓ2(k) =−εX(k+X) and the system\nis unstable with respect to infinitesimal Coulomb inter-\naction while for nonzero µ0finite strength of Coulomb\nrepulsion is needed.For the reason that the order parameter ∆ 1is set to\nzero, there is no coupling between the Γ1 band and the\nXband. The Hamiltonian of Eq. (1) is reduced to a\nmodel of two bands with one valence band (Γ2 band)\nand one conductionband (Xband). Introducing the two-\ncomponent Nambu operator, ˆψ†\nkσ= (d†\n2kσ,c†\nk+Xσ), the\nmodel Hamiltonian can be simplified as\nˆHMF=/summationdisplay\nkσˆψ†\nkσ/parenleftbigg\nεΓ(k) ∆σ\n∆σεX(k)/parenrightbigg\nˆψkσ+Himp,(7)\nwhere an impurity term has been added with the form,\nˆHimp=/summationdisplay\nk,k′,σˆψ†\nkσˆUk,k′ˆψk′σ, (8)\nwhereˆUk,k′represents a 2 ×2 matrix of the scattering\npotential associated with non-magnetic impurities. Here\nwe use ∆ σto denote ∆ σσfor short. The Green’s function\nmethodisappliedtostudythesingleimpurityeffect. The\nmatrix Greens functions are defined as\nˆGσσ(k,τ;k′,τ′) =−∝an}bracketle{tTτ[ψkσ(τ)ψ†\nk′σ(τ′])∝an}bracketri}ht,(9)\nˆGσσ(k,k′,iωn) =/integraldisplayβ\n0dτˆGσσ(k,τ;k′,0)eiωnτ(10)\nˆGσσ(k,k′,ω) =ˆGσσ(k,k′,iωn→ω+i0+).(11)\nFrom the Hamiltonian defined in Eq. (7) we can derive\nthe bare Green’s function\nˆG0\nσσ(k,ω) =/parenleftbigg\nω−εΓ(k)−∆σ\n−∆σω−εX(k)/parenrightbigg−1\n,(12)\n=˜ωˆτ0+∆σˆτ1−ε(k)ˆτ3\n˜ω2−ε(k)2−∆2σ, (13)\nwhere ˜ω=ω+µ0. ˆτ0is the 2×2unit matrix, and ˆ τ1,3are\nthe pauli matrices. The T-matrix approximation is em-\nployed to compute the Green’s function in the presence\nofimpurities. For a single impurity, the T-matrix exactly\naccountsfor the multiple scatteringoff the impurity. The\nsingle-particle Green’s function ˆGcan be obtained from\nthe following Dyson’s equation,\nˆGσσ(k,k′,ω) =ˆG0\nσσ(k,ω)δk,k′+ˆG0\nσσ(k,ω)\nˆTσσ(k,k′,ω)ˆG0\nσσ(k′,ω),(14)\nwhere the T matrix is given by\nˆTσσ(k,k′,ω) =ˆUk,k′+/summationdisplay\nk′′ˆUk,k′′ˆG0\nσσ(k′′,ω)ˆTσσ(k′′,k′,ω).\n(15)\nFor a point-like scattering potential interacting with itin-\nerant carriers just on the impurity site, the scattering\nmatrix is isotropic, ˆUk,k′′=ˆU. The above equation is\ngreatly simplified\nˆTσσ(ω) =ˆU+ˆUˆG0\nσσ(ω)ˆTσσ(ω), (16)3\nwhereˆG0\nσσ(ω) =/summationtext\nkˆG0\nσσ(k,ω). After some derivation\nwe obtain\nˆG0\nσσ(ω) =−πN0/bracketleftBigg\nα(˜ω)/radicalbig\n∆2σ−˜ω2(˜ωˆτ0+∆σˆτ1)+γ(˜ω)ˆτ3/bracketrightBigg\n,\n(17)\nwhere\nα(˜ω) =π−1/bracketleftBigg\narctan/parenleftBigg\nEc/radicalbig\n∆2σ−˜ω2/parenrightBigg\n+\narctan/parenleftBigg\nEg/radicalbig\n∆2σ−˜ω2/parenrightBigg/bracketrightBigg\n(18)\nγ(˜ω) = (2π)−1ln/parenleftbiggE2\nc+∆2\nσ−˜ω2\nE2g+∆2σ−˜ω2/parenrightbigg\n,(19)\nwithEcdenoting the high-energy cutoff and N0=\nma2/(2π¯h2)the densityofstatesperbandperspin. Note\nthatα(˜ω) andγ(˜ω) are independent of the spin index σ.\nThe first impurity model we study is the scattering-\npotential matrix with only intraband scattering terms,\ni.e.ˆU=Vimpˆτ0, which was adopted in Ref. [28] to study\neffect of many impurities. From Eq. (16), we obtain\nˆTσσ(ω) = [ˆτ0−ˆUˆG0\nσσ(ω)]−1ˆU= [V−1\nimp−ˆG0\nσσ(ω)]−1.(20)\nThe energy of the impurity bound state is determined by\nthe pole of ˆTσσ(ω), determined by det[ V−1\nimp−ˆG0\nσσ(Ω)] =\n0. Setting c≡(πN0Vimp)−1, we have the equation for\nthe energy of impurity bound state\nc2+2cα(˜Ω)˜Ω/radicalBig\n∆2σ−˜Ω2−α(˜Ω)2−γ(˜Ω)2= 0.(21)\nFor the spin triplet excitonic phase ∆ ↑=−∆↓, the\naboveequationgivesrisetoimpuritystateswiththesame\nbound energy, i.e. the impurity states are doubly degen-\nerate. Generally, the above equation has to be solved nu-\nmericallytoobtaintheboundenergy ˜Ω. However,wecan\nget some analytic results under certain approximations.\nUnder the wide-band approximation Ec,Eg≫ |∆σ|,\nα(˜Ω)≈1 andγ(˜Ω)≈γ0=π−1ln(Ec/Eg), so we have\n˜Ω\n|∆σ|= sgn(c)1−c2+γ2\n0/radicalbig\n(1−c2+γ2\n0)2+4c2,(22)\nand furthermore if the system has approximately the\nparticle-hole symmetry Ec≈Eg, thenγ0≈0 and\n˜Ω/|∆σ|= sgn(c)(1−c2)/(1+c2)fromtheaboveequation.\nFor the second impurity model, the four matrix ele-\nments of ˆUis assumed to be the same, i.e. the intra-\nand inter-band scattering terms are the same [29] with\nˆU=Vimp(ˆτ0+ ˆτ1)/2. Then the T-matrix according to\nEq. (16) is\nˆTσσ(ω) = [2V−1\nimp−ˆτ′ˆG0\nσσ(ω)]−1ˆτ′,(23)with ˆτ′= ˆτ0+ ˆτ1. The energy of the impurity bound\nstate is again determined by the pole of ˆTσσ(ω). From\ndet[2V−1\nimp−ˆτ′ˆG0\nσσ(Ω)] = 0 we have the equation for ˜Ω,\nc+˜Ω+∆ σ/radicalBig\n∆2σ−˜Ω2α(˜Ω) = 0. (24)\nFrom the above equation we find that ˜Ω is independent\nof the function of γ(˜ω) for this case, which reflects the\nparticle-hole asymmetry. Before solving the above equa-\ntion for the bound energy, we study the existence of the\nimpurity state. Because ˜Ω2<∆2\nσ,˜Ω+∆ σhas the same\nsign as that of ∆ σ. Therefore, the solution of Eq. (24)\nexists only if the sign of cis opposite to that of ∆ σ. For\nthe SDW state, i.e. the triplet excitonic phase, we have\n∆↑=−∆↓and so there is exactly one impurity bound\nstate in either the spin-up or spin-down channel. We\nmay assume ∆ ↑=−∆↓= ∆>0 as well, then for attrac-\ntive scattering Vimp<0, the impurity bound state only\nexists in the spin-up channel and its energy is given by\n˜Ω/∆ =−(1−c2)/(1+c2) under the wide-band approxi-\nmation. IfVimp>0, however, the impurity state will be\nin the spin-down channel, and ˜Ω/∆ = (1−c2)/(1+c2).\nIn general, the impurity bound-state energy is given by\n˜Ω\n∆σ= sgn(c)1−c2\n1+c2(25)\nin the valid regime of the wide-band approximation.\nTo apply the theoretical results to the iron arsenide,\nwe try to pin down the parameters of our model by ex-\ntracting them from the available experimental data for\nBaFe2As2[22].ǫΓ\n0≈16 meV and ǫX\n0≈24 meV so\nthatEg≈20 meV.mΓ≈mX≈7.0meand there-\nforeN0≈1.2 eV−1. ∆↑=−∆↓= ∆≈20 meV. The\nhigh-energycutoff is set as Ec= 500meV, which is ofthe\nsame order of magnitude as the band width. Note that\nEgextracted fromexperimental data is verysmall, which\nis in the same order of magnitude of the order parameter\n∆. Therefore, neither the wide-band approximation nor\nthe particle-hole symmetry can be applied to the present\ncase. Eqs. (21) and (24) have to be numerically solved.\nNow we examine the local characteristics induced by\nthe impurity by looking into the variation of the local\ndensity of states (LDOS), which can be probed by the\nscanning tunneling microscopy (STM). The LDOS is de-\nfined as\nN(r,ω) =−1\nπ/summationdisplay\nσIm{Tr[ˆGσσ(r,r′,ω)]},(26)\nwhereˆGσσ(r,r′,ω) the Green’s function in real space.\nApplying the T-matrix approximation we have,\nˆGσσ(r,r′,ω) =ˆG0\nσσ(r,r′,ω)+ˆG0\nσσ(r,0,ω)\nˆT(ω)ˆG0\nσσ(0,r′,ω), (27)4\nSubstituting Eq. (27) into Eq. (26) we may single out the\nvariation of LDOS due to the presence of the impurity\npotential,\nNimp(r,ω) =−1\nπ/summationdisplay\nσIm{Tr[ˆG0\nσσ(r,0,ω)ˆTσσ(ω)\nˆG0\nσσ(0,r,ω)]}. (28)\nFor the second impurity model[30], Fig. 2(a) shows the\nLDOS as a function of energy ˜ ωon the impurity site,\nnamelyN(0,˜ω), while Fig. 2(b) the impurity-induced\nLDOS at the bound energy as a function of radial dis-\ntanceroff the impurity site, i.e. Nimp(r,˜Ω).Vimphas\nbeen set as -0.36, -0.6, and -4.0 eV, giving rise to the\nimpurity bound states seen as the sharp peaks located\nrespectively at the energies ˜Ω/∆ = 0,−0.5, and−0.99 in\nFig. 2(a). The probability densities of these bound states\nexhibit a kind of exponential decay with the Friedel os-\ncillation, as seen in Fig. 2(b). Introducing two length\nscales,ξ1andξ2to characterize the oscillation and de-\ncay, we obtain the asymptotic behavior of Nimp(r,˜Ω) for\nlarger,\nNimp(r,˜Ω)∝r−1cos2(πr/ξ1)exp(−r/ξ2).(29)\nξ1/a=π//radicalbig\n4πN0Egwhich is approximately 5 .7 in con-\nsistence with the numerical results shown in Fig.2(b).\nξ2/ξ1=Eg/(π/radicalbig\n∆2−˜Ω2) = 0.32, 0.37, and 2.3 for the\nthree cases of impurity states. This explains why we see\nclear Friedel oscillation for impurity state with bound\nenergy near the gap edge.\n-15 -10 -5 0 5 10 15 20 010 20 30 40 50 60 \n-3 -2 -1 0 1 2 3010 20 30 40 50 60 \nNimp (r, Ω) \nr/a (b)  N(0, ϖ)\nϖ/Δ(a) \nFIG. 2: LDOS on the impurity site as a function of energy ˜ ω,\ni.e.N(0,˜ω) (a), and impurity-induced LDOS at the bound\nenergy˜Ω as a function of radial distance roff the impurity\nsite, i.e.N(r,˜Ω) (b).ris in unit of awithathe lattice\nconstant of Fe-Fe plane. Black, red, and blue lines correspo nd\ntoVimp=−0.36,−0.6,−4 eV, respectively.\nThis work was supported by the NSFC grand under\nGrants Nos. 10674179 and 10429401, the GRF grant of\nHong Kong.\n[1] Y. Kamihara et al., J. Am. Chem. Sco. 130, 3296 (2008).[2] C. de la Cruz et al., Nature 453, 899 (2008); J. Zhao et\nal., arXiv:0806.2528; J. Zhao et al., arXiv:0807.1077.\n[3] Y. Chen et al., Phys. Rev. B 78, 064515 (2008), see also\narXiv.org:0806.0662.\n[4] Q. Huang et al., arXiv:0806.2776.\n[5] M. A. McGuire et al., arXiv.org:0806.3878.\n[6] A. A. Aczel et al., arXiv:0807.1044.\n[7] A. I. Goldman et al., arXiv.org:0807.1525.\n[8] H. Takahashi et al., Nature 453, 376 (2008).\n[9] Z. -A. Ren et al., Europhys. Lett. 82, 57002 (2008).\n[10] H. .H. Wen et al., Europhys. Lett. 82, 17009 (2008), see\nalso arXiv:0803.3021.\n[11] X. H. Chen et al., Nature 453, 761 (2008), see also\narXiv:0803.3603.\n[12] G. F. Chen et al., Phys. Rev. Lett. 100, 247002 (2008),\nsee also arXiv:0803.3790.\n[13] J. Dong et al., Europhys. Lett. , 83, 27006 (2008); see\nalso arXiv:0803.3426.\n[14] I. I. Mazin et al., Phys. Rev. Lett. 101, 057003 (2008),\nsee also arXiv:0803.2740; arXiv.org:0806.1869.\n[15] Q. Han, Y. Chen, and Z. .D .Wang, Europhys. Lett. 82\n37007 (2008), see also arXiv:0803.4346.\n[16] V. Barzykin and L. P. Gorkov, arXiv:0806.1933.\n[17] T. Yildirim, Phys. Rev. Lett. 101, 057010 (2008), see\nalso arXiv:0804.2252.\n[18] Q. Si and E. Abrahams, arXiv:0804.2480.\n[19] F. J. Ma, Z. Y. Li, and T. Xiang, arXiv.org:0804.3370.\n[20] C. Fang et al., Phys. Rev. B 77, 224509 (2008), see also\narXiv:0804.3843.\n[21] D. J. Singh and M. H. Du, Phys. Rev. Lett. 100\n237003 (2008), see also arXiv:0803.0429; G. Xu et\nal., Europhys. Lett., 82, 67002 (2008), see also\narXiv:0803.1282; K. Haule, J. H. Shim and G. Kotliar,\nPhys. Rev. Lett. 100226402 (2008), see also\narXiv:0803.1279; F. Ma, and Z. -Y. Lu, Phys. Rev. B 78\n033111 (2008), see also arXiv:0803.3286; I. A. Nekrasov,\nZ. V. Pchelkina, M. V. Sadovskii, JETP Lett., 88, 144\n(2008), see also arXiv:0806.2630; F. Ma, Z.-Y. Lu, and\nT. Xiang, arXiv:0806.3526; D. J. Singh, arXiv:0807.2643.\n[22] L. X. Yang et al., arXiv:0806.2627.\n[23] C. Liu et al., arXiv:0806.3453.\n[24] L. Zhao et al., arXiv:0807.0398.\n[25] H. Ding et al., Europhys. Lett. 83, 47001 (2008), see also\narXiv:0807.0419; P. Richard et al., arXiv:0808.1809.\n[26] D. H. Lu et al., arXiv:0807.2009.\n[27] B. I. Halperin and T. M. Rice, Solid State Phys. 21, 125\n(1968).\n[28] J. Zittartz, Phys. Rev. 164, 575 (1967).\n[29] The impurity Hamiltonian is expressed as Himp=R\nUimp(r)φ†(r)φ(r)dr=P\nk,k′ˆψ†(k)ˆUk,k′ˆψ(k) where\nˆψ†(k) = (d†\nk,c†\nk+X)φ(r) = 2−1/2P\nkϕΓk(r)dk+\nϕXk(r)ck+X, withϕΓk(r) andϕXk(r) Bloch functions\nof quasimomentum k. In this paper the scattering po-\ntential is assumed to be pointlike, Uimp=Vimpδ(r).\nWith further simplification of the Bloch functions of the\nform exp(ik·r),ˆUk,k′iskindependent and equal to\nVimp(ˆτ0+ ˆτ1)/2.\n[30] The following conclusions are also qualitatively corr ect\nfor the first impurity model."    },    {        "title": "0809.0976v1.Comparison_of_Raman_spectra_and_vibrational_density_of_states_between_graphene_nanoribbons_with_different_edges.pdf",        "content": "arXiv:0809.0976v1  [cond-mat.mtrl-sci]  5 Sep 2008EPJ manuscript No.\n(will be inserted by the editor)\nComparison of Raman spectra and vibrational density of stat es\nbetween graphene nanoribbons with different edges\nSami Malola1,a, Hannu H¨ akkinen1,2, and Pekka Koskinen1\n1NanoScience Center, Department of Physics, FIN-40014 Univ ersity of Jyv¨ askyl¨ a, Finland\n2NanoScience Center, Department of Chemistry FIN-40014 Uni versity of Jyv¨ askyl¨ a, Finland\nReceived: date / Revised version: date\nAbstract. Vibrational properties of graphene nanoribbons are examin ed with density functional based\ntight-binding method and non-resonant bond polarization t heory. We show that the recently discovered\nreconstructed zigzag edge can be identified from the emergen ce of high-energy vibrational mode due to\nstrong triple bonds at the edges. This mode is visible also in the Raman spectrum. Total vibrational\ndensity of states of the reconstructed zigzag edge is observ ed to resemble the vibrational density of states\nof armchair, rather than zigzag, graphene nanoribbon. Edge -related vibrational states increase in energy\nwhich corroborates increased ridigity of the reconstructe d zigzag edge.\nPACS.61.46.-w Structure of nanoscale materials – 64.70.Nd Struc tural transitions in nanoscale materials\n– 63.22-m Phonons or vibrational states in low-dimensional structures and nanoscale materials – 78.30Na\nInfrared and Raman spectra, fullerenes and related materia ls\nInpastdecadescarbonnanomaterialshaveshowntheir\nrich properties in several applications[1,2]. Here graphene\nnanoribbons are not an exception, and their use in many\napplications, among which transistors in nanoelectronics,\nhavebeeninvestigatedconsideringdifferentedgestructures[3,\n4]. Low dimensionality and edges make ribbons particu-\nlarly fascinating both for theory and applications.\nThepreciseedgestructureingraphenenanoribbonsaf-\nfectsmanypropertieslikechemicalreactivity[5],electronic\nstructure[6] and vibrations[7]. Vibrational properties play\na role in structural stability [8], structure identification[9]\nandballistictransportthroughelectron-phononcoupling[9].\nInstructureidentificationscanningtunnelingmicroscopy[10,\n11] can reach near atom resolution but analysis of the full\nedge structure and properties is often ambiguous. In this\ncase Raman spectroscopy[12,13,9] is a valuable tool.\nVibrational properties of graphene nanoribbons have\nbeen recently studied for acoustic and optical phonons,\nsymmetries, and Raman activity as a function of ribbon\nwidth[12,8]. In this paper we consider recently reported\nself-passivating edge reconstruction of zigzag ribbon [14],\nandinvestigatehowthereconstructionaffectsRamanspec-\ntra and vibrational density of states (VDOS). Edge-loca-\nlized contributions to Raman spectra and vibrationalden-\nsity of states change. It turns out that high-energy modes\ndue to triple-bond vibrations, as well as overall changes in\nthe vibrational density of states and changes in rigidity of\nthe edges, make the reconstruction identifiable and visible\nin Raman spectra.\nasami.malola@phys.jyu.fi\nzigzag57\nW=20.12Å\nL=24.60Åzigzag\nW=19.88Å\nL=24.60Å\narmchair\nW=23.37Å\nL=21.30ÅL\nW\ntotal 200 atoms\nedge 80 atomstotal 200 atoms\nedge 80 atomstotal 200 atoms\nedge 80 atoms\nzx\ny\nFig. 1. Examined structures of zigzag, reconstructed zigzag\n(zigzag57) and armchair graphene nanoribbons. Selected ed ge\natoms are drawn in gray color. Coordinate axis is representa ted\nat lower right corner, where the ribbon direction (=periodi c\ndirection) is z-direction. W=width in non-periodic and L=\nlength in periodic direction.\nWe use density functional based tight-binding method\nto calculate forces[15], to optimize structure[16] and to2 S. Malola et al.: Comparison of Raman spectra and VDOS of gra phene nanoribbons with different edges\n500100015002000 500100015002000 500100015002000Intensity [arb. units]\nWavenumber [cm¯¹]armchair zigzag57 zigzagzz xz xx\n(1)\n(2)\n(L)\n(L)(L)\nFig. 2.Raman spectra of zigzag, reconstructed zigzag (zigzag57) a nd armchair graphene nanoribbons (different rows). Columns\ndenote different incident and scattered light polarization spectra (uppermost symbols xx, xz and zz). Edge-localized m odes are\nassigned with symbol (L). Symbols (1) and (2) are refered to i n the text. Raman spectra are drawn using Lorenz distributio n\nwith full width at half maximum of 5 cm−1.\ncalculate the vibrational eigenmodes for the systems. For\nRaman spectra calculationswe use non-resonantbond po-\nlarizationtheory[17,18], that has been used succesfully for\nribbons[12] and for non-identical carbon nanotubes[19,20,\n21]. The slight consistent energy shift in the high-energy\nmodes[19] does not affect the qualitative changes in Ra-\nman spectra or VDOS we want point out in this paper.\nFigure 1 shows the atom structures of zigzag, recon-\nstructedzigzag(zigzag57)andarmchairgraphenenanorib-\nbons. Widths and numbers of atoms of the ribbons are\nchosen to be easily comparable; also other sizes were sys-\ntematically studied and all qualitative results given below\nwere supported by these systematics. Edge atoms, as we\ndefine them, are drawn in gray and are used to deter-\nmine edge-weighted vibrational density of states, where\nthe edge-weight is ratio\nwµ\nedge=/summationtext\ni∈edgevµ\ni·vµ\ni/summationtext\nj∈allvµ\nj·vµ\nj(1)\nwhere(3-dimensionalvector) vµ\niisthecomponentonatom\nifor eigenmode µ. Hencewµ\nedge= 1 means complete lo-\ncalization to edge, wµ\nedge=80\n200means ”uniformly dis-\ntributed” mode, and wµ\nedge= 0 would mean a mode lo-calized in the central region. Contributions of different\ndirections are also analysed, analogously including weight\nof the components of the vibrational eigenvectors vµ\niin\neitherx-,y- orz-direction,\nwµ\nx=/summationtext\nivµ\ni,x·vµ\ni,x/summationtext\nj∈allvµ\nj·vµ\nj(2)\nand similarly for y and z. Hence wµ\nx+wµ\ny+wµ\nz= 1 and\nwµ\nx,edge+wµ\ny,edge+wµ\nz,edge=wµ\nedge;wµ\nxmeasureshowmuch\nofthe modeis in the x-directionand wµ\nx,edgemeasureshow\nmuch of the mode at the edge is in the x-direction.\nFigure2showsthexx,xzandzzpolarizedRamanspec-\ntra of the ribbons, where e.g. xz stands for incident light\npolarization in x- and scattered light polarization in z-\ndirection. The polarization pictures xx and zz show the\nsame modes, but with different intensities. Polarization\npictures that contain y-direction and out-of-plane modes\nyield tiny intensities and have been left out from figure 2.\nOrientation of the ribbons with respect to coordinate axis\nis shown in figure 1. The most pronounced modes in figure\n2 include breathing modes (intensive low-energy peaks\nin xx/zz spectra), G-band-related high-energy modes (in-\ntensive high-energy peaks in the xz and xx/zz spectra)S. Malola et al.: Comparison of Raman spectra and VDOS of grap hene nanoribbons with different edges 3\n00.500.500.5\n 0 500 1000  1500  2000  2500VDOS [1/cm-1]\nWavenumber [cm-1]zigzag\nzigzag57\narmchairy\nx\nz\ntotal\nFig. 3. Total vibrational density of states of zigzag, recon-\nstructed zigzag (zigzag57) and armchair graphene nanorib-\nbons. Contribution of different directions in vibrational s tates\nare shown in different patterned areas. Vibrational density\nof states is broadened with normal distribution with σ=\n45 cm−1, chosen to illuminate general changes in VDOS under\nreconstruction.\n00.500.500.5\n 0 500 1000  1500  2000  2500VDOS [1/cm-1]\nWavenumber [cm-1]zigzag\nzigzag57\narmchairy\nx\nz\ntotal\nFig. 4.As figure 3, but showing the edge weighted vibrational\ndensity of states of zigzag, reconstructed zigzag (zigzag5 7) and\narmchair graphene nanoribbons.and edge-localized modes (symbols (L) in xz and zz spec-\ntra). One of the main points is that only edge-localized\nmodes undergo observable changes under reconstruction,\nothersbeingwithin 2cm−1fromcorrespondingzigzagrib-\nbon modes. These changes in edge-localized Raman active\nmodes make the reconstruction visible. For zigzag57 the\nedge-localized mode of zigzag edge at 1550 cm−1disap-\npears in xz polarization spectrum in figure 2 and edge-\nlocalized triple-bond vibration becomes visible at energies\naround 2250 cm−1in zz polarization spectrum of zigzag57\nribbon, like in armchair ribbon. Intensity of the triple-\nbond vibration in reconstructed zigzag ribbon is smaller\nin our bond polarization theory than in armchair ribbon,\nmainly because of the differences in bond angles on the\nedge (the edge profile of zigzag57 is more linear).\nCouple of new Raman active modes can be noticed to\nappear for reconstructed zigzag edge, assinged with sym-\nbols (1) in xx and (2) in xz polarization spectrum in fi-\ngure 2. Mode (1) at 1500 cm−1includes stretching of edge\nbonds of pentagons. Mode (2) at 1600 cm−1is mainly a\nlongitudinal mode where first normal zigzag row of atoms\nin the zigzag57ribbon vibrates as the edge-localized mode\nof zigzag ribbon. Neither of the modes (1) nor (2) is fully\nlocalized on the edge.\nBreathing modes remain unchanged, because they are\ncollective vibrations which hold the edges internally un-\nchanged during the vibration. Similarly, because G-band-\nrelated high-energy modes vanish towards the edge, re-\nconstruction has only minor effect on them. Raman active\nout-of-plane vibrations have much lower intensity than in-\nplane vibrations and are not therefore considered here.\nNext we compare more generally the vibrational pro-\nperties of the different ribbon edges. Figure 3 shows the\ntotal vibrationaldensity of states, analysedwith contribu-\ntions from different directions. The surprising and signi-\nficant result is that zigzag57 ribbon’s vibrational density\nof states resembles armchair ribbon’s rather than zigzag\nribbon’s vibrational density of states. This means that for\nvibrational modes it is not the orientation of the under-\nlying honeycomb lattice that is important, but only the\nlocal structures of the edges . Comparison of changes bet-\nween total vibrational density of states in figure 3 and\nedge-weighted vibrational density of states in figure 4 re-\nveals that the most significant changes are seen in edge-\nlocalized modes. Triple-bond vibrations in z-direction on\nthe high energies appear as a separate peak around 2250\ncm−1in figures 3 and 4 for both reconstructed zigzag rib-\nbon and armchair ribbon. The triple-bond edge modes\nin z-direction in the zigzag57 ribbon, somewhat deplete\nmodes from the energy range 1700 −2000 cm−1of the\nG-band-related modes in z-direction. At the same time\nin the same energy range the number of edge-related in-\nplane vibrations in x-direction are increased due to recon-\nstruction - induced edge stiffening. Similar effects are seen\neven in lower energies. Some edge-localized out-of-plane\nmodes around 500 −650 cm−1are upshifted in energy\nfor zigzag57 ribbon for the same reason. Even the energy\nof the lowest energy out-of-plane modes increase, which4 S. Malola et al.: Comparison of Raman spectra and VDOS of gra phene nanoribbons with different edges\ncan dramatically alter low-temperature properties (such\nas heat capacity) of the ribbons.\nHence, due to edge stiffening, edge vibrations gene-\nrally up-shift in energy under reconstruction. This type\nof changes in mechanical properties can have an effect on\napplications such as sensors. The similarity of armchair\nand zigzag57 ribbon’s VDOS is general observation, as\nthe resemblance and also other qualitative features of the\nvibrational properties remained also for wider ribbons (up\ntoW= 40˚A).\nTo conclude, we have analysed the Raman spectra and\nvibrational density of states of the recently reported self-\npassivating edge reconstruction of zigzag ribbons. In the\nRaman spectra edge-localized triple-bond vibrations be-\ncome visible like the corresponding mode in armchair rib-\nbon’s spectra. These high-energy vibrations can be used\ntoidentify the reconstructionbyconcentratingthe Raman\nmeasurement on the edge.\nThepredictedintensitydifferencesbetweentriple-bond\nvibrations of the zigzag57ribbon and the armchair ribbon\ncould be used to separate the reconstructed edge struc-\nture from the armchair edge structure with help of scan-\nningtunneling microscopyimages.Scanningtunnelingmi-\ncroscopycanalsohelpinavoidingidentificationdifficulties\nwith ribbons that have combination of zigzag and arm-\nchair edges. For those ribbons the density of triple-bonds\nat the edges is lower and further intensity of the triple-\nbond vibrations is lower like in reconstructed zigzag rib-\nbon, which can lead to mix up between these structures.\nThe rest of the visible Raman active modes of the rib-\nbons do not change for zigzag57 and the breathing mode\nhas the same ribbon width dependence. Of course, in very\nnarrow ribbons with majority of atoms on the edge, ener-\ngies or spatial nature of the G-band-related high-energy\nmodes may still change.\nThis research is supported by the Academy of Finland through\nthe FINNANO consortium MEP (molecular electronics and\nnanoscale photonics) and project 121701. S. Malola acknow-\nledges a grant from the Finnish Cultural Foundation.\nReferences\n1. R.H. Baughman, A.A. Zakhidov, W.A. de Heer, Science\n297, 787 (2002)\n2. P. Avouris, Z. Chen, V. Perebeinos, Nature Nanotechnol-\nogy2, 605 (2007)\n3. Y. Ouyang, Y. Yoon, J. Fodor, J. Guo, Appl. Phys. Lett.\n89, 203107 (2006)\n4. Y. Yoon, J. Guo, Appl. Phys. Lett. 91, 073103 (2007)\n5. D. Jiang, B. Sumpter, S. Dai, The Journal of Chem. Phys.\n126, 134701 (2007)\n6. K. Nakada, M. Fujita, G. Dresselhaus, M. Dresselhaus,\nPhys. Rev. B 54, 17954 (1996)\n7. M. Yamada, Y. Yamakita, K. Ohno, Phys. Rev. B 77,\n054302 (2008)\n8. N. Dugan, S. Erkoc, Phys. Stat. Sol. b 245, 695 (2008)\n9. A. Ferrari, Solid State Comm. 143, 47 (2007)\n10. L. Tapaszto, G. Dobrik, P. Lambin, L. Biro, Nature Nan-\notechnology 149, 1 (2008)11. Y. Kobayashi, K. Fukui, T. Enoki, K. Kusakabe, Phys.\nRev. B73, 125415 (2006)\n12. J. Zhou, J. Dong, Appl. Phys. Lett. 91, 173108 (2007)\n13. L. Cancado, M. Pimenta, B. Neves, G. Medeiros-Ribeiro,\nT. Enoki, Y. Kobayashi, K. Takai, K. Fukui, M. Dressel-\nhaus, R. Saito et al., Phys. Rev. Lett. 93, 047403 (2004)\n14. P.Koskinen, S.Malola, H.H¨ akkinen, Phys. Rev. Lett.\n(in print), Preprint: arXiv:0802.2623 [cond-mat.mtrl-sc i]\n(2008)\n15. T. Frauenheim, G. Seifert, M. Elstner, T. Niehaus,\nC. K¨ ohler, M. Amkteutz, M. Sternbergand, Z. Hajnal,\nA.D. Carlo, S. Suhai, J. Phys.: Condens. Matter 14, 3015\n(2002)\n16. E. Bitzek, P. Koskinen, F. G¨ ahler, M. Moseler, P. Gumbsh ,\nPhys. Rev. Lett. 97, 170201 (2006)\n17. S. Guha, J. Menendez, J. Page, G. Adams, Phys. Rev. B\n53, 13106 (1996)\n18. R. Saito, T. Takeya, T. Kimura, G. Dresselhaus, M. Dres-\nselhaus, Phys. Rev. B 57, 4145 (1998)\n19. S. Malola, H. H¨ akkinen, P. Koskinen, Phys. Rev. B 77,\n155412 (2008)\n20. G. Wu, J. Zhou, J. Dong, Phys. Rev. B 72, 115411 (2005)\n21. S.Malola, H.H¨ akkinen, P.Koskinen, (submitted), Prep rint:\narXiv:0809.0769 [cond-mat.mtrl-sci] (2008)"    },    {        "title": "0811.3623v3.Superfluid_density_of_the_ultra_cold_Fermi_gas_in_optical_lattices.pdf",        "content": "arXiv:0811.3623v3  [cond-mat.other]  16 Jul 2009Superfluid-density of the ultra-cold Fermi gas in\noptical lattices\nT. Paananen1,2\n1Department of Physics, P.O. Box 64, FI-00014 University of Helsink i, Finland\n2Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, 02015\nHUT, Finland\nE-mail:tomi.paananen@helsinki.fi\nAbstract. In this paper we study the superfluid density of the two component Fermi\ngas in optical lattices with population imbalance. Three different type of phases, the\nBCS-state (Bardeen, Cooper, and Schrieffer), the FFLO-state (Fulde, Ferrel, Larkin,\nand Ovchinnikov), and the Sarma state, are considered. We show t hat the FFLO\nsuperfluid density differs from the BCS/Sarma superfluid density in a n important way.\nAlthough there are dynamical instabilities in the FFLO phase, when th e interaction is\nstrong or densities are high, on the weak coupling limit the FFLO phase is found to\nbe stable.\nSubmitted to: J. Phys. B: At. Mol. Phys.Superfluid-density of the ultra-cold Fermi gas in optical la ttices 2\n1. Introduction\nThe field of ultra-cold Fermi gases offers a great tool to study man y different problems\nof correlated quantum systems. For example, in recent experimen ts [1, 2, 3, 4, 5, 6]\npolarized Fermi gases were considered. These systems make it pos sible to study physics\nin the presence of mismatched Fermi surfaces, and non-BCS type pairing such as that\nappearing in FFLO-states [7, 8] or Sarma-states [9, 10]. These pos sibilities have been\nconsidered extensively in condensed-matter, nuclear, and high-e nergy physics [11].\nIt is also possible to study many different physical problems with close analogs in\nthe field of solid state physics using optical lattices. However, unlike solid state systems,\nultra-coldgases inoptical lattices provide a very clean environment . These systems have\nvery few imperfections and if one is interested in imperfections, the y can be imposed\neasily on the system. Optical lattices are made with lasers, thus the lattice geometry is\neasy to modify [12, 13, 14, 15] by changing theproperties ofthe int ersecting laser beams.\nFor these reasons, in optical lattices one can study various quant um many-body physics\nproblems, such as Mott insulators, phase coherence, and superfl uidity. The possibility\nof a superfluid alkali atom Fermi gas in an optical lattice has been rec ently studied both\ntheoretically [16, 17, 18, 19, 20, 21, 22], as well as experimentally [23 ].\nThe pairing predicted by the BCS theory does not always mean that t he gas is\na superfluid [24]. When a linear phase is imposed on the order paramete r, the phase\ngradient corresponds to superfluid velocity, and a part of the gas , which has superfluid\nvelocity is called superfluid, and the coefficient of the inertia of this mo ving part is\ncalled superfluid density [25, 26, 27]. However, if the gas is normal th e order parameter\nvanishes and thus the phase shift does not affect to the normal ga s. If the superfluid\ndensity is positive in all directions the gas is a superfluid. Negative sup erfluid density\nimplies dynamical instability of the gas [28, 29].\nThere are a fair number of studies about superfluid density of Ferm i gas in the free\nspace as well as in a trap [30, 31, 32, 33], and few studies about supe rfluid density in\noptical lattices [34, 29]. However, these papers have focused on p opulation balanced\ncases i.e. the densities of the components are same. In this paper w e study the\nsuperfluid density of an ultra-cold two component Fermi gas in optic al lattices at finite\ntemperatures and with finite polarization. We are motivated by the f act by calculating\nthe superfluid density we can draw more conclusions on whether the gas is actually\nsuperfluid or not. Furthermore, we investigate if the superfluid de nsity can be used\nas an indicator of different superfluid phases. The phases we consid er are the BCS\nphase,i.e. the densities of the component are equal, the one mode FF LO phase, i.e.\nthe densities are not equal and the phase of the pairing gap modulat es as a function\nof position, and Sarma phase i.e. the densities are not equal and the phase of the\npairing gap is constant. We show that there are qualitatively differen ce between the\nBCS/Sarma superfluid density and the one mode FFLO superfluid den sity. We also\nstudy the stability of different phases. We also demonstrate that t here can be dynamical\ninstabilities in the one mode FFLO phase.Superfluid-density of the ultra-cold Fermi gas in optical la ttices 3\nThis paper is organized as follows. In Sec. 2 we discuss the physical s ystem and\npresent the Hamiltonian of the system. In Subsec. 2.1 the mean-fie ld approximation and\nthe ansatzes we consider are presented. In Sec. 3 we determine t he superfluid density\ntensor. In Sec. 4 we present the numerical results and we end with some concluding\nremarks in Sec. 5.\n2. Lowest band fermions in an optical lattice\nWe consider a two component Fermi gas, whose components are tw o different hyperfine\nstates of the same isotope, and we call them ↑-state and ↓-state. The Hamiltonian of\nthe system is given by\nˆH=/summationdisplay\nσ=↑,↓/integraldisplay\ndrˆΨ†\nσ(r)/parenleftbigg\n−/planckover2pi12∇2\n2m+Vσ(r)−µσ/parenrightbigg\nˆΨσ(r) (1)\n+g/integraldisplay /integraldisplay\ndrdr′ˆΨ†\n1(r)ˆΨ†\n2(r′)δ(r−r′)ˆΨ2(r′)ˆΨ1(r)\nwhere/planckover2pi1=h/2π,his Planck’s constant, Ψ†\nσ(r) and Ψ σ(r) are the fermionic creation-\nand annihilation field operators of the component σ,µσis the chemical potential of the\ncomponent σ, andδ(r−r′) is Dirac’s δ-function. The interaction strength is related to\nthe s-wave scattering length asthrough\ng=4π/planckover2pi12as\nm.\nThe lattice potential has a cubic structure and is given by\nVσ(r) =Er/summationdisplay\nαsσ,αsin2(kxα),\nwhereEr=/planckover2pi12k2/2mis the recoil energy ( k= (π/d) wheredis a lattice constant), and\nsσ,αis the lattice depth in the α-direction for the component σ.\nWhen we assume that only the lowest band is occupied the field operat ors can be\nexpanded by using the localized Wannier functions in the following way\nˆΨσ(r) =/summationdisplay\niwσ,i(r)ˆcσ,i (2)\nˆΨ†\nσ(r) =/summationdisplay\niw∗\nσ,i(r)ˆc†\nσ,i, (3)\nwherewσ,i(r) is a Wannier function, which is localized at a lattice point i, and ˆcσ,iis an\nannihilation operator.\nThe lowest band Hubbard model is valid, when the lattice is deep enoug h. In other\nwords, it is valid, when the Wannier functions decay within a single lattic e constant,\nand when the effective interaction between atoms is much smaller tha n the bandgap\nbetween bands. These two conditions imply that one has to take into account only the\non-site interactions. In this case overlap integrals of the kinetic en ergy operator between\nthe next nearest neighbor are small compared to overlap integrals of the kinetic energy\noperator between the nearest neighbor [35], and consequently, one needs to take intoSuperfluid-density of the ultra-cold Fermi gas in optical la ttices 4\naccount only hopping between the nearest neighbours. Then the lo west band Hubbard\nHamiltonian is given by\nˆH=−/summationdisplay\nσ=↑,↓\nJσ,x/summationdisplay\n/angbracketlefti,j/angbracketrightx+Jσ,y/summationdisplay\n/angbracketlefti,j/angbracketrighty+Jσ,z/summationdisplay\n/angbracketlefti,j/angbracketrightz\nˆc†\nσ,iˆcσ,j (4)\n−/summationdisplay\ni(µ↑c†\n↑,iˆc↑,i+µ↓c†\n↓,iˆc↓,i)+U/summationdisplay\niˆc†\n↑,iˆc†\n↓,iˆc↓,iˆc↑,i,\nwhere/angb∇acketlefti,j/angb∇acket∇ightαmeans sum over the nearest neighbours in the α-direction, and hopping\nstrength is defined by\nJσ,α=−/integraldisplay\ndrw∗\nσ,i(r)/parenleftbigg\n−/planckover2pi12∇2\n2m+Vσ(r)/parenrightbigg\nwσ,i±dˆxα(r).\nThe coupling strength of the above Hubbard model is given by\nU=g/integraldisplay\ndr|w↑,i(r)|2|w↓,i(r)|2.\n2.1. Mean-field approximation\nBecause the interaction term U/summationtext\niˆc†\n↑,iˆc†\n↓,iˆc↓,iˆc↑,iis hard to handle, we can approximate\nit by using mean-field approximation. Under this approximation the int eraction part\nbecomes\n/summationdisplay\ni∆(i)ˆc†\n↑,iˆc†\n↓,i+∆∗(i)ˆc↓,iˆc↑,i−|∆(i)|2\nU,\nwhere the pairing gap ∆( i) =U/angb∇acketleftˆc↓,iˆc↑,i/angb∇acket∇ight. With respect to position dependence, we only\nconsider the case, in which only the phase of the gap ∆( i) =|∆|eiq·Ri(Riis a lattice\nvector) candependonthelatticesite i. ThisansatziscalledtheonemodeFFLO(Fulde,\nFerrell, Larkin, and Ovchinnikov) [7, 8]. This ansatz includes also the B CS-ansatz and\nthe Sarma-state ansatz as special cases. In these phases the m omentum qis simply\nzero. Under this mean.field approximation the Hamiltonian is given by\nˆH0=−/summationdisplay\nσ=↑,↓\nJσ,x/summationdisplay\n/angbracketlefti,j/angbracketrightx+Jσ,y/summationdisplay\n/angbracketlefti,j/angbracketrighty+Jσ,z/summationdisplay\n/angbracketlefti,j/angbracketrightz\nˆc†\nσ,iˆcσ,j (5)\n−/summationdisplay\ni(µ↑c†\n↑,iˆc↑,i+µ↓c†\n↓,iˆc↓,i)+/summationdisplay\ni|∆|eiq·Riˆc†\n↑,iˆc†\n↓,i+|∆|e−iq·Riˆc↓,iˆc↑,i.\nBy minimizing the free energy F0= Ω0+µ↑N↑+µ↓N↓(Ω0is the grand canonical\npotential of the mean-field Hamiltonian) with respect |∆|andq, and solving the\nnumber equations ∂F0/∂µσ= 0 simultaneously, one finds the pairing gap, the chemical\npotentials, and the momentum qas functions of the temperature and the particle\nnumbers Nσ.Superfluid-density of the ultra-cold Fermi gas in optical la ttices 5\n3. Superfluid density\nLandau’s two component model for a superfluid gas says that the s uperfluid gas consists\ntwo components the normal component and the superfluid compon ent. In the free space\nthe superfluid density is density of the superfluid component. The c ase is a little bit\ndifferent inlatticesthustheenergydifferencebetween thetwisted systemandthesystem\nwithout thetwisting phase candepend onthedirectionofthephase shift. This isrelated\nthe fact that the effective masses can depend on the direction (th e effective masses are\nrelated to the hopping strengths). Thus in the lattice superfluid de nsity is more like a\ntensor than a scalar.\nIn the free space when the superfluid gas flows the kinetic energy o f the gas is given\nby\nEk=1\n2/integraldisplay\ndr˜ρs(r)vs(r)2,\nwhere ˜ρs(r) is the superfluid density and vs(r) is the superfluid velocity. The superfluid\nvelocity is defined by\nvs(r) =/planckover2pi1\n2m∇φ(r),\nwhereφ(r) is the phase of the order parameter.\nTo impose the linear phase variation to the order parameter, one ca n impose a\nlinear phase variation Θ·Ri= (Θx/(Mxd),Θy/(Myd),Θz/(Mzd))·Ri[26, 27, 36] to the\nHamiltonian. Here Mαindicates the number of lattice sites in direction alpha, i.e., the\ntotal volume the lattice V=MxMyMzd3This variation corresponds small (Θ αis small)\nsuperfluid velocity, which is given by\nvs=/planckover2pi1\n2m(Θx/(Mxd),Θy/(Myd),Θz/(Mzd)).\nThus this imposed phase gradient gives the system a kinetic energy, which corresponds\nto the free energy difference FΘ−F0, whereFΘis the free energy within the the phase\nvariation and F0is the free energy without the phase variation. The superfluid frac tion\nin this case can be determined as [26, 27]\nραα′= lim\nΘ→01\nNFΘ−F0\n¯JxΘαΘα′=1\nN¯Jx∂2FΘ\n∂Θα∂Θα′/vextendsingle/vextendsingle/vextendsingle\nΘ=0, (6)\nwhereNis the total number of particles, and ¯Jx= (J↑,x+J↓,x)/2. This¯Jxcorresponds\nthe effective mass\nmeff=/planckover2pi12\n2¯Jxd2.\nWhen one takes the limit Θ→0 the temperature, and the number of particles (and of\ncourse the interaction strength) should be kept as constants.\nOf course, one could define the superfluid fraction as\nραα= lim\nΘ→01\nNFΘ−F0\n¯Jα,αΘαΘα′(7)Superfluid-density of the ultra-cold Fermi gas in optical la ttices 6\nwhere¯Jα,α′= (J↑,α+J↓,α)/2, but in this case it is a little bit unclear how to define the\noff-diagonal elements. This definition is not directly connected to th e energy difference\nbetween the twisted system and the untwisted system.\nBy imposing the phase variation to the order parameter the one find s\nˆHΘ=−/summationdisplay\nσ=↑,↓\nJσ,x/summationdisplay\n/angbracketlefti,j/angbracketrightx+Jσ,y/summationdisplay\n/angbracketlefti,j/angbracketrighty+Jσ,z/summationdisplay\n/angbracketlefti,j/angbracketrightz\nˆc†\nσ,iˆcσ,j (8)\n−/summationdisplay\ni(µ↑c†\n↑,iˆc↑,i+µ↓c†\n↓,iˆc↓,i)+/summationdisplay\ni|∆|ei(q+2Θ)·Riˆc†\n↑,iˆc†\n↓,i+|∆|e−i(q+2Θ)·Riˆc↓,iˆc↑,i.\nOne can make the unitary transformation [26, 27, 36], which corres ponds the local\ntransformation ˆ cσ,i→ˆcσ,ieiΘ·Ri, and the twisted Hamiltonian becomes\nˆHΘ=−/summationdisplay\ni(µ↑c†\n↑,iˆc↑,i+µ↓c†\n↓,iˆc↓,i) (9)\n+/summationdisplay\ni|∆|eiq·Riˆc†\n↑,iˆc†\n↓,i+|∆|e−iq·Riˆc↓,iˆc↑,i\n−/summationdisplay\nσ/bracketleftBig/summationdisplay\nn/parenleftBig\nJσ,x(eiΘx/Mxˆc†\nσ,nˆcσ,n+dˆx+e−iΘx/Mxˆc†\nσ,nˆcσ,n−dˆx)\n+Jσ,y(eiΘy/Myˆc†\nσ,nˆcσ,n+dˆy+e−iΘy/Myˆc†\nσ,nˆcσ,n−dˆy)\n+Jσ,z(eiΘz/Mzˆc†\nσ,nˆcσ,n+dˆz+e−iΘz/Mzˆc†\nσ,nˆcσ,n−dˆz)/parenrightBig/bracketrightBig\n.\nThetwist anglesΘ αhave tobesufficiently small toavoid effects otherthanthecollective\nflow of the superfluid component. Since Θ α/Mαis small, we can expand up to second\norder\ne±iΘα/Mα≈1±iΘα\nMα−Θ2\nα\n2M2\nα.\nIn this way we can write the twisted Hamiltonian as\nˆHΘ=ˆH0+H′≈ˆH0+/summationdisplay\nσ/bracketleftBigΘ2\nx\n2M2\nx/summationdisplay\nnJσ,x(ˆc†\nσ,nˆcσ,n+dˆx+ˆc†\nσ,nˆcσ,n−dˆx)+ (10)\nΘ2\ny\n2M2\ny/summationdisplay\nnJσ,y(ˆc†\nσ,nˆcσ,n+dˆy+ˆc†\nσ,nˆcσ,n−dˆy)+Θ2\nz\n2M2\nz/summationdisplay\nnJσ,z(ˆc†\nσ,nˆcσ,n+dˆz+ˆc†\nσ,nˆcσ,n−dˆz)\n−iΘx\nMx/summationdisplay\nnJσ,x(ˆc†\nσ,nˆcσ,i+dˆx−ˆc†\nσ,nˆcσ,n−dˆx)−iΘy\nMy/summationdisplay\nnJσ,y(ˆc†\nσ,nˆcσ,i+dˆy−ˆc†\nσ,nˆcσ,n−dˆy)\n−iΘz\nMz/summationdisplay\nnJσ,z(ˆc†\nσ,nˆcσ,i+dˆz−ˆc†\nσ,nˆcσ,n−dˆz)/bracketrightBig\n.\nIn the momentum space this formula becomes\nˆHΘ≈ˆH0+/summationdisplay\nσ,α/bracketleftBigΘ2\nα\n2M2\nα/summationdisplay\nk2Jσ,αcos(kαd)ˆc†\nσ,kˆcσ,k (11)\n+Θα\nMα/summationdisplay\nk2Jσ,αsin(kαd)ˆc†\nσ,kˆcσ,k/bracketrightBig\n=ˆH0+ˆT+ˆJ,Superfluid-density of the ultra-cold Fermi gas in optical la ttices 7\nwhere\nˆT=/summationdisplay\nσ,α/bracketleftBigΘ2\nα\n2M2\nα/summationdisplay\nk2Jσ,αcos(kαd)ˆc†\nσ,kˆcσ,k/bracketrightBig\n(12)\nand\nˆJ=/summationdisplay\nσ,α/bracketleftBigΘα\nMα/summationdisplay\nk2Jσ,αsin(kαd)ˆc†\nσ,kˆcσ,k/bracketrightBig\n. (13)\nThese two terms are proportional to the current, and number op erator of the system,\nrespectively. These two terms commute with the number operator s, thus when one\nimposes the perturbation in the system the number of particles is co nserved. Using the\nsame method, what is used in reference [30], can be shown that in th e mean-field level\nlim\n|Θ|→0∂2F(Θ)\n∂Θα∂Θα′= lim\n|Θ|→0/parenleftbigg∂2Ω(Θ)\n∂Θα∂Θα′/parenrightbigg\n∆,µ↑,µ↓. (14)\nTherefore we can use the grand canonical potential.\nThe grand potential of the system can be written with a perturbing phase gradient\nas a series [37]\nΩΘ= Ω0−∞/summationdisplay\nn=1(−1)n\nβn/planckover2pi1n/integraldisplayβ/planckover2pi1\n0dτ1···/integraldisplayβ/planckover2pi1\n0dτn/angb∇acketleftTτˆH′(τ1)···ˆH′(τn)/angb∇acket∇ight0,(15)\nwhereβ= 1/kBT,kBis Boltzmann’s constant, Ω 0is the grand canonical potential in\nthe absence of the perturbation. The symbol Tτorders the operators such a way that τ\ndecreases from left to right. The brackets /angb∇acketleft.../angb∇acket∇ight0= Tr[exp( −β(ˆH0−µ↑ˆN↑−µ↓ˆN↓))...]\nmean the thermodynamic average evaluated in the equilibrium state o f the unperturbed\nsystem at temperature T. Because all the twisted angles Θ αare small, we can safely\nignore terms of higher order than Θ2\nα. With this approximation the grand potential\nbecomes\nΩΘ≈Ω0+1\nβ/planckover2pi1/integraldisplayβ/planckover2pi1\n0dτ/angb∇acketleftTτˆT(τ)/angb∇acket∇ight0−1\n2β/planckover2pi12/integraldisplayβ/planckover2pi1\n0/integraldisplayβ/planckover2pi1\n0dτ dτ′/angb∇acketleftTτˆJ(τ)J(τ′)/angb∇acket∇ight0,(16)\nThe first perturbation term is easy to calculate and it is given by\n1\nβ/planckover2pi1/integraldisplayβ/planckover2pi1\n0dτ/angb∇acketleftTτˆT(τ)/angb∇acket∇ight0=/summationdisplay\nσ,α/bracketleftBigΘ2\nα\n2M2α/summationdisplay\nk2Jσ,αcos(kαd)Nσ,k/bracketrightBig\n, (17)\nwhereNσ,kis number of particles of the σ-component in the momentum state k. Using\nWick’s theorem the second perturbation term is given by\n−1\n2β/planckover2pi12/integraldisplayβ/planckover2pi1\n0/integraldisplayβ/planckover2pi1\n0dτ dτ′/angb∇acketleftTτˆJ(τ)ˆJ(τ′)/angb∇acket∇ight0 (18)\n=−1\n2β/planckover2pi12/integraldisplayβ/planckover2pi1\n0/integraldisplayβ/planckover2pi1\n0dτ dτ′/bracketleftBig/summationdisplay\nσ,σ′,α,α′ΘαΘα′\nMαMα′/summationdisplay\nk,k′4Jσ,αJσ′,α′sin(kαd)sin(k′\nα′d)\n×/parenleftBig\n(1−δσσ′)δk,−k′+qF(k,τ,τ′)F†(k′,τ,τ′)−δσσ′δkk′Gσ(k,τ,τ′)Gσ′(k′,τ,τ′)/parenrightBig/bracketrightBig\n,Superfluid-density of the ultra-cold Fermi gas in optical la ttices 8\nwhere\nGσ(k,τ,τ′) =−/angb∇acketleftTτˆcσ,k(τ)ˆc†\nσ,k(τ′)/angb∇acket∇ight0=1√β/planckover2pi1∞/summationdisplay\nn=−∞e−iωn(τ−τ′)Gσ(k,ωn)(19)\nF(k,τ,τ′) =−/angb∇acketleftTτˆc↑,k+q(τ)ˆc↓,−k+q(τ′)/angb∇acket∇ight0=1√β/planckover2pi1∞/summationdisplay\nn=−∞e−iωn(τ−τ′)F(k,ωn).\nThe fermionic Matsubara frequencies are given by\nωn=π(2n+1)\n/planckover2pi1β,\nwherenis an integer. The individual Green’s functions of the mean-field theo ry can be\nwritten as\nG↑(k,ωn) =|uk,q|2\ni/planckover2pi1ωn−E+,k,q+|vk,q|2\ni/planckover2pi1ωn+E−,k,q(20)\nG↓(k,ωn) =|uk,q|2\ni/planckover2pi1ωn−E−,k,q+|vk,q|2\ni/planckover2pi1ωn+E+,k,q(21)\nF(k,ωn) =uk,qv∗\nk,q\ni/planckover2pi1ωn+E−,k,q−uk,qv∗\nk,q\ni/planckover2pi1ωn−E+,k,q, (22)\nwhere the quasiparticle dispersions and amplitudes are given by\nE±,k,q=/radicalBigg/parenleftbiggǫ↑,k+q/2+ǫ↓,−k+q/2\n2/parenrightbigg2\n+|∆|2±ǫ↑,k+q/2−ǫ↓,k+q/2\n2(23)\n|uk,q|2=1\n2/parenleftbigg\n1+ǫ↑,k+q/2+ǫ↓,−k+q/2\nE+,k,q+E−,k,q/parenrightbigg\n(24)\n|vk,q|2= 1−|uk,q|2, (25)\nand furthermore ǫσ,k=/summationtext\nα2Jσ,α(1−cos(kαd))−µσ. By calculating the integrals and\nthe Matsubara summations one finds a lengthy formula\n−1\n2β/planckover2pi12/integraldisplayβ/planckover2pi1\n0/integraldisplayβ/planckover2pi1\n0dτ dτ′/angb∇acketleftTτˆJ(τ)ˆJ(τ′)/angb∇acket∇ight0 (26)\n=/summationdisplay\nα,α′ΘαΘα′\nMαMα′/summationdisplay\nk/bracketleftBig\nJ↑,αJ↑,α′sin((kα+qα/2)d)sin((kα′+qα′/2)d)\n×/parenleftBig\n2|uk,q|2|vk,q|2f(E+,k,q)+f(E−,k,q)−1\nE+,k,q+E−,k,q−\nβ|uk,q|4f(E+,k,q)(1−f(E+,k,q))−β|vk,q|4f(E−,k,q)(1−f(E−,k,q))/parenrightBig\n+2J↓,αJ↓,α′sin((kα−qα/2)d)sin((kα′−qα′/2)d)\n×/parenleftBig\n2|uk,q|2|vk,q|2f(E+,k,q)+f(E−,k,q)−1\nE+,k,q+E−,k,q−\nβ|uk,q|4f(E−,k,q)(1−f(E−,k,q))−β|vk,q|4f(E+,k,q)(1−f(E+,k,q))/parenrightBig\n+4J↑,αJ↓,α′sin((kα+qα/2)d)sin((kα′−qα′/2)d)\n×|uk,q|2|vk,q|2/parenleftBig2(f(E+,k,q)−f(E−,k,q))\nE+,k,q−E−,k,qSuperfluid-density of the ultra-cold Fermi gas in optical la ttices 9\n−2f(E+,k,q)−1\n2E+,k,q−2f(E−,k,q)−1\n2E−,k,q/parenrightBig/bracketrightBig\n,\nwheref(E) is the Fermi-Dirac distribution. In order to handle the limit lim k′→k, we\nhave implicitly assumed in equation (26) that the lattice is large, i.e, Mα≫1.\nNow the twisted grand canonical potential can be written as\nΩΘ≈Ω0+/summationdisplay\nα,α′δΩαα′ΘαΘα′\nMαMα′. (27)\nIn all the cases we consider δΩαα′= 0 when α/negationslash=α′. The off-diagonal terms can be non-\nzero only when the single particle dispersion couples the different dire ctions together.\nIn our case where the directions are independent, the ksums in different directions\nare independent, and because sine is an antisymmetric function the se sums vanishes.\nTherefore the grand potential becomes\nΩΘ≈Ω0+/summationdisplay\nαδΩααΘ2\nα\nM2\nα. (28)\nWe can now determine the components of the dimensionless superflu id fraction tensor\nas\nραα′=δΩαα′\n¯Jx(N↑+N↓).\nAs a formula these components are given by\nραα=1\nN/summationdisplay\nk,σ˜Jσ,αcos(kαd)Nσ,k (29)\n+1\nN/summationdisplay\nk/bracketleftBig\n˜J2\n↑,αsin2((kα+qα/2)d)\n×/parenleftBig\n2|uk,q|2|vk,q|2f(E+,k,q)+f(E−,k,q)−1\nE+,k,q+E−,k,q−\nβ|uk,q|4f(E+,k,q)(1−f(E+,k,q))−β|vk,q|4f(E−,k,q)(1−f(E−,k,q))/parenrightBig\n+2˜J2\n↓,αsin2((kα−qα/2)d)/parenleftBig\n2|uk,q|2|vk,q|2f(E+,k,q)+f(E−,k,q)−1\nE+,k,q+E−,k,q−\nβ|uk,q|4f(E−,k,q)(1−f(E−,k,q))−β|vk,q|4f(E+,k,q)(1−f(E+,k,q))/parenrightBig\n+4˜J↑,α˜J↓,α′sin((kα+qα/2)d)sin((kα−qα/2)d)\n×|uk,q|2|vk,q|2/parenleftBig2(f(E+,k,q)−f(E−,k,q))\nE+,k,q−E−,k,q\n−2f(E+,k,q)−1\n2E+,k,q−2f(E−,k,q)−1\n2E−,k,q/parenrightBig/bracketrightBig\nραα′= 0,\nwhereN=N↑+N↓and˜Jσ,α=Jσ,α/¯Jx. When the momentum qis non-zero the\nsuperfluid density describes the one mode FFLO phase, and when th e momentum q= 0\nthesuperfluiddensitydescribestheBCSortheSarmaphase. ρααisasuperfluidfraction,Superfluid-density of the ultra-cold Fermi gas in optical la ttices 10\nit describes what fraction of the atoms is in the superfluid state. In the limit where\nJ↑,α=J↓,α′=Jfor allα,α′andN↑=N↓, the chemical potentials are same and q= 0.\nIn this limit superfluid density tensor can be simplified as\nραα=1\nN/summationdisplay\nk,σcos(kαd)Nσ,k−4βJ\nN/summationdisplay\nksin2(kαd)f(Ek)(1−f(Ek)) (30)\nραα′= 0 (31)\nwhere\nEk=/radicalBigg/parenleftbiggǫ↑,k+ǫ↓,−k\n2/parenrightbigg2\n+|∆|2.\nIn the continuum limit i.e. d→0, where Jd2remains constant, the superfluid\nfraction becomes\nραα=ρ= 1+2/planckover2pi12\nmanV/summationdisplay\nkk2\nz∂f(Ek)\n∂Ek,\nwhere the effective mass ma=/planckover2pi12/(2Jd2),n=n↑+n↓is the total number density and\nVis the size of the system. This continuum result is precisely the Landa u’s formula for\nthe BCS superfluid fraction [38].\n4. Results\n4.1. BCS/Sarma-phase results\nIn the continuum the formula of the BCS superfluid fraction is the we ll known Landau’s\nformula. Figure1(a)showstheBCSsuperfluidfractionasafunctio nofthetemperature,\nin the uniform case (the external potential is zero), without the la ttice. In figure 1 (b)\nwe show the BCS superfluid fraction divided by |∆|2as a a function of the temperature.\nOne sees from figure 1 (a) that the BCS superfluid fraction is one at zero temperature\nand from figure 1 (b) that the superfluid fraction is almost proport ional to |∆|2,\nbut the temperature dependence of the superfluid fraction differ s somewhat from the\ntemperaturedependence of |∆|2. ThestandardBCSresult isthatthesuperfluid fraction\nis proportional to |∆|2in limitT→Tc[39]. When we calculated the gap, we used the\nrenormalization, i.e., we have cancelled out the divergent part of the gap equation, in\nreference [39] is used the cutoff energy. This is the cause of little diff erence between the\nresults.\nWhen all the hopping strengths are the same i.e. J=J↑,α=J↓,α′, and filling\nfractions nσ=Nσ/MxMyMzare same and the average filling fraction nav= (n↑+n↓)/2\ngoes to zero, the superfluid fraction should go to one at zero temp erature. For low filling\nfractions the the atoms occupy only the lowest momentum states a nd the single particle\ndispersions ǫσ,k=/summationtext\nα2Jσ,α(1−cos(kαd))−µσapproaches to form J(kd)2−µσ, which\nis the free space dispersion. Therefore the results on the limit of low filling fractions\nbecome the free space result and we know from the Landau’s formu la that the BCS\nsuperfluid fraction in the free space is one at zero temperature (s ee figure 1 (a)) .Superfluid-density of the ultra-cold Fermi gas in optical la ttices 11\nIn figure 2 (a) we show the BCS superfluid fraction as a function of t he total filling\nfraction in the lattice. As we can see, the BCS superfluid fraction ap proaches one, when\nthe filling fractions become small. In the free space the superfluid fr action is one at zero\ntemperature, but in the lattice this no longer holds. In figure 2 (b) w e showρ(n↑+n↓)\nas a function of the total filling fraction and as one can see from it, t he BCS superfluid\ndensity, which is not divided by the total filling fraction is symmetric wit h respect to\nhalf filling. The cause of this is the particle-hole symmetry of the lattic e model. The\nparticle-hole symmetry of the lattice model can be seen also from fig ures 2 (c) and 2\n(d). In figures 2 (c) and 2 (d) we show ∆ /Jand|∆|2/J2, respectively, as functions of of\nthe total filling fraction. By comparing figures 2 (b)-(d) one can no tice that ρ(n↑+n↓)\nis not directly proportional to ∆ or |∆|2.\nFigure 3 (a) shows the BCS superfluid fraction as a function of the t emperature,\nwith three different average filling fractions. When one compares th is figure to figure 1\n(a), one notices that the lattice result is very similar compared to th e free space result.\nThe superfluid fraction goes to zero, when the paring gap goes to z ero as expected. The\nfigure is consistent with figure 2 (a), in that the total filling fraction increases when the\nsuperfluid fraction decreases. In figure 3 (b) we show the BCS sup erfluid fraction as a\nfunction of the dimensionless coupling strength −U/Jat zero temperature. As we can\nsee, when −U/Jincreases the superfluid fraction decreases contrary to the fre e space\nresult where superfluid fraction is a constant as a function of the c oupling strength at\nzero temperature. Of course at finite temperatures the free sp ace superfluid fraction\ndecreases with increasing coupling strength. This is due to the fact as−U/Jincreases\nthemovement oftheatomsdecreases inthelattice. Inotherword swhen−U/Jincreases\nthen Cooper pairs become smaller and the atoms are better localized . Of course in the\nlimit−U/J→ ∞the mean-field theory fails and in large values of −U/Jthe results\nare not reliable. Figure 3 (c) shows the BCS superfluid fraction divide d by|∆|2as a\nfunction of the temperature, with three different average filling fr actions and we see that\nthe BCS superfluid fraction is almost proportional to |∆|2(compare to figure 1 (b)) i.e\nρ=c(T,n↑=n↓)|∆|2, wherec(T,n↑=n↓) depends weakly on the temperature. We\nshow the pairing gap as a function of −U/Jat zero temperature in figure 3 (d). As one\ncan notice, the BCS superfluid fraction does not behave anything lik e the pairing gap\nas a function of −U/Jat zero temperature.\nSince it is experimentally easy to study anisotropic lattices and use th ese to explore\ndimensional crossovers, let us explore superfluid fractions in aniso tropic lattices. We\nshow in figure 4 (a) the BCS superfluid fraction as a function of the t emperature in\nthe case, where the hopping strengths are different in the differen t directions. As one\ncan see from the figure the components of the superfluid fraction tensor are different\nin different directions. Furthermore, we see that the superfluid fr action is larger in the\ndirection of large hopping strength. This is easy to understand, an d is due to the free\nenergy difference in α-direction being proportional to Jαas we see from equations (28)\nand (29). If the superfluid fraction was defined as in equation (7), the components of\nthe superfluid fraction would be equal in every direction. In other w ords the effectiveSuperfluid-density of the ultra-cold Fermi gas in optical la ttices 12\nmasses are different in different directions. figure 4 (c) shows that in the in the case,\nwhere the hopping strengths are different in the different direction s,ραα/|∆|2is almost\na constant as a function of the temperature.\nFigure 4 (b) shows the BCS superfluid fraction as a function of the t emperature in\nthecase, wherethehoppingstrengthsaredifferentforthecomp onents(but J↑+J↓iskept\nconstant). As it is clear from figure 4 (b), when the ratio J↑/J↓increases the superfluid\nfraction decreases, but the pairing gap, however, does not decr ease as a function of\nJ↑/J↓at zero temperature (it does as a function of the temperature), but it remains\nalmost a constant (at zero temperature it is a constant). This can be seen from figure 4\n(d), in which is shown the pairing gaps as as functions of the tempera ture in the cases,\nwhere the hopping strengths are different for the components. O ne can also notice that\nwhenJ↑/J↓→ ∞thenρ→0 ( in this case the critical temperature also goes to zero).\nWhenJ↑/J↓increases but 2 U/(J↑+J↓) remains constant, the lattice becomes deeper\nand deeper for ↓-component. Thus the atoms of the component ↓are more localized\nand the atoms do not move easily.\nFigure 5 (a) shows the superfluid fraction as a function of polarizat ionP=\n(n↑−n↓)/(n↑+n↓), ataconstant temperature( kBT/J= 0.75). Whenthepolarizationis\nlarger thanzero the stateis called the Sarma state. One sees, tha t when thepolarization\nincreases the superfluid fraction decreases. Figure 5 (b) shows t he superfluid fraction\ndivided by |∆|2as a function of the polarization at a constant temperature. When\nPis about 0 .35 the superfluid fraction divided by |∆|2drops to zero suddenly. This\nhappens because the pairing gap vanishes at this polarization. From figure 5 (b) is seen\nthat the superfluid fraction at a constant temperature is almost p roportional to |∆|2.\nFigure 5 (c) shows the Sarma state superfluid fraction as a functio n of the temperature,\nat a constant polarization ( P= 0.10). Figure 5 (d) shows the Sarma state superfluid\nfraction divided by |∆|2(ρ/|∆|2). We notice from figures 5 (b) and (d) that ρ/|∆|2is\nalmost a constant asa functionof thetemperature andpolarizatio ni.e.ρ=c(T,P)|∆|2,\nwherec(T,P) depends weakly on the temperature and the polarization.\n4.2. Results for FFLO-phase\nWhen the lattice is cubic and all the hopping strengths are equal the one mode FFLO\nsuperfluid fraction is symmetric like the BCS/Sarma superfluid fract ion. This due the\nfollowing fact, when one varies the q, it turns out that the free-energy is minimized,\nwhen the qlies along side the axis i.e. q=qαˆxα, but the system does not favour any of\nthese axis.\nThe BCS-andthe Sarmastatesuperfluid fractionsarealmost direc tly proportional\nto|∆2|i.eρ∼ |∆2|. This changes for the one mode FFLO superfluid fraction as we will\nnow demonstrate. We show in figure 6 (a) ρxx-component of the FFLO superfluid\nfraction as a function of the total filling fraction, with three differe nt interactions\nU/J=−4.0,−5.0,−6.0, at zero temperature. figure 6 (b) shows ρxx(n↑+n↓) as a\nfunction of the total filling fraction, when U/J=−6.0, at zero temperature. The dataSuperfluid-density of the ultra-cold Fermi gas in optical la ttices 13\nwhich is used in figure 6 (a) and (b) have been calculated just above t he Clogston\nlimit [40]. The Clogston limit is the limit for the chemical potential differenc e below\nwhich one can find only the BCS type solution when one minimizes the gra nd potential.\nThe numerical value of this limit is roughly δµ≈√\n2∆0, where ∆ 0is the pairing gap at\nzero temperature when δµ= 0. As one can see from figure 6 (a) when |U/J|increases\nthe superfluid fraction decreases and eventually becomes negativ e. One can also see,\nthat as the total filling fraction increases the superfluid fraction d ecreases but does\nnot necessarily become negative. Negative superfluid fraction implie s that the gas is\nunstable. If superfluid fraction is negative the state around which we expanded is not\nthe ground state of the system, and thus the state cannot be st able. In the case where\nsuperfluid fractionis negative, the real minimum of the free energy is beyond our ansatz.\nThe real minimum of the system may be a some kind of phase separatio n or modulating\ngap state (like the two-mode FFLO state) [41, 42, 43].\nThe superfluid density i.e. ρxx(n↑+n↓) is symmetric as a function of the total filling\nfraction, with the value n↑+n↓= 1.0, this is shown in figure 6 (b). This is again due\nto the particle-hole symmetry of the lattice model.\nFigure 7 (a) shows the superfluid fraction as a function of polarizat ion at a constant\ntemperature ( kBT/J≈0.23), and figure 7 (b) shows the superfluid fraction divided by\n|∆|2as a function of polarization. There is a second order phase transit ion between the\nSarma/BCS phase and the FFLO phase, when polarization P≈0.15, this can be seen\nfrom the bend in the superfluid fraction, clearly. We see also from fig ure 7 (b) that the\nFFLO superfluid fraction is not proportional to |∆|2. When one writes ρ=c(T,P)|∆|2,\nthe polarization dependence of c(T,P) is very different in the FFLO phase compared to\nthat in the Sarma phase.\n0.010.020.030.040.050.0600.10.20.30.40.50.60.70.80.91ρ\nT/TF(a)\n0.010.020.030.040.050.0601020304050607080ρ/|∆|2\nT/TF(b)\nFigure 1. Figure (a) we show the BCS superfluid fraction as a function of the\ntemperature in a free space (absence of the lattice). Figure (b) s hows the BCS\nsuperfluid fraction divided by |∆|2as a function of temperature. The interaction\nstrength kFas=−0.30, where kFis the Fermi wave vector.\nIn figures 8 and 9 we present phase diagrams with four different inte raction\nstrengths, when the average filling fractions are nav= 0.2 andnav= 0.5, respectively.\nThe figures show that there can be unstable regions in the FFLO pha se. In theSuperfluid-density of the ultra-cold Fermi gas in optical la ttices 14\n0.5 1 1.500.10.20.30.40.50.60.70.80.91ρ\nn↑+n↓(a)\n0.5 1 1.500.050.10.150.20.25ρ(n↑+n↓)\nn↑+n↓(b)\n0.5 1 1.500.511.52∆/J\nn↑+n↓(c)\n0.5 1 1.5012345|∆|2/J2\nn↑+n↓(d)\nFigure 2. In figure (a) we show the BCS superfluid fraction ρ=ρxx=ρyy=ρzzas\na function of the total filling fraction n↑+n↓= 2n↑= 2n↓at zero temperature. In\nfigure (b) is shown ρ(n↑+n↓) as a function of the total filling fraction. Figures (c) and\n(d) show ∆ /Jand|∆|2/J2, respectively as functions of the total filling fraction. All\nthe hopping strengths are same i.e. J=J↑,α=J↓,α′, and−U/J= 6.0.\nunstable FFLO region the superfluid density is negative, which implies t hat the phase\nis dynamically unstable. However, these instabilities occur only when t he interaction\nis relative high. By comparing the two figures one notices that the un stable regions\nincrease when the density increases. Thus when the density decre ases the unstable\nregions eventually disappear, and this implies that the FFLO phase re gion is stable\nwithout the lattice. Due to the particle-hole symmetry average filling fraction 0 .5 is\nthe worst case for the stability, but even when the average densit y is 0.5, the one\nmode FFLO is stable when |U/J|<4.4 for any polarization. The induced interactions\nmight make the FFLO phase more stable, since these induced interac tions make the\neffective interaction weaker [44]. References [22, 45] considered also the possibility of\nphase separation. However, this work we do not consider phase se paration, because\nwhen there is phase separation between the BCS phase and the nor mal gas, there is a\nsuperfluid gas in a part of the lattice and normal gas in another part of the lattice. In\nthis case the superfluid density is the standard BCS superfluid dens ity and the gas is\nstable. It should be notice that the region of the phase separation does not include the\nunstable regions of the FFLO phase [22].Superfluid-density of the ultra-cold Fermi gas in optical la ttices 15\n00.2 0.4 0.6 0.8 11.200.050.10.150.20.250.30.35\nkBT/Jρ(a)\n  \nnav=0.3\nnav=0.5\nnav=0.7\n20 40 60 80 10000.050.10.150.20.250.30.350.4ρ\n−U/J(b)\n00.2 0.4 0.6 0.8 11.200.010.020.030.040.050.060.070.08\nkBT/Jρ/|∆|2(c)\n  \nnav=0.3\nnav=0.5\nnav=0.7\n20 40 60 80 10001020304050∆/J\n−U/J(d)\nFigure 3. Figure (a) shows the BCS superfluid fraction ρ=ρxx=ρyy=ρzz\nas a function of the temperature,with three different average filling fractions nav=\n(n↑+n↓)/2. Figure (b) shows the BCS superfluid fraction as a function of a\ndimensionless coupling strength −U/Jat zero temperature. In figure (c) we show\nthe BCS superfluid fraction divided by |∆|2as a function of the temperature,with\nthree different average filling fractions. Figure (d) shows the pairin g gap as a function\nof−U/Jat zero temperature. In the all figures all the hopping strengths are same\ni.e.J=J↑,α=J↓,α′. In figures (a) and (c) −U/J= 6.0, and in figures (b) and (d)\nn↑=n↓= 0.5.\n5. Conclusions\nIn this paper we have presented, at the mean-field level, the super fluid density of the\ntwo component Fermi gas in an optical lattice. We have shown that t he BCS superfluid\ndensity in optical lattices differs from the free space results. We ha ve also shown that\nthe one-mode FFLO superfluid density differs crucially from the BCS/ Sarma superfluid\ndensity. In the BCS/Sarma phase the gas is always a stable superflu id, but in the\nFFLO phase dynamical instabilities can appear. However, the FFLO p hase is stable,\nwhen|U/J|<4.0 i.e, on the BCS limit. Even in the BCS phase the superfluid density\ncan be different in different directions depending on the lattice struc ture.\nAlthough the one mode FFLO phase is stable when |U/J|<4.0, it is unlikely that\nthe one mode FFLO is the free energy minimum the system; calculation s made by using\nthe Bogoliubov-de Gennes equations show that the spatially modulat ing gap might be\nmore favorable [41, 42, 43]. Such states are also, quite likely, more s table than the oneSuperfluid-density of the ultra-cold Fermi gas in optical la ttices 16\n0.2 0.4 0.6 0.8 11.200.050.10.150.20.250.30.350.40.45\nkBT/Jxρ(a)\n  \nρxx\nρyy\nρzz\n0.20.40.60.8 11.200.050.10.150.20.25\n2kBT/(J↑+J↓)ρ(b)\n  \nJ↑/J↓=1.2\nJ↑/J↓=5.7\nJ↑/J↓=20\nJ↑/J↓=200\n0.2 0.4 0.6 0.8 11.200.020.040.060.080.10.12\nkBT/Jxρ/|∆|2(c)\n  \nρxx/|∆|2\nρyy/|∆|2\nρzz/|∆|2\n0.20.40.60.8 11.200.511.522.5\n2kBT/(J↑+J↓)2∆/(J↑+J↓)(d)\n  \nJ↑/J↓=1.2\nJ↑/J↓=5.7\nJ↑/J↓=20\nJ↑/J↓=200\nFigure 4. [Colour online] In figure (a) we show the BCS superfluid fraction as a\nfunction of the temperature in the case, where the hopping stren gths are different\nin the different directions. In figure (b) we show the BCS superfluid f raction as a\nfunction of the temperature. Figure (c) shows the BCS superfluid fraction divided\nby|∆|2as a function of the temperature. In figure (d) we show the pairing gaps\nas functions of the temperature. In figures (a) and (c) Jx= (J↑,x+J↓,x)/2,\nJ↑,y=J↓,y= 0.8J↑,x= 0.8J↓,x,J↑,z=J↓,z= 0.5J↑,x= 0.5J↓,x,n↑=n↓= 0.2,\nand−U/J↑,x= 6.0. In figures (b) and (d) Jσ,α=Jσ′,α′=Jσ,−2U/(J↑+J↓) = 6.0,\nandn↑=n↓= 0.5\nmode FFLO phase.\nThe methods which have used in this paper to calculate thesuperfluid density, work\nin principle for many different systems. For example, one could use th e methods used\nhere to calculate the superfluid density in the system where anothe r component of the\ntwo component Fermi gas occupies the first excited band [46], or in the case where an\noptical lattice is in a trap. These methods can also been used calculat ing the superfluid\ndensity of modulating gap states.\nAcknowledgments I would like to thank J.-P. Martikainen and T. K. Koponen\nfor many illuminating discussions. This work was supported by Academ y of Finland\n(Project No. 1110191).Superfluid-density of the ultra-cold Fermi gas in optical la ttices 17\n00.050.10.150.20.250.300.020.040.060.080.10.120.140.160.180.2ρ\nP(a)\n00.050.10.150.20.250.300.0050.010.0150.020.0250.030.0350.040.045ρ/|∆|2\nP(b)\n0.7 0.8 0.9 1 1.1 1.200.020.040.060.080.10.120.140.16ρ\nkBT/J(c)\n0.7 0.8 0.9 1 1.1 1.200.0050.010.0150.020.0250.030.0350.040.045ρ/|∆|2\nkBT/J(d)\nFigure 5. [Colour online] The BP/Sarma phase superfluid fraction. In figure (a ) we\nshow the superfluid fraction as a function of the polarization P= (n↑−n↓)/(n↑−n↓).\nFigure (b) shows the superfluid fraction divided by |∆|2. Figure (c) shows the\nsuperfluid fraction as a function of the temperature. In figure (d ) we show the\nsuperfluid fraction divided by |∆|2. In figures (a) and (b) kBT/J= 0.75 and in\nfigures (c) and (d) P= 0.10. In all the figures all the hopping strengths are same i.e.\nJσ,α=Jσ′,α′=J, the average filling fraction nav= 0.50,U/J=−6.0, andq= 0.\n6. References\n[1] M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle. F ermionic superfluidity with\nimbalanced spin populations. Science, 311:492, 2006.\n[2] G. B. Partridge, W. Li, R. I. Kamar, Y. Liao, and R. G. Hulet. Pairin g and phase separation in\na polarized Fermi gas. Science, 311:506, 2006.\n[3] M. W. Zwierlein, C. H. Schunck, A. Schirotzek, and W. Ketterle. D irect observation of the\nsuperfluid phase transition in ultracold Fermi gases. 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Inhomogeneous superconductiv ity in condensed matter and qcd.Superfluid-density of the ultra-cold Fermi gas in optical la ttices 18\n0.2 0.4 0.6 0.8 1−0.2−0.100.10.20.30.40.5\nn↑+n↓ρxx(a)\n  \nU/J=−6.0\nU/J=−5.0\nU/J=−4.0\n0 0.5 1 1.5 2−0.2−0.15−0.1−0.050ρ(n↑+n↓)\nn↑+n↓(b)\nFigure 6. The FFLO superfluid fraction at zero temperature. In figure (a) w e show\nthe FFLO superfluid fraction as a function of the total filling fractio n, with three\ndifferent interactions at zero temperature. In figure (b) we show ρxx(n↑+n↓) as a\nfunction of the total filling fraction at zero temperature, when U/J=−6.0. In the\nboth figures all the hopping strengths are same i.e. Jσ,α=Jσ′,α′=J,q=qxˆx, and\nP≈0.30. The horizontal lines in the figures show the value 0.\n0 0.1 0.2 0.3 0.4 0.500.10.20.30.40.5\nPρ(a)\n0 0.1 0.2 0.3 0.4 0.500.10.20.30.40.50.60.7\nPρ/|∆|2(b)\nFigure 7. Figure (a) shows the Sarma- and FFLO phase superfluid fraction as\na function of the polarization at constant temperature. 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Tosi.\nSuperfluidanddissipativedynamicsofaBose-Einsteincondensatein aperiodicopticalpotential.\nPhys. Rev. Lett. , 86:4447, August 2001.\n[15] Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J. Dalibard . Interference of an array of\nindependent Bose-Einstein condensates. Phys. Rev. Lett. , 93:180403, 2004.Superfluid-density of the ultra-cold Fermi gas in optical la ttices 19\n00.050.10.150.20.250.30.3500.050.10.150.20.250.30.35\nkBT/JP\n00.1 0.2 0.3 0.4 0.500.10.20.30.40.5\nkBT/JP\n00.10.20.30.40.50.60.700.10.20.30.40.50.6\nkBT/JP\n00.2 0.4 0.6 0.8 100.20.40.60.8\nkBT/JP\nFigure 8. [Color online] Phase diagrams with four different interaction strengt hs,\nwhen the average filling fraction nav= 0.2. The interaction strengths are from up left\nto bottom right: U/J≈ −3.7,U/J≈ −4.4,U/J≈ −5.1, andU/J≈ −6.3. All the\nhopping strengths are equal. The color are as follows:BCS/Sarma=b lue(black), stable\nFFLO=yellow(light grey), unstable FFLO=red(dark grey), normal gas=white.\n[16] W. 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BogoliubovSuperfluid-density of the ultra-cold Fermi gas in optical la ttices 20\n00.1 0.2 0.3 0.4 0.500.050.10.150.20.250.3\nkBT/JP\n00.10.20.30.40.50.60.700.10.20.30.4\nkBT/JP\n0 0.2 0.4 0.6 0.8 100.10.20.30.40.50.6\nkBT/JP\n00.20.40.60.811.200.10.20.30.40.50.6\nkBT/JP\nFigure 9. [Color online] Phase diagrams with four different interaction strengt hs,\nwhen the average filling fraction nav= 0.5. The interaction strengths are from up left\nto bottom right: U/J≈ −3.7,U/J≈ −4.4,U/J≈ −5.1, andU/J≈ −6.3. All the\nhopping strengths are equal. The color are as follows:BCS/Sarma=b lue(black), stable\nFFLO=yellow(light grey), unstable FFLO=red(dark grey), normal gas=white.\napproach to superfluidity of atoms in an optical lattice. J. Phys. B , 36:825, 2003.\n[27] R. Roth and K. Burnett. Superfluidity and interference patte rn of ultracold bosons in optical\nlattices. Phys. Rev. A , 67:031602, 2003.\n[28] B. Wu and Q. Niu. 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A , 78:023607, 2008."    },    {        "title": "0811.4513v2.Continuity_of_the_integrated_density_of_states_on_random_length_metric_graphs.pdf",        "content": "arXiv:0811.4513v2  [math.SP]  25 Apr 2009CONTINUITY OF THE INTEGRATED DENSITY OF STATES ON\nRANDOM LENGTH METRIC GRAPHS\nDANIEL LENZ, NORBERT PEYERIMHOFF, OLAF POST, AND IVAN VESEL I´C\nAbstract. We establish several properties of the integrated density of stat es for\nrandom quantum graphs: Under appropriate ergodicity and amena bility assump-\ntions, the integrated density of states can be defined using an exh austion procedure\nby compact subgraphs. A trace per unit volume formula holds, similar ly as in the\nEuclidean case. Our setting includes periodic graphs. For a model wh ere the edge\nlengths are random and vary independently in a smooth way we prove a Wegner\nestimate and related regularity results for the integrated density of states.\nThese results are illustrated for an example based on the Kagome lat tice. In the\nperiodic case we characterise all compactly supported eigenfunct ions and calculate\nthe position and size of discontinuities of the integrated density of s tates.\n1.Introduction\nQuantum graphs are Laplace or Schr¨ odinger operators on metric graphs. As struc-\ntures intermediate between discrete and continuum objects they have received quite\nsome attention in recent years in mathematics, physics and materia l sciences, see e.g.\nthe recent proceeding volume [EKK+08] for an overview.\nHere, westudyperiodicandrandomquantumgraphs. Ourresultsc oncernspectral\nproperties which are related to the integrated density of states ( IDS), sometimes\ncalled spectral distribution function. As in the case of random Schr ¨ odinger operators\nin Euclidean space, disorder may enter the operator via the potent ial. Moreover, and\nthis is specific to quantum graphs, randomness may also influence th e characteristic\ngeometric ingredients determining the operator, viz.\n•the lengths of the edges of the metric graph and\n•the vertex conditions at each junction between the edges.\nIn the present paper we pay special attention to randomness in th ese geometric\ndata. Our results may be summarised as follows. For quite wide classe s of quantum\ngraphs we establish\n•the existence, respectively the convergence in the macroscopic lim it, of the\nintegrated density of states under suitable ergodicity and amenab ility con-\nditions (see Theorem 2.6),\n•a trace per unit volume formula for the IDS (see equation (2.9)),\n•a Wegner estimate for random edge length models (assuming indepen dence\nand smoothness for the disorder) (Theorem 2.9). This implies quant itative\ncontinuity estimates for the IDS (Corollary 2.10).\nDate: October 24, 2018, File:contids-qg20.tex .\n2000Mathematics Subject Classification. 35J10; 82B44.\nKey words and phrases. integrated density of states, periodic and random operators, me tric\ngraphs, quantum graphs, continuity properties.\n12 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\nThese abstract results are illustrated by the thorough discussion of an example\nconcerning a combinatorial and a metric graph based on the Kagome lattice. In this\ncase we calculate positions and sizes of all jumps of the IDS. Our res ults show the\neffect of smoothing of the IDS via randomness.\nThe article is organised as follows: In the remainder of this section we summarise\nthe origin of results about the construction of the IDS and of Wegn er estimates and\npoint out aspects of the proofs which are different in the case of qu antum graphs\nin comparison to random Schr¨ odinger operators on L2(Rd) orℓ2(Zd). We mention\nbriefly recent results about spectral properties of random quan tum graphs which are\nin some sense complementary to ours. Finally, we point out some open problems in\nthis field of research. In the next section, we introduce the rando m length model\nand state the main results. In Section 3 we present the Kagome latt ice example. In\nSection 4 we prove Theorem 2.6 concerning the approximability of the IDS. Finally,\nin Section 5 we prove the Wegner estimate Theorem 2.9.\nIntuitively, the IDS concerns the number of quantum states per u nit volume below\na prescribed energy. From the physics point of view the natural de finition of this\nquantity is via a macroscopic limit. This amounts to approximating the ( ensemble-\naveraged) spectral distribution function of an operator on the w hole space by nor-\nmalised eigenvalue counting functions associated to finite-volume re strictions of the\noperator. For ergodic random and almost-periodic operators in Eu clidean space this\napproach has been implemented rigorously in [Pas71, Shu79], and dev eloped further\nin a number of papers, among them [KM82], [Mat93] and [HLMW01]. All t hese\noperators were stationary and ergodic with respect to a commuta tive group of trans-\nlations. For graphs and manifolds beyond Euclidean space the releva nt group is in\ngeneral no longer abelian. The first result establishing the approxim ability of the\nIDS of a periodic Schr¨ odinger operator on a manifold was [AS93]. An important\nassumption on the underlying geometry is amenability. Analogous res ults on tran-\nsitive graphs have been established e.g. in [MY02] and [MSY03]. For Sc hr¨ odinger\noperators with a random potential on a manifold with an amenable cov ering group\nthe existence of the IDS was established in [PV02], and for Laplace-B eltrami oper-\nators with random metrics in [LPV04]. For analogous results for discr ete operators\non amenable graphs see e.g. [Ves05] and [LV08]. A key ingredient of t he proofs of the\nabove results is the amenable ergodic theorem of [Lin01]. More recen ty, the question\nof approximation of the IDS uniformly with respect to the energy va riable has been\npursued, see for instance [LMV08] and the references therein.\nIndependently of the approximability by finite volume eigenvalue coun ting func-\ntions it is possible to give an abstract definition of the IDS by an avera ged trace per\nunit volume formula, see [Shu79, BLT85, Len99, LPV07]. In the ame nable setting,\nboth definitions of the IDS coincide.\nFor a certain class of metric graphs the approximability of the IDS ha s been\nestablished before. In [HV07, GLV07, GLV08] this has been carried out for random\nmetricgraphswithan Zdstructure. Astepoftheproofwhichisspecifictothesetting\nof quantum graphs concerns the influence of finite rank perturba tions on eigenvalue\ncounting functions. When one considers Laplacians on manifolds, on e would rather\nuse the principle of not feeling the boundary of heat kernels, cf. e.g . [AS93, PV02,\nLPV04], to derive the analogous step of the proof.CONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 3\nNext we discuss the literature on Wegner estimates and on the regu larity of the\nIDS. Wegner gave in [Weg81] convincing arguments for the Lipschitz continuity of\nthe IDS of the discrete Anderson model on ℓ2(Zd). The proof is based on an esti-\nmate for the expected number of eigenvalues in a finite energy inter val of a restricted\nbox Hamiltonian. A rigorous proof of the latter estimate was given in [ Kir96] (for\nthe analogous alloy-type model on L2(Rd)). However, the bound of [Kir96] was not\nsufficient to establish the Lipschitz continuity of the IDS. In [CHN01 ] tools to prove\nH¨ older continuity were supplied, see also [HKN+06]. They concern bounds on the\nspectral shift function. Up to now the most widely applicable result c oncerning\nthe Lipschitz-continuity of the IDS is given in [CHK07]. An alternative a pproach\nto derive Lipschitz continuity of the IDS goes via spectral averagin g of resolvents,\nsee [KS87, CH94]. However, this method requires more assumptions on the under-\nlying model.\nWegner’s estimate and all references mentioned so far concern th e case where\nthe random variables couple to a perturbation which is a non-negativ e operator. If\nthis is not the case, additional ideas are necessary to obtain the de sired bounds,\nsee [Klo95, Ves02, HK02, KV06, Ves08]. In our situation, where the perturbation\nconcerns the metric of the underlying space, the dependence on t he random variables\nis not monotone. This is also the case for random metrics on manifolds studied\nin [LPPV08]. To deal with non-monotonicity, the proof of the Wegner estimate\n(Theorem 2.9) takes up an idea developed in [LPPV08], which is not unre lated\nto [Klo95]. The relevant formula used in the proof is (5.2). We need also a partial\nintegration formula whose usefulness was first seen in [HK02].\nIn the context of quantum graphs it is not necessary to rely on sop histicated\nestimates on the spectral shift function. It is sufficient to adapt a finite rank pertur-\nbation bound, which was used in [KV02] for the analysis of one-dimens ional random\nSchr¨ odingeroperators. Theseestimatesarecloselyrelatedtot hefiniterankestimates\nmentioned earlier in the context of the approximability of the IDS. Fo r Schr¨ odinger\noperators on metric graphs where the randomness enters via the potential, Wegner\nestimates have been proved in [HV07, GV08, GHV08]. In the recent p reprint [KP09]\na Wegner estimate for a model with Zd-structure and random edge lengths has been\nestablished. The proof is based on different methods than we use in t he present\npaper.\nNext we want to explain an application of Wegner estimates apart fro m the con-\ntinuity of the IDS. It concerns the phenomenon of localisation of wa ves in random\nmedia. More precisely, for certain types of random Schr¨ odinger o perators on ℓ2(Zd)\nand onL2(Rd) it is well known that in certain energy intervals near spectral boun d-\naries the spectrum is pure point. There are two basic methods to es tablish this fact\n(apart form the one-dimensional situation where specific methods apply). The first\none is called multiscale analysis and was invented in [FS83]. The second a pproach\nfrom [AM93] is called fractional moment method or Aizenman-Molchan ov method.\nA certain step of the localisation proof via multiscale analysis concern s the control of\nspectral resonances of finite box Hamiltonians. A possibility to achie ve this control is\nthe use of a Wegner estimate. In fact, the Wegner estimates need ed for this purpose\nare much weaker than those necessary to establish regularity of t he IDS. This has\nbeen discussed in the context of random quantum graphs in Section 3.2 of [GHV08].4 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\nRecently localisation hasbeen proven for several types ofrandom quantum graphs.\nIn[EHS07,KP08,KP09]thishasbeendoneformodelswith Zd-structure, while[HP06]\nconsiders operators on tree-graphs. On the other hand, deloca lisation, i.e. existence\nof absolutely continuous spectrum, for quantum graph models on t rees has been\nshown in [ASW06a]. This result should be seen in the context of earlier , similar\nresults for combinatorial tree graphs [Kle96, Kle98, ASW06a, ASW0 6b, FHS06].\nNowletusdiscusssomeopenquestionsconcerningrandomquantum graphmodels.\nAs for models on ℓ2(Zd), proofs of localisation require that the random variables\nentering the operators should have a regular distribution. In part icular, if the law of\nthe variables is a Bernoulli measure, no known proof of localisation ap plies. This is\ndifferent for random Schr¨ odinger operators L2(Rd). Using a quantitative version of\nthe unique continuation principle for solutions of Schr¨ odinger equa tions, localisation\nwas established in [BK05] for certain models with Bernoulli disorder. T he proof\ndoes not carry over to the analogous model on ℓ2(Zd), since there is no appropriate\nversion of the unique continuation principle available. For random qua ntum graphs\nthesituationisevenworse, sincetheyexhibitingreatgeneralitycom pactlysupported\neigenfunctions, even if the underlying graph is Zd.\nLikeforrandom, ergodicSchr¨ odinger operatorson ℓ2(Zd)andonL2(Rd)thereisno\nproof of delocalisation for random quantum graphs with Zdstructure. In the above\nmentioned papers on delocalisation it was essential that the underly ing graph is a\ntree. An even harder question concerns the mobility edge. Based o n physical reason-\ning one expects that localised point spectrum and delocalised absolut ely continuous\nspectrumshouldbeseparatedindisjointintervalsbymobilityedges. Inthecontextof\nrandom operators where the disorder enters via the geometry th is leads to an intrigu-\ning question pointed out already in [CCF+86]. If one considers a graphover Zdwhich\nis diluted by a percolation process, the Laplacian on the resulting com binatorial or\nmetric graph has a discontinuous IDS. In fact, the set of jumps ca n be characterised\nrather explicitely and is dense in the spectrum [CCF+86, Ves05, GLV08]. Now the\nquestion is, where the eigenvalues of these strongly localised state s repell in some\nmanner absolutely continuous spectrum (if it exists at all).\nFor the interested reader we provide here references to textbo ok accounts of the\nissuesdiscussed above. Theyconcernthemoreclassical modelson L2(Rd)andℓ2(Zd),\nrather than quantum graphs. In [Ves07] one can find a detailed disc ussion and proofs\nof the approximability of the IDS by its finite volume analogues and of W egner\nestimates. The survey article [KM07] is devoted to the IDS in gener al, while the\nmultiscale proof of localisation is exposed in the monograph [Sto01]. T he theory of\nrandom Schr¨ odinger operators is presented from a broader per spective in the books\n[CL90, PF92] and in the summer school notes [Kir89, Kir07].\nAcknowledgements. The second author is grateful for the kind invitation to the\nHumboldt University of Berlin which was supported by the SFB 647. NP and OP\nalso acknowledge the financial support of the Technical University Chemnitz.\n2.Basic notions, model and results\nIn the following subsections, we fix basic notions (metric graphs, La placians and\nSchr¨ odinger operators with vertex conditions), introduce the r andom length modelCONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 5\nand state our main results. For general treatments and further references on metric\ngraphs, we refer to [EKK+08].\n2.1.Metric graphs. Since our randommodel concerns aperturbationof the metric\nstructure of a graph, we carefully distinguish between combinatorial ,topological and\nmetric graphs . Acombinatorial graph G= (V,E,∂) is given by a countable vertex\nsetV, a countable set Eof edge labels and a map ∂(e) ={v1,v2}from the edge\nlabels to (unordered) pairs of vertices. If v1=v2, we callealoop. Note that this\ndefinition allows multiple edges, but we only consider locally finite combina torial\ngraphs, i.e., every vertex has only finitely many adjacent edges. A t opological graph\nXis a topologicalmodel of a combinatorial graphtogether with a choic e of directions\non the edges:\nDefinition 2.1. A(directed) topological graph is a CW-complex Xcontaining only\n(countably many) 0- and 1-cells. The set V=V(X)⊂Xof 0-cells is called the\nset of vertices . The 1-cells of Xare called (topological) edges and are labeled by the\nelements of E=E(X) (the(combinatorial) edges ), i.e., for every edge e∈E, there is\na continuous map Φ e: [0,1]−→Xwhose image is the corresponding (closed) 1-cell,\nand Φ e: (0,1)−→Φe((0,1))⊂Xis a homeomorphism. A 1-cell is called a loopif\nΦe(0) = Φ e(1). The map ∂= (∂−,∂+):E−→V×Vdescribes the direction of the\nedges and is defined by\n∂−e:= Φe(0)∈V, ∂ +e:= Φe(1)∈V.\nForv∈Vwe define\nE±\nv=E±\nv(X) :={e∈E|∂±e=v}.\nThe set of all adjacent edges is defined as the disjointunion1\nEv=Ev(X) :=E+\nv(X)·∪E−\nv(X).\nThedegreeof a vertex v∈VinXis defined as\ndegv= degX(v) :=|Ev|=|E+\nv|+|E−\nv|.\nAtopological subgraph Λ is a CW-subcomplex of X, and therefore Λ is itself a topo-\nlogical graph with (possible empty) boundary ∂Λ := Λ∩Λc⊂V(X).\nSinceatopologicalgraphisatopologicalspace, wecanintroduceth espaceC(X)of\nC-valued continuous functions and the associated notion of measur ability. A metric\ngraph is a topological graph where we assign a length to every edge.\nDefinition 2.2. A(directed) metric graph (X,ℓ) is a topological graph Xtogether\nwith alength function ℓ:E(X)−→(0,∞). The length function induces an identifi-\ncation of the interval Ie:= [0,ℓ(e)] with the edge Φ e([0,1]) (up to the end-points of\nthe corresponding 1-cell, which may be identified in Xifeis a loop) via the map\nΨe:Ie−→X,Ψe(x) = Φe/parenleftBigx\nℓ(e)/parenrightBig\n.\n1The disjoint union is necessary in order to obtain twodifferent labels in Ev(X) for a loop.6 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\nNotethat every topological graph Xcanbe canonically regarded asa metric graph\nwhereall edges have length one . The corresponding length function /BDE(X)is denoted\nbyℓ0. In our random model, we will consider a fixed topological graph Xwith a\nrandom perturbation ℓωof this length function ℓ0.\nTo simplify matters, we canonically identify a metric graph ( X,ℓ) with the disjoint\nunionXℓof the intervals Iefor alle∈Esubject to appropriate identifications of the\nend-points of these intervals (according to the combinatorial str ucture of the graph),\nnamely\nXℓ:=·/uniondisplay\ne∈EIe/∼.\nThe coordinate maps {Ψe}ecan be glued together to a map\nΨℓ:Xℓ−→X. (2.1)\nRemark2.3.A metric graph is canonnically equpped with a metric and a measure.\nGiven the information about the lenght of edges, each path in Xℓhas a well defined\nlenght. Thedistancebetweentwo arbitrarypoints x,y∈Xℓisdefinedastheinfimum\nof the lenghts of paths joining the two points. The measure on Xℓis defined in the\nfollowing way. For each measurable Λ ⊂Xthe sets Λ ∩ψe(Ie) are measurable\nas well, and are assigned the Lebesgue measure of the preimage ψ−1\ne(Λ∩ψe(Ie)).\nConsequently, we define the volume of Λ by\nvol(Λ,ℓ) :=/summationdisplay\ne∈Eλ/parenleftbig\nψ−1\ne(Λ∩ψe(Ie))/parenrightbig\n(2.2)\nUsing the identification (2.1), we define the function space L2(X,ℓ) as\nL2(X,ℓ) :=/circleplusdisplay\ne∈EL2(Ie), f={fe}ewithfe∈L2(Ie) and\n/bardblf/bardbl2\nL2(X,ℓ)=/summationdisplay\ne∈E/integraldisplay\nIe|fe(x)|2dx.\n2.2.Operators and vertex conditions. For a given metric graph ( X,ℓ), we in-\ntroduce the operator\n(Df)e(x) = (Dℓf)e(x) =dfe\ndx(x),\nwhere the derivative is taken in the interval Ie= [0,ℓ(e)]. Note that both the\nnorm in L2(X,ℓ) andD=Dℓdepend on the length function. This observation is\nparticularly important in our random length model below, where we pe rturb the\ncanonical length function ℓ0= /BDE(X)and therefore have (a priori) different spaces\non which a function flives. Our point of view is that fis a function on the fixed\nunderlying topological graph X, and that the metric spaces are canonically identified\nvia the maps Ψ−1\nℓ0◦Ψℓ: (X,ℓ)−→(X,ℓ0). One easily checks that\n/bardblf/bardbl2\nL2(X,ℓ)=/summationdisplay\ne∈Eℓ(e)/integraldisplay\n(0,1)|fe(x)|2dx, (2.3a)\n(Dℓf)e(x) =1\nℓ(e)(Dℓ0f)e/parenleftBig1\nℓ(e)x/parenrightBig\n, (2.3b)CONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 7\nwherefeandDℓ0fon the right side are considered as functions on [0 ,1] via the\nidentification Ψ−1\nℓ0◦Ψℓ.\nNext we introduce general vertex conditions for Laplacians ∆(X,ℓ)=−Dℓ2and\nSchr¨ odinger operators H(X,ℓ)= ∆(X,ℓ)+qwith real-valued potentials q∈L∞(X).\nThemaximal ordecoupled Sobolev space of order kon (X,ℓ) is defined by\nHk\nmax(X,ℓ) :=/circleplusdisplay\ne∈EHk(Ie)\n/bardblf/bardbl2\nHkmax(X,ℓ):=/summationdisplay\ne∈E/bardblfe/bardbl2\nHk(Ie).\nNote thatDℓ:Hk+1\nmax(X,ℓ)−→Hk\nmax(X,ℓ) is a bounded operator. We introduce the\nfollowing two different evaluation maps H1\nmax(X,ℓ)−→/circleplustext\nv∈VCEv:\nfe(v) :=/braceleftBigg\nfe(0),ifv=∂−e,\nfe(ℓ(e)),ifv=∂+e,andf− →e(v) :=/braceleftBigg\n−fe(0),ifv=∂−e,\nfe(ℓ(e)),ifv=∂+e,\nandf(v) ={fe(v)}e∈Ev∈CEv,f− →(v) ={f− →e(v)}e∈Ev∈CEv. It follows from\nstandard Sobolev estimates (see e.g. [Kuc04, Lem. 8]) that these e valuation maps\nare bounded by max {(2/ℓmin)1/2,1}, provided the minimal edge length\n0<ℓmin:= inf\ne∈Eℓ(e) (2.4)\nis strictly positive. The second evaluation map is used in connection wit h the deriva-\ntiveDfof a function f∈H2\nmax(X,ℓ). Note that Df− →is independent of the orientation\nof the edge.\nAsingle-vertex condition atv∈Vis given by a Lagrangian subspace L(v) of\nthe Hermitian symplectic vector space ( CEv⊕CEv,ηv) with canonical two-form ηv\ndefined by\nηv((x,x′),(y,y′)) :=/a\\}bracketle{tx′,y/a\\}bracketri}ht −/a\\}bracketle{tx,y′/a\\}bracketri}ht,\nwhere/a\\}bracketle{t·,·/a\\}bracketri}htdenotes the standard unitary inner product in CEv. The set of all La-\ngrangian subspaces of ( CEv⊕CEv,ηv) is denoted by Lvand has a natural manifold\nstructure (see, e.g., [Har00, KS99] for more details on these notio ns). A Lagrangian\nsubspaceL(v) can uniquely be described by the pair ( Q(v),R(v)) whereQ(v) is an\northogonal projection in CEvwith range G(v) := ranQ(v) andR(v) is a symmetric\noperator on G(v) such that\nL(v) :=/braceleftbig\n(x,x′)/vextendsingle/vextendsingle(1−Q(v))x= 0, Q(v)x′=R(v)x/bracerightbig\n(2.5)\n(see e.g. [Kuc04]).\nA field of single-vertex conditions L:={L(v)}v∈Vis called a vertex condition . We\nsay thatLisbounded, if\nCR:= sup\nv∈V/bardblR(v)/bardbl<∞, (2.6)\nwherethenormistheoperatornormon G(v). Foranysuchboundedvertexcondition\nL, a bounded potential qand a metric graph ( X,ℓ) withℓmin>0, we obtain a self-\nadjoint Schr¨ odinger operator H(X,ℓ),L= ∆(X,ℓ),L+q, by choosing the domain\ndomH(X,ℓ),L:={f∈H2\nmax(X,ℓ)|(f(v),Df− →(v))∈L(v) for allv∈V}.8 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\nOf particular interest are the following vertex conditions with vanish ing vertex oper-\natorR(v) = 0 for all v∈V:Dirichlet vertex conditions (where L(v) ={0}⊕CEvor\nG(v) ={0}),Kirchhoff (also known as free) vertex conditions (where ( x,x′)∈L(v)\nif all components of xare equal and the sum of all components of x′add up to\nzero, or equivalently G(v) =C(1,...,1)) andNeumann vertex conditions (where\nL(v) =CEv⊕{0}or equivalently G(v) =CEv).\n2.3.Random length model. The underlying geometric structure of a random\nlength model is a random length metric graph. A random length metric graph is\nbased on a fixed topological graph XwithVandEthe sets of vertices and edges\nofX, a probability space (Ω ,P), and ameasurable mapℓ: Ω×E−→(0,∞), which\ndescribes the random dependence of the edge lengths. We also ass ume that there are\nω-independent constants ℓmin,ℓmax>0 such that ℓmin≤ℓω(e)≤ℓmaxfor allω∈Ω\nande∈E. We will use the notation ℓω(e) :=ℓ(ω,e).\nArandom length model associates to such a geometric structure ( X,Ω,P,ℓ) a\nrandom family of Schr¨ odinger operators Hω, by additionally introducing measurable\nmapsL(v): Ω−→Lvfor allv∈V, andq: Ω×X−→R, describing the random\ndependence of the vertex conditions and the potentials of these o perators. We will\nuse the notation Lω:={Lω(v)}v∈Vandqω(x) =q(ω,x). We assume that we have\nconstantsCR,Cpot>0 such that\n/bardblqω/bardbl∞≤Cpotand/bardblRω(v)/bardbl ≤CR (2.7)\nfor almost all ω∈Ω and allv∈V, whereRω(v) is the vertex operator associated to\nLω(v). From (2.7) and the lower length bound (2.4) it follows that the Schr ¨ odinger\noperatorsHω:= ∆ω+qωare self-adjoint and bounded from below by some constant\nλ0∈Runiformly in ω∈Ω (see Lemma 4.1). We call the tuple ( X,Ω,P,ℓ,L,q) a\nrandom length model with associated Laplacians and Schr¨ odinger operators ∆ωand\nHωand underlying random metric graphs ( X,ℓω).\n2.4.Approximation of the IDS via exhaustions. Let us describe the setting,\nfor which our first main result holds.\nAssumption 2.4. Let (X,Ω,P,ℓ,L,q) be a random length model with the following\nproperties:\n(i) The topological graph Xis non-compact and connected with underlying\n(undirected) combinatorial graph G= (V,E,∂). There is a subgroup Γ ⊂\nAut(G), acting freely on Vwith only finitely many orbits. Then Γ acts also\ncanonically on X(but does not necessarily respect the directions) by\nγΦe(x) =/braceleftBigg\nΦγe(x) if∂±(γe) =γ(∂±e),\nΦγe(1−x) if∂±(γe) =γ(∂∓e).\nThis action carries over to Γ-actions on the metric graphs ( X,ℓ0) and (X,ℓω)\nvia the identification (2.1). Note that Γ acts even isometrically on the equi-\nlateral graph ( X,ℓ0) withℓ0= /BDE. We can think of ( X,ℓ0) as acovering of\nthe compact topological graph ( X/Γ,ℓ0).\n(ii) We also assume that Γ acts ergodically on (Ω,P) by measure preserving\ntransformations with the following consistencies between the two Γ -actions\nonXand Ω:CONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 9\nMetric consistency: We assume that\nℓγω(e) =ℓω(γe) (2.8a)\nfor allγ∈Γ,ω∈Ω ande∈E. This implies that for every γ∈Γ, the\nmap\nγ: (X,ℓω)−→(X,ℓγω)\nis an isometry between two (different) metric graphs. Moreover, t he\ninduced operators\nU(ω,γ):L2(X,ℓγ−1ω))−→L2(X,ℓω)\nare unitary.\nOperator consistency: The transformation behaviour of qωandLωis\nsuch that we have for all ω∈Ω,γ∈Γ,\nHω=U(ω.γ)Hγ−1ωU∗\n(ω,γ). (2.8b)\nSuch a random length model ( X,Ω,P,ℓ,L,q) is called a random length cov-\nering model with associated operators Hωand covering group Γ.\nRemark2.5.The simplest random length covering model is given when the probabil-\nity space Ω consists of only one element with probability 1. In this case , we have only\none length function ℓ=ℓω, one vertex condition L=Lω, and one potential q=qω.\nThe corresponding family of operators consists then of a single ope ratorH=Hω.\nMoreover, the metric consistency means that Γ acts isometrically o n (X,ℓ), and the\noperator consistency is nothing but the periodicity of H, i.e., the property that H\ncommutes with the induced unitary Γ-action on L2(X,ℓ).\nNext, weintroducesomemorenotation. Let F0bearelativelycompacttopological\nfundamental domain of the Γ-action on ( X,ℓ0) such that its closure F=F0is a\ntopological subgraph. (An example of such a topological fundamen tal domain is\ngiven in Figure 2 (a) below.) There is a canonical spectral distribution function\nN(λ), associated to the family Hω, given by the trace formula\nN(λ) :=1\nE(vol(F,ℓ•))E(tr•[ /BDFP•((−∞,λ])]), (2.9)\nwhereE(·) denotes the expectation in (Ω ,P), trωis the trace on the Hilbert space\nL2(X,ℓω), andPω(I)denotesthespectralprojectionassociatedto Hωandtheinterval\nI⊂R. Moreover, the volume vol( F,ℓ•) is defined in (2.2). The function Nis called\nthe(abstract) integrated density of states with abbreviation IDS.\nIn the case of an amenable group Γ the abstract IDS can also be obt ained via\nappropriate exhaustions. This is the statement of Theorem 2.6 belo w. A discrete\ngroup Γ is called amenable , if there exist a sequence In⊂Γ of finite, non-empty\nsubsets with\nlim\nn→∞|In∆Inγ|\n|In|= 0,for allγ∈Γ. (2.10)\nA sequence Insatisfying (2.10) is called a Følner sequence .10 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\nFor every non-empty finite subset I⊂Γ, we define Λ( I) :=/uniontext\nγ∈IγF. A sequence\nIn⊂Γ of finite subsets is Følner if and only if the associated sequence Λ n= Λ(In)\nof topological subgraphs satisfies the van Hove condition\nlim\nn→∞|∂Λ(In)|\nvol(Λ(In),ℓ0)= 0. (2.11)\nTheproofofthisfactisanalogoustotheproofof[PV02, Lemma 2.4] intheRiemann-\nian manifold case. Note that (2.11) still holds if we replace ∂Λ(In) by∂rΛ(In) for any\nr≥1, where∂rΛ denotes the thickened combinatorial boundary {v∈V|d(v,∂Λ)≤\nr}andddenotes the combinatorial distance which agrees (on the set of vertices)\nwith the distance function of the unilateral metric graph ( X,ℓ0).\nA Følner sequence Inis called tempered, if we additionally have\nsup\nn∈N|/uniontext\nk≤nIn+1I−1\nk|\n|In+1|<∞. (2.12)\nTempered Følner sequences are needed for an ergodic theorem of Lindenstrauss\n[Lin01]. This ergodic theorem plays a crucial role in the proof of Theor em 2.6 pre-\nsented below. However, the additional property (2.12) is not very restrictive since it\nwas also shown in [Lin01] that every Følner sequence Inhas a tempered subsequence\nInj.\nForany compact topological subgraph Λ of X, we denote theoperator with Dirich-\nlet vertex conditions on the boundary vertices ∂Λ and with the original vertex con-\nditionsLω(v) on all inner vertices v∈V(Λ)\\∂Λ byHΛ,D\nω. The label D refers to\nthe Dirichlet conditions on ∂Λ. For a precise definition of the Dirichlet operator\nvia quadratic forms, we refer to Section 4. The spectral project ion corresponding to\nHΛ,D\nωis denoted by PΛ,D\nω. It is well-known that compactness of Λ implies that the\noperatorHΛ,D\nωhas purely discrete spectrum. The normalised eigenvalue counting\nfunction associated to the operator HΛ,D\nωis defined as\nNΛ\nω(λ) =1\nvol(Λ,ℓω)trω[PΛ,D\nω((−∞,λ])].\nThe function NΛ\nωis the distribution function of a (unique) pure point measure which\nwe denote by µΛ\nω.\nIf Λ = Λ(In) is associated to a Følner sequence In⊂Γ, we use the abbreviations\nHn,D\nω:=HΛ(In),D\nωfor the Schr¨ odinger operator with Dirichlet conditions on ∂Λ(In),\nNn\nω:=NΛ(In)\nωfor the normalised eigenvalue counting function and µn\nω:=µΛ(In)\nωfor\nthe corresponding pure point measure on Λ( In). We can now state our first main\nresult:\nTheorem 2.6. Let(X,Ω,P,ℓ,L,q)be a random length covering model as described\nin Assumption 2.4 with amenable covering group Γ. LetNbe the IDS of the operator\nfamilyHω. Then there exist a subset Ω0⊂Ωof fullP-measure such that we have,\nfor every tempered Følner sequence In⊂Γ,\nlim\nn→∞Nn\nω(λ) =N(λ)\nfor allω∈Ω0and all points λ∈Rat whichNis continuous.\nThe proof is given in Section 4.CONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 11\nRemark2.7.The proof of Theorem 2.6 yields even more. Let µdenote the measure\nassociated to the distribution function N. Then we have\nlim\nj→∞µn\nω(f) =µ(f) (2.13)\nfor allω∈Ω0and all functions fof the form f(x) =g(x)(x+1)−1with a function\ngcontinuous on [0 ,∞) and with limit at infinity. (The behaviour of g(x) forx<0\nis of no importance since the spectral measures of all operators u nder consideration\nare supported on R+= [0,∞).)\n2.5.Wegner estimate. In this subsection, we state a linear Wegner estimate for\nLaplace operators of a random length model with independently dist ributed edge\nlengths and fixed Kirchhoff vertex conditions. This Wegner estimate is linear both\nin the number of edges and in the length of the considered energy int erval. As\nmentioned in the introduction, a similar result for the case Zdwas proved recently\nby different methods in [KP09]. In contrast to the previous subsect ion, we do not\nrequire periodicity of the graph Xassociated to a group action. More precisely, we\nassume the following:\nAssumption 2.8. Let (X,Ω,P,ℓ,L,q) be a random length model with the following\nproperties:\n(i) We have q≡0, i.e., the random family of operators are just the Laplacians\n(Hω= ∆ω)andwe have norandomness inthevertex conditionby fixing Lto\nbe Kirchhoff in all vertices. Thus it suffices to look at the tuple ( X,Ω,P,ℓ).\n(ii) We have a uniform upper bound dmax<∞on the vertex degrees deg v,\nv∈V(X).\n(iii) Since the only randomness occurs in the edge lengths satisfying\n0<ℓmin≤ℓω(e)≤ℓmaxfor allω∈Ω ande∈E(X),\nwe think of the probability space Ω as a Cartesian product/producttext\ne∈E[ℓmin,ℓmax]\nwith projections Ω ∋ω/ma√sto→ωe=ℓω(e)∈[ℓmin,ℓmax]. The measure Pis\nassumed to be a product/circlemultiplytext\ne∈EPeof probability measures Pe. Moreover,\nfor everye∈E, we assume that Peis absolutely continuous with respect\nto the Lebesgue measure on [ ℓmin,ℓmax] with density functions he∈C1(R)\nsatisfying\n/bardblhe/bardbl∞,/bardblh′\ne/bardbl∞≤Ch, (2.14)\nfor a constant Ch>0 independent of e∈E.\nRecall that tr ωis the trace in the Hilbert space L2(Λ,ℓω). In the next theorem\nPΛ,D\nωdenotes the spectral projection of the Laplacian ∆Λ,D\nωon (Λ,ℓω) with Kirchhoff\nvertex conditions on all interior vertices and Dirichlet boundary con ditions on∂Λ.\nUnder these assumptions we have:\nTheorem 2.9. Let(X,Ω,P,ℓ)be a random length model satisfying Assumption 2.8.\nLetu>1andJu= [1/u,u]. Then there exists a constant C >0such that\nE(trPΛ,D\n•(I))≤C·λ(I)·|E(Λ)|\nfor all compact subgraphs Λ⊂Xand all compact intervals I⊂Ju, whereλ(I)\ndenotes the Lebesgue-measure of I, and where |E(Λ)|denotes the number of edges in12 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\nΛ. The constant C >0depends only the constants u,dmax,ℓmin,ℓmaxand the bound\nCh>0associated to the densities he(see(2.14)).\nThe proof will be given in Section 5. We finish this section with the followin g\ncorollary. Recall that the periodic situation is a special case of a ran dom length\ncovering model (see Remark 2.5):\nCorollary 2.10. Let(X,Ω,P,ℓ)be a random length covering model, satisfying both\nAssumptions 2.4 and 2.8, with amenable covering group Γ. Then the IDS Nof\nthe Laplacians ∆ωis a continuous function on Rand even Lipschitz continuous on\n(0,∞).\nProof.The Lipschitz continuity of Non (0,∞) follows immediately from Theo-\nrems 2.6 and 2.9. It remains to prove continuity of Non (−∞,0]. Note that our\nmodel is a special situation of the general ergodic groupoid setting given in [LPV07].\nThus,Nis the distribution function of a spectral measure of the direct inte gral op-\nerator/integraltext⊕\nΩ∆ωdP(ω). Since ∆ω≥0 for allω,N(λ) vanishes for all λ<0. Moreover,\nifNwould have a jump at λ= 0, then ker∆ωwould be non-trivial for almost all\nω∈Ω. But ∆ωf= 0 implies\n0 =/a\\}bracketle{tf,∆ωf/a\\}bracketri}ht=/integraldisplay\nX/vextendsingle/vextendsingle/vextendsingledf\ndx(x)/vextendsingle/vextendsingle/vextendsingle2\ndx\nsince ∆ωhas Kirchhoff vertex conditions. Thus fis a constant function. Now Xis\nconnected as well as non-compact, which implies that vol( X,ℓω) =∞by the lower\nboundℓminon the lengths of the edges. Hence constant functions are not in L2. This\ngives a contradiction. /square\nOur result on Lipschitz continuity of Non (0,∞) is optimal in the following sense:\nRemark 2.11.It is well-known that the IDS of the free Laplacian ∆RonRis pro-\nportional to the square root of the energy. Note that this does n ot change when\nadding Kirchhoff boundary conditions at arbitrary points. Therefo re, every model\nsatisfying Assumptions 2.4 and 2.8 for a metric graph isometric to Rhas in fact the\nabove IDS. Therefore, we cannot expect Lipschitz continuity of t he IDS at zero for\nrandom length models without further assumptions.\n3.Kagome lattice as an example of a planar graph\nIn this section, we illustrate the concepts of the previous section f or an explicit\nexample. We introduce a particular regular tessellation of the Euclide an plane ad-\nmitting finitely supported eigenfunctions of the combinatorial Lapla cian. We discuss\nin detail the discontinuities of the IDS of the combinatorial Laplacian and of the\nKirchhoff Laplacian of the induced equilateral metric graph. On the o ther hand, ap-\nplying Corollary2.10, we see that theIDSofa randomfamilyof Kirchho ff Laplacians\nfor independent distributed edge lengths is continuous. Thus, ran domness leads to\nan improvement of the regularity of the IDS in this example.\nWe consider the infinite planar topological graph X⊂Cas illustrated in Figure 1.\nThis graph is sometimes called Kagome lattice . Every vertex of Xhas degree four\nand belongs to a uniquely determined upside triangle. Introducing w1= 1 and\nw2= eπi/3, we can identify the lower left vertex of a particular upside triangle w ithCONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 13\nthe origin in Cand its other two vertices with w1,w2∈C. Consequently, the vertex\nset ofXis given explicitly as the disjoint union of the following three sets:\nV(X) = (2Zw1+2Zw2)·∪(w1+2Zw1+2Zw2)·∪(w2+2Zw1+2Zw2).\nA pairv1,v2∈V=V(X) of vertices is connected by a straight edge if and only if\n|v2−v1|= 1. We write v1∼v2for adjacent vertices. The above realisation of the\nplanar graph X⊂Cis an isometric embedding of the metric graph ( X,ℓ0).\nPSfrag replacements w1w2\nFigure 1. Illustration of the planar graph X(Kagome lattice).\nThe group Z2acts onXvia the maps Tγ(x) := 2γ1w1+2γ2w2+x. A topological\nfundamental domain F0ofXis thickened in Figure 2 (a). The set of vertices of the\ntopological subgraph F=F0(obtained by taking the closure of F0considered as a\nsubset of the metric space ( X,ℓ)) is given by {a,b,c,a′,b′,b′′,c′′}.\nNote that we have to distinguish carefully between a topological and a combina-\ntorial fundamental domain. Let Gdenote the underlying combinatorial graph with\nsetVof vertices and Eof combinatorial edges. The maps Tγact also on the set of\nverticesVand a combinatorial fundamental domain is given by Q={a,b,c}. We\ndenote the translates Tγ(Q) ofQbyQγ.PSfrag replacements\n2w12w2\nab c\na′b′\nb′′c′′b′′′\n(a) (b)v1v2v3\nv4v5\nw1\nw2Hγ0\nFigure 2. (a) The periodic graph with thickened topological fun-\ndamental domain F0and combinatorial fundamental domain Q=\n{a,b,c}(b)Ifγ0isvertically extremal for F, allwhiteencircled vertices\nare zeroes of F.14 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\n3.1.Spectrum and IDS of the combinatorial Laplacian. We first observe that\nGadmits finitely supported eigenfunctions of the combinatorial Laplacian ∆comb:\nChoose an arbitrary hexagon H⊂Xwith vertices {u0,u1,...,u 5}. Then there\nexists a centre w0∈CofHsuch that we have\n{u0,u1,...,u 5}={w0+ekπi/3|k= 0,1,...,5}.\nThe following function FH:V−→ {0,±1}on the vertices\nFH(v) :=/braceleftBigg\n0,ifv∈V\\{u0,...,u 5},\n(−1)k,ifv=w0+ekπi/3,(3.1)\nsatisfies\n∆combFH(v) =1\ndeg(v)/summationdisplay\nw∼v(FH(v)−FH(w)) =3\n2FH(v).\nThus, the vertices of every hexagon H⊂Xare the support of a combinatorial\neigenfunction FH:V−→R. The functions FHare the only finitely supported eigen-\nfunctions up to linear combinations:\nProposition 3.1. (a) LetF:V−→Rbe a combinatorial eigenfunction on Xwith\nfinite support suppF⊂V. Then\n∆combF=3\n2F\nandFis a linear combination of finitely many eigenfunctions FHof the above\ntype(3.1).\n(b) LetHi(i= 1,...,k)be a collection of distinct, albeit not necessarily disjoin t,\nhexagons, and Fi:=FHithe associatedcompactlysupported eigenfunctions. Then\nthe setF1,...,F kis linearly independent.\n(c) Ifg∈ℓ2(V)satisfies∆combg=µg, thenµ= 3/2.\n(d) The space of ℓ2(V)-eigenfunctions to the eigenvalue 3/2is spanned by compactly\nsupported eigenfunctions.\nProof.To prove (a), assume that F:V−→Ris a finitely supported eigenfunction.\nLetQ={a,b,c}be a combinatorial fundamental domain of Z2, as illustrated in\nFigure 2 (a) and Qγ:=Tγ(Q). LetHγbe the uniquely defined hexagon containing\nthe three vertices Qγ. Moreover, we define\nA0:={γ∈Z2|suppF∩Qγ/\\e}atio\\slash=∅}.\nLetε1= (1,0) andε2= (0,1). We say that γ0= (γ01,γ02)∈A0isvertically extremal\nforF, if the second coordinate γ02is maximal amongst all γ∈A0and ifγ0−ε1/∈A0.\nThis means that Fvanishes in the left neighbour of Qγ0and in all vertices vertically\naboveQγ0. Hence,γ0in Figure 2 (b) is vertically extremal if Fvanishes in all white\nencircled vertices and does not vanish in at least one of the black ver tices. Obviously,\nA0has always vertically extremal elements. Choosing such a γ0∈A0, we will show\nbelow that Fis an eigenfunction with eigenvalue 3 /2 and that the following facts\nhold:\n(i)γ0+ε1belongs toA0,\n(ii)γ0−ε2orγ0−ε2−ε1belong toA0,CONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 15\n(iii) adding a suitable multiple of FHγ0toF, we obtain a new eigenfunction F1\nand a setA1:={γ∈Z2|suppF1∩Qγ/\\e}atio\\slash=∅}satisfying\nγ0/∈A1, A1\\A0⊂ {γ0−ε2,γ0+ε1−ε2}.\nTo see this, let γ0∈A0be vertically extremal and v1,...,v 5,w1,w2be chosen as in\nFigure 2 (b). The eigenvalue equation at the vertices v4andv5, in whichFvanishes,\nimply that we have F(v1) =−F(v2) =F(v3)/\\e}atio\\slash= 0. Applying the eigenvalue equation\nagain, now at v2, yields that the eigenvalue of Fmust be 3/2.\nIfγ0+ε1/∈A0,Fwould vanish in w1and all its neighbours, except for v3. This\nwould contradict to the eigenvalue equation at w1and (i) is proven. Similarly, if\nγ0−ε2,γ0−ε2−ε1/∈A0, we would obtain a contradiction to the eigenvalue equation\nat the vertex w2. This proves (ii).\nBy addingF(v1)FHγ0toF, we obtain a new eigenfunction F1(again to the eigen-\nvalue 3/2) which vanishes at all vertices of Qγ0={v1,v2,v3}. Thus we have γ0/∈A1.\nButFandF1differ only in the vertices Qγ0,Qγ0+ε1,Qγ0−ε2andQγ0+ε1−ε2, estab-\nlishing property (iii).\nThe above procedure can be iteratively (from left to right) applied t o the hexagons\nin the top row of A0: Step (iii) can be applied to the function F1and a vertically\nextremal element of A1. After a finite number nof steps the top row of hexagons in\nA0is no longer in the support of the function Fn. (Note that property (i) implies\nthat when removing the penultimate hexagon form the right, one ha s simultaneously\nremoved the rightermost one, too.) Again, this procedure can be it erated removing\nsuccessively rows of hexagons. This time property (ii) guarantees that the procedure\nstops after a finite number Nof steps with FN≡0. We have proven statement (a).\nNow we turn to the proof of (b). Since the graph is connected ther e exists a vertex\nvinA:=∪k\ni=1Hiwhich is adjacent to some vertex outside A. Thenvis contained in\nprecisely one hexagon Hi0. (In the full graph each vertex is in two hexagons.) Thus\nthe condition\nk/summationdisplay\ni=1αiFi= 0αi∈C (3.2)\nevaluated at the vertex vimpliesαi0= 0. This shows that all coefficients αiin (3.2)\ncorresponding to hexagons Hilying at the boundary of Avanish. This leads to an\nequation analogous to (3.2) where the indices in the sum run over a st rict subset of\n{1,...,k}. Now one iterates the pocedure and shows that actually all coefficie nts\nα1,...,α kin (3.2) are zero. We have shown linear independence of F1,...,F k.\nTo prove (c) we recall that the IDS ∆ combis a spectral measure (see e.g. [LPV07,\nProp. 5.2]). Thus the IDS jumps at the value µ. This in turn implies by [Ves05,\nProp. 5.2] that there is a compactly supported ˜ gsatisfying the eigenvalue equation.\nNow (a) implies µ= 3/2.\nStatement (d) follows from [LV08, Thm. 2.2], cf. also the proof of Pr oposition 3.3.\n/square\nWe are primarily interested in ℓ2-eigenfunctions of ∆comb, since their eigenvalues\ncoincide with the discontinuities of the corresponding IDS. For combinatorial cov-\nering graphs with amenable covering group Γ, every ℓ2-eigenfunction Fimplies the16 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\nexistence of a finitely supported eigenfunction to the same eigenvalue which is im-\nplied, e.g., by [Ves05, Prop. 5.2] or [LV08, Thm. 2.2]. (Related, but diffe rent results\nhave been obtained before in [MY02]. If the group is even abelian, as is the case\nfor the Kagome lattice, the analogous result was proven even earlie r in [Kuc91].) It\nshould be mentioned here that the situation is very different in the sm ooth cate-\ngory of Riemannian manifolds. There, compactly supported eigenfu nctions cannot\noccur due to the unique continuation principle . In the discrete setting of graphs,\nnon-existence of finitely supported combinatorial eigenfunctions is — at present —\nonly be proved for particular examples or in the case of planar graph s of non-positive\ncombinatorial curvature; see [KLPS06] for more details. Hence, P roposition 3.1 tells\nus thatXdoes not admit combinatorial ℓ2-eigenfunctions associated to eigenvalues\nµ/\\e}atio\\slash= 3/2.\nNext, let us discuss spectral informations which can be obtained wit h the help of\nFloquet theory . Using a general result of Kuchment (see [Kuc91] or [Kuc05, Thm. 8 ])\nfor periodic finite difference operators (applying Floquet theory to such operators)\nwe conclude that the compactly supported eigenfunctions of ∆combassociated to the\neigenvalue 3 /2 are already dense in the whole eigenspace ker(∆comb−3/2). As for\nthe whole spectrum, we derive the following result:\nProposition 3.2. Denote by σac(∆comb)andσp(∆comp)the absolutely continuous\nand point spectrum of ∆combon ourZ2-periodic graph X. Then we have\nσac(∆comb) =/bracketleftBig\n0,3\n2/bracketrightBig\nandσp(∆comb) =/braceleftBig3\n2/bracerightBig\n.\nThe proof follows from standard Floquet theory (for a similar hexag onal graph\nmodel see [KP07]):\nProof.Note that we have the unitary equivalence\n∆comb∼=/integraldisplay⊕\nT2∆θ\ncombdθ,\nwhere ∆θ\ncombis theθ-equivariant Laplacian on Q,θ∈T2:=R2/(2πZ)2. This\noperator is equivalent to the matrix\n∆θ\ncomb∼=1\n4\n4−1−e−iθ2−e−iθ1−e−iθ2\n−1−eiθ24 −1−e−iθ1\n−eiθ1−eiθ2−1−eiθ1 4\n\nusingthebasis F∼=(F(a),F(b),F(c))forafunctionon Qandthefactthat F(Tγv) =\nei/a\\gbracketleftθ,γ/a\\gbracketrightF(v) (equivariance). The characteristic polynomial is\np(µ) =/parenleftBig\nµ−3\n2/parenrightBig/parenleftbigg/parenleftBig\nµ−3\n4/parenrightBig2\n−3+2κ\n16/parenrightbigg\n,\nwhereκ= cosθ1+cosθ2+cos(θ1−θ2), and the eigenvalues of ∆θ\ncombare\nµ1=3\n2andµ±=3\n4±1\n4√\n3+2κ.CONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 17\nIn particular, we recover the fact that ∆combhas an eigenfunction, since µ1is inde-\npendent of θ, onlyµ±depend onθviaκ=κ(θ). Note that we have\n−3\n2=κ/parenleftBig2π\n3,4π\n3/parenrightBig\n≤κ(θ)≤κ(0,0) = 3,\ngiving the spectral bands B−= [0,3/4] andB+= [3/4,3/2]. /square\nThe next result discusses (dis)continuity properties of the IDS as sociated to the\ncombinatorial Laplacian on X:\nProposition 3.3. LetNcombbe the (abstract) IDS of the Z2-periodic operator ∆comb,\ngiven by\nNcomb(µ) =1\n|Q|tr[ /BDQPcomb((−∞,µ])],\nwheretris the trace on the Hilbert space ℓ2(V)andPcombdenotes the spectral pro-\njection of ∆comb. ThenNcombvanishes on (−∞,0], is continuous on R\\{3/2}and\nhas a jump of size 1/3atµ= 3/2. Moreover, Ncombis strictly monotone increasing\non[0,3/2]andNcomb(µ) = 1forµ≥3/2.\nProof.The following facts are given, e.g., in [MY02, p. 119]:\n(i) the points of increase of Ncombcoincide with the spectrum σ(∆comb) and\n(ii)Ncombcan only have discontinuities at σp(∆comb).\nTogether with Proposition 3.2, all statements of the proposition fo llow, except for\nthe size of the jump at µ= 3/2.\nLet us choose a Følner sequence In⊂Z2and define Λ n=/uniontext\nγ∈InQγ. Let∂Λn\ndenote the set of boundary vertices of the combinatorial graph in duced by the vertex\nset Λn, and\n∂rΛn:={v∈V(X)|d(v,∂Λn)≤r} (3.3)\nbe the thickened (combinatorial) boundary. Let\nD(µ) :=Ncomb(µ)−lim\nε→0Ncomb(µ−ε) =1\n|Λn|tr/bracketleftbig/BDΛnPcomb({µ})/bracketrightbig\n.(3.4)\nThe last equality in (3.4) holds for all nand follows easily from the Z2-invariance of\nthe operator ∆comb. It remains to prove that D(3/2) = 1/3. Let Λ′\nn= Λn\\∂1Λnand\nDn(µ) :=1\n|Λn|dimEn(µ),\nwhereEn(µ) :={F∈ker(∆comb−µ)|suppF⊂Λ′\nn}. Arguments as in [MSY03] or\nin [LV08] show that\nD(µ) = lim\nn→∞Dn(µ). (3.5)\nFor the convenience of the reader, we outline the proof of (3.5) be low. Using part\n(b) of Proposition 3.1 one can show that dim En(µ) equals up to a boundary term\nthe number of hexagons contained in Λ′\nn. Since every translated combinatorial fun-\ndamental domain Qγuniquely determines a hexagon Hγand|Q|= 3, we conclude\nthat dimEn(µ)≈1\n3|Λn|, up to an error proportional to |∂1Λn|. The van Hove prop-\nerty (2.11) (which holds also in the combinatorial setting) then implies the desired\nresultD(3/2) = lim n→∞Dn(3/2) = 1/3.18 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\nFinally, we outline the proof of (3.5): Let E(µ) = ker(∆comb−µ) andSn(µ) =/BDΛnE(µ). Letbn:Sn(µ)−→R|∂1Λn|betheboundary map, i.e., bn(F) isthecollection\nof all values of Fassumed at the (thickened) boundary vertices ∂1Λn. Then kerbn=\nEn(µ)⊂Sn(µ), and we have\nDn(µ)≤D(µ)≤dimSn(µ)\n|Λn|=dimkerbn\n|Λn|+dimranbn\n|Λn|≤Dn(µ)+|∂1Λn|\n|Λn|,\nwhich yields (3.5), by taking the limit, as n→ ∞. /square\n3.2.Spectrum and IDS of the periodic Kirchhoff Laplacian. There is a well\nknown correspondence between the spectrum σ(∆comb) on a graph Gand the spec-\ntrumofthe(Kirchhoff)Laplacian∆0onthecorresponding (equilateral)metricgraph\n(X,ℓ0) withℓ0= /BDE(see e.g. [vB85, Nic85, Cat97, BGP08, Pos08] and the references\ntherein). Namely, any λ/\\e}atio\\slash=k2π2lies inσp(∆0) resp.σac(∆0) iffµ(λ) = 1−cos√\nλ\nlies inσp(∆comb) resp.σac(∆comb). Moreover, the eigenspace of the metric Laplacian\nis isomorphic to the corresponding eigenspace of the combinatorial Laplacian.\nLetF:V−→Cbe a finitely supported eigenfunction of ∆combas in the previous\nsection. In particular, the eigenvalue must be µ= 3/2. The above mentioned\ncorrespondence shows that, for every λ= (2k+2/3)2π2,k∈Z, (i.e.µ(λ) = 3/2)),\nthere is a Kirchhoff eigenfunction f:X−→Rof compact support associated to the\neigenvalueλ, satisfying f(v) =F(v) at all vertices v∈V. In addition, if λ=k2π2,\nthere are so-called Dirichlet eigenfunctions of ∆0, determined by the topology of the\ngraph (see e.g. [vB85, Nic85, Kuc05, LP08]), which are also generat ed by compactly\nsupported eigenfunctions.\nUsing the results [Cat97, BGP08], we conclude from Proposition 3.2:\nCorollary 3.4. Let∆0denote the Kirchhoff Laplacian of the equilateral metric gra ph\n(X,ℓ0). Letσpandσacdenote the point spectrum and absolutely continuous spectr um\nandσcompdenote the spectrum given by the compactly supported eigenf unctions. Then\nwe have\nσcomp(∆0) =σp(∆0) =/braceleftBig/parenleftBig\n2k+2\n3/parenrightBig2\nπ2/vextendsingle/vextendsingle/vextendsinglek∈Z/bracerightBig\n∪/braceleftbig\nk2π2/vextendsingle/vextendsinglek∈N/bracerightbig\nand\nσac(∆0) =/bracketleftBig\n0,/parenleftBig2\n3/parenrightBig2\nπ2/bracketrightBig\n∪/uniondisplay\nk∈N/bracketleftBig/parenleftBig\n2k−2\n3/parenrightBig2\nπ2,/parenleftBig\n2k+2\n3/parenrightBig2\nπ2/bracketrightBig\n. (3.6)\nSimilarly, as inthe discrete setting, we conclude thefollowing (dis)con tinuity prop-\nerties of the IDS:\nProposition 3.5. LetN0be the (abstract) IDS of the Z2-periodic Kirchhoff Lapla-\ncian∆0on the metric graph (X,ℓ0), given by\nN0(λ) =1\nvol(F,ℓ0)tr[ /BDFP0((−∞,λ])],\nwheretris the trace on the Hilbert space L2(X,ℓ0)andP0denotes the spectral pro-\njection of ∆0. Then all the discontinuities of N0:R−→[0,∞)are\n(i) atλ= (2k+2\n3)2π2,k∈Z, with jumps of size1\n6,\n(ii) atλ=k2π2,k∈N, with jumps of size1\n2.CONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 19\nMoreover,N0is strictly monotone increasing on the absolutely continuo us spectrum\nσac(∆0)given in (3.6)andN0is constant on the complement of σ(∆0).\nProof.Our periodic situation fits into the general setting given in [LPV07], by choos-\ning the trivial probability space Ω = {ω}with only one element. Proposition 5.2\nin [LPV07] states that N0is the distribution function of a spectral measure for the\noperator∆0. Consequently, discontinuities of N0canonly occur atthe L2-eigenvalues\nof ∆0, and the points of increase of N0coincide with the spectrum σ(∆0), which is\ngiven in Corollary 3.4. Hence, it only remains to prove the statements about the dis-\ncontinuitiesof N0. Weknowfrom[Kuc05, Theorem11]thatthecompactlysupported\neigenfunctions densely exhaust every L2-eigenspace of ∆0.\nLetIn⊂Z2be a Følner sequence. This time, we look at the corresponding\ntopological graphs Λ( In) and their thickened topological boundaries ∂rΛ(In) ={x∈\nX|d(x,∂Λ(In))≤r}, and denote them by Λ nand∂rΛn, respectively. We are\ninterested in the jumps\nD(λ) :=N0(λ)−lim\nε→0N0(λ−ε) =1\nvol(Λn,ℓ0)tr/bracketleftbig/BDΛnP0({λ})/bracketrightbig\n,\nwhere the right hand side is, again, independent of the choice of n. Let Λ′\nnbe the\nclosure of Λ n\\∂1Λnand\nDn(λ) :=1\nvol(Λn,ℓ0)dimEn(λ),\nwithEn(λ) ={f∈ker(∆0−λ)|suppf⊂Λ′\nn}. Arguments analogously to the proof\nof (3.5) yield\nD(λ) = lim\nn→∞Dn(λ). (3.7)\nFor the proof of (3.7), however, we have to define the boundary m ap\nbn:Sn(λ)−→/circleplusdisplay\nv∈∂Λn(C⊕CEv) by (bnf)v:= (f(v),Df− →(v)).\nLetλ= (2k+2/3)2π2,k∈Z. We follow the same arguments as in the proof of\nProposition 3.3. Again, dim En(λ) is equal to the number of hexagons contained in\nΛnup to a boundary term and we have vol( F,ℓ0) = 6 (see Figure 2 (a)). Therefore,\nwe derive that the corresponding jump is of size 1 /6.\nLetλ=k2π2,k∈N. We know from [vB85, Nic85] or from [LP08, Lem. 5.1\nand Prop. 5.2] that the dimension of En(λ) is (up to an error proportional to |∂Λn|)\napproximately equal to\n|E(Λn)|−|V(Λn)| ≈1\n2vol(Λn,ℓ0).\nThis implies that N0has a discontinuity at λ=k2π2of size 1/2. /square\nRemark3.6.Note that Propositions 3.3 and 3.5 hold also for general covering gra phs\nX→X0with amenable covering group Γ and compact quotient X0∼=X/Γ, once we\nhave information about the shape of the support of elementary eig enfunctions (i.e.,\neigenfunctions, which generate the eigenspace by linear combinatio ns and transla-\ntions). In our Kagome lattice example the elementary eigenfunction is supported on20 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\na hexagon. For example, the jump of size 1 /3 at the eigenvalue µ= 3/2 in the dis-\ncrete case is the number νof hexagons determined by a combinatorial fundamental\ndomain (ν= 1) divided by the number of vertices in a combinatorial fundamenta l\ndomain ( |Q|= 3).\nIn the metric graph setting, the jump at λ= (2k+ 2/3)2π2is of size 1/6 due to\nthe fact that we have six edges in one topological fundamental dom ain.\nFor the eigenvalues at λ=k2π2(also called topological , see [LP08]) we even have\na precise information for any r-regular amenable covering graph, namely\ndimEn(λ)≈ |E(Λn)|−|V(Λn)| ≈/parenleftBig\n1−2\nr/parenrightBig\n|E(Λn)|=/parenleftBig\n1−2\nr/parenrightBig\nvol(Λn,ℓ0),\nup to an error proportional to |∂Λn|, so that the jump of N0atλis (1−2/r).\n3.3.IDS of associated random length models. Finally, we impose a random\nlength structure ℓ: Ω×E−→[ℓmin,ℓmax] on the edges of ( X,ℓ0) with independently\ndistributed edge lengths, as described in Assumption 2.8. Then Coro llary 2.10 tells\nus that the associated integrated density of states N:R−→[0,∞) is continuous and\neven Lipschitz continuous on (0 ,∞). Hence, all discontinuities occurring for the IDS\nof the Kirchhoff Laplacian on the Z2-periodic graph ( X,ℓ0) disappear by introducing\nthis type of randomness.\n4.Proof of the approximation of the IDS via exhaustions\nIn this section, we prove Theorem 2.6, namely, that the non-rando m integrated\ndensity of states (2.9) can be approximated by suitably chosen nor malised eigenvalue\ncounting functions, for P-almost all random parameters ω∈Ω.\nFor the following considerations, we need the quadratic forms asso ciated to the\nSchr¨ odinger operators. Recall that for each Lagrangian subsp aceLv⊂CEv⊕CEv\ndescribing the vertex condition at v∈Vthere exists a unique orthogonal projection\nQvonCEvwith range Gv:= ranQvand a symmetric operator on Gvsuch that (2.5)\nholds.\nLet Λ⊂Xbe a topological subgraph. The quadratic form associated to the\noperator with vertex conditions given by ( Gv,Rv) at inner vertices V(Λ)\\∂Λ and\nDirichlet conditions at ∂Λ is defined as\ndomhΛ,D=/braceleftbig\nf∈H1\nmax(X,ℓ)/vextendsingle/vextendsinglef(v)∈Gv∀v∈V(Λ)\\∂Λ, f(v) = 0∀v∈∂Λ/bracerightbig\n,\nhΛ,D(f) =/bardblDf/bardbl2\nL2(Λ,ℓ)+/a\\}bracketle{tqf,f/a\\}bracketri}htL2(Λ,ℓ)+/summationdisplay\nv∈V(Λ)/a\\}bracketle{tRvf(v),f(v)/a\\}bracketri}htGv.\nIn particular, if Λ = Xis the full graph, then there is no boundary and h=hXis\nthe quadratic form associated to the operator H=H(X,ℓ),L.\nIfℓmin:= inf eℓ(e)>0,Cpot:=/bardblq/bardbl∞<∞and supv/bardblRv/bardbl=:CR<∞, thenhΛ,D\nis a closed quadratic form with corresponding self-adjoint operato rHΛ,D.CONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 21\nLemma 4.1. For any subgraph ΛofX, the quadratic form hΛ,Dis closed. Moreover,\nthe associated self-adjoint operator HΛ,Dhas domain given by\ndomHΛ,D=/braceleftBig\nf∈H2\nmax(X,ℓ)/vextendsingle/vextendsingle/vextendsinglef(v) = 0∀v∈∂V,\nf(v)∈Gv, QvDf− →(v) =Rvf(v)∀v∈V(Λ)\\∂Λ/bracerightBig\n.\nMoreover,HΛ,Disuniformly bounded from below by −C0whereC0≥0depends only\nonℓ−,CRandCpot, but not on Λ.\nProof.Thefirstassertionfollowsfrom[Kuc04, Thm.17]. The uniformlowerboundis\naconsequence of[Kuc04, Cor.10]wherethelowerboundisgivene xplicitly. Basically,\nthe statements follow from a standard Sobolev estimate of the typ e/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nv/a\\}bracketle{tRvf(v),f(v)/a\\}bracketri}ht/vextendsingle/vextendsingle/vextendsingle≤CR/summationdisplay\nv∈V(Λ)|f(v)|2≤η/bardblDf/bardbl2+Cη/bardblf/bardbl2\nforη>0, whereCηdepends only on η,CRandℓmin. /square\nThe Dirichlet operator will serve as upper bound in the bracketing ine quality (4.1)\nlater on. In order to have a lower bound we introduce a Neumann-ty pe operator\nHΛvia its quadratic form hΛ. Since the vertex conditions can be negative, we have\nto use the boundary condition ( CEv,−CR) instead of a simple Neumann boundary\ncondition ( CEv,0). The quadratic form hΛis defined by\ndomhΛ=/braceleftbig\nf∈H1\nmax(X,ℓ)/vextendsingle/vextendsinglef(v)∈Gv∀v∈V(Λ)\\∂Λ/bracerightbig\n,\nhΛ(f) =/bardblDf/bardbl2\nL2(Λ,ℓ)+/a\\}bracketle{tqf,f/a\\}bracketri}htL2(Λ,ℓ)+/summationdisplay\nv∈V(Λ)\\∂Λ/a\\}bracketle{tRvf(v),f(v)/a\\}bracketri}htGv−CR/summationdisplay\nv∈∂Λ|f(v)|2\nGv.\nNotethattheboundarycondition /tildewideRv=−CRtriviallyfulfillsthenormbound /bardbl/tildewideRv/bardbl ≤\nCR, and therefore by Lemma 4.1, the form hΛis uniformly bounded from below by\nthe same constant −C0ashΛ,D. By adding C0to the (edge) potential qwe may\nassume that w.l.o.g. HX,HΛ,DandHΛare all non-negative for all subgraphs Λ.\nWe can now show the following bracketing result:\nLemma 4.2. LetΛbe a topological subgraph of XandΛ′be the closure of the\ncomplement Λc. Then\nHΛ,D⊕HΛ′,D≥H≥HΛ⊕HΛ′≥0 (4.1)\nin the sense of quadratic forms.\nProof.Itisclearfromtheinclusions {0} ⊂Gv⊂CEvforallboundaryvertices v∈∂Λ\nthat the quadratic form domains fulfil\ndomhΛ,D⊕domhΛ′,D⊂domh⊂domhΛ⊕domhΛ′.\nMoreover, if f=fΛ⊕fΛ′is in the decoupled Dirichlet domain, then\nhΛ,D(fΛ)+hΛ′,D(fΛ′) =h(f)\nsincef(v) = 0 on boundary vertices, if f∈domh, then\nh(f)≥hΛ(fΛ)+hΛ′(fΛ′)22 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\nsinceRv≥ −CR. In particular, we have shown the inequality for the quadratic\nforms. /square\nNext, we provide a useful lemma about the spectral shift function of two opera-\ntors. For a non-negative operator Hwith purely discrete spectrum {λk(H)|k≥0}\n(repeated according to multiplicity), the eigenvalue counting funct ion is given by\nn(H,λ) := tr /BD[0,λ)(H) =/vextendsingle/vextendsingle{k≥0|λk(H)≤λ}/vextendsingle/vextendsingle.\nThespectral shift function (SSF) of two non-negative operators H1,H2with purely\ndiscrete spectrum is then defined as\nξ(H1,H2,λ) :=n(H2,λ)−n(H1,λ).\nWe have the following estimate:\nLemma 4.3. Let(X,Ω,P,ℓ)be a random length metric graph (as described in Sub-\nsection 2.3) and Λ⊂Xbe a compact topological subgraph. Let L1,L2be two vertex\nconditions differing in the vertex set Vdiff⊂V(Λ)only, and such that the opera-\ntors∆(Λ,ℓω),Liare non-negative. Let 0≤qbe a bounded measurable potential and\nHi= ∆(Λ,ℓω),Li+q. Then we have\n|ξ(H1,H2,λ)| ≤2/summationdisplay\nv∈Vdiffdegv. (4.2)\nMoreover, if ρ:R+−→Ris a monotone function with ρ′∈L1(R+), then\n/vextendsingle/vextendsingletr[ρ(H1)−ρ(H2)]/vextendsingle/vextendsingle≤2|ρ(∞)−ρ(0)|/summationdisplay\nv∈Vdiffdegv, (4.3)\nwhere the trace is taken in the Hilbert space L2(Λ,ℓω).\nProof.LetD0= domH1∩domH2. Then D0has finite index in dom Hi, bounded\nabove by twice the number of all edges adjacent to vertices v∈Vdiff. This im-\nplies dim(dom Hi/D0)≤2/summationtext\nv∈Vdiffdegv. Inequality (4.2) follows now from [GLV07,\nLemma 9]. The second inequality (4.3) follows readily from Krein’s trace identity\n|trρ(H1)−ρ(H2)| ≤/integraldisplay∞\n0|ρ′(λ)|·|ξ(H1,H2,λ)|dλ. (4.4)\n/square\nThe following uniform resolvent boundedness holds in every random le ngth cover-\ning model:\nLemma 4.4. Let(X,Ω,P,ℓ,L,q)be a random length covering model with covering\ngroupΓ, as described in Assumption 2.4, and λ>0. Then there is a constant Cλ>0\nsuch that we have\ntr(HΛ\nω+λ)−1≤Cλvol(Λ,ℓ0)\nfor all compact subgraphs Λ⊂(X,ℓ)and allω∈Ω.CONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 23\nProof.LetHΛ,0\nωdenote the restriction on Λ with Dirichlet vertex conditions at all\nvertices. ThenHΛ,0\nω=/circleplustext\ne∈E(Λ)He,D\nω, where we identify the edge ewith the topo-\nlogical subgraph consisting of this edge and its end vertices in X. From (4.2) of\nLemma 4.3 we conclude that\n|tr(HΛ,0\nω+λ)−1−(HΛ\nω+λ)−1| ≤4\nλ|E(Λ)|=4\nλvol(Λ,ℓ0).\nSince (He,D\nω+λ)−1is bounded from above by (∆e,D\nω+λ)−1, and since the edges\nare uniformly bounded from above by ℓmax, there is a constant cλ>0 such that\ntr(He,D\nω+λ)−1≤cλfor alle∈E(Λ) andω∈Ω. This implies the desired estimate\nwith constant Cλ= 4λ−1+cλ. /square\nThe proof of Theorem 2.6 will now be given in four lemmata. All of these lemmata\narebasedonagivenrandomlengthcovering model( X,Ω,P,ℓ,L,q)withan amenable\ncovering group Γ and a fixed tempered Følner sequence Inwith associated compact\ntopological graphs Λ n:= Λ(In).\nInthe first lemma, we prove the convergence (2.13) for a special f amilyof functions\nfλassociated to resolvents of the operators. Here, we need to app ly an ergodic\ntheorem of Lindenstrauss [Lin01].\nIn later lemmata we show that the convergence (2.13) carries over to the uniform\nclosure of finite linear combinations of the functions fλ, identify this closure with the\nhelp of the Stone-Weierstrass Theorem, and finally conclude the de sired convergence\nfor characteristic functions /BD[0,λ]at continuity points λ>0 of the IDS.\nLemma 4.5. Letλ >0andfλ: [0,∞)−→R,fλ(x) =1\nx+λ. Then there exists a\nsubsetΩ0⊂Ωof fullP-measure such that\nlim\nn→∞1\nvol(Λn,ℓω)tr[fλ(Hn,D\nω)] =1\nE(vol(F,ℓ•))E(tr[ /BDFfλ(H•)])\nfor allω∈Ω0.\nProof.We first consider a fixed ω∈Ω and a fixed Λ = Λ( In) and suppress the\nparameters ωandnin the notation. Recall the definitions of HΛ,DandHΛwith\nquadratic form domains given below. Let Λ′denote the closure of the complement\nΛcin the metric graph ( X,ℓ). By Lemma 4.2 we have (4.1) in the sense of quadratic\nforms. Since taking inverses is operator monotone, this implies\n(HΛ,D⊕HΛ′,D+λ)−1≤(H+λ)−1≤(HΛ⊕HΛ′+λ)−1\nfor allλ >0. In particular, we obtain inequalities for the following restricted qu a-\ndratic forms: Set ( H+λ)−1\nΛ=pΛ(H+λ)−1iΛ, whereiΛandpΛdenote the canonical\ninclusions and projections between L2(Λ,ℓ) andL2(X,ℓ). Then\n(HΛ,D+λ)−1≤(H+λ)−1\nΛ≤(HΛ+λ)−1. (4.5)\nConsequently, ( H+λ)−1\nΛ−(HΛ,D+λ)−1is non-negative and we have\n0≤trL2(Λ,ℓ)/bracketleftbig\n(H+λ)−1\nΛ−(HΛ,D+λ)−1/bracketrightbig\n≤trL2(Λ,ℓ)/bracketleftbig\nfλ(HΛ)−fλ(HΛ,D)/bracketrightbig\n≤2\nλdmax|∂Λ|,24 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\nusing Lemma 4.3, where dmaxis a finite upper bound on the vertex degree of X,\nwhich exists due to the Γ-periodicity of X. Using the van Hove property (2.11) and\nthe estimate\nℓminvol(Λ,ℓ0)≤vol(Λ,ℓω)≤ℓmaxvol(Λ,ℓ0),\nwe conclude that\nlim\nn→∞1\nvol(Λn,ℓω)/parenleftbig\ntr[(Hω+λ)−1\nΛn]−tr[fλ(Hn,D\nω)]/parenrightbig\n= 0. (4.6)\nUsing additivity of the trace and the operator consistency (2.8b), we obtain\ntrL2(Λn,ℓω)(Hω+λ)−1\nΛn=/summationdisplay\nγ∈IntrL2(γF,ℓω)(Hω+λ)−1\nγF=/summationdisplay\nγ∈I−1\nngλ(γω),\nwhere\ngλ(ω) = trL2(F,ℓω)[(Hω+λ)−1\nF] = tr[ /BDFfλ(Hω)]. (4.7)\nSince, by monotonicity (4.5) and Lemma 4.4,\n0≤gλ(ω)≤trL2(F,ℓω)[(HF\nω+λ)−1]≤Cλvol(F,ℓ0),\nweconclude that gλ∈L1(Ω). Now, weargueasintheproofofTheorem 7in[LPV04]:\nApplying Lindenstrauss’ ergodic theorem separately to both expr essions\n1\n|In|/summationdisplay\nγ∈I−1\nngλ(γω) and1\n|In|/summationdisplay\nγ∈I−1\nnvol(F,ℓγω),\nwe conclude that\nlim\nn→∞1\nvol(Λn,ℓω)tr[(Hω+λ)−1\nΛn] =1\nE(vol(F,ℓ•))E(tr[ /BDFfλ(H•)]) (4.8)\nfor almost all ω∈Ω. The lemma follows now immediately from (4.6) and (4.8). /square\nLetusdenote by Ltheset offunctions {x/ma√sto→fλ(x) = (x+λ)−1|λ>0}andby A\nthe/bardbl·/bardbl∞-closure of the linear span of Land the constant function /BD: [0,∞)−→R,/BD(x) = 1. Note that, by monotonicity (4.5) and Lemma 4.4, both express ionsµn\nω(f1)\nandµ(f1) = (E(vol(F,ℓ•)))−1E(g1) (withg1defined in (4.7)) are bounded by a\nconstantK >0, independent of ωandn. Let Ω 0⊂Ω be the set of full P-measure\nfrom Lemma 4.5.\nLemma 4.6. Letω∈Ω0. Setνn=f1·µn\nω(forn∈N) andν=f1·µ. Then we\nhave, for all g∈A,\nlim\nn→∞νn(g) =ν(g).\nProof.By Lemma 4.5 we know that the statement holds for the function g= /BD. We\nnote thatfλ·f1=1\nλ−1(fλ−f1) forλ/\\e}atio\\slash= 1. Thus, by linearity and Lemma 4.5, the\nconvergence holds also for all functions g=fλwithλ>0,λ/\\e}atio\\slash= 1. To deal with the\ncaseλ= 1 note that f1+εconverges to f1uniformly, as ε→0. Thus\n|νn(f1)−νn(f1+ε)| ≤ /bardblf1−f1+ε/bardbl∞νn( /BD)≤Kε.\nAn analogous statement holds for νnreplaced by ν. Thus\n|ν(f1)−νn(f1)| ≤2Kε+/vextendsingle/vextendsingleν(f1+ε)−νn(f1+ε)/vextendsingle/vextendsingle→2Kε, (4.9)CONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 25\nasn→ ∞. Sinceε >0 was arbitrary, we conclude that lim n→∞νn(f1) =ν(f1).\nBy linearity, the convergence statement of the Lemma holds for all functionsgin\nthe linear span of L∪ { /BD}. To show that is holds for all functions in the closure\nA, as well, one uses uniform approximation and an estimate of the same type as\nin (4.9). /square\nThe next lemma identifies the space Aexplicitly:\nLemma 4.7. The function space Acoincides with the set of continuous functions\non[0,∞)which converge at infinity.\nProof.The statement of the lemma is equivalent to A=C([0,∞]), where [0 ,∞] is\nthe one-point-compactification of [0 ,∞). We want to apply the Stone-Weierstrass\nTheorem. Any fλwithλ>0 separates points and /BDis nowhere vanishing in [0 ,∞].\nBy definition Ais a linear space. To show that it is an algebra we use again the\nformulafλ1·fλ2=1\nλ2−λ1(fλ1−fλ2), which shows that fλ1·fλ2∈Aforλ1/\\e}atio\\slash=λ2.\nSince Ais closed in the sup-norm, we can use an approximation as in the proof of\nthe Lemma 4.6 to show f2\nλ∈A. A similar argument shows that the product of two\nlimit points f,gof the linear span of L∪{ /BD}is inA. /square\nWe have established the convergence µn\nω(g)→µ(g) for all functions of the form\ng·f1withg∈A. The following lemma shows that this is sufficient to conclude the\nalmost sure convergence Nn\nω(λ)→N(λ) at continuity points λ, finishing the proof\nof Theorem 2.6. One has only to observe that every continuous fun ction of compact\nsupport on R+= [0,∞) can be written as g·f1, with an element g∈A.\nLemma 4.8. Forn∈N, letρn,ρbe locally finite measures on R+. Then\nlim\nn→∞ρn(g) =ρ(g)\nfor all continuous functions gof compact support implies that\nlim\nn→∞ρn([0,λ]) =ρ([0,λ])\nfor allλ>0which are not atoms of ρ.\nProof.The proof is standard. First note that locally finiteness of ρimplies\nlim\nε→0ρ([λ−ε,λ+ε]) =ρ({λ}) = 0.\nNow choose monotone functions g��\nε,g+\nε∈Cc(R+) satisfying/BD[0,λ−ε]≤g−\nε≤ /BD[0,λ]≤g+\nε≤ /BD[0,λ+ε].\nThen\nρ([0,λ])−ρn([0,λ])≤ρ(g+\nε)−ρ(g−\nε)+ρ(g−\nε)−ρn(g−\nε)\n≤ρ([λ−ε,λ+ε])+ρ(g−\nε)−ρn(g−\nε).\nFor anyδ >0 one can choose ε>0 such that ρ([λ−ε,λ+ε])<δ. Sinceδ >0 was\narbitrary, we have shown ρ([0,λ])≤liminf n→∞ρn([0,λ]). The opposite inequality is\nshown similarly. /square26 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\n5.Proof of the Wegner estimate\nThis section is devoted to the proof of Theorem 2.9. Let ( X,Ω,P,ℓ) be a random\nlength model satisfying Assumption 2.8. We first introduce a new mea surable map\nα: Ω×E−→[ω−,ω+] withω−= lnℓmin,ω+= lnℓmax, defined by αω(e) :=α(ω,e) =\nlnℓω(e). The random variables α(·,e),e∈E, are independently distributed with\ndensity functions ge(x) = exhe(ex), and we have\n/bardblg′\ne/bardbl∞≤ℓmax/bardblhe/bardbl∞+ℓ2\nmax/bardblh′\ne/bardbl∞≤(ℓmax+ℓ2\nmax)Ch=:Dh<∞.(5.1)\nThus, we can re-identify Ω with the Cartesian product/producttext\ne∈E[ω−,ω+], and the maps\nα(·,e) are simply projections to the component with index e. The measure Pis\nnow given as the product/circlemultiplytext\ne∈E/tildewidePeof marginal measures /tildewidePewith density functions\nge∈C1(R) satisfying the above estimate (5.1). The advantage of the new “r escaled”\nidentification Ω =/producttext\ne∈E[ω−,ω+] is the following property of the eigenvalues of the\nLaplacian on any compact subgraph (Λ ,ℓω):\nλi(∆Λ,D\nω+s /BD) = e−2sλi(∆Λ,D\nω). (5.2)\nHere, the eigenvalues λiare counted with multiplicity and ω+s /BDdenotes the ele-\nment{ωe+s}e∈E(X)∈Ω. Property (5.2) is an immediate consequence of (2.3b),\nℓω+s /BD(e) = eαω(e)+s= esℓω(e), and the fact that a rescaling of all lengths by a fixed\nmultiplicative constant does not change the domain of the Kirchhoff L aplacian with\nDirichlet boundary conditions on ∂Λ. Property (5.2) is of crucial importance for the\nproof of the Wegner estimate.\nHenceforth, weusethisnewinterpretationofΩandrename /tildewidePebyPe, forsimplicity.\nLet Λ⊂Xbe a compact topological subgraph, λ∈Randε >0. We write\nthe interval Ias [λ−ǫ,λ+ǫ] and start with a smooth function ρ:R−→[−1,0]\nsatisfyingρ≡ −1 on (−∞,−ε], 0≤ρ′≤1/ε,ρ≡0 on [ε,∞). Moreover, we set\nρλ(x) =ρ(x−λ). Then we have/BD[λ−ε,λ+ε](x)≤ρλ(x+2ε)−ρλ(x−2ε) =/integraldisplay2ε\n−2ερ′\nλ(x+t)dt.\nUsing the spectral theorem, we obtain\nPΛ,D\nω([λ−ε,λ+ε]) = /BD[λ−ε,λ+ε](∆Λ,D\nω)≤/integraldisplay2ε\n−2ερ′\nλ(∆Λ,D\nω+t)dt,\nand, consequently,\ntrPΛ,D\nω([λ−ε,λ+ε])≤/integraldisplay2ε\n−2εtrρ′\nλ(∆Λ,D\nω+t)dt.\nDenote by (Ω(Λ) ,PΛ) the space Ω(Λ) =/producttext\ne∈E(Λ)[ω−,ω+] with probability measure\nPΛ=/circlemultiplytext\ne∈E(Λ)Pe, andEΛ(·) denote the associated expectation. E(·) means expecta-\ntion with respect to the full space (Ω ,P). Applying expectation yields\nE(trPΛ,D\n•([λ−ε,λ+ε])) =EΛ(trPΛ,D\n•([λ−ε,λ+ε]))\n≤/integraldisplay\nΩ(Λ)/integraldisplay2ε\n−2εtrρ′\nλ(∆Λ,D\nω+t)dtdPΛ(ω).(5.3)CONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 27\nUsing the chain rule and scaling property (5.2), we obtain\n/summationdisplay\ne∈E(Λ)∂\n∂ωeρλ(λi(∆Λ,D\nω)+t) =ρ′\nλ(λi(∆λ,D\nω)+t)d\nds/vextendsingle/vextendsingle/vextendsingle\ns=0/parenleftbig\ns/ma√sto→λi(∆Λ,D\nω+s /BD)/parenrightbig\n=−2ρ′\nλ(λi(∆λ,D\nω)+t)λi(∆Λ,D\nω)≤0.\nNow, we use that [ λ−ε,λ+ε]⊂Ju= [1/u,u]. Since supp ρ′\nλ⊂[λ−ε,λ+ε], we\nderive\n0≤trρ′\nλ(∆Λ,D\nω+t)≤ −u\n2/parenleftBig/summationdisplay\ne∈E(Λ)∂\n∂ωetrρλ(∆Λ,D\nω+t)/parenrightBig\n. (5.4)\nFore∈E(Λ), denote by Λ ethe topological subgraph with vertex set Ve:=V(Λ)\nand edge set Ee:=E(Λ)\\{e}. Using the estimate (5.4), we obtain from (5.3)\nE(trPΛ,D\n•([λ−ε,λ+ε]))\n≤ −u\n2/summationdisplay\ne∈E(Λ)/integraldisplay\nΩ(Λe)/integraldisplay2ε\n−2ε/integraldisplayω+\nω−/parenleftBig∂\n∂ωetrρλ(∆Λ,D\n(ω′,x)+t)/parenrightBig\nge(x)dxdtdPΛe(ω′) (5.5)\nwith(ω′,x)∈Ω(Λe)×[ω−,ω+] = Ω(Λ). Next, wewanttocarryoutpartialintegration\nwith respect to xin (5.5). Before doing so, it is useful to observe, for fixed c∈\n[ω−,ω+],\n∂\n∂ωetrρλ(∆Λ,D\n(ω′,x)+t) =∂\n∂ωe/parenleftBig\ntrρλ(∆Λ,D\n(ω′,x)+t)−trρλ(∆Λ,D\n(ω′,c)+t)/parenrightBig\n.(5.6)\nUsing (5.6) and applying partial integration, we obtain\n/vextendsingle/vextendsingle/vextendsingle/integraldisplayω+\nω−/parenleftBig∂\n∂ωetrρλ(∆Λ,D\n(ω′,x)+t)/parenrightBig\nge(x)dx/vextendsingle/vextendsingle/vextendsingle\n≤ /bardblg′\ne/bardblL1sup\nc′∈[ω−,ω+]/vextendsingle/vextendsingletrρλ−t(∆Λ,D\n(ω′,c′))−trρλ−t(∆Λ,D\n(ω′,c))/vextendsingle/vextendsingle.(5.7)\nFor notational convenience, we identify the compact topological g raph consisting\nonly of the edge eand its end-points with e, and we denote by ∆e,D\ncbe the Dirichlet-\nLaplacian on the metric graph ( e,ℓc) defined by ℓc(e) = exp(c). Using (4.3) in\nLemma 4.3, we conclude that\n/vextendsingle/vextendsingletrρλ−t(∆Λ,D\n(ω′,c))−trρλ−t(∆Λe,D\nω′⊕∆e,D\nc)/vextendsingle/vextendsingle≤2|ρ(∞)−ρ(t−λ)|2dmax≤4dmax,\nforallvalues c∈[ω−,ω+]. Consequently, sup/vextendsingle/vextendsingletrρλ−t(∆Λ,D\n(ω′,c′))−trρλ−t(∆Λ,D\n(ω′,c))/vextendsingle/vextendsinglein(5.7)\ncan be estimated from above by\n8dmax+/vextendsingle/vextendsingletrρ(∆e,D\nc′+t−λ)−trρ(∆e,D\nc+t−λ)/vextendsingle/vextendsingle.\nNote that all eigenfunctions of the Dirichlet operator ∆e,D\ncare explicitly given sine\nfunctions. Therefore, since λ∈[1/u+ε,u−ε] andt∈[−2ε,2ε], there is a constant\nCu,ℓmax>0, depending only on u,ℓmax, such that\n/vextendsingle/vextendsingletrρ(∆e,D\nc+t−λ)/vextendsingle/vextendsingle≤Cu,ℓmax,28 D. LENZ, N. PEYERIMHOFF, O. POST, AND I. VESELI ´C\nfor all exp(c)∈[ℓmin,ℓmax]. 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Veseli´ c, Wegner estimates for sign-changing single site potentials , arXiv:0806.0482\n(2008).CONTINUITY OF THE IDS ON RANDOM LENGTH METRIC GRAPHS 31\n[vB85] J. von Below, A characteristic equation associated to an eigenvalue prob lem onC2-\nnetworks , Linear Algebra Appl. 71(1985), 309–325.\n[Weg81] F. Wegner, Bounds on the DOS in disordered systems , Z. Phys. B 44(1981), 9–15.\n(D. Lenz) Friedrich-Schiller-Universit ¨at Jena, Fakult ¨at f¨ur Mathematik & Infor-\nmatik, Mathematisches Institut, 07737 Jena, Germany\nE-mail address :daniel.lenz@uni-jena.de\nURL:www.tu-chemnitz.de/mathematik/mathematische physik/\n(N.Peyerimhoff) Department of Mathematical Sciences, Durham University, S cience\nLaboratories South Road, Durham, DH1 3LE, Great Britain\nE-mail address :norbert.peyerimhoff@durham.ac.uk\nURL:www.maths.dur.ac.uk/~dma0np/\n(O. Post) Institut f ¨ur Mathematik, SFB 647 “Space – Time – Matter”, Humboldt-\nUniversit ¨at zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany\nE-mail address :post@math.hu-berlin.de\nURL:www.math.hu-berlin.de/~post/\n(I. Veseli´ c) Fakult¨at f¨ur Mathematik,TU Chemnitz, D-09107 Chemnitz, Germany,\n& Emmy-Noether Programme of the DFG\nURL:www.tu-chemnitz.de/mathematik/schroedinger/members. php"    },    {        "title": "0812.2231v1.Supersolid_state_in_fermionic_optical_lattice_systems.pdf",        "content": "arXiv:0812.2231v1  [cond-mat.supr-con]  11 Dec 2008Supersolid state in fermionic optical lattice systems\nAkihisa Koga,1Takuji Higashiyama,2Kensuke Inaba,2Seiichiro Suga,2and Norio Kawakami1\n1Department of Physics, Kyoto University, Kyoto 606-8502, J apan\n2Department of Applied Physics, Osaka University, Suita, Os aka 565-0871, Japan\n(Dated: November 8, 2018)\nWe study ultracold fermionic atoms trapped in an optical lat tice with harmonic confinement by\ncombiningthereal-space dynamicalmean-fieldtheorywitha two-site impuritysolver. Bycalculating\nthe local particle density and the pair potential in the syst ems with different clusters, we discuss\nthe stability of a supersolid state, where an s-wave superfluid coexists with a density-wave state of\ncheckerboard pattern. It is clarified that a confining potent ial plays an essential role in stabilizing\nthe supersolid state. The phase diagrams are obtained for se veral effective particle densities.\nI. INTRODUCTION\nSince the successful realization of Bose-Einstein con-\ndensation in a bosonic87Rb system1, ultracold atomic\nsystems have attracted considerable interest.2,3,4One of\nthe most active topics in this field is an optical lattice\nsystem,5,6,7,8which is formed by loading the ultracold\natoms in a periodic potential. This provides a clean sys-\ntem with quantum parameters which can be tuned in a\ncontrolledfashionfromweaktostrongcouplinglimits. In\nfact, remarkable phenomena have been observed such as\nthe phase transition between a Mott insulator and a su-\nperfluid in bosonic systems9. In addition, the superfluid\nstate10and the Mott insulating state11,12have been ob-\nserved in the fermionic optical lattices, which stimulates\ntheoretical investigations on the quantum states in the\noptical lattice systems. Among them, the possibilty of\nthesupersolidstatehasbeendiscussedasoneoftheinter-\nesting problems in optical lattice systems. The existence\nof the supersolid state was experimentally suggested in a\nbosonic4He system,13and was theoretically discussed in\nthestronglycorrelatedsystemssuchasbosonicsystems14\nand Bose-Fermi mixtures15. As for fermionic systems, it\nis known that a density wave (DW) state and an s-wave\nsuperfluid (SSF) state are degeneratein the half-filled at-\ntractiveHubbardmodelonthebipartitelatticeexceptfor\nonedimension,16,17whichmeansthatthesupersolidstate\nmight be realizable in principle. However, the degener-\nate ground states are unstable against perturbations. In\nfact, the hole doping immediately drives the system to a\ngenuine SSF state. Therefore, it is difficult to realize the\nsupersolid state in the homogeneous bulk system. By\ncontrast, in the optical lattice, an additional confining\npotential makes the situation different.18In our previ-\nous paper,19we studied the attractive Hubbard model\non square lattice with harmonic potential to clarify that\nthe supersolid state is indeed realized at low tempera-\ntures. However, we were not able to systematically deal\nwith large clusters to discuss how the supersolid state de-\npends on the particle density, the system size, etc. This\nmight be important for experimental observations of the\nsupersolid state in the optical lattice.\nIn this paper, we address this problem by combining\nthe real-space dynamical mean-field theory (R-DMFT)with atwo-site impurity solver. We then discuss how sta-\nble the DW, SSF and supersolid states are in the optical\nlattice system. We also clarify the role of the confining\npotential in stabilizing the supersolid state.\nThe paper is organized as follows. In Sec. II, we in-\ntroduce the model Hamiltonian and explain the detail of\nR-DMFT and its impurity solver. We demonstrate that\nthe supersolid state is indeed realized in a fermionic op-\ntical lattice with attractive interactions in Sec. III. In\nSec. IV, we discuss the stability of the supersolid state\nin large clusters. We also examine how the phase dia-\ngram depends on the particle number. A brief summary\nis given in Sec. V.\nII. MODEL HAMILTONIAN AND METHOD\nLetusconsiderultracoldfermionicatomsinthe optical\nlattice with confinement, which may be described by the\nfollowing attractive Hubbard model,16,17,20,21,22,23,24\nH=−t/summationdisplay\n/angbracketleftij/angbracketrightσc†\niσcjσ−U/summationdisplay\nini↑ni↓+/summationdisplay\niσv(ri)niσ,(1)\nwhereciσ(c†\niσ) annihilates (creates) a fermion at the ith\nsite with spin σandniσ=c†\niσciσ.t(>0) is a near-\nest neighbor hopping, U(>0) an attractive interaction,\nv(r) [=V(r/a)2] a harmonic potential and the term /an}bracketle{tij/an}bracketri}ht\nindicates that the sum is restricted to nearest neighbors.\nriis a distance measured from the center of the system\nandais lattice spacing. Here, we define the characteris-\ntic length of the harmonic potential as d= (V/t)−1/2a,\nwhich satisfies the condition v(d) =t.\nThe ground-state properties of the Hubbard model on\ninhomogeneous lattices have theoretically been studied\nby various methods such as the Bogoljubov-de Gennes\nequations25, the Gutzwiller approximation26, the slave-\nboson mean-field approach27, variational Monte Carlo\nsimulations28, local density approximation.29Although\nmagnetically ordered and superfluid states are described\nproperly in these approaches, it may be difficult to de-\nscribe the coexisting phase like a supersolid state in the\ninhomogeneous system. The density matrix renormal-\nization group method30and the quantum Monte Carlo\nmethod31are efficient for one-dimensional systems, but2\nit may be difficult to apply them to higher dimensional\nsystems with large clusters. We here use R-DMFT32,\nwhere local particle correlations are taken into account\nprecisely. This treatment is formally exact for the ho-\nmogeneous lattice model in infinite dimensions32and the\nmethod has successfully been applied to some inhomo-\ngeneous correlated systems such as the surface33or the\ninterface of the Mott insulators34, the repulsive fermionic\natoms35,36. Furthermore, it has an advantage in treating\nthe SSF state and the DW state on an equal footing in\nthe strong coupling regime, which allows us to discussthe supersolid state in the optical lattice.\nIn R-DMFT, the lattice model is mapped to an effec-\ntive impurity model, where local electron correlationsare\ntaken into account precisely. The lattice Green function\nisthen obtainedviaself-consistentconditionsimposedon\ntheimpurityproblem. When onedescribesthe superfluid\nstate in the framework of R-DMFT,32the lattice Green’s\nfunction for the system size Lshould be represented in\nthe Nambu-Gor’kov formalism. It is explicitly given by\nthe (2L×2L) matrix,\n/bracketleftBig\nˆG−1\nlat(iωn)/bracketrightBig\nij=−tδ/angbracketleftij/angbracketrightˆσz+δij/bracketleftBig\niωnˆσ0+{µ−v(ri)}ˆσz−ˆΣi(iωn)/bracketrightBig\n, (2)\nwhere ˆσα(α=x,y,z) is theαth component of the (2 ×\n2) Pauli matrix, ˆ σ0the identity matrix, µthe chemical\npotential, ωn= (2n+1)πTtheMatsubarafrequency, and\nTthe temperature. The site-diagonal self-energy at ith\nsite is given by the following (2 ×2) matrix,\nˆΣi(iωn) =/parenleftbigg\nΣi(iωn)Si(iωn)\nSi(iωn)−Σ∗\ni(iωn)/parenrightbigg\n,(3)\nwhere Σ i(iωn) [Si(iωn)] is the normal (anomalous) part\nof the self-energy. In R-DMFT, the self-energy at the\nith site is obtained by solving the effective impurity\nmodel, which is explicitly given by the following Ander-\nson Hamiltonian,23,24\nHimp,i=/summationdisplay\nkσEika†\nikσaikσ+/summationdisplay\nk(Dikaik↑aik↓+h.c.)\n+/summationdisplay\nkσVik/parenleftBig\nc†\niσaikσ+a†\nikσciσ/parenrightBig\n+ǫi/summationdisplay\nσc†\niσciσ−Uc†\ni↑ci↑c†\ni↓ci↓, (4)\nwhereaikσ(a†\nikσ)annihilates(creates)afermionwithspin\nσin the effective bath and ǫiis the impurity level. We\nhave here introduced the effective parameters in the im-\npurity model such as the spectrum of host particles Eik,\nthe pair potential Dikand the hybridization Vik. By\nsolving the effective impurity model eq. (4) for each\nsite, we obtain the site-diagonal self-energy and the lo-\ncal Green’s function. The R-DMFT self-consistent loop\nof calculations is iterated under the condition that the\nsite-diagonal component of the lattice Green’s function\nis equal to the local Green’s function obtained from the\neffective impurity model as/bracketleftBig\nˆGlat(iωn)/bracketrightBig\nii=ˆGimp,i(iωn).\nWhen R-DMFT is applied to our inhomogeneous sys-\ntem, it is necessary to solve the effective impurity mod-\nelsLtimes by iteration. Therefore, numerically powerful\nmethods such as quantum Monte Carlo simulations, theexact diagonalization method, and the numerical renor-\nmalization group method may not be efficient since they\nrequire long time to perform R-DMFT calculations. In\nthis paper, we use a two-site approximation38,39, where\nthe effective bath is replaced by only one site. In spite\nof this simplicity, it has an advantage in taking into ac-\ncount both low- and high-energy properties reasonably\nwell within restricted numerical resources34,38.\nIn the two-site approximation, a non-interacting\nGreen’s function for the impurity model at the ith site is\nsimplified as,\n/bracketleftBig\nˆG0\nimp,i(iωn)/bracketrightBig−1\n=iωnˆσ0−ǫiˆσz\n−Viˆσz1\niωnˆσ0−Eiˆσz−DiˆσxViˆσz,(5)\nwhere the index kwas omitted. The effective parameters\n{Ei,Di,Vi,ǫi}should be determined self-consistently so\nthat the obtained results properly reproduce the original\nlattice problem. Here, we use the following equations,\nǫi=−Re/bracketleftBig\nˆG0\nimp,i(iωn)/bracketrightBig−1\n11/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nn→∞(6)\nVi=/radicalBig\n(a−1)(π2T2+E2\ni+D2\ni) (7)\nDi=b\n1−a, (8)\nwherea= Im[ˆG0\ni(iω0)]−1\n11/πTandb= Re[ˆG0\ni(iω0)]−1\n12.\nFurthermore, the number of particles is fixed in the non-\ninteracting Green’s function, as\nn(i)\n0= 2T/summationdisplay\nn=0Re/bracketleftBig\nˆG0\nimp,i(iωn)/bracketrightBig\n11+1\n2.(9)\nWecandeterminetheeffectiveparameters {Ei,Di,Vi,ǫi}\nin terms of these equations.\nHere, the effective particle density is defined as ˜ ρ=\nN/πd2, whereNis the total number of particles. This3\ndensity relates the systems with different sites, num-\nber of particles, and curvatures of the confining po-\ntentials in the same way as the particle density does\nfor periodic systems with different sites. We set t\nas a unit of energy and calculate the density profile\n/an}bracketle{tniσ/an}bracketri}ht= 2T/summationtext\nn=0Re[Giσ(iωn)]+1\n2and the distribution\nof the pair potential ∆ i= 2T/summationtext\nn=0Re[Fi(iωn)], where\nGiσ(iωn)[Fi(iωn)] is the normal (anomalous) Green’s\nfunction for the ith site. Note that ∆ irepresents the\norder parameter for the SSF state for the ith site.\nIn the following, we consider the attractive Hubbard\nmodel on square lattice with harmonic confinement as a\nsimple model for the supersolid. In this case, it is known\nthat the symmetry of the square lattice is not broken in\nthe DW, SSF, and supersolid states19,25,30. Therefore,\nthe point group C4vis useful to deal with the system\non the inhomogeneous lattice. For example, when the\nsystem with 5513 sites ( r <42.0) is treated, one can\ndeal with only 725 inequivalent sites. This allows us to\ndiscuss the low temperature properties in larger clusters,\nin comparison with those with ( d/a= 6.5,N∼300)\ntreated in our previous paper.\nIII. LOW TEMPERATURE PROPERTIES\nBy means of R-DMFT with the two-site impurity\nsolver,we obtainthe resultsforthe system with d/a= 10\nandN∼720(˜ρ∼2.3). Figures 1 and 2 show the profiles\nof the local density and the pair potential at T/t= 0.05.\nIn the non-interacting case ( U/t= 0), fermionic atoms\naresmoothlydistributed upto r/a∼21, asshownin Fig.\n1 (a). Increasing the attractive interaction U, fermions\ntend to gather around the bottom of the harmonic po-\ntential, as seen in Fig. 1 (b). In these cases, the pair po-\ntential is not yet developed, as shown in Figs. 2 (a) and\n(b), and thereby the normal metallic state with short-\nrange pair correlations emerges in the region ( U/t<∼2).\nFurther increase in the interaction Uleads to different\nbehavior, where the pair potential ∆ iis induced in the\nregion with /an}bracketle{tniσ/an}bracketri}ht /ne}ationslash= 0. Thus, the SSF state is induced by\nthe attractive interaction, which is consistent with the\nresults obtained from the Bogoljubov-de Gennes equa-\ntion25. In the case with U/t= 3, another remarkable\nfeature is found around the center of the harmonic po-\ntential (r/a <7), where a checkerboard structure ap-\npears in the density profile /an}bracketle{tniσ/an}bracketri}ht, as shown in Fig. 1\n(c). This implies that the DW state is realized in the\nregion. On the other hand, the pair potential ∆ iis not\nsuppressed completely even in the DW region, as shown\nin Fig. 2 (c). This suggests that the DW state coex-\nists with the SSF state, i.e. a supersolid state appears\nin our optical lattice system. The profile characteris-\ntic of the supersolid state is clearly seen in the case of\nU/t= 5. Figures 1(d) and 2(d) show that the DW state\nof checkerboard structure coexists with the SSF state in\nthe doughnut-like region (5 < r/a < 15). By contrast,\nthe genuine SSF state appears inside and outside of the\nFIG. 1: (Color online) The density profile /angbracketleftniσ/angbracketrightin the optical\nlattice system with d/a= 10 at T/t= 0.05 when U/t=\n0.0,2.0,3.0,5.0,7.0 and 10 .0 (from the top to the bottom).\nregion (r/a <5,15< r/a < 17). Further increase in\nthe interaction excludes the DW state out of the cen-\nter since fermionic atoms are concentrated around the\nbottom of the potential for large U. In the region, two\nparticles with opposite spins are strongly coupled by the\nattractive interaction to form a hard-core boson, giving\nrise to the band insulator with /an}bracketle{tniσ/an}bracketri}ht ∼1, instead of the\nSSF state. Therefore, the SSF state survives only in the\nnarrow circular region surrounded among the empty and\nfully occupied states. We see such behavior more clearly\nin Figs. 1 (f) and 2 (f). Note that in the strong coupling\nlimitU/t→ ∞, all particles are condensed in the region\nr < rc=/radicalbig\n˜ρ/2d∼1.07d= 10.7a.\nIn this section, wehavestudied the attractiveHubbard4\nFIG. 2: (Color online) The pair potential ∆ iin the optical\nlattice system with d/a= 10 at T/t= 0.05 when U/t=\n0.0,2.0,3.0,5.0,7.0 and 10 .0 (from the top to the bottom).\nmodel with the harmonic potential to clarify that the\nsupersolid state is realized in a certain parameter region.\nHowever, it is not clear how the supersolid state depends\non the system size and the number of particles. To make\nthis point clear, we deal with largeclusters to clarify that\nthe supersolid state is indeed realized in the following.\nIV. STABILITY OF THE SUPERSOLID STATE\nInthissection,wediscussthestabilityofthesupersolid\nstate in fermionic optical lattice systems, which may be\nimportant for experimental observations. First, we clar-\nifyhowlow-temperaturepropertiesdependonthesystemsize, byperformingR-DMFTforseveralclusterswithdif-\nferentd. We here fix U/t= 5 and µ/t∼ −1.58 to obtain\nthe profiles of the local particle density and the pair po-\ntential, which are shown in Fig. 3. We note that the\nS\u0010E \nS\u0010E \nFIG. 3: Profiles of particle density /angbracketleftniσ/angbracketrightand pair potential\n∆ias a function of r/dwith fixed d= 12.5,17.5 and 22 .5,\nwhenU/t= 5.\ndistance ris normalized by din the figure. It is found\nthat/an}bracketle{tniσ/an}bracketri}htand ∆ idescribe smooth curves for r/d<∼0.6\nand 1.4<∼r/d<∼1.7, where the genuine SSF state is re-\nalized. On the other hand, for 0 .6<∼r/d<∼1.4, two dis-\ntinct magnitudes appear in /an}bracketle{tniσ/an}bracketri}ht, reflecting the fact that\nthe DW state with two sublattices is realized. Since the\npair potential is also finite in the region, the supersolid\nstate is realized. In this case, we deal with finite systems,\nand thereby all data are discrete in r. Nevertheless, it is\nfound that the obtained results are well scaled by dal-\nthough some fluctuations appear due to finite-size effects\nin the small dcase. The effective particle density ˜ ρis\nalmost constant in the above cases. Therefore, we con-\nclude that when ˜ ρ∼2.3, the supersolid state discussed\nhere is stable in the limit with N,d→ ∞. This result\ndoes not imply that the supersolid state is realized in\nthe homogeneous system with arbitrary fillings. In fact,\nthe supersolid state might be realizable only at half fill-\ning16,17,20,21,23. Therefore, we can say that a confining\npotential is essential to stabilize the supersolid state in\nthe optical lattice system.\nNext, we focus on the system with U/t= 5 and\nd/a= 10 to discuss in detail how the supersolid state\ndepends on the effective particle density ˜ ρ. The DW\nstate is characterized by the checkerboard structure in5\nthe density profile /an}bracketle{tniσ/an}bracketri}ht, so that the Fourier transform\nnqatq= (π,π) is appropriate to characterize the exis-\ntence of the DW state. On the other hand, the Fourier\ntransform ∆ qatq= (0,0) may represent the rigidity of\nthe SSF state in the system. In Fig. 4, we show the\nʪO ππ ʫʪO \u0011\u0011 ʫ\nʪ∆\u0011\u0011 ʫʪO \u0011\u0011 ʫ \u0016\nρ∼\nFIG. 4: (Color online) /angbracketleftnππ/angbracketright//angbracketleftn00/angbracketrightand ∆ 00/5/angbracketleftn00/angbracketrightas a func-\ntion of the effective particle density ˜ ρ(=N/πd2) whenU= 5t\nandT= 0.05t. A broken line represents the local particle\ndensity at the center of the lattice in the noninteracting ca se.\nsemilog plots of the parameters normalized by /an}bracketle{tn00/an}bracketri}ht. It\nis found that the normalized parameter ∆ 00is always fi-\nnite although the increase in the attractive interaction\nmonotonically decreases it. This implies that the SSF\nstate appears in the system with the arbitrary particle\nnumber. In contrast to this SSF state, the DW state\nis sensitive to the effective particle density as shown in\nFig. 4. These may be explained by the fact that in the\nsystem without a harmonic confinement ( V0= 0), the\nDW state is realized only at half filling ( n= 0.5), while\nthe SSF state is always realized. To clarify this, we also\nshow the local particle density in the noninteracting case\nat the center of the system as the broken line in Fig. 4.\nIt is found that when the quantity approaches half-filling\n(∼0.5),/an}bracketle{tnππ/an}bracketri}ht//an}bracketle{tn00/an}bracketri}httakes its maximum value, where the\nSSF state coexists with the DW state. Therefore, we can\nsay that the supersolid state is stable around this condi-\ntion. Increasing the effective particle density, the band\ninsulating states become spread around the center, while\nthe DW and SSF states should be realized in a certain\ncircular region surrounded among the empty and fully-\noccupied regions. Therefore, the normalized parameters\n/an}bracketle{tnππ/an}bracketri}htand ∆ 00are decreased with increase in ˜ ρ. On the\nother hand, in the case with low density ˜ ρ<∼0.3, the\nlocal particle density niat each site is far from half fill-\ning even when U/t= 5. Therefore, the DW state does\nnot appear in the system, but the genuine SSF state is\nrealized. These facts imply that the condition n∼0.5\nis still important to stabilize the supersolid state even in\nfermionic systems confined by a harmonic potential.18\nBy performing similar calculations for the systems\nwith low, intermediate, and high particle densities (˜ ρ∼\n0.63,2.3 and 8.9), we end up with the phase diagrams, asshown in Fig. 5. We find that increasing the attractive\nFIG. 5: (Color online) The phase diagram of the attractive\nHubbard model on the optical lattice with ˜ ρ∼0.63,2.3 and\n8.9. The density plot represents the profiles of the s-wave pair\npotential as a function of the attractive interaction U/t. The\nDW state is realized in the shaded area. The broken lines\ngive a guide to eyes which distinguishes the region with a\nfractional particle density from the empty and fully-occup ied\nregions.\ninteraction, fermionic particles gradually gather around\nthe center of the system, where the empty state is stabi-\nlized away from the center and the band insulating state\nwith fully occupied sites is stabilized. It is found that\nthe region surrounded among these states strongly de-\npend on the effective particle density ˜ ρ. The increase in\nthe effective particle density shrinks the region, which af-\nfects the stability of the SSF, DW and their coexisting\nstates. In particular, the DW region, which is shown as\nthe shaded area in Fig. 5, is sensitive to the effective par-\nticle density, as discussed above. Namely, the local pair6\npotential ∆ itakes its maximum value around U/t∼15,\nwhich may give a rough guide for the crossover region\nbetween the BCS-type and the BEC-type states. We\nnote that the DW state appears only in the BCS region\n(U/t∼5). This implies that the condition n∼0.5 is not\nsufficient, but necessary to stabilize the supersolid state\nin the attractive Hubbard model with an inhomogeneous\npotential.\nWe wish to comment on the conditions to observe the\nsupersolid state in the fermionic optical lattice system.\nNeedless to say, one of the most important conditions\nis the low temperature.19Second is the tuning of the\neffective particle density ˜ ρ(∼1), which depends on the\ncurvature of the harmonic potential as well as the total\nnumber of particles. This implies that a confined poten-\ntial play a crucial role in stabilizing the supersolid state\nin fermionic optical lattice systems. In addition to this,\nan appropriate attractive interaction is necessary to sta-\nbilize the DW state in the BCS-type SSF state. When\nthese conditions are satisfied, the supersolid state is ex-\npected to be realized at low temperatures.\nV. SUMMARY\nWe haveinvestigatedthe fermionic attractiveHubbard\nmodel in the optical lattice with harmonic confinement.By combining R-DMFT with a two-site impurity solver,\nwe have obtained the rich phase diagram on the square\nlattice, which has a remarkable domain structure includ-\ning the SSF state in the wide parameter region. By per-\nforming systematic calculations, we have then confirmed\nthat the supersolid state, where the SSF state coexists\nwith the DW state, is stabilized even in the limit with\nN→ ∞,V0→0 and ˜ρ∼const. 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Ichikawa3\n1Division of Mathematical Physics, LTH, Lund University, P. O. Box 118, S-221 00 Lund, Sweden\n2Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA\n3RIKEN Nishina Center, RIKEN, Wako, Saitama 351-0198, Japan\nA microscopic nuclear level-density model is presented. Th e model is a completely combinatorial\n(micro-canonical) model based on the folded-Yukawa single -particle potential and includes explicit\ntreatment of pairing, rotational and vibrational states. T he microscopic character of all states\nenables extraction of level distribution functions with re spect to pairing gaps, parity and angular\nmomentum. The results of the model are compared to available experimental data: neutron separa-\ntion energy level spacings, data on total level-density fun ctions from the Oslo method, cumulative\nlevel densities from low-lying discrete states, and data on parity ratios.\nI. INTRODUCTION\nNuclear many-body level-density models arekey ingre-\ndients in nuclear reaction theories, where they, for exam-\nple, govern the rates and decay patterns of astrophysi-\ncal processes and nuclear fission. In statistical methods,\nfor example the Hauser-Feshbach formalism [1] for de-\nscribing nuclear reactions, a knowledge of the level den-\nsity is crucial [2, 3, 4]. How to calculate the nuclear\nlevel density (NLD) has been a long-standing challenge\n[5, 6, 7, 8, 9, 10]. Recently it has been subject to renewed\ninterest, theoretically as well as experimentally.\nThe simplest type of model is the Fermi-gas model,\nwhich is based on the partition-function method. It pro-\nvidessimpleanalyticalformulasfortheNLD[11]. Several\nphenomenologicalextensionshavebeenproposedinorder\nto reproduce experimental data. By adjusting free model\nparameters to data these models give unprecedented ac-\ncuracy in the region of the parameter fit [4, 12]. Also\nsemi-classical methods have been used to obtain expres-\nsions for the level density [13]. However, the Fermi-gas\nmodels are unreliable outside these regions, eg. when\nthey are extrapolated to higher excitation energies or\nto nucleon numbers far from stability. Ideally nuclear\nstructure should be included in NLD models. Several\ncombinatorial models based on nuclear mean-field the-\nory have been proposed, see eg. Refs. [14, 15]. Beyond\nmean-field methods have also been used to model the\nNLD, eg. the Shell-Model Monte-Carlo method [16, 17]\nand the interacting shell model [18]. These latter models\ntake into account effective nucleon-nucleon interactions,\nbut at the same time suffer from limitations due to the\nlimited size of the Hilbert space, and hence are presently\nunable to provide global predictions for level spacings at\nthe neutron-separation energy.\nExperimentally the NLD has been subject to renewed\ninterest in the last decade partly due to the development\nof the Oslo method, which has provided new experimen-\ntal data [19]. The Oslo method provides the level density\nover extended regions of excitation energy as opposed\nto the neutron-separation level-spacings data which only\nprovide one data point at relatively high excitation en-\nergy. Also recent measurements of separate level densi-tiesof2+and2−states[20]challengetheorytoreproduce\nthese observed parity ratios.\nFew nuclear-structure models have been used to si-\nmultaneously globally describe nuclear masses, fission\nbarriers, ground-state spins and decay rates. One such\nmodel is the microscopic-macroscopic FRLDM model\nwhich has previously been used to model these observ-\nables [3, 21, 22, 23], and here serve as the starting point\nfor calculating the nuclear level density. In this paper\na combinatorial (micro-canonical) nuclear level-density\nmodel based on the folded-Yukawa single-particle model\nis presented. The model is fully microscopic with pairing\ncorrelations, vibrations, and rotational excitations cal-\nculated for each many-particle-many-hole excited state.\nPresently, the lowest ten million or so states can be ac-\ncounted for. This implies that the excitation energy re-\ngion from the ground state to well above the neutron res-\nonance region is included. No additional parameters are\nintroduced in the model, and no refitting of parameters\nof the FRLDM is performed.\nIn Sec. II the combinatorial folded-Yukawa (CFY)\nlevel-density model is described. The model allows ex-\nplicit tracking of quantum numbers, and distributions\nof pairing gaps, parity and angular momentum are dis-\ncussed in Sec. III. Results from the CFY model are com-\npared to experimental data in Sec. IV, and in Sec. V the\nCFY model is compared to other theoretical NLD mod-\nels. Finally, a short summary is given in Sec. VI.\nII. THE COMBINATORIAL MODEL OF NLD\n(CFY)\nThe NLD is calculated by means of a combinatorial\ncountingofexcitedmany-particle-many-holestatesasde-\nscribed in Sec. IIA. In Sec. IIB we present how pairing\nis taken into account for excited states by explicitly solv-\ning the BCS equations for all individual configurations.\nRotations are taken into account combinatorially with\na pairing-dependent moment of inertia (Sec. IIC). The\nvibrational contribution to the NLD is investigated by\nincluding microscopically described phonons using the\nQuasi-particle Tamm-Dancoff Approximation (QTDA),2\nseeSec.IID. InSec.IIEitisdescribedhowwebrieflyac-\ncount for a general residual interaction causing a smear-\ning of the level-density distribution at higher excitation\nenergies.\nAll produced nuclearlevels, calculatedasdescribedbe-\nlow, are sorted into a binned level density, where the typ-\nical bin size is in the range ∆ E= 30–50 keV. The level\ndensity is calculated by counting the number of levels in\nthe energy bin,\nρ(Eb,I,π) =1\n∆E/integraldisplayEb+∆E\n2\nEb−∆E\n2/summationdisplay\niδ(E−Ei(I,π))dE,(1)\nwhereEbis the energy center of bin bandEi(I,π)\ndenotes the calculated state with energy Ei(given by\nEq. (15)), angular momentum Iand parity π. The total\nlevel density at a given excitation energy Eis\nρtot(E) =/summationdisplay\nI,πρ(E,I,π). (2)\nA. Combinatorial intrinsic level density\nA high-quality combinatorial level-density model re-\nquires a realistic description of the single-particle ener-\ngies. We shall here utilize a well-tested mean-field model,\nnamely the microscopic-macroscopic finite range liquid\ndrop model (FRLDM) [21], that provides a good global\ndescriptionofseveralnuclear-structurepropertiessuchas\nmasses, fission barriers and beta-decay properties. The\ngood agreement between ground-state spins calculated\nin the FRLDM model and experiment implies that the\nsingle-particle spectrum close to the Fermi surface is well\ndescribed [22].\nThe single-particle energies are thus obtained by solv-\ning the one-body Schr¨ odinger equation,\n(T+VFY(¯ε))|ν/an}bracketri}ht=eν|ν/an}bracketri}ht, (3)\nfor protons as well as for neutrons, where VFYis the\nFolded-Yukawasingle-particle potential. The parameters\nof the potential, as well as the deformation parameters,\n¯ε= (ε2,ε3,ε4,...), are taken from an extensive calcula-\ntion of nuclear masses [21]. All parameters of the single-\nparticle model are thus fixed from other studies. One\naim of the present study is to see how well highly excited\nstates (in terms of level-densities) are described for all\nnuclei heavier than16O.\nThe many-body ground state (the many-body particle\nvacuum) |0/an}bracketri}ht, isobtainedbyfillingthe lowest N(Z)states\n(including time-reversed states),\n|0/an}bracketri}ht=N(Z)/summationdisplay\nν=1a+\nν|−/an}bracketri}ht, (4)\nwhere|−/an}bracketri}htis the one-body vacuum. Intrinsic excitations\nare obtained by many-particle-many-hole excitations onthe many-body particle vacuum,\n|i/an}bracketri}ht=n/productdisplay\nα=1a+\nναaν′α|0/an}bracketri}ht, (5)\nwhereνandν′span all single-particle states (as well as\ntime-reversed states) in the potential. We include all n-\nparticle-n-hole states with n <10 for protons as well as\nfor neutrons. Corresponding energies are given by\nEi=E0+n/summationdisplay\nα=1(eνα−eν′\nα), (6)\nwhereE0is the ground-state energy. (When pairing\nis included this expression is modified as described in\nSec. IIB.) Nuclei are assumed to have constant defor-\nmation for all excitation energies. For spherical and\nwell-deformed nuclei this assumption is expected to be\na good approximation for excitation energies here con-\nsidered (mainly below the neutron separation energy).\nFor transitional nuclei this approximation might, how-\never, introduce some inaccuracies in the calculations.\nIn addition to the energy and the number of unpaired\nnucleons (seniority), we also register the total parity and\nK-quantum number for each excited state, where Kas\nusualisthe angularmomentum projectiononan intrinsic\nsymmetry axis. The identification of a K-quantum num-\nber can be done for all nuclei by assuming an infinites-\nimal deformation for spherical nuclei. The K-quantum\nnumber of the individual unpaired nucleons are allowed\nto couple in all different ways to build up the total K-\nquantum number.\nB. Pairing\nThe many-body wave-function of the excited states is\napproximated by the BCS wave-function with excited\nquasi-particles\n|τ/an}bracketri}ht=/productdisplay\nν′′∈τ2(−Vν′′+Uν′′a†\nν′′a†\n¯ν′′)×\n×/productdisplay\nν′∈τ1a†\nν′/productdisplay\nν∈τ0(Uν+Vνa†\nνa†\n¯ν)|0/an}bracketri}ht, (7)\nwhereτ2,τ1andτ0denote the spaces of double, single\nand zero quasi-particle excitations, respectively, used to\nbuild the many-body state iof Eq. 5. UνandVνare the\nstandard BCS vacancy and occupation factors and |0/an}bracketri}htis\nthe particle vacuum, see e.g. Ref. [24]. For the excited\npairs in the group τ2the effect (compared to zero-quasi-\nparticle states) is simply\nUν→ −Vν, V ν→Uν. (8)\nThepairinggap∆andtheFermienergy λareobtained3\nby solving the BCS-equations\n∆ =G/bracketleftBigg/summationdisplay\nν∈τ0UνVν−/summationdisplay\nν′′∈τ2Uν′′Vν′′/bracketrightBigg\n,(9)\nN= 2/summationdisplay\nν∈τ0V2\nν+/summationdisplay\nν′∈τ11+2/summationdisplay\nν′′∈τ2U2\nν′′(10)\nfor each state.\nThe pairing strength in the BCS model is governed\nby a single, primary BCS model parameter, namely the\nparameter of the effective-interaction pairing gap ( rin\nEq.˜(47) in [25] or the equivalent cpin Eq. (49) and\ncnin Eq. (50) in [26]). In an adjustment of macro-\nscopic, pairing, and other model parameters to optimize\na nuclear mass model r= 4.8 MeV was obtained for\nthe standard BCS pairing model [21]. Because the BCS\nmodel we use here, in contrast to Ref. [21, 25], includes\nblocking and particle-number-projection it is necessary\nto use a pairing strength optimized for this method. In\nRef. [26] cp=cn= 4.95 MeV were obtained in an ad-\njustment of calculated pairing gaps to odd-even mass\ndifferences. However, it has been pointed out [21, 25]\nthat odd-even mass differences are subject to numerous\nnon-smooth contributions, for example from deformation\nchanges and from irregularities in the microscopic level\nstructure, not just odd-even staggering due to pairing. It\nis thereforebetter to determine the pairing strength from\na full nuclear mass calculation that includes the particu-\nlar pairing model under consideration and the associated\nadjustment of all model parameters [21]. In the folded-\nYukawa macroscopic-microscopic mass model this yields\nthe value 4.5 MeV for the optimized strength parameter\nof the particle-number-projected BCS model [26]. The\npairing strength G, which is used to calculate pairing\ngaps for all excited states in the level-density calculation,\nis determined from this average pairing gap for each nu-\nclear system; for details see Ref. [26].\nIt follows that the excitation energy of the intrinsic\nmany-body configuration for one nucleon type is\nEmb,t= 2/summationdisplay\nν∈τ0eνV2\nν+/summationdisplay\nν′∈τ1eν+2/summationdisplay\nν′′∈τ2eνU2\nν′′−\nG/summationdisplay\nν∈τ0V4\nν−G\n2/summationdisplay\nν′∈τ11−G/summationdisplay\nν′′∈τ2U4\nν′′−∆2\nG−E0\nt,(11)\nwheretdenotes protons or neutrons and E0\ntis the proton\nor neutron part of the ground-state energy. The pairing\ngap and Fermi level are calculated by solving the BCS\nequationsEqs.(9)and(10). Thetotalintrinsicexcitation\nenergy of a many-body state iis\nEi\nmb=Emb,p+Emb,n. (12)\nThe combinatorial approach to calculate the level den-\nsity involves calculating all possible many-body states,\nEi\nmb, which thus means that the BCS equations are\nsolved about 107times for each nucleus.C. Rotations\nA general feature of deformed nuclei is the existence of\nrotational bands built on the ground state as well as on\nexcited states. The question is how to incorporate these\nstates in the NLD. In principle, rotational states may be\nmicroscopically treated by solving the cranking Hamil-\ntonian,hω=h0−ωrotjx, whereωrotis the rotational\nfrequency and jxis the angular momentum operator for\nrotations around the x-axis. Matrix elements of the jx-\noperator correspond to energy excitations of the order\n∆E=ε2¯hω≈ε2·41A−1/3MeV for a nucleus with mass\nnumberAand quadrupole deformation ε2[27]. This en-\nergy can be compared to a typical energy of lowest ro-\ntational state, E2+=¯h2\n2J2(2 +1) ≈90A−5/3MeV, and\nwe see that ∆ E >> E 2+for all considered nuclei. Since\nthecollectivestateandthe correspondingmatrixelement\nare well separated in energy, the risk of double-counting\nstates by adding a rotational band on a microscopically\ncalculated band-head is quite small, cf Ref. [7]. This is\ntrue as long as the excitation energies are smaller than\nenergies corresponding to the temperature, T= ∆E, i.e.\nfor excitation energies smaller than about 10 A1/3MeV\n(forε2= 0.25), giving about 30 MeV for mass A=25\nand 55 MeV for A=160. For higher excitation energies a\nsaturation of the rotational enhancement should set in.\nIn this study we concentrate on lower energies, say up to\nabout10MeV. It is thereforeareasonableapproximation\nto account for rotational enhancement by simply adding\na rotational band on each band head; double-counting of\nstates would not occur.\nRotational states are consequently taken into account\nbyaddingarotationalbandontopofeachintrinsicband-\nhead for deformed nuclei (here defined as nuclei with cal-\nculated quadrupole deformation |ε2| ≥0.05). The rota-\ntional energy of the different angular momentum states,\nI, in the rotational band is given by\nEi\nrot(I,Ki) =I(I+1)−K2\ni\n2J⊥(ε2,∆ip,∆in), (13)\nwhereKiis the angular momentum projection on the\nsymmetry axis of the intrinsic state iupon which the\nrotational band is built. The quadrupole deformation is\ndenoted by ε2while ∆i\npand ∆i\nndenote the proton and\nneutron pairinggapsof the intrinsic state i. The moment\nof inertia around an axis perpendicular to the symmetry\naxisJ⊥(ε2,∆i\np,∆i\nn) is approximated by the rigid-body\nmoment of inertia with deformation ε2, modified by the\ncalculated pairing gaps for the considered state, as given\nin Ref. [28]. Given the angular momentum projection\nKand parity πof the band-head the rotational band\nincludes the following levels\nIπ=\n\nKπ,(K+1)π,(K+2)π,...ifK/ne}ationslash= 0,\n0+,2+,4+,... ifK= 0+,\n1−,3−,5−,... ifK= 0−.(14)\nThe Coriolis anti-pairing effect is neglected and no vir-\ntual crossings of rotational bands are taken into account.4\n0 1 2 3 4 5 6 7 8\nExcitation Energy Eexc [MeV]012345678910Rotational enhancement Krot\n0 1 2 3 4 5 6 78\nExcitation Energy Eexc [MeV]0204060Krot\n162Dy\nFIG. 1: (Color online) Rotational enhancement, Krot, for the\nnucleus162Dy as a function of excitation energy. The insert\nshows the enhancement compared to the simple enhancement\nmodel of Ref. [29] (red dashed line).\nThus, the pairinggap and moment ofinertia areassumed\nto be unchanged from the band-head pairing gap for all\nstates in the rotational band. This approximation is rea-\nsonable since mainly low-spin states play a role in the\npresent study.\nThe total energy of the calculated level is thus given\nby\nEi=Ei\nmb+Ei\nrot(I,Ki) (15)\nwhereEi\nmbis the energy, Eq. (12), of the intrinsic many-\nbody configuration i, andEi\nrot(I,Ki) is the rotation en-\nergy, Eq. (13), of the level with angular momentum I\nbuilt from the intrinsic many-body configuration with\nangular momentum projection Ki. These energies are\nused to calculate the level density according to Eq. (1).\nFig. 1 shows the rotational enhancement, Krot, for the\nwell-deformed nucleus ( ε2=0.26)162Dy (cf. Fig. 12) cal-\nculated as the ratio of the level density when rotations\nare included or excluded. For low excitation energies\nthere are large fluctuations which are artifacts of the low\nlevel density combined with the smoothing procedure of\nSec. IIE. For higher excitation energies ( >∼3 MeV) the\nrotational enhancement is a slowly increasing function,\nof the order of a factor 5 at the neutron separation en-\nergy. This prediction is compared to the SU(3) model\nof Ref. [29], which is shown in the insert of Fig. 1. The\nSU(3) model gives almost an order of magnitude larger\nenhancement for excitation energies in the region of the\nneutron separation energy. We note that a combinatorial\nlevel-density model based on the Nilsson potential gives\na rotational enhancement for162Dy similar to our results\n[30].D. Vibrations\nMany nuclei exhibit low-lying states of vibrational\ncharacter, which are usually of quadrupole or octupole\ntype. Such low-lying vibrational states appear at exci-\ntation energies not much lower than the 2qp excitations\nwhich in a coherent way build up the collective state. We\ntherefore believe that the vibrational enhancement of the\nlevel density, contraryto the rotationalenhancement dis-\ncussed above, must be described microscopically.\nIn order to describe vibrational states the Quasi-\nparticle-Tamm-Dancoff-Approximation (QTDA) is used.\nAccording to the Brink-Axel hypothesis [31, 32] phonons\nare built on every intrinsic many-body configuration\nEi\nmb. The QTDA equation is solved for each state iin\norder to get all possible phonon excitation energies and\nwave-functions. Thismeans solvingthe QTDAequations\nmillions of times for every single nucleus.\nThe residual interaction is approximated by the dou-\nble stretched (isoscalar) Quadrupole-Quadrupole inter-\naction. This interaction is well defined in the case of a\nharmonicoscillatorpotential. Inthecaseofafinite-depth\npotential as the folded-Yukawa potential the interaction\nshould take into account additional finite size effects, for\nexample as is done in Ref. [33]. In the present work the\nfinite-depth effects are ignored and the double stretched\napproach is used as defined in Refs. [34, 35].\nThe QTDA secular equation can be written [36]\n1\nχ2K=/summationdisplay\nµν/vextendsingle/vextendsingle/angbracketleftbig\nµ/vextendsingle/vextendsingle¯Q2K/vextendsingle/vextendsingleν/angbracketrightbig/vextendsingle/vextendsingle2(UµVν+VµUν)2\n(Eqp,i\nµ+Eqp,i\nν)−(¯hω)i\nj,(16)\nwhere the effect of Eq. (8) has not been explicitly written\nout.¯Q2Kis the double stretched quadrupole operator,\nwhere the components K= 0 and K= 2 are considered.\nThe set of roots {(¯hω)i\nj}of this equation is the excitation\nenergies of the vibrational phonons jon top of the intrin-\nsic state i, whereas the poles Eqp,i\nµ=/radicalbig\n(eµ−λ)2+∆2\ni\nare the unperturbed two-quasiparticle excitations on the\nmany-body configuration Ei\nmbwith pairing gap ∆ ias\ncalculated by Eqs. (9) and (10), and eµare the single-\nparticle energies.\nThe self-consistent coupling strength is given by [34]\nχ2K=8π\n5Mω2\n0\nA/an}bracketle{t¯r2/an}bracketri}ht+g2K/radicalBig\n4π\n5A/angbracketleftbig¯Q20/angbracketrightbig,(17)\nwhereg20= 1 and g22=−1. The expectation-values/angbracketleftbig\n¯r2/angbracketrightbig\nand/angbracketleftbig¯Q20/angbracketrightbig\nare calculated in double stretched co-\nordinates [34]. To test that this gives overall reason-\nable result, we have verified for a large number of nuclei\nthat for the K= 0 and the K= 2 components of the\niso-scalar Giant Quadrupole Resonances the calculations\nagree well with the systematics of the Giant Resonance\nenergies ¯ hωGQR= 58A−1/3MeV [34].\nThe phonons are never repeated and double counting\nis explicitly avoided by the following procedure. The5\nphonon wave functions are given by\nO†=/summationdisplay\nµ,νXµ,νa†\nµaν, (18)\nwhereXµ,νare the wave-function components of all ex-\ncited quasi-particle states a†\nµaνon top of the configura-\ntionEi\nmb. The level density is increased by one state at\nthe energy of the phonon (¯ hω)i\nj, and decreased by the\namount given by the wave-function component X2\nµ,νat\nthe energy of the corresponding pole ( Eqp,i\nµ+Eqp,i\nν). The\nchange in level density due to the phonon jon top of in-\ntrinsic state iis thus\nδρ(E) =δ(E−Ei\nmb−(¯hω)i\nj)−\n/summationdisplay\nµ,νX2\nµ,νδ/parenleftbig\nE−Ei\nmb−(Eqp,i\nµ+Eqp,i\nν)/parenrightbig\n,(19)\nwhereEis the excitation energy relative to the ground-\nstate. This procedure increases the level density by re-\nducing the strength of the pure quasi-particle excitations\nand adding strength by means of the phonons (which are\npushed down in energy due to the QQ-interaction).\nMost phonons have little collectivity in the QTDA ap-\nproximation, as most of the phonon excitations lie very\nclose to a pure quasi-particle excitation, and hence the\nwave-function is completely dominated by that single\ncomponent. Even if there are a few very collective low-\nlying phonon states the vibrational enhancement of the\nlevel density becomes small, since most excited states\nprovide quite non-collective phonon excitations. Due to\nthis non-collectivityofmostphonons thereis no waythat\nthe phonons, in general, can be repeated to form two- or\neven three-phonon excitations.\nThe vibrational enhancement factor in this method is\nin general quite small, of the order of a few percent. This\nissubstantiallylowerthanpredictionsfromothermodels.\nFor example, the attenuated phonon method [12, 14, 37]\nand the Boson partition function method [38] both give\nup to an order of magnitude enhancement at the neu-\ntron separation energy. Fig. 2 shows the vibrational en-\nhancement as a function of excitation energy for162Dy.\nThe effect is very small, close to 1 % at 7 MeV excita-\ntion energy. For the same nucleus the attenuated phonon\nmethod gives an enhancement factor of about 3 as shown\nin the insert of Fig. 2.\nThe level-density enhancement due to quadrupole vi-\nbrations is thus found to be very small. Higher multipole\nvibrations (as octupole vibrations) are expected to con-\ntribute with enhancements of the same order of magni-\ntude or less and are neglected in the CFY model.\nE. Many-body damping width\nIn the mean-field approach all excited many-body\nstates are treated as non-interacting. A residual two-\nbody interaction will mix the many-body states obtained0 1 2 3 4 5 6 7 8\nExcitation Energy Eexc [MeV]11.0051.011.0151.02Vibrational enhancement Kvib\n0 1 2 3 4 5 6 78\nExcitation Energy [MeV]11.522.533.5Kvib162Dy\nFIG. 2: (Color online) Vibrational enhancement, Kvib, us-\ning the QTDA method for the nucleus162Dy as a function\nof excitation energy. The insert shows the enhancement com-\npared to the enhancement of the attenuated phonon model of\nRef. [12] (red dashed line).\nfrom the combinatorics. Smearing effects from the resid-\nual interaction can approximately be taken into account\nby assuming a spreading width of all excited states. The\nspreading width is implemented in terms of a Gaussian\nenvelope with width σ, i.e. the delta functions in Eq. 1\nare replaced by Gaussians. Estimates of the spreading\nwidth FWHM gives [39]\nΓ = 0.039/parenleftbiggA\n160/parenrightbigg−1/2\nE3/2MeV,(20)\nwhere the FWHM is related to the Gaussian spreading\nwidthσ=Γ\n2√\n2ln2.\nAssuming that all excited many-body states in an en-\nergy bin ∆ Eare uniformly distributed, the level density\nbecomes\nρ(Eb) =/summationdisplay\naρ(Ea)1\n2/bracketleftbigg\nerf/parenleftbiggEa+∆E/2−Eb√\n2σ/parenrightbigg\n−\n−erf/parenleftbiggEa−∆E/2−Eb√\n2σ/parenrightbigg/bracketrightbigg\n,(21)\nfor bin-point b. The method implies a smearing out of\nlevel-densitypropertiesovera rangeΓ, which is smoothly\nincreasing with excitation energy. As a consequence fluc-\ntuations of energies and wave-functions in the range Γ\nfollow GOE statistics of Random Matrix theory, that is\noften denoted as quantum chaos in the nucleus, see e.g\nRefs. [40, 41].\nIII. LEVEL DISTRIBUTIONS\nIn the present combinatorial approach it is possible\nto extract different distributions exhibiting details of the\nlevel-density function. We present here microscopically6\ncalculated distributions for pairing gaps (Sec. IIIA), par-\nity (Sec. IIIB), and angular momentum (Sec. IIIC).\nA. Pairing-gap distribution\nThe BCS equations, Eqs. (9) and (10), are solved for\nall individual many-body configurations and hence pair-\ning gaps for all states are obtained. In Fig. 3 the dis-\ntribution of proton pairing gaps for162Dy is shown for\na number of excitation energies. For the lowest excita-\ntion energies only the ground-state and the states in the\nground-state rotational band exist. The pairing gap of\nthe levels in the rotational band is fixed to be the same\nas the pairing gap of the band-head, see Sec. IIC. As the\nexcitation energy increases levels with reduced pairing\ngaps appear. However, no transition to a completely un-\npaired system is observed, and levels with non-collapsed\ngaps (∆>0) surviveto the highest considered excitation\nenergies. At the highest considered excitation energy for\n162Dy,Eexc= 8.4MeV, 36%ofthe levelsstill havea non-\nzeroprotonpairinggap,seeFig.3. Themeanvalueofthe\nproton pairing gap /an}bracketle{t∆/an}bracketri}htat different excitation energies is\nshown in Fig. 4 for162Dy. Between 2 MeV and 3.5 MeV\nexcitation energy there is a rapid decrease in the mean\npairing gap. At higher excitation energies the decrease\nis slower. At 8.4 MeV excitation energy the mean value\nis/an}bracketle{t∆/an}bracketri}ht= 0.2 MeV. The reason why pairing may survive\nand be of substantial size in states at these high ener-\ngies can be explained as follows: Several highly excited\nstates are built by exciting particles and holes far from\nthe Fermi energy, where blocking in the BCS equations\nhas no effect on the pairing gap.\nThe non-collapsed pairing gaps influence the moment\nof inertia and keep it reduced as compared to the rigid-\nbody value. This implies that even at excitation energies\nin the region of the neutron separation energy the mo-\nment of inertia is on average smaller than the rigid-body\nvalue. A similar observation is made in Ref. [15].\nB. Parity distribution\nIn Fermi-gas level-density models there is an implicit\nassumption of equal number of states with different par-\nity at any given excitation energy. However, models that\ntake into account microscopic effects show clear struc-\nture in the parity ratio, see eg. Refs. [16, 42, 43]. In\nconnection with astrophysical reaction rates the parity\nratio may play an important role, and can be included in\nthe Hauser Feshbach formalism, see eg Refs. [44, 45].\nFrom the single-particle point of view it is clear that\nthe parityratioshould showstructure. For84\n38Sr46the sit-\nuationisilluminating. Thenucleusisclosetosphericalin\nits ground-state ( ε2= 0.05) [21]. It has 2 proton holes in\nthepfshell and 6 neutrons in the g9/2shell. Since there\nare large gaps in the single-particle spectrum the energy\nto excite nucleons across the gaps, which cause changes0246\n 67 % Eexc= 0.7 MeV\n024 71 % Eexc= 1.8 MeVDistribution162Dy\n024 59 % Eexc= 2.9 MeV\n0 0.25 0.5 0.75024 49 % Eexc= 4.0 MeV 47 % Eexc= 5.1 MeV\n 42 % Eexc= 6.2 MeV\n 39 % Eexc= 7.3 MeV\n0 0.25 0.5 0.75 1 36 % Eexc= 8.4 MeV\nPairing Gap ∆ [MeV]\nFIG. 3: (Color online) Proton pairing-gap distributions at\ndifferent excitation energies Eexcfor162Dy shown in 20 keV\npairing-gap bins. The proportion of paired states (∆ >0) are\nshown in percent in the boxes.\n0 1 2 3 4 5 6 7 8 9\nExcitation Energy Eexc [MeV]00.10.20.30.40.50.6Average Proton Pairing Gap < ∆> [MeV]162Dy\nFIG.4: Average protonpairinggap /angbracketleft∆/angbracketrightat differentexcitation\nenergies Eexcfor162Dy. The mean values are given by the\ndistributions in Fig. 3.\nin parity, is quite large. The parity ratio displays long-\nrangeoscillations,seeupperleft panelofFig.5, whichare\ndirectly connected to the large single-particle gaps. The\nsingle-particle gaps effectively decrease with increasing\ndeformation, and the oscillations are therefore expected\nto decrease with deformation. To test these ideas the\nparityratio in84Sr is calculated at different deformations\nand shown in Fig. 5. Indeed, for larger deformations the\noscillatory pattern vanishes and the parity ratio equili-\nbrates to one.\nIn Fig. 6 the parity ratios for56Fe,60Ni and68Zn are\nshownandcomparedtotwocalculationsofRef.[16]. The\nsolid blue lines show the CFY model, the red dashed\nlines show the statistical parity projection model and\nthe black dots show calculations using the Shell-Model\nMonte-Carlo method. The parity ratio in the Fe and7\n0123\nε2=0.0584Sr ε2=0.20\n012Parity Ratio ρ-/ρ+\nε2=0.10 ε2=0.25\n0 5 10012 ε2=0.15\n0 5 10 15ε2=0.30\nExcitation Energy Eexc [MeV]\nFIG. 5: (Color online) Parity ratio versus excitation energ y\nfor84Sr calculated with different deformations. Large defor-\nmations imply a parity distribution close to 1 even at low\nexcitation energies. For small deformations the parity non -\nequilibrium can survive to high excitation energies.\n00.511.52\n56Fe, ε2=0.00\n60Ni, ε2=0.025\n68Zn, ε2=-0.1500.511.5Parity Ratio ρ-/ρ+\n5 10 15 20\nExcitation Energy Eexc [MeV]00.511.5\nFIG. 6: (Color online) Parity ratio versus excitation energ y.\nThe solid blue lines are calculated with the CFY model. The\nred dashed line and black dots are given by a statistical mode l\nand a Monte-Carlo method, respectively [16].\nNi isotopes is not equilibrated below 15 MeV in any of\nthe calculations, while68Zn equilibrates at much lower\nenergies (below 10 MeV). For60Ni and68Zn there is\na clear oscillatory behavior prior to equilibration in the\nCFY model, and in the case of Ni there is a good agree-\nment between the Monte-Carlo method and the CFY\nmodel, especially in the region of 8–12 MeV excitation\nenergy. Note, however, that fluctuations seen in the\npresent micro-canonical approach may be smeared out\nby the grand-canonical approach as used in Ref. [16].\nThe parity ratio has been calculated for Sr-isotopes\nwithin a statistical method in Ref. [45] where the im-\npact on astrophysical reaction rates was investigated and\nfound to be small. The statistical method gives at most\none oscillatorymaximum before it equilibrates, asseen in0123\n76Sr, ε2=0.37582Sr, ε2=0.05\n012Parity Ratio ρ-/ρ+\n78Sr, ε2=0.37584Sr, ε2=0.05\n0 5 10 1501280Sr, ε2=0.05\n0 5 10 15 2086Sr, ε2=0.05\nExcitation Energy Eexc [MeV]\nFIG. 7: (Color online) Parity ratio versus excitation energ y\nfor Sr isotopes. The solid blue lines are obtained from the\nCFY model and the dashed red lines are obtained from the\nstatistical parity projection of Ref. [45].\nFig. 7 (dashed lines). In the CFY model the parity ratio\nhassubstantiallymorestructure(solidlines). Nucleiwith\nsmall ground-state deformations show long-range oscil-\nlations which survive to high excitation energies before\nequilibration. The overall results are quite different from\nthe model of Ref. [45].\nC. Angular momentum distribution\nIn Fermi-gas models the distribution of angular mo-\nmentum is given by the Gaussian envelope in the spin\ncutoff model, see eg. Ref. [4],\nF(U,I) =2I+1\n2σ2exp/parenleftbigg−I(I+1)\n2σ2/parenrightbigg\n,(22)\nwhich is obtained by random coupling of uncorrelated\nspins of the nucleons. In Eq.(22) the spin cutoff factor,\nσ, is defined by\nσ2=Jrigid\n¯h2/radicalbigg\nU\na, (23)\nwhereJrigidis the rigid-body moment of inertia, U=\nE−δis the effective excitation energy shifted by the\nback-shift δ, andais the level-density parameter.\nIn Fig. 8 the angular-momentumdistributions for68Zn\nand162Dy are shown for several excitation-energy re-\ngions. The black lines with dots show the CFY model\nresults while the red solid lines show the Gaussian dis-\ntribution of Eq. (22) with the spin cutoff factors given in\nRef. [4], and the blue dashed lines show Gaussian distri-\nbutions fitted to the combinatorial calculation. For low\nexcitation energies there are clear deviations from the\nGaussian profiles while for higher excitations the combi-\nnatorial distribution tends to the Gaussian profile. How-\never, for68Zn the deviations from a Gaussian angular8\n00.050.10.150.20.25\nEexc = 3.02 MeV Eexc = 4.86 MeV\n0 5 10 1500.050.10.150.2\nEexc = 6.76 MeVAngular-momentum distribution\n0 5 10 15 20Eexc = 8.67 MeV\nAngular Momentum I68Zn\n00.050.10.150.2\nEexc = 2.63 MeV Eexc = 4.28 MeV\n0 5 10 1500.050.10.15\nEexc = 5.94 MeV\n0 5 10 15 20Eexc = 7.60 MeVAngular-momentum distribution\nAngular Momentum I162Dy\nFIG. 8: (Color online) Angular-momentum distribution for\n68Zn and162Dy in 4 different excitation-energy regions. The\nblack lines with dots show results from the CFY model. The\nred solid lines and blue dashed lines show the Gaussians give n\nby Eq. (22) using the spin cutoff factors from Ref. [4] and a\ndirect fit to CFY, respectively.\nmomentum distribution remain up to about the neutron\nseparation energy ( Sn=6.48 MeV).\nForlowexcitationenergiesin68Znthereisanodd-even\nspin staggering which is not explained by the spin cutoff\nmodel. This effect has also been observed in Fe-isotopes\nin the Shell Model Monte-Carlo Method [17].\nFig. 9 shows the spin cutoff factor deduced from the\nCFY model (black solid line) and from the statistical\nmodel [4] (red dashed line), as a function of excitation\nenergy for162Dy. For this particular nucleus the spin\ncutoff factor for excitation energies >∼3 MeV is similar in\nshape but ∼10 % larger than in the statistical model [4].\nThe fact that the statistical model is smaller than the\nCFY model, despite the presence of pairing in the CFY\nmodel, is accidental for this nucleus. Since both the rigid\nbody moment of inertia and the level density parameter\nare fitted to experiments using only three parameter, the\ndetailed description for a particular nucleus might give\nthis result when compared to a very different model as01234567Spin cutoff factor σ\n0 1 2 3 4 5 6 7 8\nExcitation Energy Eexc [MeV] 0.50.7511.25σCFY/σStat162Dy\nFIG. 9: (Color online) The top panel shows the spin cutoff\nfactor as a function of excitation energy for162Dy in the CFY\nmodel (black solid line) and the statistical model of Ref. [4 ]\n(red dashed line). The bottom panel shows the ratio of the\ncombinatorial and statistical spin cutoff factors of the top\npanel.\nthe CFY model.\nIV. COMPARISON WITH EXPERIMENTAL\nDATA\nA. Neutron separation-energy level spacings\nThe s-wave neutron resonance spacings constitute the\nmost comprehensive experimental database for compar-\nison with NLD calculations [46]. This database serves\nas a benchmark for all large-scale level-density models\n[4, 14, 15, 47].\nThe s-wave neutron resonance spacing D0at the\nneutron separation energy Snof the compound nu-\ncleus (Z,N) is obtained from calculated level densities\nρ(E,I,π) as,\n1\nD0=\n/braceleftbigg\nρ(Sn,I0+1/2,π0)+ρ(Sn,I0−1/2,π0) forI0>0,\nρ(Sn,1/2,π0) for I0= 0,\n(24)\nwhereI0is the ground-state spin and π0is the ground-\nstate parity of the target nucleus ( Z,N−1). In Fig. 10\nwe study the ratio of calculated and experimental level\nspacings at the neutron separation energy, DTh/DExp,\nfor all nuclei in the database.\nThe quality of a level-density model can be estimated\nby the rms-factor [15]\nfrms= exp/bracketleftBigg\n1\nNeNe/summationdisplay\ni=1ln2Di\nTh\nDi\nExp/bracketrightBigg1/2\n,(25)9\n0 50 100 150 200 250 300\nMass Number A0.010.1110100DTh/DExp\nFIG. 10: Ratio between theoretical and experimental level\nspacings at neutron separation energy versus mass number\nA. The rms-factor is frms= 4.2. The experimental data are\ntaken from Ref. [46].\nand the mean factor\nm= exp/bracketleftBigg\n1\nNeNe/summationdisplay\ni=1lnDi\nTh\nDi\nExp/bracketrightBigg\n, (26)\nwhereDi\nThandDi\nExpare the theoretical and experimen-\ntal level spacings and Neis the number of nuclei in the\ndatabase. The CFY model gives frms= 4.2 andm= 1.1,\nsee Fig. 10 of the error ratio versus A. These results are\ncompared to other statistical and combinatorial models\nin Table II and discussed in Sec. V.\nThereseems to be a clearresidual shellstructure in the\nlevel spacings, as seen in Fig. 10, especially in the doubly\nmagic208Pb region. A similar effect can be observed\nfor the Gogny model in Ref. [15], and in results of the\nSkyrme model with the BSk9 interaction, as shown in\nRef. [14]. In addition, there seems to be a drift with\nmass number where especially nuclei in the rare-earth\nandactinideregionsover-estimatetheneutronseparation\nlevel spacings. A similar drift can be seen in the Gogny\nmodel of Ref. [15] while it seems not to be present in the\nSkyrme-HFB model in Ref. [14].\nThe rare-earth and actinide regions, where too large\nlevel spacings are calculated, mainly correspond to de-\nformed nuclei. On the other hand, nuclei corresponding\nto regions with particularly low ratio DTh/DExpseen in\nFig. 10(around A=32, 132and 208)correspondto spher-\nical nuclei. The overestimation of Dfor deformed nuclei\n(too low calculated level density), and underestimation\nofDfor spherical nuclei (too high calculated level den-\nsity) seems to be approximatively systematic, as seen in\nFig. 11, where the ratio between theoretical and experi-\nmental level spacings is shown versus the absolute value\nof the deformation.\nHowever, as seen in the lower panel of Fig. 11 there\nis no clear correlation with the microscopic energy Emic\nof Ref. [21] as might be expected from the residual shell\nstructure seen in Fig. 10.0 0.1 0.2 0.3 0.4\nDeformation | ε2|0.010.1110100DTh/DExp\n-15 -10 -5 0 5\nMicroscopic Energy Emic [MeV]0.010.1110100DTh/DExp\nFIG.11: (Color online)Thetoppanelshowstheratiobetween\ntheoretical and experimental level spacings at neutron sep a-\nration energy versus the absolute value |ε2|of the calculated\nquadrupole deformation. The bottom panel shows this ratio\nas a function of the microscopic energy. The solid red line\nshows an exponential fit to illustrate the correlation. The m i-\ncroscopic energies and the quadrupole deformations are tak en\nfrom Ref. [21]. The experimental data are from Ref. [46].\nB. Detailed level-density functions\nThe Oslo method is a commonly used experimental\nmethod for extracting detailed level-density functions for\nlarge ranges of excitation energy [19]. It provides a valu-\nable test for NLD models. However, the approach is\nmodel dependent since the level density is extracted by\nuse of a back-shifted Fermi Gas approximation and a\nspin-cutoff factor, see also discussion in Ref. [38].\nIn Figs. 12 and 13 level densities obtained from the\nCFY model are compared to data for a number of nuclei\nwhere experimental data are available [48, 49, 50, 51, 52,\n53].\nIn Fig. 12 we show level densities of the rare-earth nu-\nclei148,149Sm,161,162Dy,166,167Er, and170,171,172Yb as\nfunctions of excitation energy. The data are in general\nwell reproduced by the CFY model, with an error of less\nthan a factor of 2. The good agreement between the cal-\nculatedand experimentalslopesindicatesthat the single-\nparticle structure and the moments of inertia in the rota-\ntional bands are sound. However, an observable trend in\nthismassregionisthatthe leveldensityforeven-evennu-\nclei is slightly over-estimated and for odd nuclei the level\ndensity is slightly under-estimated. The effective back-\nshift is to a large extent controlled by the ground-state\npairing gap. By fine-tuning the pairing gaps it is possi-\nble to get an almost perfect agreement with experiment.\nHowever, in this paper no efforts are made to adjust pa-\nrameters of the model to fit measured level-density data.\nIndeed, the ground-state pairing gaps are given by the\nmass model, see Sec. IIB. Neither are any renormaliza-\ntions of calculated level densities to fit data performed,\nas is sometimes done in other models, see Refs. [14, 38].10\n100102104106108\n100102104106\n100102104106\n0 1 2 3 4 5 6 78\nExcitation Energy Eexc [MeV]100102104106149Sm\n148Sm\n161Dy\n162Dy\n167Er\n166Er\n171Yb\n170Yb, 172YbLevel density ρ [MeV-1]\nFIG. 12: (Color online) Level densities ρas functions of ex-\ncitation energy for Sm, Dy, Er and Yb isotopes. The black\nsolid lines show the CFY model and the red dots show the\nexperimental data [50, 51, 52, 53].\nExperimental level-density functions are available also\nfor lighter mass regions, and in Fig. 13 data for V and\nMo isotopes are shown. The overall agreement between\nthe model and these experimental data is somewhat in-\nferior to what was obtained in the rare-earth region. For\nthe Mo isotopes in Fig. 13 the CFY model is roughly a\nfactor of 3 too large for high excitation energies while\nfor the V isotopes the over-estimation is roughly a fac-\ntor of 4. However, these errors are consistent with the\noverall results from the neutron separation level spac-100102104106Level density ρ [MeV-1]\n0 1 2 3 4 5 6 78\nExcitation Energy exc [MeV]100102104Level density ρ [MeV-1]50V\n51V\n97Mo\n98Mo\nFIG. 13: (Color online) Level densities as functions of exci ta-\ntion energy for50,51V and97,98Mo. The black solid lines show\nthe CFY model and the red dots show the experimental data\n[48, 49].\ning, see Fig. 10, which have an overall frms= 4.2. Also,\nthe slopes and the detailed structures are not as well de-\nscribed in these nuclei as in the rare-earths. Especially\n50V exhibits an oscillatory pattern in the CFY model,\nwhich is an effect of the small ground-state deformation\nε2= 0.05 [21]. A corresponding oscillatory pattern is not\nas clearly present in the experimental data. In51V the\nexperimental data shows oscillatory behavior while the\nCFY model is more smooth. The ground-state deforma-\ntion is slightly larger with ε2= 0.083 [21], which in the\nCFY model means that since the moment of inertia is\nlarger the rotational states have a larger influence and\nsmooths the level density.\nC. Low-lying discrete levels\nThe amount of experimental information on low-lying\ndiscrete energy levels is indeed substantial. In addition\nto the energy parity and angular momentum are often\nknown. In Ref. [22] a global comparison is made between\nmeasuredand FRLDM calculatedground-statespinsand\nparities for odd-mass nuclei, and the agreement was\nfound to be very good. The collection of low-energy data11\nprovides important information for level-density models,\nsince the accuracy of the model can be studied at the\nvery lowest energies. Cumulative distributions of known\nlevels can thus be constructed, and compared to calcula-\ntions. In Ref. [14] a selection of 15 different nuclei was\nmade to represent all different kinds of nuclear aspects,\nsuch as light-heavy, spherical-transitional-deformed, as\nwellasodd-odd,even-evenandodd-evennuclei(alsoused\nin Ref. [38]). We therefore compare our model low-lying\nlevel densities to experimental data for the identical set\nof 15 nuclei.\nThe cumulative level density for the 15 nuclei are\nshown in Fig. 14. The CFY model (solid blue line) is\ncompared to experimental data (black dash-dotted line)\nand to the HFB model (red dashed line) of Ref. [38]. For\na good agreement with data, the calculated line should\nfollow as precise as possible the experimental curve at\nthe lowest energies, and should then smoothly deviate,\nalways larger than data.\nIn general, the theoretical models seem to give very\ncomparable results for deformed nuclei while differences\nare larger for transitional and spherical nuclei. For the\nlight spherical nuclei, like42K and56Fe there are pro-\nnounced single-particle gap effects which cause jumps in\nthe level density. These effects are smoothly disappear-\ning at higher energies. A quite drastic deviation between\nour calculation and data is seen for208Pb. The calcu-\nlated (cumulative) level density is systematically larger\nstarting already at about 3 MeV excitation energy (in\nthe figure seen at 0.5 MeV; the energy scale is shifted by\n2.5 MeV for this nucleus). This is also seen in the to-\ntal level density at the neutron separation energy, and in\nTable I the ratio between theoretical and measured level\nspacingsat the neutron separationenergyis listed for the\n9 nuclei. For208Pb this ratio is 0.02, i.e. the calculated\nlevel density is about 50 times larger than experiment!\nWhen comparing to the HFB model calculation of the\nlevel density [38] one should note that interpolations be-\ntween different deformations are included in this model,\nwhile in our model the ground-state deformation is kept\nfixed for all excited states. In particular, this somewhat\nphenomenological way to treat deformation changes has\nstrong effects on the results for127Te.\nNuclDTh/DExpNuclDTh/DExpNuclDTh/DExp\n42K 0.9456Fe 0.5560Co 0.62\n94Nb 2.50107Cd 0.77127Te 0.26\n148Pm 1.72155Eu 2.95161Dy 2.61\n162Dy 1.27172Yb 3.10194Ir 4.58\n208Pb 0.02237U 5.18239Pu 3.46\nTABLE I: Table of neutron separation energy spacings com-\npared to experimental data for the nuclei showed in Fig. 14.100101102103104\n42K56Fe60Co\n100101102103\n94Nb107Cd127Te\n100101102103Accumulated Level Density N148Pm155Eu161Dy\n100101102103\n162Dy172Yb194Ir\n0 1 2 3 4 5 6100101102103\n208Pb(-2.5 MeV)\n0 1 2 3 4 5 6\nExcitation Energy [MeV]237U\n0 1 2 3 4 5 67239Pu\nFIG. 14: (Color online) Accumulated level density as a func-\ntion of excitation energy. The blue solid and red dashed line s\nshow the present CFY model and the HFB model [38]. The\nblack dash-dotted line shows experimental data collected i n\nRef. [38].\nD. Parity ratio\nThe parity ratio has been measured experimentally by\nKalmykov et.al. [20] for the two spherical nuclei58Ni\nand90Zr. For these nuclei the I= 2 angular-momentum\ncomponentofthe leveldensityismeasuredand separated\ninto parity components. In the top panel of Fig. 15(a)\nthe level-density component ρ(E,I= 2,π=±1) for90Zr\nis compared to CFY model calculations. The level den-\nsities from the CFY model are in good agreement with\nexperimental data, and in the lower panel of Fig. 15(a)\nthe parity ratio is shown. Predictions for the total level\ndensityaswellasthe I= 2componentareshownasblack\nsolid and red dashed lines, respectively. The difference\nbetweentheparityratioforthetotalleveldensityandthe\nI= 2 component decreases with excitation energy, be-\ncause the spin cutoff model becomes more realistic when\nthe excitation energy increases, see Sec. IIIC. For the\nparity ratio the CFY model result is within the experi-\nmental error-barsfor all excitation energies and seems to\nshow a similar pattern as the experiments: a high par-\nity ratio (excess of negative parity states) at 8 MeV and\na low ratio (excess of positive parity states) at 11 MeV\nexcitation energy. The blue dot-dashed line shows the12\n100102104106  Level density ρ [MeV-1]\n2+\n2-\n5 7.5 10 12.5 15\nExcitation Energy Eexc [MeV]012Parity Ratio ρ-/ρ+  90Zr\n90Zr     I = 2\n100102104106   Level Density [MeV-1]\n2+\n2-\n5 7.5 10 12.5 15 17.5 20\nExcitation Energy Eexc [MeV]00.511.5Parity Ratio ρ-/ρ+58Ni\n58Ni      I = 2\nFIG. 15: (Color online) The top panels show CFY calculated\ncomponents of the 2+(black solid) and 2−(red dashed) level\ndensities for90Zr (top part of figure) and58Ni (lower part\nof figure), as functions of excitation energy. Experimental\ndata from Ref. [20] are shown as black dots and red diamonds\nwith error-bars for 2+and 2−, respectively. The lower panels\nshowforeachnucleustheparityratioversusexcitationene rgy.\nThe solid black and dashed red lines show the CFY model\nresults for the total level density and the I=2 component,\nrespectively. The blue dot-dashed lines show the Skyrme-\nHFB model of Ref. [14]. Experimental data from Ref. [20] are\nshown with error-bars.\nSkyrme-HFB model of Ref. [14]. It is also within the\nexperimental error-bars for all excitation energies except\nat 9 MeV where the HFB model gives a large positive\nratio (∼3) while the experiments and the CFY model\nare close to unity.\nTheI= 2 component of the level density in58Ni is\nshown in the top panel of Fig. 15(b). The agreement\nbetween experimental data and the CFY model is some-\nwhat inferior to what we obtained for90Zr. For both\nparities the level density seems to increase faster in the\nCFY model than what is seen in the experimental data.\nAt 14 MeV excitation energy the model overestimates\nthe level density by roughly a factor of 6. The parity\nratio is shown in the lower panel of Fig. 15(b). The ex-5 6 7 8 9 10\nExcitation Energy Eexc [MeV]102103104105Level density ρ [MeV-1]90Nb       Iπ = 1+\nFIG. 16: (Color online) Level density of the 1+component of\nthe level density as a function of excitation energy for90Nb.\nThe black solid and red dashed lines show the CFY model\nand Skyrme-HFB model of Ref. [14], respectively. Data from\nRef. [20] are shown as black dots with error-bars.\nperimental parity ratio is close to unity at 8 MeV and\nthen decreases with increasing excitation energy. The\nCFY model shows a different pattern. It gives a very\nlow parity ratio at low excitation energies with an al-\nmost monotonic increase with increasing excitation en-\nergy. The ratio only becomes close to unity at 20 MeV.\nThe Skyrme-HFB model of Ref. [14] shows a similar pat-\ntern as the CFY model but with an even lower ratio for\nexcitation energies below 20 MeV.\nThe 1+component of the level density of90Nb has\nalso been measured in Ref. [20]. Fig. 16 shows the ex-\nperimental data compared to the CFY and Skyrme-HFB\nmodels. In the experimental data there is a clear os-\ncillating structure. The CFY model (black solid line)\nover-estimates the level density and the oscillating struc-\nture is much less pronounced. In the Skyrme-HFB model\n(red dashed line) there are long-range oscillations simi-\nlar to what is seen in the experimental data, but the\nenergy separation between consecutive minima is larger.\nThe level density is under-estimated in the Skyrme-HFB\nmodel, whereas the CFY model over-estimates the level\ndensity by a similar factor.\nV. COMPARISON WITH OTHER MODELS\nThe CFY model is here compared to other statistical\nand combinatorialNLD models that provideneutron res-\nonance level spacings. In general, the statistical models\nlistedinTableIIhaveanrmsdeviationjustbelow2. This\nlow rms deviation is obtained because severalparameters\nin the level-density formulas are directly adjusted to the\nneutron separation-energy level spacings and low-lying\ndiscrete energy levels.\nThe back-shifted Fermi model of Ref. [47] is based on\na simple Fermi-gas formula whereas the Constant Tem-13\nStatistical Models frmsRef.\nBack-shifted Fermi Model 1.71 [47]\nConst. Temp. Model 1.77 [47]\nBack-shifted Fermi + Const. Temp. Model 1.7 [4]\nGeneralized Superfluid Model 1.94 [47]\nCombinatorial Models frmsRef.\nSkyrme-HFB 2.35 [47]\nGogny-HFB 4.55 [15]\nCFY 4.2 Present\nTABLE II: Table of rms-factors frmsfor statistical and com-\nbinatorial models for neutron separation energy level spac ings\n[4, 15, 47].\n12345678\nAttenuated Phonon Method\n0 50 100 150 200 250 300\nMass Number A11.11.2 QTDAVibrational enhancement Kvib\nFIG. 17: Vibrational enhancements at the neutron separatio n\nenergy in the attenuated phonon method (top panel) and the\nQTDA method (bottom panel) versus of mass number A.\nperature model is based on the approach of Gilbert and\nCameron [10]. Both models contain explicit enhance-\nment factors for rotations and vibrations. The model\nby Rauscher, Thielemann and Kratz [4] is also based on\nthe approach of Gilbert and Cameron. In contrast to\nthe first two models this model has no explicit enhance-\nment factors for rotations and vibrations. It accounts for\nshell effects and thermal damping in terms of an effective\nlevel-density parameter. The model uses the microscopic\nenergy corrections of Ref. [21] together with three free\nparametersin the fitting procedure. The GeneralizedSu-\nperfluid Model is similar to the models mentioned above.\nIn addition it takesinto account howpairingevolveswith\nincreasing excitation energy. It also incorporates explicit\nrotational and vibrational enhancements [47].\nPredictions by statistical models in regions outside\nthe fitting region probably give much less accurate re-\nsults than in the adjustment region. On the other hand,\nsince the combinatorial models are all based on calcu-\nlated single-particle spectra they could in principle have\nbetter predictive power, in particular in regions wherethe single-particle model is sound. In contrast, they are\nnot equally flexible in terms of parameter fits to the level\nspacings database. Therefore, the rms deviation factor\nof combinatorial models with respect to known data is\nlarger than for the statistical models. The frms= 4.2\nfor the CFY model is about a factor of 2 larger than in\nstatistical models. However, the latter result is obtained\nwithnoparameters specifically fitted to the level-density\ndata. The mean deviation factor m= 1.1 indicates that\ntheleveldensityisonaveragereasonable. However,there\nseems to be a drift with mass number, as seen in Fig. 10,\npresent in the CFY model which is not seen in the HFB\nmodels of Refs. [15, 47].\nIn the lower part of Table II the CFY model is com-\npared to two other large-scale combinatorial NLD mod-\nels based on the HFB method. The model of Ref. [15]\nis based on the Gogny D1S interaction for the mean-\nfield and incorporates combinatorial rotations (similar to\nSec. IIC but no pairing dependence of the moment of\ninertia), and the attenuated phonon method is applied\nto account for vibrational states. Pairing is included\nexplicitly for the ground-state, and excited states are\nback-shiftedbyan energy-dependentgapprocedure. The\nmodel gives frms= 4.55 for the subset of even-even axi-\nally deformed nuclei. The erroris slightly largerthan the\nCFY model.\nFor the Skyrme-HFB NLD model the deviation factor\nisfrms= 2.35 [47], which is ���35% larger than the sta-\ntistical models and 44% smaller that in the CFY model.\nThis model is based on a Skyrme-HFB mean-field to-\ngether with combinatorial rotations and the attenuated\nphonon method for modeling vibrational states. In ad-\ndition a phenomenological deformation change from de-\nformed to spherical shape at some specified excitation\nenergy is incorporated.\nThe attenuated phonon method is a phenomenological\nway to model nuclear vibrations, see eg. Refs. [12, 14]. It\nassumes that the quadrupole and octupole phonon states\nin nuclei can be modeled by a gas of non-interacting\nbosons. The vibrational excitation energies are taken\nfrom systematics of the lowest non-rotational 2+and 3−\nstates. Themodel is formulatedasa multiplicativefactor\nKvibr= exp[δS−δU/T] which enhances the level den-\nsity.Tis the nuclear temperature and the δSandδUare\nthe entropy and internal energy change induced by the\nbosons. The occupation probabilities of the bosons are\ndescribed by damped Bose statistics, where the damping\nsets in at considerably higher energies than the neutron\nseparation energy. At excitation energies close to or be-\nlow the neutron separation energy the damping has neg-\nligible effect which implies that the phonons are allowed\nto be repeated several times.\nThe vibrational enhancement at the neutron separa-\ntion energy is quite small in the QTDA method as com-\npared to the attenuated phonon method, see Fig. 17.\nThe QTDA gives an enhancement of the order of a few\npercent compared to up to a factor 7 in the attenuated\nphonon method. The largest vibrational enhancements14\nare obtained for the A=75 region and for Cd isotopes in\ntheA=115 region, which is consistent with the strong\n(quadrupole) vibrational character of nuclei in these re-\ngions.\nThe QTDA phonons are microscopically built from\nquasi-particle excitations Eqp,i\nµwith energies not much\ndifferent than the QTDA phonon energies (¯ hω)i\nj. The\nwayto fully accountfordouble-countingofphononstates\n(Eq. 19), thus implies a small vibrational enhancement\nsince the existence of a few low-lying collective phonons\nwill not impact the level density if all other phonons\nare very non-collective. The non-collective character of\nmost of the phonons also implies that they cannot be re-\npeated. Phononsin theattenuatedphononmodelarenot\ndescribed microscopically, and double-counting of states\nmay clearly appear. In addition, it assumes that each\nphonon can be repeated several times.\nVI. SUMMARY\nA combinatorial model for the nuclear level density\nis presented. The model is based on the folded-Yukawa\nsingle-particlemodelwithground-statedeformationsand\nparameters from Ref. [21]. The model is used to calcu-\nlate the neutron resonancelevelspacings, yieldingan rms\ndeviation of frms= 4.2. It also compares favorably with\nexperimental level density data versus excitation energy\nfor several nuclei in the rare-earth region as measured\nby the Oslo method, as well as with the cumulative level\ndensity extracted from low-energy spectra.\nThe role of collective enhancements has been investi-\ngated in detail. Pairing is incorporated for each individ-\nual many-body configuration, and the distribution of the\npairinggapsisinvestigated. Nosharppairingphasetran-\nsition is observed. Instead, even at the highest excitation\nenergy considered, a non-negligible fraction of the states\nhave a considerable pairing gap.Rotational states are included combinatorially by a\nsimple rotor model with a moment of inertia dependent\non the deformation and pairing gap. Vibrational states\nare included using a Quasi-particle-Tamm-Dancoff Ap-\nproximation. It is found that the vibrational enhance-\nment in the QTDA model is very small, on the order of\na few percent at the neutron separation energy.\nThe parity distribution in the CFY model shows large\noscillatory patterns for nuclei which have large gaps in\nthe single-particle spectrum, separating shells with dif-\nferent parities. This is often the case for nuclei with\nsmalldeformations. Thepatternsarequitedifferent from\nthe smooth pattern of the statistical parity distribution\nmodel of Ref. [16]. The CFY model is compared to other\nmodels and to experimental data when available.\nThe angular-momentum distribution in the CFY\nmodel is compared to the spin cutoff model in Sec. IIIC.\nIt is found that the Gaussian envelope of the spin cutoff\nmodel is in good agreement with the CFY model for high\nexcitation energies.\nAcknowledgments\nWe acknowledge discussions with R. Capote, M. Gut-\ntormsen, K.-L. Kratz, T. Rauscher, A. Richter and F.-K.\nThielemann. S. Hilaire and S. Goriely are acknowledged\nfor providing data underlying Fig.14.\nH. U. is grateful for the hospitality of the Los Alamos\nNational Laboratory during several visits. S. ˚A. and H.\nU. thank the Swedish national research council (VR) for\nsupport. This work was supported by travel grants for\nP. M. to JUSTIPEN (Japan-U. S. Theory Institute for\nPhysics with Exotic Nuclei) under grant number DE-\nFG02-06ER41407 (U. Tennessee). 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Kratz, Atomic Data and\nNuclear Data Tables, 66(1997) 131.\n[24] T. Døssing et. al., Phys. Rev. Lett. 75(1995) 1276.\n[25] P. M¨ oller and J.R. Nix, Nucl. Phys. A 536(1992) 20.\n[26] H. Olofsson, R. Bengtsson and P. M¨ oller, Nucl. Phys. A\n784(2007) 104.\n[27] S.G. Nilsson, Kgl. Danske Videnskab. Selskab. Mat.-Fy s.\nMedd.29: No. 16 (1955).\n[28] R. Bengtsson and S. ˚Aberg, Phys. Lett. B 172(1986)\n277.\n[29] G. Hansen and A. S. Jensen, Nucl. Phys. A 406(1983)\n236.\n[30] M. Guttormsen, private comm. (2009).\n[31] D. M. Brink, PhD-thesis (Oxford University, 1955).\n[32] P. Axel, Phys. Rev. 126(1962) 671.\n[33] V. O. Nesterenko, W. Kleinig, V. V. Gudkov and J.\nKvasil, Phys. Rev. C 53(1996) 1632.\n[34] H. Sakamoto and T. Kishimoto, Nucl. Phys. A 501\n(1989) 205.\n[35] S.˚Aberg, Phys. Lett. B 157(1985) 9.\n[36] D. J. Rowe, Nuclear Collective Motion - Models and The-\nory(Butler & Tanner Ltd, Frome and London, 1970).[37] A. V. 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C 68(2003) 064306.\n[52] E. Melby et. al., Phys. Rev. C 63(2001) 044309.\n[53] A. Schiller et. al., Phys. Rev. C 63(2001) 021306(R)."    },    {        "title": "0902.1850v1.Accurate_calculation_of_the_local_density_of_optical_states_in_inverse_opal_photonic_crystals.pdf",        "content": "arXiv:0902.1850v1  [physics.optics]  11 Feb 2009Accurate calculation of the local density of optical states in\ninverse-opal photonic crystals\nIvan S. Nikolaev1, Willem L. Vos1,2, and A. Femius Koenderink1,∗\n1Center for Nanophotonics, FOM Institute for Atomic and Mole cular Physics (AMOLF), Kruislaan 407, NL-1098SJ\nAmsterdam, The Netherlands\n2Complex Photonic Systems, MESA+Institute for Nanotechnology, University of Twente, The Ne therlands\nCorresponding author: f.koenderink@amolf.nl\nSubmitted to OSA Dec 23, 2008\nWe have investigated the local density of optical states (LD OS) in titania and silicon inverse opals –\nthree-dimensional photonic crystals that have been realiz ed experimentally. We used the H-field plane-wave\nexpansion method to calculate the density of states and the p rojected local optical density of states, which\nare directly relevant for spontaneous emission dynamics an d strong coupling. We present the first quantitative\nanalysis of the frequency resolution and of the accuracy of t he calculated local density of states. We have\ncalculated the projected LDOS for many different emitter pos itions in inverse opals in order to supply a\ntheoretical interpretation for recent emission experimen ts and as reference results for future experiments and\ntheory by other workers. The results show that the LDOS in inv erse opals strongly depends on the crystal\nlattice parameter as well as on the position and orientation of emitting dipoles.\nOCIS codes: 160.5298,260.2510,270.5580,290.4210\n1. Introduction\nPhotonic crystals are metamaterials with periodic varia-\ntions of the dielectric function on length scales compara-\nble to the wavelength of light. These dielectric compos-\nites are of a keen interest for scientists and engineers\nbecause they offer exciting ways to manipulate pho-\ntons [1,2]. Of particular interest are three-dimensional\n(3D) photonic crystals possessing a photonic bandgap,\ni.e.a frequency range where no photon modes exist at\nall.Photonicbandgapmaterialspossessagreatpotential\nfor drastically changing the rate of spontaneous emission\nand for achieving localization of light [3–6]. Control over\nspontaneousemissionis importantformanyapplications\nsuch as miniature lasers [7], light-emitting diodes [8] and\nsolar cells [9]. According to Fermi’s golden rule, the rate\nof spontaneous emission from quantum emitters such as\natoms, molecules or quantum dots is proportional to the\n‘local radiative density of states’ (LDOS) [10,11], which\ncounts the number of electromagneticstates at given fre-\nquency, location and orientation of the dipolar emitters.\nIn addition, nonclassical effects can occur that are be-\nyond Fermi’s Golden Rule [2,11]. These strong coupling\nphenomena, such as fractional decay, rely on coherent\ncoupling of the quantum emitter to a sharp feature in\na highly structured LDOS. Whether these effects are in\nfact observable in real photonic crystals depends on how\nrapid the LDOS changes within a small frequency win-\ndow [12]. Therefore local density of states calculations\nfor experimentally realized photonic crystals are essen-\ntial to assess current spontaneous experiments and fu-\nturestrongcouplingexperimentsalike,providedthatthe\ncalculationsareaccurateandhaveacontrolledfrequency\nresolution.LDOS effects on spontaneous emission in photonic\ncrystals have been experimentally demonstrated in a\nvariety of systems. Since only three-dimensional (3D)\ncrystals promise full control over all optical modes with\nwhich elementary emitters interact, many groups have\npursued 3D photonic crystals. Fabrication of such pe-\nriodic structures with high photonic strengths is, how-\never, a great challenge [2,13]. Inverse opals are among\nthe most stronglyphotonic 3Dcrystalsthat can be fabri-\ncated relatively easily using self-assembly methods. Such\ncrystals consist of fcclattices of close-packed air spheres\nin a backbone material with a high dielectric constant\n[14–18]. In these inverse opals, continuous-wave exper-\niments on light sources with a low quantum efficiency\nrevealed inhibited radiative emission rates [19,20]. En-\nhanced and inhibited time-resolved emission rates have\nbeen observed from highly-efficient emitters in 3D in-\nverse opals [21]. Simultaneously, several groups have re-\nalized that emission enhancement and partial inhibition\ncan also be obtained in 2D slab structures [22–26].\nFollowing the first calculations by Suzuki and Yu [27],\nseveral papers have reported on calculations of the lo-\ncal density of optical states in photonic crystals using\nboth time domain [28–30] and frequency domain meth-\nods[31–34].Unfortunately,analysisofexperimentaldata\nfor emitters in 3D crystalsin terms ofthese LDOS calcu-\nlations is compounded by several problems in literature:\n1. Most prior calculations were performed for model\nsystems that don’t correspond to structures used in\nexperiments, and for emitter positions that are not\nprobed in experiments.\n2. The accuracy of the reported LDOS has never been\n1discussed, hampering comparison to experiments.\n3. The frequency resolution of the reported LDOS has\nremained unspecified. Therefore sharp features of\nrelevance for non-classical emission can not be as-\nsessed [11,12].\n4. ManypreviouslyreportedLDOScalculationsareer-\nroneous for symmetry reasons, as pointed out by\nWanget al.[33].\nIn this paper we aim to overcome all these problems.\nWe benchmarkthe accuracyandfrequencyresolutionfor\nour LDOS calculations to allow quantitative comparison\nwith experiments and with quantum optics strong cou-\npling requirements. Since prior LDOS calculations are\nscarce (and partly erroneous, item 1 above) we present\nsets of LDOS’s for experimentally relevant structures,\nand for spatial positions where sources can be practi-\ncally placed. Specifically, we model the spatial distribu-\ntion of the dielectric function ǫ(r) in such a way that it\nclosely resembles ǫ(r) in titania (TiO 2) [14,15] and sili-\ncon (Si) inverse-opal photonic crystals [17,18]. For these\ntwo structures we calculated the LDOS at various posi-\ntions in the crystal unit cell and for specific orientations\nof the transition dipoles. The results on the TiO 2inverse\nopals are relevant for interpreting recent emission exper-\niments [21,35].Toaidotherworkerstointerprettheir ex-\nperiments and to benchmark their codes, we make data\nsets that we report in figuresthroughout this manuscript\navailable as online material.\nThe paper is arranged as follows: in Section 2, we\npresent a detailed description of the method by which\nwe have calculated the photonic band structures and the\nLDOS.We discussthe accuracyandfrequencyresolution\nof our calculations. In Section 3 we compare our compu-\ntations with the known DOS in vacuum and with earlier\nresults on the DOS and LDOS [33] in 3D periodic struc-\ntures.Section4describestheLDOSininverseopalsfrom\nTiO2, and in Section 5 we present results of the LDOS\nin Si inverse opal photonic band gap crystals.\n2. Calculation of local density of states\nA. Introduction\nThe local radiative density of optical states is defined as\nN(r,ω,ed) =1\n(2π)3/summationdisplay\nn/integraldisplay\nBZdkδ(ω−ωn,k)|ed·En,k(r)|2,\n(1)\nwhere integration over k-vector is performed over the\nfirst Brillouin zone, nis the band index and edis the\norientation of the emitting dipole. The total density of\nstates (DOS) is the unit-cell and dipole-orientation av-\nerage of the LDOS defined as N(ω) =/summationtext\nn/integraltext\nBZdkδ(ω−\nωn,k). The important quantities that determine the\nLDOS are the eigenfrequencies ωn,kand electric field\neigenmodes En,k(r) for each k-vector. Calculation of\nthese parameters will be discussed below in Section B.The expression for the LDOS contains the term |ed·\nEn,k(r)|that depends on the dipole orientation ed. It is\nimportant to realize that in photonic crystals the vec-\ntor fields En,k(r) are not invariant under the lattice\npoint-group operations α, as first reported in Ref. [33].\nExplicitly, this means that the projection of the field\nof a mode at wave vector kon the dipole orientation\ned(i.e.|ed·En,k(r)|)is not identical to the projection\n|ed·En,α[k](r)|ofthe symmetry related modes with wave\nvectorsα[k] on the same ed. As a consequence one can\nnot calculate the LDOS for a specific dipole orientation\nby restricting the integral in Eq. (1) over the irreducible\npart (1/48th) of the Brillouin zone, since symmetry re-\nlated wave vectors do not give identical contributions.\nUnfortunately, in many previous reports on the LDOS,\nthis reduced symmetry for vector modes as compared to\nscalar quantities was overlooked, resulting in erroneous\nresults [33]. In general, the only symmetry that can be\ninvoked to avoid using the full Brillouin zone for LDOS\ncalculations is inversion symmetry, which correspondsto\ntime reversalsymmetry.Consequently,correctresultsre-\nquire that exactly half of the Brillouin zone is considered\nfor LDOS calculations, rather than the irreducible part\nof the Brillouin zone that was used in most previous lit-\nerature. We have explicitly verified that our implemen-\ntation (using the kz≥0 half of the Brillouin zone) re-\nsults in the same LDOS on symmetry related positions,\nprovided that one also takes the concomitant symmetry\nrelateddipole orientationintoaccount.Furthermore,our\ncalculations confirm the claim by Wang et. al. that this\nrequired symmetry is only recovered upon integration\nover half the Brillouin zone, rather than over just the\nirreducible part as considered in, e.g., Ref. [31,32].\nB. Plane-wave expansion\nWe use the H-field inverted plane wave expansion\nmethod [31,36,37] to solve for the electromagnetic field\nmodes in photonic crystals. For nonmagnetic materials,\nit is most convenient to solve the wave equation for the\nH(r) field [38]\n∇×/bracketleftbig\nǫ(r)−1∇×H(r)/bracketrightbig\n=ω2\nc2H(r).(2)\nbecause the operator ∇ ×ǫ(r)−1∇×is Hermitian, and\nconsequentlyhas real eigenvalues ω2/c2[1,31,36,37].Be-\ncause of the periodicity of the dielectric function ǫ(r) in\nphotonic crystals, the field modes Hk(r) of the eigen-\nvalue problem Eq. (2) satisfy the Bloch theorem [39]:\nHk(r) =eik·ruk(r). (3)\nTheseBlochmodes arefully described bythe wavevector\nkand the periodic function uk(r), which has the period-\nicity of the crystal lattice so that uk(r) =uk(r+R). To\nsolve the wave equation, the inverse dielectric function\nand the Bloch modes are expanded in a Fourier series\nover the reciprocal-lattice vectors G:\nǫ(r)−1=η(r) =/summationdisplay\nGηGeiG·rand (4)\n2Hk(r) =/summationdisplay\nGuk\nGei(k+G)·r, (5)\nwhereηGanduk\nGare the 3D Fourier expansion coeffi-\ncients of respectively η(r) anduk(r). Substituting these\nexpressions into the H-field wave equation in Eq. (2), we\nobtain a linear set of eigenvalue equations:\n−/summationdisplay\nG′ηG−G′(k+G)×[(k+G′)×un,k\nG′] =ω2\nn(k)\nc2un,k\nG.\n(6)\nThis infinite equation set with the known parameters G\nandηG−G′determines all allowed frequencies ωn(k) for\neach value of the wave vector k, subject to the transver-\nsality requirement ∇·Hk(r) = 0. Due to the periodicity\nofuk(r), we can restrict kto the first Brillouin zone. For\neach wave vector k, there is a countably infinite num-\nber of modes with discretely spaced frequencies. All the\nmodes are labeled with the band number nin order of\nincreasing frequency and are described as a family of\ncontinuous functions ωn(k) ofk.\nTo compute the eigenfrequencies ωn(k) and the ex-\npansion coefficients of the eigenmodes un,k\nG, the infinite\nequation set is truncated. By restricting the number of\nreciprocal-latticevectors Gto a finite set GwithNGele-\nments, Eq. (6) is limited to a 3 NGdimensional equation\nset. In ourimplementation wechoosethe truncatedset G\nto correspond to the set of all reciprocal lattice vectors\nwithin a sphere centered around the origin of k-space.\nThe transversality of the H-field gives an additional con-\ndition on the eigenmodes: ( k+G)·un,k\nG= 0, which elim-\ninates one vector component of un,k\nG. Following Ref. [31],\nforeachk+Goneneedstofindtwoorthogonalunitvec-\ntorse1,2\nk+Gthat form an orthogonaltriad with k+G. By\nexpressing the eigenmode expansion coefficients in the\nplane normal to k+Gasun,k\nG=un,k\nG,1e1\nk+G+un,k\nG,2e2\nk+G,\nwe remove one third of the unknowns. Then, Eq. (6) be-\ncomes\n/summationdisplay\nG′∈GηG−G′|k+G||k+G′|·/bracketleftBigg\n/parenleftbigge2\nk+G·e2\nk+G′−e2\nk+G·e1\nk+G′\n−e1\nk+G·e2\nk+G′e1\nk+G·e1\nk+G′/parenrightbigg/parenleftBigg\nun,k\nG′,1\nun,k\nG′,2/parenrightBigg/bracketrightBigg\n=ω2\nn(k)\nc2/parenleftBigg\nun,k\nG,1\nun,k\nG,2/parenrightBigg\n,∀G∈ G.\n(7)\nTo find the matrix of Fourier coefficients ηG−G′, we used\nthe method of Refs. [31,36]. The coefficients are com-\nputed by first Fourier-transforming the dielectric func-\ntionǫ(r), andthen truncatingandinvertingthe resulting\nmatrix. As first noted by Ho, Chan and Soukoulis [36],\nusing the inverse of ǫG−G′rather than ηG−G′dramati-\ncally improves the (poor) convergence of the plane-wave\nmethod that is associated with the discontinuous nature\nof the dielectric function [37]. A rigorous explanation\nfor this improvement was put forward by Li [40], whostudied the presence of Gibbs oscillations in the trun-\ncated Fourier expansion of products of functions with\ncomplementary jump discontinuities. Using the H-field\ninverted matrix plane wave method, the frequencies ob-\ntained with N G= 725 (for fccstructures) deviate by\nless than 0.5 % from the converged band structures [31].\nSolving Eqs. (7) gives the frequencies ωn(k) and the\nFourier expansion coefficients for the H-field eigenmodes\nHn,k(r) needed to calculate the LDOS in the photonic\ncrystal. The required E-fields En,k(r) are obtained using\nthe Maxwell equation ∂D/∂t=∇×H:\nEn,k(r) =1\nωn(k)ǫ0/summationdisplay\nG,G′∈GηG′−G|k+G|·/bracketleftbigg\n/parenleftBig\nun,k\nG,1e2\nk+G−un,k\nG,2e1\nk+G/parenrightBig\nei(k+G′)·r/bracketrightbigg\n.\n(8)\nFrom the orthonormality of the eigenvectors of Eq. (7)\nit follows that the Bloch functions Hn,k(r) andEn,k(r)\ndefined above satisfy the orthonormality relations:\n/integraldisplay\nBZHn,k(r)·H∗\nn′,k′(r)dr=δ(k−k′)δn,n′,(9)\n/integraldisplay\nBZǫ(r)En,k(r)·E∗\nn′,k′(r)dr=δ(k−k′)δn,n′.(10)\nIt should be noted in the definition of En,kthat the mul-\ntiplication by 1 /ǫ(r) to calculate EfromDis not in the\ndenominator in front of the Fourier expansion. Rather it\nappears as a matrix multiplying the D-field in Fourier\nspace, i.e., within the sum over reciprocal lattice vec-\ntors. This ordering ensures that complementary jumps\ninDand 1/ǫ(r) cancel, even for the truncated Fourier\nseries, as can be easily checked by plotting calculated\nmode profiles for TM modes in 2D crystals.\nC. Frequency resolution and accuracy of LDOS\nWe are not aware of any previous report that bench-\nmarks the accuracy of the calculated local radiative den-\nsity of states or that specifies the frequency resolution.\nMotivated by the requirements for accurate results for\ncomparison with experiments and for judging the utility\nof crystals for non-classical emission dynamics, we con-\nsider the accuracy and resolution of our approximation\nto the LDOS integral in Eq. (1). The main approxima-\ntion is to replace the integration over wave vector kby\nan appropriately weighted summation over a discrete set\nof wave vectors on a discretization grid [41]. Either in-\nterpolationschemes [31] orsimple histogramming(‘root-\nsampling’) methods are used to compute the LDOS. The\nk-point grid density sets the number of k-points in the\nBrillouin zone, Nk. The accuracy of the resulting LDOS\napproximation is set by the the density of grid points\nthat is used to discretize the wave vector integral. Due\nto the transparent relation between the accuracy of the\nDOS,thefrequencyresolutionandthe k-vectorsampling\nresolution, we will focus on the simple histogramming\n3Fig. 1. (Color online) DOS per volume in units 4 /a2cfor\nvacuum modeled as an empty fcccrystal. The calculated\nDOS shown by red histogram bars is compared to the\nanalytically derived ω2behavior (curve). In vacuum the\nDOS per volume equals the dipole-averaged LDOS.\nFig. 2. Averageabsolute deviation ofthe calculated DOS\nfrom the exact total DOS N(ω) for an ‘empty crystal.’\nThe average runs over the frequency range 0 < ω <\n2πc/aand the deviation is in units of the DOS N(ω)\nper volume at ωa/2πc= 1, i.e., in units 4 /a2c. In accor-\ndance with Eq. (11), the error is inversely proportional\nto the ratio ∆ ω/∆kof the histogram bin width ∆ ωto\nthe integration grid spacing ∆ k. Symbols correspond to\nintegration using the number of k-points Nk= 280, 770,\n1300, 2480, 2992 and 3570 ( /square,/squaresolid,◦,•,♦,/diamondsolid) respectively,\nwith various ∆ ω.\napproach. For the LDOS computations one chooses a\ncertain frequency bin width ∆ ωto build an LDOS his-\ntogram. For a desired frequency resolution ∆ ωand a\ndesired accuracy for the LDOS content N(r,ω,ed)∆ωin\neach frequency bin, one needs to choose an appropriate\nk-vector spacing ∆ kthat depends on the steepness of\nthe sampled dispersion relation ω(k). Indeed, the useful\nfrequency resolution ∆ ωof a histogram of the DOS and\nLDOS is limited by the resolution ∆ kof the grid in kspace to be\n∆ω≈∆k|∇kω|, (11)\nas detailed in Ref. [42] for the electronic DOS. This cri-\nterion relates the separation between adjacent k-grid\npoints to their approximate frequency spacing via the\ngroup velocity. If histogram bins are chosen too narrow\ncompared to the expected frequency spacing between\ncontributions to the discretized LDOS integral, unphysi-\ncal spikes appear in the approximation, especially in the\nlimit of small ω, where the group velocity |∇kω|is usu-\nally largest. Apart from full gaps in the LDOS, photonic\ncrystals also promise sharp lines at which the LDOS is\nenhanced, which are important for non-classical emis-\nsion dynamics. Hence it is especially important to dis-\ntinguish sharp spikes that are due to histogram binning\nnoise from true features. Unfortunately, many reports in\nliterature feature sharp spikes that are evidently binning\nnoise (wave vector undersampling errors), as they occur\nin the long wavelength limit, below any stop gap.\nTo improve the resolution without adding time-\nconsuming diagonalizations, several interpolation\nschemes have been suggested [42]. Within the his-\ntogramming approach, an interpolation scheme essen-\ntially improves binning statistics by adding histogram\ncontributions from intermediate grid points on the\nassumption that quantities vary linearly between grid\npoints. While interpolation decouples the binning noise\nfromthe frequencybin width ∆ ω, resultingin arbitrarily\nsmooth LDOS approximations, it has the disadvantage\nof obscuring the intrinsic relation between frequency\nresolution and wave vector resolution in Eq. (11).\nAgoodbenchmarkforthebinning noiseofthe k-space\nintegration,independent ofthe convergenceofthe plane-\nwave method, is to calculate the LDOS or DOS of an\n‘empty’ crystal, with uniform dielectric constant equal\nto unity. Such an ‘empty’ crystal represents a limit of\nzero photonic strength and the maximum possible group\nvelocity|∇kω|.InFigure1weshowtheDOSinanempty\nfcccrystal. As expected for a crystal with zero dielectric\ncontrast,thecalculatedDOSisindependentofdipolepo-\nsition andorientation.In agreementwith the well-known\nDOS in vacuum the calculated DOS increases paraboli-\ncally with frequency. Fluctuations around the parabola\nare due to binning noise associated with the finite k-\nspace discretization.\nWe have calculated the relative root-mean-square er-\nror in the calculated density of states (DOS) for an\nfcc‘empty’ crystal averaged over all histogram bins\nin the frequency range 0 < ωa/2πc <1 for several\ncombinations of histogram binwidth and k-grid resolu-\ntion, as specified in the caption of Fig. 2. As predicted\nby Eq. (11), the deviation of the approximation from\nthe analytic result is inversely proportional to the ratio\n∆ω/∆k. In most cases, one wants to calculate the LDOS\nwithagivenfrequencyresolution∆ ω,i.e.N(r,ω,ed)∆ω,\nto within a predetermined accuracy. For instance, calcu-\nlating the vacuum DOS for frequencies 0 < ωa/2πc <1\n4with a desired absolute accuracy better than 0 .01·4/a2c\n(1% of the vacuum DOS at ωa/2πc= 1) and a desired\nfrequencyresolutionof∆ ω= 0.01(2πc/a),requiresusing\n∆k∼∆ω/0.3c. The number of k-points corresponding\nto this wave vector sampling equals 2480 k-points in the\nirreducible wedge of the fccBrillouin zone, 59520 in the\nrequiredhalfBrillouinzone,orequivalently Nk= 119040\nin the full Brillouin zone. Photonic crystals with nonzero\nindex contrast cause a pronounced frequency structure\nof the LDOS. Due to the flattening of bands compared\nto the dispersion bands of vacuum-only crystals, the k-\nspace integration itself is at least as accurate, assuming\nthattheeigenfrequenciesandthefield-modepatternsare\nknown with infinite accuracy. In practice, one needs to\nadjust the number of plane waves to obtain all eigen-\nfrequencies to within the desired frequency resolution\n∆ω. In our computations for fcccrystals, we represented\nthe k-space of a half of the first Brillouin zone by an\nequidistant grid consisting of Nk/2 = 145708 k-points.\nThe frequency resolution of our LDOS histograms is\n∆ω= 0.01·2πc/a, and the spatial resolution of the\nLDOSis approximately a/40judgingfromthe maximum\n|G−G′| ≈120/ainvolved in the expansion with NG=\n725 plane waves in Eq. (4).\nD. Computation time required for LDOS\nCalculating the LDOS in 3D periodic structures is a\ntime-consuming task. The chosen degree of k-space dis-\ncretization ( Nk/2 = 145708 k-points) and the number\nof plane-waves ( NG= 725) are the result of a trade-off\nbetween the desired accuracy and tolerated duration of\nthe calculations. Essentially the computation consists of\nsolving for the lowest neigenvalues and eigenvectors of\na real symmetric 2 NG×2NGmatrix for each of the Nk\nk-points independently. To accomplish this calculation,\nwe use the standard Matlab ”eigs” implementation [43]\nof an implicitly restarted Arnoldi method (ARPACK)\nthat takes approximately 2.2 seconds to find the low-\nest 20 eigenvalues and eigenvectors of a single Hermitian\n1500x1500 (i.e., NG≈750 plane waves) matrix on a\n3 GHz Intel Pentium 4 processor, or on a 2.4 GHz In-\ntel Core Duo processor. For 145708 k-points the result-\ning computation time is hence on the order of 90 hours\n(4 days). Since the Matlab ARPACK routine is already\nhighlyoptimized,wedonotexpect thatthe computation\ntime per k-point can be significantly reduced for LDOS\ncalculations based on the standard H-field inverted ma-\ntrix plane wave method. In terms of the number of plane\nwavesNG, the computation time scales as N3\nG. As in\nRef. [44], the algorithm can be accelerated by realizing\nthat the iterative eigensolver does not require the full\nmatrixHmultiplying the vector uin Eq. 7, but rather\na function that quickly computes the image Huof any\ntrial vector u. Since calculating Hurepeatedly involves\na slow matrix-vector multiplication with ηG−G′, the al-\ngorithm is only accelerated for NG>1000.\nFig. 3. (Color online)DOS per volume in an fcccrys-\ntal consisting of spheres with ǫ= 7.35 in a medium\nwithǫ= 1.77 with a filling fraction of the spheres of\n25 vol %. The solid dotted curve represents calculations\nfrom Ref. [31]. Our result is plotted as a histogram(red).\n3. Comparison with previous results\nTo test the computations, we compare our results with\nearlier reports. We have calculated the DOS and LDOS\nin anfcccrystal consisting of dielectric spheres with ǫ1\n= 7.35 (TiO 2) in a medium with ǫ2= 1.77 (water) –\nthe same structure as was analyzed in Refs [31,33]. The\nspheresoccupy25vol%ofthe crystal.Figure3showsthe\ntotal DOS in this photonic crystal calculated by us and\nby Busch and John [31] . We reproduce the earlier cal-\nculations of the total DOS: both results shown in Fig. 3\nare in excellent agreement, with deviations less than 2%\nthroughout the frequency range 0 < ωa/2πc <1. In Fig-\nure 4 (Media 1)we demonstrate the LDOS in the same\nphotonic crystal at a specific location: at a point equidis-\ntant from two nearest-neighbor spheres. In this calcu-\nlation, we used the same number of reciprocal-lattice\nvectors N G= 965 as in the only available benchmark\npaper by Wang et al. [33] that does not contain sym-\nmetry errors. We find that our calculations (empty cir-\ncles) are in good agreement up to a/λ= 0.85 with the\nLDOS reported previously (solid curve): the deviations\nare smaller than 1%. Deviations at higher frequencies\nare either due to a difference in accuracy of k-space\nintegration (frequency binning resolution and k-space\nsampling density), or to a difference in accuracy of the\nplane wave methods (eigenmodes and eigenfrequencies).\nUnfortunately, neither the accuracy nor the frequency\nresolution is specified in Ref. [33]. The k-space sampling\ndensity in our work is approximately twice the density\nspecified in Ref. [33]. Based on the excellent agreement\nof our total DOS calculations with those of Busch and\nJohn[31],andonthefactthatWangetal.usedak-space\ndensity and plane wave number comparable to that of\nBusch and John, it is unlikely that inaccuracy of the k-\nspaceintegrationineithercalculationisthesourceofdis-\ncrepancy. We therefore surmise that deviations are due\nto a difference in the method of evaluation of En,k(r).\nConversion of DtoEby multiplication with 1 /ǫ(r) in\n5Fig. 4.(Coloronline) Dipole-averagedLDOS in the same\nphotonic crystal as in Fig. 3 at a position (1\n4,1\n4,0). Red\nhistogram: our calculations (Media 1). Solid curve: re-\nsults from Ref. [33]. This relative LDOS is the ratio of\nthe LDOS to that in vacuum at a/λ= 0.495.\nreal space as proposed in [33] can cause incorrect mode\namplitudes due to Gibbs oscillations, as opposed to the\nFourier space conversion using Eq. (8). In general our\ncalculations confirm the result by Wang et. al. [33] that\ntheLDOSisonlycorrectlycalculatedbyintegrationover\nhalf the Brillouin zone, rather than over the irreducible\npart as was incorrectly used in earlier reports.\n4. LDOS in TiO 2inverse opals\nIn recent time-resolved experiments, enhanced and in-\nhibited emission rates were demonstrated for quantum\ndots embedded inside strongly photonic TiO 2inverse\nopals [19,21,35]. In the framework of these experiments,\nit is highly relevant to calculate the LDOS inside such\ninverse-opal photonic crystals, especially for the source\npositions occupied in experiments. We model the posi-\ntion dependence of the dielectric function ǫ(r) as shown\nin Figure 5. This model assumes an infinite fcclattice\nof air spheres with radius r= 0.25√\n2a(ais the cubic\nlattice parameter). The spheres are covered by overlap-\nping dielectric shells ( ǫ= 6.5) with outer radius 1.09 r.\nNeighboringairspheresareconnectedbycylindricalwin-\ndows of radius 0.4 r. The resulting volume fraction of\nTiO2is equal to about 10.7%. These structural param-\neters are inferred from detailed characterization of the\ninverse opals using electron microscopy and small an-\ngle X-ray scattering [14,15]. Moreover, the stop gaps in\nthe photonic band structure (Fig. 6 (Media 2 and Me-\ndia 3)) calculated using this model agree well with re-\nflectivity measurements in the ranges of both the first-\norder (a/λ= 0.7) and second-order Bragg diffraction\n(a/λ= 1.2) [45].\nHenceforth, we will consider the relative LDOS, which\nis the ratio of the LDOS in a photonic crystal to that in\na homogeneous medium with the same volume-averaged\ndielectric function. This scaling is motivated by exper-\nimental practice, in which emission rate modifications\nare judged by normalizing measured rates to the emis-\nsion rate of the same emitter in crystals with much\nsmaller lattice constant a[19,21,35]. In these reference\nFig.5.Renderingofthedielectricfunctioninone fccunit\ncell that models the TiO 2inverse-opal structure in sec-\ntion 4: an fcclattice of air spheres of radius r= 0.25√\n2a\nwithabeing the cubic lattice parameter. The spheres\nare covered by shells with ǫ= 6.5 and outer radius\n1.09r. Neighboring air spheres are connected by win-\ndows of radius 0.4 r. The letters ( a–d) indicate four dif-\nferent positions at the TiO 2-air:a= (1,0,0)/(2√\n2),b=\n(1,1,2)/(4√\n3),c= (1,1,1)/(2√\n6) andd= (0.33,0.13,0)\n(points shown are symmetry-equivalents). The dash-\ndotted line shows the main diagonal of the cubic unit\ncell.\nFig. 6.Left: Photonic band structure (Media 2)for the\nTiO2inverse opal shown in Fig. 5. The grey rectangles\nindicate stopgaps in the ΓL direction and one stopgap\nin the ΓX direction for the inverse opal. The stopgaps\nresult in the decreased DOS ( right: circles, Media 3) at\ncorrespondingfrequencies compared to the DOS in a ho-\nmogeneous medium with nav= 1.27 (right: solid line).\n6Fig. 7. Relative LDOS in the inverse opal shown in Fig 5\nat three different positions: (a, Media 4) r= (0,0,0) [the\ncenterofan airsphere, solidcurve], r=1\n4(1,1,1)[among\nthree air spheres, dash-dotted curve] and r= (1\n2,0,0)\n[midwaybetweentwospheresalong[1,0,0]direction,dot-\nted curve]; (b, Media 5) r=1\n4(1,1,0) [in the window\nbetween two spheres] projected on [1,1,0], [-1,1,0] and\n[0,0,1] directions shown by solid, dash-dotted and dot-\nted curves, respectively.\nsystems, emission frequencies correspond to the effective\nmedium limit a/λ≪0.5 quantified by an average index\nnav=√ǫav= 1.27 for the TiO 2crystals [19]. In units\nof 4/a2c, the LDOS in a homogeneous medium is equal\ntonav(a/λ)2/3. In Figure 7a (Media 4) we plot the re-\nsulting LDOS at three positions in the unit cell: at r=\n(0,0,0),1\n4(1,1,1) and (1\n2,0,0). Due to the high symmetry\nof these points, the LDOS does not depend on the dipole\norientation, as we verified explicitly. A first main obser-\nvation from Fig. 7a (Media 4) is that the LDOS differs\nconsiderablybetween these threepositions at all reduced\nfrequencies. This observation illustrates the well-known\nstrong dependence of the LDOS on position within the\nunit cell of photonic crystals [10,27,31].\nA secondmain observationin Fig. 7a (Media 4) is that\ntheLDOSstronglyvarieswithreducedfrequency,reveal-\ning troughs and peaks caused by the pseudogap near\na/λ= 0.7, which is related to 1st-order stop gaps such\nas the L-gap. The effects of 2nd-order stopgaps appear\nbeyonda/λ >1.15 [45]. In the middle of the air region\nat position r= (0,0,0) there is a sharp, factor-of-three\nenhancement at a/λ≈1.35 within a narrow frequency\nrange. This feature could be probed by resonant atoms\ninfiltrated in the crystals [46]. At position r= (1\n2,0,0) in\nan interstitial, the mode density has a broad trough near\na/λ= 1.25 that will lead to strongly inhibited emission.\nA third main observation is that at spatial positions\nwith low symmetry, the LDOS clearly depends on the\norientation of the transition dipole moment. Figure 7b\n(Media 5) shows the frequency dependent LDOS for\nthree perpendicular dipole orientations at a position in\nthe center of a window that connects two neighboring\nair spheres (see Figure 5). The LDOS differs for all three\norientations, and is thus anisotropic. At low frequency\n(a/λ= 0.3), the emission rate is highest for a dipole\npointing in the (1 ,1,0) direction, with increasing fre-\nquency the highest rate shifts to the (0 ,0,1) and then to\nthe (−1,1,0) orientation. We emphasize that for emit-\nterswith fixedorslowlyvaryingdipole orientations,such\nFig. 8. Relative LDOS at four key frequencies, a/λ=\n0.535, 0.725, 0.865 and 1.295, in the inverse opal as a\nfunctionofposition rfrom(0,0,0)inthe[1,1,1]direction.\nThe hatched boxes indicate the position of the dielectric\nTiO2shell. The LDOS is projected on two dipole ori-\nentations: (a) [1,1,1] perpendicular to the dielectric-air\ninterface, and (b) [-1,1,0] parallel to the interface. The\nLDOS projected on the [-1,-1,2] and [-1,1,0] directions is\nequal. For r/a∈[√\n3\n2,√\n3] the LDOS is mirror-symmetric\nto that in the region [0 ,√\n3\n2].\nas dye molecules or quantum dots on solid interfaces, the\nemissionrateisdeterminedbytheopticalmodesthatare\nprojected on the dipole orientation. Therefore, knowl-\nedge of the projected LDOS is important for controlling\nspontaneous-emissionratesaswellasforinterpretingthe\ndata from experiments on emitters in photonic metama-\nterials.\nA fourth main observation from Fig. 7a and b (Media\n4 and 5) is that at low frequencies a/λ <0.5, the rela-\ntive LDOS hardly varies with frequency, which means\nthat the mode density is proportional to ω2, as in homo-\ngeneous media. Interestingly, there is a clear dependence\nofthe LDOS onboth the position andthe dipole orienta-\ntion even at these low frequencies, i.e., long wavelengths\nrelative to the crystal periodicity. The reason for these\neffects is that the photonic Bloch modes exhibit local\nvariations of the electric field related to local variations\nof the dielectric function in order to satisfy the conti-\nnuity equations at dielectric boundaries for the parallel\nEor perpendicular Dfield, respectively [34,47]. Conse-\nquently, the LDOS strongly varies on length scales much\nless than the wavelength, i.e., even in electrostatic or ef-\nfective medium limit. While such behavior may appear\nsurprising, its origin in electrostatic depolarization ef-\nfects has been discussed before [47,48].\nTo gain more insight in the spatial dependence of the\nLDOS in the inverse opals, we have performed calcula-\ntions for dipoles positioned along a characteristic axis in\nthe unit cell. Figure 8 shows the LDOS at four key fre-\nquencies for dipoles placed on the body diagonal of the\ncubic unit cell, that is, from (0 ,0,0) in the [1 ,1,1] direc-\ntion. The LDOS has clear maxima and minima along the\ndiagonal, varying by more than 10 ×. Figure 8(a) shows\nthat for a dipole oriented in the [1 ,1,1] direction (per-\npendicular to the dielectric-air interface), the LDOS is\nstrongly ( >5x) suppressed near the dielectric TiO 2shell\n(r/a≈0.353) at all frequencies. In contrast, no strong\n7/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s101\n/s100/s32/s61/s32/s91/s49/s44/s49/s44/s50/s93/s101\n/s100/s32/s61/s32/s91/s45/s49/s44/s45/s49/s44/s49/s93\n/s101\n/s100/s32/s61/s32/s91/s45/s49/s44/s49/s44/s48/s93/s80/s111/s115/s105/s116/s105/s111/s110/s32 /s98 /s32/s82/s101/s108/s46/s32/s76/s68/s79/s83/s32\n/s32/s32\n/s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s101\n/s100/s32/s61/s32/s91/s45/s49/s44/s49/s44/s48/s93\n/s101\n/s100/s32/s61/s32/s91/s49/s44/s49/s44/s49/s93/s82/s101/s108/s46/s32/s76/s68/s79/s83\n/s97/s47\n/s32/s32\n/s80/s111/s115/s105/s116/s105/s111/s110/s32 /s99/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s101\n/s100/s32/s61/s32/s91/s48/s44/s49/s44/s48/s93\n/s101\n/s100/s32/s61/s32/s91/s49/s44/s48/s44/s48/s93/s82/s101/s108/s46/s32/s76/s68/s79/s83\n/s32/s32\n/s80/s111/s115/s105/s116/s105/s111/s110/s32 /s97\n/s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54/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/s101\n/s100/s32/s61/s32/s91/s48/s46/s51/s51/s44/s48/s46/s49/s51/s44/s48/s93/s101\n/s100/s32/s61/s32/s91/s45/s48/s46/s49/s51/s44/s48/s44/s48/s46/s51/s51/s93/s101\n/s100/s32/s61/s32/s91/s48/s44/s48/s44/s49/s93\n/s97/s47/s80/s111/s115/s105/s116/s105/s111/s110/s32 /s100/s82/s101/s108/s46/s32/s76/s68/s79/s83/s32\n/s32/s32\n/s40/s100/s41/s40/s99/s41/s40/s98/s41 /s40/s97/s41\nFig.9.RelativeLDOSintheinverseopalatfourdifferent\npositions on the TiO 2-air interface shown in Fig. 5. At\neach position the LDOS is projected on three mutually\northogonal dipole orientations ed. (a, Media 6) point a\nfored= [1,0,0] and [0,1,0]. LDOS at ed= [0,0,1] and\n[0,1,0] is the same. (b, Media 7) point bfored= [1,1,2],\n[-1,1,0] and [-1,-1,1]. (c, Media 8) point cfored= [1,1,1]\nand [-1,1,0]. LDOS at ed= [-1,-1,2] is equal to that at\ned= [-1,1,0]. (d, Media 9) point dfored= [0.33,0.13,0],\n[-0.13,0,0.33] and [0,0,1].\nsuppression or enhancement occurs for dipole orienta-\ntions parallel to the dielectric interface, see Fig. 8(b).\nThe strongly differing LDOS for dipoles near the dielec-\ntric rationalize the broad distributions of emission rates\nthat were recently observed for quantum dots in inverse\nopals [35], see below. Enhancements and inhibitions also\noccur in the air regions: the LDOS is enhanced at r/a≈\n0.28 and 0.57 by up to 2.5 to 3 times, respectively, see\nFigure 8(a). At point r/a= 1/√\n3≈0.57 that lies in the\n(111) lattice plane in the air region, the LDOS is inhib-\nited at all frequencies for the dipole orientations [-1,1,0]\nand [-1,-1,2] that are perpendicular to the (111) plane\n(see Fig. 8(b)). Finally, for dipoles parallel to the body\ndiagonal (Figure 8(a)) the mode density shows pseudo-\noscillatory behavior on length scales much less than the\nwavelength, e.g., a period of 0 .2aat frequency 0 .535a/λ\n(period corresponds to 1 /10thof a wavelength). This ob-\nservation confirms that photonic crystals are bona fide\nmetamaterials where optical properties strongly vary on\nlength scales much less than the wavelength.\nIn recent time-resolved experiments [21,35], emission\nfrom quantum dot light sources distributed at the in-\nternal TiO 2-air interfaces of the inverse opals was in-\nvestigated. To analyze the experimental data, we have\ncalculated the LDOS at several symmetry-inequivalent\npositions on the TiO 2shells: at points a,b,candd\n(see Fig. 5) for three mutually orthogonal orientations\nof the emitting dipole, where the first orientation is cho-\nsen along the vector pointing from (0,0,0) toward the\ncorresponding point. Figures 9(a) through 9(d) (Media\n6 through 9) show that at all these positions the LDOSstrongly varies with reduced frequency and position, as\nexpected, and also with orientation of the dipole. In\nbroad terms, for a dipole parallel to the interface the\nLDOS is near 1 at low frequency, increases to a peak\nenhancement of 2 .5×ata/λ= 1.22 before strongly\nvarying at high frequencies. For reference, the frequency\na/λ= 1.22 is in the range where a band gap is expected\nfor more strongly interacting crystals. The plots also re-\nveal that for dipoles perpendicular to the TiO 2-air in-\nterface, the LDOS is strongly inhibited (more than 10 ×)\nover broad frequency ranges. For instance, Figure 9(a)\nshows that the emission rate for a dipole perpendicular\ntothe interfaceis16-foldinhibited toalevelof0.06,with\na maximum of 0.22 (5-fold inhibition) at a/λ= 1.22.\nIt is striking that the strong inhibition occurs over a\nbroad frequency range from 0.3 to 1.6, i.e., more than\ntwo octaves in frequency. While the inhibition is not\ncomplete, the bandwidth is much larger than the maxi-\nmum bandwidth of 12% for a full 3D bandgap in inverse\nopals [49] and far exceeds the bandwidth of the 2D gap\nforTEmodesinmembranephotoniccrystals,thatallows\na seven-fold inhibition [26,30] over a 30% bandwidth.\nIn the frequency range up to a/λ= 1.1, the depen-\ndence of the LDOS on frequency is quite similar at all\nthe positions and dipole orientations. This remarkable\nresult agrees with the experimental observation from\nRef. [35]: the complex decay curves of light sources in\ntheinverseopalsaredescribedbyoneand thesamefunc-\ntional shape (log-normal) of the decay-rate distribution\nfor all reduced frequencies studied.\nFor the interpretation of the time-resolved experi-\nments on ensembles of emitters in the inverse-opal pho-\ntonic crystals, the results presented above mean that\n1. thedecayrateofanindividualemitterisdetermined\nbyits frequency,positionandalsobytheorientation\nof its transition dipole in the photonic crystal;\n2. the measured spontaneous-emission decay depends\non how the emitters are distributed in the crystal;\n3. even in the low-frequency regime, ensemble\nmeasurements will reveal non-exponentional decay\ncurves;\n4. the similar shape of the reduced-frequency de-\npendence of the LDOS allows modeling of the\nnon-exponentional decay curves with a single type\nof decay-rate distributions. In other words, one\ndistribution function can be successfully used to\nmodel the multi-exponentional decay curves meas-\nured from crystals with different lattice parameters.\n5. LDOS in silicon inverse opals\nA complete inhibtion of spontaneous emission may only\nbe achieved in photonic crystals with a 3D photonic\nband gap. Therefore,there has been much effort to fabri-\ncate inverse opals from silicon rather than titania, since\nthe higher index of silicon allows for a photonic band\n8Fig. 10. Left: Photonic band structure (Media 10) for\nan inverse opal from silicon ( ǫ= 11.9). Right: The to-\ntal DOS in the Si inverse opal (Media 11). The DOS is\nstrongly depleted for frequencies near ΓL and ΓX stop-\ngaps (grey rectangles). A photonic band gap (grey bar)\noccurs between bands 8 and 9, as also reflected in the\nvanishing DOS.\nFig. 11. Relative LDOS in a Si inverse opal at: (a, Me-\ndia 12)r= (0,0,0) [the center of an air sphere, solid\ncurve],r=1\n4(1,1,1) [among three air spheres, dash-\ndotted curve] and r= (1\n2,0,0) [midway between two\nspheres along [1,0,0] direction, dotted curve]; (b, Media\n13)r=1\n4(1,1,0)[inthe windowbetween twoairspheres]\nprojected on [1,1,0], [-1,1,0] and [0,0,1] directions shown\nby solid, dash-dotted and dotted curves, respectively.\ngap [31,37]. For the calculation of the LDOS in such Si\ninverse opals, the dielectric function ǫ(r) was modeled\nsimilarly to that in the inverse opals shown in Fig. 5.\nFrom SEM observations we inferred the following struc-\ntural parameters [17,18]: the outer radius of the over-\nlapping dielectric shells with ǫ= 11.9 is about 1.15 r\n(wherer= 0.25√\n2ais the air-sphere radius, ais the\nlattice parameter). The cylindrical windows connecting\nneighboring air spheres have a radius of 0.2 r. The larger\nouter radius and the smaller window size compared to\nthe TiO 2inverse opals are commensurate with a higher\nvolume fraction of about 23% Si [18].\nThe band structure and total DOS for this system are\nshown in Fig. 10 (Media 10 and Media 11). The DOS is\nstrongly depleted in a pseudo-gap between the 2ndand\n3rdbands [36], at frequencies near 0.55. Near frequency\n0.85, both the band structures and the DOS reveal a\n3D photonic band gap of relative width ∆ ω/ω≈3 %between the 8thand 9thbands [31]. Compared to the\nTiO2structure, both the lowest-order L-gap and the 8th\nand 9thbands are shifted to lower reduced frequencies.\nThisshift isdue to the highereffective refractiveindex of\nthe Si inverse opals ( nav= 1.88), on account of a higher\nindex of the backbone and a higher filling fraction.\nFigure 11(a) (Media 12) presents the relative LDOS\nat three high-symmetry positions in the unit cell r=\n(0,0,0),1\n4(1,1,1) and (1\n2,0,0). The LDOS varies much\nmorestronglywith frequencythan in TiO 2inverseopals,\nas a result of the larger dielectric contrast that leads to\nstrongly modified dispersion relations and Bloch mode\nprofiles.While themaximaupto3.2arenotmuchhigher\nthan in TiO 2inverse opals, the minima in the mode den-\nsity are reduced and the slopes are steeper. As expected,\nthe LDOS is completely inhibited at all positions in the\nfrequency range of the band gap. Previously, it has been\nsuggested that the LDOS could be inhibited at salient\npositions in the unit cell for frequencies outside a com-\nplete gap. While Figure 11(a) (Media 12) reveals that\nthe mode density is strongly reduced above the band\ngap, e.g., at r= (1\n2,0,0), it is not truly inhibited. In\nfact, in the course of our study, we have not encoun-\ntered any ”sweet spots” where the LDOS is completely\ninhibited at frequencies outside a 3D photonic band gap.\nFigure 11(b) (Media 13) shows the LDOS at a lower\nsymmetry position r=1\n4(1,1,0), in the window between\ntwo nearest-neighbor air spheres. The mode density has\nbeen calculated for three perpendicular dipole orienta-\ntions. Figure 11(b) (Media 13) shows that the LDOS is\nhighly anisotropic since it differs for all three orienta-\ntions: at low frequencies ( a/λ <0.5), the mode density\nis highest for the ( −1,1,0) orientation, intermediate for\nthe (0,0,1) orientation, and lowest for the (1 ,1,0) ori-\nentation. With increasing frequency, the highest LDOS\nalso changes to other orientations, with the (0 ,0,1) ori-\nentation having the highest LDOS above the pseudo-gap\nand even a narrow peak at frequency 0.95. Therefore, if\nit is possible to orient a quantum emitter with its dipole\nparallel to (0 ,0,1), this frequency range is conducive to\nstrong emission enhancement and perhaps even QED ef-\nfects beyond weak-coupling. Interestingly, these frequen-\ncies are slightly higher than the upper edge of the band\ngap where strong coupling effects have recently been dis-\ncussed [12].\n6. Conclusions\nWe have performed intensive calculations of the local\ndensity ofstates in TiO 2and Si inverseopalswith exper-\nimentally relevant structural parameters. Since conflict-\ning and incorrect reports have appeared on the LDOS in\nphotoniccrystals,wehavesetouttovalidateourmethod\nof choice, i.e., the H-field plane-wave expansion method.\nThisvalidationreliedoncomparisontoliteratureresults,\non the explicit verification of required symmetries that\nprevious reports failed to satisfy, and on quantitative\nconsiderations of resolution and accuracy. 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ISBN 90-9016903-2, available from\nwww.photonicbandgaps.com\n11"    },    {        "title": "0902.1902v2.Local_density_of_states_of_electron_crystal_phases_in_graphene_in_the_quantum_Hall_regime.pdf",        "content": "arXiv:0902.1902v2  [cond-mat.mes-hall]  14 Feb 2009\n/C4/D3 \r/CP/D0 /CS/CT/D2/D7/CX/D8 /DD /D3/CU /D7/D8/CP/D8/CT/D7 /D3/CU /CT/D0/CT\r/D8/D6/D3/D2/B9\r/D6/DD/D7/D8/CP/D0 /D4/CW/CP/D7/CT/D7 /CX/D2 /CV/D6/CP/D4/CW/CT/D2/CT /CX/D2 /D8/CW/CT /D5/D9/CP/D2 /D8/D9/D1 /C0/CP/D0/D0/D6/CT/CV/CX/D1/CT/C7/BA /C8 /D3/D4/D0/CP /DA/D7/CZ/DD/DD/B8\n/BD/B8 /BE/B8 /BF /B8∗/C5/BA /C7/BA /BZ/D3 /CT/D6/CQ/CX/CV/B8\n/BE/CP/D2/CS /BV/BA /C5/D3/D6/CP/CX/D7 /CB/D1/CX/D8/CW\n/BF/BD/BY /CP\r/D9/D0/D8/DD /D3/CU /C5/CP/D8/CW/CT/D1/CP/D8/CX\r/D7/B8 /CF/CX/D0/CQ /CT/D6/CU/D3/D6 \r /CT /CA /D3 /CP/CS/B8 /CD/D2/CX/DA/CT/D6/D7/CX/D8/DD /D3/CU /BV/CP/D1/CQ/D6/CX/CS/CV/CT/B8 /BV/BU/BF /BC/CF /BT/B8 /CD/D2/CX/D8/CT /CS /C3/CX/D2/CV/CS/D3/D1/BE/C4 /CP/CQ /D3/D6 /CP/D8/D3/CX/D6 /CT /CS/CT 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/CT/BA/CV/BA /CS/D9/CT /D8/D3 /CP /D7/D9Ꜷ\r/CX/CT/D2 /D8/D0/DD /D0/CP/D6/CV/CT /CI/CT/CT/B9/D1/CP/D2 /CT/AR/CT\r/D8/BA /CC/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /C0/CP/D6/D8/D6/CT/CT/B9/BY /D3 \r /CZ /C0/CP/D1/CX/D0/B9/D8/D3/D2/CX/CP/D2 /CU/D3/D6 /D8/CW/CT /BE/BW/BX/BZ /CX/D2 /BZ/CP/BT/D7 /CW/CP/D7 /CQ /CT/CT/D2 /CT/DC/D8/CT/D2/D7/CX/DA /CT/D0/DD /CS/CX/D7/B9\r/D9/D7/D7/CT/CS /CX/D2 /D8/CW/CT /D0/CX/D8/CT/D6/CP/D8/D9/D6/CT/BA\n/BG/BC/B8/BG/BD/C1/D2 /CV/D6/CP/D4/CW/CT/D2/CT/B8 /D8/CW/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /CU/D3/D6 /D8/CW/CT /BE/BW/BX/BZ /CX/D7 /D7/CX/D1/CX/D0/CP/D6 /D8/D3 /D8/CW/CP/D8 /CX/D2 /BZ/CP/BT/D7/B8 /CP/D0/B9/CQ /CT/CX/D8 /DB/CX/D8/CW /CS/CX/AR/CT/D6/CT/D2 /D8 /CU/D3/D6/D1 /CU/CP\r/D8/D3/D6/D7 /CS/D9/CT /D8/D3 /D8/CW/CT /D7/D4/CX/D2/D3/D6/CX/CP/D0 /CU/D3/D6/D1/D3/CU /D8/CW/CT /DB /CP /DA /CT /CU/D9/D2\r/D8/CX/D3/D2/D7/BA\n/BE/BI/B8/BG/BE/CC/CW/CX/D7 /D7/CX/D1/CX/D0/CP/D6/CX/D8 /DD /CP/D0/D0/D3 /DB/D7 /D3/D2/CT /D8/D3/D9/D7/CT /D8/CW/CT /D7/CP/D1/CT /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0 /D1/CT/D8/CW/D3 /CS/D7 /DB/CW/CX\r /CW /DB /CT/D6/CT /D9/D7/CT/CS /D4/D6/CT/B9/DA/CX/D3/D9/D7/D0/DD /D8/D3 /D7/D8/D9/CS/DD /D8/CW/CT /BE/BW/BX/BZ /CX/D2 /BZ/CP/BT/D7/B8 /DB/CX/D8/CW /D8/CW/CT /CX/D1/D4 /D3/D6/D8/CP/D2 /D8/CS/CX/AR/CT/D6/CT/D2\r/CT /D8/CW/CP/D8 /DB /CT /D2/CT/CT/CS /D8/D3 /D8/CP/CZ /CT /CX/D2 /D8/D3 /CP\r\r/D3/D9/D2 /D8 /D8/CW/CT /D8 /DB /D3/CU/D3/D0/CS/DA /CP/D0/D0/CT/DD /CS/CT/CV/CT/D2/CT/D6/CP\r/DD /CX/D2 /D8/CW/CT /CU/D3/D6/D1 /D3/CU /CP/D2 /CB/CD/B4/BE/B5 /D4/D7/CT/D9/CS/D3/D7/D4/CX/D2/CS/CT/CV/D6/CT/CT /D3/CU /CU/D6/CT/CT/CS/D3/D1/B8 β=±1 /BA /C8/D6/D3 /DA/CX/CS/CT/CS /D8/CW/CP/D8 /CX/D2 /D8/CT/D6/B9/C4/C4 /D8/D6/CP/D2/B9/D7/CX/D8/CX/D3/D2/D7 /CP/D6/CT /D2/CT/CV/D0/CT\r/D8/CT/CS/B8 /DB /CT /D1/CP /DD /DB/D6/CX/D8/CT /D8/CW/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /D4/CP/D6/D8/D3/CU /D8/CW/CT /CU/D9/D0/D0 /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /CU/D3/D6 /D8/CW/CT /BE/BW/BX/BZ /D3/CU /D7/D4/CX/D2/D0/CT/D7/D7 /CT/D0/CT\r/D8/D6/D3/D2/D7/CX/D2 /CV/D6/CP/D4/CW/CT/D2/CT /CP/D7\n/BE/BI\nˆHint=VC\n2/summationdisplay\nβ,β′, /D51\n| /D5|[FN( /D5)]2ˆρβ,β(− /D5)ˆρβ′,β′( /D5), /B4/BD/B5/DB/CW/CT/D6/CTVC≡e2/lBǫ /CX/D7 /D8/CW/CT /BV/D3/D9/D0/D3/D1 /CQ /CT/D2/CT/D6/CV/DD /D7\r/CP/D0/CT/B8 /DB/CX/D8/CW\nlB=/radicalbig\n/planckover2pi1/eB /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /D0/CT/D2/CV/D8/CW/B8B /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS/B8/CP/D2/CSǫ /D8/CW/CT /CS/CX/CT/D0/CT\r/D8/D6/CX\r /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D3/CU /D8/CW/CT /D1/CT/CS/CX/D9/D1/B8 /CP/D2/CS/D5≡(qx,qy) /CX/D7 /CP /BE/BW /DB /CP /DA /CT/DA /CT\r/D8/D3/D6/BA /CC/CW/CT /B4/CV/D9/CX/CS/CX/D2/CV \r/CT/D2 /D8/CT/D6/B5/CS/CT/D2/D7/CX/D8 /DD /D3/D4 /CT/D6/CP/D8/D3/D6 /CX/D2 /D8/CW/CT /C4/CP/D2/CS/CP/D9 /CV/CP/D9/CV/CT /D6/CT/CP/CS/D7\nˆρβ1,β2( /D5) =N−1\nφ/summationdisplay\nXexp/parenleftbigg\n−iqxX−il2\nBqxqy\n2/parenrightbigg\n׈c†\nX,β1ˆcX+l2\nBqy,β2. /B4/BE/B5/BF/C0/CT/D6/CT/B8Nφ=S/2πl2\nB\n/D1/CT/CP/D7/D9/D6/CT/D7 /D8/CW/CT /C4/C4 /CS/CT/CV/CT/D2/CT/D6/CP\r/DD /B8 /DB/CX/D8/CW/D8/CW/CT /D7/D5/D9/CP/D6/CT /CP/D6/CT/CPS /D3/CU /D8/CW/CT /BE/BW/BX/BZ /D7/CP/D1/D4/D0/CT/B8ˆcX,β\n/CP/D2/CSˆc†\nX,β\n/CP/D6/CT/D8/CW/CT /CT/D0/CT\r/D8/D6/D3/D2/B3/D7 /CS/CT/D7/D8/D6/D9\r/D8/CX/D3/D2 /CP/D2/CS \r/D6/CT/CP/D8/CX/D3/D2 /D3/D4 /CT/D6/CP/D8/D3/D6/D7/B8 /D6/CT/D7/D4 /CT\r/B9/D8/CX/DA /CT/D0/DD /B8 /DB/CW/CT/D6/CTX /CS/CT/D2/D3/D8/CT/D7 /D7/CX/D2/CV/D0/CT/B9/D4/CP/D6/D8/CX\r/D0/CT /D5/D9/CP/D2 /D8/D9/D1 /D7/D8/CP/D8/CT/D7/DB/CX/D8/CW/CX/D2 /D8/CW/CTN /B9/D8/CW /C4/C4/BA /BY/CX/D2/CP/D0/D0/DD /B8 /CX/D2 /BX/D5/BA /B4/BD /B5 /D8/CW/CT /CV/D6/CP/D4/CW/CT/D2/CT/CU/D3/D6/D1 /CU/CP\r/D8/D3/D6FN( /D5) /D6/CT/CP/CS/D7\n/BE/BI /B8/BG/BE\nFN( /D5) =\n\n1\n2/bracketleftBig\nL|N|/parenleftBig\nq2\n2/parenrightBig\n+L|N|−1/parenleftBig\nq2\n2/parenrightBig/bracketrightBig\ne−q2\n4, N∝negationslash= 0;\ne−q2\n4, N = 0,/B4/BF/B5/DB/CW/CT/D6/CTq≡ | /D5| /B8 /CP/D2/CSLn(x) /CX/D7 /D8/CW/CT /C4/CP/CV/D9/CT/D6/D6/CT /D4 /D3/D0/DD/D2/D3/D1/CX/CP/D0 /D3/CU/D3/D6/CS/CT/D6n /BA /CF /CT /D2/D3/D8/CT /D8/CW/CP/D8 /D8/CW/CT /BE/BW/BX/BZ /CU/D3/D6/D1 /CU/CP\r/D8/D3/D6 /CX/D2 /BZ/CP/BT/D7 /CX/D7/CV/CX/DA /CT/D2 /CQ /DD\n/BG/BC\nFN( /D5) =LN/parenleftbiggq2\n2/parenrightbigg\ne−q2/4. /B4/BG/B5/C1/D8 /CX/D7 /CP/D4/D4/CP/D6/CT/D2 /D8 /CU/D6/D3/D1 /BX/D5/D7/BA /B4/BF/B5 /CP/D2/CS /B4/BG/B5 /D8/CW/CP/D8 /D8/CW/CT /CV/D6/CP/D4/CW/CT/D2/CT/B4/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX\r /BE/BW/BX/BZ/B5 /CU/D3/D6/D1 /CU/CP\r/D8/D3/D6 /CX/D7 /D7/CX/D1/D4/D0/DD /CP /D0/CX/D2/CT/CP/D6 \r/D3/D1/B9/CQ/CX/D2/CP/D8/CX/D3/D2 /D3/CU /CU/D3/D6/D1 /CU/CP\r/D8/D3/D6/D7 /CU/D3/D6 /CP/CS/CY/CP\r/CT/D2 /D8 /C4/C4/B3/D7 /D3/CU /D8/CW/CT /D2/D3/D2/B9/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX\r /BE/BW/BX/BZ /CX/D2 /BZ/CP/BT/D7/BA /CC/CW/CX/D7 /D4 /CT\r/D9/D0/CX/CP/D6 /CU/CP\r/D8 /D6/CT/D7/D9/D0/D8/D7/CU/D6/D3/D1 /D1/CX/DC/CX/D2/CV /D8/CW/CT /BW/CX/D6/CP\r /D4/CP/D6/D8/CX\r/D0/CT /DB /CP /DA /CT/CU/D9/D2\r/D8/CX/D3/D2/D7 /CQ /CT/D8 /DB /CT/CT/D2/D8/CW/CT /D7/CX/D8/CT/D7 /D3/CU /D8 /DB /D3 /D7/D9/CQ/D0/CP/D8/D8/CX\r/CT/D7 /CX/D2 /CV/D6/CP/D4/CW/CT/D2/CT/B8 /CP/D2/CS /CX/D7 /CP/D0/D7/D3 /CP\r/D3/D2/D7/CT/D5/D9/CT/D2\r/CT /D3/CU /D8/CW/CT /D7/D4/CX/D2/D3/D6/CX/CP/D0 /D2/CP/D8/D9/D6/CT /D3/CU /D8/CW/CT/D7/CT /DB /CP /DA /CT/CU/D9/D2\r/B9/D8/CX/D3/D2/D7/BA /BT/D4/CP/D6/D8 /CU/D6/D3/D1 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2\r/CT /CX/D2 /CU/D3/D6/D1 /CU/CP\r/D8/D3/D6/D7 /CV/CX/DA /CT/D2 /CQ /DD/BX/D5/D7/BA /B4/BF/B5 /CP/D2/CS /B4/BG/B5/B8 /D8/CW/CT /BE/BW/BX/BZ /CX/D2 /BZ/CP/BT/D7 /CP/D2/CS /CV/D6/CP/D4/CW/CT/D2/CT /CX/D7 /CS/CT/B9/D7\r/D6/CX/CQ /CT/CS /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8/D0/DD /B8 /CP/D7 /CU/D3/D0/D0/D3 /DB/D7 /CU/D6/D3/D1 /D8/CW/CT /D7/CP/D1/CT /CP/D2/CP/D0/DD/D8/CX\r/CP/D0/D7/D8/D6/D9\r/D8/D9/D6/CT /D3/CU /D8/CW/CT /BV/D3/D9/D0/D3/D1 /CQ /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /D8/CT/D6/D1 /CV/CX/DA /CT/D2 /CQ /DD /BX/D5/BA/B4/BD/B5/BA/BY/CX/D2/CP/D0/D0/DD /B8 /DB /CT /D2/D3/D8/CT /D8/CW/CP/D8 /D8/CW/CT /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /CX/D2 /BX/D5/BA /B4/BD/B5 /CX/D7/CB/CD/B4/BE/B5/B9/CX/D2 /DA /CP/D6/CX/CP/D2 /D8 /DB/CX/D8/CW /D6/CT/D7/D4 /CT\r/D8 /D8/D3 /D8/CW/CT /DA /CP/D0/D0/CT/DD /D4/D7/CT/D9/CS/D3/D7/D4/CX/D2/BA/C1/D2 \r/D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /D8/CW/CT /D4/CW /DD/D7/CX\r/CP/D0 /CT/D0/CT\r/D8/D6/D3/D2 /D7/D4/CX/D2/B8 /D8/CW/CX/D7 /CB/CD/B4/BE/B5 /D7/DD/D1/B9/D1/CT/D8/D6/DD /CX/D7 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /CB/CD/B4/BE/B5/B9/D7/DD/D1/D1/CT/D8/D6/DD /CQ/D6/CT/CP/CZ/B9/CX/D2/CV /D8/CT/D6/D1/D7 /CP/D6/CT /D7/D9/D4/D4/D6/CT/D7/D7/CT/CS /D0/CX/D2/CT/CP/D6/D0/DD /CX/D2a/lB≪1 /DB/CW/CT/D6/CT\na= 0.14 /D2/D1 /CX/D7 /D8/CW/CT \r/CP/D6/CQ /D3/D2/B9\r/CP/D6/CQ /D3/D2 /CS/CX/D7/D8/CP/D2\r/CT /CX/D2 /CV/D6/CP/D4/CW/CT/D2/CT/CP/D2/CSlB= 26//radicalbig\nB[T] /D2/D1/B8 /CX/BA/CT/BA /CP/D8 /CP/D2 /CT/D2/CT/D6/CV/DD /D7\r/CP/D0/CT /D8/CW/CP/D8/CX/D7 /DB /CT/D0/D0 /CQ /CT/D0/D3 /DB /D8/CW/CT /CS/CX/D7/D3/D6/CS/CT/D6 /CQ/D6/D3/CP/CS/CT/D2/CX/D2/CV /D3/CU /D8/CW/CT /C4/C4/B3/D7/BA\n/BE/BI/B8/BG/BF/CC/CW/CX/D7 /D4/CW /DD/D7/CX\r/CP/D0 /D1/D3 /CS/CT/D0 /CX/D7 /D7/CX/D1/CX/D0/CP/D6 /D8/D3 /CP/D2/D3/D8/CW/CT/D6 /D8 /DB /D3/B9\r/D3/D1/D4 /D3/D2/CT/D2 /D8/D5/D9/CP/D2 /D8/D9/D1 /C0/CP/D0/D0 /D7/DD/D7/D8/CT/D1 /AL /CX/CU /D3/D2/CT /D6/CT/D4/D0/CP\r/CT/D7 /CX/D2 /BX/D5/BA /B4/BD/B5FN( /D5)/CQ /DD /D8/CW/CT /D2/D3/D2/B9/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX\r /CU/D3/D6/D1/B9/CU/CP\r/D8/D3/D6 FN( /D5) /B8 /D3/D2/CT /D3/CQ/D8/CP/CX/D2/D7/D8/CW/CT /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /CU/D3/D6 /D8/CW/CT /D2/D3/D2/B9/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX\r /BE/BW/BX/BZ /CX/D2\r/D0/D9/CS/CX/D2/CV/D8/CW/CT /CT/D0/CT\r/D8/D6/D3/D2/D7/B3 /D7/D4/CX/D2 /CX/D2 /D8/CW/CT /CP/CQ/D7/CT/D2\r/CT /D3/CU /CP /D4 /D3/D0/CP/D6/CX/DE/CX/D2/CV /CI/CT/CT/D1/CP/D2/CT/AR/CT\r/D8/BA /BT/D0/D8/CT/D6/D2/CP/D8/CX/DA /CT/D0/DD /B8 /D8/CW/CX/D7 /D1/D3 /CS/CT/D0 /D1/CP /DD /CS/CT/D7\r/D6/CX/CQ /CT /CP /D5/D9/CP/D2 /D8/D9/D1/C0/CP/D0/D0 /CQ/CX/D0/CP /DD /CT/D6 /CX/D2 /D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0 /D0/CX/D1/CX/D8 /D3/CU /DE/CT/D6/D3 /D0/CP /DD /CT/D6 /D7/CT/D4/CP/D6/CP/B9/D8/CX/D3/D2/B8 /DB/CW/CT/D6/CT /D8/CW/CT /D8 /DB /D3 /AG/D7/D4/CX/D2/AH /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2/D7 /CS/CT/D2/D3/D8/CT /D8/CW/CT /D8 /DB /D3/CS/CX/AR/CT/D6/CT/D2 /D8 /D0/CP /DD /CT/D6/D7/BA\n/BG/BD/C7/D2/CT /D1/CP /DD /CU/D9/D6/D8/CW/CT/D6 /D7/CX/D1/D4/D0/CX/CU/DD /D8/CW/CT /D1/D3 /CS/CT/D0 /CX/D2/BX/D5/BA /B4/BD/B5 /CQ /DD /D3/D1/CX/D8/D8/CX/D2/CV /D8/CW/CT /DA /CP/D0/D0/CT/DD /D4/D7/CT/D9/CS/D3/D7/D4/CX/D2 /CS/CT/CV/D6/CT/CT /D3/CU /CU/D6/CT/CT/B9/CS/D3/D1/B8 /CX/D2 /DB/CW/CX\r /CW \r/CP/D7/CT /D3/D2/CT /D4/D6/CT/D7/D9/D4/D4 /D3/D7/CT/D7 /CP \r/D3/D1/D4/D0/CT/D8/CT /DA /CP/D0/D0/CT/DD/D4 /D3/D0/CP/D6/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CT/D0/CT\r/D8/D6/D3/D2/CX\r /D4/CW/CP/D7/CT/D7/B8 /DB/CW/CX\r /CW /DB /D3/D9/D0/CS /D1/CP/DC/CX/B9/D1/CP/D0/D0/DD /D4/D6/D3/AS/D8 /CU/D6/D3/D1 /D8/CW/CT /CT/DC\r /CW/CP/D2/CV/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/BA /CC/CW/CX/D7 /CT/AR/CT\r/D8/CX/DA /CT/CD/B4/BD/B5 /D1/D3 /CS/CT/D0 /CX/D7 /CS/CT/D7\r/D6/CX/CQ /CT/CS /CQ /DD /D8/CW/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /D8/CT/D6/D1\nˆHint=VC\n2/summationdisplay/D51\n| /D5|[FN( /D5)]2ˆρ(− /D5)ˆρ( /D5), /B4/BH/B5/DB/CW/CT/D6/CT /D8/CW/CT /CS/CT/D2/D7/CX/D8 /DD /D3/D4 /CT/D6/CP/D8/D3/D6 /D3/CU /D7/D4/CX/D2/D0/CT/D7/D7 /CT/D0/CT\r/D8/D6/D3/D2/D7 ˆρ( /D5) /CX/D7/D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /BX/D5/BA /B4/BE/B5 /CQ /DD /D2/CT/CV/D0/CT\r/D8/CX/D2/CV /D8/CW/CT /D4/D7/CT/D9/CS/D3/D7/D4/CX/D2 /CX/D2/B9/CS/CX\r/CT/D7/BA /CC/CW/CX/D7 /D7/CX/D1/D4/D0/CX/AS/CT/CS /CD/B4/BD/B5 /D1/D3 /CS/CT/D0 /D3/CU /CU/D9/D0/D0/DD /DA /CP/D0/D0/CT/DD/B9/D4 /D3/D0/CP/D6/CX/DE/CT/CS\n/CV/D6/CP/D4/CW/CT/D2/CT /DB/CW/CX\r /CW /CX/D7 /CS/CT/D7\r/D6/CX/CQ /CT/CS /CQ /DD /BX/D5/BA /B4/BH/B5 /CX/D7 \r/CP/D0/D0/CT/CS /CD/B4/BD/B5/B9/CV/D6 /CP/D4/CW/CT/D2/CT /CX/D2 /D8/CW/CT /D6/CT/D1/CP/CX/D2/CS/CT/D6 /D3/CU /D8/CW/CT /D4/CP/D4 /CT/D6/BA /C6/D3 /DB/B8 /CX/CU /D3/D2/CT/D7/D9/CQ/D7/D8/CX/D8/D9/D8/CT/D7 /CX/D2 /D8/D3 /BX/D5/BA /B4/BH /B5 /D8/CW/CT /D2/D3/D2/B9/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX\r /CU/D3/D6/D1/B9/CU/CP\r/D8/D3/D6\nFN( /D5) /B8 /D3/D2/CT /D3/CQ/D8/CP/CX/D2/D7 /D8/CW/CT /D9/D7/D9/CP/D0 /D7/CX/D2/CV/D0/CT/B9/D0/CP /DD /CT/D6 /D5/D9/CP/D2 /D8/D9/D1 /C0/CP/D0/D0/BE/BW/BX/BZ /CU/D3/D6 /D7/D4/CX/D2/B9/D4 /D3/D0/CP/D6/CX/DE/CT/CS /CT/D0/CT\r/D8/D6/D3/D2/D7 /CX/D2 /BZ/CP/BT/D7/BA/CC/CW/CT /C0/CP/D6/D8/D6/CT/CT/B9/BY /D3 \r /CZ /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /CP/D4/D4/D0/CX/CT/CS /D8/D3 /D8/CW/CT/CV/D6/CP/D4/CW/CT/D2/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 /D8/CT/D6/D1 /CX/D2 /BX/D5/BA /B4/BD/B5 /DD/CX/CT/D0/CS/D7\n/BF/BD/B8/BG/BD\nˆH(HF)\nint=NφVC/summationdisplay\nβ, /C9/braceleftBig/bracketleftbig\nH( /C9)−Xββ( /C9)/bracketrightbig\nˆρβ,β( /C9)\n−Xβ¯β( /C9)ˆρ¯β,β( /C9)/bracerightBig\n, /B4/BI/B5/DB/CW/CT/D6/CT¯β=−β /B8 /CP/D2/CS /C9 /B3/D7 /CP/D6/CT /D8/CW/CT /D6/CT\r/CX/D4/D6/D3 \r/CP/D0 /DB /CP /DA /CT/DA /CT\r/D8/D3/D6/D7 /D3/CU/D8/CW/CT /CF /BV /D0/CP/D8/D8/CX\r/CT/BA /CC/CW/CT /C0/CP/D6/D8/D6/CT/CT /CP/D2/CS /BY /D3 \r /CZ /CT/AR/CT\r/D8/CX/DA /CT /CX/D2 /D8/CT/D6/CP\r/B9/D8/CX/D3/D2 /D4 /D3/D8/CT/D2 /D8/CX/CP/D0/D7 /D6/CT/CP/CS/B8 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /B8\nH( /C9) =e−Q2/2\nQ|FN( /C9)|2ρ(− /C9)(1−δ/C9,0), /B4/BJ/B5\nXββ′( /C9) =/integraldisplay∞\n0dxe−x2\n2|FN( /C9)|2J0(xQ)ρβ,β′(− /C9),/B4/BK/B5/DB/CW/CT/D6/CTQ≡ | /C9| /B8J0\n/CX/D7 /CP /BU/CT/D7/D7/CT/D0 /CU/D9/D2\r/D8/CX/D3/D2/B8 /CP/D2/CS /D8/CW/CT /CS/CT/D2/D7/CX/D8 /DD/CP /DA /CT/D6/CP/CV/CT/D7 /CP/D6/CTρβ,β′( /C9) =∝angbracketleftˆρβ,β′( /C9)∝angbracketright /B8ρ( /C9) =/summationtext\nβρβ,β( /C9) /BA/CF /CT /CP/D7/D7/D9/D1/CT /CP /D8/D6/CX/CP/D2/CV/D9/D0/CP/D6 /CT/D0/CT\r/D8/D6/D3/D2 /D0/CP/D8/D8/CX\r/CT /CU/D3/D6 /D8/CW/CT /CQ/D6/D3/CZ /CT/D2/B9/D7/DD/D1/D1/CT/D8/D6/DD /D7/D8/CP/D8/CT/B8 /DB/CX/D8/CW /D6/CT\r/CX/D4/D6/D3 \r/CP/D0 /D0/CP/D8/D8/CX\r/CT /DA /CT\r/D8/D3/D6/D7 /CV/CX/DA /CT/D2 /CQ /DD/C9=Q0/parenleftBigg\nn\n2,n\n2+m√\n3\n2/parenrightBigg\n, n,m ∈Z. /B4/BL/B5/C0/CT/D6/CTQ0\n/CX/D7 /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /CQ/CP/D7/CX/D7 /DA /CT\r/D8/D3/D6 /D3/CU /D8/CW/CT /D6/CT\r/CX/D4/D6/D3 \r/CP/D0/D0/CP/D8/D8/CX\r/CT/B8\nQ0=l−1\nB/parenleftbigg4πνN√\n3Me/parenrightbigg1/2\n, /B4/BD/BC/B5\nνN\n/CX/D7 /D8/CW/CT /AS/D0/D0/CX/D2/CV /CU/CP\r/D8/D3/D6 /D3/CU /D8/CW/CT /D0/CP/D7/D8 /D4/CP/D6/D8/CX/CP/D0/D0/DD /AS/D0/D0/CT/CS /C4/C4/B8/CP/D2/CSMe\n/CX/D7 /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /CT/D0/CT\r/D8/D6/D3/D2/D7 /D4 /CT/D6 /D7/CX/D8/CT /B4Me= 1\r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /CF /BV/B8 /CP/D2/CSMe≥2 /D8/D3 /CP/D2 /CT/D0/CT\r/D8/D6/D3/D2/B9/CQ/D9/CQ/CQ/D0/CT \r/D6/DD/D7/D8/CP/D0 /DB/CX/D8/CWMe\n/CT/D0/CT\r/D8/D6/D3/D2/D7 /D4 /CT/D6 /CQ/D9/CQ/CQ/D0/CT/B5/BA /CC/CW/CT /D7/CX/D2/CV/D0/CT/B9/D4/CP/D6/D8/CX\r/D0/CT /BZ/D6/CT/CT/D2/B3/D7 /CU/D9/D2\r/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /CX/D1/CP/CV/CX/D2/CP/D6/DD/B9/D8/CX/D1/CT /C5/CP/D8/D7/D9/CQ/B9/CP/D6/CP /CU/D3/D6/D1/CP/D0/CX/D7/D1\n/BG/BG/D6/CT/CP/CS/D7\nGβ1,β2( /C9,iωn) =−N−1\nφ/integraldisplay/planckover2pi1/kBT\n0dτexp(iωnτ)\n×/summationdisplay\nXexp/bracketleftbigg\n−iQxX+l2\nBQxQy\n2/bracketrightbigg\n×∝angbracketleftTτˆcX−l2\nBQy,β1(τ)ˆc†\nX,β2(0)∝angbracketright, /B4/BD/BD/B5/DB/CW/CT/D6/CTT /CX/D7 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/B8 kB\n/CX/D7 /D8/CW/CT /BU/D3/D0/D8/DE/D1/CP/D2/D2\r/D3/D2/D7/D8/CP/D2 /D8/B8 Tτ\n/CS/CT/D2/D3/D8/CT/D7 /CX/D1/CP/CV/CX/D2/CP/D6/DD/B9/D8/CX/D1/CT /D3/D6/CS/CT/D6/CX/D2/CV/B8 /CP/D2/CS\nωn=π(2n+ 1)kBT//planckover2pi1 /CP/D6/CT /D8/CW/CT /C5/CP/D8/D7/D9/CQ/CP/D6/CP /CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7/BA\nGβ1,β2( /C9,iωn) /D1/CP /DD /CQ /CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /D7/CT/D0/CU/B9\r/D3/D2/D7/CX/D7/D8/CT/D2 /D8/D0/DD /CU/D6/D3/D1/D8/CW/CT /D5/D9/CP/CS/D6/CP/D8/CX\r /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /B4/BI/B5 /CQ /DD /D9/D7/CX/D2/CV /D8/CW/CT /C0/CT/CX/D7/CT/D2 /CQ /CT/D6/CV/CT/D5/D9/CP/D8/CX/D3/D2/D7 /D3/CU /D1/D3/D8/CX/D3/D2 /DB/CX/D8/CW/CX/D2 /D8/CW/CT /CX/D8/CT/D6/CP/D8/CX/DA /CT/B9/D7/D3/D0/D9/D8/CX/D3/D2 /D1/CT/D8/CW/D3 /CS/D4/D6/D3/D4 /D3/D7/CT/CS /CX/D2 /CA/CT/CU/BA /BG/BH /DB/CW/CX\r /CW /DB /CT /CP/CS/D3/D4/D8 /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /D8 /DB /D3/D6/CZ/BA/BG/BT/CU/D8/CT/D6 /CP/D2/CP/D0/DD/D8/CX\r \r/D3/D2 /D8/CX/D2 /D9/CP/D8/CX/D3/D2 /D8/D3 /D6/CT/CP/D0 /CU/D6/CT/D5/D9/CT/D2\r/CX/CT/D7 iωn→\nω+i0+/B8Gβ1,β2( /C9,iωn) /DD/CX/CT/D0/CS/D7 /D8/CW/CT /D6/CT/D8/CP/D6/CS/CT/CS /BZ/D6/CT/CT/D2/B3/D7 /CU/D9/D2\r/B9/D8/CX/D3/D2 /DB/CW/CX\r /CW /D1/CP /DD /CQ /CT /D9/D7/CT/CS /D8/D3 \r/CP/D0\r/D9/D0/CP/D8/CT /D8/CW/CT /BW/C7/CBg(ω) /B8\ng(ω) =−N−1\nφπ/summationdisplay\nβ\n/C1/D1Gβ,β( /C9= 0,iωn→ω+i0+),/B4/BD/BE/B5/CP/D2/CS /D8/CW/CT /C4/BW/C7/CBA( /D6,ω) /B8\nA( /D6,ω) =−N−1\nφπ/summationdisplay\nβ\n/C1/D1Gβ,β( /D6,iωn→ω+i0+), /B4/BD/BF/B5/DB/CW/CT/D6/CT /D8/CW/CT /BZ/D6/CT/CT/D2/B3/D7 /CU/D9/D2\r/D8/CX/D3/D2 /CX/D2 /D6/CT/CP/D0 /D7/D4/CP\r/CT /D6/CT/CP/CS/D7\nGβ1,β2( /D6,iωn) = (2πl2\nB)−1/summationdisplay/C9,βexp(−i /C9· /D6)FN(− /C9)\n×Gβ1,β2( /C9,iωn). /B4/BD/BG/B5/C1 /C1 /C1/BA /CA/BX/CB/CD/C4 /CC/CB /BT/C6/BW /BW/C1/CB/BV/CD/CB/CB/C1/C7/C6/CB/C1/D2 /D8/CW/CX/D7 /D7/CT\r/D8/CX/D3/D2/B8 /DB /CT /CS/CX/D7\r/D9/D7/D7 /D8/CW/CT /D7/D4 /CT\r/D8/D6/D3/D7\r/D3/D4/CX\r /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7/D3/CU /D8/CW/CT /CT/D0/CT\r/D8/D6/D3/D2/B9/D7/D3/D0/CX/CS /D4/CW/CP/D7/CT/D7 /CU/D3/D6 /D8/CW/CT /C4/C4N= 2 /BA /BT/D7 /CP/D0/B9/D6/CT/CP/CS/DD /D1/CT/D2 /D8/CX/D3/D2/CT/CS /CX/D2 /D8/CW/CT /CX/D2 /D8/D6/D3 /CS/D9\r/D8/CX/D3/D2/B8 /DB /CT \r/D3/D2\r/CT/D2 /D8/D6/CP/D8/CT /D3/D2/D8/CW/CX/D7 /C4/C4 /CU/D3/D6 /CX/D0/D0/D9/D7/D8/D6/CP/D8/CX/D3/D2 /D4/D9/D6/D4 /D3/D7/CT/D7 /CP/D2/CS /CQ /CT\r/CP/D9/D7/CT /D8/CW/CT/DD /CP/D6/CT/D1/D3/D7/D8/D0/DD /D7/CX/CV/D2/CX/AS\r/CP/D2 /D8 /CU/D6/D3/D1 /D8/CW/CT /D4/CW /DD/D7/CX\r/CP/D0 /D4 /D3/CX/D2 /D8 /D3/CU /DA/CX/CT/DB/BA /CF /CT/CW/CP /DA /CT /D3/CQ/D8/CP/CX/D2/CT/CS /D7/CX/D1/CX/D0/CP/D6 /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6N= 1 /B8 /BF /CP/D2/CS /BG /B4/D2/D3/D8/CS/CX/D7\r/D9/D7/D7/CT/CS /CW/CT/D6/CT/B5/BA/CF /CT /CW/CP /DA /CT \r /CW/D3/D7/CT/D2 /D8/CW/D6/CT/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /CT/D0/CT\r/D8/D6/D3/D2/B9/D7/D3/D0/CX/CS /D0/CP/D8/D8/CX\r/CT/D7/DB/CW/CX\r /CW /CW/CP /DA /CT /D8/CW/CT /D7/CP/D1/CT /D6/CP/D8/CX/D3νN/Me= 0.14≈1/7 /B8 /CP/D2/CS/CW/CT/D2\r/CT /D8/CW/CT /D7/CP/D1/CT /D0/CP/D8/D8/CX\r/CT /D4 /CT/D6/CX/D3 /CS /CV/CX/DA /CT/D2 /CQ /DD /BX/D5/BA /B4/BD/BC /B5 /BA /C7/D9/D6\r /CW/D3/CX\r/CT /CU/D3/D6 /D8/CW/CTνN/Me\n/D6/CP/D8/CX/D3 /CX/D7 /D6/CP/D8/CW/CT/D6 /CP/D6/CQ/CX/D8/D6/CP/D6/DD /BA /CF /CT /D2/D3/D8/CT/D8/CW/CP/D8 /D8/CW/CTMe= 1 /CP/D2/CSMe= 2 /D7/D8/CP/D8/CT/D7 /DD/CX/CT/D0/CS /CX/D2 /CV/D6/CP/D4/CW/CT/D2/CT /D8/CW/CT/CV/D0/D3/CQ/CP/D0 /CT/D2/CT/D6/CV/DD /D1/CX/D2/CX/D1/CP/B8 /DB/CW/CX/D0/CT /D8/CW/CTMe= 3 /CS/D3 /CT/D7 /D2/D3/D8/BA /C1/D8 /CW/CP/D7/CQ /CT/CT/D2 /D7/CW/D3 /DB/D2 /CX/D2 /CA/CT/CU/BA /BF/BD /D8/CW/CP/D8 /D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /D3/CU /CV/D6/CP/D4/CW/CT/D2/CT/CP/D8νN≤0.43 /CX/D7 /CP/D2 /CP/D2/CX/D7/D3/D8/D6/D3/D4/CX\r /CF/CX/CV/D2/CT/D6 \r/D6/DD/D7/D8/CP/D0 /DB/CW/CT/D6/CT/CP/D7 /CP/D8\n0.28≤νN≤0.43 /D8/CW/CT /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CX/D7 /D8/CW/CTMe= 2 /CQ/D9/CQ/CQ/D0/CT\r/D6/DD/D7/D8/CP/D0/B8 /CP/D2/CS /CP/D8νN≤0.28 /D8/CW/CT /CF/CX/CV/D2/CT/D6 \r/D6/DD/D7/D8/CP/D0 /DD/CX/CT/D0/CS/D7 /D8/CW/CT/D0/D3 /DB /CT/D7/D8 /CT/D2/CT/D6/CV/DD /B4Me= 1 /B5/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT/B8 /D8/CW/CTMe= 3 /D4/CW/CP/D7/CT/CX/D7 /D2/D3/D8 /D8/CW/CT /D0/D3 /DB /CT/D7/D8/B9/CT/D2/CT/D6/CV/DD /D7/D8/CP/D8/CT /CX/D2 /CV/D6/CP/D4/CW/CT/D2/CT/BN /D2/CT/DA /CT/D6/D8/CW/CT/D0/CT/D7/D7/B8/CX/D8 /CX/D7 /D9/D7/CT/CU/D9/D0 /D8/D3 /CP/D2/CP/D0/DD/DE/CT /D3/D2 /D8/CW/CT /D7/CP/D1/CT /CU/D3 /D3/D8/CX/D2/CV /CP/D0/D0 /D8/CW/D6/CT/CT \r/CP/D7/CT/D7\nMe= 1,2,3 /BA/BY /D3/D6 \r/D3/D1/D4/D0/CT/D8/CT/D2/CT/D7/D7/B8 /DB /CT /D1/CT/D2 /D8/CX/D3/D2 /D8/CW/CP/D8 /DB /CT /CW/CP /DA /CT /CP/D0/D7/D3 \r/CP/D0/B9\r/D9/D0/CP/D8/CT/CS /D8/CW/CT \r/D3/CW/CT/D7/CX/DA /CT /CT/D2/CT/D6/CV/CX/CT/D7 /D3/CU /D3/D8/CW/CT/D6 /D8 /DD/D4 /CT/D7 /D3/CU /CT/D0/CT\r/D8/D6/D3/D2/B9\r/D6/DD/D7/D8/CP/D0 /D4/CW/CP/D7/CT/D7 /B4/D2/D3/D8 /D3/D2/D0/DD /D8/D6/CX/CP/D2/CV/D9/D0/CP/D6 /CQ/D9/CQ/CQ/D0/CT /D4/CW/CP/D7/CT/D7/B8 /CQ/D9/D8/CP/D0/D7/D3 /CP/D2/CX/D7/D3/D8/D6/D3/D4/CX\r /CF/CX/CV/D2/CT/D6 \r/D6/DD/D7/D8/CP/D0/D7/B5/BA /C7/D9/D6 /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6 /D8/CW/CT/CT/D2/CT/D6/CV/CX/CT/D7 \r/D3/CX/D2\r/CX/CS/CT /DB/CX/D8/CW /D8/CW/D3/D7/CT /D3/CU /CA/CT/CU/BA /BF/BD /DB/CX/D8/CW /CT/DC\r/CT/D0/D0/CT/D2 /D8 /CP\r/B9\r/D9/D6/CP\r/DD /CP/D2/CS/B8 /D8/CW/CT/D6/CT/CU/D3/D6/CT/B8 \r/D3/D6/D6/D3/CQ /D3/D6/CP/D8/CT /D8/CW/CT /BW/C7/CB /CP/D2/CS /C4/BW/C7/CB/D6/CT/D7/D9/D0/D8/D7 /CS/CX/D7\r/D9/D7/D7/CT/CS /CQ /CT/D0/D3 /DB/BA/BT/BA /B4/C1/D2 /D8/CT/CV/D6/CP/D8/CT/CS/B5 /BW/CT/D2/D7/CX/D8 /DD /D3/CU /D7/D8/CP/D8/CT/D7/C7/D9/D6 /D6/CT/D7/D9/D0/D8/D7 /CU/D3/D6 /D8/CW/CT /BW/C7/CB /CX/D2 /CV/D6/CP/D4/CW/CT/D2/CT /CP/D8N= 2 /CP/D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS /CX/D2 /BY/CX/CV/BA /BD/BA/CF /CT /AS/D2/CS /D8/CW/CP/D8 /D8/CW/CT /BW/C7/CB \r/D3/D2/D7/CX/D7/D8/D7 /D3/CU /D8 /DB /D3 /DB /CT/D0/D0/B9/D7/CT/D4/CP/D6/CP/D8/CT/CS\r/D0/CP/D7/D7/CT/D7 /D3/CU /D4 /CT/CP/CZ/D7/BM /DB /CT/D0/D0/B9/CS/CT/AS/D2/CT/CS /D0/D3 /DB/B9/CT/D2/CT/D6/CV/DD /D4 /CT/CP/CZ/D7 /CP/D6/CT /CU/D3/D9/D2/CS/CQ /CT/D0/D3 /DB /D8/CW/CT \r /CW/CT/D1/CX\r/CP/D0 /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 µ /B8 /DB/CW/CX\r /CW /CX/D7 /D7/CW/CX/CU/D8/CT/CS /D8/D3 /DE/CT/D6/D3\n/CT/D2/CT/D6/CV/DD /B8 /DB/CW/CT/D6/CT/CP/D7 /D8/CW/CT /D0/CP/D6/CV/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D4 /CT/CP/CZ/D7 /CP/CQ /D3 /DA /CTµ /CP/D6/CT/D2/D3/D8 /D8/CW/CP/D8 /CT/CP/D7/CX/D0/DD /CS/CX/D7/D8/CX/D2/CV/D9/CX/D7/CW/CT/CS/BA /CF /CT /D2/D3/D8/CT /CW/CT/D6/CT /D8/CW/CP/D8 /CX/D2/CV/D6/CP/D4/CW/CT/D2/CT /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D0/D3 /DB/B9/CT/D2/CT/D6/CV/DD /D4 /CT/CP/CZ/D7 /CX/D2 /CP/D0/D0 \r/CP/D7/CT/D7/CX/D7 /CT/D5/D9/CP/D0 /D8/D3Me\n/B8 /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /CT/D0/CT\r/D8/D6/D3/D2/D7 /CX/D2 /CP /CQ/D9/CQ/CQ/D0/CT/BA/CC/CW/CT /D7/CP/D1/CT /D6/CT/D7/D9/D0/D8 /CW/CP/D7 /CQ /CT/CT/D2 /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /CT/CU/D3/D6/CT /CX/D2 /C0/CP/D6/D8/D6/CT/CT/B9/BY /D3 \r /CZ /D7/D8/D9/CS/CX/CT/D7 /D3/CU /D8/CW/CT /D7/CX/D1/D4/D0/CT/D6 /D7/CX/D2/CV/D0/CT/B9/D0/CP /DD /CT/D6 /BE/BW /D5/D9/CP/D2 /D8/D9/D1/C0/CP/D0/D0 /D7/DD/D7/D8/CT/D1/BA\n/BK/CF /CT \r /CW/CT\r /CZ /CT/CS /D8/CW/CP/D8 /D8/CW/CT /D7/CP/D1/CT /D4/D6/D3/D4 /CT/D6/D8 /DD /CW/D3/D0/CS/D7/D8/D6/D9/CT /CP/D0/D7/D3 /CU/D3/D6 /CD/B4/BD/B5/B9/CV/D6/CP/D4/CW/CT/D2/CT/B8 /CP/D2/CS /CX/D2 /D2/D3/D2/B9/D6/CT/D0/CP/D8/CX/DA/CX/D7/D8/CX\r /D8 /DB /D3/B9\r/D3/D1/D4 /D3/D2/CT/D2 /D8 /D5/D9/CP/D2 /D8/D9/D1 /C0/CP/D0/D0 /D7/DD/D7/D8/CT/D1/D7/B8 /D7/D9\r /CW /CP/D7 /CP /CQ/CX/D0/CP /DD /CT/D6 /DB/CX/D8/CW/DE/CT/D6/D3 /D0/CP /DD /CT/D6 /D7/CT/D4/CP/D6/CP/D8/CX/D3/D2/BA/C1/D2 /D8/CW/CT /D7/CX/D1/D4/D0/CT/D6 /D7/CX/D2/CV/D0/CT/B9/D0/CP /DD /CT/D6 /BE/BW /D5/D9/CP/D2 /D8/D9/D1 /C0/CP/D0/D0 /D7/DD/D7/D8/CT/D1/CX/D2 /BZ/CP/BT/D7/B8 /D8/CW/CT /BW/C7/CB /CP/D8Me= 1 /CT/DC/CW/CX/CQ/CX/D8/D7 /D8/CW/CT /CU/CT/CP/D8/D9/D6/CT/D7 /D3/CU/D8/CW/CT /C0/D3/CU/D7/D8/CP/CS/D8/CT/D6 /CQ/D9/D8/D8/CT/D6/AT/DD /D7/D8/D6/D9\r/D8/D9/D6/CT/BA\n/BK/C1/D8 /D1/CT/CP/D2/D7 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/B9/CX/D2/CV/BM /CV/CX/DA /CT/D2 /D8/CW/CP/D8 /D8/CW/CT /AS/D0/D0/CX/D2/CV /CU/CP\r/D8/D3/D6 /D1/CP /DD /CQ /CT /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CT/CS /CQ /DD/CP /D6/CP/D8/CX/D3 /D3/CU /D8 /DB /D3 /CX/D2 /D8/CT/CV/CT/D6/D7 p, q /DB/CX/D8/CW/D3/D9/D8 /CP \r/D3/D1/D1/D3/D2 /CS/CX/DA/CX/D7/D3/D6/B8\nνN=p/q /B8 /D8/CW/CT/D7/CT /CX/D2 /D8/CT/CV/CT/D6/D7p, q /CS/CT/D8/CT/D6/D1/CX/D2/CT /D8/CW/CT/D2 /D8/CW/CT /D7/D8/D6/D9\r/B9/D8/D9/D6/CT /D3/CU /D8/CW/CT /D7/CX/D2/CV/D0/CT/B9/D4/CP/D6/D8/CX\r/D0/CT /CT/D2/CT/D6/CV/DD /D7/D4 /CT\r/D8/D6/D9/D1 /D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1/BN/D2/CP/D1/CT/D0/DD /B8 /D8/CW/CT/D6/CT /D7/CW/D3/D9/D0/CS /CT/DC/CX/D7/D8p /D0/D3 /DB/B9/CT/D2/CT/D6/CV/DD /D0/CT/DA /CT/D0/D7 /CP/D2/CSq−p/CW/CX/CV/CW/B9/CT/D2/CT/D6/CV/DD /D0/CT/DA /CT/D0/D7 /B4/C0/D3/CU/D7/D8/CP/CS/D8/CT/D6 /CQ/D9/D8/D8/CT/D6/AT/DD \r/D3/D9/D2 /D8/CX/D2/CV /D6/D9/D0/CT/B5/BA/C1/D2 /D8/CW/CT /BW/C7/CB/B8 /DB/CW/CX\r /CW /CX/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /CU/D6/CT/D5/D9/CT/D2\r/DD /B8 /D8/CW/CT/D7/CT /CT/D2/B9/CT/D6/CV/DD /D0/CT/DA /CT/D0/D7 /CP/D6/CT /D6/CT\r/D3/CV/D2/CX/DE/CT/CS /CP/D7 /D7/D1/D3 /D3/D8/CW/CT/CS /D4 /CT/CP/CZ/D7/BA /CC/CW/CT /C0/D3/CU/D7/B9/D8/CP/CS/D8/CT/D6 /CQ/D9/D8/D8/CT/D6/AT/DD \r/D3/D9/D2 /D8/CX/D2/CV /D6/D9/D0/CT /DB /CP/D7 \r/D3/D2/AS/D6/D1/CT/CS /CU/D3/D6Me= 1/CX/D2 /D8/CW/CT /D7/CX/D2/CV/D0/CT/B9/D0/CP /DD /CT/D6 /BE/BW/BX/BZ /CX/D2 /BZ/CP/BT/D7/B8 /DB/CW/CX/D0/CT /CU/D3/D6Me≥2 /CX/D8 /CX/D7\r/D0/CP/CX/D1/CT/CS /D8/CW/CP/D8 \r/D3/D9/D2 /D8/CX/D2/CV /D3/CU /D8/CW/CT /D7/CX/D2/CV/D0/CT/B9/D4/CP/D6/D8/CX\r/D0/CT /D0/CT/DA /CT/D0/D7 /CX/D7 /CS/CX/CU/B9/CU/CT/D6/CT/D2 /D8/BM /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D0/D3 /DB/B9/CT/D2/CT/D6/CV/DD /D4 /CT/CP/CZ/D7 /CX/D7 /CT/D5/D9/CP/D0 /D8/D3Me\n/B8/DB/CW/CT/D6/CT/CP/D7 /D2/D3/D8/CW/CX/D2/CV /CX/D7 /CZ/D2/D3 /DB/D2 /CP/CQ /D3/D9/D8 /DB/CW/CP/D8 /CX/D7 /D8/CW/CT /D4/D6/CT\r/CX/D7/CT /D6/D9/D0/CT/CU/D3/D6 \r/D3/D9/D2 /D8/CX/D2/CV /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /CW/CX/CV/CW/B9/CT/D2/CT/D6/CV/DD /D4 /CT/CP/CZ/D7/BA\n/BK/B4/BT/B5 /B4/BU/B5 /B4/BV/B5/BU/D3/D9/D2/CS /D7/D8/CP/D8/CT/D7\n−0.4 −0.2 0.0\nω     /CB/CX/D2/CV/D0/CT/B9/D4 /CP/D6/D8/CX\r/D0/CT /CT/DC\r/CX/D8/CP/D8/CX/D3/D2/D7\n/BY/C1/BZ/BA /BD/BM /C4/D3/CV/CP/D6/CX/D8/CW/D1/CX\r /D4/D0/D3/D8/D7 /CU/D3/D6 /D8/CW/CT /CS/CT/D2/D7/CX/D8 /DD /D3/CU /D7/D8/CP/D8/CT/D7g(ω) /D3/CU/CV/D6/CP/D4/CW/CT/D2/CT /CX/D2 /D8/CW/CTN= 2 /C4/CP/D2/CS/CP/D9 /D0/CT/DA /CT/D0 /CP/D8 /B4/BT/B5Me= 1, νN=\n0.14 /BN /B4/BU/B5Me= 2, νN= 0.28 /BN /B4/BV/B5Me= 3, νN= 0.42 /BA/CC/CW/CT /CU/D6/CT/D5/D9/CT/D2\r/DD ω /CX/D7 /CV/CX/DA /CT/D2 /CX/D2 /D9/D2/CX/D8/D7 /D3/CU /D8/CW/CT /BV/D3/D9/D0/D3/D1 /CQ /D7\r/CP/D0/CTVC\n/BA/CC/CW/CTω= 0 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/D6/CT/CP/D0/B9/D7/D4/CP\r/CT /CT/D0/CT\r/D8/D6/D3/D2 /CS/CT/D2/D7/CX/D8 /DD /D4/D6/D3/AS/D0/CT\nn( /D6) =1\n2πl2\nB/summationdisplay/C9exp(−i /C9· /D6)FN(− /C9)ρ( /C9) /B4/BD/BH/B5/CU/D3/D6 /D8/CW/CT /D7/CP/D1/CT \r /CW/D3/CX\r/CT/D7 /D3/CU /B4νN, Me\n/B5 /CP/D7 /CX/D2 /BY/CX/CV/BA /BD /BA /CC/CW/CT /D6/CT/D7/D9/D0/D8/D7/B8/DB/CW/CX\r /CW /CP/D6/CT /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BE/B8 /CP/CV/D6/CT/CT /DB/CX/D8/CW /D4/D6/CT/DA/CX/D3/D9/D7 \r/CP/D0\r/D9/D0/CP/B9\n/D8/CX/D3/D2/D7 /CU/D3/D6 /CV/D6/CP/D4/CW/CT/D2/CT /D4 /CT/D6/CU/D3/D6/D1/CT/CS /CQ /DD /CI/CW/CP/D2/CV /CP/D2/CS /C2/D3/CV/D0/CT/CZ /CP/D6/BA\n/BF/BD/B4/BT/B5 /B4/BU/B5 /B4/BV/B5\n[0.0,2.1] [0 .3,1.7] [0 .6,2.4]\n/BY/C1/BZ/BA /BE/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /CA/CT/CP/D0/B9/D7/D4/CP\r/CT /CS/CT/D2/D7/CX/D8 /DD /D4/D6/D3/AS/D0/CTn( /D6) /CX/D2/CV/D6/CP/D4/CW/CT/D2/CT /CX/D2 /D8/CW/CTN= 2 /C4/C4/BA /BV/CW/D3/CX\r/CT/D7 /B4/BT/B5/B9/B4/BV/B5 /CP/D6/CT /D8/CW/CT /D7/CP/D1/CT/CP/D7 /CX/D2 /BY/CX/CV/BA /BD /BA 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/D8/CW/CT /D6 /CT/D7/D9/D1/D1/CT /CS/BI/B4/BT/B5/BD /BE /BF /BG /BH\n/BI /BJ /BK /BL /BD/BC\n/BD/BD /BD/BE /BD/BF\n/BY/C1/BZ/BA /BF/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /C4/BW/C7/CBA( /D6,ω) /CU/D3/D6 /CV/D6/CP/D4/CW/CT/D2/CT /CP/D8νN=\n0.14, Me= 1 /CJ\r/CP/D7/CT /B4/BT/B5℄/BA /BV/D3/D2 /D8/D3/D9/D6 \r/D3/D0/D3/D9/D6/D7 /CP/D6/CT /CV/D6/CP/CS/CT/CS /CX/D2 /D8/CW/CT/D7/CP/D1/CT /DB /CP /DD /CP/D7 /CS/CT/AS/D2/CT/CS /CX/D2 /BY/CX/CV/BA /BE /BA /CC/CW/CT \r/D3/D2 /D8/D3/D9/D6 /D4/D0/D3/D8/D7 /CP/D6/CT /D3/D6/CS/CT/D6/CT/CS/DB/CX/D8/CW /D6/CT/D7/D4 /CT\r/D8 /D8/D3 /D8/CW/CT /CX/D2/CS/CT/DC /D3/CU /BW/C7/CB /D4 /CT/CP/CZ/D7 /CJ/CX/D2/CS/CX\r/CP/D8/CT/CS /CP/CQ /D3 /DA /CT /D8/CW/CT/D4/D0/D3/D8/D7℄/BA /CC/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /CT/DC/D8/D6/CP\r/D8/CT/CS /BW/C7/CB /D4 /CT/CP/CZ/D7 /CX/D7Np\n/BP/BD/BF/BA/B4/BU/B5/BD /BE /BF /BG /BH\n/BI /BJ /BK /BL /BD/BC\n/BD/BD /BD/BE /BD/BF /BD/BG /BD/BH\n/BY/C1/BZ/BA /BG/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /C4/BW/C7/CBA( /D6,ω) /CU/D3/D6 /CV/D6/CP/D4/CW/CT/D2/CT /CP/D8νN=\n0.28, Me= 2 /CJ\r/CP/D7/CT /B4/BU/B5℄/BANp= 15 /BA/C4/BW/C7/CB˜A( /D6,ω) /B8 /CS/CT/AS/D2/CT/CS /CU/D3/D6 /CP /AS/DC/CT/CS /D7/CX/D2/CV/D0/CT/B9/D4/CP/D6/D8/CX\r/D0/CT /CT/D2/CT/D6/CV/DD\nωi\n/CP/D7 /CP /D7/D9/D1 /D3/CU /CP/D0/D0 /C4/BW/C7/CB /D4/CP/D8/D8/CT/D6/D2/D7 /CP/D8 /D7/D1/CP/D0/D0/CT/D6 /D4 /CT/CP/CZ /CT/D2/CT/D6/CV/CX/CT/D7/B8\n˜A( /D6,ωi) =i/summationdisplay\nj=1A( /D6,ωj). /B4/BD/BI/B5/BZ/CX/DA /CT/D2 /D8/CW/CT /CT/DC\r/CT/D0/D0/CT/D2 /D8 \r/D3/CX/D2\r/CX/CS/CT/D2\r/CT /D3/CU /D8/CW/CT /C4/BW/C7/CB /D4/CP/D8/D8/CT/D6/D2/D7/D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BI /DB/CX/D8/CW /D8/CW/CT /D6/CT/CP/D0/B9/D7/D4/CP\r/CT /CS/CT/D2/D7/CX/D8/CX/CT/D7 /CX/D2 /BY/CX/CV/BA /BE /B8/D3/D2/CT /D1/CP /DD /CT/D1/D4/CX/D6/CX\r/CP/D0/D0/DD /DB/D6/CX/D8/CT\nn( /D6,Me=i)↔˜A( /D6,ωi), i= 1,2,3, /B4/BD/BJ/B5\n/B4/BV/B5/BD /BE /BF /BG /BH\n/BI /BJ /BK /BL /BD/BC\n/BD/BD /BD/BE /BD/BF /BD/BG\n/BY/C1/BZ/BA /BH/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 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Man'ko,1Giuseppe Marmo2and E. C. George\nSudarshan3\n1P. N. Lebedev Physical Institute, Leninskii Prospect 53, Moscow 119991,\nRussia\n2Dipartimento di Scienze Fisiche, Universit\u0012 a \\Federico II\" di Napoli\nand Istituto Nazionale di Fisica Nucleare, Sezione di Napoli\nComplesso Universitario di M. S. Angelo, Via Cintia, I-80126 Napoli, Italy\n3Department of Physics, University of Texas, Austin, Texas 78712, USA\nE-mail: mmanko@sci.lebedev.ru\nAbstract. The quasidistributions corresponding to the diagonal representation\nof quantum states are discussed within the framework of operator-symbol\nconstruction. The tomographic-probability distribution describing the quantum\nstate in the probability representation of quantum mechanics is reviewed. The\nconnection of the diagonal and probability representations is discussed. The\nsuperposition rule is considered in terms of the density-operator symbols. The\nseparability and entanglement properties of multipartite quantum systems are\nformulated as the properties of the density-operator symbols of the system states.\nPACS numbers: 03.65.-w, 03.65.-Wj\n1. Introduction\nThe pure quantum states are traditionally associated with the wave function [1] or\na vector in the Hilbert space [2]. The mixed quantum states are described by the\ndensity matrix [3] or the density operator [4]. There exist several representations of\nquantum states in terms of the quasidistribution functions like the Wigner function [5]\nand the Husimi{Kano function [6, 7]. The diagonal representation of quantum states\nwas suggested in [8] (see also [9]). It was studied and applied in [10, 11]. In this\nrepresentation, a quantum state is represented in terms of weighted sum of coherent-\nstatejziprojectors. The properties of all the quantum-state representations considered\nare associated with the properties of the density operator which is Hermitian, trace-\nclass nonnegative operator. This means, in particular, that all the eigenvalues of the\ndensity operators must be nonnegative. In the quantum domain, the multipartite\nsystems have a speci\fc property connected with strong correlations of the quantum\nsubsystems. This property provides the entanglement phenomenon [12].\nIn the diagonal representation of the density states, the weight function \u001e(z)\nis an analog of the probability-distribution function in the phase space. For some\nclass of states, this function is identical to the probability-distribution function like in\nzbased on the invited talk presented by one of us (V.I.M.) at the XV Central European Workshop\non Quantum Optics (Belgrade, Serbia, 30 May { 3 June 2008).arXiv:0902.4351v1  [quant-ph]  25 Feb 2009Diagonal and probability representations 2\nclassical statistical mechanics. In [13], the tomographic-probability representation of\nquantum states, where the quantum state is associated with the so-called symplectic\ntomogram, was introduced. The tomogram is a fair probability distribution containing\nthe same information on quantum state that the density operator does (or such its\ncharacteristics as the Wigner or Husimi{Kano functions). The aim of this work is to\n\fnd the explicit formulae realizing the connection of the diagonal and tomographic\nprobability representations. In [14], a review of the star-product-quantization schemes\nwas given in a uni\fed form. According to this scheme, the functions like the Wigner\nfunction, Husimi{Kano function and tomographic-probability-distribution function\nare considered as symbols of the density operators of a corresponding star-product\nscheme. The other goal of our work is to discuss in detail the diagonal representation\nwithin the framework of the star-product scheme along the lines of construction\ngiven in [14] and to \fnd mutual relations between the tomographic-probability\nrepresentation and the diagonal representation in this context. Using formulation\nof the superposition rule in terms of the density operator [15, 16, 17], we consider it\nwithin the framework of the density-state symbols. We focus on the superposition\nrule given in terms of tomograms and in terms of weight functions of the diagonal\nrepresentation where explicit kernels of the corresponding star-products are employed\nto obtain the addition rules for the tomograms and weight functions. We discuss also\nthe formulation of the separability and entanglement properties of composed system\nin the tomographic probability and diagonal representations.\nThe paper is organized as follows.\nIn Section 2, symplectic tomograms and the diagonal representation of quantum\nsates are reviewed. In Section 3, the superposition rule is considered. In Section 4, the\ndiagonal representation and the star-product formalism are compared. In Section 5,\nthe superposition rule for tomograms is presented. In Section 6, the entanglement\nin the tomographic and diagonal representations is studied. Conclusions are given in\nSection 7.\n2. Symplectic tomogram and diagonal representation\nBelow we review the approach where the quantum state associated with tomographic\nsymbol (called symplectic tomogram) of the density operator (density state) ^ \u001areads\n(see, for example, [17])\nw(X;\u0016;\u0017 ) = Tr ^\u001a\u000e(X^1\u0000\u0016^q\u0000\u0017^p) (\u0016h= 1): (1)\nHereX; \u0016; \u00172R, ^qand ^pare the position and momentum operators, respectively.\nFor the pure state, tomogram is expressed in terms of the wave function [18]\nw(X;\u0016;\u0017 ) =1\n2\u0019j\u0017j\f\f\f\fZ\n (y) exp\u0012i\u0016\n2\u0017y2\u0000iXy\n\u0017\u0013\ndy\f\f\f\f2\n: (2)\nThe tomogram is nonnegative normalized probability distribution function of a random\nvariableX, i.e.,\nw(X;\u0016;\u0017 )\u00150;Z\nw(X;\u0016;\u0017 )dX= 1: (3)\nIn the diagonal representation, the density state ^ \u001areads [8]\n^\u001a=Z\n\u001e(z)jzihzjdRez dImz; (4)Diagonal and probability representations 3\nwherejzi=^D(z)j0iis the coherent state ^ ajzi=zjziand the displacement operator\n^D(z) = exp\u0000\nz^ay\u0000z\u0003^a\u0001\nis called Weyl system. Here ^ a= 2\u00001=2(^q+i^p) is the\nboson annihilation operator and zis a complex number. The probability distribution\nw(X;\u0016;\u0017 ) is expressed in terms of the weight function \u001e(z) as follows:\nw(X;\u0016;\u0017 ) =Z\n\u001e(z)hzj\u000e(X^1\u0000\u0016^q\u0000\u0017^p)jzidRez dImz: (5)\nUsing in (1) the Fourier decomposition of delta-function and taking the density state\n^\u001ain form (4), we obtain for tomogram\nw(X;\u0016;\u0017 ) =1\n2\u0019Z\n\u001e(z)hzjeik(X\u0000\u0016^q\u0000\u0017^p)jzidRez dImz; (6)\nwhere the diagonal matrix element of the operator in the integral can be considered\nas the Weyl-system matrix element, i.e.,\nhzje\u0000k(i\u0016^q+i\u0017^p)jzi= exp\u0014\nz\u0003\u000b\u0000\u000b\u0003z\u0000j\u000bj2\n2\u0015\n; (7)\nwith\n\u000b=kp\n2(\u0017\u0000i\u0016): (8)\nEvaluating Gaussian integral (6), we arrive at\nw(X;\u0016;\u0017 ) =1p\n\u0019(\u00162+\u00172)\n\u0002Z\n\u001e(z) exp8\n><\n>:\u0000h\nX\u00002\u00001=2\u0010\nz\u0003(\u0016+i\u0017) +z(\u0016\u0000i\u0017)\u0011i2\n\u00162+\u001729\n>=\n>;dRez dImz: (9)\nThe above formula provides the relation of the weight function of the diagonal\nrepresentation of the density state and symplectic tomogram of the quantum state.\nFor example, the vacuum state j0ih0jhas the weight function\n\u001e0(z) =\u000e(Rez)\u000e(Imz):\nFormula (9) provides tomogram of the ground state\nw0(X;\u0016;\u0017 ) =1p\n\u0019(\u00162+\u00172)exp\u0012\n\u0000X2\n\u00162+\u00172\u0013\n: (10)\nThis expression can be obtained also by means of formula (2) with\n o(y) =\u0019\u00001=4exp\u0000\n\u0000y2=2\u0001\n:\n3. Superposition rule for density operators\nFor two orthogonal pure states j 1iandj 2i, the superposition rule provides the state\nj i=pp1j 1i+ei\u001epp2j 2i; (11)\nwhich can be realized in the nature as a Schr odinger cat state. Here the positive\nnumbersp1andp2satisfy the equality p1+p2= 1 and the phase factor ei\u001edeterminesDiagonal and probability representations 4\nthe interference picture. The density states ^ \u001a1=j 1ih 1jand ^\u001a2=j 2ih 2jprovide\nthe state ^\u001a=j ih j, if one uses the nonlinear addition rule [15]\n^\u001a=p1^\u001a1+p2^\u001a2+pp1p2^\u001a1^P0^\u001a2+ ^\u001a2^P0^\u001a1r\nTr\u0010\n^\u001a1^P0^\u001a2^P0\u0011; (12)\nwhere the operator ^P0is a projector (Tr ^P0= 1) which corresponds to the phase term\nei\u001ein (11).\nThe superposition rule can be formulated for any symbol of pure density states\n^\u001a1, ^\u001a2and ^\u001a.\n4. Diagonal representation and star-product formalism\nThe diagonal representation of density operators can be considered within the\nframework of star-product scheme [14]. Let us construct two families of operators,\nwhich are called dequantizer\n^U(z) =1\n\u00192Z\nexp\u00121\n2juj2+z\u0003u\u0000zu\u0003\u0013\n^D(u)dReu dImu (13)\nand quantizer\n^D(z) =jzihzj; (14)\nwhere ^D(u) is the Weyl system and z=x+iyis a complex number. One can check\nthat\nTrbU(z)bD(z0) =\u000e(x\u0000x0)\u000e(y\u0000y0): (15)\nIn view of this, one can construct the symbol of a density operator ^ \u001ain the diagonal\nrepresentation\n\u001e(z) = TrbU(z)b\u001a=1\n\u00192Z\nexp\u00121\n2juj2+z\u0003u\u0000zu\u0003\u0013\nTrb\u001abD(u)dReu dImu (16)\nand the reconstruction formula for the density operator reads\nb\u001a=Z\n\u001e(z)jzihzjdRez dImz: (17)\nAccording to [19, 20], one can construct dual symbol of the operator b\u001a\n\u001e(d)(z) = Trb\u001ajzihzj=hzjb\u001ajzi (18)\nand dual reconstruction formula\nb\u001a=Z\n\u001e(d)(z)bU(z)dRez dImz\n=1\n\u00192Z\n\u001e(d)(z) exp\u00121\n2juj2+z\u0003u\u0000zu\u0003\u0013\nbD(u)dReu dImu dRez dImz: (19)\nIf in (16) the operator b\u001ais replaced by some operator bA, the corresponding symbol\n\u001eA(z) provides the diagonal representation of the operator. The dual symbol (18)\nprovides the Husimi{Kano function Q(z). The reconstruction formula for the density\nstate in terms of the Husimi{Kano function is just formula (19) with the replacement\n\u001e(d)(z)!Q(z). The duality relation of the diagonal representation of the density\nstateb\u001aand the Husimi{Kano function was discussed in [19, 21].Diagonal and probability representations 5\nUsing the connection of an operator symbol with its dual [14], one has the\nconnection formula\n\u001e(z) =1\n\u00193Z\nQ(z1) exp\u0002\njuj2+ (z\u0003\u0000z\u0003\n1)u\u0000(z\u0000z1)u\u0003\u0003\ndReu dImu: (20)\nThe inverse formula reads\nQ(z) =Z\n\u001e(z)e\u0000jz1\u0000zj2dRez1dImz1: (21)\nAccording to the general formalism [14], the star-product of symbols related to the\ndiagonal representation is determined by the kernel\nK(z1;z2;z) =1\n\u00192Tr\u0014Z\njz1ihz1jjz2ihz2jbD(u) exp\u00121\n2juj2+z\u0003u\u0000zu\u0003\u0013\ndReu dImu\u0015\n;\n(22)\nwhich is generalized function of the form\nK(z1;z2;z) =1\n\u00192Z\nexp\u0010\n\u0000(x2\u0000x1)2\u0000(y2\u0000y1)2+ (x2\u0000x1)a+ (y2\u0000y1)b\n\u0000i(2y+y1+y2)a+i(2x+x1+x2)b\u0011\nda db; (23)\nwhere\nz=x+iy; z 1=x1+iy1; z 2=x2+iy2:\nThe star-product of symbols of arbitrary operators bAandbBin the diagonal\nrepresentation reads\n(\u001eA\u0003\u001eB)(z) =Z\nK(z1;z2;z)\u001eA(z1)\u001eB(z2)dx1dy1dx2dy2: (24)\nFor example, for the vacuum-state projector b\u001a0=j0ih0jwith the weight function {\nsymbol\u001e0(z) =\u000e(z), one has\n(\u001e0\u0003\u001e0)(z) =Z\n\u000e(z1)\u000e(z2)K(z1;z2;z)dx1dy1dx2dy2=\u000e(z); (25)\nwhich is equal to \u001e0(z) and corresponds to the pure-vacuum-state property b\u001a2\n0=b\u001a0.\nTomogram w(X;\u0016;\u0017 ) of the density state b\u001aprovides the following formula for the\ndiagonal representation of the density operator\n\u001e(z) =1\n2\u00192Z\nw(X;\u0016;\u0017 ) exp\u0014\niX+\u00162+\u00172\n4+z(\u0017+i\u0016)p\n2\u0000z\u0003(\u0017\u0000i\u0016)p\n2\u0015\ndX d\u0016 d\u0017:\n(26)\nFor example, the vacuum-state tomogram\nw0(X;\u0016; 0) =1\n\u0019(\u00162+\u00172)exp\u0012\n\u0000x2\n\u00162+\u00172\u0013\nprovides, by means of the above formula, the symbol of the state in the diagonal\nrepresentation, i.e., \u000e(z).Diagonal and probability representations 6\n5. Superposition rule for tomograms\nThe superposition of two pure states with their symbols \u001e1(z) and\u001e2(z) is described\nby the function\n\u001e(z) =p1\u001e1(z) +p2\u001e2(z) +pp1p2(\u001e1\u0003\u001e0\u0003\u001e2)(z) + (\u001e2\u0003\u001e0\u0003\u001e1)(z)qR\n(\u001e1\u0003\u001e0\u0003\u001e2\u0003\u001e0)(z)dxdy: (27)\nThe star-product in (27) is determined by the kernel (23).\nThe result obtained can be repeated also for tomographic symbols of the density\nstates. Thus, the addition rule of two tomographic probabilities of two pure states\nj 1iandj 2ireads\nw(X;\u0016;\u0017 ) =p1w1(X;\u0016;\u0017 ) +p2w2(X;\u0016;\u0017 )\n+pp1p2(w1\u0003w0\u0003w2)(X;\u0016;\u0017 ) + (w2\u0003w0\u0003w1)(X;\u0016;\u0017 )qR\n\u000e(\u0016)\u000e(\u0017)d\u0016d\u0017d\u00160d\u00170R\neiX(w1\u0003w0\u0003w2\u0003w0)(X;\u0016;\u0017 )dX: (28)\nThe kernel of tomographic star-product is given in [19, 20]. The order of integration\nin the denominator term is essential to obtain the correct result. In (27) and (28), \u001e0\nandw0are the corresponding symbols of projector bP0which determines the relative\nphase of statesj 1iandj 2iin their superposition.\n6. Entanglement in the diagonal and tomographic-probability\nrepresentations\nGiven bipartite system of a two-mode \feld.\nThe tomographic probability distribution is determined as follows:\nw(X1;\u00161;\u00171;X2;\u00162;\u00172) = Trh\nb\u001a(1;2)\u000e(X1b1\u0000\u00161bq1\u0000\u00171bp1)\u000e(X2b1\u0000\u00162bq2\u0000\u00172bp2)i\n:\n(29)\nThe density matrix in the diagonal representation is determined by the symbol of the\ndensity state b\u001a(1;2)\n\u001e(z1;z2) =1\n\u00194Z\nexp\u00141\n2\u0000\nju1j2+ju2j2\u0001\n+z\u0003\n1u1\u0000z1u\u0003\n1+z\u0003\n2u2\u0000z2u\u0003\n2\u0015\n\u0002Trb\u001a(1;2)bD1(u1)bD2(u2)dReu1dImu1dReu2dImu2: (30)\nThe stateb\u001a(1;2) is separable, if the density state can be written as a convex sum\nb\u001a(1;2) =X\nkPkb\u001ak(1)\nb\u001ak(2); Pk\u00150;X\nkPk= 1: (31)\nIn view of linearity property of tomographic map, one has the de\fnition of separability\nin terms of the state tomogram, i.e., the state is separable, if\nw(X1;\u00161;\u00171;X2;\u00162;\u00172) =X\nkPkw(1)\nk(X1;\u00161;\u00171)w(2)\nk(X2;\u00162;\u00172): (32)\nTomogram is the joint probability-distribution function of two random variables\nX1;X22R. Thus, the condition of the state separability is formulated as the above\nproperty (32) of the joint probability distribution.\nIf tomogram cannot be written as convex sum (32), the state is entangled. The\nseparability condition can be reformulated, in view of the standard characteristicDiagonal and probability representations 7\nfunction for the tomographic probability distribution (32). In fact, if the characteristic\nfunction can be written as\n\u001f(k1;\u00161;\u00171;k2;\u00162;\u00172) =X\nkPk\u001f(1)\nk(k1;\u00161;\u00171)\u001f(2)\nk(k2;\u00162;\u00172) (33)\nthe state is separable. Here \u001f(1)\nk(k1;\u00161;\u00171) and\u001f(2)\nk(k2;\u00162;\u00172) are the characteristic\nfunctions for tomographic probabilities w(1)\nk(X1;\u00161;\u00171) andw(2)\nk(X2;\u00162;\u00172), respec-\ntively.\nAn analogous de\fnition of the separability and entanglement of the density state\nb\u001a(1;2) can be formulated in the diagonal representation.\nThus the state is separable, if the function which is symbol of the density state\nin the diagonal representation can be written as\n\u001e(z1;z2) =X\nkPk\u001e(1)\nk(z1)\u001e(2)\nk(z2): (34)\nThus we formulated the problem of separability and entanglement in the diagonal\nrepresentation of the density state b\u001a(1;2). One can easily extend the de\fnition of\nseparable and entangled states to multipartite systems in both the tomographic and\ndiagonal representations of density states.\n7. Conclusions\nTo conclude, we resume the main results of this work.\nWe reviewed the diagonal and probability representations of quantum states using\nthe standard star-product scheme. We found mutual relations of the weight function\nof the diagonal representation and the tomographic-probability distribution associated\nwith the quantum state. We obtained the kernel of star-product of operator symbols in\nthe diagonal representation. The duality relation between the diagonal representation\nof the weight function and the Husimi{Kano function was obtained in the explicit\nform. The superposition rule was formulated in both the diagonal representation\nand probability representation of the density states. The problem of separability and\nentanglement was formulated in both the diagonal and probability representations.\nAcknowledgments\nV.I.M. thanks the Russian Foundation for Basic Research for partial support under\nProjects Nos. 07-02-00598 and 08-02-90300 and the Organizers of the XV Central\nEuropean Workshop on Quantum Optics (Belgrade, Serbia, 30 May { 3 June 2008)\nfor kind hospitality.\nReferences\n[1] Schr odinger E 1926 Ann. Phys. (Liepzig) 79489\n[2] Dirac P A M 1058 Principles of Quantum Mechanics 4th ed (London: Oxford University Press)\n[3] Landau L D 1927 Z. Physik 45430\n[4] von Neumann J 1932 Matematische Grundlagen der Quantenmechanyk (Berlin: Springer)\n[5] Wigner E 1932 Phys. Rev. 40749\n[6] Husimi K 1940 Proc. Phys. Math. Soc. Jpn 23264\n[7] Kano Y 1956 J. Math. Phys. 61913\n[8] Sudarshan E C G 1963 Phys. Rev. Lett. 10277\n[9] Glauber R J 1963 Phys. Rev. Lett. 1084Diagonal and probability representations 8\n[10] Klauder J R and Sudarshan E C G 1968 Fundamentals of Quantum Optics (New York:\nBenjamin)\n[11] Mehta C L and Sudarshan E C G 1965 Phys. Rev. 138B274\n[12] Schr odinger E 1935 Naturwissenchaften 23823\n[13] Mancini S, Man'ko V I and Tombesi P 1996 Phys. Lett. A 2131\n[14] Man'ko O V, Man'ko V I and Marmo G 2002 J. Phys. A: Math. Gen. 35699\n[15] Man'ko V I, Marmo G, Sudarshan E C G and Zaccaria F 2002 J. Phys. A: Math. Gen. 357137\n[16] Man'ko V I, Marmo G, Sudarshan E C G and Zaccaria F 2003 J. Russ. Laser Res. 24507\n[17] Man'ko V I, Marmo G, Simoni A, Sudarshan E C G and Ventriglia F 2008 Rep. Math. Phys.\n61337\n[18] Man'ko V I and Mendes R V 1999 Phys. Lett. A 26353\n[19] Man'ko V I, Marmo G and Vitale P 2005 Phys. Lett. A 3341\n[20] Man'ko O V, Man'ko V I, Marmo G and Vitale P 2007 Phys. Lett. A 360522\n[21] Klauder J R and Scagerstam B-S K 2007 J Phys. A: Math. Theor. 402093"    },    {        "title": "0903.2903v2.Measuring_Qutrit_Qutrit_Entanglement_of_Orbital_Angular_Momentum_States_of_an_Atomic_Ensemble_and_a_Photon.pdf",        "content": "arXiv:0903.2903v2  [quant-ph]  12 Sep 2009Measuring Qutrit-Qutrit Entanglement of Orbital Angular M omentum States\nof an Atomic Ensemble and a Photon\nR. Inoue1,∗T. Yonehara1, Y. Miyamoto2, M. Koashi3, and M. Kozuma1,4\n1Department of Physics, Tokyo Institute of Technology,\n2-12-1 O-okayama, Meguro-ku, Tokyo 152-8550, Japan\n2Department of Information and Communication Engineering, The University of Electro-Communications,\n1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan\n3Division of Materials Physics, Graduate School of Engineer ing Science, Osaka University,\n1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan and\n4CREST, Japan Science and Technology Agency, 1-9-9 Yaesu, Ch uo-ku, Tokyo 103-0028, Japan\n(Dated: December 6, 2018)\nThree-dimensional entanglement of orbital angular moment um states of an atomic qutrit and a\nsingle photon qutrit has been observed. Their full state was reconstructed using quantum state\ntomography. The fidelity to the maximally entangled state of Schmidt rank 3 exceeds the threshold\n2/3. This result confirms that the density matrix cannot be deco mposed into ensemble of pure\nstates of Schmidt rank 1 or 2. That is, the Schmidt number of th e density matrix must be equal to\nor greater than 3.\nPACS numbers: 03.65.Wj, 03.67.Mn, 32.80.-t, 42.50.Dv\nAn essential requirement towardpractical quantum in-\nformation systems is the capability to generate entangled\nstates between distant sites over quantum networks [1].\nInspired bya schemeto createlong-livedentanglementin\nscalable quantum networks proposed by Duan et al.[2],\nvariousexperimental resultsusingatomicensembles have\nbeen reported (such as [3]). Recently, d-dimensional\nquantum systems, or qudits, have been studied, and it\nhas been pointed out that qudits are better adapted for\ncertain purposes. As an example, they enable more effi-\ncientuseofcommunicationchannelsinquantumcryptog-\nraphy [4]. Following the pioneering experiment on para-\nmetric down-conversion [5], various protocols have been\ndemonstrated using orbital angular momentum (OAM)\nstates of photons (such as [6]). Photons are a promis-\ning carrier of quantum information. However, they are\ndifficult to store for appreciable periods of time. The re-\nalization of massive qudits and the ability to characterize\ntheir entanglement are therefore critical for applications.\nAnother recent landmark is the demonstration of the\nuseofOAMtogeneratearbitrarysuperpositionofatomic\nrotational states with the coherent transfer the OAM of\nlight to atoms in Bose-Einstein condensate [7, 8]. The\nexperiment illustrates the potential of OAM as a tool to\ngenerate and to control the atomic qudits.\nPrevious work [9] demonstrated entanglement associ-\nated with OAM in an ensemble of atoms and a photon.\nThe atomic OAM state is linked to the spatial degree\nof freedom of collective atomic excitations, and, in the\ncase of photons, the OAM state correspondsto Laguerre-\nGaussian (LG) modes. This result suggests that atomic\nensembles can be used as nodes of qudit-based quan-\ntum networks. However, previous observations were lim-\nited to two-qubit entanglement. In this letter, we report\nhigher-dimensionalityoftheentanglementofOAMstatesof an atomic ensemble and a photon, as confirmed by es-\ntimating the Schmidt number [10] of the reconstructed\ntwo-qutrit (i.e., qudits with d= 3) density matrix. For\npure states, the dimension of the range of the marginal\nstate is called the Schmidt rank, which describes how\nmany local levels are involved in the entanglement. Ex-\ntending this notion, the Schmidt number of a bipartite\nmixed state is defined to be the minimum Schmidt rank\nthat must appear in any decomposition of the state as a\nmixture of pure states. For example, any decomposition\nof a mixed state with Schmidt number 3 must include a\npure state of Schmidt rank 3 or greater.\nThe LG modes constitute a complete basis for describ-\ning the paraxial propagation of light [11, 12]. The in-\ntensity and phase distributions of several LG modes and\nsuperpositions of them are shown in Fig. 1. An LG mode\nis characterized by its two indices pandm, and by the\nGaussian beam waist w0. The integers pandmare the\nradial and azimuthal mode index, respectively, and the\nphase variation for a closed path around the optical axis\nis 2mπ. A single photon in the LGp,mmode carriesa dis-\nFIG.1: The intensity(left panelofeach pair)andphase(rig ht\npanel) distribution of several LG modes and their superposi -\ntions.2\ncrete OAM of m/planckover2pi1along its propagation direction. Here\nwe define |L/an}bracketri}ht,|G/an}bracketri}ht, and|R/an}bracketri}htto be the single photon states\nwith OAM of −/planckover2pi1, 0, and + /planckover2pi1, respectively.\nConsider an ensemble of atoms having a three-level\nstructure |a/an}bracketri}ht,|b/an}bracketri}ht, and|c/an}bracketri}ht, as shown in Fig. 2(a). Initially\nallofthe atomsarepreparedinlevel |a/an}bracketri}ht. Aclassicalwrite\npulsetunedtothe |a/an}bracketri}ht → |c/an}bracketri}httransitionwithproperdetun-\ning ∆ is incident on the atomic ensemble. In this process,\nthe|c/an}bracketri}ht → |b/an}bracketri}httransitionis stimulatedand a Stokesphoton\nis generated. The write pulse is weak and its interaction\ntime is short. Therefore the excitation probability of a\nStokes photon into a specified mode is much less than\nunity per pulse. The collective atomic excitation retains\nthe spatial distribution of the relative phase between the\nFIG. 2: (Color online) (a) Energy levels of87Rb, and the\nassociated laser frequencies. (b) Schematic of experimen-\ntal setup: SLM, spatial light modulator; SMF, single-mode\nfiber; SPCM, single photon counting module. Circularly\npolarized write and read pulses illuminate the87Rb MOT\n(Magnet-Optical Trap), and circularly polarized Stokes an d\nanti-Stokes photons are selectively directed onto the SPCM s\npassing through quarter-wave plates and polarizers. The pa ir\nof SLM and SMF serves as a spatial mode filter. (c) Co-\nincidence rate and time-resolved coincidence (inset) betw een\nthe Stokes and anti-Stokes photons of the LG0,0mode as a\nfunction of the time delay.write pulse and the Stokes photon. According to angular\nmomentum conservation, the state of the Stokes photon\nand collective atomic excitation will be entangled. We\nuse|l/an}bracketri}ht,|g/an}bracketri}ht, and|r/an}bracketri}htto denote the states of the collective\natomic excitation with OAM of −/planckover2pi1, 0, and + /planckover2pi1, respec-\ntively. In the present work, our measurement is sensitive\nto only the three-dimensional photonic and atomic OAM\nstates.\nWhen the write pulse carries zero OAM, The resultant\natoms-photon state will be\n|φ/an}bracketri}ht=CL|L/an}bracketri}ht|r/an}bracketri}ht+CG|G/an}bracketri}ht|g/an}bracketri}ht+CR|R/an}bracketri}ht|l/an}bracketri}ht,\nwhereCL,CG, andCRare the relative complex ampli-\ntudes. Theamplitudesdepend onthespatialshapeofthe\nregion where the write pulse interacts with the atoms.\nTheatoms-photonentanglementcanbetestedbymap-\nping the state of the atoms to that of an anti-Stokes\nphoton by illuminating the atomic ensemble with a laser\npulse (read pulse) resonant with the |b/an}bracketri}ht → |c/an}bracketri}httransition.\nThe efficiency of this transfer can be nearly unity be-\ncauseit correspondsto the retrievalprocessofthe atomic\nquantum memory based on electromagnetically induced\ntransparency [13].\nA schematic of the experimental setup is shown in\nFig. 2(b). An optically thick (optical depth of about\n5) cold atomic cloud is created using a magneto-optical\ntrap (MOT) for87Rb. Levels |a/an}bracketri}htand|b/an}bracketri}htcorrespond to\n5S1/2F= 1 and 2, respectively, and level |c/an}bracketri}htcorresponds\nto 5P1/2F′= 2. One cycle in the experiment comprises\na 6-ms loading period and a 4-ms measurement period.\nDuring the loading period, a gas of cold87Rb atoms is\ncooledandtrapped, andtheyareopticallypumped tothe\n5S1/2F= 1 level using a 50 µs depumping pulse tuned to\nthe 5S1/2F= 2→5P3/2F′= 2 transition. After the\nloading period, the magnetic field and the radiation re-\nsponsible for the cooling, trapping, and depumping are\nshut off. The vacuum cell is magnetically shielded using\na single-layer permalloy. The coil jig is non-metallic, and\nthus eddy currents, which prolong the decay of the mag-\nneticfield[9], aresuppressed. Theresidualmagneticfield\nis about 1 mG during the measurement period. During\nthat period, read and write pulses illuminate the atomic\nensemble with a 400-nsrepetition cycle. The 15-nsGaus-\nsian write pulse is tuned to the |a/an}bracketri}ht → |c/an}bracketri}httransition with\n10-MHz detuning and comprises 4 ×104photons. After a\n40-nsdelay, a 200-nsrectangularread pulse of300 µW in-\ntensity illuminates the MOT. As shown in Fig. 2(b), the\nreadandwritepulses,whosespatialmodesarecleanedup\nby passage through single-mode fibers SMF1 and SMF2,\nare counter-propagated along the same axis. The out-\nput beam from the fiber is focused into the MOT with\na Gaussian beam waist of 400 µm. Similarly, the Stokes\nandanti-Stokesphotonsshareasinglespatialmodewhen\ntheir conversions are appropriately chosen using spatial\nlight modulators SLM1 and SLM2 (Hamamatsu model\nX8267). Their Gaussian beam waist at the center of the3\nMOT is adjusted to 275 µm. The angle between the axis\nof the Stokes or anti-Stokes photons and that of SMF1\nor SMF2 is ∼3◦in order to spatially separate the weak\nStokes(anti-Stokes)photonsfromthe strongwrite (read)\npulses. The incident write and read pulses are circularly\npolarized, as are the Stokes and anti-Stokes photons.\nThe Stokes and anti-Stokes photons coupled into\nSMF3 and SMF4 are directed onto the single-photon-\ncounting modules SPCM1 and SPCM2 (Perkin-Elmer\nmodel SPCM-AQR-14). From the Stokes photon count-\ning, theexcitationprobabilityisestimated tobe5 ×10−4.\nTheir outputs are then fed into the start and stop inputs\nof the time interval analyzer. The experimental results\nfor the coincidence rate between the Stokes and anti-\nStokes photons in an LG0,0mode are displayed versus\ntime delay in Fig. 2(c), from which the normalized cross-\nintensity correlation is estimated to be g(2)\ns,as= 74.6±7.4,\nconfirming that the excitation probability of a Stokes\nphoton into a specified mode is much less than unity for\neach pulse. The coincidence count rate was 5 .2 s−1.\nTo determine the full state of the atoms and Stokes\nphoton, two-qutrit state tomography [14, 15] was per-\nformed, wherethe density matrixwasreconstructedfrom\nthe set of 81 measurements represented by the operators\nˆµi⊗ˆµj(withi,j= 0,1,···,8) and where ˆ µk≡ |k/an}bracketri}ht/an}bracketle{tk|.\nThe ket|k/an}bracketri}htforthe Stokesphoton waschosenfromamong\n{|L/an}bracketri}ht,|G/an}bracketri}ht,|R/an}bracketri}ht,(|G/an}bracketri}ht+|L/an}bracketri}ht)/√\n2,(|G/an}bracketri}ht+|R/an}bracketri}ht)/√\n2,(|G/an}bracketri}ht+\ni|L/an}bracketri}ht)/√\n2,(|G/an}bracketri}ht −i|R/an}bracketri}ht)/√\n2,(|L/an}bracketri}ht+|R/an}bracketri}ht)/√\n2,(|L/an}bracketri}ht+\nFIG. 3: (Color online) Gaussian components of applied phase\nmodulations T(x,y) to an incoming Gaussian beam of waist\nw0= 2.2 mm. (a) T(x,y) =eiarg(x−x0+i(y−y0)). The\nleft panel plots the simulation while the right panel shows\nthe experimental result. (b) T(x,y) =eiarg(x−x0+iy). (c)\nT(x,y) =eiπ\n2sgn(x−x0). In panels (b) and (c), the dots are\nexperimental results and the solid curves are obtained from\nthe numerical simulation.i|R/an}bracketri}ht)/√\n2}, and for the collective atomic excitation from\namong{|l/an}bracketri}ht,|g/an}bracketri}ht,|r/an}bracketri}ht,(|g/an}bracketri}ht+|l/an}bracketri}ht)/√\n2,(|g/an}bracketri}ht+|r/an}bracketri}ht)/√\n2,(|g/an}bracketri}ht−\ni|l/an}bracketri}ht)/√\n2,(|g/an}bracketri}ht+i|r/an}bracketri}ht)/√\n2,(|l/an}bracketri}ht+|r/an}bracketri}ht)/√\n2,(|l/an}bracketri}ht−i|r/an}bracketri}ht)/√\n2}.\nThese measurements are implemented using SLMs and\nSMFs; the SLMs produce spatial phase modulation and\nthe SMFs filter the LG0,0mode [16]. As reported in [17]\nin detail, arbitrary superpositions of an LG0,0and an\nLG0,±1mode can be converted into an LG0,0mode by\napplication of the phase modulation Tcorresponding to\nthe relative phase difference between the superposition\nmode and the LG0,0mode. As is clear from Fig. 1, such\nconversions can be achieved by moving the singularity in\nthe phase modulation to a particular location. Our re-\nflective SLMs have an active region of 768px ×768px.\nEven if the phase modulation is discrete, the fractional\nintensitydiffractedintohigherorderscanbedecreasedby\nadding the blazed phase grating structure. The spatial\nperiod of the grating is 4 px ∼100µm with a diffrac-\ntion efficiency of 25 %. In order to check the SLMs, the\nGaussian components of a beam diffracted with a spatial\nphase modulation of T(x,y) =eiarg(x−x0+i(y−y0))were\nmeasured. Here arg( z) is the argument of the complex\nnumberz. The results are plotted in Fig. 3(a) and (b),\nand are in good agreementwith numerical calculationsof\nthe superposition modes. The location of the singularity\nthatconvertsthesuperpositionmodeintoaGaussiancan\ntherefore be determined. At position ( x0,y0) = (0,0),\nwhere an incoming Gaussian mode is converted into an\nLG0,±1mode, the normalized intensity was measured to\nbe 3×10−3, indicating a high extinction ratio. Simi-\nlarly, the superposition mode ( LG0,−1+eiθLG0,+1)/√\n2\ncan be converted into an LG0,0mode by applying a dis-\ncontinuousphasemodulation. TheGaussiancomponents\nof the beam diffracted by an SLM with a spatial phase\nmodulation of T(x,y) =eiπ\n2sgn(x−x0)was also measured,\nwhere sgn( x) is the sign of x. The experimental results\nare shown in Fig. 3(c), and are again in good agreement\nwith numerical calculations.\nFig. 4 shows the graphical representation of density\nmatrixρexpreconstructed from the 81 coincidences. The\ntypical coincidence rate was roughly 5 s−1, and the data\nacquisition time of each measurement was 100 s. From\nthe density matrix, the fidelity to a maximally entangled\nstateFexp≡ /an}bracketle{tMES|ˆρexp|MES/an}bracketri}ht= 0.74±0.02 was ob-\ntained. Here |MES/an}bracketri}htwaschosenfromthesetofmaximally\nentangled states ( eiαπ|L/an}bracketri}ht|r/an}bracketri}ht+|G/an}bracketri}ht|g/an}bracketri}ht+eiβπ|R/an}bracketri}ht|l/an}bracketri}ht)/√\n3\nso as to maximize the fidelity, where the values of αand\nβwere 0.019πand−0.058π, respectively. The error in\nFexpis calculated by using Monte-Carlo method from\nthe statistical uncertainties in the coincidences. An op-\ntimal witness operator of Schmidt number 3 in C3⊗C3\nis given by ˆW3= 1−3|MES/an}bracketri}ht/an}bracketle{tMES|/2 [18], resulting in\nTr(ˆW3ˆρ)<0⇔ /an}bracketle{tMES|ˆρ|MES/an}bracketri}ht>2/3. The experimen-\ntal result Fexp>2/3 therefore confirms that the Schmidt\nnumber of the mixed state of the atomic ensemble and\nthe photon is greater than or equal to 3.4\nFIG. 4: (Color online) Graphical representation of the den-\nsity matrix ρexpof a state as estimated by quantum state\ntomography from the experimentally obtained coincidences .\nThe upper plot is the real part, and the lower plot is the\nimaginary part.\nThe OAM measurements were achieved using mode\nconversion by SLMs and mode filtering by SMFs in this\nexperiment. However, as frequently occurs in experi-\nments utilizing a spatial phase modulation, the mea-\nsurement bases cannot be realized completely accurately,\nresulting in unwanted radial and azimuthal components\ncomprising up to 20%. While this systematic effect in-\ncreases the error in Fexp, the resultant fidelity is never-\ntheless 0 .74+0.06\n−0.07, which is larger than 2 /3 even at the\nlowest error bound.\nThe major diagonal elements of the recon-\nstructed density matrix are /an}bracketle{tL|/an}bracketle{tr|ρexp|L/an}bracketri}ht|r/an}bracketri}ht=\n0.25,/an}bracketle{tG|/an}bracketle{tg|ρexp|G/an}bracketri}ht|g/an}bracketri}ht= 0.37,and/an}bracketle{tR|/an}bracketle{tl|ρexp|R/an}bracketri}ht|l/an}bracketri}ht=\n0.26. The summation of remaining elements\n1−0.25−0.37−0.26 = 0.12 suggests that compo-\nnents of non-zero total OAM, which may originate\nmainly from stray light, are one of the dominant factors\nlimiting the fidelity. The normalized cross intensity\ncorrelation g(2)\ns,as= 74.6±7.4 is significantly smaller thanthe value 2000 expected from the measured excitation\nprobability of 5 ×10−4. Therefore, stray light appears\nto have strongly affected the photon statistics and\ndecreased the fidelity. The imbalance of the diagonal\nelements may also affect the fidelity. However, even\nsupposing that the Stokes photons are locally filtered\nso as to balance the relative amplitude, the fidelity is\nstill expected to be 0 .74. This result confirms that the\nimbalance is not the dominant factor decreasing the\nfidelity in our case. Note that the diagonal elements can\nbe balanced by changing parameters such as the beam\nwaist of the write pulse since the relative amplitude\nis dependent on the spatial shape of the effective\ninteraction volume [19]. The other factor limiting the\nfidelity is the decoherence caused by the environmental\nnoise, such as the Larmor precession of the ground-state\nZeeman sublevels and the ballistic expansion of the\natomic ensemble.\nIn conclusion, higher-dimensionality of the entangle-\nment of OAM states has been observed for an atomic\nensemble and a photon by estimating the Schmidt num-\nber of the reconstructed density matrix. The experiment\ndescribed here enables one to communicate quantum in-\nformation encoded in the spatial degrees of freedom of a\nphoton and an atomic ensemble [20].\nWe gratefully acknowledge M. Ueda, K. Usami, and\nN. Kanai for valuable comments and stimulating dis-\ncussions. R.I. was supported by a Grant-in-Aid from\nJSPS. This work was supported by the Global Center\nof Excellence Program by MEXT, Japan through the\nNanoscience and Quantum Physics Project of the Tokyo\nInstitute of Technology, and by MEXT through a Grant-\nin-Aid for Scientific Research (B).\n∗inoue.r.aa@m.titech.ac.jp\n[1] H. J. Kimble, Nature 453, 1023 (2008).\n[2] L. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Nature\n414, 413 (2001).\n[3] K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, Nature\n452, 67 (2008).\n[4] S. P. Walborn, D. S. Lemelle, M. P. Almeida, and\nP. H. Souto Ribeiro, Phys. Rev. Lett. 96, 090501 (2006).\n[5] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature\n412, 313 (2001).\n[6] N. K.Langford et al., Phys.Rev.Lett. 93, 053601 (2004).\n[7] M. F. Andersen et al., Phys. Rev. Lett. 97, 170406\n(2006).\n[8] K. Helmerson et al., Nucl. Phys. A, 790, 705c-712c\n(2007).\n[9] R. Inoue et al., Phys. Rev. A 74, 053809 (2006).\n[10] B. M. Terhal and P. Horodecki, Phys. Rev. A 61, 040301\n(2000).\n[11] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and\nJ. P. Woerdman, Phys. Rev. A 45, 8185 (1992).\n[12] G. F. Calvo, A. Picon, and E. Bagan, Phys. Rev. A 73,\n013805 (2006).5\n[13] M. Fleischhauer and M. D. Lukin, Phys. Rev. A 65,\n022314 (2002).\n[14] D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G.\nWhite, Phys. Rev. A 64, 052312 (2001).\n[15] R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro,\nPhys. Rev. A 66, 012303 (2002).\n[16] E. Yao et al., Optics Express 14, 13089 (2006).\n[17] A. Vaziri, G. Weihs, and A. Zeilinger, Journal of OpticsB: Quantum and Semiclassical Optics 4, S47 (2002).\n[18] A. Sanpera, D. Bruß, and M. Lewenstein, Phys. Rev. A\n63, 050301(R) (2001).\n[19] C. I. Osorio, S. Barreiro, M. W. Mitchell, and J. P. Tor-\nres, Phys. Rev. A 78, 052301 (2008).\n[20] G. Molina-Terriza, J. P. Torres, and L. Torner, Nature\nPhysics3, 305 (2007)."    },    {        "title": "0904.2396v2.Carrier_Density_and_Magnetism_in_Graphene_Zigzag_Nanoribbons.pdf",        "content": "arXiv:0904.2396v2  [cond-mat.mes-hall]  25 Jun 2009CarrierDensity and Magnetism inGraphene ZigzagNanoribbo ns\nJ. Jung1,∗and A. H. MacDonald1\n1Department of Physics, University of Texas at Austin, USA\n(Dated: November 6, 2018)\nThe influence of carrier density on magnetism in a zigzag grap hene nanoribbon is studied in a π-orbital\nHubbard-model mean-field approximation. Departures from h alf-filling alter the magnetism, leading to states\nwith charge density variation across the ribbon and paralle l spin-alignment on opposite edges. Finite carrier\ndensities cause the spin-density near the edges to decrease steadily, leading eventually to the absence of mag-\nnetism. At low doping densities the system shows a tendency t o multiferroic order in which edge charges and\nspins are simultaneously polarized.\nINTRODUCTION\nGraphene sheets and related carbon based nanomaterials\nhave attracted attention recently after seminal experimen ts\n[1,2]revealednovelphysicsrelatedtotheiruniqueelectr onic\nstructure[3]. In graphene nanoribbons[4, 5, 6, 7, 8, 9, 10,\n11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] lateral\nconfinementleadstosizequantizationandtoone-dimension al\nconductionchannelswhosepropertiesdependqualitativel yon\nedgeterminationcharacter. Neutralzigzagterminatedrib bons\nhave attracted particular attention because they have a flat\nband, perflectly flat in simple π-band models, pinned to the\nFermi level. In self-consistent field (SCF) theories, inclu ding\nab initio spin-density-functional theories, the flat band leads\ntorobustmagneticorder. Ferromagneticalignmentofspins at\nthe zigzagedgesis predictedalso in treatmentsgoingbeyon d\nmean field [7, 8]. Although the reliability of SCF theories is\nuncertainandnot yet tested experimentally,interest in zi gzag\nedge magnetismhasremainedstrong becauseof potential for\ninterestingapplicationsin nano-electronics[18].\nMoststudiesoftheelectronicstructureofzigzagterminat ed\ngraphene ribbons have focused on properties of the neutral\nsystem or systems with substitutional doping [25]. We study\ntheroleofgatevoltageinducedchangesincarrierdensity, i.e.\ngate doping. A related work in the low carrier dopingregime\nwith an additional neutralizing background charge explore d\nthepossibilityofstable non-collinearmagneticstates[2 6]. In\nneutral systems, SCF theories predict edge magnetization i n\ngraphenenanoribbonswithoppositespinpolarizationsono p-\nposite edges [4, 9, 15]. In theoretical studies of locally ga ted\nzigzagribbonjunctionsusuallythe non-interactingelect ronic\nstructureisassumed[27,28,29,30,31,32,33],neglecting the\npossibility of doping-dependentinteraction-drivenrear range-\nments. Inthisworkweshowthatgatedopingleadstochanges\nin charge distribution, spin configuration, and total net sp in\npolarization, which are accompanied by important modifica-\ntions in electronic structure. Our study is based on the π-\norbital Hubbard-model SCF theory for the magnetic proper-\nties of graphenenanostructures[17, 34, 35, 36, 37], in whic h\nan electron of spin σin siteiexperiences a repulsive inter-\naction proportional to the density of opposite-spin electr ons\nniσ. The Hubbard-model SCF theory is broadly consistent\nwith DFT calculations when the interaction parameter UisW (˚A)δn\n100 90 80 70 60 50 40 30 20 0.6 \n0.4 \n0.2 \n0AFFb P\nF\nAFb NC \nFIG. 1: (Color online) Hubbard model SCF-theory phase diagr am\nas a function of doping per length δn(defined in the text) and rib-\nbon width Wfor nearest-neighbor hopping γ0=2.6eV, The on-site\nrepulsion strength was chosen to have a value U=2eVwhich re-\nproducetheribbonbandgapsobtainedintheLDA-DFTcalcula tions\nin reference [17]. We used 1200 k-points for Brillouin-zone sam-\npling. The energetic preference for opposite spins (AF) on o pposite\nedges isreplacedbya preference forparallel( F)spinsatlargerdop-\ning. Above a critical doping δn∼0.7 the SCF calculation does not\nfind magnetic states. Solutions at finite doping sometimes (A Fband\nFb)breaktheinversionsymmetryoftheribbon. Whennon-col linear\nspin (NC) is allowed canted spin solutions midway between AF and\nFconfiguration become energeticallyfavored atlow doping.\nchosen appropriately. π-orbital Hartree-Fock theory reduces\ntotheHubbardmodelwhenonlytheon-siteCoulombinterac-\ntions are retained. We have chosen to use a Hubbard interac-\ntionparameter U=2eVwhichreproducesintheundopedcase\ntheband-gapsobtainedbymicroscopicdensityfunctionalt he-\nory in the local density approximation. This value is smalle r\nthanotherestimates[34], buthasbeenadoptedwith asimila r\nmotivationin someotherrecentwork[35,36].\nThe Hubbard model mean-field Hamiltonian for each spin\nσis\nHσ=−γ0∑\n/angbracketlefti,j/angbracketrightc†\niσcjσ+U∑\niniσniσc†\niσciσ+vext∑\nic†\niσciσ2\nEP−EAFEFb−EAFEF−ENCEF−EAFEF−EAFb\nδn∆E (meV)\n0.5 0.4 0.3 0.2 0.1 04\n2\n0\n-2\nFIG. 2: (Color online) Total energy differences per edge ato m be-\ntweentheAF(orAFb)andF,Fb,Pstatesasafunctionof doping δn\nforarelativelynarrowribbonwith N=8atompairsperunitcell. For\nlowdopingAFbtypesolutionswithbrokenchargesymmetryar een-\nergeticallyfavored over AF solutions although the energy d ifference\nis very small. The non-collinear (NC) spin solutions are low est in\nenergy inthe weakly doped regime.\nconsist of a nearest neighbor tight-binding term with hop-\npingγ0=2.6eVconnecting lattice sites iandj, the Hubbard\ntermrepresentingelectron-electroninteractions,andan exter-\nnalpotentialtermaccountingfortheinteractionwiththec on-\nstant positivebackgroundchargeproportionalto a coeffici ent\nwe choose to be vext=−U. Given the uncertainty of predic-\ntionsimpliedbyparticularversionsofSCF-theory,theadv an-\ntagesofthisrelativelysimplemodeloftenoutweighdisadv an-\ntages. Because themagnetisminzigzagribbonsisessential ly\none-dimensional,we measuredoping δnin unitsof the num-\nberofexcesselectronsperrepeatdistance a=2.46˚Aalongthe\nedge. The corresponding areal density δn2D=δn/Wwhere\nthe ribbon width W=√\n3Na/2 andNis the number of atom\npairsperribbonunitcell.\nSCFSOLUTIONSAT FINITEDOPING\nThe main players in zigzag edge magnetism are the flat\nband states which occupy one-third of the one-dimensional\nribbonBrillouin-zone(BZ)andarelocalized[11]nearther ib-\nbonedges, most stronglyso nearthe BZ boundary |k|∼π/a.\nIn the undoped SCF ground state, electrons of opposite-spin\nare localized near opposite edgesof the ribbon and a gap[19]\nΔ ∝W−1separates occupied valence and empty conduction\nband ribbon states. By appealing to particle-hole symme-\ntry we can limit our discussion of doping to the n-type case\nin which electrons start filling the conduction band. Dop-\ning causes charge-density variation across the ribbon and t o\na complicated competition between band and interaction en-\nergies manifested by the variety of SCF equation solutions\nclassified below. We label solutions as AF (opposite) or F\n(parallel) to indicate the relative alignment of spins on op po-\nsite edges. The label NC is used indicate non-collinear spin\nsolutions. Thelabelbisappliedforsolutionswhichbreaki n-Fb FAFb \nδnζ\n0.4 0.2 00.03 \n0.02 \n0.01 \n0\nFIG. 3: (Color online) Net spin polarization obtained from t he total\nelectron spin densities ζ=/parenleftbig\nn↑−n↓/parenrightbig\n//parenleftbig\nn↑+n↓/parenrightbig\nfor AFb, F and Fb\nsolutions as a function of doping. AFb solutions collapse in to AF\nsolutions with zero net spin polarization for high enough do ping. F\nand Fb configurations also progressively lose net spin polar ization\nas they approach the non-magnetic Plimit. The shaded region rep-\nresents the doping regime where non-collinear solutions ar e favored\nenergetically.\nversion symmetry across the ribbon in a way which will be\nexplained in more detail later. Finally we use the letter P to\ndesignateaparamagneticstatewithnolocalspin-polariza tion.\nThephasediagraminFig. (1)illustratesthesequenceoftra n-\nsitions AF →NC→F→Fb→P in narrower ribbons. In\nwider ribbons we find an additional AF state region between\ntheF andFbregimes.\nThe total energy per unit cell consist of a sum over all the\noccupied single-particle eigenvalues εkmσlabeled with kand\nmthe band index divided by NKthe total number of k-points\nminusa termto accountforthe doublecountingcorrectionin\ntheinteraction\nE=1\nNKocc\n∑\nkmσεkmσ−U\n2∑\niσnlocal\niσnlocal\niσ\nwheretheoccupations nlocal\niσareevaluatedinthelocalframeat\nlatticesite iwherespinisdiagonal. Theirdifferencesbetween\ndifferentself-consistent solutionsare shown in Fig. ( 2) f or a\nparticular ( N=8) ribbon width when only collinear spin so-\nlutionsare considered. In the collinearscheme the energya s-\nsociatedwithbreakinginversionsymmetryacrosstheribbo ns\nisalwayssmallandthemaintrendisacrossoverfromantifer -\nromagnetic solutions at small δnto ferromagnetic solutions\nforδn/greaterorsimilar0.04 to non-magnetic solutions for δn/greaterorsimilar0.4. For\nthe F type solutions and those with broken charge symmetry\nthesystemhasanonzeronetspinpolarizationasafunctiono f\ndoping density. The doping dependence of spin-polarizatio n\nis illustrated for the same N=8 ribbon width in Fig. ( 3).\nEach of the solution types identified in Fig. (1) is associate d\nwith particular electronic structure features which are il lus-\ntrated in Fig. (4). For the AF solution, finite doping require s\nthatstatesabovetheinteractioninducedgapbeoccupied. F or\nsmalldopingelectronsstartoccupyingstatesnearthecond uc-\ntion band minima. (See Fig. (4).) These additional electron s3\nE↓E↑\nAFbδn= 0.03\nkaE↑/↓(eV)\nπ/2 2 π/3 π0.2\n0\n-0.2\n-0.4AFbδn= 0.03\nkaE↑/↓(eV)\nπ/2 2 π/3 π0.2\n0\n-0.2\n-0.4E↓E↑\nAFδn= 0.03\nka2π/3 πAFδn= 0.03\nka2π/3 πE↓E↑\nFδn= 0.1\nka2π/3 πFδn= 0.1\nka2π/3 πE↓E↑\nFbδn= 0.4\nka2π/3 πFbδn= 0.4\nka2π/3 πE↓E↑\nPδn= 0.46\nka2π/3 πPδn= 0.46\nka2π/3 π\nn↓n↑\nAFbδn= 0.03\ny(˚A)n↑/↓(y)\n16128400.6\n0.5\n0.4n↓n↑\nAFδn= 0.03\ny(˚A)1612840n↓n↑\nFδn= 0.1\ny(˚A)1612840n↓n↑\nFbδn= 0.4\ny(˚A)1612840n↓n↑\nPδn= 0.46\ny(˚A)1612840\nFIG.4: (Color online) Upper row. Bandstructures corresponding toAFb, AF,F,Fband Pspincol linear solutions of the Hubbard-model SCF\nequations forazigzagnanoribbon with N=8atompairsintheunitcell. Atfinitedopingtheenergygaind ue tothegappresent inthetheAFb\nand AF solutions is reduced, favoringthe Fsolution which do es not have a gap. Lower row. Up anddown spinelectron occupation per lattice\nsite in the unit cell across the ribbon. The AFb configuration has broken charge distribution symmetry relative to the rib bon center due to an\nunequal occupation of up and down spin bands. All mean-field b ands are invariant under k→−k. We show only the portion of the 1D BZ\nwithstatesclose tothe Fermilevel.\nsuffera largeenergypenaltyduetothe neutralsolutionban d-\ngapandhavelowerenergywhenspin-polarized. Theresultin g\nhalf metallic solution in the spin-collinear scheme implie s a\nnon-zerooverallspinpolarizationinthesystemandisacco m-\npanied by a breakingof chargedistributionsymmetry around\nthe ribbon center. This asymmetric charge distribution is a\ncombinedeffect of the net spin polarizationand the charact er\nof the AF solution at the neutrality point, in which electron s\nwith opposite spin polarizationsare concentratedon oppos ite\nedges[19]. Ifbothupanddownspinbandswereequallyoccu-\npiedtherewouldbenochargedistributionasymmetryaround\ntheribboncenter.\nIn the low doping regime a non-collinear spin-order that\ncontinuously bridges the intermediate situation between t he\nneutralAFconfigurationandmostlyFconfigurationathigher\ndoping is [26] a possibility. In the version of the Hubbard\nmodel mean-field theory which allows for non-collinear spin\ndenisties we must allow for the possibility that the average\nspin polarization on different lattice sites points in diff erent\ndirections [38]. This allows a larger variational space wit hin\nasingleSlaterdeterminantapproximationandcanpotentia lly\nleadtolowerenergysolutions,butthespinlabelbecomesun -\ndefined for each single-particle wave function. We verifiedthatnon-collinearspinsolutionsarefavoredenergetical ly[26]\nin the Hubbardmodel calculationsfor low dopingregionand\nthatthetransitiontoFconfigurationhappensatdopingdens i-\nties typically about 20% higher than when only collinear so-\nlutions are considered. The angle between the spin densitie s\non opposite edges and the band structure of the non-collinea r\nstate arerepresentedin Fig. 5.\nIn the intermediate doping regime the total energy is min-\nimized by solutions which are more similar to the F neutral-\nribbon configuration[19] which do not have an energy gap,\nand are thereforefavored by doping. This transition to F-li ke\nsolutions occurs already at a relatively small value of dopi ng\nδn≃0.06. The states that are occupied first at finite dop-\ning are those near the valley points |k|=2π/3athat are[19]\nspread across the ribbon and therefore control the exchange\ncouplingbetween opposite edges. The W-scaling rulesof the\nenergy bands near the valley points [19] are consistent with\ntheW−1decaylaw of the thresholddopingat which the tran-\nsition to F-like transition occurs in our numerical phase di a-\ngram. InelectronicstructureswithdominantlyFcharacter the\nchargedistributionsymmetryaroundtheribboncenterispr e-\nserved. In this case every occupied states, up or down spin\nand valenceor conductionedge band,has a symmetric distri-4\nnlocal \n↓nlocal \n↑\nE= 0 δn= 0 .05 \ny(˚A)nlocal ↑/↓(y)\n16 12 8400.6 \n0.5 \n0.4 NC δn= 0 .05 \nka E(eV )\nπ 2π/3 π/20.2 \n0\n-0.2 \n-0.4 NC δn= 0 .05 \nka E(eV )\nπ 2π/3 π/20.2 \n0\n-0.2 \n-0.4 \ny(˚A)nlocal ↑(y)−nlocal ↓(y)\n16 12 8 4 00.2 \n0.1 \n0δn= 0 .05 \nFIG. 5: (Color online) In the weakly doped region canted spin ori-\nentations develop in order to minimize the total energy when non-\ncollinear solutions are allowed. Upper row. Band structure and\nspinresolvedelectronoccupationper latticeforazigzagr ibbonwith\nN=8 andδn=0.05. The occupation and spin polarization at each\nlattice site are represented in a local frame where the spin i s diago-\nnal.Lowerrow. Spinpolarizationandrelativeorientationofthe spin\ndirection between different lattice sites represented wit h the arrow\nheads.\nbutionofelectrondensityaroundtheribboncenter. Whenth e\ndopingissufficientlylarge,however,we finda brokencharge\nsymmetry solution that we label as Fb. In this state one of\ntheoccupiedconductionbandshasAF(unbalancedacrossthe\nribbon) rather than F (balanced across the ribbon) characte r.\nIn addition to these solutions, we find that for wide ribbons\nthere is an intermediate doping region in which AF solutions\nhavelower total energythanthe F solutionsbeforethe Fb so-\nlutionisstabilized. Thedifferenceinenergybetweendiff erent\nmagneticsolutionsissmall atintermediateandlargedopin g.\nInthehighdopingregimethe magneticfeaturesofthe sys-\ntem progressively disappear as the edge state bands become\nfilled. The ribbon is found to turn paramagneticabove a crit-\nical value that increases with the ribbon width and saturate s\naroundδnc∼0.7. Considering that edge localized states in\nthe conduction bands with k-points near 2 π/3a≤ |k| ≤π/a\nspan approximately 1 /3 of the whole Brillouin zone we find\nthat the total amountof dopingelectronsrequiredto fill com -\npletelytheedgeforbothupanddownspinsis2 /3,anamount\nthat can be surpassed near the mentioned doping saturation\nlimit.DISCUSSION\nThe AF state of zigzag nanoribbons has the unusual fea-\nture that inversion symmetry across the ribbon is broken in\noppositesensesinthetwospinsub-systems[17,19]. Ourcal -\nculationsuggestthatinlowdopingregimethesystemcaneas -\nily develop solutions with a charge density that is distribu ted\nasymmetrically across the ribbon, creating an interesting and\nunusually strong type of multiferroic behavior [17, 39] in\nwhich spin polarization and charge density are coupled. We\nexpect that transport properties can correspondingly be ma -\nnipulated in interesting interrelated ways by both externa l\nmagneticfieldsandexternalelectric fieldsdirectedacross the\nribbon. Edgetransportshouldbe stronglysuppressed,fore x-\nample, when a transverse electric field is applied which has\noppositeorientationsonoppositeendsofa ribbon.\nAbove a certain critical doping density, which is inversely\nproportional to the ribbon width W−1, we find that the sys-\ntem undergoes a transition to a F configuration in which op-\nposite edgeshave parallel spin polarizations. Whendoping is\nincreased further the spin-configuration is altered yet aga in,\nrestoring inversion symmetry breaking across the ribbon. I n\nthis high-doping regime the total magnetic condensation en -\nergy is small and the energy differences between different\nmagnetic configurations is small. Eventually at sufficientl y\nhigh doping the Hubbard-model SCF equations have only\nparamagneticsolutions.\nTheSiO2substrates on which exfoliatedgraphenesamples\nare usually prepared have electron density inhomogeneitie s\n[40] of the order of nfluc∼1011cm−2that can extend over\nlengths of the order of L≃1µm. In the limiting case of rib-\nbonswiththissamewidthasthepuddlesizesaroughestimate\nofdopingperunitlatticeconstant awithineachpuddlecanbe\nevaluatedwiththeproductofthese twoquantities\nδnfluc∼L·nfluc∼0.25/a\nThis amount of doping can influence the spin configurations\nin the system and the presence of these randomperturbations\nis therefore expected to appreciably weaken the tendency to -\nwards magnetic order, especially for wide ribbons. For this\nreason we should expect a better chances of detecting edge\nmagnetism in suspended ribbons which have much weaker\nelectrondensityfluctuations.\nThe long-rangedcharacter of the Coulomb interaction, ne-\nglectedinthepresentwork,isexpectedtointroduceimport ant\nchangesin the details of electronic structure especially i n the\nregions in which states with different charge and spin con-\nfigurations compete closely. The discrepancies can be more\nacutethanintheneutralcasebecausetheinadequacyofshor t\nranged screening can be more relevant when the occupation\nofeachlatticesiteintheunitcellbecomesinhomogeneousa s\nwe depart from half filling. Nevertheless, it is also likely t hat\nseveral qualitative features of the solutions are still cor rectly\ncapturedbytheHubbardmodelandthereforecanprovideuse-\nfulhintsontheactualbehaviorofthemagneticconfiguratio ns5\nin ribbons as a function of doping. 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Sme t,\nK. vonKlitzing,and A.Yacobi, Nature Phys.4, 144 (2008)."    },    {        "title": "0905.1442v1.On_the_density_matrix_for_the_kink_ground_state_of_higher_spin_XXZ_chain.pdf",        "content": "arXiv:0905.1442v1  [cond-mat.stat-mech]  11 May 2009Typeset with jpsj2.cls <ver.1.2.2b > Short Note\nOn the density matrix for the kink\nground state of higher spin XXZ chain\nKohei Motegi\nInstitute of Physics, University of Tokyo, Meguro-ku, Toky o\n153\nKEYWORDS: XXZ chain, kink ground state, correlation func-\ntion\nThe exact computation of the correlation functions\nof 1D quantum integrable models has been one of the\nchallenging problems. For the spin-1/2 XXZ chain, the\ncorrelation functions in the antiferromagnetic regime\nwere found to be expressed in the multiple integral\nform.1,2)However, the exact evaluation of them is still\na hard work, and has been only successful for some spe-\ncial anisotropy parameters.3)\nIn theferromagnetic regime, there is a class of non-\ntranslationally invariant ground state which should be\ncalled as kinkground state4,5). We have studied the cor-\nrelation function of the kinkground state, and exactly\ncalculated the density matrix6)(see also ref 7).\nThe kink ground state also exists for arbitrary spin\nand dimension. In this paper, we extend the analysis de-\nveloped in our previous paper to higher spin 1D XXZ\nchain, and calculate the density matrix (note that the\nmodel we consider in this paper is not the integrable\nhigher spin XXZ chain8).\nTheHamiltonianofthe spin S1Dinfinite XXZ chain\nis\nH=−/summationdisplay\nm∈Z(Sx\nmSx\nm+1+Sy\nmSy\nm+1+∆(Sz\nmSz\nm+1−S2)).\n(1)\nWe shall consider the ferromagnetic regime ∆ >1. For\nlater convenience, we parametrize ∆ as ∆ = ( q1\n2+\nq−1\n2)/2.Then ∆ >1 corresponds to 0 < q <1. A kink\nground state is the superposition of kinks which have\nthe same center. For a (normalized) kink state ⊗x∈Z|mx/an}bracketri}ht\n(mx∈ {−S,···,S}), the center j−1/2 (j∈Z) is the\nposition where\n/summationdisplay\nx<j(S−mx) =/summationdisplay\nx>j(mx+S),\nholds. Let us denote the kink ground state whose center\nis atj−1\n2by|Ψj/an}bracketri}ht, and introduce the generatingfunction\nof|Ψj/an}bracketri}ht9)\n|Ψ(z)/an}bracketri}ht=/circlemultiplydisplay\nx∈Z<0(S/summationdisplay\nmx=−S(zq1\n2(1\n2+x))mx−Sc(mx)1\n2|mx/an}bracketri}ht)⊗/circlemultiplydisplay\ny∈Z≥0(S/summationdisplay\nmy=−S(zq1\n2(1\n2+y))my+Sc(my)1\n2|my/an}bracketri}ht),\n(2)\nwherec(mx) = (2S)!/(S−mx)!(S+mx)!.|Ψj/an}bracketri}htis the\ncoefficient of zjof the expansion of |Ψ(z)/an}bracketri}ht, i.e,\n|Ψ(z)/an}bracketri}ht=/summationdisplay\nj∈Zzj|Ψj/an}bracketri}ht. (3)\nLet us focus on one of the kink ground states |Ψ0/an}bracketri}ht, and\ncalculate the density matrix\n/an}bracketle{tn/productdisplay\nj=1Eǫ′\njǫj\nxj/an}bracketri}ht:=/an}bracketle{tΨ0|/producttextn\nj=1Eǫ′\njǫj\nxj|Ψ0/an}bracketri}ht\n/an}bracketle{tΨ0|Ψ0/an}bracketri}ht,(4)\nwhereEǫ′\nxǫx\nx|mx/an}bracketri}ht=δmx,ǫx|ǫ′\nx/an}bracketri}ht.xjis the position of the\nsite where the operator Eǫ′\njǫj\nxjacts on, and is assumed\nto bexj/ne}ationslash=xkforj/ne}ationslash=k. We only consider the case\nwhen/summationtextn\nj=1(ǫj−ǫ′\nj) = 0 is satisfied: otherwise, the\ndensity matrix is zero. We calculate /an}bracketle{tΨ(z)|Ψ(z)/an}bracketri}htand\n/an}bracketle{tΨ(z)|/producttextn\nj=1Eǫ′\njǫj\nxj|Ψ(z)/an}bracketri}htto obtain the exact expression\nof the density matrix since\n/an}bracketle{tΨ(z)|Ψ(z)/an}bracketri}ht=∞/summationdisplay\nj=−∞z2j/an}bracketle{tΨj|Ψj/an}bracketri}ht, (5)\n/an}bracketle{tΨ(z)|n/productdisplay\nj=1Eǫ′\njǫj\nxj|Ψ(z)/an}bracketri}ht=∞/summationdisplay\nj=−∞z2j/an}bracketle{tΨj|n/productdisplay\nj=1Eǫ′\njǫj\nxj|Ψj/an}bracketri}ht,\n(6)\nholds, and /an}bracketle{tΨ0|Ψ0/an}bracketri}ht,/an}bracketle{tΨ0|/producttextn\nj=1Eǫ′\njǫj\nxj|Ψ0/an}bracketri}htcan be ex-\ntracted from /an}bracketle{tΨ(z)|Ψ(z)/an}bracketri}htand/an}bracketle{tΨ(z)|/producttextn\nj=1Eǫ′\njǫj\nxj|Ψ(z)/an}bracketri}ht,\nwhich are easier to calculate. Let us first calculate\n/an}bracketle{tΨ(z)|Ψ(z)/an}bracketri}ht. Using the Jacobi triplet product identity\n(q;q)∞(−xq1\n2;q)∞(−x−1q1\n2;q)∞=∞/summationdisplay\nj=−∞xjqj2\n2,(7)\nwe have\n/an}bracketle{tΨ(z)|Ψ(z)/an}bracketri}ht=(−wq1\n2;q)2S\n∞(−w−1q1\n2;q)2S\n∞\n=1\n(q;q)2S∞/parenleftBig∞/summationdisplay\nj=−∞wjqj2\n2/parenrightBig2S\n=1\n(q;q)2S∞∞/summationdisplay\nj=−∞Ajwj, (8)\nwherew=z2, (a;q)∞:=/producttext∞\nj=0(1−aqj) and\nAj=/summationdisplay\nPjk=j,jk∈Z2S/productdisplay\nk=1qj2\nk\n2.\n1J. Phys. Soc. Jpn. Short Note\nUsing eq. (7) and the following identity6)\nn/productdisplay\nj=11\n1+xuj=∞/summationdisplay\nj=0(−x)jn/summationdisplay\nl=1uj+n−1\nl/producttext\ni/negationslash=l(ul−ui),(9)\n/an}bracketle{tΨ(z)|/producttextn\nj=1Eǫ′\njǫj\nxj|Ψ(z)/an}bracketri}htcan be calculated as\n/an}bracketle{tΨ(z)|n/productdisplay\nj=1Eǫ′\njǫj\nxj|Ψ(z)/an}bracketri}ht\n=n/productdisplay\nj=1{(wζj)[ǫ′\njǫj]c(ǫj)c(ǫ′\nj)}n/productdisplay\nj=11\n(1+wζj)2S\n×(−wq1\n2;q)2S\n∞(−w−1q1\n2;q)2S\n∞\n=1\n(q;q)2S∞n/productdisplay\nj=1{(wζj)[ǫ′\njǫj]c(ǫj)c(ǫ′\nj)}\n×∞/summationdisplay\nj=−∞/parenleftBig/summationdisplay\nj1∈Z,j2∈Z≥0\nj1+j2=jAj1Bj2/parenrightBig\nwj, (10)\nwhereζj=q1\n2+xj,/bracketleftbig\nǫ′\njǫj/bracketrightbig\n= (ǫj+ǫ′\nj+2S)/2 and\nBj=/summationdisplay\nPjk=j,jk∈Z≥02S/productdisplay\nk=1/braceleftBigg\n(−1)jkn/summationdisplay\nl=1ζjk+n−1\nl/producttext\ni/negationslash=l(ζl−ζi)/bracerightBigg\n.\nFrom eqs. (8) and (10), one has\n/an}bracketle{tΨ0|Ψ0/an}bracketri}ht=A0\n(q;q)2S∞, (11)\n/an}bracketle{tΨ0|n/productdisplay\nj=1Eǫ′\njǫj\nxj|Ψ0/an}bracketri}ht=1\n(q;q)2S∞n/productdisplay\nj=1{ζ[ǫ′\njǫj]\njc(ǫj)c(ǫ′\nj)}\n×/summationdisplay\nj1∈Z,j2∈Z≥0\nj1+j2=−Pn\nj=1[ǫ′\njǫj]Aj1Bj2,(12)\nleading to the exact expression of the density matrix\n/an}bracketle{tn/productdisplay\nj=1Eǫ′\njǫj\nxj/an}bracketri}ht=n/productdisplay\nj=1{ζ[ǫ′\njǫj]\njc(ǫj)c(ǫ′\nj)}\n×/summationdisplay\nj1∈Z,j2∈Z≥0\nj1+j2=−Pn\nj=1[ǫ′\njǫj]Aj1Bj2\nA0.(13)\nOne can check that for S= 1/2, eq. (13) recovers the\nresult in ref. 6. However, the expression of eq. (13) gets\nmore complicated as the spin becomes higher.\nConcentratingon the S= 1 case,we can derive some\nslightly easier expression by using\nn/productdisplay\nj=11\n(1+xuj)2=∞/summationdisplay\nj=0(−x)jXn,j,Xn,j= lim\nui+n→ui,i=1,···n2n/summationdisplay\nl=1uj+2n−1\nl/producttext\ni/negationslash=l(ul−ui),\nwhich is a special case of eq. (9). The result is\n/an}bracketle{tn/productdisplay\nj=1Eǫ′\njǫj\nxj/an}bracketri}ht=n/productdisplay\nj=1{ζ[ǫ′\njǫj]\njc(ǫj)c(ǫ′\nj)}\n×∞/summationdisplay\nj=0(−1)jXn,jq1\n4(j+Pn\nk=1[ǫ′\nkǫk])2\n×(δeven\nj+Pn\nk=1[ǫ′\nkǫk]+Cδodd\nj+Pn\nk=1[ǫ′\nkǫk]),(14)\nwhereC= 2/summationtext∞\nk=1q(k−1\n2)2/(1 + 2/summationtext∞\nk=1qk2). From eq.\n(14), the magnetization and the spin-spin correlation\nfunctions can be easily calculated.\n/an}bracketle{tSz\nx/an}bracketri}ht=∞/summationdisplay\nj=0(−1)j(j+1)ζjqj2\n4\n×(ζ2qj+1−1)(δeven\nj+Cδodd\nj), (15)\n/an}bracketle{tSz\nx1Sz\nx2/an}bracketri}ht=∞/summationdisplay\nj=0(−1)jX2,j(δeven\nj+Cδodd\nj)\n×(ζ2\n1ζ2\n2q(j+4)2\n4+qj2\n4−(ζ2\n1+ζ2\n2)q(j+2)2\n4),(16)\n/an}bracketle{tS+\nx1S−\nx2/an}bracketri}ht= 4ζ1\n2\n1ζ1\n2\n2∞/summationdisplay\nj=0(−1)jX2,j\n×{(δeven\nj+Cδodd\nj)(ζ1+ζ2)q(j+2)2\n4\n+(δodd\nj+Cδeven\nj)(q(j+1)2\n4+ζ1ζ2q(j+3)2\n4)},\n(17)\nwhere\nX2,j=(j+1)(ζj+3\n1−ζj+3\n2)−(j+3)ζ1ζ2(ζj+1\n1−ζj+1\n2)\n(ζ1−ζ2)3.\nFrom these expressions, one can easily show that the\nspin-spin correlation functions decay exponentially for\nlarge distances.\n/an}bracketle{tSz\nx1Sz\nx2/an}bracketri}ht−/an}bracketle{tSz\nx1/an}bracketri}ht/an}bracketle{tSz\nx2/an}bracketri}ht ∼Azz(x1)qx2+1\n2forx2≫1,\n(18)\nAzz(x1) = 2∞/summationdisplay\nj=0(−1)jqj2\n4(1−ζ2\n1qj+1)\n×(jζj−1\n1+(j+1)q1\n4Cζj\n1)(δeven\nj+Cδodd\nj),\n/an}bracketle{tS+\nx1S−\nx2/an}bracketri}ht ∼A+−(x1)q1\n2(x2+1\n2)forx2≫1, (19)\nA+−(x1) = 4ζ1\n2\n1∞/summationdisplay\nj=0(−ζ1)j(j+1)\n2/3J. Phys. Soc. Jpn. Short Note\n×{(δeven\nj+Cδodd\nj)ζ1q(j+2)2\n4+(δodd\nj+Cδeven\nj)q(j+1)2\n4}.\nAcknowledgment\nThe author thanks K. Sakai for useful discussions\nand comments on this work. This work was partially\nsupported by Global COE Program (Global Center of\nExcellence for Physical Sciences Frontier) from MEXT,\nJapan.\n1) M. Jimbo, K. Miki, T. Miwa and A. Nakayashiki, Phys. Lett.A168(1992) 256.\n2) N. Kitanine, J.M. Maillet, N.A. Slavnov and V. Terras, J Ph ys.\nA35(2002) L385.\n3) J. Sato and M. Shiroishi, Nucl.Phys. B 729(2005) 441.\n4) C.-T. Gottstein and R.F. Werner, e-print cond-mat/95011 23.\n5) F.C. Alcaraz, S.R. Salinas and W.F. Wreszinski, Phys. Rev .\nLett.75(1995) 930.\n6) K. Motegi and K. Sakai, Phys. Rev. E 79, (2009) 031108.\n7) R. Dijkgraaf, D. Orlando and S. Reffert, Nucl. Phys. B 811\n(2009) 463.\n8) L. A. Takhtajan, Phys. Lett. A 87(1982) 479.\n9) T. Komaand B.Nachtergaele, Adv.Theor.Math.Phys.2(199 8)\n533.\n3/3"    },    {        "title": "0905.2312v1.Spatial_distribution_of_local_density_of_states_in_vicinity_of_impurity_on_semiconductor_surface.pdf",        "content": "arXiv:0905.2312v1  [cond-mat.mes-hall]  14 May 20096 pages, 4 figures\nSpatial distribution of local density of states in vicinity of impurity on semiconductor\nsurface\nV.N.Mantsevich∗and N.S.Maslova†\nMoscow State University, Department of Physics, 119991 Mos cow, Russia\n(Dated: December 4, 2018)\nWe present the results of detailed theoretical investigati ons of changes in local density of total\nelectronic surface states in 2 Danisotropic atomic semiconductor lattice in vicinity of im purity atom\nfor a wide range of applied bias voltage. We have found that ta king into account changes in density\nof continuous spectrum states leads to the formation of a dow nfall at the particular value of applied\nvoltage when we are interested in the density of states above the impurity atom or even to a series\nof downfalls for the fixed value of the distance from the impur ity. The behaviour of local density of\nstates with increasing of the distance from impurity along t he chain differs from behaviour in the\ndirection perpendicular to the chain.\nPACS numbers: 71.55.-i\nKeywords: D. Non-equilibrium effects; D. Many-particle int eraction; D. Tunneling nanostructures\nI. INTRODUCTION\nInfluence of different impurities on the semiconduc-\ntor local density of surface states was widely studied\nexperimentally and theoretically. Most of the experi-\nments were carried out with the help of scanning tun-\nneling microscopy/spectroscopy technique [1], [2], [3].\nTheoretical investigations of single impurities and clus-\nters influence on density of surface states deals with\nGreen’s functions formalism [4], [5] or based on the\ntotal-energy density-functional calculations using first-\nprinciple pseudo-potential [6]. Numerical calculations\nbased on the tight-binding model are also carried out\n[7].\nMost of the theoretical calculations don’t take into ac-\ncount modification of local density of continuous spec-\ntrum states due to the influence of impurity atom on\nsemiconductor surface which we consider to play an im-\nportant role in the formation of peculiarities in the local\ndensity of total surface states . So in the present work\nwe suggest a simple model of anisotropic atomic lattice\nwith impurity atom and we pay special attention to the\nchanges of local density of continuous spectrum states.\nThis model suits well for theoretical investigation of π-\nbonded chains on the reconstructed Ge or Si surfaces\n[8]. It can be also used for investigation of the sublat-\ntices on the cleaved planes of AIIIBVsemiconductors.\nWe have found that the view of local density of surface\nstates (amount of downfalls and their shape) strongly\ndiffers depending on the value of the distance from the\nimpurity atom position and from the direction of obser-\nvation (along the atomic chain or perpendicular to the\natomic chain). It will be shown that downfall in the res-\n∗vmantsev@spmlab.phys.msu.ru\n†Electronic address: spm@spmlab.phys.msu.ruonance when we are interested in the density of states\nabove the impurity atom can transforms to a series of\ndownfalls or even to a peak for different values of the\ndistance from the impurity.\nII. THE SUGGESTED MODEL AND MAIN\nRESULTS\nWe shall analyze 2 Danisotropic atomic lattice formed\nby the similar atoms with energy levels ε1and similar\ntunneling transfer amplitudes between the atoms talong\nthe atomic chain. The interaction between atomic chains\nis described by tunneling amplitude T, which has the\nsame value for all the similar atoms in the chain (Fig. 1).\nDistance between the atoms in the atomic chain is equal\ntoa, distance between the atoms in the neighboring\nchains is equal to b. Atomic lattice includes impurity\natom with energy level εd, tunneling transfer amplitude\nfrom impurity atom to the nearest atoms in the atomic\nchainτand to the nearest atoms in the neighbor chains\nℑ.\nThe model system can be described by the Hamilto-\nnian:ˆH:\nˆH=ˆH0+ˆHimp+ˆHtun\nˆH0=/summationdisplay\niεic+\nici+/summationdisplay\n<i,j>tc+\nicj+/summationdisplay\n<k,l>Tc+\nkcl+h.c.\nˆHtun=/summationdisplay\ni,dτc+\nicd+/summationdisplay\nk,dℑc+\nkcd+h.c.\nˆHimp=/summationdisplay\ndεdc+\ndcd\n(1)\nˆH0is a typical Hamiltonian for atomic lattice with hop-\npings without any impurities. ˆHtundescribes transitions2\nbetween impurity atom and neighboring atoms of the\natomic lattice. ˆHimpcorresponds to the electrons in the\nlocalizedstateformedbytheimpurityatomintheatomic\nchain.\nIndexes i,j correspond to the direction along the chain;\nindexes k,l correspond to the direction perpendicular to\nthe chain.\nFIG. 1: Schematic diagram of 2 Datomic lattice with impu-\nrity atom.\nWe shall use diagramm technique in our investigation\nof atomic chain local density of states.\nThe dependence of local density of states on the dis-\ntance alongthe atomic chainand in the directionperpen-\ndicular to the atomic chain in the presence of impurity\natom is described by the equation:\nρ(ω,/vector r) =−1\nπSp/parenleftBig\nIm/summationdisplay\n/vector κ,/vector κ1ˆGR(/vector κ, /vector κ1,ω)ei/vector κ/vector rei/vector κ1/vector r/parenrightBig\n(2)\nWhere/vector r= (x,y),/vector κ= (κx,κy) and/vector κ1= (κx1,κy1).\nGreen function ˆGR(/vector κ, /vector κ1,ω) corresponds to the electron\ntransition from the impurity to the semiconductor con-\ntinuum states and can be found from the system of equa-\ntions:\nGR\nκx0dd=G0R\nκx0κx0τGR\ndd+G0R\nκx0κx0T/summationdisplay\nkyGR\nκxκydd\nGR\n0κydd=G0R\n0κy0κyℑGR\ndd+G0R\n0κy0κyt/summationdisplay\nkxGR\nκxκydd\nGR\ndd=G0R\ndd+G0R\nddτ/summationdisplay\nkxGR\nκx0dd+G0R\nddℑ/summationdisplay\nkyGR\n0κydd\nGR\n/vector κdd=G0R\n/vector κ/vector κτGR\nκx0dd+G0R\n/vector κ/vector κℑGR\n0κydd\nGR\n/vector κ/vector κ1=G0R\n/vector κ/vector κ+G0R\n/vector κ/vector κτGR\ndd/vector κ1+G0R\n/vector κ/vector κℑGR\ndd/vector κ1\nWhere zero Green function is evaluated for the 2 D\natomic lattice without any impurities and has the form:\nG0R\nκxκy(ω) =1\nω−ε1−2t·cos(kxa)−2T·cos(kyb)\n(3)\nSubstituting the expression for Green function\nˆGR(/vector κ, /vector κ1,ω) obtained from system into equation ( 2) and\nperforming summarization over wave vectors kxandkx1(kyandky1) we get the final expression for the local den-\nsity of continuous spectrum states along (perpendicular)\nthe atomic chain ρvolume(x) (ρvolume(y)):\nρvolume(ω,x) =ρ0(ω)·(ω−εd)2+γ2·(1−f(2kx(ω)x)\n(ω−εd)2+γ2\nρvolume(ω,y) =ρ0(ω)·(ω−εd)2+γ2·(1−f(2ky(ω)y)\n(ω−εd)2+γ2(4)\nWhere parameter γ= (τ2+ℑ2)·ρ0(ω) corresponds\nto relaxation rate of electron distribution at the local-\nized state formed by impurity atom, ρ0(ω) is a local den-\nsity of states for the atomic chain without any impuri-\nties. Functions f(2kx(ω)x) andf(2ky(ω)y) are periodi-\ncal and have the property: f(2kx(ω)x) = 1 ifx= 0 and\nf(2ky(ω)y) = 1 if y= 0. If both directions are equiv-\nalentf(2kx(ω)) =f(2ky(ω)y) =J0(2kx(ω)).Expression\nforkx(ω) orky(ω) can be found from the dispersion law\nof the 2Datomic lattice which has the form.\nω(kx,ky) = 2t·cos(kxa)+2T·cos(kyb)\n(5)\nImpurity atom density of states has lorentzian form\nline shape and can be evaluated as:\nρimpurity(ω) =−1\nπ·Im/summationdisplay\ndGR\ndd(ω) =γ2\n(ω−εd)2+γ2\n(6)\nLocaldensityoftotalsurfacestatesistheresultofsum-\nmarization between local density of continuous spectrum\nstates and impurity atom density of states.\nρ(ω) =ρvolume(ω)+ρimpurity(ω)\n(7)\nLet’s start from the 1 Dcase of the atomic chain. In this\ncase it is necessary to put in the Hamiltonian: b= 0,\nT= 0 and ℑ= 0. Final expression for the local density\nof continuous spectrum states ρvolumewill have the form:\nρvolume(ω) =��0(ω)·(ω−εd)2+γ2·(1−cos(2k(ω)r)\n(ω−εd)2+γ2\n(8)\nExpressionfor k(ω) can be found from the dispersion law\nof the 1Datomic chain.\nTypical numerical results for local density of total\nsurface states and local density of continuous spectrum\nstates calculated above the impurity atom in 1 Dcase\n(distance value is equal to zero ( r= 0)) are shown on\n(Fig. 2a,b). For the local density of continuous spectrum\nstates (Fig. 2a) a downfall exist in the resonance when\nenergy is equal to the impurity atom energy level depo-\nsition (ω=εd). Width of the downfall depends on the3\nFIG. 2: a) Local density of continuous spectrum states in the case of the distance from the impurity atom along the atomic\nchain equal to zero. b) Local density of surface spectrum sta tes in the case of the distance from the impurity atom along th e\natomic chain equal to zero. c)-f) Local density of continuou s spectrum states (black line) and local density of surface s pectrum\nstates (grey line) for the different values of the distance fr om the impurity atom along the atomic chain. For all the figure s\nvalues of the parameters a= 1,t= 1,5,εd= 0,6 are the same.\nparameters of the atomic chain, such as relaxation rate\nor tunneling transfer amplitude, it rises with the increas-\ning of tunneling transfer amplitude from impurity atom\nto the neighbor atoms of the atomic chain. Local den-\nsity of surface states (Fig. 2b) has lorentzian form with\na downfall in the resonance. With the increasing of re-\nlaxation rate (increasing of τ) the downfall depth at the\ntop of the peak decreases, resonance peak shape spreads\nand it’s amplitude falls down.\nNow let’s start to analyze the dependence of local den-\nsity of continuous spectrum states and local density of\nsurfacestatesatthefixedvalueofthedistance ralongthe\natomicchain fromthe impurity atomposition (Fig. 2c-f).\nWe shall again start from the local density of continuous\nspectrum states (black lines on Fig. 2).\nWhen the value of a distance is not equal to zero a se-\nries of downfalls in the local density of continuous spec-\ntrum exists. Amount of downfalls increases with the in-\ncreasing of distance value and downfalls amplitude de-\ncreases when energy aspire to the edges of the band. The\nmost significant amplitude of the downfalls corresponds\nto the vicinity of the resonance region. It is clearly evi-\ndent that positions of the downfalls on the energy scale\ncan be found from the equation 2 πn= 2k(ω)rwhere\nnis an integer number. This means that numerator ofthe equation ( 8) is equal to zero. When the distance is\nnot equal to zero not only a downfall in the resonance\n(Fig. 2d,f) but also a peak (Fig. 2c,e) can exist in the\nlocal density of continuous spectrum states. At the fixed\nparameters of the atomic chain existance of a downfall\nor a peak in the resonance is determined by the value\nof the distance. Local density of surface states is shown\nby the grey line on Fig. 2c-f. Comparison between local\ndensity of surface states and local density of continuous\nspectrum states for the fixed value of rmake it clearly\nevident that impurity atom density of states can drasti-\ncally influence on the local density of continuous spec-\ntrum states. Result depends on the value of tunneling\ntransfer amplitude from impurity atom to the nearest\natoms of the chain. We have found that the most signifi-\ncant influence corresponds to the situation when tunnel-\ning transfer amplitude between the atoms in the chain\ntsignificantly exceeds tunneling transfer amplitude from\nimpurity atom to the atoms of the chain τ. In this case\ndownfall in the resonance becomes a peak with a small\ndownfallonthe top ofthe peak(Fig. 2e). So onlyonesig-\nnificant downfall exists in the resonance region and there\nare no downfalls at the energies different from the reso-\nnance value (Fig. 2c)in the local density of surface states,\nwhileinthecontinuousspectrumdensityofstatesaseries4\nFIG. 3: a) Local density of continuous spectrum states (blac k line) and local density of surface spectrum states (grey li ne) for\nthe different values of the distance from the impurity atom al ong the atomic chain. For all the figures values of the paramet ers\na= 1,b= 2,t= 1,5,T= 1,2,τ= 0,6,ℑ= 0,3,εd= 0,6 are the same.\nof downfalls exists. With increasing of τlocal density of\nsurface states differs from the local density of continuous\nspectrum states only by the amplitude of the downfalls\n(Fig. 2e,f). In this case number of downfalls is the same\nin comparison with the continuous spectrum density of\nstates, downfalls don’t change their shape or position on\nthe energy scale and contribution to the local density of\nsurface states from the impurity atom can be considered\nto be a background. This effect can be qualitatively un-\nderstoodbythefollowingway: withincreasingoftransfer\namplitude from impurity atom to the atom of the chain\nincreases relaxation rate and lorentzian form peak of im-\npurity atom density of states spreads and it’s amplitude\ndecreases.\nNow let’s start to analyze 2 Datomic lattice. Nu-\nmerical results for local density of total surface states\nand local density of continuous spectrum states calcu-\nlated above the impurity atom in perpendicular direc-\ntions (along the atomic chain and perpendicular to the\natomic chain) are shown on Fig. 3a, Fig. 4a. It’s clear\nthat in this case all the results are equal for both direc-\ntions and downfall in the continuous spectrum density\nof states or a peak in the local density of surface states\nposesjust thesamepropertiesasin thecaseof1 Datomic\nchain.\nLet’s analyze the situation when the value of the dis-tances from the impurity atom position along the atomic\nchain (Fig. 3b-f) and perpendicular to the atomic chain\n(Fig. 4b-f) are not equal to zero. We shall start from the\nlocal density of continuous spectrum states (black lines\non Fig. 3,4).\nIn this case number of downfalls,their shape and po-\nsition on the energy scale in each of the perpendicular\ndirections can be found from the dispersion law just in\nthe same way as for 1 Datomic chain. When the distance\nis not equal to zero not only a downfall in the resonance\n(Fig. 3d,e; Fig. 4e,f) but also a peak (Fig. 3c,f; Fig. 4c,d)\ncan exists in the local density of continuous spectrum\nstates.\nWe have found distance interval for both directions\n(along the atomic chain and perpendicular to the atomic\nchain) where exists replacement of a peak by a down-\nfall (Fig. 3c-f; Fig. 4c-f) in local density of continuous\nspectrum states (and also in the local density of sur-\nface states). Moreover peak in one direction can cor-\nresponds to a downfall in the another direction. Let’s\nanalyze this interval carefully. We shall start from the\ndistance value when peaks in the resonance in local den-\nsity ofcontinuousspectrum states forboth directions can\nbe seen ( x(y) = 15·a) (Fig. 3c; Fig. 4c). With increasing\nof the distance value ( x(y) = 20·a) a resonance peak in\nthe direction perpendicular to the atomic chain still ex-5\nFIG. 4: Local density of continuous spectrum states (black l ine) and local density of surface spectrum states (grey line ) for\nthe different values of the distance from the impurity atom pe rpendicular to the atomic chain. For all the figures values of the\nparameters a= 1,b= 2,t= 1,5,T= 1,2,τ= 0,6,ℑ= 0,3,εd= 0,6 are the same.\nists (Fig. 4d) and in the direction along the atomic chain\na downfall appears (Fig. 3d). Further increasing of the\ndistance value ( x(y) = 25·a) shows that in the direction\nalong the atomic chain a downfall still exists (Fig. 3e)\nand in the perpendicular direction a downfall substitutes\npeak (Fig. 4e). Finally when the value of distances in\nboth directions becomes equal to x(y) = 30·ain the per-\npendicular direction a downfall still exists (Fig. 4f) and\nin the direction along the atomic chain a resonance peak\ncan be seen (Fig. 3f).\nLocal density of total surface states is shown by the\ngrey lines on Fig. 3,4. It is clearly evident that for the\nstudied parametersof the system taking into account im-\npurity atom density of states slightly changes local den-\nsity of total surface states in comparison with local den-\nsity of continuous spectrum states for both directions. In\nthis case number of downfalls is the same in comparison\nwith the continuousspectrum densityofstates, downfalls\ndon’t change their shape or position on the energy scale.\nIII. CONCLUSION\nIn this work we have shown that taking into account\nchangesofthelocaldensityofcontinuousspectrumstatesformed by the presence of the impurity atom in the 2 D\nanisotropic atomic lattice or even in the 1 Datomic chain\nleads to significant modification of the total local density\nof surface states and consequently to the modification of\nSTS spectra. We have found that a downfall exists in\nthe STS spectra measured just above the impurity when\nimpurity atom energy level is equal to the applied bias\nvoltage. With changingofthe distancefromthe impurity\na series of downfalls is formed on the energy scale both in\nthelocaldensityofcontinuousspectrumstatesandin the\nlocal density of total surface states. Number of downfalls\nand their position are determined by the atomic lattice\ndispersion law. It was shown that at some values of the\ndistance from the impurity a peak can exist in the reso-\nnance region instead of a downfall. We have found that\nbehaivour of local density of surface states depends on\nthe direction of the observation. Switching on and off\nof impurity atom in both directions was found. This ef-\nfect can be well observed experimentally with the help of\nSTM/STS technique.\nThis work was supported by RFBR grants and by the\nNational Grants for technical regulation and metrology\n01.648.12.3017 and 154 −6/259/4−08.6\n[1] R. Dombrowski, C. Wittneven, M. Morgenstern et al.,\nAppl. Phys. A ,66, S203-S206, (1998)\n[2] J. Inglesfield, M. Boon, S. Crampin, Condens. Matter ,12,\nL489-L496, (2000)\n[3] K. Kanisawa, M. Butcher, Y. Tokura, Phys. Rev. Letters ,\n87, 196804, (2001)\n[4] N. Sivan, N. Wingreen, Phys. Rev. B ,54, 11622, (1996)\n[5] V. Madhavan, W. Chen, M. Crommie et al., Phys. Rev.B,64, 165412, (2001)\n[6] M. Qian, M. Gothelid, B. Johnsson, Phys. Rev. B ,66,\n155326, (2002)\n[7] F. Marczinowski ,J. Weibe, J. Tang et al., Phys. Rev. Let-\nters,99, 1572002, (2007)\n[8] K. Pandey, Phys. Rev. Letters ,47, 1913, (1981)"    },    {        "title": "0905.2766v1.Gaps_and_tails_in_graphene_and_graphane.pdf",        "content": "arXiv:0905.2766v1  [cond-mat.dis-nn]  17 May 2009Gaps and tails in graphene and graphane\nB. D´ ora1and K. Ziegler2\n1Max-Planck-Institut f¨ ur Physik Komplexer Systeme,\nN¨ othnitzer Str. 38, 01187 Dresden, Germany\n2Institut f¨ ur Physik, Universit¨ at Augsburg\nE-mail:klaus.ziegler@physik.uni-augsburg.de\nAbstract. We study the density of states in monolayer and bilayer graphene in t he\npresence of a random potential that breaks sublattice symmetrie s. While a uniform\nsymmetry-breaking potential opens a uniform gap, a random symm etry-breaking\npotential also creates tails in the density of states. The latter can close the gap again,\npreventing the system to become an insulator. However, for a suffi ciently large gap\nthe tails contain localized states with nonzero density of states. Th ese localized states\nallow the system to conduct at nonzero temperature via variable-r ange hopping. This\nresult is in agreement with recent experimental observations in gra phane by Elias et\nal..2\n1. Introduction\nGraphene is a single sheet of carbon atoms that is forming a honeyco mb lattice. A\ngraphene monolayer as well as a stack of two graphene sheets (i.e. a graphene bilayer)\nare semimetals with remarkably good conducting properties [1, 2, 3]. These materials\nhave been experimentally realized with external gates, which allow fo r a continuous\nchange in the charge-carrier density. There exists a non-zero min imal conductivity at\nthe charge neutrality point. Its value is very robust and almost una ffected by disorder\nor thermal fluctuations [3, 4, 5, 6].\nManypotentialapplicationsofgraphenerequireanelectronic gapt oswitch between\nconductingandinsulatingstates. Asuccessful stepinthisdirectio nhasbeenachieved by\nrecent experiments with hydrogenated graphene (graphane) [7] and with gated bilayer\ngraphene [8, 9, 10]. These experiments take advantage of the fac t that the breaking of\na discrete symmetry of the lattice system opens a gap in the electro nic spectrum at the\nFermi energy. In the case of monolayer graphene (MLG), a stagg ered potential that\ndepends on the sublattice of the honeycomb lattice plays the role of such symmetry-\nbreakingpotential(SBP).Forbilayergraphene(BLG)agatepote ntialthatdistinguishes\nbetween the two graphene layers plays a similar role.\nWith these opportunities one enters a new field in graphene, where o ne can switch\nbetween conducting and insulating regimes of a two-dimensional mat erial, either by a\nchemical process (e.g. oxidation or hydrogenation) or by applying a n external electric\nfield [11].\nTheopeningofagapcanbeobservedexperimentallyeitherbyadirec tmeasurement\nof the density of states (e.g., by scanning tunneling microscopy [12]) or indirectly by\nmeasuring transport properties. In the gapless case we observe a metallic conductivity\nσ∝ρD, whereDis the diffusion coefficient (which is proportional to the scattering\ntime)and ρisthedensityofstates(DOS).Thisgivestypicallyaconductivityoft heorder\nofe2/h. The gapped case, on the other hand, has a strongly temperatur e-dependent\nconductivity due to thermal activation of charge carriers [13]\nσ(T) =σ0e−T0/T(1)\nwith some characteristic temperature scale T0which depends on the underlying model.\nA different behavior was found experimentally in the insulating phase o f graphane [7]:\nσ(T)≈σ0e−(T0/T)1/3, (2)\nwhich is known as 2D variable-range hopping [14]. This behavior indicat es the\nexistence of well-separated localized states, even at the charge- neutrality point, where\nthe parameter T0depends on the DOS at the Fermi energy EFasT0∝1/ρ(EF).\nThe experimental observation of a metal-insulator transition in gra phane raises two\nquestions: (i) what are the details that describe the opening of a ga p and (ii) what is\nthe DOS in the insulating phase? In this paper we will focus on the mech anism of the\ngap opening due to a SBP in MLG and BLG. It is crucial for our study th at the SBP\nis not uniform in the realistic two-dimensional material. One reason fo r the latter is3\nthe fact that graphene is not flat but forms ripples [15, 16, 17]. An other reason is the\nincomplete coverage of a graphene layer with hydrogen atoms in the case of graphane\n[7]. The spatially fluctuating SBP leads to interesting effects, including a second-order\nphase transition due to spontaneous breaking of a discrete symme try and the formation\nof Lifshitz tails.\n2. Model\nQuasiparticles in MLG or in BLG are described in tight-binding approxima tion by a\nnearest-neighbor hopping Hamiltonian\nH=−/summationdisplay\n<r,r′>tr,r′c†\nrcr′+/summationdisplay\nrVrc†\nrcr+h.c., (3)\nwherec†\nr(cr) are fermionic creation (annihilation) operators at lattice site r. The\nunderlying lattice structure is either a honeycomb lattice (MLG) or t wo honeycomb\nlattices with Bernal stacking (BLG) [11, 18]. We have an intralayer ho pping rate tand\nan interlayer hopping rate t⊥for BLG. There are different forms of the potential Vr,\ndepending on whether we consider MLG or BLG. Here we begin with pot entials that are\nuniform on each sublattice, whereas random fluctuations are cons idered in subsection\n2.4.\n2.1. MLG\nVris a staggered potential with Vr=mon sublattice A and Vr=−mon sublattice\nB. This potential obviously breaks the sublattice symmetry of MLG. Such a staggered\npotential can be the result of chemical absorption of non-carbon atoms in MLG (e.g.\noxygen or hydrogen [7]). A consequence of the symmetry breaking is the formation of\na gap ∆ g=m: The spectrum of MLG consists of two bands with dispersion\nEk=±/radicalBig\nm2+ǫ2\nk, (4)\nwhere\nǫ2\nk=t2[3+2cos k1+4cos(k1/2)cos(√\n3k2/2)] (5)\nfor lattice spacing a= 1.\n2.2. BLG\nVris a biased gate potential that is Vr=m(Vr=−m) on the upper (lower) graphene\nsheet. The potential in BLG has been realized as an external gate v oltage, applied to\nthe two layers of BLG [8]. The spectrum of BLG consists of four band s [11] with two\nlow-energy bands\nE−\nk(m) =±/radicalbigg\nǫ2\nk+t2\n⊥/2+m2−/radicalBig\nt4\n⊥/4+(t2\n⊥+4m2)ǫ2\nk, (6)4\nwhereǫkis the monolayer dispersion of Eq. (5), and two high-energy bands\nE+\nk(m) =±/radicalbigg\nǫ2\nk+t2\n⊥/2+m2+/radicalBig\nt4\n⊥/4+(t2\n⊥+4m2)ǫ2\nk. (7)\nThe spectrum of the low-energy bands has nodes for m= 0 where E−\nk(0) vanishes\nin a (k−K)2manner, where Kis the position of the nodes, which are the same as\nthose of a single layer. For small m≪t⊥, a mexican hat structure develops around\nk=K, with local extremum in the low-energy band at E−\nk(m) =±m, and a global\nminimum/maximum inthe upper/lower low energy bandat E−\nk(m) =mt⊥//radicalbig\nt2\n⊥+4m2.\nFor small gating potential Vr=±mwe can expand E−\nk(m) under the square root\nnear the nodes and get\nE−\nk(m)∼ ±/radicalBig\n[1−4ǫ2\nkt⊥(t2\n⊥+4ǫ2\nk)−1/2]m2+E−\nk(0)2. (8)\nt⊥apparently reduces the gap. Very close to the nodes we can appro ximate the\nfactor in front of m2by 1 and obtain an expression similar to the dispersion of MLG:\nE−\nk(m)∼ ±/radicalbig\nm2+E−\nk(0)2. Here we notice the absence of the mexican hat structure\nin this approximation. The resulting spectra for MLG and BLG are sho wn in Fig. 1.\nPSfrag replacements\n|k|−KEk\nbilayermonolayer\nFigure 1. The energy spectra of MLG (blue) and BLG (red) are shown, with an d\nwithout a gap (dashed and solid line, respectively) for positive energ ies. Note the\ncharacteristic mexican hat structure of gapped BLG.\n2.3. Low-energy approximation\nThe two bands in MLG and the two low-energy bands in BLG represent a spinor-1/2\nwave function. This allows us to expand the corresponding Hamiltonia n in terms of\nPauli matrices σjas\nH=h1σ1+h2σ2+mσ3. (9)5\nNear each node the coefficients hjread in low-energy approximation [19]\nhj=i∇j(MLG), h1=∇2\n1−∇2\n2, h2= 2∇1∇2(BLG), (10)\nwhere (∇1,∇2) is the 2D gradient.\n2.4. Random fluctuations\nIn a realistic situation the potential Vris not uniform, neither in MLG nor in BLG,\nas discussed in the Introduction. As a result, electrons experienc e a randomly varying\npotential Vralong each graphene sheet, and min the Hamiltonian of Eq. (9) becomes a\nrandomvariableinspaceaswell. ForBLGitisassumed thatthegatevo ltageisadjusted\nat the charge-neutrality point such that in average mris exactly antisymmetric with\nrespect to the two layers: ∝angb∇acketleftm1∝angb∇acket∇ightm=−∝angb∇acketleftm2∝angb∇acket∇ightm.\nAt first glance, the Hamiltonian in Eq. (3) is a standard hopping Hamilto nian\nwith random potential Vr. This is a model frequently used to study the generic case of\nAnderson localization [20]. The dispersion, however, is special in the c ase of graphene\ndue to the honeycomb lattice: at low energies it consists of two node s (or valleys) K\nandK′[17, 19]. It is assumed here that randomness scatters only at small momentum\nsuch that intervalley scattering, which requires large momentum at least near the nodal\npoints (NP), is not relevant and can be treated as a perturbation. Then each valley\ncontributes separately to the DOS, and the contribution of the tw o valleys to the DOS\nρis additive: ρ=ρK+ρK′. This allows us to consider the low-energy Hamiltonian in\nEqs. (9), (10), even in the presence of randomness for each valle y separately. Within\nthis approximation the term mris a random variable with mean value ∝angb∇acketleftmr∝angb∇acket∇ightm= ¯mand\nvariance ∝angb∇acketleft(mr−¯m)(mr′−¯m)∝angb∇acket∇ightm=gδr,r′. The following analytic calculations will be\nbased entirely on the Hamiltonian of Eqs. (9),(10) and the numerical calculations on\nthe lattice Hamiltonian of Eq. (3). In particular, the average Hamilto nian∝angb∇acketleftH∝angb∇acket∇ightmcan be\ndiagonalized by Fourier transformation and is\n∝angb∇acketleftH∝angb∇acket∇ightm=k1σ1+k2σ2+ ¯mσ3 (11)\nfor MLG with eigenvalues Ek=±√\n¯m2+k2. For BGL the average Hamiltonian is\n∝angb∇acketleftH∝angb∇acket∇ightm= (k2\n1−k2\n2)σ1+2k1k2σ2+ ¯mσ3 (12)\nwith eigenvalues Ek=±√\n¯m2+k4.\n2.5. Symmetries\nLow-energy properties are controlled by the symmetry of the Ham iltonian and of the\ncorresponding one-particle Green’s function G(iǫ) = (H+iǫ)−1. In the absence of\nsublattice-symmetry breaking (i.e. for m= 0), the Hamiltonian H=h1σ1+h2σ2has a\ncontinuous chiral symmetry\nH→eασ3Heασ3=H (13)6\nwith a continuous parameter α, sinceHanticommutes with σ3. The term mσ3breaks\nthe continuous chiral symmetry. However, the behavior under tr ansposition hT\nj=−hj\nfor MLG and hT\nj=hjfor BLG in Eq. (10) provides a discrete symmetry:\nH→ −σnHTσn=H , (14)\nwheren= 1 for MLG and n= 2 for BLG. This symmetry is broken for the one-\nparticle Green’s function G(iǫ) by the iǫterm. To see whether or not the symmetry is\nrestored in the limit ǫ→0, the difference of G(iǫ) and the transformed Green’s function\n−σnGT(iǫ)σnmust be evaluated:\nG(iǫ)+σnGT(iǫ)σn=G(iǫ)−G(−iǫ). (15)\nFor the diagonal elements this is the DOS at the NP ρ(E= 0)≡ρ0in the limit ǫ→0.\nThus the order parameter for spontaneous symmetry breaking is ρ0. According to the\ntheory of phase transitions, the transition from a nonzero ρ0(spontaneously broken\nsymmetry) to ρ0= 0 (symmetric phase) is a second-order phase transition, and sho uld\nbe accompanied by a divergent correlation length at the transition p oint. Since our\nsymmetry is discrete, such a phase transition can exists in d= 2 and should be of Ising\ntype. A calculation, using the SCBA of ρ0, gives indeed a second-order transition at the\npoint where ρ0vanishes with a divergent correlation length ξfor the DOS fluctuations\nξ∼ξ0(m2\nc−¯m2)−1\nfor ¯m2∼m2\ncwith a finite coefficient ξ0[21]. Whether or not this transition is an\nartefact of the SCBA or represents a physical effect due to the a ppearence of two types\nof spectra (localized for vanishing SCBA-DOS and delocalized for non zero SCBA-DOS)\nis not obvious here and requires further studies.\n2.6. Density of states\nOur focus in the subsequent calculation is on the DOSof MLG and BLG. In the absence\nof disorder, the DOS of 2D Dirac fermions opens a gap ∆ ∝¯mas soon as a nonzero\nterm ¯mappears in the Hamiltonian of Eq. (9), since the low-energy dispersio n is\nEk=±√\n¯m2+k2for MLG and Ek=±√\n¯m2+k4for BLG, respectively (cf Fig. 2).\nHere we evaluate the DOS of MLG and BLG in the presence of a uniform gap. Given\nthe energy spectrum, the DOS is defined as\nρ(E) =/summationdisplay\nkδ(E−Ek). (16)\nBy using the MLG dispersion, this reduces to\nρ(E) =|E|Θ(|E|−m), (17)\nwhere Θ( x) is the Heaviside function. For BLG, this gives\nρ(E) =|E|\n2√\nE2−m2Θ(|E|−m), (18)7\n00.511.522.533.5400.511.522.533.54\nPSfrag replacements\nE/mmρ(E)\nBLGMLG\n0 1 2 3 4 5012345678910\nPSfrag replacements\nE/m\nmρ(E)\nBLG\nMLG\nt⊥=∞t⊥= 2m2t⊥=m\nE/mt⊥ρ(E)\nFigure 2. Density of states for a uniform symmetry-breaking potential for monolayer\ngraphene and bilayer graphene is shown in the left panel. The density of states for a\nuniform symmetry-breaking potential for BLG is shown for severa l values of t⊥. For\nsmallt⊥, the mexican hat structure influences the DOS by shifting the gap t o lower\nvalues, and by developing a kink at E=m.\nwhich are shown in Fig. 2. By retaining the full low-energy spectrum f or BLG, E−\nk, the\nDOS can still be evaluated in closed form, with the result\nρ(E) =|E|×\n\n(t2\n⊥+4m2)√\n(t2\n⊥+4m2)E2−t2\n⊥m2form >|E|>mt⊥√\nt2\n⊥+4m2/parenleftbigg\n(t2\n⊥+4m2)\n2√\n(t2\n⊥+4m2)E2−t2\n⊥m2+1/parenrightbigg\nfor|E|> m.(19)\nIn the limit of t⊥≫(E,m), this reduces to Eq. (18) after dividing it by t⊥, which was\nset to 1 in the low-energy approximation, and the DOS saturates to a constant value\nafter the initial divergence. For finite t⊥, however, the Dirac nature of the spectrum\nappears again, and the high energy DOS increases linearly even for t he BLG, similarly\nto the MLG case. For m= 0, and E≪t⊥, this lengthy expression gives\nρ(E≪t⊥) =t⊥\n2. (20)\nAn interesting question, from the theoretical as well as from the e xperimental point\nof view, appears here: What is the effect of random fluctuations ar ound ¯m? Previous\ncalculations, based on the self-consistent Born approximation (SC BA), have revealed\nthat those fluctuations can close the gap again, even for an avera ge SBP term ¯ m∝negationslash= 0\n[22]. Only if ¯ mexceeds a critical value mc(which depends on the strength of the\nfluctuations), an open gap was found in these calculations. This des cribes a special\ntransition from metallic to insulating behavior. In particular, the DOS at the Dirac\npointρ0vanishes with ¯ mlike a power law\nρ0(¯m)∼/radicalbig\n��m−m2\nc. (21)8\nThe exponent 1/2 of the power law is probably an artefact of the SC BA, similar to the\ncritical exponent in mean-field approximations.\n3. Self-consistent Born approximation\nThe average one-particle Green’s function can be calculated from t he average\nHamiltonian ∝angb∇acketleftH∝angb∇acket∇ightmby employing the self-consistent Born approximation (SCBA)\n[23, 24, 25]\n∝angb∇acketleftG(iǫ)∝angb∇acket∇ightm≈(∝angb∇acketleftH∝angb∇acket∇ightm+iǫ−2Σ)−1≡G0(iη,ms). (22)\nThe SCBA is also known as the self-consistent non-crossing approx imation in the Kondo\nand superconducting community. The self-energy Σ is a 2 ×2 tensor due to the spinor\nstructure of the quasiparticles: Σ = −(iησ0+msσ3)/2. Scattering by the random SBP\nproduces an imaginary part of the self-energy η(i.e. a one-particle scattering rate) and\na shiftmsof the average SBP ¯ m(i.e., ¯m→m′≡¯m+ms). Σ is determined by the\nself-consistent equation\nΣ =−gσ3(∝angb∇acketleftH∝angb∇acket∇ightm+iǫ−2Σ)−1\nrrσ3. (23)\nThe symmetry in Eq. (14) implies that with Σ also\nσnΣσn=−(iησ0−msσ3)/2 (24)\nis a solution (i.e. ms→ −mscreates a second solution).\nThe average DOS at the NP is proportional to the scattering rate: ρ0=η/2gπ.\nThis reflects that scattering by the random SBP creates a nonzer o DOS at the NP if\nη >0.\nNow we assume that the parameters ηandmsare uniform in space. Then Eq. (23)\ncan be written in terms of two equations, one for the one-particle s cattering rate ηand\nanother for the shift of the SBP ms, as\nη=gIη, m s=−¯mgI/(1+gI). (25)\nIis a function of ¯ mandηand also depends on the Hamiltonian. For MLG it reads with\nmomentum cutoff λ\nIMLG=1\n2πln/bracketleftbigg\n1+λ2\nη2+(¯m+ms)2/bracketrightbigg\n(26)\nand for BLG\nIBLG∼1\n4/radicalbig\nη2+(¯m+ms)2(λ∼ ∞). (27)\nA nonzero solution ηrequiresgI= 1 in the first part of Eq. (25), such that ms=−¯m/2\nfrom the second part. Since the integrals Iare monotonically decreasing functions for\nlarge ¯m, a real solution with gI= 1 exists only for |¯m| ≤mc. For both, MLG and BLG,\nthe solutions read\nη2= (m2\nc−¯m2)Θ(m2\nc−¯m2)/4, (28)9\na)\nb)E\nEDOS\nDOS\nFigure 3. Schematic shape of the density of states: full curves are the bulk\ndensity of states for uniform symmetry-breaking potential, dott ed curves represent the\nbroadening by disorder. The broadened density of states can ove rlap inside the gap\nfor ¯m < m c(a) or not for ¯ m > m c(b), depending on the average symmetry-breaking\npotential ¯ m.mcis given in Eq. (29).\nwhere the model dependence enters only through the critical ave rage SBP mc:\nmc=/braceleftBigg\n(2λ/√\ne2π/g−1)∼2λe−π/gMLG\ng/2 BLG. (29)\nmcis much bigger for BGL, a result which indicates that the effect of diso rder is much\nstrongerinBLG.Thisisalsoreflectedbythescatteringrateat ¯ m= 0whichis η=mc/2.\nA central assumption of the SCBA is a uniform self-energy Σ. The ima ginary\npart of Σ is the scattering rate η, created by the random fluctuations. Therefore, a\nuniformηmeans that effectively random fluctuations are densely filling the latt ice. If\nthe distribution of the fluctuations is too dilute, however, there is n o uniform nonzero\nsolution of Eq. (23). Nevertheless, a dilute distribution can still cre ate a nonzero DOS,\naswewill discuss inthefollowing: westudy contributions totheDOSdu etorareevents,\nleading to Lifshitz tails.\n4. Lifshitz tails\nInthesystem withuniformSBPthegapcanbedestroyed locallybyalo calchangeofthe\nSBPm→m+δmrdue to the creation of a bound state. We start with a translational-\ninvariant system and add δmron siter. To evaluate the corresponding DOS from the\nGreen’s function G= (H+iǫ+δmσ3)−1, using the Green’s function G0= (H+iǫ)−1\nwith uniform m, we employ the lattice version of the Lippmann-Schwinger equation [2 6]\nG=G0−G0TSG0= (1−G0Tr)G0 (30)\nwith the 2 ×2 scattering matrix\nTr= (σ0+δmrσ3G0,rr)−1σ3δmr. (31)10\nIn the case of MLG we have\nG0= [(E+iǫ)σ0−mσ3]1\n2π/integraldisplayλ\n0k\n(ǫ−iE)2+m2+k2dk (32)\n∼(Eσ0−mσ3)1\n4πlog[1+λ2/(m2−E2)]+o(iǫ)≡(g0+iǫs)σ0+g3σ3.(33)\n(remark: the DOS of BLG has the same form.) Then the imaginary par t of the Green’s\nfunction reads\nIm[G(η)] =−/parenleftBigg\nδǫs(g0+g3+δmr) 0\n0 δǫs(g0−g3−δmr)/parenrightBigg\n(34)\nwith\nδǫs(x) =ǫs\nx2+ǫ2s2. (35)\nThus the DOS is the sum of two Dirac delta peaks\nρr∝δǫs(g0+g3+δmr)+δǫs(g0−g3−δmr). (36)\nThe Dirac delta peak appears with probability ∝exp(−(g0±g3)2/g) for a Gaussian\ndistribution. This calculation can easily be generalized to δmron a set of several\nsitesr[26]. Then the appearance of the several such Dirac delta peaks de creases\nexponentially. Moreover, these contributions are local and form lo calized states. For\nstronger fluctuations δmr(i.e., for increasing g) the localized states can start to overlap.\nThis is a quantum analogue of classical percolation.\nThe localized states in the Lifshitz tails can be taken into account by a\ngeneralization of the SCBA to non-uniform self-energies. The main id ea is to search\nfor space-dependent solutions Σ rof Eq. (23). In general, this is a diffult problem.\nHowever, we have found that this problem simplifies essentially when w e study it in\nterms of a 1 /¯mexpansion. Using a Gaussian distribution, this method gives Lifshitz\ntails of the form [27]:\nρ0(¯m)∼¯m4\n32√πg5/2e−¯m2/4g. (37)\n5. Numerical approach\nTo understand to behavior of random gap fluctuations in graphene , and also the\nlimitations of the SCBA, we carried out extensive numerical simulation s on the\nhoneycomb lattice, allowing for various random gap fluctuations on t op of a uniform\ngapm. These fluctuations are simulated by box and Gaussian distributions . From the\nSCBA, the emergence of a second-order phase transition at a crit ical mean mcis obvious\nfor a given variance. This is best manifested in the behavior of the DO S, which stays\nfinite for ∝angb∇acketleftm∝angb∇acket∇ight< mc, and vanishes afterwards, and serves as an order parameter. D oes\nthis picture indeed survive, when higher order corrections in the flu ctuations are taken\ninto account?11\nTo start with, we take a fix random mass configuration with a given va riance and\nthe honeycomb lattice (HCL) with the conventional hoppings ( t), represented by H0.\nThen, wetakeaseparateHamiltonian, responsiblefortheuniform, non-fluctuatinggaps,\ndenoted by Hgap, and study the evolution of the eigenvalues of H0+mHgapby varying\nm for a 600x600 lattice. By using Lanczos diagonalization, we focus o ur attention\nonly to the 200 eigenvalues closest to the NP. Their evolution is shown in Fig. 4.\nThis supports the existence of a finite mc, but since it originates from a single random\ndisorder configuration, rare events can alter the result. As a pos sible definition of the\nrigid gap, we also show the maximum of the energy level spacing for th ese eigenvalues\nas a function of m. As seen, it starts to increase abruptly at a given value of m, which\ncan define mc.\n00.511.52−101−0.500.5−0.500.5\nPSfrag replacements\nm/teigenvalues /t\ng= 0.62g= 0.82g= 1\nFigure 4. (Color online) The evolution of the 200 lowest eigenvalues is shown for a\ngiven random mass configuration with Gaussian distribution (with var iance g) on a\n600x600 HCL, by varying the uniform gap. The red line denotes the m aximum of the\nlevel spacing of these eigenvalues, a possible definition of the avera ge gap.\nTo investigate whether a finite critical mcsurvives, we take smaller systems and\nevaluate the averaged DOSdirectly frommany disorder realizations . To achieve this, we\ntake a 200x200 HCL, and evaluate the 200 closest eigenvalues to th e NP, and count their\nnumber in a given small interval, ∆ E(smaller than the maximal eigenvalues) around\nzero. This method was found to be efficient in studying other types o f randomness [28].\nWe mention that large values of ∆ Etake contribution from higher energy states into\naccount, while too small values are sensitive to the discrete lattice a nd consequently the12\n0 0.5 1 1.5 200.020.040.060.080.10.12\n0.511.511.522.5\nPSfrag replacements\nm/tρ(0)t\ngexponent ( c)\nFigure 5. (Color online) The density of states at the NP is plotted for Gaussian\nrandom mass for a 200x200 HCL for g= 0.92, 1, 1.12, 1.22and 1.32from bottom to\ntop after 400 averages. The symbols denote the numerical data, solid lines are fits\nusingaexp(−bmc). The inset shown the obtained exponents, c, as a function of g,\nwhich is close to 1.5.\ndiscrete eigenvalue structure of the Hamiltonian. For lattices cont aining a few 104−105\nsites, ∆E/t∼10−2−10−4are convenient.\nThe resulting DOS is plotted in Figs. 5 and 6 for Gaussian (with variance g) and\nbox distribution (within [ −W..W], variance g=W2/3). This does not indicate a sharp\nthreshold, but rather the development of long Lifshitz tails due to r andomness, as we\nalready predicted in the previous section. To analyze them, we fitte d the numerical data\nby assuming exponential tails of the form\nρ(0) =texp(−a−bmc) (38)\nfor a Gaussian and\nρ(0) =texp(−a−b/|m−W|c) (39)\nfor a box distribution, as suggested by Ref. [29]. The obtained cvalues are visualized in\nthe insets of Figs. 5 and 6. Given the good agreement, we believe tha t the DOS at the\nNP is made of states that are localized in a Lifshitz tail. We mention that these results\nare not sensitive to finite size scaling at these values of the disorder and uniform gap,\nonly smaller systems (like the 30x30 HCL) require more averages ( ∼104), whereas for\nlarger ones (such as the 200x200 with 400 averages) fewer avera ges are sufficient.\nIn Fig. 7, the energy dependent DOS is shown for Gaussian random m ass with\ng= 1 and for several uniform gap values. With increasing m, the DOS dimishes rapidly\nat low energies, and develops a pseudogap. The logarithmic singularit y atE=tis\nwashed out for g= 1. We also show the inverse of the DOS, proportional to T0, the13\ncharacteristic temperature scale of variable range hopping as a fu nction of the carrier\ndensity (which is proportional to E2).\n00.20.40.60.8 11.200.020.040.060.080.10.120.14\n0.511.51.41.61.82\nPSfrag replacements\nm/tρ(0)t\ngexponent ( c)\nFigure 6. (Color online) The density of states at the NP is plotted for box distr ibuted\n([−W..W]) randomness for a 200x200 HCL for W= 1.5, 1.7 and 2 ( g=W2/3) from\nbottom to top after 400 averages. The symbols denote the numer ical data, solid lines\nare fits using aexp(−b/|m−W|c). The inset shown the obtained exponents, c, as a\nfunction of g.\n6. Discussion\nMLG and BLG consist of two bands that touch each other at two nod al points (or\nvalleys). Near the nodes the spectrum of MLG is linear (Dirac-like) an d the spectrum\nof BLG is quadratic. The application of a uniform SBP opens a gap in the DOS for both\ncases. For a random SPB, however, the situation is less obvious. Fir st of all, it is clear\nthat randomness leads to a broadening of the bands. If we have tw o separate bands\ndue to a small uniform SPB, randomness can close the gap again due t o broadening (cf.\nFig. 2a). The broadening of the bands depends on the strength of the fluctuations of\nthe random SBP. In the case of a Gaussian distribution there are en ergy tails for all\nenergies.\nNow we focus on the NP, i.e. we consider E= 0 and ρ0. Then we have two\nparameters in order to change the gap structure: the average S BP∝angb∇acketleftm∝angb∇acket∇ight ≡¯mand the\nvariance g. ¯mallows us to broaden the gap and ghas the effect of closing it due to\nbroadening of the two subbands. Previous calculations have shown that at the critical\nvaluemc(g) of Eq. (29) the metallic behavior breaks down for ¯ m > m c(g) [22]. On\nthe other hand, Gaussian randomness creates tails at all energies . Consequently, there\nare localized states for |¯m| ≥mc(g) at the NP, and there are delocalized states for14\n00.5 11.5 22.5 300.050.10.150.20.250.30.350.4\nPSfrag replacements\nE/tρ(E)t\n(E/t)2∼n\n1/ρ(E)t∼T0−0.1 −0.05 0 0.05 0.105101520253035404550\nPSfrag replacements\nE/t\nρ(E)t\n(E/t)2∼n1/ρ(E)t∼T0\nFigure 7. (Color online) The energy dependent density of states is plotted fo r\nGaussiandistributed randommassfora30x30HCLafter 104averagesfor g= 1,m= 2\n(cyan), 1 (blue), 0.5 (red), 0.3 (black), 0.2 (magenta) and 0 (gree n) in the left panel.\nThe right panel visualizes the inverse of the density of states, bein g proportional to T0\nin the variable range hopping model as a function of the energy squa red (proportional\nto the carrier density).\n|¯m|< mc(g) at the NP. The localized states in the tails are described, for instan ce, by\nthe Lippmann-Schwinger equation (30) . The SCBA with uniform self- energy is not able\nto produce the localized tails. An extension of the SCBA with non-unif orm self-energies\nprovides localized tails though, as an approximation for large ¯ mhas shown [27]. This is\nalso in good agreement with our exact diagonalization of finite system s up to 200 ×200\nsize.\nA possible interpretation of these results is that there are two diffe rent types of\nspectra. In a special realization of mrthe tails of the DOS represent localized states.\nOn the other hand, the DOS at the NP E= 0, obtained from the SCBA with uniform\nself-energy, comes from extended states [22]. The localized andth e delocalized spectrum\nseparate at the critical value mc, undergoing an Anderson transition.\nConductivity : Transport, i.e. the metallic regime, is related to the DOS trough the\nEinstein relation σ∝ρD, whereDis the diffusion coefficient. The latter was found in\nRef. [22] for E∼0 as\nD=ag/radicalbig\nm2c−¯m2\n2πm2cΘ(m2\nc−¯m2), (40)\nwherea= 1 (a= 2) for MLG (BLG). Together with the DOS ρ0=η/2gπand the\nscattering rate ηin Eq. (28), the Einstein relation gives us at the NP\nσ(ω∼0)∝ρ0De2\nh≈a\n8π2/parenleftbigg\n1−¯m2\nm2c/parenrightbigg\nΘ(m2\nc−¯m2)e2\nh. (41)\nIn the localized regime (i.e. for |¯m| ≥mc) the conductivity is nonzero only for\npositive temperatures T >0. Then we can apply the formula for variable-range hopping15\ninEq. (2), whichfitswell theexperimental result ingraphaneofRef . [7]. Theparameter\nT0is related to the DOS at the Fermi level as [14]\nkBT0∝1\nξ2ρ(EF), (42)\nwhereξis the localization length. T0has its maximum at the NP EF= 0, as shown in\nFig. 7 and decreases monotonically with increasing carrier density, a s in the experiment\non graphane [7].\nInconclusion, wehavestudiedthedensityofstatesinMLGandBLGa tlowenergies\nin the presence of a random symmetry-breaking potential. While a un iform symmetry-\nbreaking potential opens a uniform gap, a random symmetry-brea king potential also\ncreates tails in the density of states. The latter can close the gap a gain, preventing the\nsystem to become an insulator at the nodes. However, for a sufficie ntly large gap the\ntails contain localized states with nonzero density of states. These localized states allow\nthe system to conduct at nonzero temperature via variable-rang e hopping. This result\nis in agreement with recent experimental observations [7].\nAcknowledgement: This work was supported by a grant from the Deutsche\nForschungsgemeinschaft and by the Hungarian Scientific Researc h Fund under grant\nnumber K72613.\n[1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson , I.V. Grigorieva, S.V.\nDubonos, A.A. Firsov, Nature 438, 197 (2005)\n[2] Y. Zhang, Y.-W. Tan, H.L. Stormer, P. Kim, Nature 438, 201 (2005)\n[3] A.K. Geim and K.S. Novoselov, Nature Materials, 6, 183 (2007)\n[4] Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E.H. Hwang, S. Das Sarma, H.L. Stormer,\nP. Kim, Phys. Rev. Lett. 99, 246803 (2007)\n[5] J.H. Chen, C. Jang, M.S. Fuhrer, E.D. Williams, M. Ishigami, Nature P hysics4, 377 (2008)\n[6] S.V. Morozov, K.S. Novoselov, M.I. Katsnelson, F. Schedin, D.C. E lias, J.A. Jaszczak, A.K. Geim,\nPhys. Rev. Lett. 100, 016602 (2008)\n[7] D.C. Elias, R.R. Nair, T.M.G. Mohiuddin, S.V.Morozov, P. Blake, M.P.H alsa ll, A.C. Ferrari,\nD.W. Boukhvalov, M.I. Katsnelson, A.K. Geim, K.S. and Novoselov, Scie nce323, 610 (2009)\n[8] O. Taisuke, A. Bostwick, T. Seyller, K. Horn, E. Rotenberg, Scie nce18, Vol. 313, 951\n[9] J.B. Oostinga, H.B. Heersche, X. Liu, A.F. Morpurgo, L.M.K. Vande rsypen, Nature Materials 7,\n151 (2008)\n[10] R.V. Gorbachev, F.V. Tikhonenkoa, A.S. Mayorova, D.W. Horsella and A.K. Savchenkoa, Physica\nE40, 1360 (2008)\n[11] E.V. Castro, N.M.R. Peres, J.M.B. Lopes dos Santos, F. Guinea, a nd A.H. Castro Neto, J. Phys.:\nConf. Ser. 129 012002 (2008)\n[12] G. Li, A. Luican, E.Y. Andrei, Phys. Rev. Lett. 102, 176804 (2009)\n[13] N.F. Mott, Metal-Insulator Transitions , (Taylor & Francis, London, 1990)\n[14] N.F. Mott, Philos. Mag. 19, 835 (1969)\n[15] S.V. Morozov et al., Phys. Rev. Lett. 97, 016801 (2006)\n[16] J.C. Meyer et al., Nature 446, 60 (2007)\n[17] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A .K. Geim, Rev. Mod. Phys.\n81, 109 (2009)\n[18] E. McCann and V.I. Fal’ko, Phys. Rev. Lett. 96, 086805 (2006); E. McCann, Phys. Rev. B 74,\n161403(R) (2006)\n[19] E. McCann et al., Phys. Rev. Lett. 97, 146805 (2006)16\n[20] P.W. Anderson, Phys. Rev. 109, 1492 (1958)\n[21] K. Ziegler, Phys. Rev. B 55, 10661 (1997)\n[22] K. Ziegler, Phys. Rev. Lett. 102, 126802 (2009); arXiv:0903.0740\n[23] H. Suzuura and T. Ando, Phys. Rev. Lett. 89, 266603 (2002)\n[24] N.M.R. Peres, F. Guinea, and A.H. Castro Neto, Phys. Rev. B 73, 125411 (2006)\n[25] M. Koshino and T. Ando, Phys. Rev. B 73, 245403 (2006)\n[26] K. Ziegler, J. Phys. A 18, L801 (1985)\n[27] S. Villain-Guillot, G. Jug, K. Ziegler, Ann. Phys. 9, 27 (2000)\n[28] K. Ziegler, B. D´ ora, P. Thalmeier, arXiv:0812.2790\n[29] J. L. Cardy, J. Phys. C: Solid State Phys. 11, L321 (1978)."    },    {        "title": "0905.3207v1.Density_functional_theory_of_two_component_Bose_gases_in_one_dimensional_harmonic_traps.pdf",        "content": "arXiv:0905.3207v1  [cond-mat.str-el]  20 May 2009Density-functional theory of two-component Bose gases in o ne-dimensional harmonic\ntraps\nYajiang Hao1and Shu Chen2,∗\n1Department of Physics, University of Science and Technolog y Beijing, Beijing 100083, China\n2Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n(Dated: November 13, 2018)\nWe investigate the ground-state properties of two-compone nt Bose gases confined in one-\ndimensional harmonic traps in the scheme of density-functi onal theory. The density-functional\ncalculations employ a Bethe-ansatz-based local-density a pproximation for the correlation energy,\nwhich accounts for the correlation effect properly in the ful l physical regime. For the binary Bose\nmixture with spin-independent interaction, the homogeneo us reference system is exactly solvable\nby the Bethe-ansatz method. Within the local-density appro ximation, we determine the density\ndistribution of each component and study its evolution from Bose distributions to Fermi-like dis-\ntribution with the increase in interaction. For the binary m ixture of Tonks-Girardeau gases with\na tunable inter-species repulsion, with a generalized Bose -Fermi transformation we show that the\nBose mixture can be mapped into a two-component Fermi gas, wh ich corresponds to exact soluble\nYang-Gaudin model for the homogeneous system. Based on the g round-state energy function of the\nYang-Gaudin model, the ground-state density distribution s are calculated for various inter-species\ninteractions. It is shown that with the increase in inter-sp ecies interaction, the system exhibits\ncomposite-fermionization crossover.\nPACS numbers: 67.85.-d, 67.60.Bc, 03.75.Mn\nI. INTRODUCTION\nExperimental realization of the atomic mixtures and\nprogress in manipulating cold atom systems have opened\nexciting new possibilities to study many-body physics of\nlow-dimensional quantum gases beyond the mean-field\ntheory. In comparison with the single-component sys-\ntems, mixtures of quantum degenerate atoms may form\nnovel quantum many-body systems with richer phase\nstructures. Particularly, two-component Bose gases have\nbeen under intensive studies both experimentally [1, 2,\n3, 4, 5] and theoretically [6, 7, 8, 9, 10]. Due to the com-\npetition among the inter-species and intra-species inter-\nactions, many interesting phenomena such as phase sep-\naration, new quantum states and phase transitions arise\nin two-component cold atomic gases [6, 7, 8, 9, 10]. With\nstrong anisotropic magnetic trap or two-dimensional op-\ntical lattice the radial degrees of freedom of cold atoms\nare frozen and the quantum gas is described by an ef-\nfective one-dimensional (1D) model [11, 12, 13]. As\na textbook example the 1D Tonks-Girardeau (TG) gas\n[14] was observed firstly [12, 13]. Since then, not only\nthe single-component bosonic gas but also 1D multi-\ncomponent atomic mixtures have become the current re-\nsearch focus. In principle, both the intra-component and\ninter-component interactions can be tuned via the mag-\nnetic Feshbach resonance, which allows us to study the\nbosonic mixture in the whole interaction regime. Very\nrecently, two-species Bose gases with tunable interaction\n∗Electronic address: schen@aphy.iphy.ac.cnhave been successfully produced [4, 5].\nIn general, 1D quantum systems exhibit fascinating\nphysics significantly different from its three-dimensional\ncounterpart because of the enhanced quantum fluctua-\ntions in 1D[15, 16, 17, 18, 19, 20]. The mean-field theory\ngenerallyworksnotwellforthe1Dsystemsexceptinvery\nweakly interacting regime. When interaction is strong\nenough, non-perturbation methods, for instance, Bose-\nFermi mapping (BFM), Bethe ansatz and Bosonization\nmethod, have to be exerted to properly characterize the\nfeaturesofthesystem[9,21, 22, 23, 24, 25, 26,27, 28, 29].\nFor a single-component Bose gas, it has been shown that\nthe density profiles continuously evolve from Gaussian-\nlike distribution of bosons to shell-structured distribu-\ntion of fermions [30, 31, 32, 33] with the increase in re-\npulsion strength. Similar behavior of composite fermion-\nization has been observed in the 1D Bose-Bose mixtures\n[27, 28, 29]. For two-component Bose mixture with spin-\nindependent repulsions, the normalized density distribu-\ntion of each component displays the same behavior up to\na normalized constant which is proportional to respec-\ntive atomic number of each component [27]. While the\nextended Bose-Fermi mappings [21, 22, 23] are restricted\nto the infinitely repulsive limit, some sophisticate meth-\nods, such as multi-configuration self-consistent Hartree\nmethod [28], numerical diagonalization method [29], and\nBethe ansatz method [24, 25, 26, 27], have been applied\nto study the two-component Bose gases in the whole re-\npulsive regime. Nevertheless, the above methods either\nsuffer the small-size restriction [28, 29] or only limit to\nthe integrable models [24, 25, 26, 27] which generally\ndo not cover the realistic system confined in an external\nharmonic trap.2\nIn this work, we use the basic idea of density func-\ntionaltheory(DFT)toinvestigatetheground-stateprop-\nerties of the two-component Bose gases trapped in har-\nmonic potential in the full interacting regime. It is well\nknown that the DFT can handle the system with large\nsizes and will not encounter the exponential wall that\nthe above traditional multiparticle wave-function meth-\nods faced. The effects ofthe externalpotential will be ac-\ncounted by using the local-density approximation (LDA)\n[19, 34, 35, 36, 37, 38, 39, 40, 41, 42]. The combinationof\nDFTandLDAhadbeenappliedtodealwiththeconfined\n1D single-component Bose gas [19, 34, 35, 36] and Fermi\ngas [37, 38, 39, 40, 41, 42] successfully. However, the\ntwo-component Bose gas has rarely been studied. The\npresentworkwillstudytheground-statepropertiesofthe\nconfined Bose mixture and focus on two cases where the\ngroundstateenergyfunctionforthehomogeneoussystem\ncan be obtained exactly. In the first case, we consider the\ntwo-component Bose gas with spin-independent interac-\ntion. In the absence of external potential, this model is\nexactlysolvablebytheBethe-ansatzmethodandthusthe\nground-state energy density function can be extracted\nfromtheBethe-ansatzsolution[24, 25,26, 27]. Inthesec-\nond case, we consider the two-component Bose gas with\ninfinitely repulsive intra-component interactions and a\ntunable inter-component interaction. We will show that\nsuchamodelcanbemappedintotheYang-Gaudinmodel\nwith generalizedBose-Fermi transformationand thus the\nground-state energy function can be obtained from the\nBethe-ansatz solution of the Yang-Gaudin model [43]. In\nthe scheme of DFT, the ground-state density profiles are\nobtained by numerically solving the coupled nonlinear\nSchr¨ odinger equations (NLEs) for both the equal-mixing\nBose mixture and polarized Bose mixture.\nThe present paper is organized as follows. Sec-\ntion II investigates two-component Bose gas with spin-\nindependent interaction and introduce the method. Sec-\ntion III is devoted to the mixture of two TG gases with\ntunable inter-species repulsion. A summary is given in\nthe last section.\nII. BINARY BOSE MIXTURE WITH\nSPIN-INDEPENDENT INTERACTION\nWe consider a two-component Bose gas confined in the\nexternal potential Vext(x) independent of the species of\natoms, where the atomic mass of the i-component is mi.\nTheatomnumbersineachcomponentare N1andN2and\nN=N1+N2is the total atom number. The many-body\nHamiltonian in second quantized form can be formulatedas\nH=/integraldisplay\ndx/summationdisplay\ni=1,2/braceleftbigg\nˆΨ†\ni(x)/bracketleftbigg\n−¯h2\n2mi∂2\n∂x2+Vext(x)/bracketrightbigg\nˆΨi(x)\n+gi\n2ˆΨ†\ni(x)ˆΨ†\ni(x)ˆΨi(x)ˆΨi(x)/bracerightBig\n+g12/integraldisplay\ndxˆΨ†\n1(x)ˆΨ†\n2(x)ˆΨ2(x)ˆΨ1(x), (1)\nin which gi(i= 1,2) andg12denote the effective\nintra- and inter-species interaction that can be controlled\nexperimentally by tuning the corresponding scattering\nlengthes a1,a2anda12, respectively [4, 5]. Ψ†\ni(x)\n(Ψi(x)) is the field operator creating (annihilating) an i-\ncomponent atom at position x. Here we will consider the\ntwo-component Bose gas composed of two internal states\nofsamespeciesofatomssuchthatwehavethesamemass\nfor all atoms m1=m2=m. When the intra- and inter-\ncomponent interactions are equal ( g1=g2=g12=g),\nthe system is governed by the spin-independent Hamil-\ntonian with the first quantized form:\nH=N/summationdisplay\nj=1/bracketleftBigg\n−∂2\n∂x2\nj+Vext(xj)/bracketrightBigg\n+2c/summationdisplay\nj<lδ(xj−xl),(2)\nwherec=mg/¯h2. The model in the absence of exter-\nnal trap is integrable and can be diagonalized via Bethe\nansatz method [24, 26].\nWe first focus on the homogeneous case by setting\nVext(x) = 0. The effects of the external potential will\nbe discussed latter in the scheme of the LDA and DFT.\nBeforethe application of DFT based on the Bethe-ansatz\nsolution, we first give a brief summary of the Bethe-\nansatz solution for the integrable two-component Bose\ngas. For the eigenstate with total spin S=N/2−M\n(0≤M≤N/2), the Bethe ansatz equations for the inte-\ngrable two-component Bose gas take the following form\n[24, 26]\nkjL= 2πIj−N/summationdisplay\nl=1tan−1kj−kl\nc+M/summationdisplay\nα=1tan−1kj−Λα\nc/2,\nN/summationdisplay\nj=1tan−1Λα−kj\nc/2= 2πJα+M/summationdisplay\nβ/negationslash=αtan−1Λα−Λβ\nc,(3)\nwherekjand Λ αare the quasi-momentum and spin ra-\npidity, respectively, the quantum numbers IjandJα\ntake integer or half-integer values, depending on whether\nN−Mis odd or even. In term of the quasi-momentums,\nthe energy eigenspectrum is given by E=/summationtextN\nj=1k2\nj. In\nthe thermodynamic limit, i.e., N,L→ ∞withN/Land\nM/Lfinite, if one introduces density of krootsρ(k) and\ndensity of Λ roots σ(k) as well as the corresponding hole\nroot densities ρh(k) andσh(λ), the Bethe-ansatz equa-3\ntions thus is simplified to two coupled integral equations\nρ(k)+ρh(k) =1\n2π+1\n2π/integraldisplayQ\n−Q2cρ(k′)\nc2+(k−k′)2dk′\n−1\n2π/integraldisplayB\n−Bcσ(λ)\nc2/4+(k−λ)2dλ(4)\nσ(λ)+σh(λ) =1\n2π/integraldisplayQ\n−Qcρ(k)\nc2/4+(λ−k)2dk\n−1\n2π/integraldisplayB\n−B2cσ(λ)\nc2+(λ−λ′)2dλ′.(5)\nTheintegrationlimits QandBaredeterminedby N/L=\nρ=/integraltextQ\n−Qρ(k)dkandM/L=/integraltextB\n−Bσ(λ)dλ.\nAt zero temperature, the ground state corresponds\nto the case with M= 0 [24, 25, 26, 27], where Ij=\n(N+1)/2−jandJαis an empty set. In other words,\nthe ground state correspondsto the configuration σ(λ) =\nρh(k) = 0, i.e., no holes in the charge degrees of free-\ndom and no quasiparticles in the spin degrees of free-\ndom. Consequently, the ground states with S=N/2\nare(N+1)-folddegenerateisospin‘ferromagnetic’states\n[24, 27, 44, 45]. Therefore the Bethe ansatz equation for\nthe ground state reduces to [24, 27]\nρ(k) =1\n2π+1\n2π/integraldisplayQ\n−Q2cρ(k′)\nc2+(k−k′)2dk′,\nor\ng(x) =1\n2π+1\n2π/integraldisplay1\n−12λg(x′)\nλ2+(x−x′)2dx′,(6)\nwherewehavemadethereplacementofvariables k=Qx,\nc=Qλandg(x) =ρ(Qx) accordingto [15]. The ground-\nstate energy density can be expressed as\nǫ(ρ) =¯h2\n2mρ2e(γ) (7)\nwhereρ=/summationtext\niρiis the total density and\ne(γ) =γ3\nλ3/integraldisplay1\n−1g(x)x2dx (8)\nwith the dimensionless parameter γ=c\nρ. In terms of\nnew variables, the equation ρ=/integraltextQ\n−Qρ(k)dkis rewritten\nas\nλ=γ/integraldisplay1\n−1g(x)dx. (9)\nThe expression of ground state energy is identical to\nthat of Lieb-Liniger Bose gas [15] except that now ρ\nshould be replaced by the total density. Here e(γ) can\nbe obtained by numericallysolvingthe integralequations\n(6), (8) and (9) [15, 19] and we find that it can be fitted\nvery accurately by the rational function\ne(γ) =γ/parenleftbig\n1+p1γ+p2γ2π2/3/parenrightbig\n1+q1γ+q2γ2+p2γ3(10)/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s49/s50/s51\n/s32/s69/s120/s97/s99/s116/s32/s114/s101/s115/s117/s108/s116\n/s32/s50/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s115\n/s32/s52/s32/s112/s97/s114/s97/s109/s101/s116/s101/s114/s115\n/s32/s101/s40 /s41/s61\n/s32/s101/s40 /s41/s61/s50\n/s47/s51\n/s32/s32/s32\n/s51 /s52 /s53 /s54 /s55 /s56/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48/s50/s46/s50/s101/s40 /s41\n/s32\n/s32/s32\nFIG. 1: (color online) Fit function with four parameters and\ntwo parameters.\nwithp1= 2.05737,p2= 0.0688097, q1= 3.16861 and\nq2= 0.898297. In Fig. 1 we compare the fit function\nwith the exact numerical result, where the exact match\nbetween them is displayed. In the weakly interacting TF\nregime (γ≪1)e(γ) is approximated as e(γ) =γand\nin the strongly interacting TG regime ( γ≫1) we have\ne(γ) =π2/3 such that in these two limits the energy\ndensity take the form of\nǫ(ρ) ={¯h2\n2mcρ=g\n2ρ, γ≪1\nρ2π2¯h2\n6m, γ≫1.\nAlternatively, e(γ)canalsobe formulatedbythe rational\nfunctional with onlytwo parameters p=−5.0489470and\nq=−20.8604983\ne(γ) =γ/parenleftbig\n1+pγπ2/3/parenrightbig\n1+qγ+pγ2, (11)\nwhich also match well with the exact numerical result\naccording to Fig. 1 although the accuracy is less perfect\nthan Eq. (10).\nIn order to deduce the bulk properties of non-uniform\nsystems, we have to combine the analytical solution of\nthe homogeneous system with local density approxima-\ntion. Within the local density approximation, one can\nassume that the system is in local equilibrium at each\npointxin the external trap, with local energy per parti-\ncle provided by Eq. (7). Therefore the energy functional\nof an inhomogeneous system can be formulated as\nE(ρ) =/integraldisplay\ndx/bracketleftBigg\n¯h2\n2m/vextendsingle/vextendsingle/vextendsingle/vextendsingled\ndx√ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+Vext(x)ρ+ρǫ(ρ)/bracketrightBigg\n(12)\nwhereǫ(ρ) takes the form of Eq.(7) which is the exact\nground-state energy density of the homogenous system\n[19, 34, 36, 46, 47]. The first gradientterm representsad-\nditionalkineticenergyassociatedwiththeinhomogeneity4\nof the gas that is not accounted for by the “local” kinetic\nenergy included in the homogenous energy [35, 47]. Next\nwe shall consider the case with external potential being a\nweak harmonic trap Vext(x) =1\n2mω2x2. For the conve-\nnience of latter calculation, we introduce Φ i=√ρiand\nthus we have ρ=|Φ1|2+|Φ2|2. In terms ofΦ i, the energy\nfunctional can be represented as\nE=/integraldisplay\ndx\n/summationdisplay\ni=1,2Φ∗\ni/parenleftbigg\n−¯h2\n2md2\ndx2+Vext/parenrightbigg\nΦi+ρǫ(ρ)\n.\n(13)\nThe ground state corresponds to the minimization of\nthe energy functional Eq. (13). For the two-component\nbosonic system, the particle number of each component\nis conserved. In order to obtain the ground state from\na global minimization of Ewith the constraints on both\nNi, we introduce Lagrange multiplier chemical potential\nµ1to conserve N1andµ2to conserve N2. The ground\nstate is then determined by a minimization of the free-\nenergyfunctional F=E−µ1N1−µ2N2. AsshowninRef.\n[48], the minimization of energy functional corresponds\nof the ground state of the modified coupled nonlinear\nSchr¨ odinger equations\n/parenleftbigg\n−¯h2\n2md2\ndx2+Vext/parenrightbigg\nΦ1+˜F(ρ)Φ1=µ1Φ1,\n/parenleftbigg\n−¯h2\n2md2\ndx2+Vext/parenrightbigg\nΦ2+˜F(ρ)Φ2=µ2Φ2.(14)\nwith the normalization condition/integraltext\ndx|Φi(x)|2=Ni,\nwhere\n˜F(ρ) =∂\n∂ρ[ρǫ(ρ)] =¯h2ρ2\n2m[3e(γ)−γ∂γe(γ)].\nIn the weakly interacting regime ( γ≪1) and strongly\ninteracting regime ( γ≫1) the explicit forms are\n˜F(ρ) ={gρ γ≪1,\nπ2¯h2ρ2/2m γ≫1,\nwhile in the middle regime which can be obtained by Eq.\n(10) or Eq. (11).\nBy numerically solving the coupled nonlinear\nSchr¨ odinger equations Eq.(14) we can obtain the ground\nstate density profiles of each component in the full\ninteracting regime, which are displayed in Fig. 2 for the\nsystem with the atomic number N1= 20 and N2= 15.\nHere the dimensionless interacting constant U=g/l¯hω\nand the length xis in unit of l=/radicalbig\n¯h/mω. In order to\ncompare the distribution of each component, both of\nthem are normalized to one, i.e.,ni(x) =ρi(x)/Niwhere/integraltext\nρi(x)dx=Ni. It is shown that in the full physical\nregime the density distribution of two components\nmatch exactly. That is to say, the density distribution\nofi-component fulfills a simple relation with the total\ndistribution\nρi(x) =Ni\nNρtot(x) (15)/s45/s49/s48 /s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s32/s32/s110\n/s105/s40/s120/s41\n/s120\nFIG. 2: (color online) Normalized ground-state density dis -\ntributions for N1= 20 (dashed lines) and N2= 15 (dotted\nlines). From the top to the bottom: U= 0.2,2.0,20,100.\nUnit ofni(x):Ni/l; Unit of x:l.\nwithρtot(x) =ρ1(x)+ρ2(x). Similar to the case of single\nspecies ofBosegas, all ofthem exhibit the evolution from\nBose distribution to Fermi-like one with the increase in\ninteraction constant. In the weakly interacting regime\nthe density profiles display the Gauss-like Bose distri-\nbution, while in the strongly interacting regime Bose\natomsbehavelikefermions, whichdistributeuniformlyin\na more extensive area and the density profiles decrease\nto zero rapidly at the boundary. In order to compare\nthe results obtained from the density-functional calcula-\ntions to the exact results obtained by Bose-Fermi map-\nping method, in Fig. 3 we show the density profiles\nof each component and the total density profile in the\nstrongly interacting limit for U= 500. Here each com-\nponents are normalized to their particle numbers and\nclearly they satisfy the relation of eq.(15). The total\ndensity profile obtained by BFM method is given by\nρTG(x) =/summationtextN−1\nn=0|φn(x)|2which is also shown in Fig. 3.\nHereφn(x) denotes the eigenstate of the single particle\nHamiltonian of harmonic oscillator. It turns out that the\nresults obtained from both methods agree well with each\nother in the full space except that the density profile ob-\ntained by BFM displays oscillations. In the limit of large\nparticle number, the differences become imperceptible.\nIII. MIXTURE OF TG GASES WITH TUNABLE\nINTER-COMPONENT INTERACTION\nIn this section, we focus on the mixture of two TG\ngases with infinitely strong intra-species interaction, i.e.,\ngi=∞(i= 1,2) and a tunable inter-species interaction\ng12=g. In this limit, the original Hamiltonian Eq. (1)5\n/s45/s56 /s45/s52 /s48 /s52 /s56/s48/s49/s50/s51\n/s32\n/s49/s40/s120/s41\n/s32\n/s50/s40/s120/s41\n/s32\n/s116/s111/s116/s40/s120/s41\n/s32\n/s84/s71/s40/s120/s41\n/s32/s32/s105/s40/s120/s41\n/s120\nFIG. 3: (color online) Ground-state density distributions for\nN1= 20 (dashed lines) and N2= 15 (dotted lines) with\nU= 500. Unit of ρi(x): 1/l; Unit of x:l.\nis simplified to\nHTG=/integraldisplay\ndx/summationdisplay\ni=1,2/braceleftbigg\nˆΨ†\ni(x)/bracketleftbigg\n−¯h2\n2mi∂2\n∂x2+Vext(x)/bracketrightbigg\nˆΨi(x)\n+gˆΨ†\n1(x)ˆΨ†\n2(x)ˆΨ2(x)ˆΨ1(x)/bracerightBig\n, (16)\nwith the hard-core constraints ˆΨ†\ni(x)ˆΨ†\ni(x) =\nˆΨi(x)ˆΨi(x) = 0 and/braceleftBig\nˆΨi(x),ˆΨ†\ni(x)/bracerightBig\n= 1. For the\nmixture of TG gases, it is convenient to introduce the\ngeneralized Bose-Fermi transformations [49]\nˆΨ1(x) = exp/bracketleftbigg\niπ/integraldisplayx\n−∞nF\n1(z)dz/bracketrightbigg\nΨF\n1(x) (17)\nˆΨ2(x) = exp/bracketleftbigg\niπ/integraldisplay∞\n−∞nF\n1(z)dz/bracketrightbigg\n×exp/bracketleftbigg\niπ/integraldisplayx\n−∞nF\n2(z)dz/bracketrightbigg\nΨF\n2(x) (18)\nwhere ΨF†\ni(x) and ΨF\ni(x) are the creation and annihi-\nlation operator at location xfori-component fermions\nandnF\ni(x) = ΨF†\ni(x)ΨF\ni(x). With the above transfor-\nmations, the Hamiltonian (16) is mapped into the two-\ncomponent fermionic gas with the Hamiltonian given by\nHF=/integraldisplay\ndx/summationdisplay\ni=1,2/braceleftbigg\nˆΨF†\ni/bracketleftbigg\n−¯h2\n2mi∂2\n∂x2+Vext(x)/bracketrightbigg\nˆΨF\ni\n+gˆΨF†\n1(x)ˆΨF†\n2(x)ˆΨF\n2(x)ˆΨF\n1(x)/bracerightBig\n. (19)\nThe first term on the right side of eq. (18) is intro-\nduced to enforce the fermionic annihilation operators\nˆΨF\n1(x) andˆΨF\n2(x) fulfilling the anti-commutation rela-\ntions{ˆΨF\n1(x),ˆΨF\n2(x)}= 0.Since the bosonic model of (16) is related to the\nfermionic model of (19) by an unitary transformation,\nthey have the same energy spectrum. Furthermore the\nground-statepropertieswhichareindependent ofthe sta-\ntistical properties of bosonic system, for example, the\nground-state density distribution, can be obtained by\nconsulting for the corresponding fermionic model. Obvi-\nously the homogeneous system of (19) with Vext(x) = 0\nis the Yang-Gaudin model which can be solved exactly\nby means of the Bethe ansatz method [43]. The Bethe-\nansatz equations take the form of\nkjL= 2πIj−2M/summationdisplay\nα=1tan−1kj−Λα\nc/2\nN/summationdisplay\nj=1tan−1Λα−kj\nc/2=πJα+M/summationdisplay\nβ=1tan−1Λα−Λβ\nc(20)\nfor the quasi-momentum {kj}and rapidity {Λα}with\nthe quantum number IjandJα. While in the thermo-\ndynamic limit the equations (20) reduce to two coupled\nintegral equations\nρ(k) =1\nπ/integraldisplayB\n−BdΛc′σ(Λ)\nc′2+(k−Λ)2+1\n2π,\nσ(Λ) =−1\nπ/integraldisplayB\n−BdΛ′cσ(Λ′)\nc2+(Λ−Λ′)2\n+1\nπ/integraldisplayQ\n−Qdkc′ρ(k)\nc′2+(Λ−k)2(21)\nwithc′=c/2, where QandBare determined by N/L=/integraltextQ\n−Qρ(k)dkandM/L=/integraltextB\n−Bσ(λ)dλ. If we make the\nreplacement of variables x=k/Q,y= Λ/B,gc(x) =\nρ(xQ) =ρ(k) andgs(y) =σ(yB) =σ(Λ) , the above\nintegral equations (21) can be formulated as\ngc(x) =1\n2π+1\n2πλs/integraldisplay1\n−1dygs(y)\n1/4+(x/λc−y/λs)2\ngs(y) =1\n2πλc/integraldisplay1\n−1dxgc(x)\n1/4+(y/λs−x/λc)2\n−1\nπλs/integraldisplay1\n−1dy′gs(y′)\n1+(y−y′)2/λ2s\nwithλc=γ/integraltext1\n−1dxgc(x) andλs=2γ\n1−ζ/integraltext1\n−1dygs(y). Af-\nter solving the integral equations the energy density of\nground state can be evaluated by\nεGS=¯h2N2γ3\n2mL2λ3c/integraldisplay1\n−1x2gc(x)dx.\nFor the system with total density ρ=ρ1+ρ2and po-\nlarization ζ= (ρ1−ρ2)/ρ, the energy density of ground\nstate can be parameterized as\nεGS(ρ,ζ) =π2¯h2ρ2\n8m/bracketleftbig/parenleftbig\n1/3+ζ2/parenrightbig\n+f(χ,ζ)/bracketrightbig\n(22)6\nwith\nf(χ,ζ) = [e(χ)−1/3]/braceleftbig\n1+α(χ)ζ2+β(χ)ζ4\n−[1+α(χ)+β(χ)]ζ6/bracerightbig\n, (23)\nwhereχ= 2γ/π, ande(χ),α(χ) andβ(χ) are given in\nRef. [39, 40]\ne(χ) =4χ2/3+apχ+bp\nχ2+cpχ+dp,\nα(χ) =−χ2+aαχ+bα\nχ2+cαχ−bα,\nβ(χ) =aβχ\nχ2+bβχ+cβ(24)\nwithap= 5.780126, bp=−(8/9)ln2 +πap/4,cp=\n(8/π)ln2 + 3ap/4,dp= 3bp,aα=−1.68894,bα=\n−8.0155,cα= 2.74347,aβ=−1.51457,bβ= 2.59864\nandcβ= 6.58046.\nThus for the mixture of two-component TG gas, the\nenergydensity ǫ(ρ)inenergyfunctional(13)shouldbere-\nplaced by ǫGS(ρ,ζ)|ρ→ρ(x),ζ→ζ(x)[38, 39]. With the same\nprocedure as the former section, the coupled Schr¨ odinger\nequations take the same formula as Eq. (14) and ˜F(ρ) in\nthe equation about Φ i(i= 1,2) should be replaced with\n˜Fi(ρ,ζ) =π2¯h2\n8m∂\n∂ρiρ3/bracketleftbig/parenleftbig\n1/3+ζ2/parenrightbig\n+f(χ,ζ)/bracketrightbig\n.(25)\nFor the unpolarized case, it is reduced to\n˜F(ρ) =π2¯h2\n8m/bracketleftbigg\n3ρ2e(χ)+ρ3∂e\n∂ρ/bracketrightbigg\n.(26)\nBy numerically solving the coupled NLEs, we can ob-\ntain the ground-state density distributions for different\ninteracting constants U(U=g/l¯hω).\nWe first consider the unpolarized Bose mixture with\nN1=N2, for which we always have ρ1(x) =ρ2(x). In\nFig. 4, we show the density profiles ρ1(x) for various\ninter-species interactions, which are normalized to the\natom number N1. When the inter-species interaction is\nzero, no correlation exists between two TG gases and\nρ1(x) is given by the density distribution of the single-\ncomponent TG gas, which is identical to the density\ndistribution of a polarized Fermi gas with N1fermions\n[14]. When the inter-species interaction is weak, the den-\nsity distribution of each component does not deviate far\nfrom the distribution of the single-component TG gas.\nWith the increase in inter-species interaction, the den-\nsity profiles become broader and broader, and continu-\nously evolve to the limit of strong inter-species interac-\ntion. As shown in Fig.4, the density profile for U= 500\nalreadyagreesverywell with the exactprofilein the limit\nofU=∞except the oscillation peaks. In this limit, the\ndensityprofilefor eachcomponent fulfils the relation(15)\nand displays Npeaks, which can be exactly calculated\nby using the Bose-Fermi mapping method [21]./s45/s56 /s45/s52 /s48 /s52 /s56/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s32/s85/s61/s48/s46/s48\n/s32/s85/s61/s48/s46/s49\n/s32/s85/s61/s49/s46/s48\n/s32/s85/s61/s49/s48/s46/s48\n/s32/s85/s61/s53/s48/s46/s48\n/s32/s85/s61/s53/s48/s48/s46/s48\n/s32/s85/s61\n/s32/s32/s49/s40/s120/s41\n/s120/s45/s50 /s45/s49 /s48 /s49 /s50/s49/s46/s51/s56/s49/s46/s52/s48/s49/s46/s52/s50/s49/s46/s52/s52\n/s32/s32\n/s65\nFIG. 4: (color online) Ground state density distribution fo r\nunpolarized two-component TG gases with N1=N2= 20.\nUnit ofρ1(x): 1/l; Unit of x:l.\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48\n/s32\n/s32/s32/s49\n/s120/s40/s97/s41\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48\n/s32/s85/s61/s48\n/s32/s85/s61/s48/s46/s48/s50\n/s32/s85/s61/s49/s46/s48\n/s32/s85/s61/s49/s48/s46/s48\n/s32/s85/s61/s49/s48/s48/s46/s48\n/s32/s85/s61/s51/s48/s48/s46/s48\n/s32/s85/s61/s53/s48/s48/s46/s48\n/s32/s85/s61\n/s32/s32\n/s120/s50/s40/s98/s41\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s32/s32/s49/s43\n/s50\n/s120/s40/s99/s41\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50\n/s32/s32/s49/s45\n/s50\n/s120/s40/s100/s41\nFIG. 5: (color online) Ground state density distribution fo r\npolarized two-component TG gases with N1= 15 and N2=\n10. Unit of ρi(x): 1/l; Unit of x:l.\nFor the imbalance trapped mixture of TG gases, the\ndistributions of each component will display different be-\nhaviors in the full interacting regime except in the limit\nof strong interspecies interaction, where the distributions\ndisplay same behaviors only with different normalization\ncondition, i.e.,n1(x) =n2(x) withni(x) =ρi(x)/Ni.\nThe density profiles of each components and the total\ndensity profiles for the polarized mixture of TG gases are\nexhibited in Fig. 5a, Fig. 5b and Fig. 5c, respectively.\nThe profiles for the case of vanished inter-species interac-\ntion and of infinitely strong inter-species interaction are\nalso plotted (oscillating lines) in the figure, which are ob-\ntained by means of BFM method. It is shown that in the7\nregime close to these two limits the distributions agree\nwellwiththeresultsofBFMmethod. Itisfeasibleevenin\nthesituationoffiniteweakorstronginter-speciesinterac-\ntion. With the increase in inter-species interaction both\ncomponents distribute in more extensive regime and the\ndensity profiles of i-component evolve from the behavior\nof single-component TG gas with ρi(x) =ρTG(x)|N=Ni\nto the behavior of the fermionized Bose mixture with\nρi(x) =Ni\nNρTG(x)|N=N1+N2. In Fig. 5d, we show the\nspin density distribution defined as the density differ-\nence between two components. For U= 0, we have\nρ1(x)−ρ2(x) =/summationtextN1−1\nn=N2|φn(x)|2. In the weakly and\nintermediately interacting regimes, the spin density dis-\ntributions display two explicit peaks at the location far\naway from the center of trap and at the locations near\nthe middle of trap spin density profiles are almost flat.\nThat is to say, two weakly interacting TG gases form the\nshell structure of the coexisted regime oftwo components\nsurrounded by the fully polarized majority component.\nWith the increase in interspecies interaction, the peaks\nbecome smaller and smaller and disappear finally. In the\nlimit of strong inter-species interaction, two components\ncoexist in the full spaces and the spin density profile ex-\nhibits similar behavior to the total density profile. This\ncould be understood because in the infinite- Ulimit one\nhasρ1(x)−ρ2(x) =N1−N2\nNρtot(x) according to (15).\nIV. SUMMARY\nIn summary, the ground-state properties of two-\ncomponent Bose gases trapped in 1D harmonic trapshave been studied in the scheme of DFT. On the ba-\nsis of Bethe-ansatz solution, we calculate the ground-\nstate density distributions of the confined systems for\neach component by combining the local density approx-\nimation with the exact solution of homogeneous system.\nFor the case with spin-independent interaction, the dis-\ntribution of each component is Ni/Nof the total density\ndistribution in the full repulsive regime. With the in-\ncrease in the interaction the density distributions show\nevolution from Gauss-like Bose distributions to Fermi-\nlike distributions. For the mixture of two TG gases with\ntunable inter-species interaction, we show that it can be\nmappedintoatwo-componentFermi gasbyageneralized\nBose-Fermi transformation. Taking advantage of the ex-\nact ground-state energy of Yang-Gaudin model for the\nhomogeneous system, we calculate the density profiles of\nthe confined systems. 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Lett. 85, 60004\n(2009)."    },    {        "title": "0905.3795v1.A_stochastic_optimal_velocity_model_and_its_long_lived_metastability.pdf",        "content": "arXiv:0905.3795v1  [cond-mat.stat-mech]  23 May 2009A stochastic optimal velocity model and its long-lived meta stability\nMasahiro Kanai\nGraduate School of Mathematical Sciences, University of To kyo, 3-8-1 Komaba, Tokyo 153-8914, Japan.\nKatsuhiro Nishinari\nDepartment of Aeronautics and Astronautics, Faculty of Eng ineering,\nUniversity of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan.\nTetsuji Tokihiro\nGraduate School of Mathematical Sciences, University of To kyo, 3-8-1 Komaba, Tokyo 153-8914, Japan.\n(Dated: October 26, 2018)\nIn this paper, we propose a stochastic cellular automaton mo del of traffic flow extending two\nexactly solvable stochastic models, i.e., the asymmetric s imple exclusion process and the zero range\nprocess. Moreover it is regarded as a stochastic extension o f the optimal velocity model. In the fun-\ndamental diagram (flux-density diagram), our model exhibit s several regions of density where more\nthan one stable state coexists at the same density in spite of the stochastic nature of its dynamical\nrule. Moreover, we observe that two long-lived metastable s tates appear for a transitional period,\nand that the dynamical phase transition from a metastable st ate to another metastable/stable state\noccurs sharply and spontaneously.\nTraffic dynamics has been attracting much attention\nfrom physicists, engineers and mathematicians as a typi-\ncal example of non-equilibrium statistical mechanics of\nself-driven many particle systems for the last decade\n[1, 2, 3]. Statistical properties of traffic phenomena are\nstudied empirically by using the fundamental diagram ,\nwhich displays the relation of the flux (the averageveloc-\nity of vehicles multiplied by the density of them) to the\ndensity. We have been found an emergence of more than\nonedifferent flux atthe samedensity in thetransit region\nfrom free to congested phase is almost universal in real\ntraffic [4]. Recent experimental studies also show that\nthe phase transition occurs discontinuously against the\ndensity and structurally complex states appear around\nthe critical density [5, 6, 7].\nWhile traffic models with many parameters and com-\nplex rules may reproduce empirical data, a strong math-\nematical support, if any, allows a direct connection be-\ntween microscopic modelling and the universal feature\nextracted from various traffic flows. In this paper, we\ntherefore propose a cellular automaton (CA) model ex-\ntending two significant stochastic processes, i.e., the\nasymmetric simple exclusion process (ASEP) and the\nzero range process (ZRP) as described later. More-\nover, we find that our model is supported by a success-\nful traffic model, the optimal velocity (OV) model [8, 9].\nThe OV model, which is a continuous and determinis-\ntic model, is expressed by coupled differential equations;\nd2xi/dt2=a[V(xi+1−xi)−dxi/dt], where xi=xi(t) is\nthe position of the i-th vehicle at time tand the func-\ntionVis called the optimal velocity function, which gives\nthe optimal speed of a vehicle according to its headway\nxi+1−xi(thei-thvehiclefollowsthe( i+1)-thinthesame\nlane). The intrinsic parameter aindicates the driver’s\nsensitivity to traffic situations and governs the stability\nof flow. Our stochatic CA model is, however, different\nfrom a noisy OV model [10] as described below.First of all, we explain the general framework of our\nstochastic CA model for one-lane traffic. Nvehicles are\nmoving on a single-lane road which is divided into a one-\ndimensional array of Lsites. Each site contains one ve-\nhicle at most, and collision and overtaking are thus pro-\nhibited (the so-called hard-core exclusion rule ). In this\npaper, parallel updating is adopted, i.e., all the vehicles\nattempt to move at each step. We introduce a probabil-\nity distribution function wt\ni(m) which gives the probabil-\nity, or the driver’s intention , of thei-th vehicle hopping\nm(m= 0,1,2,...) sites ahead at time t. Then assuming\nthat the next intention wt+1\ni(m) is determined by a func-\ntionfidepending on wt\ni(0),wt\ni(1),...and the positions\nxt\n1,xt\n2,...,xt\nN, the configuration of vehicles is recursively\nupdated according to the following procedure:\n•Foreachvehicle, calculatethe nextintentiontohop\nmsites with the configuration xt\n1,xt\n2,...,xt\nNand\nthe intention wt\ni(0),wt\ni(1),...;\nwt+1\ni(m) =fi/parenleftbig\nwt\ni(0),wt\ni(1),...;xt\n1,...,xt\nN;m/parenrightbig\n.(1)\n•Determine the number of sites Vt+1\niat which a ve-\nhicle moves (i.e. the velocity) probabilistically ac-\ncording to the intention wt+1\ni. In other words, the\nprobability of Vt+1\ni=mis equal to wt+1\ni(m).\n•Each vehicle moves avoiding a collision;\nxt+1\ni=xt\ni+min(∆ xt\ni,Vt+1\ni), (2)\nwhere ∆ xt\ni=xt\ni+1−xt\ni−1 defines the headway,\nand the vehicles thus move at either Vt+1\nior ∆xt\ni\nsites. IfVt+1\ni>∆xt\ni, a vehicle must stop at the cell\nxt\ni+1−1.\nIn what follows, we assume that the maximum allowed\nvelocity is equal to 1, i.e., if m/greaterorequalslant2,wt\ni(m) = 0. Then,2\n✇/archrightdownp\n✇/archrightdownp\n✇✇\n(a)ASEP✇/archrightdownp(3)\n✇/archrightdownp(1)\n✇✇\n(b)ZRP\nFIG. 1: Schematic view of the tagged-particle model for\nASEP(a) and ZRP(b). The hopping probability of a parti-\ncle depends on the gap size in front of it in ZRP, while it is\nalways constant in ASEP. In both cases, hopping to an occu-\npied cell is prohibited.\nputtingvt\ni≡wt\ni(1) (accordingly wt\ni(0) = 1 −vt\ni), we\npropose a special type of fiin (1) as\nvt+1\ni= (1−ai)vt\ni+aiVi(∆xt\ni) (∀t≥0,∀i),(3)\nwhereai(0≤ai≤1) is a parameter and the function\nViis restricted to values in the interval [0 ,1] so that vt\ni\nalso should be within [0 ,1]. The intrinsic parameter ai,\na weighting factor of the optimal velocity Vi(∆xt\ni) to the\nintention wt+1\ni,correspondstothedriver’ssensitivitytoa\ntraffic condition. As long as the vehiclesmove separately,\nwe can also rewrite (2) simply as xt+1\ni=xt\ni+ 1 with\nprobability vt+1\ni. (Note that the original OV model does\nnot support the hard-core exclusion [8, 9].) Therefore,\nvt\nican be regarded as the average velocity, i.e., /an}bracketle{txt+1\ni/an}bracketri}ht=\n/an}bracketle{txt\ni/an}bracketri}ht+vt+1\niin the sense of expectation values. We call\nthe model expressedby (3) the stochastic optimal velocity\n(SOV) model because ofits formal similarityto a discrete\nversion of the OV model (or a coupled map lattice[11])\nxt+1\ni=xt\ni+vt+1\ni∆t, (4)\nvt+1\ni=(1−a∆t)vt\ni+(a∆t)V(∆xt\ni),(5)\nwhere ∆ tis a time interval and the OV model is recov-\nered in the limit ∆ t→0. In this special case that the\nmaximum allowed velocity equals to 1, we thus have an\nobvious correspondence of our stochastic CA model to\nan existent traffic model. As a matter of convenience, we\nsetai=aandVi=V(∀i) hereafter.\nFrom the viewpoint of mathematical interest, the SOV\nmodel includes two significant stochastic models. When\na= 0, (3) becomes vt+1\ni=vt\ni, i.e., the model reduces to\nASEP [12, 13, 14, 15] with a constant hopping probabil-\nityp≡v0\ni. When a= 1, (3) becomes vt+1\ni=V(∆xt\ni),\ni.e., the model reduces to ZRP [16] considering the head-\nways{∆xt\ni}as the stochastic variables of ZRP. In ZRP,\nthe hopping probability of a vehicle is determined ex-\nclusively by its present headway. Fig.1 illustrates the\ntwo stochastic models schematically. ASEP and ZRP are\nboth known to be exactly solvable in the sense that the\nprobability distribution of the configuration of vehicles\nin the stationary state can be exactly calculated [17, 18],\nand thus our model admits an exact calculation of the\nfundamental diagram in the special cases.\nIn order to investigate a phenomenological feature, we\ntake a realistic form of the OV function as\nV(x) =tanh(x−c)+tanh c\n1+tanh c, (6)00.050.10.150.2\n00.20.40.60.8100.050.10.150.2\n00.20.40.60.81t=10 t=100\nt=1000 t=10000\nDensityFlux\nFIG. 2: The fundamental diagram of the SOV model with the\nOV function (6) (c=1 .5 anda= 0.01) plotted at each time\nstaget, starting from uniform/random states with p(≡v0\ni) =\n0.5, including the exact curve(gray) of ASEP for comparison\n[13]. The system size is L= 1000.\nwhich was investigated in [8]. We find that the funda-\nmental diagrams simulated with (6) have a quantitative\nagreement with the exact calculation of ZRP ( a= 1)\nup toa∼0.6. In contrast, the fundamental diagram of\nthe SOV model does not come closer to that of ASEP\nasaapproaches to 0 although the SOV model coincides\nwith ASEP at a= 0. Fig. 2 shows that a curve sim-\nilar to the diagram of ASEP appears only for the first\nfew steps ( t= 10) and then changes the shape rapidly\n(t= 100,1000). Surprisingly, when the diagram be-\ncomes stationary, it allows a discontinuous point and two\noverlapping stable states around the density ρ∼0.14\n(t= 10000).\nLet us study the discontinuity of the flux in detail.\nFig. 3 shows the fundamental diagram expanded around\nthe discontinuous point. We have three distinct branches\nand then call them as follows: free-flow phase (vehicles\nmove without interactions), congested phase (a mixture\nof small clusters and free vehicles), and jam phase (one\nbig stable jam transmitting backward). These branches\nappearinthefundamentaldiagram,respectivelyasaseg-\nment of line with slope 1(free-flow), as a thick curve with\naslightpositiveslope(congested), andasathickline with\na negative slope(jam). Note that the congested and jam\nlines shows some fluctuations due to a randomness of\nthe SOV model. Fig. 3(a) shows snapshots of the flux\natt= 1000 and 5000. There exists midstream flux be-\ntween free-flow and congested phases at t= 1000, and\nbetween congested and jam phases at t= 5000. This\nsuggests that the high-density free-flow states can hold\nuntilt= 1000 but not until t= 5000, and that the\nhigh-density congested states have already started to de-\ncay into jam states before t= 5000. We have thereby\nrevealed the existence of two metastable branches lead-\ning out of the free-flow or congested lines. Comparing\nFig.3(a) with Fig. 3(b), we have six qualitatively dis-\ntinct regions of density(Fig.3(b)); free region S1(includ-3\n00.050.10.150.20.250.3\nDensity0.020.040.060.080.10.120.140.16Flux(a)\nFreeflow\nCongested\nJamS1B1T1T2B20.020.040.060.080.10.120.140.16Flux\n(b)\nFIG. 3: (a)The expanded fundamental diagram of the SOV\nmodel with a= 0.01 att= 1000 (gray) and t= 5000 (black)\nstarting from two typical states; the uniform state with equ al\nspacing of vehicles and p(≡v0\ni) = 1, and the random state\nwith random spacing and p= 1. We observe three distinct\nbranches, which we call the free-flow, congested, and jam\nbranch. They survive even in the stationary state, which is\nplotted at t= 50000 in (b). (b)The stationary states (black)\nand the averaged three branches (gray lines) are plotted at\nt= 50000. The vertical dotted lines distinguish the regions o f\ndensity from S1toB2. The arrows in T2indicate the trace of\na metastable free-flow state decaying to the lower branches.\n(see also Fig. 4).\ning only free-flow phase), bistable region B1(including\nfree-flow and congested phases, which are both stable),\ntristable region T1(including all the three phases, which\nare all stable), tristable region T2(including free-flow,\ncongested and jam phases. The former two phases are\nstable and the last one is metastable), bistable region\nB2(including congested and jam phases. The former is\nmetastable, and the latter is stable), and jam region S2\n(including only stable jam phase, which is not displayed\nhere). We stress that the tristable region T1is a novel\nand remarkable characteristic of many-particle systems,\nand that the successive phase transitions from a free-flow\nstate to a jam state via a congested state occur respec-\ntively on a short time scale.\nLet us study the dynamical phase transition especially\nat the density ρ= 0.14, indicated by successive arrowsin\nFig. 3(b). Fig. 4(upper) shows the flux plotted against\ntime. It is striking that there appear three plateaus\nwhich respectively correspond to a free-flow state, a con-\ngested state, and a jam state, and that the flux changes\nsharply from one plateau to another. In other words, the\nmetastable states have a remarkably long lifetime before\nundergoing a sudden phase transition. Stochastic mod-\nFIG. 4: The time evolution of flux at the density ρ= 0.14\nstarting from the uniform state. We observe two plateaus at\nthefluxQ= 0.14 with alifetime T≃5000, and Q≃0.08 with\nT≃7000 before reaching the stationary jam state(upper).\nThe lower figure shows the corresponding spatio-temporal di -\nagram, where vehicles (black dots) move from bottom up.\n(Note that the periodic boundary condition is imposed.)\nels, in general, arenot anticipated to havesuch long-lived\nmetastable states because stochastic fluctuations break a\nstability of states very soon [7]. Fig. 4(lower) shows the\nspatio-temporal diagram corresponding to the dynami-\ncal phase transition. Starting from a free-flow state, the\nuniform configuration stochastically breaks down at time\nt∼5000, and then the free-flow state is rapidly replaced\nby a congested state where a lot of clusters are forming\nand dissolving, moving forward and backward. Fig. 5\nshowsthe distributionofheadwaysatseveraltimestages.\nWe find that the distribution of headways changes signif-\nicantly after each phase transition. In particular, the\nratio of vehicles with null headway increases. As for a\ncongested state, it is meaningful to evaluate the average\nsize of clusters from a distribution of headways. If the\nratio of the vehicles with null headway is b0, the average\nclustersize ℓcanbeevaluatedas1 /(1−b0). Inthepresent\ncase, the average size of clusters is about 1 .2∼1.4 dur-\ningt= 6000∼12000. We also have some remarks, from\na microscopic viewpoint, on a single cluster: Since the\nsensitivity parameter ais set to a small value, the inten-\ntionvt\nidoes not change a lot before the vehicle catches\nup with the tail of a cluster(i.e. vt\ni∼1). Accordingly,\nthe aggregation rate αis roughly estimated at the den-\nsity of free region behind the cluster; α∼(1−b0)ρ. In\nthe present case, the aggregation rate averaged over the\nwhole clustersofa congestedstate is 0 .10∼0.12. For the\nsame reason, we can estimate the average velocity of the\nfront vehicle of a cluster at (1 −a)ℓ/δ, whereδdenotes\nthedissolutionrateand ℓ/δindicatesthe durationofcap-4\nt=40000.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0t=6000\n0510 15 202530\nHeadwayt=14000\n0510 15 2025300.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0Ratiot=12000\nFIG. 5: The distribution of the headways with which the\nvehicles move at each time stage t. It changes a lot after\nphase transitions( t∼5000,12000). The traffic states, free-\nflow(t <5000), congested(5000 < t <12000), and jam( t >\n12000) respectively show their specific pictures.\nture. Since δis also equivalent to the average velocity of\nthe front vehicle, it amounts roughly to 1 −aℓafter all.\nIn the present case, the dissolution rateaveragedoverthe\nwhole clusters of a congested state is 0 .86∼0.88. Then,\nthe average lifetime of the clusters ℓ/(δ−α) is estimated\nat 1.54∼1.89. The above estimations are appropriate\nonly when clusters are small and spaced apart. As many\nclusters arise everywhere and gather, a vehicle out of a\ncluster tends to be caught in another cluster again be-\nfore it recovers its intention at full value. Consequently,the clusters arising nearby reduce their dissolution rates,\nand finally grow into a jam moving steadily backward.\nThe steady transmitting velocity (the aggregation rate)\nis estimated at −0.055, coinciding with Fig. 4.\nIn this paper, beginning with a general scheme, we\nhave proposed a stochastic CA model to which we in-\ntroduce a probability distribution function of the vehi-\ncle’s velocity. It includes two exactly solvable stochastic\nprocesses, and it is also regarded as a stochastic gen-\neralization of the OV model. Moreover, it exhibits the\nfollowing features: In spite of a stochastic model, the\nfundamental diagram shows that there coexist two or\nthree stable phases (free-flow, congested, and jam) in\na region of density. As the density increases, the free-\nflow and congested states lose stability and change into\nmetastable states which can be observed only for a tran-\nsitional period. Moreoverthe dynamical phase transition\nfrom a metastable state to another metastable/stable\nstate, which is triggered by stochastic perturbation, oc-\ncurs sharply and spontaneously. We consider that the\nmetastable state may be relevant to the transient con-\ngested state observed in the upstream of on-ramp [5, 19].\nSuch a dynamical phase transition has not been observed\nin previous works [8, 9, 10] or among existent many-\nparticle systems. Further studies, e.g., on another choice\noftheOVfunction, underopenboundaryconditions, and\non the general (i.e. multi-velocity) version will be given\nin subsequent publications [20].\n[1] D. Helbing, Rev. Mod. Phys. 73, 1067 (2001).\n[2] D. Chowdhury, L. Santen and A. Schadschneider, Phys.\nRep.329, 199 (2000).\n[3] T. Nagatani, Rep. Prog. Phys. 65, 1331 (2002).\n[4] K. Nishinari and M. Hayashi, Traffic statistics in Tomei\nexpress way , (The Mathematical Society of Traffic Flow,\n1999, Nagoya).\n[5] B. S. Kerner, The Physics of Traffic , Springer-Verlag,\nBerlin, New York 2004.\n[6] M. Treiber, A. Hennecke and D. Helbing, Phys. Rev. E,\n62, 1805 (2000).\n[7] K. Nishinari, M. Fukui and A. Schadschneider, J. Phys.\nA37, 3101 (2004).\n[8] M. Bando, K. Hasebe, A. Nakayama, A. Shibata and\nY. Sugiyama, Phys. Rev. E 51, 1035 (1995).\n[9] M. Bando, K. Hasebe, K. Nakanishi, A. Nakayama,\nA. Shibata and Y. Sugiyama, J. Phys. I France 5, 1389\n(1995).\n[10] D. Helbing and M. Schreckenberg, Phys. Rev. E 59,R2505 (1999).\n[11] S. Yukawa and M. Kikuchi, J. Phys. Soc. Jpn. 64, 35\n(1995).\n[12] B. Derrida, E. Domany, and D. Mukamel, J. Stat. Phys.\n69, 667 (1992).\n[13] B. Derrida, M. R. Evans, V.Hakim, and V.Pasquier, J.\nPhys. A 26, 1493 (1993).\n[14] N. Rajewsky, L. Santen, A. Schadschneider, and M.\nSchreckenberg, J. Stat. Phys. 92, 151 (1998).\n[15] G. M. Sh¨ utz, J. Phys. A 36, R339 (2003).\n[16] F. Spitzer, Adv. Math. 5, 246 (1970).\n[17] M. R. Evans, J. Phys. A 30, 5669 (1997).\n[18] O. J. O’Loan, M. R. Evans, and M. E. Cates, Phys. Rev.\nE58, 1404 (1998).\n[19] N. Mitarai and H. Nakanishi, Phys. Rev. Lett. 85, 1766\n(2000).\n[20] M. Kanai, K. Nishinari and T. Tokihiro, to be published."    },    {        "title": "0905.4229v1.High_Precision_Thermodynamics_and_Hagedorn_Density_of_States.pdf",        "content": "arXiv:0905.4229v1  [hep-lat]  26 May 2009MIT-CTP 4041\nHigh-Precision Thermodynamics and Hagedorn Density of Sta tes\nHarvey B. Meyer\nCenter for Theoretical Physics\nMassachusetts Institute of Technology\nCambridge, MA 02139, U.S.A.\n(Dated: May 26, 2009)\nWe compute the entropy density of the confined phase of QCD wit hout quarks on the lattice to\nvery high accuracy. The results are compared to the entropy d ensity of free glueballs, where we\ninclude all the known glueball states below the two-particl e threshold. We find that an excellent,\nparameter-free description of the entropy density between 0.7TcandTcis obtained by extending\nthe spectrum with the exponential spectrum of the closed bos onic string.\nPACS numbers: 12.38.Gc, 12.38.Mh, 25.75.-q\nI. INTRODUCTION\nThe phase diagram of quantum chromodynamics\n(QCD) is being actively studied in heavy ion collision\nexperiments as well as theoretically. A form of mat-\nter with remarkable properties [1] has been observed\nin the Relativistic Heavy Ion Collider (RHIC) experi-\nments [2, 3, 4, 5]. It appears to be a strongly coupled\nplasma of quarks and gluons (QGP), but no consensus\non a physical picture that accounts for both equilibrium\nand non-equilibrium properties has been reached yet. On\nthe other hand, below the short interval of temperatures\nwhere the transition from the confined phase to the QGP\ntakes place [6, 7, 8, 9], it is widely believed that the most\nprominent degrees of freedom are the ordinary hadrons.\nFrom this point of view, the zeroth order approximation\nto the properties of the system is to treat the hadrons\nas infinitely narrow and non-interacting. We will refer to\nthis approximation as the hadron resonance gas model\n(HRG). The HRG predictions were compared with lat-\ntice QCD thermodynamics data in [6, 10], and lately they\nhave been used to extrapolate certain results to zero tem-\nperature [7]. The HRG is also the basis of the statistical\nmodel currently applied to the analysis of hadron yields\nin heavy ion collisions [11], and recently the transport\nproperties of a relativistic hadron gas have been studied\nin detail [12].\nSince any heavy ion reaction ends up in the low-\ntemperature phase of QCD, it is important to under-\nstand its properties in detail in order to extract those of\nthe high-temperature phase with minimal uncertainty. In\nthis Letter we study whether the HRG model works in the\nabsence of quarks, in other words in the pure SU( N= 3)\ngauge theory, where the low-lying states are glueballs.\nThere are reasons to believe that if the HRG model is\nto work at any quark content of QCD, it is in the zero-\nflavor case. Firstly, the mass gap in SU(3) gauge theory\nis very large, M0/Tc≃5.3. As we shall see, the thermo-\ndynamic properties up to quite close to Tcare dominated\nby the states below the two-particle threshold, which\nare exactly stable. Furthermore, because of their large\nmass, neglecting their thermal width should be a good\napproximation. Secondly, the scattering amplitudes be-tween glueballs are parametrically 1 /N2suppressed while\nthose between mesons are only 1 /Nsuppressed [13]. This\nmeans that the glueballs should be free to a better ap-\nproximation than the hadrons of realistic QCD.\nAn additional motivation to study the thermodyna-\nmics of the confined phase of SU(3) gauge theory is that it\nis a parameter-free theory, simplifying the interpretatio n\nof its properties. Its spectrum is known quite accurately\nup to the two-particle threshold [14, 15]. By contrast,\nin full QCD calculations, lattice data calculated at pion\nmasses larger than in Nature are often compared out of\nnecessity to the HRG model based on the experimental\nspectrum [6, 7]. Finally, calculations in the pure gauge\ntheory are at least two orders of magnitude faster, which\nallows us to reach a high level of control of statistical and\nsystematic errors; in particular, we are able to perform\ncalculations in very large volumes.\nII. LATTICE CALCULATION\nWe use Monte-Carlo simulations of the Wilson action\nSg=1\ng2\n0/summationtext\nx,µ,νTr{1−Pµν(x)}for SU(3) gauge the-\nory [16], where Pµνis the plaquette. The lattice spacing\nis related to the bare coupling through g2\n0∼1/log(1/aΛ).\nWe calculate the thermal expectation value of θ≡Tµµ,\nthe (anomalous) trace of the energy-momentum tensor\nTµν, and of θ00≡T00−1\n4θ. In the thermodynamic limit,\nTs=e+p=4\n3/an}bracketle{tθ00/an}bracketri}htT, e−3p=/an}bracketle{tθ/an}bracketri}htT− /an}bracketle{tθ/an}bracketri}ht0.(1)\nHeree, p, s are respectively the energy density, pressure\nand entropy density. The operator θ00=1\n2(−Ea·Ea+\nBa·Ba) requires no subtraction, because its vacuum\nexpectation value vanishes. The choice of of θ00and\nθas independent linear combinations is convenient be-\ncause they both renormalize multiplicatively. We use the\n‘HYP-clover’ discretization of the energy-momentum ten-\nsor introduced in [17, 18]. The normalization of the θ00\noperator differs from its naive value by a factor that we\nparametrize as Z(g0)χ(g0). The factor Z(g0) is taken\nfrom [19] and rests on the results of [20]; its accuracy\nis about one percent. The factor χ(g0) is obtained by2\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8\n 4  5  6  7  8  9 10s / T3\nL TFinite-volume effects at 0.985Tc and 0.929Tc\n8x643     6/go2 = 6.053\n12x803    6/go2 = 6.3238\n8x643     6/go2 = 6.018\n12x803    6/go2 = 6.2822\nAsymptot. vol. correction\n0.239 + 7.4 exp(-0.82 LT)\nFIG. 1: Finite volume effects on the entropy density close to\nthe deconfining temperature Tc.\ncalibrating our discretization to the ‘bare plaquette’ dis -\ncretization in the deconfined phase at Nt= 6 [17]. We\nfind, for 6 /g2\n0between 5.90 and 6.41, χ(g0) = 0 .1306·\n(6/g2\n0)−0.1865 with an accuracy of half a percent. For\nthe lattice beta-function that renormalizes θ, we use the\nparametrization [21] of the data in [22] and the same ca-\nlibration method.\nOur results for the entropy density from Nt= 8 and\nNt= 12 simulations are shown on Fig. (3). The displayed\nerror bars do not contain the uncertainty on the normal-\nization factor, which is much smaller and would introduce\ncorrelation between the points. This factor varies by only\n7% over the displayed interval and so to a first approxi-\nmation amounts to an overall normalization of the curve.\nOur data is about five times statistically more accurate\nthan that of previous thermodynamic studies [23, 24],\nwhich were primarily focused on the deconfined phase.\nJust as importantly, we kept the finite-spatial-volume ef-\nfects under good control, in particular very close to Tc.\nFigure (1) shows the size of finite-volume effects. For\ninstance, at 0 .985Tcthe conventional choice LT= 4 leads\nto an overestimate of the entropy density by a factor\nthree. The fact that the Nt= 12 data fall on the same\nsmooth curve as the Nt= 8 is strong evidence that dis-\ncretization errors are small. We parametrize the volume\ndependence empirically by a A+Be−cLTcurve, and use it\nto convert the Nt= 12 data to LT= 8. At 0 .929Tc, there\nis no statistically significant difference between LT= 6\nand 8 and we do not apply any correction. It is the cor-\nrected Nt= 12 data that is then displayed on Fig. (3).\nIn [25], formulas for the leading finite-volume effects\non the thermodynamic potentials were derived in terms\nof the energy gap of the theory defined on a (1 /T)×L×L\nspatial hypertorus. Close to Tc, this gap corresponds to\nthe mass of the ground state flux loop winding around\nthe cycle of length 1 /T. Ifδs(T, L)≡s(T,∞)−s(T, L), 0 1 2 3 4 5 6\n 0  0.2  0.4  0.6  0.8  1(m(T) T / Tc2)2 \n(T / Tc)2Flux-Loop Mass (Nc=3, Nf=0)\nNt > 11\nNt = 8\nNt = 6\nNt = 5\nFit\nFIG. 2: The mass of the temporal flux loop as calculated from\nPolyakov loop correlators, and the fit (5). The Nt>11 data\nare from [26], the Nt= 5 data from [27].\nthe formula then reads\nδs(T, L) =e−m(T)L\n2πL/bracketleftbig\nm2(T) +3\n2T∂Tm2(T)/bracketrightbig\n.(2)\nUsing the calculation of m(T) described in the next sec-\ntion, the predicted asymptotic approach to the infinite-\nvolume entropy density for 0 .985Tcis displayed on\nFig. (1). While the sign is correct, the magnitude of the\nfinite-volume effects is not reproduced for LT≤8. We\nconclude that the asymptotic approach to infinite volume\nsets in for very large values of LT. Since m(T)Lis only\nabout 4 when LT= 6, it is not implausible that flux-loop\nstates with high multiplicity dominate the finite-volume\neffects at that box size.\nNext we obtain the correlation length ξ(T) of the\norder parameter for the deconfining phase transition,\nthe Polyakov loop. The method consists in computing\nthe two-point function of zero-momentum operators, de-\nsigned to have large overlaps with the ground state flux\nloop, along a spatial direction. We fit the lattice data for\nm(T)≡1/ξ(T) displayed on Fig. (2) with the formula\n/parenleftBig\nm(T)T\nT2c/parenrightBig2\n=a0−a1/parenleftBig\nT\nTc/parenrightBig2\n−a2/parenleftBig\nT\nTc/parenrightBig4\n(3)\nand find, either fitting a2or setting it to zero,\na0= 5.76(15) , a1= 4.97(65) , a2= 0.55(54) (4)\na0= 5.90(9), a1= 5.62(10) , a2= 0 (5)\nwith in both cases a χ2/dof of about 0.3. We remark\nthat the aiare not far from the Nambu-Goto string [28]\nvalues a1=2π\n3σ\nT2\nc= 5.02(5) [27] and a2= 0 (σis the ten-\nsion of the confining string). We extract the ‘Hagedorn’\ntemperature, defined as in [29] by m(Th) = 0, from the\nsecond fit,\nTh/Tc= 1.024(3) . (6)3\nThis extraction amounts to assuming mean-field expo-\nnents near Th(it is not clear which universality class\nshould be used [30]). The result is stable if the fit in-\nterval is varied, and also if a2is fitted with a0anda1\nconstrained to the known values of ( σ/T2\nc)2and2π\n3σ\nT2c.\nAs a check on the normalization of the operators θ00\nandθ, we calculate the latent heat in two different ways.\nThe latent heat is the jump in energy density at Tc. Since\nthe pressure is continuous, we obtain it instead from the\ndiscontinuity in entropy density or the ‘conformality mea-\nsure’e−3p. We obtain sande−3pon either side of\nTcby extrapolating LT= 10 data from the confined (de-\nconfined) phase towards Tc. The result is\n∆s\nT3c= 1.45(5)(5) ,∆(e−3p)\nT4c= 1.39(4)(5) ,(7)\nwhere the first error is statistical and the second comes\nfrom the uncertainty in the extrapolation (taken to be\nthe difference between a linear and quadratic fit). The\ncompatibility between these two estimates of Lh/T4\ncis\nstrong evidence that we control the normalization of our\noperators. They are in good agreement with previous cal-\nculations of the latent heat performed on coarser lattices\n[27, 31]. We have also verified more generally that the\nthermodynamic identity T∂T(s/T3) = (1 /T3)∂T(e−3p)\nis satisfied within statistical errors.\nIII. INTERPRETATION\nIn infinite volume the pressure associated with a single\nnon-interacting, relativistic particle species of mass M\nwithnσpolarization states reads\np=nσ\n2π2M2T2∞/summationdisplay\nn=11\nn2K2(nM/T ) (8)\nwhere K2is a modified Bessel function. By linearity,\nthe knowledge of the glueball spectrum leads to a simple\nprediction for the pressure and entropy density s=∂p\n∂T,\nwhich is expected to become exact in the large- Nlimit.\nSince only the low-lying spectrum of glueballs is known,\nit is useful to consider how the density of states might be\nextended above the two-particle threshold 2 M0, where\nM0is the mass of the lightest (scalar) glueball. The\nasymptotic closed bosonic string density of states in four\ndimensions is given by [32]\nρ(M) =(2π)3\n27Th/parenleftbiggTh\nM/parenrightbigg4\neM/T h. (9)\nIn the string theory, the Hagedorn temperature This re-\nlated to the string tension, T2\nh=3σ\n2π, corresponding to\nTh/Tc= 1.069(5) [33]. Below we use this value as an\nalternative to the more direct determination (6).\nOn Fig. 3, we show the entropy contribution of the\nglueballs lying below the two-particle threshold 2 M0. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4\n 0.7  0.75  0.8  0.85  0.9  0.95  1s / T3\nT / TcEntropy of the confined phase (Nc=3, Nf=0)\n8 x 643\n12 x 803\nGlueballs (<2M0) + Hagedorn (>2M0)\nidem, with Th2 = 3σ/2π\nGlueballs (<2M0)\n0++ & 2++ contribution\nFIG. 3: The entropy density in units of T3forLT= 8. We\napplied a (modest) volume-correction to the Nt= 12 data.\nThe curve is just about consistent with the smallest tem-\nperature lattice data point, but clearly fails to reproduce\nthe strong increase in entropy density as T→Tc. The\nfigure also illustrates that the two lowest-lying states, th e\nscalar and tensor glueballs, account for about three quar-\nters of the stable glueballs’ contribution. We have used\nthe continuum-extrapolated lattice spectrum [14, 34].\nAdding the Hagedorn spectrum contribution, Eq. (9)\nwithThgiven by Eq. (6), leads to the solid curve on\nFig. 3. It describes the direct calculation of the entropy\ndensity surprisingly well, particularly close to Tc. The\ncurve tends to underestimate somewhat the entropy den-\nsity at the lower temperatures. This is likely to be a cut-\noff effect. Indeed, at fixed Ntlower temperatures corre-\nspond to a coarser lattice spacing, and the scalar glueball\nmass in physical units is known to be smaller on coarse\nlattices with the Wilson action [35]. If we use the stable\nglueball spectrum calculated at g2\n0= 1 instead of the con-\ntinuum spectrum, the agreement of the non-interacting\nglueball + Hagedorn spectrum with the lattice data at\nthe lower four temperatures is again excellent. This dif-\nference provides an estimate for the size of lattice effects.\nTo summarize, we have computed to high accuracy the\nentropy of the confined phase of QCD without quarks.\nThe low-lying states of the theory are therefore bound\nstates called glueballs, and their spectrum is well deter-\nmined [14, 15]. If the size Nof the gauge group is in-\ncreased, the interactions of the glueballs are expected to\nbe suppressed [13]. To what extent the glueballs really\nare weakly interacting at N= 3 is not known precisely.\nSome evidence for the smallness of their low-energy in-\nteractions was found some time ago by looking at the\nfinite-volume effects on their masses [26]. But it seems\nunlikely that glueballs well above the two-particle thresh -\nold would have a small decay width. We have neverthe-\nless compared the entropy density data to the entropy\ndensity of a gas of non-interacting glueballs. While re-4\nstricting the spectral sum to the stable glueballs leads to\nan underestimate by at least a factor two of the entropy\ndensity near Tc, extending the spectral sum with an ex-\nponential spectrum ρ(M)∼exp(M/T h), suggested long\nago by Hagedorn [36], leads to a prediction in excellent\nagreement with the lattice data for the entropy density\n(Fig. 3). This is remarkable, since the analytic form of\nthe asymptotic spectrum is completely predicted by free\nbosonic string theory, including its overall normalizatio n\n(Eq. 9). Therefore, since we also separately computed\nthe temperature (identified with Th) where the flux loop\nmass vanishes, no parameter was fitted in the comparison\nwith the thermodynamic data. By contrast, the entropy\ndensity is not nearly as well described if the Nambu-Goto\nvalue of This used, see Fig. (3).\nThe success of the non-interacting string density of\nstates in reproducing the entropy density suggests that\nonce the Hagedorn temperature has been determined di-\nrectly from the divergence of the flux-loop correlation\nlength, the residual effects of interactions on the thermo-\ndynamic potentials are small. It may be that thermody-\nnamic properties in general are not strongly influenced by\ninteractions when a large number of states are contribut-\ning. A well-known example is provided by the N= 4\nsuper-Yang-Mills theory, whose entropy density at very\nstrong coupling is only reduced by a factor 3/4 with re-spect to the free theory [37]. In this interpretation, the\nmain effect of interactions among glueballs on thermo-\ndynamic properties is to slightly shift the value of the\nHagedorn temperature Thwith respect to its free-string\nvalue. A possible mechanism is that the string tension\nthat effectively determines This an in-medium string ten-\nsion that is ∼8% lower than at T= 0.\nReturning to full QCD, our results lend support to the\nidea that the hadron resonance gas model can largely\naccount for the thermodynamic properties of the low-\ntemperature phase. Whether the open string density of\nstates reproduces the entropy calculated on the lattice\ncan also be tested at quark masses not necessarily as\nlight as in Nature using a simple open string model [38].\nAcknowledgments\nI thank B. Zwiebach for a discussion on the bosonic\nstring density of states. The simulations were done on\nthe Blue Gene L rack and the desktop machines of the\nLaboratory for Nuclear Science at M.I.T. This work was\nsupported in part by funds provided by the U.S. Depart-\nment of Energy under cooperative research agreement\nDE-FG02-94ER40818.\n[1] B. Mueller, Prog. Theor. Phys. Suppl. 174, 103 (2008).\n[2] I. Arsene et al. (BRAHMS), Nucl. Phys. A757 , 1 (2005),\nnucl-ex/0410020.\n[3] B. B. Back et al., Nucl. Phys. A757 , 28 (2005), nucl-\nex/0410022.\n[4] K. Adcox et al. (PHENIX), Nucl. Phys. A757 , 184\n(2005), nucl-ex/0410003.\n[5] J. Adams et al. (STAR), Nucl. Phys. A757 , 102 (2005),\nnucl-ex/0501009.\n[6] M. Cheng et al., Phys. Rev. D77, 014511 (2008),\n0710.0354.\n[7] A. Bazavov et al. (2009), 0903.4379.\n[8] Y. Aoki, Z. Fodor, S. D. Katz, and K. K. Szabo, Phys.\nLett.B643 , 46 (2006), hep-lat/0609068.\n[9] Y. Aoki et al. (2009), 0903.4155.\n[10] F. Karsch, K. Redlich, and A. Tawfik, Eur. Phys. J. C29,\n549 (2003), hep-ph/0303108.\n[11] A. Andronic, P. Braun-Munzinger, and J. Stachel, Phys.\nLett.B673 , 142 (2009), 0812.1186.\n[12] N. Demir and S. A. Bass (2008), 0812.2422.\n[13] E. Witten, Nucl. Phys. B160 , 57 (1979).\n[14] H. B. Meyer (2004), hep-lat/0508002.\n[15] Y. Chen et al., Phys. Rev. D73, 014516 (2006), hep-\nlat/0510074.\n[16] K. G. Wilson, Phys. Rev. D10, 2445 (1974).\n[17] H. B. Meyer and J. W. Negele, Phys. Rev. D77, 037501\n(2008), 0707.3225.\n[18] H. B. Meyer and J. W. Negele, PoS LATTICE2007 ,\n154 (2007), 0710.0019.\n[19] H. B. Meyer, Phys. Rev. D76, 101701 (2007), 0704.1801.\n[20] J. Engels, F. Karsch, and T. Scheideler, Nucl. Phys.B564 , 303 (2000), hep-lat/9905002.\n[21] S. Duerr, Z. Fodor, C. Hoelbling, and T. Kurth, JHEP\n04, 055 (2007), hep-lat/0612021.\n[22] S. Necco and R. Sommer, Nucl. Phys. B622 , 328 (2002),\nhep-lat/0108008.\n[23] G. Boyd et al., Nucl. Phys. B469 , 419 (1996), hep-\nlat/9602007.\n[24] Y. Namekawa et al. (CP-PACS), Phys. Rev. D64, 074507\n(2001), hep-lat/0105012.\n[25] H. B. Meyer (2009), 0905.1663.\n[26] H. B. Meyer, JHEP 03, 064 (2005), hep-lat/0412021.\n[27] B. Lucini, M. Teper, and U. Wenger, JHEP 02, 033\n(2005), hep-lat/0502003.\n[28] J. F. Arvis, Phys. Lett. B127 , 106 (1983).\n[29] B. Bringoltz and M. Teper, Phys. Rev. D73, 014517\n(2006), hep-lat/0508021.\n[30] L. G. Yaffe and B. Svetitsky, Phys. Rev. D26, 963 (1982).\n[31] B. Beinlich, F. Karsch, and A. Peikert, Phys. Lett. B390 ,\n268 (1997), hep-lat/9608141.\n[32] B. Zwiebach (2004), A First Course in String Theory,\nCambridge, UK: Univ. Pr. 558 p.\n[33] B. Lucini, M. Teper, and U. Wenger, JHEP 01, 061\n(2004), hep-lat/0307017.\n[34] H. B. Meyer, JHEP 01, 071 (2009), 0808.3151.\n[35] B. Lucini and M. Teper, JHEP 06, 050 (2001), hep-\nlat/0103027.\n[36] R. Hagedorn, Nuovo Cim. Suppl. 3, 147 (1965).\n[37] S. S. Gubser, I. R. Klebanov, and A. W. Peet, Phys. Rev.\nD54, 3915 (1996), hep-th/9602135.\n[38] A. Selem and F. Wilczek (2006), hep-ph/0602128."    },    {        "title": "0906.1036v1.Quantum_Hall_Systems_Studied_by_the_Density_Matrix_Renormalization_Group_Method.pdf",        "content": "arXiv:0906.1036v1  [cond-mat.str-el]  5 Jun 20091\nQuantum Hall Systems Studied by the Density Matrix\nRenormalization Group Method\nNaokazu Shibata\nDepertment of Physicas, Tohoku University, Aoba, Aoba-ku, Sendai, Miyagi\n980-8578\nThe ground-state and low-energy excitations of quantum Hal l systems are studied by the\ndensity matrix renormalization group (DMRG) method. From t he ground-state pair corre-\nlation functions and low-energy excitions, the ground-sta te phase diagram is determined,\nwhich consists of incompressible liquid states, Fermi liqu id type compressible liquid states,\nand many kinds of CDW states called stripe, bubble and Wigner crystal. The spin transition\nand the domain formation are studied at ν= 2/3. The evolution from composite fermion\nliquid state to an excitonic state in bilayer systems is inve stigated at total filling factor ν= 1.\n§1. Introduction\nIntwodimensionalsystems, applyingperpendicularmagnet icfieldstronglymod-\nifies the wave function of electrons leading to many interest ing phenomena at low\ntemperatures. The fractional quantum Hall effects1)are typical example, where in-\ncompressible ground states are realized only at some fracti onal fillings of Landau\nlevels.2),3)Since fractional quantum Hall effects are observed only in hig h quality\nsamples, the Coulomb interaction between the electrons is t hought to be essential\nrather than random potentials from impurities. This is cont rasted with the case of\ninteger quantum Hall effect where random potentials are essen tial. The importance\nof the Coulomb interaction in high magnetic field is followed from the increase in\nthe energy scale of the Coulomb interaction. The wave functi on is scaled by the\nmagnetic length ℓ=/radicalbig\n/planckover2pi1/eB, which is equivalent to the classical cyclotron radius rc\nin the lowest Landau level. The increase in the magnetic field decreases the mag-\nnetic length and enhances the energy scale of the Coulomb int eraction between the\nelectrons, e2/εℓ.\nAt typical magnetic field of 10T, ℓis about 8nm, which is still much larger\nthan the atomic length of 0.1nm. Since the conduction electr ons are on the positive\nbackground charge from ions over length scale of ℓ, the positive charge may be\nsimplified to be uniform. Then the system is equivalent to the electron gas in a\nmagnetic field, and ℓbecomes unique length scale of the system.\nIn the magnetic field, the kinetic energy is scaled by the cycl otron frequency\nωc=eB/m, which is determined by the magnetic field B. The quantization of the\nwave function discretizes the classical cyclotron radius rc, which also discretizes the\nkinetic energy and makes Landau levels E=/planckover2pi1ωc(n+ 1/2). This means that the\nmacroscopic number of electrons have the same energy in each Landau level, and\nlarge-scale degeneracy appears in the ground state. This ma croscopic degeneracy is\nlifted by the Coulomb interaction between the electrons and various types of liquid\ntypeset using PTPTEX.cls/angbracketleftVer.0.9/angbracketright2 Naokazu Shibata\nstates4),5),6),7),8)and CDW states9),10),11),12)are realized depending on the filling of\nthe Landau levels.\nSince the ground state has macroscopic degeneracy in the lim it of weak Coulomb\ninteraction, standard perturbation theories are not usefu l. Thus numerical diagonal-\nizations of the many body Hamiltonian have been used to study this system. Since\nnumerical representation of the Hamiltonian needs complet e set of many body basis\nstates, we divide the system into unit cells with finite numbe r of electrons in each\ncell. The properties of the infinite system are obtained by th e finite size scalings.\nHowever, the number of many body basis states increases expo nentially with the\nnumber of electrons. For example, when we study the ground st ate atν= 1/3\nwith 18 electrons, each unit cell has 54 degenerated orbital s. The number of many\nbody basis states is given by the combination of occupied and unoccupied orbitals,\n54C18∼1014, which is practically impossible to manage by using standar d numerical\nmethod such as exact diagonalization.\nTo study systems with typically more than 10 electrons, we ne ed to reduce the\nnumberofmanybodybasisstates. For thispurpose,weusethe density matrixrenor-\nmalization group (DMRG) method, which was originally devel oped by S. White in\n1992.13),14)This method is a kind of variational method combined with a re al space\nrenormalization group method, which enables us to obtain th e ground-state wave\nfunction of large-size systems with controlled high accura cy within a restricted num-\nber of many body basis states. The DMRG method has excellent f eatures compared\nwith other standard numerical methods. In contrast to the qu antum Monte Carlo\nmethod, theDMRG method is freefrom statistical errors and t henegative sign prob-\nlem, whichinhibitconvergence ofphysical quantities atlo w temperatures. Compared\nwith the exact diagonalization method, the DMRG method has n o limitation in the\nsize of system. The error in the DMRG calculation comes from r estrictions of the\nnumber of basis states, which is systematically controlled by the density matrix cal-\nculated from the ground-state wave function, and the obtain ed results are easily\nimproved by increasing the number of basis states retained i n the system.\nThe application of the DMRG method to two-dimensional quant um systems\nis a challenging subject and many algorithms have been propo sed. Most of them\nuse mappings on to effective one-dimensional models with long -range interactions.\nHowever, the mapping from two-dimensional systems to one-d imensional effective\nmodels is not unique and proper mappingis necessary to keep h igh accuracy. In two-\ndimensional systems under a perpendicular magnetic field, a ll the one-particle wave\nfunctions ΨNX(x,y) are identified by the Landau level index Nand the x-component\nof the guiding center, X, in Landau gage. The guiding center is essentially the cente r\ncoordinate of the cyclotron motion of the electron and it is n atural to use Xas a\none-dimensional index of the effective model. More important ly,Xis discretized in\nfinite unit cell of Lx×Lythrough the relation to y-momentum, X=kyℓ2, which\nis discretized under the periodic boundary condition, ky= 2πn/Lywithnbeing\nan integer. Therefore, the two-dimensional continuous sys tems in magnetic field are\nnaturally mapped on to effective one-dimensional lattice mod els, and we can apply\nthe standard DMRG method.15)\nThis method was first applied to interacting electron system s in a high LandauQuantum Hall Systems Studied by the DMRG Method 3\nlevel and the ground-state phase diagram, which consists of various CDW states\ncalled stripe, bubble and Wigner crystal, has been determin ed.16),17)The ground\nstate and low energy excitations in the lowest and the second lowest Landau levels\nhave also been studied by the DMRG and the existence of variou s quantum liquid\nstates such as Laughlin state and charge ordered states call ed Wigner crystal have\nbeen confirmed and new stripe state has been proposed.18),19)\nIn the following, we first explain the effective one-dimension al Hamiltonian used\nin the above studies and then show the results obtained for th e spin polarized single\nlayer system. We next review recent study on the spin transit ion and domain forma-\ntion atν= 2/3,20)and finally explain the results on bilayer quantum Hall syste ms\natν= 1,21)where crossover from a Fermi liquid state to an excitonic inc ompressible\nstate occurs.\n§2. DMRG method\nHere we briefly describe how the effective 1D Hamiltonian is obt ained from 2D\nquantum Hall systems.15)To describe the many body Hamiltonian for a interacting\nsystem, we first need to define one-particle basis states. Her e, we use the eigenstates\nof free electrons in a magnetic field as one-particle basis st ates and represent the\nwave function ΨNX(x,y) in Landau gauge:\nΨNX(x,y) =CNexp/bracketleftbigg\nikyy−(x−X)2\n2ℓ2/bracketrightbigg\nHN/bracketleftbiggx−X\nℓ/bracketrightbigg\n, (2.1)\nwhereHNare Hermite polynomials and CNis the normalization constant. Then\nall the eigenstates ΨNX(x,y) are specified using two independent parameters Nand\nX;Nis the Landau level index and Xis thex-component of the guiding center\ncoordinates of the electron. Since the guiding center Xis related to the momentum\nkyasX=kyℓ2, andkyis discretized under the periodic boundary conditions, the\nguiding center Xtakes only discrete values\nXn= 2πℓ2n/Ly, (2.2)\nwhereLyis the length of the unit cell in the y-direction.\nIfwefixtheLandaulevel index N, all theone-particle states arespecifiedbyone-\ndimensional discrete parameter Xn. Since many body basis states are product states\nof one-particle states, they are also described by the combi nations of Xnof electrons\nin the system. Thus the system can be mapped on to an effective on e-dimensional\nlattice model.\nThe macroscopic degeneracy in the ground state of free elect rons in partially\nfilled Landau level is lifted by the Coulomb interaction\nV(r) =e2\nǫr. (2.3)\nThe Coulomb interaction makes correlations between the ele ctrons and stabilizes\nvarious types of ground states depending on the filling νof Landau levels. When4 Naokazu Shibata\nM M−1 n=1 243 n=1 2 7 8 65\n3M−2\n4 5\nM=8\nBL BR(a) (b) M=8\nBL BR\nFig. 1. Schematic diagrams for (a) infinite system algorithm and (b) finite system algorithm of the\nDMRG method. •represents a one-particle orbital in a given Landau level. B Land B Rare left\nand right blocks, respectively.\nthe magnetic field is strong enough so that the Landau level sp litting is sufficiently\nlarge compared with the typical Coulomb interaction e2/(ǫℓ), the electrons in fully\noccupied Landau levels are inert and the ground state is dete rmined only by the\nelectrons in the top most partially filled Landau level.\nThe Hamiltonian is then written by\nH=S/summationdisplay\nnc†\nncn+1\n2/summationdisplay\nn1/summationdisplay\nn2/summationdisplay\nn3/summationdisplay\nn4An1n2n3n4c†\nn1c†\nn2cn3cn4, (2.4)\nwhere we have imposed periodic boundary conditions in both x- andy-directions,\nandSis the classical Coulomb energy of Wigner crystal with a rect angular unit cell\nofLx×Ly.22)c†\nnis the creation operator of the electron represented by the w ave\nfunction defined in equation (2 .1) withX=Xn.An1n2n3n4are the matrix elements\nof the Coulomb interaction defined by\nAn1n2n3n4=δ′\nn1+n2,n3+n41\nLxLy/summationdisplay\nqδ′\nn1−n4,qyLy/2π2πe2\nǫq\n×/bracketleftbig\nLN(q2ℓ2/2)/bracketrightbig2exp/bracketleftbigg\n−q2ℓ2\n2−i(n1−n3)qxLx\nM/bracketrightbigg\n,(2.5)\nwhereLN(x) are Laguerre polynomials with Nbeing the Landau level index.17)\nδ′\nn1,n2= 1 when n1=n2(modM) withMbeing the number of one-particle states\nin the unit cell, which is given by the area of the unit cell 2 πMℓ2=LxLy.\nInordertoobtaintheground-statewavefunctionweapplyth eDMRGmethod.15)\nAs shown in Fig. 1 (a), we start from a small-size system consi sting of only four\none-particle orbitals whose indices nare 1, 2, M−1, andM, and we calculate the\nground-state wave function. We then construct the left bloc k containing one-particle\norbitals of n= 1 and 2, and the right block containing n=M−1 andMby using\neigenvectors of the density matrices which are calculated f rom the ground-state wave\nfunction. We then add two one-particle orbitals n= 3 and M−2 between the two\nblocks and repeat the above procedure until Mone-particle orbitals are includedQuantum Hall Systems Studied by the DMRG Method 5\n-10-8-6-4-20\n0 10010\n1010101010\n200 300Ne=18  M=54\n2D system\nw\nFig. 2. Eigenvalues wαof the density matrix for two-dimensional system of 54 orbit als with 18\nelectrons. Sumof wαis equivalentto the norm of theground-state wave function a nd normalized\nto be unity.\nin the system. We then apply the finite system algorithm of the DMRG shown in\nFig. 1 (b) to refine the ground-state wave function. After we h ave obtained the\nconvergence, we calculate correlation functions to identi fy the ground state.\nThe ground-state pair correlation function g(r) in guiding center coordinates is\ndefined by\ng(r) =LxLy\nNe(Ne−1)/angbracketleftΨ|/summationdisplay\ni/negationslash=jδ(r−Ri+Rj)|Ψ/angbracketright, (2.6)\nwhereRiis the guiding center coordinate of the ith electron, and it is calculated\nfrom the following equation\ng(r) =1\nNe(Ne−1)/summationdisplay\nq/summationdisplay\nn1,n2,n3,n4exp/bracketleftbigg\niq·r−q2ℓ2\n2−i(n1−n3)qxLx\nM/bracketrightbigg\n×\nδ′\nn1−n4,qyLy/2π/angbracketleftΨ|c†\nn1c†\nn2cn3cn4|Ψ/angbracketright, (2.7)\nwhereΨis the ground state and Neis the total number of electrons.\nThe accuracy of the results depends on the distribution of ei genvalues of the\ndensity matrix. A typical example of the eigenvalues of thed ensity matrix for system\nofM= 54 with 18 electrons is shown in Fig. 2, which shows an expone ntial decrease\nof eigenvalues wα. In this case accuracy of 10−4is obtained by keeping more than\none hundred states in each block.\n§3. Single layer system\nHere we present diverse ground states obtained by the DMRG me thod applied\nto the single layer quantum Hall systems. In the limit of stro ng magnetic field,\nthe electrons occupy only the lowest Landau level N= 0. In this limit, fractional\nquantum Hall effect (FQHE) has been observed at various fracti onal fillings.3)The\nFQHE state is characterized by incompressible liquid with a finite excitation gap.1)6 Naokazu Shibata\n 0 0.02 0.04 0.06 0.08\n 0.3  0.4  0.52/53/7\n4/9\n5/11\nFig. 3. The lowest excitation gap at various νin the lowest Landau level. Relatively large excitation\ngap is obtained at fractional fillings ν=n/(2n+1). The excitation gap is in units of e2/(ǫℓ).\nThese FQHE states are confirmed by the DMRG calculations, whe re relatively\nlarge excitation gaps are obtained at various fillings betwe enν= 1/2 and 3/1018)\nas shown in Fig. 3. We clearly find large excitation gaps at fra ctional fillings ν=\n1/3,2/5,3/7,4/9 and 5/11, which correspond to primary series of the FQHE at\nν=n/(2n+ 1). The pair correlation function at ν= 1/3 is presented in Fig. 4,\nwhich shows a circularly symmetric correlation consistent with the Laughlin’s wave\nfunction.1)\nIn the limit of low filling ν→0, mean separation between the electrons becomes\nmuch longer than the typical length-scale of the one-partic le wave function. In this\nlimit the quantum fluctuations are not important and electro ns behave as classical\npoint charges. The ground state is then expected to be the Wig ner crystal. The\nformation of the Wigner crystal is also confirmed by the DMRG c alculations at low\nfillings as shown in Fig. 5 (a). The ν-dependence of the low energy spectrum shows\nthat the first-order transition to Wigner crystal occurs at ν∼1/7.18)\nWith decreasing magnetic field, electrons occupy higher Lan dau levels. In high\nLandau levels, the one-particle wave function extends over space leading to effective\nlong range exchange interactions between the electrons. Th e long range interaction\nstabilizes CDW ground states and various types of CDW states called stripe and\nbubble are predicted by Hartree-Fock theory.9)These CDW states are confirmed by\nthe DMRG calculations as shown in Figs. 5 (b) and (c), where tw o-electron bubble\nstate and stripe state are obtained at ν= 8/27 and 3 /7, respectively, in the N= 2\nLandau level. Although the CDW structures are similar to tho se obtained in the\nHartree-Fock calculations, the ground state energy and the phase diagram are sig-\nnificantly different.16)The DMRG results are consistent with recent experiments,10)\nandthediscrepancyis dueto thequantumfluctuations neglec ted intheHartree-Fock\ncalculations.\nThe ground-state phase diagram obtained by the DMRG is shown in Fig. 6. In\nthe lowest Landau level, we find many liquid states at fractio nal fillings and around\nν= 1/2. Nevertheless, CDW states dominate over the whole range of filling inQuantum Hall Systems Studied by the DMRG Method 7\nhigher Landau levels. This difference in the ground state phas e diagram comes from\ndifferent effective interactions between the electrons. In the lowest Landau level,\nthe one particle wave function is localized within the magne tic length ℓ, that yields\nstrong short-range repulsion between the electrons. Since quantum liquid states such\nas Laughlin state are stabilized by the strong short-range r epulsion, liquid states are\nrealized in the lowest Landau level. In higher Landau levels , however, the wave\nfunction extends over space with the increase in the classic al cyclotron radius rc.\nThus the short-range repulsion is reduced and liquid states become unstable. As\nshown in Fig. 7 (a), the real space effective interaction betwe en the electrons in\nhigher Landau levels has a shoulder structure around the dis tance twice the classi-\ncal cyclotron radius. This structure of effective interactio n makes minimum in the\nCoulomb potential near the guiding center of the electron as shown in Fig. 7 (b) and\nstabilizes the clustering of electrons. This is the reason w hy stripe and bubble states\nare realized in higher Landau levels.19)\n0\n3.76-3.76\n7.52-7.520\n-3.763.76\n-7.527.52\nx\nyLaughlin state\n01.0\nFig. 4. Pair correlation function g(r) atν= 1/3 in the lowest Landau level. The length is in units\nofℓ.\n9.264.630-4.63-9.26\n-7.12-3.56\n0\n3.567.12(c)  stripe\nx\ny12.706.350-6.35-12.70\n-6.68-3.34\n0\n3.34\n6.68(b)  two-electron  \n   bubble (a)\nx x\nyy-11.65\n-5.82\n0 \n5.85\n11.65\n-9.71-4.85\n0 \n4.859.71Wigner crystal\nFig. 5. Pair correlation functions g(r) in guiding center coordinates. (a) Wigner crystal realize d in\nan excited state at ν= 1/6 in the lowest Landau level. The number of electrons in the un it\ncellNeis 12. (b) Two-electron bubble state at ν= 8/27 inN= 2 Landau level. Ne= 16. (c)\nStripe state at ν= 3/7 inN= 2 Landau level. Ne= 18.8 Naokazu Shibata\nstripe I2-electron \nbubble N=2Wigner crystal\nWigner crystal\nWigner crystal\n0 0.5 0.4 0.3 0.2 0.1stripe I N=1 pairing stripe II\n0 0.5 0.4 0.3 0.2 0.1FQHE\n1/5FQHE\n1/3N=0 liquidFQHEFQHE\nNN1/32/7\n2/5FQHE\n3/74/95/11\n0 0.5 0.4 0.3 0.2 0.11/5  stripe II\nFig. 6. The ground state phase diagram obtained by the DMRG me thod.Nis the Landau level\nindex and νNin the filling factor of the Nth Landau level.\n 0 1\n 0  2  4  6  8  10\nrVeff (r)\nN=0 231N = 1N = 0\nN = 2\nN = 3pure Coulomb\n2Rc 0 1\n 0  2  4  6xVeff(x) (b) (a)\nN=2\nx\nx x xx x\nFig. 7. (a) Effective interaction between the electrons in th eNth Landau level. Rcis the classical\ncyclotron radius. (b) Coulomb potential made by two electro ns separated by ∆x.\n§4. Spin transitions\nIn two dimensional systems, strong perpendicular magnetic field completely\nquenches the kinetic energy of electrons. Since the kinetic energy is independent\nof the spin polarization, the exchange Coulomb interaction easily aligns the electron\nspin. The ferromagnetic ground state at ν= 1/q(qodd) is thus realized even in\nthe absence of the Zeeman splitting.23)At the filling ν= 2/3 and 2/5, however, the\nparamagnetic ground states compete with the ferromagnetic state, and the Zeeman\nsplitting ∆z=gµBBinduces a spin transition.24)Such aspin transition in fractional\nquantum Hall states has been naively explained by the compos ite fermion theory.25)\nThe composite fermions are electrons coupled with even numb er of fluxes. These\nfluxes effectively reduces external magnetic field and the ν=p/(2p±1) fractional\nquantum Hall effect (FQHE) state is mapped on to the ν′=pinteger QHE state\nof composite fermions. The spin transitions at ν= 2/3 and 2/526)correspond to\nthe spin transition at ν= 2, where the Zeeman splitting corresponds to the effectiveQuantum Hall Systems Studied by the DMRG Method 9\nFig. 8. Lowest energies for fixed polarization ratio Pas a function of magnetic field Bat filling\nfactorν= 2/3 in units of e2/(ǫℓ). The total number of electron is 20. The aspect ratio is fixed\nat 2.0. The g-factor is 0.44.\nFig. 9. Charge gap of ν= 2/3 spin polarized states ( /square), unpolarized states ( •), and partially\npolarized states ( ◦) for various Neand aspect ratios Lx/Ly.∆cis in units of e2/(ǫℓ).\nLandau level separation, and theenergy levels of the minori ty spinstate in thelowest\nLandau level and the majority spin state in the second lowest Landau level coincide.\nExtensive experimental27),28),29),30),31),32),33),34),35)and theoretical36),37),38),39)\nstudies have been made on this transition. Nevertheless, th ere is no clear theoretical\nconsensus on this issue. This is due to the difficulties of stud ies in this system. A\nnumber of states possibly compete in energy, and large enoug h systems are needed to\nsee non-uniform structures in the partially polarized stat es. Here we use the DMRG\nmethod,15)and study the spin transition and the spin structures in larg e system to\nclarify the nature of the spin transition at ν= 2/3.\nWe first calculate the energy at various polarization Pas a function of the\nZeeman splitting, ∆z=gµB. The obtained results are shown in Fig. 8. In the\nabsence of the Zeeman splitting, the unpolarized state ( P= 0) is the lowest. The\nenergy of the polarized state ( P >0) monotonically increases as Pincreases. With10 Naokazu Shibata\n(a)\n(b)(c)\n(d)\nFig. 10. Pair correlation functions for minority spins g↓↓atν= 2/3 for several polarization ratios\n(a)P= 0.8, (b)P= 0.6, (c)P= 0.5, and (d) P= 0.4.\nthe increase in Zeeman splitting ∆z, however, the energy of polarized state decreases\nand the fully polarized state ( P= 1) becomes the lowest. Figure 8 shows that the\ntransition from the unpolarized state to the fully polarize d state occurs at B≃6T\nwhich is roughly consistent to the earlier work done in a sphe rical geometry.24)In\nthe present calculation on a torus, all partially polarized states (0 < P <1) are\nhigher in energy than the ground states ( P= 0 or 1). This feature is independent of\nthe size of the system and the aspect ratio Lx/Ly, and indicates phase separations\nofP= 0 and P= 1 in partially polarized states.\nThe unpolarized state of P= 0 and the fully polarized state of P= 1 are both\nquantum Hall states with finite charge excitation gap, which is defined by\n∆c(P) =E(Nφ+1,P)+E(Nφ−1,P)−2E(Nφ,P), (4.1)\nwhereNφis the number of one-particle states in the lowest Landau lev el. The filling\nfactorνis then given by Ne/Nφ. The charge gap ∆cfor various Nφand aspect\nratios of the unit cell is presented in Fig. 9. In this figure, t he gap∆cseems to\nvanish for partially polarized state P∼1/2 in the limit of Ne→ ∞. This result\nclearly indicates that partially polarized state with P∼1/2 is a compressible state\nin contrast to the incompressible states at P= 0 and 1, where ∆cremains to be\nfinite in the limit of Ne→ ∞.\nTo study the spin structure in the partially polarized state s, we next calculate\nthe pair-correlation function defined by\ngσσ(r) =LxLy\nNσ(Nσ−1)/angbracketleftΨ|/summationdisplay\nnmδ(r+Rσ,n−Rσ,m)|Ψ/angbracketright, (4.2)\nwhereσ=±1/2 is the spin index and Nσis the number of electrons with spin\nσ. The spin structures in partially spin polarized states are clearly shown in the\npair correlation function between minority spins. Namely, if unpolarized regions\nare formed in the partially polarized states, then electron s with minority spins are\nconcentrated in the unpolarized regions. This concentrati on of the minority spins isQuantum Hall Systems Studied by the DMRG Method 11\nFig. 11. Local densities of up spin, and down spin electrons f or various polarization ratios Pat\nν= 2/3. The number of electrons is 20.\nshown in Fig. 10, which shows g↓↓(x,y) for partially polarized states at (a) P= 0.8,\n(b)P= 0.6, (c)P= 0.5, and (d) P= 0.4. When Pis close to 1, for example P= 0.8\nshown in Fig. 10(a), a pair of minority spins is found only nea r the origin. As the\npolarization ratio Pdecreases, minority spins make a domain around the origin, a nd\ntwo domain walls along the y-direction are formed. These domain walls move along\nx-direction and the domain of minority spin finally covers ent ire unit cell in the limit\nofP= 0. This change in the size of the domain is consistent with th e expectation\nthat the domain in Fig. 10 corresponds to the unpolarized spi n singlet region where\nthe density of up-spin electrons and the down-spin electron s are the same.\nTo confirm the separation of the unpolarized and polarized sp in regions, we\nnext consider the local electron density of up-spin electro nsν↑(x) and down-spin\nelectrons ν↓(x). Figure 11 shows ν↑(x) andν↓(x) for partially polarized states with\nP= 0.2,0.4,0.6 and 0.7. Here ν↑(x) andν↓(x) are scaled to be the local filling\nfactor of the lowest LL. Thus, the total local electron densi tyν↑(x)+ν↓(x) is almost\n2/3. In this figure the separation to two regions is clearly seen ; the unpolarized spin\nregion around Lx/2, where both ν↑andν↓are close to 1/3, and the fully polarized\nspin region around x∼0 or equivalently x∼Lx, whereν↑is almost 2/3 while ν↓\nis close to 0. These results confirm the separation of the unpo larized and polarized\nspin regions as expected from the pair correlation function s shown in Fig. 11.\nThe polarized and unpolarized spin regions are separated by the domain walls\nwhose width is about 4 ℓ. This means that the phase separation is realized only\nfor systems whose size of the unit cell Lx,(Ly) is larger than twice the width of\ndomain wall; Lx,(Ly)>8ℓ. Indeed, exact diagonalization studies up to Ne= 8\nelectrons have never found the phase separation at ν= 2/3.39)We have found the\nphase separation only for large systems with Ne>12. We note that above behavior\nis generic over the aspect ratio. In an ideal system, the two s tates separate into\ntwo regions even when the system size is infinitely large. In e xperimental situations,\nhowever, multi-domain structuresarerealized duetothein homogeneity andcoupling\nwith randomly distributed nuclear spins.12 Naokazu Shibata\nThe DMRG study on the ground state energy for various polariz ationPshows\nthat the ground state at ν= 2/3 evolves discontinuously from the unpolarized P= 0\nstate to the fullypolarized P= 1state as the Zeeman splittingincreases. In partially\npolarized states 0 < P <1, the electronic system separates spontaneously into two\nstates; the P= 0 and the P= 1 states. These two states are separated by the\ndomain wall of width4 ℓ. Sincetheenergy of thedomain wall is positive, thepartial ly\npolarized states always have higher energy than that of P= 1 orP= 0 states. We\nthink this is the reason of the direct first order transition f romP= 0 toP= 1 state\nin the ground state.\nIt is useful to compare our result with the spin transition at ν= 2 which occurs\nwhen minority spin states in the lowest LL and majority spin s tates in the second\nlowest LL cross by varying the ratio of the Zeeman and Coulomb energy. The\nground state at ν= 2 is thus a fully polarized state or a spin singlet state. In\nanalogous to the case of ν= 2/3 the transition between them is first order,40)\nand spin domain walls have been found in high energy states.41)This analogy can\nbe expected, because the ν= 2 states and the ν= 2/3 states are connected in the\ncompositefermiontheory,25),26)althoughtheeffective interaction betweencomposite\nfermions is different from that for electrons.\n§5. Bilayer system\nThepropertiesofquantumHall systemssensitively dependo nthemagneticfield,\nand various types of ground states including incompressibl e liquids,1)compressible\nliquids,42),43)spin singlet liquid, CDW states called stripes, bubbles, an d Wigner\ncrystal are realized depending on the filling νof Landau levels. In bilayer quantum\nHall systems, additional length scale of the layer distance d, and the degrees of\nfreedom of layers make the ground state much more diverse and interesting.44)\nExcitonic phase, namely Haplerin’s Ψ1,1,1state, is one of the ground states real-\nized in bilayer quantum Hall systems at total filling ν= 1 at small layer separation\nd, where electrons and holes in different layers are bound with e ach other due to\nstrong interlayer Coulomb interaction. This excitonic sta te has recently attracted\nmuch attention because a dramatic enhancement of zero bias t unneling conductance\nbetween the two layers,45)and the vanishing of the Hall counterflow resistance are\nobserved.46),47)As the layer separation dis increased, the excitonic phase vanishes,\nand at large enough separation, composite-fermion Fermi-l iquid state is realized in\neach layer.\nSeveral scenarios have been proposed for the transition of t he ground state as\nthelayer separation increases.48),49),50),51),52),53),54)However howtheexcitonic state\ndevelops into independent Fermi-liquid state has not been f ully understood. In this\nsection we investigate the ground state of ν= 1 bilayer quantum Hall systems\nby using the DMRG method.15)We calculate energy gap, two-particle correlation\nfunction g(r) and excitonic correlation function for various values of l ayer separation\nd, and show the evolution of the ground state with increasing d.Quantum Hall Systems Studied by the DMRG Method 13\nNe=24 Lx/Ly=1.6Ne=18 Lx/Ly=1.0Ne=12 Lx/Ly=1. 5\nd/lgex(n)n=M /2\nn=M /2−1\n 0 0.2 0.4 0.6 0.8 1\n 0  1  2\nFig. 12. The exciton correlation of bilayer quantum Hall sys tems atν= 1. The solid line represents\ngex(M/2). The dashed line represents gex(M/2−1).\nThe Hamiltonian of the bilayer quantum Hall systems is writt en by\nH=/summationdisplay\ni<j/summationdisplay\nqV(q) e−q2ℓ2/2eiq·(R1,i−R1,j)\n+/summationdisplay\ni<j/summationdisplay\nqV(q) e−q2ℓ2/2eiq·(R2,i−R2,j)\n+/summationdisplay\ni,j/summationdisplay\nqV(q) e−qde−q2ℓ2/2eiq·(R1,i−R2,j), (5.1)\nwhereR1,iare the two-dimensional guiding center coordinates of the ith electron\nin the layer-1 and R2,iare that in the layer-2. The guiding center coordinates\nsatisfy the commutation relation, [ Rx\nj,Ry\nk] = iℓ2δjk.V(q) = 2πe2/(ǫq) is the Fourier\ntransform of the Coulomb interaction and the wave function i s projected on to the\nlowest Landau level. We consider uniform positive backgrou nd charge to cancel\nthe component at q= 0. We will assume zero interlayer tunneling and fully spin\npolarized ground state.\nIn the limit of d= 0, electrons in different layers can not occupy the same\nposition because of the strong interlayer Coulomb repulsio n. The strong interlayer\nrepulsionmakes electron-hole pairs, whichiscalled excit ons whosedegreesoffreedom\nare represented by interlayer dipoles or pseudo-spins at to tal filling ν= 1. The\nCoulombexchangeinteraction aligns theinterlayer dipole s(thepseudo-spins)leading\nto the macroscopic coherence of the excitons and Haplerin’s Ψ1,1,1state is realized.\nTo confirm the coherence of the excitons, we calculate the exc iton correlation\ndefined by\ngex(n)≡2M−1\nN1N2/angbracketleftΨ|c†\n1,nc2,nc†\n2,0c1,0|Ψ/angbracketright, (5.2)\nwhere|Ψ/angbracketrightis the ground state and c†\n1,n(c†\n2,n) is the creation operator of the electrons14 Naokazu Shibata\n 0 0.02 0.04 0.06 0.08\n 0  0.02  0.04  0.06  0.08  0.1Ne=18\nNe=24\nNe=32\nNe=40\nd=0.2d=0.6\n1/Lx∆ps\nFig. 13. The pseudo-spin excitation gap ∆psof bilayer quantum Hall systems at the total filling\nfactorν= 1. The dashed lines are guide for the eye.\nin thenth one-particle state defined by\nφn(r) =1/radicalBig\nLyπ1/2ℓexp/braceleftbigg\nikyy−(x−Xn)2\n2ℓ2/bracerightbigg\n(5.3)\nin the layer-1 (layer-2). Xn=nLx/M=kyℓ2is thex-component of the guiding\ncenter coordinates and Lxis the length of the unit cell in the xdirection. Mis the\nnumber of one-particle state in each layer. N1andN2are the number of electrons\nin the layer-1 and layer-2, respectively, and we impose the s ymmetric condition\nofN1=N2. Sincegex(n) represents the correlations between the two excitons at\nX= 0 andX=Xn, limn→∞gex(n)/negationslash= 0 indicates existence of macroscopic coherence\nof excitons.\nAs is shown in Fig. 12, gex(n) tends to 1 as d→0, that confirms the macroscopic\ncoherence of excitons at d= 0. Indeed, Haplerin’s Ψ1,1,1state has the macroscopic\ncoherence of excitons and gex(n) = 1 independent of n. In this figure we have\nshowngex(M/2) instead of lim n→∞gex(n), because the largest distance between the\ntwo excitons is Lx/2 in the finite unit cell of Lx×Lyunder the periodic boundary\nconditions. In order to check the size effect, we also plot gex(M/2−1) with the\ndashed line. Since the difference between gex(M/2) andgex(M/2−1) is small, we\nexpectgex(M/2) well represents the macroscopic coherence in the limit of N→ ∞.\nWith increasing d/ℓ, the excitonic correlation decreases monotonically and fin ally\nfalls down to negligible value at d/ℓ∼1.6.\nThepresenceofmacroscopiccoherenceofexcitonsshownint heFig.12meansthe\nexistence of ferromagnetic order of the interlayer dipoles (the pseudo-spins). Since\ninteraction between the interlayer dipoles has SU(2) symme try atd= 0, corrective\ngapless excitations called pseudo-spin waves are expected . Even in the case of finite\nlayer distance d, the Hamiltonian has continuous XY symmetry, and gapless ps eudo-\nspin wave excitations are still expected. This is confirmed b y the size dependence of\nthe pseudo-spin excitation gap shown in Fig. 13, where the ps eudo-spin excitationQuantum Hall Systems Studied by the DMRG Method 15\n 0 0.2 0.4 0.6 0.8 1\n 0  0.02  0.04  0.06  0.08  0.1  0.12\n1/LxNe=12Ne=18\n∆cd=0\nd=0.2\nd=0.6\nd=1.0\nFig. 14. The charge excitation gap ∆cof bilayer quantum Hall systems at the total filling factor\nν= 1. The dashed lines are guide for the eye.\ngap∆ps=E(N1+ 1,N2−1,M)−E(N1,N2,M) in finite system decreases as a\nfunction of 1 /Lx.\nIn contrast to the pseudo-spin excitation gap ∆ps, the charge excitation gap ∆c\ndefined by ∆c=E(N1,N2,M−1) +E(N1,N2,M+ 1)−2E(N1,N2,M) seems to\nbe finite even in the limit of Lx→ ∞for small das shown in Fig. 14. The charge\nexcitation brakes at least one electron-hole pair and it nee ds energy of order V(1,2)\n0\nwhich is the pseudopotential between the electrons in differe nt layers whose relative\nangular momentum is 0. This pseudopotential decreases with the increase in the\nlayer distance d, and thus the charge gap decreases with the increase in d.\nWe next see the lowest excitation gap in a fixed size of system. Figure 15 shows\nthe result for Ne= 24. The aspect ratio Lx/Ly= 1.8 is chosen from the minimum\nof the ground state energy with respect to Lx/Lyaroundd/ℓ= 1.8, where minimum\nstructure appears in the ground state energy.\nWe can see clear excitation gap of finite system for d/ℓ <1.2, where excitonic\ngroundstateisexpectedboththeoretically andexperiment ally.45),46),47),48),49),50),51),52),53),54)\nThe excitation gap rapidly decreases with increasing d/ℓfrom 1.2, and it becomes\nvery small for d/ℓ >1.6. This behavior is consistent with experiments.55)Although\nthe excitation gap for d/ℓ >1.7 is not presented in the figure for Ne= 24 because\nof the difficulty of the calculation of excited states in large system, we do not find\nany sign of level crossing in the ground state up to d/ℓ∼4, where two layers are\nalmost independent. These results suggest that the exciton ic state at small d/ℓcon-\ntinuously crossovers to compressible state at large d/ℓ, that is consistent with the\nbehaviors of exciton correlations gex(M/2) in Fig. 12, which shows gex(M/2) contin-\nuously approaches zero around d/ℓ∼1.6. In the present calculation it is difficult to\nconclude whether the gap closes at finite d/ℓ∼1.6 in the thermodynamic limit or\nexcitonic state survives with exponentially small finite ga p even for large d/ℓ >1.6.\nWe have calculated the excitation gap in different size of syst ems and aspect ratios,\nand obtained similar results as shown in the inset of Fig. 15.16 Naokazu Shibata\n 0 0.02 0.04 0.06 0.08\n 0 0.4  0.8  1.2  1.6  2  2.2\nd/ld/lNe=24\nNe=18\n 0 0.1\n 0  1  2\nFig. 15. The lowest excitation gap ∆of bilayer quantum Hall systems at the total filling factor\nν= 1.Ne= 24 and Lx/Ly= 1.6. The inset shows the result for Ne= 18 and Lx/Ly= 1.0.\nConcerning the first excited state, however, Fig. 15. shows a level crossing at\nd/ℓ∼1.2, where we can see sudden decrease in the excitation gap. We e xpect that\nthe lowest excitation at d/ℓ <1.2 is the pseudo-spin excitation whose energy gap\ndecreases with the increase in the size of system and tends to zero in the limit of\nlarge system. On the other hand, the lowest excitation at d/ℓ >1.2 shown in Fig. 15\nis expected to be the excitation to the roton minimum which co rresponds to the\nbound state of quasiparticle and quasihole excitatins, who se energy increases with\nthe decrease in d/ℓ. This change in the character of the low energy excitations a t\nd/ℓ∼1.2 will be confirmed in a clear change in correlation functions in the excited\nstate as shown later. We note that the position of the level cr ossing in the first\nexcited state itself depends on the size of system because th e pseudo-spin excitation\ngap decreases with the increase in the system size. However, the change in the\ncharacter of the low energy excitations of finite systems is e xpected to remain even\nin the limit of large system, since the spectrum weight of pse udo-spin waves transfers\nto high energy with the increase in d.\nWe next calculate pair correlation functions of the electro ns to see detailed evo-\nlution of the ground-state wave function. The interlayer pa ir correlation functions\nare defined by\ng12(r)≡LxLy\nN1N2/angbracketleftΨ|/summationdisplay\nn mδ(r+R1,n−R2,m)|Ψ/angbracketright, (5.4)\nwhere|Ψ/angbracketrightis the ground state. We present ∆g12(r) in Fig. 16, which is defined by\n∆g12(r) =/integraldisplay\n(g12(r′)−1)δ(|r′|−r) dr′, (5.5)\nwherer′is the two-dimensional position vector in each layer. ∆g12(r) represents the\ndifference from the uniform correlation of independent elect rons.Quantum Hall Systems Studied by the DMRG Method 17\n-0.3-0.2-0.1 0 0.1 0.2\n 0  1  2  3  4d/l=0.1d/l=0.10.2 0.40.60.8\n0.61.01.2\n1.21.41.61.81.82.02.22.4\n2.42.63.0\nr/lr/lNe=24\nNe=24\n-1 0\n 0  1  2  3  4\nFig. 16. The inter-layer pair correlation function of elect rons in the ground state of bilayer quantum\nHall systems at ν= 1.Ne= 24 and Ly/Lx= 1.6.\nAtd/ℓ= 0 we find clear negative ∆g12(r) around r/ℓ= 1, which is a characteris-\ntic feature of the excitonic state made by the binding of elec trons and holes between\nthe two layers. The binding of one hole means the exclusion of one electron caused\nby the strong interlayer Coulomb repulsion. The increase in the layer separation\nweakens Coulomb repulsion between the two layers and reduce s|∆g12(r)|around\nr/ℓ= 1.\nThe decrease in the interlayer correlation |∆g12(r)|opens space to enlarge corre-\nlation hole in the same layer and reduce the Coulomb energy be tween the electrons\nwithin the layer. This is shown in Fig. 17, which shows the pai r correlation functions\nof the electrons in the same layer defined by\ng11(r)≡LxLy\nN1(N1−1)/angbracketleftΨ|/summationdisplay\nnmδ(r+R1,n−R1,m)|Ψ/angbracketright, (5.6)\n∆g11(r) =/integraldisplay\n(g11(r′)−1)δ(|r′|−r) dr′. (5.7)\nThe obtained results indeed show that the correlation hole i n the same layer around\nr/ℓ∼1 is enhanced with the increase in d/ℓcontrary to the decrease in size of\ninterlayer correlation hole in Fig. 16. The correlation hol e in the same layer mono-\ntonically increases in size up to d/ℓ= 1.8, and then it becomes almost constant.\nThe correlation function g11(r) ford/ℓ >1.8 is almost the same to that of ν= 1/2\nmonolayer quantum Hall systems realized in the limit of d/ℓ=∞. This is consistent\nwith the almost vanishing excitation gap and exciton correl ation atd/ℓ >1.8 shown\nin Figs. 15 and 12.\nFigure 17 also shows that the growing of the correlation hole aroundr/ℓ∼1.5\nis accompanied with the increase in ∆g11(r) around r/ℓ∼4. The distance 4 ℓis\ncomparable to the approximate mean distance between the ele ctrons 3.54ℓestimated\nfrom (LxLy/N1)1/2= (2πL/N1)1/2ℓ. This means the electrons in the same layer18 Naokazu Shibata\n-0.4-0.2 0 0.2 0.4\n 0  1  2  3  4d/l=0.10.2\n0.4\n0.6\n0.8\n1.0\n1.2\n1.4\n1.61.82.02.22.42.63.0\nr/lNe=24 0\n -1 0  1  2  3  4r/ld/l=0.1\n0.6\n1.2\n1.8\n2.4Ne=24\nFig. 17. The intra-layer pair correlation function of elect rons in the ground state of bilayer quantum\nHall systems at ν= 1.Ne= 24 and Ly/Lx= 1.6.\ntend to keep distance of about 4 ℓfrom other electrons with the large correlation hole\naroundr/ℓ∼1.5 ford/ℓ>∼1. This is consistent with the formation of composite\nfermions at d=∞, where two magnetic flux quanta are attached to each electron ,\nwhich is equivalent to enhance the correlation hole around e ach electron in the same\nlayer to keep distance from other electrons.\nThe large correlation hole in g11(r) attracts electrons in the other layer as shown\nin Fig. 16, where we find a clear peak in ∆g12(r) atr/ℓ∼3. This peak at r/ℓ∼3\nis comparable to the neighboring correlation hole at r/ℓ∼1, which suggests that\nthe electrons excluded from the origin by strong interlayer Coulomb repulsion are\ntrapped by the correlation hole in g11(r) within r/ℓ∼4. Since the intra-layer\ncorrelation g11ford/ℓ >1.6 is almost the same to that of composite-fermion liquid\nstate,∆g12(r) represents the correlation of composite fermions between the layers.\nThe almost same amplitude of ∆g12(r) atr/ℓ∼1 and 3 for d/ℓ >1.6 actually shows\nthat the electrons in the other layer bind holes to form compo site fermions.\nWith decreasing d/ℓfrom infinity, the correlations of composite fermions in\ndifferent layers monotonically increases down to d/ℓ∼1.2 as shown in the enhance\nof|∆g12(r)|atr/ℓ∼1 and 3. But further decrease in d/ℓbroadens the peak at\nr/ℓ∼3 in∆g12(r) with the decrease in the correlation hole in ∆g11(r), and the peak\natr/ℓ∼3 in∆g12(r) finally disappears. This change in the correlation functio n\nshows how the composite-fermion liquid state evolves into e xcitonic state: The large\ncorrelation hole in the same layer, which is a characteristi c feature of the composite\nfermions, is transfered into the other layer to form exciton ic state. The correlation\nfunctions in Figs. 16 and 17 are continuously modified with th e decrease in d/ℓfrom\n∞to 0, which supports continuous transition from the compres sible liquid state to\nthe excitonic state. Fig. 16 also shows that the peak in ∆g12(r) atr/ℓ∼3 made\nby the binding of an electron to the hole around the origin gra dually disappears\nwith decreasing d/ℓfrom 1.2. This means the gradual break down of the concept of\ncomposite fermions.\nThebreakdownof thecompositefermionsaround d/ℓ∼1.2 affects thecharacterQuantum Hall Systems Studied by the DMRG Method 19\n-0.01-0.005 0 0.005 0.01\n 0  2  4  6  8d/l=0.2\nd/l=0.20.4\n0.41.3\n1.31.4\n1.40.6\n0.60.8\n0.81.0\n1.01.2\n1.2ij\nr/lNe=18\nFig. 18. The change in correlation function through the exci tation from the ground state to the\nfirst excited state. ν= 1 and Ne= 18 with Ly/Lx= 1.0.\nof the lowest excitations, which is clearly shown in the leve l crossing of the excited\nstate at d/ℓ∼1.2. The change in the character of excitation is confirmed by th e\ncorrelation functions in the excited state. Figure 18 shows the difference in the pair\ncorrelation functions gij(r) between the ground state and first excited state defined\nby\nδgij(r) =/integraldisplay\n(gE\nij(r′)−gG\nij(r′))δ(|r′|−r) dr′, (5.8)\nwheregG\nij(r) andgE\nij(r) are the pair correlation functions in the ground state and t he\nfirst excited state, respectively. δgij(r) in Fig. 18 show that there is a discontinuous\ntransition between d/ℓ= 1.2 and 1.3, which supports the level crossing in the first\nexcited state.\nBelowd/ℓ∼1.2,δg(r) have large amplitude at r/ℓ∼2 and 6, which shows elec-\ntrons are transfered between the inside of r/ℓ∼4 and its outside. Small singularity\natr/ℓ∼5.5 is due to finite size effects of square unit cell. Above d/ℓ∼1.2, only\nδg12(r) have large amplitude at r/ℓ∼1 and 2, which shows the electrons within\nr/ℓ∼4 in different layers are responsible for the lowest excitatio n. This result sug-\ngests that the low energy excitations are made by composite f ermions in different\nlayers for d/ℓ >1.2.\n§6. Conclusions\nInthis paperwehave reviewedthegroundstateandlow energy excitations of the\nquantum Hall systems studied by the DMRG method. We have appl ied the DMRG\nmethodtotwodimensionalquantumsystemsinmagneticfieldb yusingamappingon\ntoaneffective one-dimensional lattice model. SincetheCoul ombinteraction between\nthe electrons is long-range, all the electrons in the system interact with each other.\nThis fact seems to severely reduce the accuracy of the DMRG ca lculations. However,\nin the magnetic field, one-particle wave functions are local ized within the magnetic20 Naokazu Shibata\nlengthℓ, and the overlap of the one-particle wave functions exponen tially decreases\nwith increasing the distance between the two electrons. Thi s means the quantum\nfluctuations are restricted to short-range and the effective H amiltonian is suited for\nthe DMRG scheme. This is the reason why relatively small numb er of keeping states\nis enough for quantum Hall systems compared with usual two di mensional systems.\nIn quantum Hall systems, filling νof Landau levels is determined by ν=Ne/Nφ,\nwhereNφis the number of flux quanta and related to the magnetic field as Nφ=\n(e/h)LxLyB. Thus so many types of the ground state are realized only by ch ang-\ning the uniform magnetic field B. Since the ground state of free electrons in par-\ntially filled Landau level has macroscopic degeneracy, Coul omb interaction drasti-\ncally changes the wave function. The character of the ground state is sensitive to\nthe Landau level index Nand the filling ν, which modify the effective interaction\nand the mean distance between the electrons. 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Moussa\npierre.moussa@cea.fr, Institut de Physique Th´ eorique,\nDSM, CE Saclay, F-91191 Gif/Yvette, France\nOctober 28, 2018\nAbstract\nWe generalize a “one eigenstate” theorem of Levy, Perdew and Sahni\n(LPS) [1] to the case of densities coming from eigenmixture d ensity op-\nerators. The generalization is of a special interest for the radial density\nfunctional theory (RDFT) for nuclei [2], a consequence of th e rotational\ninvariance of the nuclear Hamiltonian; when nuclear ground states (GSs)\nhave a finite spin, the RDFT uses eigenmixture density operat ors to sim-\nplify predictions of GS energies into one-dimensional, rad ial calculations.\nWealsostudySchr¨ odingerequationsgoverningspineigend ensitymatrices.\nThe theorem of Levy, Perdew and Sahni [1] may be described as follo ws:\ni) letHAbe a Hamiltonian for Aidentical particles, with individual mass m,\nHA=A/summationdisplay\ni=1[−¯h2∆/vector ri/(2m)+u(/vector ri)]+A/summationdisplay\ni>j=1v(/vector ri,/vector rj), (1)\nii) consider a GS eigenfunction of HA, ψ(/vector r1,σ1,/vector r2,σ2,...,/vector rA,σA),whereσide-\nnotes the spin state of the particle with space coordinates /vector ri,\niii) use a trace of |ψ∝an}b∇acket∇i}ht∝an}b∇acketle{tψ|upon all space coordinates but the last one, and upon\nall spins, to define the density,\nρ(/vector r) =A/summationdisplay\nσ1...σA/integraldisplay\nd/vector r1d/vector r2...d/vector rA−1|ψ(/vector r1,σ1,/vector r2,σ2,...,/vector rA−1,σA−1,/vector r,σA)|2,(2)\niv) then there exists a local potential veff(/vector r) so that,\n[−¯h2∆/vector r/(2m)+veff(/vector r)]/radicalbig\nρ(/vector r) = (EA−EA−1)/radicalbig\nρ(/vector r), (3)\n1where the eigenvalue is the difference of the GS energy EAof theA-particle\nsystem and that, EA−1,of the (A−1)-particle one.\nCan this theorem be generalized for densities derived from eigenope rators,\nD ∝/summationtextN\nn=1wn|ψn∝an}b∇acket∇i}ht∝an}b∇acketle{tψn|,corresponding to cases where Hhas several ( N>1)\ndegenerate GSs ψn? The degeneracy situation is of a wide interest in nuclear\nphysics for doubly odd nuclei, the GSs of which often have a finite spin , and,\nif only because of Kramer’s degeneracy, for odd nuclei. In particula r, because\nof the rotational invariance of the nuclear Hamiltonian, the density operator of\ninterestforthe RDFT [2]reads, D=/summationtext\nM|ψJM∝an}b∇acket∇i}ht∝an}b∇acketle{tψJM|/(2J+1),whereJandM\nare the usual angular momentum numbers of a degenerate magnet ic multiplet\nof GSsψJM.Actually, more generally, it will easily be seen that the argument\nwhich follows holds for a degenerate multiplet of excited states as we ll.\nThis paper proves the generalization, by closely following the argume nt used\nfor one eigenstate only [1]. Furthermore, there is no need to assum e identi-\ncal particles. No symmetry or antisymmetry assumption for eigenf unctions is\nneeded. Let /vector piand/vector σibe the momentum and spin operators for the ith particle,\nat position /vector ri.Single out the A-th particle, with its degrees of freedom labelled\n/vector rand/vector σrather than /vector rAand/vector σA.For a theorem of maximal generality, with dis-\ntinct masses, one-body and two-body potentials, our Hamiltonian m ay become,\nHA=HA−1+VA+hA,with\nHA−1=A−1/summationdisplay\ni=1[−¯h2∆/vector ri/(2mi)+ui(/vector ri,/vector pi,/vector σi)]+A−1/summationdisplay\ni>j=1vij(/vector ri,/vector pi,/vector σi,/vector rj,/vector pj,/vector σj),\nVA=A−1/summationdisplay\nj=1vAj(/vector r,/vector rj,/vector pj,/vector σj), hA=−¯h2∆/vector r/(2mA)+uA(/vector r). (4)\nThe potentials acting upon the first ( A−1) particles may be non local and spin\ndependent, but, for a technical reason which will soon become obv ious, those\npotentials acting upon the A-th particle in VAandhAmust be strictly local\nand independent of the A-th spin. For notational simplicity, we choose units so\nthat ¯h2/(2mA) = 1 from now on.\nAs in the one eigenstate case [1] we select situations where there ex ists a\nrepresentation in which, simultaneously, the Hermitian Hamiltonian HAand all\nthe eigenfunctions ψ(/vector r1,σ1,...,/vector rA,σA) under consideration are real. This reality\ncondition does not seem to be restrictive, in view of time reversal inv ariance.\nLetEAbe a degenerate eigenvalue of HA.The degeneracy multiplicity being\nlarger than 1 ,selectN ≥2 of the corresponding eigenfunctions ψn,orthonor-\nmalized. Their set may be either complete or incomplete in the eigensub space.\nThe density operators,\nD=N/summationdisplay\nn=1|ψn∝an}b∇acket∇i}htwn∝an}b∇acketle{tψn|,N/summationdisplay\nn=1wn= 1, (5)\nwith otherwise arbitrary,positive weights wn,arenormalizedto unity, Tr D= 1,\nin theA-body space. They are eigenoperators of HA,namelyHAD=EAD.\n2The partial trace of a Dupon the first ( A−1) coordinates and all Aspins,\nτ(/vector r) =N/summationdisplay\nn=1wn/summationdisplay\nσ1...σA−1σ/integraldisplay\nd/vector r1...d/vector rA−1[ψn(/vector r1,σ1,...,/vector rA−1,σA−1,/vector r,σ)]2,(6)\ndefines a “density” τ, normalized so that/integraltext\nd/vector r τ(/vector r) = 1.Let nowφn/vector rσbe, in\nthe space of the first ( A−1) particles, an auxiliary wave function defined by,\nφn/vector rσ(/vector r1,σ1,...,/vector rA−1,σA−1) =ψn(/vector r1,σ1,...,/vector rA−1,σA−1,/vector r,σ)//radicalbig\nτ(/vector r).(7)\nNote that this auxiliary wave function depends on the choice of the w eightswn.\nNow the density operator in the space of the first ( A−1) particles, D′\n/vector r=/summationtext\nnσ|φn/vector rσ∝an}b∇acket∇i}htwn∝an}b∇acketle{tφn/vector rσ|,is normalized, Tr′D′\n/vector r= 1,where the symbol Tr′means\nintegration upon the first ( A−1) coordinates and sum upon the first ( A−1)\nspins. Since this normalization of D′\n/vector rin the (A−1)-particle space does not\ndepend on/vector r,two trivial consequences read, ∇/vector rTr′D′\n/vector r= 0 and ∆ /vector rTr′D′\n/vector r= 0.\nMore explicitly, this gives,\n/summationtext\nnσσ1...σA−1/integraltext\nd/vector r1...d/vector rA−1wnφn/vector rσ(/vector r1,σ1,...,/vector rA−1,σA−1)×\n∇/vector rφn/vector rσ(/vector r1,σ1,...,/vector rA−1,σA−1) = 0,(8)\nand\n/summationdisplay\nnσσ1...σA−1/integraldisplay\nd/vector r1...d/vector rA−1wn{[∇/vector rφn/vector rσ(/vector r1,σ1,...,/vector rA−1,σA−1)]2+\nφn/vector rσ(/vector r1,σ1,...,/vector rA−1,σA−1)∆/vector rφn/vector rσ(/vector r1,σ1,...,/vector rA−1,σA−1)}= 0.(9)\nThen one can rewrite the eigenstate property, ( HA−EA)ψn= 0,into,\n(HA−1+VA+hA−EA)√τ φn/vector rσ= 0. (10)\nThis also reads,\n√τ(HA−1+VA+uA−EA)φn/vector rσ−(∆/vector r√τ)φn/vector rσ=\n2(∇/vector r√τ)·(∇/vector rφn/vector rσ)+√τ(∆/vector rφn/vector rσ). (11)\nThe right-hand side (rhs) of Eq. (11) occurs because the Laplacia n, ∆/vector r,present\ninhA,acts also upon the parameter, /vector r,ofφn/vector rσ.This is where the local, multi-\nplicative nature of uAandvAjin the last particle space is used and avoids the\noccurrence of further terms, that would induce a somewhat unwie ldy theory.\nDefine, for any integrand Ψ n/vector rσ,the following expectation value in the first\n(A−1)-particle space,\n∝an}b∇acketle{t∝an}b∇acketle{tΨn/vector rσ∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht=/summationdisplay\nσ1...σA−1/integraldisplay\nd/vector r1...d/vector rA−1Ψn/vector rσ(r1,σ1,...,rA−1,σA−1).(12)\n3Multiply Eq. (11) by φn/vector rσand integrate out the first ( A−1) coordinates and\nspins, to obtain,\n∝an}b∇acketle{t∝an}b∇acketle{tφ2\nn/vector rσ∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht[Eexc\nnσ(/vector r)+EA−1+Unσ(/vector r)+uA(/vector r)−EA−∆/vector r]/radicalbig\nτ(/vector r)\n= 2∝an}b∇acketle{t∝an}b∇acketle{tφn/vector rσ(∇/vector rφn/vector rσ)∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht·[∇/vector r/radicalbig\nτ(/vector r)]+∝an}b∇acketle{t∝an}b∇acketle{tφn/vector rσ(∆/vector rφn/vector rσ)∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht/radicalbig\nτ(/vector r),(13)\nwhereEexc\nnσ(/vector r) is defined from,\n∝an}b∇acketle{t∝an}b∇acketle{tφ2\nn/vector rσ∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht[Eexc\nnσ(/vector r)+EA−1] =/summationtext\nσ1...σA−1/integraltext\nd/vector r1...d/vector rA−1\nφn/vector rσ(r1,σ1,...,rA−1,σA−1) [HA−1φn/vector rσ](r1,σ1,...,rA−1,σA−1),(14)\nandUnσ(/vector r) results from,\n∝an}b∇acketle{t∝an}b∇acketle{tφ2\nn/vector rσ∝an}b∇acket∇i}ht∝an}b∇acket∇i}htUnσ(/vector r) =/summationtextA−1\nj=1/summationtext\nσ1...σA−1/integraltext\nd/vector r1...d/vector rA−1\nφn/vector rσ(r1,σ1,...,rA−1,σA−1)vAj(/vector r,/vector rj,/vector pj,/vector σj)φn/vector rσ(r1,σ1,...,rA−1,σA−1).(15)\nThe square norm of φn/vector rσin the (A−1)-particle space results from Eq. (12)\nwith Ψ =φ2.In Eq. (14) the expectation value of HA−1is explicited as the\nsum of the GS energy EA−1ofHA−1and a positive, excitation energy Eexc\nnσ(/vector r).\nFrom Eq. (15), the Hartree nature of the potential Unσ(/vector r) is transparent.\nKeeping in mind that, ∀/vector r,the density operator D′is normalized to unity,\nnamely, that/summationtext\nnσwn∝an}b∇acketle{t∝an}b∇acketle{tφ2\nn/vector rσ∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht= 1,multiply Eq. (13) by wnand perform the\nsum uponnandσ.This gives,\n[Uexc(/vector r)+EA−1+UHrt(/vector r)+uA(/vector r)−EA−∆/vector r]√τ=/summationtext\nnσwn[2∝an}b∇acketle{t∝an}b∇acketle{tφn/vector rσ(∇/vector rφn/vector rσ)∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht·(∇/vector r√τ)+∝an}b∇acketle{t∝an}b∇acketle{tφn/vector rσ(∆/vector rφn/vector rσ)∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht√τ],(16)\nwhere the “mixture excitation potential”,\nUexc(/vector r) =/summationdisplay\nnσwn∝an}b∇acketle{t∝an}b∇acketle{tφ2\nn/vector rσ∝an}b∇acket∇i}ht∝an}b∇acket∇i}htEexc\nnσ(/vector r), (17)\nis local and positive and the “mixture Hartree-like potential”,\nUHrt(/vector r) =/summationdisplay\nnσwn∝an}b∇acketle{t∝an}b∇acketle{tφ2\nn/vector rσ∝an}b∇acket∇i}ht∝an}b∇acket∇i}htUnσ(/vector r), (18)\nis also local. Because of the frequent dominance of attractive term s invAj,\nit may show more negative than positive signs. Then notice that, bec ause of\nEqs. (8), and Eq. (12) with Ψ = φ∇φ,the sum in the rhs of Eq. (16),/summationtext\nnσwn∝an}b∇acketle{t∝an}b∇acketle{tφn/vector rσ(∇/vector rφn/vector rσ)∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht,vanishes. Note also, from Eqs. (9), and Eq. (12)\nwith Ψ =φ∆φ,that, again for the rhs of Eq. (16), the following equality holds,\n−/summationdisplay\nnσwn∝an}b∇acketle{t∝an}b∇acketle{tφn/vector rσ(∆/vector rφn/vector rσ)∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht=/summationdisplay\nnσwn∝an}b∇acketle{t∝an}b∇acketle{t(∇/vector rφn/vector rσ)·(∇/vector rφn/vector rσ)∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht,(19)\nwhere, again, the symbol ∝an}b∇acketle{t∝an}b∇acketle{t ∝an}b∇acket∇i}ht∝an}b∇acket∇i}htdenotes the trace T r′,an integration upon the\nfirst (A−1) coordinates together with summation upon their spins. The rhs o f\nthis equation, Eq. (19), defines a positive, local potential,\nUkin(/vector r) =/summationdisplay\nnσwn∝an}b∇acketle{t∝an}b∇acketle{t(∇/vector rφn/vector rσ)·(∇/vector rφn/vector rσ)∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht. (20)\n4Finally, according to Eq. (16), the sum of local potentials, Ueff=Uexc+\nUHrt+Ukin+uA,drives a Schr¨ odinger equation for√τ,\n[−∆/vector r+Ueff(/vector r)]/radicalbig\nτ(/vector r) = (EA−EA−1)/radicalbig\nτ(/vector r). (21)\nThis is the expected generalizationofthe LPS theorem. Note that, if the (A−1)\nparticles are not identical, then EA−1,the GS energy of HA−1, means here\nthe mathematical, absolute lower bound of the operator in all subsp aces of\narbitrarysymmetry or lack of symmetry. In practice, however, m ost cases imply\nsymmetriesin the ( A−1)-space, and EA−1means the groundstate energyunder\nsuch symmetries.\nFor nuclear physics, this generalization can be used in two ways:\ni) The first one consists in considering hypernuclei or mesonic nuclei, where the\nA-th particle is indeed distinct. Theoretical calculations with local inte ractions\nforthedistinctparticlemaybeattemptedwhilenonlocaland/orspin dependent\ninteractionsforthe A−1 nucleons areuseful, ifnot mandatory. Then, obviously,\nthe density τrefers to the hyperon or the meson and, given the neutron and\nproton respective numbers NandZ,wave functions ψnandφn/vector r,σbelong to\nbothN- andZ-antisymmetric subspaces. The energy EA−1is the GS energy of\nnucleus{N,Z},a fermionic GS energy, rather than the absolute lower bound\nof the mathematical operator HA−1in all subspaces.\nii) The secondone consists in setting all Aparticlesto be nucleons, at the cost of\nrestricting theoretical models to local interactions. Such models a re not without\ninterest indeed, although interactions which are spin dependent ar e certainly\nmore realistic. The antisymmetric properties of the functions ψnare requested\nin both N- and Z-spaces. If the singled out, A-th coordinate is a neutron one,\nthe density τdefined by Eq. (6) is the usual neutron density, divided by N; the\nfunctionsφn/vector rσareantisymmetricinthe( N−1)-neutronspaceandthe Z-proton\nspace; the energy EA−1now means the fermionic GS of nucleus {N−1,Z},not\nthat absolute, mathematical lower bound of operator HN−1,Z.Conversely,if the\nA-th coordinate is a proton one, then, mutatis mutandis ,τis the usual proton\ndensity, divided by Z,andEA−1is the GS energy of nucleus {N,Z−1}.\nIn both cases, the Hamiltonians to be used are scalars under the ro tation\ngroup, and, therefore [2], the density operators Dconsidered by the RDFT are\nalso scalars. Hence, all calculations defining Ueffreduce to calculations with a\nradial variable ronly.\nWe shall now extend our previous results to the case where we allow s pin\ndependence for all interactions, a most useful feature if all Aparticles are nucle-\nons. Polarized eigenmixtures are also interesting and need also be co nsidered.\nHence, a generalisation of our approach, which uses the “spin-den sity matrix”\n(SDM) formalism [3] [4], is in order. The Hamiltonian may become,\nA/summationdisplay\ni=1[−¯h2∆/vector ri/(2mi)+ui(/vector ri,/vector σi)]+A/summationdisplay\ni>j=1vij(/vector ri,/vector σi,/vector rj,/vector σj).(22)\nIt allows subtle differences between neutrons and protons, beside s the Coulomb\ninteractions between protons. More explicitly, there can be two dis tinct one-\n5body potentials, un,up,namely one forneutrons and one forprotons, but within\ntheneutronspacethefunction un(/vector ri,σi)obviouslywillnotread uni(/vector ri,σi).Simi-\nlarly in the proton space, the Hamiltonian contains terms up(/vector ri,σi) rather than\nupi(/vector ri,σi.The same subtlety allows terms vpp(/vector ri,σi,/vector rj,σj), vpn(/vector ri,σi,/vector rj,σj),\nvnp(/vector ri,σi,/vector rj,σj) andvpp(/vector ri,σi,/vector rj,σj),rather than vppij(/vector ri,σi,/vector rj,σj),... etc.\n(Of course, vpn=vnp.) But non localities of potentials and interactions, in the\nsense of explicit dependences upon momenta pi,remain absent.\nThen theA-th particle is again singled out, with degrees of freedom again\nlabelled/vector rand/vector σ,and the Hamiltonian is split as a sum, KA−1+WA+kA,some-\nwhat similar to the split described by Eqs. (4). For simplicity, we shall u se short\nnotations, K,Wandk,rather than KA−1,WAandkA.With two spin states,\nσ=±1/2,for theA-th nucleon, we represent eigenstates ψnas column vec-\ntors,ψn=/bracketleftbigg\nψn+\nψn−/bracketrightbigg\n,and operators as matrices such as, W=/bracketleftbigg\nW++W+−\nW−+W−−/bracketrightbigg\n,\nk=/bracketleftbigg\nk++k+−\nk−+k−−/bracketrightbigg\nand ¯u=/bracketleftbigg\nuA++uA+−\nuA−+uA−−/bracketrightbigg\n.The matrix, K=/bracketleftbigg\nK0\n0K/bracketrightbigg\n,is a\nscalar in spin space, since Kdoes not act upon the A-th particle.\nThe spin density matrix, ρn,results from an integration and spin sum over\nthe (A−1)-space of the matrix, ψn×ψT\nn,where the superscriptTdenotes\ntransposition,\nρn(/vector r) =∝an}b∇acketle{t∝an}b∇acketle{t/bracketleftbigg\nψn+\nψn−/bracketrightbigg\n×[ψn+ψn−]∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht=∝an}b∇acketle{t∝an}b∇acketle{t/bracketleftbigg\n(ψn+)2ψn+ψn−\nψn−ψn+(ψn−)2/bracketrightbigg\n∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht.(23)\nIt depends on the last coordinate, /vector r,and its matrix elements are labelled by\ntwo values, {σ,σ′},of the last spin. For an eigenmixture one defines, obviously,\n¯θ(/vector r) =/summationtext\nnwnρn(/vector r),and the trace in the last spin space, [ θ++(/vector r)+θ−−(/vector r)],is\nthat density, τ(/vector r),defined by Eq. (6).\nThe SDM, ¯θ,is symmetric andpositive semidefinite, ∀/vector r.Exceptfor marginal\nsituations, it is also invertible, in which case there exists a unique inver se square\nroot, also symmetric and positive. Define, therefore, a column vec torφn/vector rof\nstates in the ( A−1)-space according to,\nφn/vector r=¯θ−1\n2(/vector r)ψn. (24)\nThen the following property,\n/summationdisplay\nnwn∝an}b∇acketle{t∝an}b∇acketle{tφn/vector r×φT\nn/vector r∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht=¯θ−1\n2(/vector r)/parenleftBigg/summationdisplay\nnwn∝an}b∇acketle{t∝an}b∇acketle{tψn×ψT\nn∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht/parenrightBigg\n¯θ−1\n2(/vector r) =¯1,(25)\nholds∀/vector r.Here¯1denotes the identity matrix. Hence, the following gradient and\nLaplacian properties also hold,\n/summationdisplay\nnwn∝an}b∇acketle{t∝an}b∇acketle{t/parenleftbigg\n∇/vector r/bracketleftbigg\nφn/vector r+\nφn/vector r−/bracketrightbigg/parenrightbigg\n×[φn/vector r+φn/vector r−]∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht+\n/summationdisplay\nnwn∝an}b∇acketle{t∝an}b∇acketle{t/bracketleftbigg\nφn/vector r+\nφn/vector r−/bracketrightbigg\n×(∇/vector r[φn/vector r+φn/vector r−] )∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht= 0,∀/vector r,(26)\n6and\n/summationtext\nnwn∝an}b∇acketle{t∝an}b∇acketle{t/parenleftbigg\n∆/vector r/bracketleftbigg\nφn/vector r+\nφn/vector r−/bracketrightbigg/parenrightbigg\n×[φn/vector r+φn/vector r−]∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht+\n2/summationtext\nnwn∝an}b∇acketle{t∝an}b∇acketle{t/parenleftbigg\n∇/vector r/bracketleftbigg\nφn/vector r+\nφn/vector r−/bracketrightbigg/parenrightbigg\n·(∇/vector r[φn/vector r+φn/vector r−])∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht+\n/summationtext\nnwn∝an}b∇acketle{t∝an}b∇acketle{t/bracketleftbigg\nφn/vector r+\nφn/vector r−/bracketrightbigg\n×(∆/vector r[φn/vector r+φn/vector r−] )∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht= 0,∀/vector r . (27)\nThe eigenvector property,/parenleftbig\nK+W+k−EA¯1/parenrightbig\nψn= 0,also reads,\n/parenleftbig\nK+W+ ¯u−EA¯1/parenrightbig¯θ1\n2(/vector r)φn/vector r−/bracketleftBig\n∆/vector r¯θ1\n2(/vector r)/bracketrightBig\nφn/vector r=\n2/bracketleftBig\n∇/vector r¯θ1\n2(/vector r)/bracketrightBig\n·/parenleftbig\n∇/vector rφn/vector r/parenrightbig\n+¯θ1\n2(/vector r)/parenleftbig\n∆/vector rφn/vector r/parenrightbig\n. (28)\nRight-multiply Eq. (28) by the row vector, φT\nn/vector r,integrate and sum over the\n(A−1)-space, weigh the result by wnand sum upon n.Because of Eq. (25),\nthe weighted sum of averages over the ( A−1)-space simplifies into,\n/bracketleftBig\nUexc(/vector r)+(EA−1−EA)¯1+UHrt(/vector r)+ ¯u(/vector r)−∆/vector r/bracketrightBig\n¯θ1\n2(/vector r) =/summationtext\nnwn×\n/braceleftBig\n2/bracketleftBig\n∇/vector r¯θ1\n2(/vector r)/bracketrightBig\n· ∝an}b∇acketle{t∝an}b∇acketle{t/parenleftbig\n∇/vector rφn/vector r/parenrightbig\n×φT\nn/vector r∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht+¯θ1\n2(/vector r)∝an}b∇acketle{t∝an}b∇acketle{t/parenleftbig\n∆/vector rφn/vector r/parenrightbig\n×φT\nn/vector r∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht/bracerightBig\n,(29)\nwith\nUexc(/vector r) =/summationdisplay\nnwn∝an}b∇acketle{t∝an}b∇acketle{t/parenleftbig\nKφn/vector r/parenrightbig\n×φT\nn/vector r∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht−EA−1¯1, (30)\nand\nUHrt(/vector r) =/summationdisplay\nnwn∝an}b∇acketle{t∝an}b∇acketle{t/parenleftbig\nWφn/vector r/parenrightbig\n×φT\nn/vector r∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht. (31)\nThe rhs of Eq. (29) can be simplified, but less than that of Eq. (16). Indeed\nEq. (26) does not imply that the coefficient of ∇/vector r¯θ1\n2(/vector r) vanishes. In fact this\ncoefficient is,\n/summationdisplay\nnwn∝an}b∇acketle{t∝an}b∇acketle{t/parenleftbig\n∇/vector rφn/vector r/parenrightbig\n×φT\nn/vector r∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht=/summationdisplay\nnwn/bracketleftbigg\n0 ∝an}b∇acketle{t∝an}b∇acketle{t(∇/vector rφn/vector r+)φn/vector r−∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht\n∝an}b∇acketle{t∝an}b∇acketle{t(∇/vector rφn/vector r−)φn/vector r+∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht 0/bracketrightbigg\n,\n(32)\nand Eq. (26) shows that the matrix on the rhs is antisymmetric. In t urn, from\nEq. (27), the “Laplacian induced coefficient” in the rhs of Eq. (29) m ay be\nlisted as,\n/summationtext\nnwn∝an}b∇acketle{t∝an}b∇acketle{t/parenleftbig\n∆/vector rφn/vector r/parenrightbig\n×φT\nn/vector r∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht=\n−/summationtext\nnwn∝an}b∇acketle{t∝an}b∇acketle{t/parenleftbigg\n∇/vector r/bracketleftbigg\nφn/vector r+\nφn/vector r−/bracketrightbigg/parenrightbigg\n·(∇/vector r[φn/vector r+φn/vector r−])∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht+1\n2/summationtext\nnwn×\n∝an}b∇acketle{t∝an}b∇acketle{t/bracketleftbigg/parenleftbigg\n∆/vector r/bracketleftbigg\nφn/vector r+\nφn/vector r−/bracketrightbigg/parenrightbigg\n×[φn/vector r+φn/vector r−]−/bracketleftbigg\nφn/vector r+\nφn/vector r−/bracketrightbigg\n×(∆/vector r[φn/vector r+φn/vector r−])/bracketrightbigg\n∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht.(33)\n7In the rhs of Eq. (33), the similarity of its first term with potential Ukin,Eq.\n(19), is transparent. Also transparent is the antisymmetry of th e second term.\nWith the definitions,\nUkin(/vector r) =/summationdisplay\nnwn∝an}b∇acketle{t∝an}b∇acketle{t/parenleftbig\n∇/vector rφn/vector r/parenrightbig\n·/parenleftBig\n∇/vector rφT\nn/vector r/parenrightBig\n∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht, (34)\nUant(/vector r) =1\n2/summationdisplay\nnwn∝an}b∇acketle{t∝an}b∇acketle{t/bracketleftBig/parenleftbig\n∆/vector rφn/vector r/parenrightbig\n×φT\nn/vector r−φn/vector r×/parenleftBig\n∆/vector rφT\nn/vector r/parenrightBig/bracketrightBig\n∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht,(35)\nand\nUgrd(/vector r) = 2/summationdisplay\nnwn/bracketleftbigg\n0 ∝an}b∇acketle{t∝an}b∇acketle{t(∇/vector rφn/vector r+)φn/vector r−∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht\n∝an}b∇acketle{t∝an}b∇acketle{t(∇/vector rφn/vector r−)φn/vector r+∝an}b∇acket∇i}ht∝an}b∇acket∇i}ht 0/bracketrightbigg\n,(36)\nthe Schr¨ odinger equation for the square root of the spin density matrix reads,\n/bracketleftBig\nUexc+UHrt+ ¯u+Ukin−Uant−∆/bracketrightBig\n¯θ1\n2−Ugrd·∇¯θ1\n2= (EA−EA−1)¯θ1\n2.(37)\nThis concludes our generalizations of the LPS theorem. On the one h and,\nsee Eq. (21), we obtained for the square root density of an eigenm ixture a\nSchr¨ odinger equation, most similar to the LPS equation. On the oth er hand,\nat the cost of a slightly less simple result, we also obtained, see Eq. (3 7), an\nLPS-like equation that drives the square root of the spin density ma trix.\nThis opens a completely new zoology of local, effective potentials, of w hich\nvery little is known, but the interest of which is obvious, since they dr ive a\nreasonably easily measurable observable, the square root of an eig enmixture\ndensity, which depends on one degree of freedom /vector ronly. A connection of such\npotentials with optical potentials, or rather their real parts, is like ly, but yet\nremains an open problem.\nReferences\n[1] Mel Levy, John P. Perdue and Viraht Sahni, Phys. Rev. A 302745 (1984)\n[2] B. G. Giraud, Phys. Rev. C 78014307 (2008)\n[3] O. Gunnarson and B. J. Lundqvist, Phys. Rev. B 13, 4274 (1976)\n[4] A. G¨ orling, Phys. Rev. A 47, 2783 (1993)\n8"    },    {        "title": "0906.3300v1.Optimality_of_log_Hölder_continuity_of_the_integrated_density_of_states.pdf",        "content": "arXiv:0906.3300v1  [math.SP]  17 Jun 2009OPTIMALITY OF LOG H ¨OLDER CONTINUITY OF THE\nINTEGRATED DENSITY OF STATES\nZHENG GAN AND HELGE KR ¨UGER\nAbstract. We construct examples, that log H¨ older continuity of the in te-\ngrated density of states cannot be improved. Our examples ar e limit-periodic.\n1.Introduction\nWe investigate optimality of the log H¨ older continuity of the integrat ed density\nof states. Let (Ω ,µ) be a probability space, T: Ω→Ω an invertible ergodic\ntransformation, and f: Ω→Ra bounded measurable function. Define a potential\nVω(n) =f(Tnω). The Sch¨ odinger operator Hω:ℓ2(Z)→ℓ2(Z) is defined by\n(1.1) Hωu(n) =u(n+1)+u(n−1)+Vω(n)u(n),\nand the integrated density of states kby\n(1.2) k(E) = lim\nN→∞/integraldisplay\nΩ/parenleftbigg1\nNtr(P(−∞,E)(Hω,[0,N−1]))/parenrightbigg\ndµ(ω),\nwhereHω,[0,N−1]denotes the restriction of Hωtoℓ2([0,N−1]). Using the Thouless\nformula, Craig and Simon showed that\nTheorem 1.1 (Craig and Simon, [4]) .There exists a constant C=C(/bardblf/bardbl∞)such\nthat\n(1.3) |k(E)−k(˜E)| ≤C\nlog|E−˜E|−1\nfor|E−˜E| ≤1\n2.\nThis is what is well known as log H¨ older continuity . We will be interested in the\noptimality of this statement in the sense that ε/mapsto→1\nlog(ε−1)cannot be replaced by\nanother function, which goes to zero faster. It was shown by Cra ig in [3], that the\nregularity cannot be improved to\nε/mapsto→1\nlog(ε−1)log(log(ε−1))β,\nwhereβ >1. However, in the case of specific dynamical systems (Ω ,µ,T), there\nexist many results, which improve the Craig–Simon result. We just me ntion two.\nFor quasi-periodic Schr¨ odinger operators, Goldstein and Schlag h ave shown in [7]\nthat the integrated density of states is H¨ older continuous and co mputed the H¨ older\nexponent, and shown that the integrated density of states is almo st everywhere\nDate: October 29, 2018.\n2000Mathematics Subject Classification. Primary 47B36; Secondary 47B80, 81Q10.\nKey words and phrases. Integrated density of states, limit periodic potentials.\nH. K. was supported by NSF grant DMS–0800100.\n12 Z. GAN AND H. KR ¨UGER\nLipschitz. For random Schr¨ odinger operators, the integrated d ensity of states is\neven everywhere Lipschitz. This is known as the Wegner estimate wh ich can be\nfound for example in the exposition of Kirsch in [6].\nOur interest in the question of optimality of the Craig–Simon results c omes from\nthe importance of the Wegner estimate in multiscale analysis (see the exposition of\nKirsch). If one could improve the result to a continuity of the form\nε/mapsto→1\nlog(ε−1)β\nfor some large enough β >1, one would be able to use this for multiscale analysis\n(see for example Theorem 3.12 in [8]). Already Craig’s result shows tha t this is\nimpossible, however one could hope that a combination of an improved continuity\nresultandanimprovementofmultiscaleanalysismightremovetheWeg nerestimate\nassumption. However, we will show that the continuity of integrate d density of\nstates cannot be improved for all potentials beyond log H¨ older con tinuity.\nA potential V∈ℓ∞(Z) is called almost-periodic, if the closure Ω of its translates\nis compact in the ℓ∞norm. Furthermore, then Ω can be made into a compact\ngroup, with an unique invariant Haar measure. For these our previo us definition of\nthe integrated density of states (1.2) can be replaced by\n(1.4) kV(E) = lim\nN→∞1\nNtr(P(−∞,E)(H[0,N−1])),\nwhereH[0,N−1]denotes now the restriction of ∆+ Vtoℓ2([0,N−1]) with ∆ the\ndiscrete Laplacian. This can be found for example as Theorem 2.9 in Av ron–Simon\n[2].\nNext,Vis calledpperiodic, if its p-th translate is equal to V. Furthermore, V\nis limit-periodic if it is the limit in the ℓ∞norm of periodic potentials. We denote\nbyσ(∆+V) the spectrum of the operator ∆+ V.\nTheorem 1.2. Given any increasing continuous function ϕ:R+→R+with\n(1.5) lim\nx→0ϕ(x) = 0\nand a constant C0>0, there is a limit-periodic Vsatisfying /bardblV/bardbl∞≤C0such that\nits integrated density of states satisfies\n(1.6) limsup\nE→E0|kV(E)−kV(E0)|log(|E−E0|−1)\nϕ(|E−E0|)=∞,\nfor anyE0∈σ(∆+V).\nThis result tells us, that with ϕas in the previous theorem, we cannot have\n|kV(E)−kV(E0)| ≤C·ϕ(|E−E0|)\nlog(|E−E0|−1)\nfor anyC >0 and allV. The proof of this theorem essentially happens in two\nparts. Given a periodic V0andε >0 satisfying /bardblV0/bardbl ≤C0−ε, we construct a\nsequenceVjof periodic potentials, with the following properties\n(i)Vjispj-periodic.\n(ii) The Lebesgue measure of σ(∆+Vj)\n(1.7) εj=|σ(∆+Vj)|OPTIMALITY OF LOG H ¨OLDER CONTINUITY OF THE IDS 3\nsatisfies\n(1.8) log( ε−1\nj)≥pj−1·pj·ϕ(2εj).\n(iii) We have that\n(1.9) /bardblVj−Vj−1/bardbl ≤min(ε,ε1,...,ε j−1)\n2j.\nHere/bardbl./bardbldenotes the ℓ∞norm. The construction of these Vjwill be given in the\nnext section and uses the tools developed by Avila in [1]. Before proce eding with\nthe proof of Theorem 1.2, recall that positivity of the trace implies t hat\nkW(E−/bardblV−W/bardbl)≤kV(E)≤kW(E+/bardblV−W/bardbl)\nfor any potentials VandW.\nProof of Theorem 1.2. By (1.9), we see that there exists a limiting potential V,\nsuch that for each j, we have that\n/bardblVj−V/bardbl ≤εj=|σ(∆+Vj)|.\nFurthermore, since /bardblV0−V/bardbl ≤ε, we have that /bardblV/bardbl ≤C0.\nNext, fixE0∈σ(∆+V) and letj≥1. By the previous equation, we have that\nthere exists E1∈σ(∆+Vj) such that\n|E0−E1| ≤εj=|σ(∆+Vj)|.\nDenote the band of σ(∆+Vj) containing E1by [E−,E+]. By a general fact about\nperiodic Schr¨ odinger operators, we know that\nkVj(E+)−kVj(E−) =1\npj.\nWe thus get that\nkV(E++εj)−kV(E−−εj)≥1\npj\nFurthermore, the interval [ E−−εj,E++εj] containsE0and we can choose Ej∈\n{E−−εj,E++εj}such that\n|kV(E0)−kV(Ej)| ≥1\n2pj\nand\n|E0−Ej| ≤2εj.\nThis implies the claim by (1.8), since jwas arbitrary. /square\nOne can slightly improve the above theorem, by for example showing t hat there\nis not only one V, that satisfies the conclusion, but that in fact the set is dense in\nthe limit-periodic operators. However, we have not done so, to kee p the statement\nas simple as possible.4 Z. GAN AND H. KR ¨UGER\n2.Construction of the periodic potentials\nWe will need the machinery developed by Avila in [1], in order to prove our\nresults. In the following, we let Ω be a totally disconnected compact g roup, known\nasCantor group . We furthermore let T: Ω→Ω be a minimal translation on this\ngroup. There is a decreasing sequence of Cantor subgroups\nX1⊇X2⊇...\nsuch that the quotients\nΩ/Xk\ncontainpkelements. We let Pkbe the subset of the continuous functions C(Ω) on\nΩ, which only depend on Ω /Xk.fis calledn-periodic if f(Tnω) =f(ω) for every\nω∈Ω. The elements of Pkwill bepkperiodic.\nWe now fix ω∈Ω. We have that {f(Tnω)}n∈Z∈ℓ∞(Z) is limit-periodic, since\nthe periodic fare dense in C(Ω). For a finite subset Fof the periodic potentials\nP=/uniontext\nk≥1Pk, we introduce the averaged Lyapunov exponent L(E,F) as\n(2.1) L(E,F) =1\n#F/summationdisplay\nf∈FL(E,f),\nwhere #Fdenotes the number of elements of F(with multiplicities) and\nL(E,f) = lim\nN→∞1\nNlog/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1/productdisplay\nn=N/parenleftbiggf(Tnω)−E−1\n1 0/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nis the Lyapunov exponent of the periodic potential. For f∈C(Ω), we denote by\nΣ(f) the spectrum of the operator ∆ + f(Tnω). We will use the following two\nlemmas of Avila [1], see also [5].\nLemma 2.1 (Lemma 3.1. in [1]) .LetBbe an open ball in C(Ω), letF⊂P∩B\nbe finite, and let 0<ε<1. Then there exists a sequence FK⊂P∩Bsuch that\n(i)L(E,FK)>0wheneverE∈R,\n(ii)L(E,FK)→L(E,F)uniformly on compacts.\nLemma 2.2 (Lemma3.2. in[1]) .LetBbe an open ball in C(Ω), and letF⊂Pk∩B\nbe a finite family of sampling functions. Then for every N≥2andKsufficiently\nlarge, there exists FK⊂PK∩Bsuch that\n(i)L(E,FK)→L(E,F)uniformly on compacts,\n(ii)The diameter of FKis at mostp−10\nK,\n(iii)For everyλ∈R, if\n(2.2) inf\nE∈RL(E,F)≥δ#Fpk,\nthen for every f∈FK, the spectrum Σ(f)has Lebesgue measure at most\ne−δpK/2.\nThe construction of the Vjwill be accomplished by\nProposition 2.3. Given a continuous function ψ:R+→R+satisfying\n(2.3) lim\nx→0ψ(x) = 0,\nandp-periodicfandε>0, then there exists a ˜p-periodic function ˜f, such that\n(2.4) /bardblf−˜f/bardbl∞≤ε,OPTIMALITY OF LOG H ¨OLDER CONTINUITY OF THE IDS 5\nand\n(2.5) log( |Σ(˜f)|−1)≥˜p·ψ(|Σ(˜f)|).\nProof.By Lemma 2.1, we can find a finite family of p1-periodic potentials F1within\nB1\n2ε(f) such that\nδ1=1\n#F1p1L(E,F1) =1\n#F1p1·1\n#F1/summationdisplay\nf∈F1L(E,f)>0.\nApplying Lemma 2.2 to F1, we can get a finite family F2⊂Bε(f) of ˜p-periodic\npotentials, where we might require ˜ pto be arbitrarily large. Let ˜fbe any element\nofF2. By Lemma 2.2 (iii), we have that\n|Σ(f2)|<e−δ1˜p/2.\nHence, (2.5) turns into\n1\n2δ1˜p≥˜pψ(e−δ1˜p/2).\nThe claim now follows from the fact, that ψ(x)→0 asx→0. /square\nIt now remains to construct the sequence of potentials Vj. We proceed by in-\nduction. By possibly modifying (Ω ,T), we can assume that\nV0(n) =f0(Tnω)\nfor somef0∈C(Ω). Assume now that, we are given V1=f1◦Tn,...,V j−1=\nfj−1◦Tnandwewishtoconstruct Vj=fj◦Tn. Wecanchoosenow εintheprevious\nproposition to be the right hand side of (1.9). We choose ψ(x) =pj−1ϕ(2x) and\nthe claim follows.\nReferences\n[1] A. Avila, On the spectrum and Lyapunov exponent of limit periodic Schr ¨ odinger operators ,\nComm. Math. Phys. 288:3(2009), 907–918\n[2] J. Avron, B. Simon, Almost periodic Schr¨ odinger operators II. The integrated density of\nstates, Duke Math. J. 50:1(1983), 369–391.\n[3] W. Craig, Pure point spectrum for discrete almost periodic Schr¨ odin ger operators , Comm.\nMath. Phys. 88:1(1983), 113–131.\n[4] W. Craig, B. Simon, Subharmonicity of the Lyaponov index , Duke Math. J. 50:2(1983),\n551–560.\n[5] D. Damanik, Z. Gan, Limit-periodic Schr¨ odinger operators in the regime of pos itive Lya-\npunov exponents , preprint.\n[6] M. Disertori, W. Kirsch, A. Klein, F. Klopp, V. Rivasseau ,Random Schr¨ odinger Operators ,\nPanoramas et Synthses 25(2008), xiv + 213 pages.\n[7] M. Goldstein, W. Schlag, Fine properties of the integrated density of states and a qua n-\ntitative separation property of the Dirichlet eigenvalues , Geom. Funct. Anal. 18:3(2008),\n755–869.\n[8] H.Kr¨ uger, Multiscale Analysis for Ergodic Schr¨ odinger operators an d positivity of Lyapunov\nexponents ,. preprint.\nDepartment of Mathematics, Rice University, Houston, TX 7700 5, USA\nE-mail address :zheng.gan@rice.edu\nURL:http://math.rice.edu/ ∼zg2/\nDepartment of Mathematics, Rice University, Houston, TX 7700 5, USA\nE-mail address :helge.krueger@rice.edu\nURL:http://math.rice.edu/ ∼hk7/"    },    {        "title": "0906.3721v2.Overhauser_s_spin_density_wave_in_exact_exchange_spin_density_functional_theory.pdf",        "content": "arXiv:0906.3721v2  [cond-mat.mtrl-sci]  29 Sep 2009Overhauser’s spin-density wave in exact-exchange spin den sity functional theory\nS. Kurth1,2,3and F. G. Eich4,5,3\n1Nano-Bio Spectroscopy Group, Dpto. de F´ ısica de Materiale s, Universidad del Pa´ ıs Vasco UPV/EHU,\nCentro F´ ısica de Materiales CSIC-UPV/EHU, Av. Tolosa 72, E -20018 San Sebasti´ an, Spain\n2IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao , Spain\n3European Theoretical Spectroscopy Facility (ETSF)\n4Institut f¨ ur Theoretische Physik, Freie Universit¨ at Ber lin, Arnimallee 14, D-14195 Berlin, Germany\n5Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, D-14195 Berlin, Germany\n(Dated: December 4, 2018)\nThe spin density wave (SDW) state of the uniform electron gas is investigated in the exact\nexchange approximation of noncollinear spin density funct ional theory (DFT). Unlike in Hartree-\nFock theory, where the uniform paramagnetic state of the ele ctron gas is unstable against formation\nof the spin density wave for all densities, in exact-exchang e spin-DFT this instability occurs only\nfor densities lower than a critical value. It is also shown th at, although in a suitable density range it\nis possible to find a non-interacting SDW ground state Slater determinant with energy lower than\nthe corresponding paramagnetic state, this Slater determi nant is not a self-consistent solution of the\nOptimized Effective Potential (OEP) integral equations of n oncollinear spin-DFT. A selfconsistent\nsolution of the OEP equations which gives an even lower energ y can be found using an excited-state\nSlater determinant where only orbitals with single-partic le energies in the lower of two bands are\noccupied while orbitals in the second band remain unoccupie d even if their energies are below the\nFermi energy.\nPACS numbers: 71.10.Ca,71.15.Mb,75.30.Fv\nI. INTRODUCTION\nThe ab-initio description of non-collinear magnetic\nphenomena such as spin-density waves (SDWs) is typ-\nically based on an extension1of the original Kohn-Sham\ndensity functional theory (DFT)2. As always in DFT,\nits success crucially relies on the accuracy of approxima-\ntions for the exchange-correlation energy. An important\nstep for the application of non-collinear DFT to real sys-\ntems proved to be the construction of a non-collinear\nversion of the local spin density approximation (LSDA)\nby K¨ ubler and coworkers3. An alternative density func-\ntional formalism for the description of SDWs and anti-\nferromagnetism was proposed by Capelle and Oliveira4,5.\nIn their work the system is not described in terms of its\ndensity and magnetization density as in usual spin-DFT\n(SDFT) but instead in terms of the density and the so-\ncalled “staggered density” where the latter is a nonlocal\nquantity introduced to capture the nonlocal physics of\nSDWs and antiferromagnetic systems.\nHowever, nonlocality may also be captured within the\nframework of usual SDFT if one abandons the local ap-\nproximation to the exchange-correlation energy. Orbital\nfunctionals, i.e., functionals which explicitly depend on\nthe single-particle orbitals rather than on the density (or\ndensities), can be highly nonlocal. In DFT (or SDFT),\nthe Optimized Effective Potential (OEP) method6,7,8\nprovides a framework to treat orbital functionals. This\nmethodology has recently been generalized to the case of\nnoncollinear magnetism9.\nIn the present work we use the noncollinear OEP\nmethod to study a very simple model system, the uni-\nform electron gas. This model is of paramount impor-tance in many-body physics10. Moreover, in his semi-\nnal work Overhauser11,12showed analytically that this\nsimple model, if treated within the Hartree-Fock (HF)\napproximation, leads to an instability of the paramag-\nnetic phase with respect to formation of a spin-density\nwave. In SDFT, the approximation analogous to HF is\nthe exact-exchange (EXX) approximation which is the\nHF total energy functional but evaluated with orbitals\nwhich come from both a localsingle-particle potential as\nwell as a localmagnetic field. Here we use the EXX ap-\nproximationforanumericalinvestigationofOverhauser’s\nSDW state in the framework of SDFT. This is com-\nplementary to another work13where Overhauser’s SDW\nstate is investigated numerically within HF and reduced\ndensity-matrix functional theory.\nThe paper is organized as follows: in Section II we\nbriefly review the formalism of noncollinear SDFT and\nthe corresponding OEP method. In Section III we mini-\nmize the EXXtotal energyfor agiven ansatz ofthe SDW\nstate. In Section IV we investigate if the chosen ansatz\nis selfconsistent in the framework of noncollinear SDFT\nbefore we provide our conclusions in Section V.\nII. NON-COLLINEAR SPIN DENSITY\nFUNCTIONAL THEORY\nIn non-collinear SDFT, a system of interacting elec-\ntrons with ground state |Ψ0/an}b∇acket∇i}htmoving in an external elec-\ntrostatic potential v0(r) (typically the electrostatic po-\ntential due to the nuclei) and magnetic field B0(r) is\ndescribed through its particle density\nn(r) =/an}b∇acketle{tΨ0|ˆΨ†(r)ˆΨ(r)|Ψ0/an}b∇acket∇i}ht (1)2\nand its magnetization density\nm(r) =−µB/an}b∇acketle{tΨ0|ˆΨ†(r)σˆΨ(r)|Ψ0/an}b∇acket∇i}ht. (2)\nHere,ˆΨ(r) is the field operator for Pauli spinors, µBis\nthe Bohr magneton and σis the vector of Pauli matrices\n(atomic units are used throughout). For given external\npotentials, the total ground state energyof such a system\ncan be written as a functional of these two densities,\nEv0,B0[n,m] =Ts[n,m]+/integraldisplay\nd3rv0(r)n(r)\n−/integraldisplay\nd3rB0(r)m(r)+U[n]+Exc[n,m]+Eion(3)\nwhereTs[n,m] is the non-interacting kinetic energy,\nU[n] =1\n2/integraldisplay\nd3r/integraldisplay\nd3r′n(r)n(r′)\n|r−r′|(4)\nis the classical electrostatic energy of the electrons and\nEionis the classical electrostatic energy of the ions.\nExc[n,m] is the (unknown) exchange-correlation energy\nfunctional which has to be approximated in practice.\nOnce an approximation for this functional is specified,\nthe densities n(r) andm(r) can be obtained as (ground\nstate) densities of a non-interacting system whose or-\nbitals are given by self-consistent solution of the Kohn-\nSham (KS) equation\n/parenleftbigg\n−∇2\n2+vs(r)+µBσBs(r)/parenrightbigg\nΦi(r) =εiΦi(r) (5)\nwhere the Φ i(r) are single-particle Pauli spinors. The\neffective potentials are given by\nvs(r) =v0(r)+/integraldisplay\nd3r′n(r′)\n|r−r′|+vxc(r) (6)\nand\nBs(r) =B0(r)+Bxc(r) (7)\nwith the exchange-correlation potentials\nvxc(r) =δExc[n,m]\nδn(r)(8)\nand\nBxc(r) =−δExc[n,m]\nδm(r), (9)\nrespectively.\nIn this work we will use the EXX energy functional\nas an approximation to the exchange-correlation energy\nwhich can be expressed in terms of the single-particle\nspinors Φ ias\nEEXX[{Φl}] =−1\n2occ/summationdisplay\ni,j\n/integraldisplay\nd3r/integraldisplay\nd3r′[Φ†\ni(r)·Φj(r)][Φ†\nj(r′)·Φi(r′)]\n|r−r′|(10)Since the EXX functional explicitly depends on the\n(spinor) orbitals but only implicitly on the densities,\nthe calculation of the exchange-correlation potentials (8)\nand (9) has to be performed by means of the OEP\nmethod6,7,8. A formulation of this technique for the\ncase of non-collinear magnetism has recently been given\nin Ref. 9. The coupled OEP integral equations for the\nexchange-correlation potentials can be obtained by ap-\nplying the chain rule of functional derivatives\nδExc\nδvs(r)=/integraldisplay\nd3r′/parenleftbigg\nvxc(r′)δn(r′)\nδvs(r)−Bxc(r′)δm(r′)\nδvs(r)/parenrightbigg\n=occ/summationdisplay\ni/integraldisplay\nd3r′/parenleftbiggδExc\nδΦi(r′)δΦi(r′)\nδvs(r)+h.c./parenrightbigg\n(11)\nδExc\nδBs(r)=/integraldisplay\nd3r′/parenleftbigg\nvxc(r′)δn(r′)\nδBs(r)−Bxc(r′)δm(r′)\nδBs(r)/parenrightbigg\n=occ/summationdisplay\ni/integraldisplay\nd3r′/parenleftbiggδExc\nδΦi(r′)δΦi(r′)\nδBs(r)+h.c./parenrightbigg\n(12)\nThe functional derivatives of the spinor orbitals and the\ndensities with respect to the potentials can be computed\nfrom first-order perturbation theory and, following the\nnotation of Ref. 14, the OEP equations can be written in\ncompact form as\nocc/summationdisplay\ni/parenleftBig\nΦ†\ni(r)Ψi(r)+h.c./parenrightBig\n= 0 (13)\n−µBocc/summationdisplay\ni/parenleftBig\nΦ†\ni(r)σΨi(r)+h.c./parenrightBig\n= 0 (14)\nwhere we have defined the orbital shifts7,15,16\nΨi(r) =/summationdisplay\nj\nj/negationslash=iDijΦj(r)\nεi−εj(15)\nwith\nDij=/integraldisplay\nd3r′Φ†\nj(r′)\n/parenleftBigg\n(vxc(r′)+µBσBxc(r′))Φi(r′)−δExc\nδΦ†\ni(r′)/parenrightBigg\n.(16)\nEqs. (5) - (10) and (13) - (14) constitute our formal\nframework to investigate the spin density wave in the\nuniform electron gas in exact exchange.\nIII. UNIFORM ELECTRON GAS WITH\nSPIN-DENSITY WAVE: DIRECT\nMINIMIZATION OF ENERGY\nWe study a uniform electron gas, i.e., a system of\nelectrons with spatially constant density moving in the3\nelectrostatic potential created by a neutralizing, uniform\ndensity of positive background charge.\nIn the following, rather than calculating the Kohn-\nSham potentials selfconsistently, we assumethat the\nKohn-Sham electrostatic potential vs(r) is a constant\n(which we set to zero)and that the Kohn-Shammagnetic\nfieldBs(r) forms a spiral with amplitude Band wavevec-\ntorq=qezwhereezis a unit vector in z-direction, i.e.,\nBs(r) = (Bcos(qz),Bsin(qz),0). Moreover, we also as-\nsume that the Kohn-Sham magnetic field is entirely due\nto its exchange-correlation part, i.e., we study the sys-\ntem without external magnetic field. Of course, at some\npoint we have to verify that our assumptions are consis-\ntent within the SDFT framework. This question will be\nstudied in Section IV.\nWith the effective potentials given above, the Kohn-\nShamequation(5)canbesolvedanalytically. Acomplete\nset of quantum numbers is given by a band index b= 1,2\nand a wavevector k. The corresponding single-particle\neigenstates and eigenenergies for the first band are given\nby\nΦ(1)\nk(r) =exp(ikr)√\nV/parenleftbigg\ncosϑkz\nsinϑkzexp(iqz)/parenrightbigg\n(17)\nwhereVis the system volume (which tends to infinity)\nand\nε(1)\nk=k2\n/bardbl\n2+ε(1)\nκ (18)\nwherek/bardbl=/radicalBig\nk2x+k2y,κ=kz+q\n2and\nε(1)\nκ=κ2\n2+q2\n8−/radicalbigg\nq2\n4κ2+µ2\nBB2.(19)\nThe angle ϑkzis defined through the relation\ntan2ϑκ=−2α (20)\nwith\nα=µBB\nqκ. (21)\nForthe secondband theeigenstatesandeigenenergiesare\nΦ(2)\nk(r) =exp(ikr)√\nV/parenleftbigg\n−sinϑkz\ncosϑkzexp(iqz)/parenrightbigg\n(22)\nand\nε(2)\nk=k2\n/bardbl\n2+ε(2)\nκ (23)\nwith\nε(2)\nκ=κ2\n2+q2\n8+/radicalbigg\nq2\n4κ2+µ2\nBB2.(24)\nIn order for the definition of Φ(1)\nk(r) and Φ(2)\nk(r) to be\nunique, the angle ϑκhas to be restricted to an interval oflengthπ/2. Assumingthat B≥0andq >0,wefindfrom\nEq. (20) that −π/2< ϑκ≤0. Using a trigonometric\nidentity we can transform Eq. (20) to\ntanϑκ=1\n2α/parenleftBig\n1−/radicalbig\n1+4α2/parenrightBig\n(25)\nFrom Eq. (20) we see that for finite Bandκ= 0 the\nangleϑκ=0=−π\n4. In order for ϑκto be a continuous\nfunction of κwith values in the correct range we invert\nEq. (25) as\nϑκ=/braceleftbigg\narctan/parenleftbig1\n2α/parenleftbig\n1−√\n1+4α2/parenrightbig/parenrightbig\nforκ≥0\n−π\n2+arctan/parenleftbig1\n2α/parenleftbig\n1−√\n1+4α2/parenrightbig/parenrightbig\nforκ <0\n(26)\nWith the single-particle states fully defined we can write\ndown the uniform electronic (ground state) density as\nn=/summationdisplay\nbn(b)=/summationdisplay\nb/integraldisplayd3k\n(2π)3θ(εF−ε(b)\nk)\n=1\n4π2/summationdisplay\nb/integraldisplay\ndκθ(εF−ε(b)\nκ)(εF−ε(b)\nκ) (27)\nwheren(b)is the density contribution of band b,θ(x) is\nthe Heaviside step function and the trivial integrals have\nbeen carried out in the last step. Using Eqs. (18) and\n(23), the integration limits can be determined analyti-\ncally and the remaining integral can easily be solved in\nclosed form but we refrain from giving the explicit ex-\npression here.\nSimilarly, we can compute the magnetization density.\nWe obtain for the x−andy−components\nmx(r) =m0cos(qz) (28)\nand\nmy(r) =m0sin(qz) (29)\nwhere the amplitude of the spin-density wave is\nm0=−µB\n2π2/summationdisplay\nbsign(b)\n/integraldisplay\ndκ θ(εF−ε(b)\nκ) (εF−ε(b)\nκ) sinϑκcosϑκ(30)\nand we have defined\nsign(b) =/braceleftbigg\n+1 forb= 1\n−1 forb= 2. (31)\nUsing the symmetry relation θ−κ=−π\n2−θκ(see\nEq. (26)), the z−component of the magnetization den-\nsity can be shown to vanish identically,\nmz(r) = 0. (32)\nWe point out that the vector of the magnetization den-\nsity here is parallel to the Kohn-Sham magnetic field.\nThis certainly is a consequence of the simplicity of the4\nsystem under study here. For more complicated systems\nit was shown in Ref. 9 that these quantities need not be\nparallel in noncollinear SDFT in EXX. This is an impor-\ntant difference to the noncollinear LSDA formulation of\nRef.3wherethemagnetizationdensityandtheexchange-\ncorrelationmagneticfieldarelocallyparallelbyconstruc-\ntion.\nWe now turn to the evaluation of the energy of Eq. (3)\nand note that for an electrically neutral system with uni-\nformionicandelectronicdensitiesthesumoftheionicen-\nergyEion, the electronic interactionwith the ionicpoten-\ntial/integraltext\nd3r v0(r)n(r), and the Hartree energy U[n] exactly\ncancels out. We study the system at vanishing external\nmagnetic field, B0(r)≡0, and use the exact exchange\nenergy of Eq. (10) as an approximation to the exchange-\ncorrelation energy functional. It is expected10,17that\ninclusion of correlation leads to SDW states higher in\nenergy than the paramagnetic states.\nIn our case the total energy per unit volume only con-\nsists of a kinetic and an exchange contribution, i.e.,\n˜etot=˜ts+ ˜eEXX. (33)\nThe kinetic energy per unit volume has contributions\nfrom the two bands,\n˜ts=Ts\nV=/summationdisplay\nb˜t(b)\ns, (34)\nwhere the contribution of the first band is given by\n˜t(1)\ns=1\n8π2/integraldisplay\ndκθ(εF−ε(1)\nκ)/parenleftBig\nεF−ε(1)\nκ/parenrightBig\n/parenleftbigg\nεF−ε(1)\nκ+κ2+q2\n4+2qκsin2ϑκ/parenrightbigg\n(35)\nwhile the contribution of the second band is\n˜t(2)\ns=1\n8π2/integraldisplay\ndκθ(εF−ε(2)\nκ)/parenleftBig\nεF−ε(2)\nκ/parenrightBig\n/parenleftbigg\nεF−ε(2)\nκ+κ2+q2\n4+2qκcos2ϑκ/parenrightbigg\n.(36)\nInserting the orbitals (17) and (22) into Eq. (10), the\nexchange energy per unit volume, ˜ eEXX, can also be ex-\npressed as sum of two terms,\n˜eEXX= ˜e(1)\nEXX+ ˜e(2)\nEXX. (37)\nThe first term which describes intraband exchange is,\nafter carrying out the angular integrals, given by\n˜e(1)\nEXX=−1\n32π3\n/summationdisplay\nb/integraldisplay\ndκθ(εF−ε(b)\nκ)/integraldisplay\ndκ′θ(εF−ε(b)\nκ′)\ncos2(ϑκ−ϑκ′)I(y(b)(κ),y(b)(κ′),(κ−κ′)2) (38)while the second term, describing interband exchange,\nreads\n˜e(2)\nEXX=−1\n16π3/integraldisplay\ndκθ(εF−ε(1)\nκ)/integraldisplay\ndκ′θ(εF−ε(2)\nκ′)\nsin2(ϑκ−ϑκ′)I(y(1)(κ),y(2)(κ′),(κ−κ′)2).(39)\nwhere we have defined\ny(b)(κ) = 2(εF−ε(b)\nκ). (40)\nIn Eqs. (38) and (39) we also have used the integral\nI(y1,y2,a) =/integraldisplayy1\n0dy/integraldisplayy2\n0dy′ 1/radicalbig\n(y−y′)2+2(y+y′)a+a2(41)\nwhich can be solved in closed form by transforming to\nnew integration variables z=y−y′andz′= (y+y′)/2\nand changing the integration limits accordingly. There-\nfore the calculation of the total energy only requires the\nnumerical calculation of a two-dimensional integral.\nWe have calculated the total energy per particle\netot=˜etot\nn(42)\nin the following way: we start by numerically calculating\nthe Fermi energy for a given value nof the density or,\nequivalently, the Wigner-Seitz radius\nrs=/parenleftbigg3\n4πn/parenrightbigg1/3\n, (43)\nandgivenvaluesoftheparameters BandqfromEq.(27).\nThe Fermi energy thus becomes a function of these three\nparameters,\nεF=εF(rs,q,B), (44)\nwhich is then used to evaluate the total energy per par-\nticle for these parameter values. We then have, for fixed\nrs, minimized etotas a function of the parameters qand\nBnumerically.\nIn Fig. 1 we show the total energy per electron at rs=\n5.4 for a few values of Bas function of q/kFwhere\nkF=/parenleftbigg9π\n4/parenrightbigg1/31\nrs(45)\nis the Fermi wavenumber of the uniform electron gas in\nthe paramagnetic state. The value rs= 5.4 was chosen\nbecause then i) the SDW phase is lower in energy than\nboth the paramagnetic and ferromagnetic phases and ii)\nthe amplitude of the SDW (or the KS magnetic field) is\nrelatively high such that the resulting energy differences\ncaneasilyberesolvednumerically. Weclearlyseethatfor\nthe given values of Bfor wavenumbers between q/kF≈5\n0 0.5 1 1.5 2\nq / kF-0.047-0.0469-0.0468-0.0467-0.0466etot (a.u.)µB B = 0.010 a.u.\nµB B = 0.011 a.u.\nµB B = 0.012 a.u.\nparamagnetic\n1.6 1.65 1.7 1.75\nq / kF-0.046976-0.046972-0.046968etot (a.u.)\nFIG. 1: Total energy per particle in EXX for the electron\ngas atrs= 5.4 with spin density wave as function of q/kF\nand different values of the amplitude Bof the Kohn-Sham\nmagnetic field. The straight line corresponds to the total\nenergy per particle of the paramagnetic state at this densit y.\nThe inset shows a magnification close to the minimum.\n1.5 andq/kF≈1.75the energyofthe SDW state is lower\nthan the energy of the paramagnetic state. The lowest\nenergy for this value of rsis achieved for the parameters\nµBB= 0.011 a.u. and q/kF= 1.68.\nIn Fig. 2 we show the KS single-particle dispersions\nof Eqs. (18) and (23) as well as the HF single particle\ndispersions. To obtain the latter ones we first calculate\nthe HF self energy (which is a 2 ×2 matrix in spin space)\nas\nΣHF(r,r′) =−/summationdisplay\nb/summationdisplay\nkθ(εF−ε(b)\nk)Φ(b)\nk(r)⊗Φ(b)†\nk(r′)\n|r−r′|\n(46)\nand then diagonalize the resulting HF Hamiltonian\nˆhHF=−∇2\n2+/integraldisplay\nd3r′ΣHF(r,r′)... (47)\nwhere the second term is a to be read as an integral oper-\nator. We would like to emphasize that we use the KS or-\nbitals and orbital energies to evaluate the HF self energy,\ni.e., we do notperform a selfconsistent HF calculation\nhere.\nIn Fig. 2 we show the KS and HF dispersions only\nfor theκ-coordinate, i.e., we set k/bardbl= 0. As expected,\nclose to κ/kF= 0 a direct gap opens up in the KS\nsingle-particledispersionsduetothe presenceofthe spin-\ndensity wave. The position of the Fermi energy is such\nthat not only states of the lower ( b= 1) band but also\nstates of the second ( b= 2) KS band are occupied in the\nground state.\nThe HF bands in Fig. 2 have been rigidly shifted by\na constant such that the lower HF band ( b= 1) and\nthe lower KS band equal the Fermi energy for the same\nvalue of κ/kF. It is evident that, as expected, the HF\nsingleparticledirectbandgapat κ/kF= 0ismuchlarger-2 -1 0 1 2\nκ / kF-0.100.10.20.30.40.5ε(b)(κ) (a.u.)KS band b=1\nKS band b=2\nHF band b=1\nHF band b=2\nFermi energy\nFIG. 2: Single-particle KS and HF bands at k/bardbl= 0 for the\noptimized parameter values ( µBB= 0.011 a.u. and q/kF=\n1.68 atrs= 5.4) mimizing the EXX total energy per particle\nin Fig. 1. The straight line indicates the Fermi energy and\nshows that close to κ/kF= 0 states of the second KS band\n[dashed (green) line] are occupied in the ground state. The\nKS orbitals have been used to compute the HF Hamiltonian\nand the resulting HF bands have been shifted rigidly such\nthat the lower HF and KS bands equal εFat the same value\nof|κ/kF|. The relative position of the second HF band [dash-\ndash-dotted (purple) line] indicates that also in HF states in\nboth bands will be occupied.\nthan the corresponding KS gap. Moreover, the position\nof the second HF band indicates that also in the HF\ncase there will be occupied states in the second band.\nWhile here we have calculated the HF bands using the\nDFT orbitals and orbital energies, we have confirmed13\nthat the above statement is true also for a HF energy\nminimization and the resulting HF bands are very close\nto the ones presented here.\nThe occupation of states in both single-particle bands\nis sometimes excluded in works on the SDW in the\nHartree-Fock approximation10,12and also in a numerical\ninvestigation13we have found that for the global energy\nminimum in Hartree-Fock only the lowest single-particle\nband is occupied. This has motivated us to do the mini-\nmization of the total energy in EXX also under the addi-\ntional constraint that only states of the lowest subband\nare occupied.\nSimilar to Fig. 1, in Fig. 3 we show the total energy\nper electron at rs= 5.4 for a few values of Bas function\nofq/kF. Of course, the constrained minimization leads,\nfor a given value of rs, to different optimized parameter\nvalues. Surprisingly, however, we found that the mini-\nmization constraining the occupation to the lower sub-\nbandleads to lower total energies than the ones obtained\nwith a two-band minimization. Moreover, this lower to-\ntal energy is achieved with a Slater determinant which\nhas empty states below the Fermi level.\nThis can be seen in Fig. 4 where we show the KS and\nHF energy bands at rs= 5.4 for this “one-band” mini-6\n0 0.5 1 1.5 2\nq / kF-0.047-0.0465-0.046-0.0455etot (a.u.)\nµB B = 0.019 a.u.\nµB B = 0.020 a.u.\nµB B = 0.021 a.u.\nparamagnetic1.25 1.3 1.35 1.4\nq / kF-0.04714-0.047136-0.047132etot (a.u.)\nFIG. 3: Same as Fig. 1 except that now only states in the\nlower band are allowed to be occupied. The total energy per\nparticle at the minimum is lower than when states in both\nbands are allowed to be occupied.\nmization for the optimized parameter values of µBB=\n0.020 a.u. and q/kF= 1.33. We see that there are states\nin the second KS band below the Fermi energy which,\ndue to the constraint in the minimization, remain un-\noccupied. We also note that for the one-band case the\namplitude of the minimizing Kohn-Sham magnetic field,\nand therefore also the “gap” between the two KS bands\natκ/kF= 0, is almost twice as large as in the two-band\ncase. Compared to Fig. 2, the intersection of the Fermi\nenergy with the bands ε(b)(κ) is shifted to a lower value\nof|κ|.\nTheHFbandsagainhaverigidlybeenshiftedsuchthat\nthe lower HF and KS bands intersect the Fermi energy\nat the same κ. Again, the direct HF gap at κ/kF= 0 is\nsignificantly larger than the KS gap. In contrast to the\ntwo-band case, the second HF band now is energetically\nhigher than the Fermi energy and the corresponding HF\nstate would, unlike the EXX state, have no unoccupied\nsingle-particle states below εF. Again here we have done\nonly a post-hoc evaluation of the HF bands but we have\nchecked that the statement remains valid for a selfconsis-\ntent HF calculation as well13.\nWe have optimized the EXX total energy per particle\nfor a range of rs-values once for single-particle occupa-\ntions in both energy bands and once for occupations re-\nstrictedtothelowerband. InFig.5weshowtheresulting\nphase diagram in the relevant density range. When al-\nlowing occupations in both bands, the SDW state (which\nis then a ground state Slater determinant) is lower in en-\nergy than both the paramagnetic and the ferromagnetic\nphase for rsin the range 5 .0<∼rs<∼5.46. In this case\nthe energiesareveryclosetothe energiesofthe paramag-\nnetic phase (energy differences of less than 4 ×10−5a.u.,\nsee lower panel of Fig. 5) and therefore the transition\nto the ferromagnetic phase occurs at an value of rsonly\nslightly higherthan the rs-value whereparamagneticand\nferromagnetic phase are degenerate.-2 -1 0 1 2\nκ / kF-0.100.10.20.30.40.5ε(b)(κ) (a.u.)KS band b=1\nKS band b=2\nHF band b=1\nHF band b=2\nFermi energy\nFIG. 4: Same as Fig. 2 except that the optimized parame-\nters are used which result from a minimization with occupied\nstates in the lower KS band only. For rs= 5.4 these values\nareµBB= 0.020 a.u. and q/kF= 1.33. Again, the straight\nline indicates the Fermi energy. Note that the states of the\nsecond KS band [dashed (green) line] remain unoccupied in\nthis calculation, even if their single-particle energies a re be-\nlow the Fermi level, i.e., the resulting Slater determinant is\nnota ground state of the Kohn-Sham problem. On the other\nhand, the post-hoc evaluation of the HF bands (for details se e\ncaption of Fig. 2 and the main text) indicates that the second\nHF band [dash-dash-dotted (purple) line] will remain unoc-\ncupied and the resulting HF wavefunction will be a ground\nstate Slater determinant.\nOn the other hand, restricting the single-particle occu-\npation to the lowest band, the SDW state is more stable\nthan para- and ferromagnetic state for 4 .78<∼rs<∼5.54.\nIn this case the energy differences between the param-\nagnetic and the SDW phase range to almost 4 ×10−4\na.u. (lower panel of Fig. 5), almost an order of magni-\ntude larger than in the two-band case. However, for all\nrsvalues in the stability range of the SDW phase, the\nminimizing Slater determinant in the one-band case is\nnota ground state of the Kohn-Sham problem.\nBoththe one-andtwo-bandcasesin EXXhavein com-\nmon that they predict the SDW phase to be lower in en-\nergy than the paramagnetic phase only for a restricted\nrangeofrsvalues. ThisisdifferentfromtheHartree-Fock\ncase10,12where the SDW phase is more stable than the\nparamagnetic phase for allvalues of rs. This is not com-\npletely surprising since due to the additional constraint\nof local Kohn-Sham potentials vsandBsin the EXX\nminimization, the resulting energies have to be higher\nthan the Hartree-Fock total energies. Since for small val-\nues ofrsthe SDW total energies in HF are extremely\nclose to the total energies of the paramagnetic phase13,\nthe higher EXX total energies can easily lead to a more\nstable paramagnetic phase.\nIn Fig.6 weshowthe SDW parameters q(upper panel)\nandB(middle panel) forwhich the EXXtotal energyper\nparticle is minimized in the one- and two-band cases for\nthosers-values for which the SDW phase is more stable7\n-0.0475-0.04725-0.047-0.04675etot (a.u.)paramagnetic\nferromagnetic\nSDW: two bands occupied\nSDW: one band occupied\n4.8 5 5.2 5.4\nrs (a0)1e-071e-061e-050.0001∆etot (a.u.)ePM-eSDW (2 bands)\nePM-eSDW (1 band)\nFIG. 5: Upper panel: Total energy per particle in EXX for\ndifferent phases of the uniform electron gas as function of\nWigner-Seitz radius rs. In the SDW phase, in one case the\noccupation of single-particle states in both bands is allow ed\nwhile in the other case the occupied states are restricted to\nthe lower band. Lower panel: energy difference between the\ntotal energies per particle of the paramagnetic phase and th e\nSDW phase for SDWs with occupied states in one and two\nbands. For the two-band case, the SDW phase is lower in\nenergy than both the paramagnetic and the ferromagnetic\nphase for 5 .0<∼rs<∼5.46. For the one-band case the range of\nstability of the SDW phase is 4 .78<∼rs<∼5.54.\nthan both paramagnetic and ferromagnetic phases. For\nthe one-band case, the wavenumber qof the spin-density\nwave covers almost the whole range between kFand 2kF\nwhile for the two-band case this range is much narrower.\nThe amplitudes Bandm0of the Kohn-Sham magnetic\nfield (middle panel) andthe magnetizationdensity (lower\npanel) of the SDW are significantly smaller in the two-\nband case as for the case with occupied single-particle\nstates in one band only. It is sometimes assumed10that\nthe wavenumber of the SDW is close to 2 kF. Our results\nshow that this need not be the case, as in the one-band\ncaseqapproaches kFfor densities at the lower end of the\nstability range of the SDW phase. However, neither in\nEXX nor in HF13have we ever found a stable SDW state\nwith wavenumber lower than kF.\nIV. SELFCONSISTENCY\nIn the previous Section we have used an ansatz for\nthe Kohn-Sham orbitals in the SDW phase which de-\npends on two parameters and then minimized the EXX\ntotal energy per particle with respect to these param-\neters. We have done this minimization once allowing\nsingle-particle states in both bands to be occupied and\nonce for occupations only in the lower band. This is dif-\nferent from the usual way of applying DFT where one\ncalculates the exchange-correlation potentials and solves\nthe Kohn-Sham equation self-consistently. In our case,11.251.51.75q/kFone band occupied\ntwo bands occupied\n012µB B (10-2 a.u.)\n4.8 5 5.2 5.4\nrs (a0)024m0 (10-4 a.u.)\nFIG. 6: Optimized values for the parameters q(upper panel)\nandB(middle panel) for which the EXXtotal energy perpar-\nticle of the SDW phase is minimized. The results are shown\nover the range of rs-values for which the SDW phase is lower\nin energy than both the paramagnetic and the ferromagnetic\nphases for the cases when both bands or only one band are\noccupied. Lower panel: amplitude of the SDW (Eq. (30)) for\nthe one- and two-band case.\nthe calculation of the EXX potentials requires solution\nof the OEP equations (13) and (14). In the present Sec-\ntion we still use our ansatz for the Kohn-Sham orbitals\nandinvestigateifitis consistentwith the OEPequations.\nWe start by calculating the orbital shifts of Eq. (15) in\nthe EXX approximation. Inserting our ansatz after some\nstraightforward algebra we obtain for the orbital shift of\nthe first band\nΨ(1)\nk(r) =G(k)\nε(1)\nκ−ε(2)\nκΦ(2)\nk(r) (48)\nwhile the shift for the second band reads\nΨ(2)\nk(r) =−G(k)\nε(1)\nκ−ε(2)\nκΦ(1)\nk(r) (49)\nwhere we have defined\nG(k) =−qκsinϑκcosϑκ+F(k) (50)\nas well as\nF(k) =/summationdisplay\nbsign(b)/integraldisplayd3k1\n(2π)3θ(εF−ε(b)\nk1)\n4π\n|k−k1|2sin(ϑκ1−ϑκ)cos(ϑκ1−ϑκ).(51)\nInserting the orbital shifts (Eqs. (48) and (49)) as well as\nthe orbitals (Eqs. (17) and (22)) it is straigthforward to\nsee that the first OEP equation (13) is satisfied by our\nansatz, i.e.,\n/summationdisplay\nb/summationdisplay\nkθ(εF−ε(b)\nk)/parenleftBig\nΦ(b)†\nk(r)Ψ(b)\nk(r)+c.c./parenrightBig\n= 0.(52)8\nThis can easily be understood from the physical content\nof the OEP equations: if we start from the KS Hamilto-\nnianasnon-interactingreferenceandperformaperturba-\ntion expansion of the interacting density in the perturba-\ntionˆWClb−ˆVxc−µBσˆBxc, whereˆWClbis the operatorof\nthe electron-electroninteraction and ˆVxc+µBσˆBxcis the\noperator of the KS exchange-correlation potentials, then\nthe OEP equation in EXX simply says that the density\nremains unchangedto first order. In our case we keep the\ndensity fixed and therefore the OEP equation (13) holds.\nAsimilarargumentcanbe usedfor the z-componentof\nthe OEP equation (14) which says that the z-component\nm(r) of the magnetization density remains unchanged\nunderthe sameperturbationtofirstorder. Thisequation\nreads explicitly\n/summationdisplay\nb/summationdisplay\nkθ(εF−ε(b)\nk)/parenleftBig\nΦ(b)†\nk(r)σzΨ(b)\nk(r)+c.c./parenrightBig\n=−2/summationdisplay\nbsign(b)\n/integraldisplayd3k\n(2π)3θ(εF−ε(b)\nk)sinϑκcosϑκG(k)\nε(1)\nκ−ε(2)\nκ= 0 (53)\nwhere the last equality can most easily be seen by noting\nthat the integrand is an odd function under the transfor-\nmationκ→ −κandalltheintegralsareoverasymmetric\nrange around κ= 0.\nFor thex−andy−component of Eq. (14) we obtain\n/summationdisplay\nb/summationdisplay\nkθ(εF−ε(b)\nk)/parenleftBig\nΦ(b)†\nk(r)σxΨ(b)\nk(r)+c.c./parenrightBig\n=J(εF,q,B)cos(qz) = 0 (54)\nand\n/summationdisplay\nb/summationdisplay\nkθ(εF−ε(b)\nk)/parenleftBig\nΦ(b)†\nk(r)σxΨ(b)\nk(r)+c.c./parenrightBig\n=J(εF,q,B)sin(qz) = 0 (55)\nwhere\nJ(εF,q,B) = 2/summationdisplay\nbsign(b)\n/integraldisplayd3k\n(2π)3θ(εF−ε(b)\nk)/parenleftbig\ncos2ϑκ−sin2ϑκ/parenrightbig\nG(k)\nε(1)\nκ−ε(2)\nκ(56)\nSince Eqs. (54) and (55) have to be satisfied for all values\nofz, we obtain only the condition\nJ(εF,q,B) = 0, (57)\ni.e., the two OEP equations are not independent.\nIn Fig. 7 we show Jof Eq. (56) for rs= 5.4 as function\nofq/kFfor different values of Bboth for the case of\noccupations in both bands (upper panel) as well as for\noccupations restricted to the lower band (lower panel).\nNote that for the latter case the sum over bands bboth\nin Eq. (56) as well as in Eq. (51) only extends over the\nlower band, b= 1.-2.5-2-1.5-1-0.50J (10-5 a.u.)\nµB B = 0.010 a.u.\nµB B = 0.011 a.u.\nµB B = 0.012 a.u.\n1.25 1.3 1.35 1.4\nq / kF-0.500.5J (10-5 a.u.)\n0 0.5 1 1.5 2\nq / kF-2.502.557.5J (10-5 a.u.)\nµB B = 0.019 a.u.\nµB B = 0.020 a.u.\nµB B = 0.021 a.u.\nFIG. 7: Upper panel: Jof Eq. (56) for rs= 5.4 as function of\nq/kFanddifferentvalues of Bfor thecase with occupations in\ntwo bands. The parameter values are the same as in Fig. 1 for\nwhich a minimum in the total energy per particle was found.\nSinceJnever crosses zero, in this case the minimum of the\ntotal energy is not consistent with the solution of the OEP\nequation. Lower panel: same as above but now for occupied\nsingle particle states only in the lower band. The parameter\nvalues are the same as in Fig. 3. In contrast to the two-\nband case, now Jnot only crosses zero but also does so at\nthose values of q/kFfor which a local minimum was found\nin Fig. 3 (see inset for magnification around the intersectio ns\nwith the zero axiss). Therefore, the OEP equation in this\ncase is consistent with the minimization of the total energy\nper particle.\nIn the upper panel of Fig. 7 we choose the same values\nfor the parameter Bas used in Fig. 1 which all had local\nminima for some value of q <2kF. For these values of\nB, however, Eq. (57) is not satisfied for any value of q\nin that range. We therefore have to conclude that in the\ntwo-band case the energy minimization is not consistent\nwith the OEP equations .\nIn the lower panel of Fig. 7 where only single-particle\nstates of the lower band are occupied we choose the pa-\nrameters as in Fig. 3. In this case, Jnot only crosses\nzero but also does so exactly for those values of q/kF\nfor which we found local minima in the total energy per\nparticle in Fig. 7 (see inset for a magnification of the re-\ngion where Jcrosses zero). We therefore conclude that\nin the one-band case the minimization of the total energy\nis consistent with the OEP equation, i.e., our ansatz is\nselfconsistent in this case. Again we emphasize that the\nresulting Slater determinant is notthe ground state of\nthe KS system.\nIt has been shown18that in unrestricted HF theory\nall the single-particle levels are fully occupied up to the\nFermi energy. To the best of our knowledge, a similar\nstatement has not been proven for DFT (even in EXX\napproximation)and our results indicate that it might not\nbe true in EXX. On the other hand, the proof of Ref. 18\nholds for the true, unrestricted HF ground state while in\nour case we have restricted the symmetry of our problem9\nto the SDW symmetry. It is quite conceivable that the\nfact that we find an excited-state Slater determinant as\nenergy-minimizing wavefunction hints towards an insta-\nbility of the SDW phase against further reduction of the\nsymmetry.\nV. SUMMARY AND CONCLUSIONS\nWe have investigated the SDW state of the uniform\nelectron gas within the EXX approximation of non-\ncollinear SDFT. While in the Hartree-Fock approxima-\ntion the SDW state is energetically more stable than the\nparamagneticstateforallvaluesof rs, inEXXthisisonly\ntrue for values of rslarger than a critical value. Using an\nexplicit ansatz for the spinor orbitals in the SDW state,\nwe have performed the energy minimization of the EXX\ntotal energy in two ways: (i) in the first case we used\nas non-interacting reference wavefunction a ground state\nSlater determinant with occupied single-particle orbitals\nbelonging to both single-particleenergy bands, as long as\ntheir energy is below the Fermi energy. Then the SDW\nphase is more stable than both paramagnetic and ferro-\nmagnetic phases for 5 .0<∼rs<∼5.46. (ii) In the second\ncase we required all the occupied single-particle orbitals\nin the Slater determinant to belong to the lower band.The minimizing Slater determinant in this case turns out\nto be an excited state, since orbitals with orbital ener-\ngiesbelowtheFermienergybelongingtothe secondband\nremain unoccupied. Nevertheless, for a given rsthe to-\ntal energies of the minizing SDW states are significantly\nlower than in case (i). The range of stabilty of the SDW\nphase with respect to both paramagnetic and ferromag-\nnetic phases is extended to 4 .78<∼rs<∼5.54.\nWe then have investigated if the self-consistency con-\nditions provided by the OEP equations for non-collinear\nSDFT are satisfied with our ansatz for the single-particle\norbitals. We have found that for case (i) the parameter\nvalues minimizing the EXX total energy are notconsis-\ntentwithasolutionoftheOEPequations. Incase(ii), on\nthe other hand, for the same parameter values for which\nthe EXX total energy is minimized also the OEP equa-\ntions are satisfied. In this case, the solution we found is\ntherefore selfconsistent.\nAcknowledgments\nWe would like to acknowledge useful discussions with\nIlya Tokatly, Giovanni Vignale, and Nicole Helbig.\nWe acknowledge funding by the ”Grupos Consolidados\nUPV/EHU del Gobierno Vasco” (IT-319-07).\n1U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972).\n2W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965).\n3J. K¨ ubler, K.-H. H¨ ock, J. Sticht, and A.R. Williams,\nJ. Phys. F 18, 469 (1988).\n4K. Capelle and L.N. Oliveira, Europhys. Lett. 49, 376\n(2000).\n5K. Capelle and L.N. Oliveira, Phys. Rev. B 61, 15228\n(2000).\n6J.D. Talman and W.F. Shadwick, Phys. Rev. A 14, 36\n(1976).\n7T. Grabo, T. Kreibich, S. Kurth, and E.K.U. Gross, in\nStrong Coulomb Correlations in Electronic Structure Cal-\nculations: Beyond Local Density Approximations , edited\nby V. Anisimov (Gordon and Breach, Amsterdam, 2000),\np. 203.\n8S. K¨ ummel and L. Kronik, Rev. Mod. Phys. 80, 3 (2008).\n9S. Sharma, J.K. Dewhurst, C. Ambrosch-Draxl, S. Kurth,\nN. Helbig, S. Pittalis, S. Shallcross, L. Nordstr¨ om, and\nE.K.U. Gross, Phys. Rev. Lett. 98, 196405 (2007).\n10G. Giuliani and G. Vignale, Quantum Theory of theElectron Liquid (Cambridge University Press, Cambridge,\n2005).\n11A.W. Overhauser, Phys. Rev. Lett. 4, 462 (1960).\n12A.W. Overhauser, Phys. Rev. 128, 1437 (1962).\n13F.G. Eich, S. Kurth, C.R. Proetto, S. Sharma, and\nE.K.U. Gross, in preparation.\n14S. Kurth and S. Pittalis, in Computational Nanoscience:\nDo It Yourself! , Vol. 31 of NIC Series , edited by J. Gro-\ntendorst, S. Bl¨ ugel, and D. Marx (John von Neumann In-\nstitute for Computing, J¨ ulich, 2006), p. 299.\n15S. K¨ ummel and J.P. Perdew, Phys. Rev. Lett. 90, 043004\n(2003).\n16S. K¨ ummel and J.P. Perdew, Phys. Rev. B 68, 035103\n(2003).\n17G.F. Giuliani and G. Vignale, Phys. Rev. B 78, 075110\n(2008).\n18V. Bach, E.H. Lieb, M. Loss, and J.P. Solovej, Phys. Rev.\nLett.72, 2981 (1994)."    },    {        "title": "0907.5129v1.Quantum_measures_for_density_correlations_in_optical_lattices.pdf",        "content": "arXiv:0907.5129v1  [quant-ph]  29 Jul 2009Quantum measures for density correlations\nin optical lattices\nF. Benattia,b, R. Floreaniniband G. G. Guerreschia\naDipartimento di Fisica Teorica, Universit` a di Trieste, 34 014 Trieste, Italy\nbIstituto Nazionale di Fisica Nucleare, Sezione di Trieste, 34014 Trieste, Italy\nAbstract\nThedensity-densitycorrelationprofilesobtainedsuperim posingabsorptionimagesfrom\natomic clouds freely expanding after the release of the confi ning optical lattice can be\ntheoretically described in terms of a generalized quantum m easure based on coherent-\nlike states. We show that the corresponding density pattern s differ in a testable way\nfromthosecomputedusingstandardmany-bodymeanvalues, u suallyadoptedinfitting\nexperimental data.\n1 Introduction\nA standard technique used in experiments for extracting informat ion on the behavior of\nultracold atomic gases trapped in optical lattices1is based on analysis of interference phe-\nnomena (see [3]-[14] and references therein). Since the measure o f relevant observables inside\nthe optical lattice is problematic, the usual adopted procedure co nsists in the release of the\nconfining optical potentials, followed by the (free) expansion of th e atomic cloud up to meso-\nscopic sizes. At this point, the cloud is illuminated by a laser beam and th e corresponding\nabsorption image collected. The absorption process is destructive ; nevertheless, many pic-\ntures can be obtained by starting each time with a new system, prep ared in the same initial\nstate. By superimposing the various obtained pictures, informatio n on the atom density at\nthe moment of trap release can be inferred.\nTheimagethatisobtainedbysuperimposingallthese“photographs ”isusuallyinterpreted\nas the average of the density operator over the state of the sam ple. An alternate theoretical\ndescription is however possible. Indeed, in quantum mechanics, any averaging procedure\nobtained through a measuring process corresponds to a generaliz ed quantum measure, i.e.a\nso-called Positive Operator Valued Measure (POVM) [15]-[17]. In general, the choice of the\n1For recent reviews on the study of quantum many-body effects in t hese systems, see [1, 2].\n1POVMtobeusedissuggestedbytheexperimental evidences. Inth epresent caseofultracold\natoms in optical lattices, this evidence comes from experiments invo lving a two-well trapping\npotential [18]. The actual data show that interference effects ap pear in a single absorption\nimage even when the system is prepared in a totally incoherent state : more precisely, the\ninterference pattern seen in single pictures seems to always confo rm to what is expected for\na condensed, fixed-phase state, a state in which all atoms share t he same single particle wave\nfunction. By constructing a POVM in terms of these coherent-like s tates, one finds that the\ncorresponding generalized quantum measurement process leads t o predictions that, at least\nin line of principle, differ from those obtained through the simple avera ge of the density\noperator [19].2\nThe aim of the present investigation is to analyze a possible physical s cenario in which\nthose differences may become visible and experimentally detectable. To this end, we shall\nstudy the behavior of a system of cold atoms in bichromatic optical la ttices, where a second,\nlow-intensity laser is superimposed to the one forming the periodic po tential with the aim\nof obtaining an unbalanced filling of the lattice sites [20]-[23]. By exploit ing the properties\nof the density-density correlation function [24], one can show that there are experimentally\nrelevant instances in which the differences in the predictions of the t wo above mentioned\ntheoretical interpretations can be revealed. This result has been supported by a numerical\nsimulation, reproducing the situation of an actual experimental se tup. We are confident that\nthese results will stimulate further direct analysis and tests.\n2 Cold bosonic gases in optical lattices\nWe shall study the behavior of Nbosons confined in one-dimensional lattice with Msites,\neach separated by a fixed distance d.3In a suitable approximation, i.e.for a large enough\ninter-site barriers, their dynamics can be described by a Bose-Hub bard Hamiltonian [1, 2],\nH=−J/summationdisplay\n<i,j>ˆb†\niˆbj+/summationdisplay\niǫiˆni+1\n2U/summationdisplay\niˆni(ˆni−1), i,j= 1,2,...,M , (1)\nwhere< i,j >means nearest neighbor, ˆb†\ni,ˆbiare creation and annihilation operator for an\natom in site i, ˆni=ˆb†\niˆbiis the number operator on site i, whileJ,Uandǫiare parameters\n2The standard theoretical intepretation of these experimental f acts explains the appearence of the density\ninterference fringes as an “emergent phenomena”[7]-[11]; althoug h phenomenological in character, also this\naveraging procedure can be interpreted in terms of a quantum gen eralized measure, albeit through a rather\ninvolved, unnatural POVM.\n3In the actual experimental setups, the bosons are really confine d in an three-dimensional harmonic trap\nover which a periodic potential along say the xdirection is superimposed. Since the confining potentials are\nseparable, the single particle wave function describing the state of the atoms factorizes in a part depending\nonly on the variable xtimes a piece depending on the couple ( y,z). The dependence on the transverse\ncoordinates is irrelevant for the dynamics in the lattice, that there fore can be effectively described by a\none-dimensional Hamiltonian.\n2thatquantifythehopping, repulsionandsingle-sitedepthenergy, respectively. Theoperators\nˆb†\ni,ˆbiobey the standard Bose commutation relations: [ ˆbi,ˆb†\nj] =δij.\nThe totalnumber ofparticles Nis conserved by thedynamics generated by (1). Therefore,\nthe Hilbert space of the system is/parenleftbigN+M−1\nN/parenrightbig\n-dimensional and can be be spanned by the set\nof Fock states, |/vectork;N/an}bracketri}ht, describing the situation in which the occupation number of each site\nis fixed; the M-dimensional vector /vectork= (k1,k2,...,k M), with/summationtextM\ni=1ki=N, represents a\npossible distributions of the Natoms in the Msites. These states are obtained by acting\nwith the creation operators on the vacuum/vextendsingle/vextendsingle0/angbracketrightbig\n; explicitly, one has\n/vextendsingle/vextendsingle/vectork;N/angbracketrightbig\n≡1/radicalbig\n/vectork!(ˆb†\n1)k1(ˆb†\n2)k2...(ˆb†\nM)kM/vextendsingle/vextendsingle0/angbracketrightbig\n, (2)\nwhere in short /vectork! =k1!k2!...kM!.\nA different basis in the system Hilbert space is given by the collection of fixed-phase,\ncoherent-like states,\n/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N/angbracketrightbig\n≡1√\nN!/parenleftBiggM/summationdisplay\nj=1eiϕj/radicalbig\nξjˆb†\nj/parenrightBiggN/vextendsingle/vextendsingle0/angbracketrightbig\n, (3)\nwhere/vectorξ,/vector ϕareM-dimensional vectors whose components ξj,ϕj,j= 1,2,...,M, represent sets\nof real parameters such that ϕj∈[0,2π],ξj∈[0,1] with/summationtext\njξj= 1. These states describe a\nphysical situation in which all Natoms are in a coherent superposition, where ξimeasures\nthe probability of finding an atom in the i-th site, while ϕigives the corresponding phase.\nOnly relative phases are relevant, so that one can arbitrarily fix the value of one of the ϕi.\nThe Fock states form an orthonormal set, while the coherent one s become orthogonal only\nin the large Nlimit;4nevertheless, they form an overcomplete set of states [19]:/BD=(N+M−1)!\nN!/integraldisplay2π\n0dϕ1\n2π...dϕM−1\n2π/integraldisplay1\n0dξ1.../integraldisplay1−ξ1−ξ2...−ξM−2\n1dξM−1/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N/angbracketrightbig/angbracketleftbig/vectorξ,/vector ϕ;N/vextendsingle/vextendsingle.\n(4)\nBy expanding a coherent state over the Fock basis, one finds:\n/vextendsingle/vextendsingle/vectorξ;/vector ϕ;N/angbracketrightbig\n=/summationdisplay\n/vectork/radicalBigg\nN!\n/vectork!ei/vectork·/vector ϕ/parenleftBiggM/productdisplay\nj=1(ξj)kj\n2/parenrightBigg\n/vextendsingle/vextendsingle/vectork;N/angbracketrightbig\n, (5)\nwhere the sum runs over all possible M-vectors/vectork, whose components kiobey the constraint/summationtext\niki=N. The overlap between a Fock and a coherent state is then given by:\n/angbracketleftbig/vectork;N/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N/angbracketrightbig\n=/radicalBigg\nN!\n/vectork!ei/vectork·/vector ϕ/parenleftBiggM/productdisplay\nj=1(ξj)kj\n2/parenrightBigg\n. (6)\n4Indeed, for an M-site lattice, one finds that: /an}bracketle{t/vectorξ, /vector ϕ;N|/vectorξ′, /vector ϕ′;N/an}bracketri}ht=/parenleftBig/summationtextM\ni=1/radicalbig\nξiξ′\niei(ϕi−ϕ′\ni)/parenrightBigN\n; using the\nCauchy-Schwartz inequality, one easily sees that the modulus of su m in the bracket is always less than one,\nunless/vectorξ=/vectorξ′and/vector ϕ=/vector ϕ′; as a consequence, its N-th power become vanishingly small as Nbecome large.\n3As well known [1, 2], the Hamiltonian (1) describes a cross-over bet ween a superfluid and\ninsulator phases, which becomes a true quantum phase transition, with order parameter\ndepending on the ratio J/U, in the limit of an infinite number Mof wells. Neglecting the\nshiftsǫi, for small J/U, the ground state of the system is given by a Fock state (Mott\ninsulator phase). On the other hand, for large J/U, the system shows phase coherence; all\nNparticles are in the same superposition and the ground state of (1) can be approximated\nby/vextendsingle/vextendsingle/vectorξ;/vector ϕ;N/angbracketrightbig\n, with definite relative phases and occupation probabilities (superflu id phase).\n3 Many-body states\nIn a typical experimental setup, the system of Natoms is first cooled to very low tempera-\ntures, of the order of few tens of nanokelvin, and then trapped in the optical lattice. Since\nmeasures of relevant observables directly in the lattice are difficult, indirect information on\nthe dynamics of theatoms areusually obtained by switching off the pe riodicconfining poten-\ntial and letting the atom gas expand freely up to mesoscopic dimensio ns. Absorption images\nof the expanded sample are then collected by shining it with a probe las er; by superimposing\nthe various obtained pictures, information on the atom density at t he moment of trap release\ncan be inferred.\nIn order to theoretically describe this process of measure, it is con venient to use a second\nquantized many-body formalism. Let us first introduce the field ope ratorˆψ†(x), creating\nfrom the vacuum an atom at position x,ˆψ†(x)|0/an}bracketri}ht=|x/an}bracketri}ht; it can be decomposed as\nˆψ†(x) =∞/summationdisplay\ni=1¯wi(x)b†\ni, (7)\nin terms of a complete set of single-particle wave functions wi(x)≡ /an}bracketle{tx|wi/an}bracketri}ht=/an}bracketle{tx|ˆb†\ni|0/an}bracketri}ht.\nAlthough the states |wi/an}bracketri}ht ≡ˆb†\ni|0/an}bracketri}ht,i= 1,2...,M, obtained by the action of the creation\noperators ˆb†\nionthevacuumareenoughtodescribeanatomconfinedinthelattice , acomplete,\ninfinite set is needed to properly represent its state when it moves f reely in space. These\nstates are orthonormal and therefore one can invert (7) and wr ite\nˆb†\ni=/integraldisplay\ndx wi(x)ˆψ†(x), (8)\nso that from [ ˆbi,ˆb†\nj] =/an}bracketle{twi|wj/an}bracketri}ht=δijone recovers the standard bosonic (equal-time) commu-\ntation relation,/bracketleftBig\nˆψ(x),ˆψ†(x′)/bracketrightBig\n=δ(x−x′).\nAssume now that the confining potential is released at time t= 0 and denote by ˆUtthe\nunitary operator that evolves freely in time the initial one-particle s tates:\n/vextendsingle/vextendsinglewi,t/angbracketrightbig\n≡ˆUt/vextendsingle/vextendsinglewi/angbracketrightbig\n=ˆUtˆb†\ni/vextendsingle/vextendsingle0/angbracketrightbig\n=ˆb†\ni(t)/vextendsingle/vextendsingle0/angbracketrightbig\n,ˆb†\ni(t)≡ˆUtˆb†\niˆU†\nt. (9)\n4The corresponding wave function is given by wi(x;t)≡ /an}bracketle{tx|wi(t)/an}bracketri}ht=/an}bracketle{tx|ˆb†\ni(t)|0/an}bracketri}ht, and coincides\nwith a transformed wi(x) under a ballistic expansion. At the moment of the release of the\nlattice, the wi(x,0),i= 1,2,...,M, are wave functions localized at the lattice sites xi:\nthey can be identified with one-dimensional Wannier functions. Afte r a large enough free\nexpansion time t, sufficient for the clouds coming from the various sites to overlap, o ne finds\nthrough (9) that these functions have a common envelope, differin g only by a phase [3, 4, 12]:\nwi(x,t) =|w(x,t)|eim\n2/planckover2pi1t(x−xi)2; (10)\nindeed, for those times, the scale over which |wi(x,t)|varies is larger than the product Md,\ngiving the original dimension of the lattice.\nSince the dynamics is free, every particle in a many-body state will ev olve independently\nwithˆUt; therefore, the evolution up to time tof thet= 0 coherent state/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N/angbracketrightbig\nwill simply\nbe given by\n/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N,t/angbracketrightbig\n≡1√\nN!/parenleftBiggM/summationdisplay\nj=1eiϕj/radicalbig\nξjˆb†\nj(t)/parenrightBiggN/vextendsingle/vextendsingle0/angbracketrightbig\n, (11)\nand analogously for Fock states (2)\n/vextendsingle/vextendsingle/vectork;N,t/angbracketrightbig\n=1/radicalbig\n/vectork!(ˆb†\n1(t))k1(ˆb†\n2(t))k2...(ˆb†\nM(t))kM/vextendsingle/vextendsingle0/angbracketrightbig\n. (12)\nIn this picture, only states evolve in time while operators, like ˆψ(x), remain fixed. Since\nat each instant of time tthe collection {|wi(t)/an}bracketri}ht}∞\ni=1is a complete set of single particle states\nobtained from the vacuum by the action of the creation operators ˆb†\ni(t),ˆψ(x) can be equiv-\nalently decomposed as ˆψ(x) =/summationtext∞\ni=1wi(x;t)ˆbi(t) for all times. Using this, one can compute\nthe action of the field operator ˆψ(x) on the two class of states, obtaining:\nˆψ(x)/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N,t/angbracketrightbig\n=√\nN/parenleftBigg/summationdisplay\nj/radicalbig\nξjeiϕjwj(x,t)/parenrightBigg\n/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N−1,t/angbracketrightbig\n, (13)\nˆψ(x)/vextendsingle/vextendsingle/vectork;N,t/angbracketrightbig\n=/summationdisplay\nj/radicalbig\nkjwj(x,t)/vextendsingle/vextendsinglek1,...,kj−1,...,kM;N−1,t/angbracketrightbig\n. (14)\n4 Generalized quantum measures\nIn order to apply the previous formalism to the theoretical interpr etation of the above men-\ntioned procedure of measuring density profiles, it is useful to reca ll some results deduced\nfrom the experiment.\nWhenM= 2 and the atoms in the lattice just before the release of the confin ing potential\nare prepared in a superfluid state described by (3), the picture th at is obtained at time\ntafter a free expansion shows a high visibility interference pattern, with fringe spacing\n5mediated by the wave vector Q=md//planckover2pi1t, wheremis the atom mass [18]. This is to be\nexpected, since in the superfluid phase essentially all Nparticles occupy the same quantum\nstate. The roughness and imperfection of the interference figur e in a single image, beside to\nexperimental errors, has to be ascribed to the finiteness of the p article number N. Indeed,\nin many-body physics, one can assimilate ensemble averages with mea n values with respect\nto macroscopically occupied many-body states, provided the numb er of particles involved is\nlarge enough. Therefore, the larger the number Nof atoms the system contains, the better\nasingleabsorption image will model the average n/vectorξ,/vector ϕ(x,t) of the density operator at point\nx,\nˆn(x)≡ˆψ†(x)ˆψ(x), (15)\nin the state/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N,t/angbracketrightbig\n. Using (13), one can easily compute the theoretically predicted\ndensity profile to obtain\nn/vectorξ,/vector ϕ(x,t)≡/angbracketleftbig/vectorξ,/vector ϕ;N,t/vextendsingle/vextendsingleˆn(x)/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N,t/angbracketrightbig\n=N/vextendsingle/vextendsingle/summationdisplay\nj/radicalbig\nξjeiϕjwj(x,t)/vextendsingle/vextendsingle2\n=N|w(x,t)|2/braceleftBig\n1+2/summationdisplay\nj<l/radicalbig\nξjξlcos(θjl(x,t)−ϕj+ϕl)/bracerightBig\n,(16)\nwhere, recalling (10), the relative dynamical phases take the explic it formθjl(x,t) =\nm\n2/planckover2pi1t[−(x−xj)2+ (x−xl)2]; since for the lattice site positions one has xj=jd, it can\nalso be rewritten as θjl(x,t) =Q(j−l)(x−d\n2(j+l)), thus reproducing (for M= 2) the\nobserved interference pattern mediated by the wave vector Q=md//planckover2pi1t.\nSince taking the picture of the expanded gas is a destructive opera tion (the sample is lost),\none usually repeats many times the whole measuring process, prepa ring the system in the\nsame superfluid initial state. In the case M= 2, all subsequent absorption pictures show the\nsame interference figure (modulo experimental uncertainties), w ith the same fringe spatial\npositions. Therefore, by superimposing all these pictures, one ob tains an average image that\nis indistinguishable from any of the single shot pictures, and therefo re it is again described\nby the pattern (16).\nIf instead one starts at t= 0 with a system prepared in an incoherent state, like the Fock\nstates in (2), with no definite relative phase relations, no interfere nce pattern in the absorp-\ntion images is expected to appear. Surprisingly, this is not the case: in any experiment so far\nperformed, single shot images always show interference figures co mpatible with the pattern\nin (16), irrespectively from the state of the sample at the moment o f trap release. However,\nunless the sample is prepared in a superfluid state, the absolute spa tial position of fringes in\nrepeated single shot images is seen to vary randomly; as a conseque nce, by superimposing\nmany single shot images, the interference pattern indeed complete ly disappears.\nTheusualtheoreticalinterpretationsoftheseexperimentalfa ctsisbasedontheassumption\nthat the superposition of single shot images reproduces operator averages. More precisely,\nthe procedure of repeatedly preparing the system at t= 0 in a generic many body state\ndescribed by the density matrix ρ, and then superimposing all the single shot pictures taken\n6after a free evolution of the system upto time t,ρ/ma√sto→ρ(t), is commonly believed to reproduce\nthe quantum average\nnρ(x,t) = Tr/bracketleftbig\nˆn(x)ρ(t)/bracketrightbig\n. (17)\nFor instance, for generic M, starting from a Fock state (2), ρ=/vextendsingle/vextendsingle/vectork;N/angbracketrightbig/angbracketleftbig/vectork;N/vextendsingle/vextendsingle, in this way\none would obtain\nn/vectork(x,τ)≡/angbracketleftbig/vectork;N,t/vextendsingle/vextendsingleˆn(x)/vextendsingle/vextendsingle/vectork;N,t/angbracketrightbig\n=M/summationdisplay\nj=1kj|wj(x,t)|2, (18)\nwhich shows no interference figures.\nAn alternate, different theoretical formalization of the experimen tally used measurement\ntechnique is nevertheless possible. It takes into account the prev iously mentioned result\nthat, forM= 2, single shot images always present interference patterns comp atible with\nthe profile in (16), irrespectively from the initial state ρ, and extends it to the case M >2.\nIn other terms, the procedure of “taking a photograph” of the e xpanded sample seems to\nselect a coherent-like state/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N,t/angbracketrightbig\nfor the state of the system, with definite amplitudes\nξiand phases ϕi, and as a consequence produce the profile n/vectorξ,/vector ϕ(x,t) in (16) for the average\ndensity.5As seen in experiments, amplitudes and phases nevertheless vary f rom shot to\nshot.6The occurrence of given values /vectorξand/vector ϕfor such parameters in a single shot will be\ndetermined by the initial state ρ. More precisely, the distribution of ξiandϕiover many\nabsorption pictures will be determined by the probability/angbracketleftbig/vectorξ,/vector ϕ;N,t/vextendsingle/vextendsingleρ(t)/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N,t/angbracketrightbig\n, which\ngives the weight of the configuration/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N,t/angbracketrightbig/angbracketleftbig/vectorξ,/vector ϕ;N,t/vextendsingle/vextendsinglein the expansion of ρ(t) in the\nbasis of coherent states.\nAs a result, in this scheme, the mean value of the density ˆ n(x), as for any other observable,\nis given by the sum over all possible configurations {/vectorξ,/vector ϕ}of the average n/vectorξ,/vector ϕ(x,t) weighted\nwith the above mentioned probability; explicitly, one then should write :\n˜nρ(x,t) =/integraldisplay\ndµ(ϕ)/integraldisplay\ndµ(ξ)/angbracketleftbig/vectorξ,/vector ϕ;N,t/vextendsingle/vextendsingleρ(t)/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N,t/angbracketrightbig\nn/vectorξ,/vector ϕ(x,t), (19)\nwhere for the the properly normalized volume and integration measu re, we have used the\nshorthand notation [ cf.(4)]:\n/integraldisplay\ndµ(ϕ)/integraldisplay\ndµ(ξ)≡(N+M−1)!\nN!/integraldisplay2π\n0dϕ1\n2π...dϕM−1\n2π/integraldisplay1\n0dξ1.../integraldisplay1−ξ1−ξ2...−ξM−2\n0dξM−1.\n(20)\nThe expression (19) for the averaged density corresponds to a q uantum mechanical gen-\neralized measure. It can be described by an operation of trace of ˆ n(x) over the density\n5From a different perspective, this empirical fact has recently been the object of various investigations\n[3]-[14].\n6Unless, as mentioned before, one starts at t= 0 already with a coherent state/vextendsingle/vextendsingle/vectorξ, /vector ϕ;N/angbracketrightbig\n,i.e.in a\nsuperfluid phase.\n7matrix ˜ρobtained from the starting state ρthrough the action of a (completely positive)\nmap; explicitly:\nρ(t)/ma√sto→˜ρ(t) =/integraldisplay\ndµ(ϕ)/integraldisplay\ndµ(ξ)V(/vectorξ,/vector ϕ;N,t)ρ(t)V(/vectorξ,/vector ϕ;N,t), (21)\nwith\nV(/vectorξ,/vector ϕ;N,t) =/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N,t/angbracketrightbig/angbracketleftbig/vectorξ,/vector ϕ;N,t/vextendsingle/vextendsingle; (22)\nit is a realization of a Positive Operator Valued Measure (POVM) [15]-[17].\nNoticethatthemeanvaluefortheatomdensitygivenin(19)obtaine dthroughthePOVM,\ni.e.\n˜nρ(x,t) = Tr/bracketleftBig\nˆn(x) ˜ρ(t)/bracketrightBig\n, (23)\nis clearly distinct from that obtained using the definition (17); never theless, they both repro-\nduce the profile (16) in the case of a superfluid initial state, ρ=/vextendsingle/vextendsingle/vectorξ,/vector ϕ;N/angbracketrightbig/angbracketleftbig/vectorξ,/vector ϕ;N/vextendsingle/vextendsingle, thanks\nto the (large N) orthogonality of the coherent states. On the other hand, in the case of a\nFock state, ρ/vectork=/vextendsingle/vextendsingle/vectork;N/angbracketrightbig/angbracketleftbig/vectork;N/vextendsingle/vextendsingle, the explicit evaluation of (23) gives\n˜n/vectork(x) =N\nN+MM/summationdisplay\nj=1(kj+1)|wj(x,t)|2, (24)\nwhich differs from the result (18) obtained applying (17) by an overa ll normalization factor,\nand in the weight assigned to the contribution of every single lattice s ite. Although in line\nof principle these differences might have experimental relevance, t heir actual detection looks\ntechnically problematic, except perhaps in the case of a double-well potential (M= 2) [19].\nInstead, as we will see in the next Section, density correlations app ear more suitable for\nstudying the predictions of the POVM prescription.\n5 Correlation functions\nBesides for density estimations, absorption images can also be used to extract information\non density correlations: by analyzing the absorption figure in distinc t points one can study\ncorrelations in atom positions. Let us then introduce the two-point correlation function as\nthe average of the following two-point operator7\nˆn(x,x′)≡ˆψ†(x)ˆψ†(x′)ˆψ(x)ˆψ(x′), (25)\nthat, according to the two measuring interpretations discussed in the previous Section, can\nbe either computed using the analog of the standard trace formula (17),\nnρ(x,x′,t) = Tr/bracketleftbig\nˆn(x,x′)ρ(t)/bracketrightbig\n, (26)\n7It differs from the density-density correlation operator by a δ(x−x′) contribution; on average, this term\nis suppressed by a factor 1 /Nand therefore negligible in the large Nlimit [24].\n8or by assuming that a single shot picture effectively projects the sy stem state into a coherent\none, so that operator averages should be taken using the transf ormed density matrix ˜ ρ(t) of\n(21), thus giving\n˜nρ(x,x′,t) = Tr/bracketleftbig\nˆn(x,x′) ˜ρ(t)/bracketrightbig\n. (27)\nIn order to make an easier comparison with the actual experimenta l data, a further in-\ntegration with respect to the barycenter coordinate, R= (x+x′)/2, is usually performed,\nso that the resulting integrated correlation function depends only on the relative coordinate\nr=x′−x[24]. Then, with a suitable normalization, one is lead to study the behav iour of\nthe following two functions\nGρ(r,t)≡/integraltext\ndR nρ(R−r\n2,R+r\n2,t)/integraltext\ndR nρ(R−r\n2,t)nρ(R+r\n2,t), (28)\nand, by replacing the standard averages with the POVM ones,\n˜Gρ(r,t)≡/integraltext\ndR˜nρ(R−r\n2,R+r\n2,t)/integraltext\ndR˜nρ(R−r\n2,t) ˜nρ(R+r\n2,t). (29)\nThese functions measure the conditional probability of finding two a toms in points separated\nby a distance r, averaged over all positions. In absence of correlations, they ta ke a constant\nvalue equal to one, while values greater than one would signal the te ndency of the atoms to\naggregate, a typical behaviour for bosons. As in the case of the a verage density discussed in\nthe previous Section, the expressions (28) and (29) take a partic ularly simple and compact\nformwhenthesystemisinitiallypreparedinaFockstate, ρ=/vextendsingle/vextendsingle/vectork;N/angbracketrightbig/angbracketleftbig/vectork;N/vextendsingle/vextendsingle. Indeed, recalling\n(10), that gives the free evolution of the Wannier functions, one e xplicitly finds\nGρ(r,t) =N(N−1)\nN2/braceleftbigg\n1 +1\nN(N−1)M/summationdisplay\ni/negationslash=j=1kikjeiQ(i−j)r/bracerightbigg\n, (30)\nwhile\n˜Gρ(r,t) =N(N−1)\nN2/braceleftbigg\n1 +1\n(N+M)(N+M−1)M/summationdisplay\ni/negationslash=j=1(ki+1)(kj+1)eiQ(i−j)r/bracerightbigg\n,(31)\nwhere, as before, kirepresents the initial occupation number of the i-th lattice site and\nQ=md//planckover2pi1t. It turns out that for a given initial state, the overall profile of (3 0) and (31)\nis very similar: both functions are essentially constant and close to o ne almost everywhere,\nexcept at the origin and at positions multiple of 2 π/Qwhere sharp peaks occur.\nNevertheless, the two expressions do differ in the normalization of t he oscillator terms, as\nwell as in their dependence on the occupation numbers. One can che ck that the effects of\nthese differences become more and more visible as the initial configur ation differs from that\ndescribed by a balanced Fock state, the one with an equal number N/Mof atoms in each\n9lattice site. Indeed, for such a state, up to terms of order 1 /N, one finds that both functions\n(30) and (31) reduce to:\nG(r,t) = 1 +sin2(πNQr)\nN2sin2(πQr)(32)\n≃1 ++∞/summationdisplay\nj=−∞δ/parenleftbiggQr\n2π−j/parenrightbigg\n. (33)\nIn order to appreciate the differences between the two theoretic al predictions Gρ(r,t) and\n˜Gρ(r,t), one has therefore to prepare the system in an unbalanced Fock state. The way\nin which this can be experimentally realized is by superimposing a second weaker periodic\npotential to the original one; this is done through the introduction of a second laser directed\nalong the optical lattice, whose wavevector κ2is chosen to be incommensurate with respect\ntoκ1=π/d, the wave vector of the first, original one [20]-[23]. When the amplitu de of the\nsecond laser is weak, thus introducing only small perturbations to t he original potential, the\ndynamics of the atoms in the lattice can still be described by the Bose -Hubbard Hamiltonian\n(1), where however thesitedepthenergy ǫigetsafurthersitedependent contributionoforder\nV2sin2(iκ2π/κ1), proportional to the strength V2of the second laser.\nThen, in order to prepare the system in a Fock state, one gradually increases the intensity\nof the lasers, so that the hopping term in the Hamiltonian (1) become s negligible. In this\nway, one drives the system into a ground state of the form/vextendsingle/vextendsingle/vectork;N/angbracketrightbig\n, characterized by the\nfilling configuration /vectork, minimizing the total energy E=/summationtext\njǫjkj+U\n2/summationtext\njkj(kj−1) with the\nconstraint/summationtext\njkj=N, forkjinteger. The advantage of working with a bichromatic lattice is\nnow apparent: it allows to change the distribution of the Natoms in the Mwells by varying\nthe amplitude V2of the second potential.\nFor a given value of the characteristic, physical parameters ente ring the Hamiltonian (1),\nthe configuration /vectorkthat minimizes the energy can be efficiently obtained using a numerical\nsimulation. We have used a Monte Carlo method implementing a simulated annealing algo-\nrithm, that better conforms to the experimentally adopted proce dure, since it considers all\natomsin the latticeat once.8Indeed, ina typical experiment, theatoms arefirst cooled inan\nharmonic potential, at the center of which the optical lattice is then slowly raised. Initially,\nthe hopping dynamics allows a redistribution of the atoms in the variou s sites; however, this\nbecomes highly suppressed at regime, leaving the system in a Fock gr ound state.\nTo be as closer as possible to an actual experimental situation, we h ave chosen to work\nwith the physical parameters that define the apparatus describe d in Ref.[21]. Although the\nprincipal lattice is in that case three-dimensional, the second, weak er potential is switched\non only along one direction, making the whole system effectively behav ing as a collection of\nindependent, separate one-dimensional bichromatic lattices. The whole system is filled with\n8This simulation procedure should be contrasted with the one adopte d in [22], where the Natoms are\ninserted in the lattice one by one, while minimizing the energy at each st ep.\n10about 3×105particles, but any single one-dimensional lattice is formed by about M= 130\nsites, filled with roughly N= 170 ultracold atoms.\nThe result of the simulation is summarized in Figure 1: it shows the plots of the corre-\nsponding correlation functions Gρ(r,t) (green line) and ˜Gρ(r,t) (red line). As expected, these\nfunctions are almost everywhere equal to one, except for the pr esence of periodic bumps.\nThe difference between the two is particularly visible in the lower, seco ndary peaks, whose\nheight is highly suppressed when the correlations are computed usin g the POVM prescrip-\ntion. This difference become more and more evident as the strength of laser giving rise to\nthe second potential is increased, as clearly shown in Figure 2. As a r esult, the suppression\nof the secondary peaks can be made visible well beyond any statistic al error, thus becoming\nexperimentally relevant.\n 0.8 1 1.2 1.4 1.6 1.8 2\n-250 -200 -150 -100 -50  0  50  100  150  200  250G(r)\nrG(r) with and without POVM\nG_POVM(r)\nG_without(r)\nFigure 1: Behavior of G(r,t) (green line) and of ˜G(r,t) (red line) with N= 170,M= 130,V2=\nh×9.9 kHz.\n6 Outlook\nThe measuring procedure commonly used in experiments with ultraco ld gases, consisting\nin extracting density profiles from absorption images taken after t he release of the optical\nlattice, suggests a theoretical interpretation in terms of genera lized quantum measurement\nprocesses. The experimental evidence regarding the presence o f interference fringes in single\n11 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45\n 0  1  2  3  4  5  6  7  8  9height of the secondary peak\nV2 in units of energy of (h * 1.98 kHz)height of the secondary peak\nh_peak_POVM(r)\nh_peak_without(r)\nFigure 2: Height of the secondary peaks as a function of the intensity o f the secondary laser with\n(red line) and without POVM (green line).\nshot absorption pictures irrespective from the initial state of the system suggests the use of\na POVM based on coherent-like, fixed phase states. Within this fram ework, the averages\nof system observables in general differ from those obtained throu gh mean values of the\ncorresponding operator in the system state, the usually adopted paradigm in interpreting\nexperimental data.\nAs discussed in Section 4, these differences are hardly visible in the ca se of density mea-\nsures, since they are of the order of the inverse total number of atoms in the sample. Instead,\nthe situation appears quite different for density-density correlat ions. As explicitly discussed\nin the previous Section, by preparing the system in a suitable Fock st ate, the profile of the\nintegrated and normalized correlation function is characterized by a double series of peaks,\nthe smaller of which appear to be much lowered when the correlations are computed using\nthe POVM prescription. This effect is very pronounced and surely we ll beyond statistical\nerrors for situations close to the actual experiment: this may ope n the way to a direct test\nof the POVM assumption.\nAcknowledgements\nThis work is supported by the MIUR project “Quantum Noise in Mesos copic Systems”.\n12References\n[1] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen and U. Sen,Adv. in Phys.\n56(2007) 243\n[2] I. Bloch, J. Dalibard and W. Zwerger, Rev. Mod. Phys. 80(2008) 885\n[3] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University Press,\nOxford, 2003)\n[4] C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases , (Cambridge\nUniversity Press, Cambridge, 2004)\n[5] S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms, Cavities and Photons ,\n(Oxford University Press, Oxford, 2006)\n[6] A.J. Leggett, Quantum Liquids , (Oxford University Press, Oxford, 2006)\n[7] J. Javanainen and Sun Mi Ho, Phys. Rev. Lett. 76(1996) 161\n[8] M. 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Phys. A 42(2009)\n035306\n[20] L. Fallani, L.E. Lye, V. Guarrera, C. Fort and M. Inguscio, Phys. Rev. Lett. 98, 130404\n(2007)\n[21] V. Guarrera, L. Fallani, J.E. Lye, C. Fortand M. Inguscio, New J. Physics 9, 107(2007)\n[22] V. Guarrera, N. Fabbri, L. Fallani, C. Fort, K.M.R. van der Stam a nd M. Inguscio,\nPhys. Rev. Lett. 100, 250403 (2008)\n[23] L. Fallani, C. Fort and M. Inguscio, Bose-Einstein condensates in disordered potentials,\narXiv:0804.2888\n[24] S. F¨ olling, F. Gerbier, A. Widera, O. Mandel, T. Gericke and I. Blo ch,Nature434, 481\n(2005)\n14"    },    {        "title": "0908.2743v1.Conductance_characteristics_of_current_carrying_d_wave_weak_links.pdf",        "content": "arXiv:0908.2743v1  [cond-mat.supr-con]  19 Aug 2009Conductance characteristics of current-carrying d-wave weak links\nS.N. Shevchenko1\n1B. Verkin Institute for Low Temperature Physics and Enginee ring, 47 Lenin Ave., 61103, Kharkov, Ukraine\n(Dated: December 4, 2018)\nThe local quasiparticle densityofstates in thecurrent-ca rryingd-wavesuperconductingstructures\nwas studied theoretically. The density of states can be acce ssed through the conductance of the\nscanning tunnelling microscope. Two particular situation s were considered: the current state of the\nhomogeneous film and the weak link between two current-carry ingd-wave superconductors.\nPACS numbers: 74.50.+r, 74.78.-w, 74.78.Bz, 85.25.Cp\nINTRODUCTION\nUnconventional superconductors exhibit different fea-\nturesinterestingbothfromthefundamentalpointofview\nand for possible applications [1]. In particular, double\ndegenerated state can be realized in d-wave Josephson\njunctions [2]. If the misorientation angle between the\nbanks of the junction χis takenπ/4, the energy minima\nof the system appear at the order parameterphase differ-\nenceφ=±π/2. These degenerate states correspond to\nthe counter flowing currents along the junction bound-\nary. Such characteristics make d-wave Josephson junc-\ntions interesting for applications, such as qubits [3]. Our\nproposition was to make these qubits controllable with\nthe externallyinjected alongthe boundarytransportcur-\nrent [4]. It was shown that the transport current and the\nspontaneous one do not add up – more complicated in-\nterference of the condensate wave functions takes place.\nThis is related to the phenomena, known as the param-\nagnetic Meissner effect [1].\nIt was demonstrated both experimentally [5] and the-\noretically [6, 7] that at the boundary of some high- Tc\nsuperconductors placed in external magnetic field the\ncurrent flows in the direction opposite to the diamag-\nnetic Meissner supercurrent which screens the external\nmagnetic field. This countercurrent is carried by the\nsurface-induced quasiparticle states. These nonthermal\nquasiparticles appear because of the sign change of the\norder parameter along the reflected quasiparticle trajec-\ntory. Such a depairing mechanism is absent in the homo-\ngeneous situation. Note that in a homogeneous conven-\ntional superconductor at zero temperature the quasipar-\nticles appear only when the Landau criterion is violated,\natvs>∆0/pF. Herevsis the superfluid velocity which\nparameterizes the current-carrying state, ∆ 0stands for\nthe bulk order parameter, and pFis the Fermi momen-\ntum. The appearance of the countercurrent can be un-\nderstood as the response of the weak link with negative\nself-inductance to the externally injected transport su-\npercurrent. The state of the junction in the absence of\nthe transport supercurrent at zero temperature is unsta-\nble atφ=πfrom the point of view that small deviations\nδφ=±0 changethe Josephsoncurrent from 0 to its max-imal value [8]. The response of the Josephson junction\nto small transport supercurrent at φ=πproduces the\ncountercurrent [9]. It is similar to the equilibrium state\nwith the persistent current in 1D normal metal ring with\nstrong spin-orbit interaction: there is degeneracy at zero\ntemperature and φ=π, and the response of the ring is\ndifferent at δφ/negationslash= 0 orB/negationslash= 0, where Bis the effective\nmagnetic field which enters in the Hamiltonian through\nthe Zeeman term (which breaks time-reversal symmetry)\n[10]. The degeneracy is lifted by small effective magnetic\nfield so that the persistent current rapidly changes from\n0 to its maximum value. In the case of the weak link\nbetween two superconductors in the absence ofthe trans-\nport supercurrent there is degeneracy between + pyand\n−pyzero-energy states; both the time-reversal symme-\ntry breaking by the surface (interface) order-parameter\nand the Doppler shift (due to the transport supercurrent\nor magnetic field) lift the degeneracy and result in the\nsurface (interface) current [5].\nIn recent years mesoscopic superconducting structures\ncontinue to attract attention because of the possible ap-\nplication as qubits, quantum detectors etc. (e.g. [3],\n[11]). In particular, such structures can be controlled\nby the transport supercurrent and the magnetic flux\n(through the phase difference on Josephson contact).\nThis was in the focus of many recent publications, e.g.\n[4, 12, 13, 14, 15, 16]. Here we continue to study the\nmesoscopic current-carrying d-wave structures. Particu-\nlarly, we study the impact of the transport supercurrent\non the density of states in both homogeneous film and in\nthe film which contains a weak link.\nMODEL AND BASIC EQUATIONS\nWeconsideraperfectcontactbetweentwocleansinglet\nsuperconductors. The external order parameter phase\ndifference φis assumed to drop at the contact plane at\nx= 0. The homogeneoussupercurrent flows in the banks\nof the contact along the y-axis, parallel to the boundary.\nThe sample is assumed to be smaller than the London\npenetration depth so that the externally injected trans-\nport supercurrent can indeed be treated as homogeneous2\nfar from the weak link. The size of the weak link is as-\nsumed to be smaller than the coherence length. Such\na system can be quantitatively described by the Eilen-\nberger equation [8]. Taking transport supercurrent into\naccount leads to the Doppler shift of the energy variable\nbypFvs. The standard procedure of matching the solu-\ntions of the bulk Eilenberger equations at the boundary\ngives the Matsubara Green’s function /hatwideGω(0) at the con-\ntact atx= 0 [4]. Then for the component G11\nω≡g(ω,r)\nof/hatwideGω, which defines both the current density and the\ndensity of states (see below), we obtain in the left ( L)\nand right ( R) banks of the junction:\ngL,R(r) =gL,R(∞)+[g(0)−gL,R(∞)]exp/parenleftbigg\n−2|r|ΩL,R\n|vx|/parenrightbigg\n,\n(1)\ngL,R(∞) =/tildewideω\nΩL,R, (2)\ng(0) =/tildewideω(ΩL+ΩR)−isgn(vx)∆L∆Rsinφ\nΩLΩR+/tildewideω2+∆L∆Rcosφ.(3)\nHereω=πT(2n+ 1) are Matsubara frequencies, ∆ L,R\nstands for the order parameter in the left (right) bank,\nand\n/tildewideω=ω+ipFvs,ΩL,R=/radicalBig\n/tildewideω2+∆2\nL,R.(4)\nThe direction-dependent Doppler shift pFvsresults in\nthe modification of current-phase dependencies and in\ntheappearanceofthecountercurrentalongtheboundary.\nThe function g(ω,r) defines the current density, as fol-\nlowing:\nj= 4πeN0vFT/summationdisplay\nωn>0/angbracketleft/hatwidevImg/angbracketrightbv. (5)\nHereN0is the density of states at the Fermi level, /angbracketleft.../angbracketrightbv\ndenotes averaging over the directions of Fermi velocity\nvF,/hatwidev=vF/vFis the unit vector in the direction of vF.\nAnalytic continuation of g(ω),i.e.\ng(ε) =g(ω→ −iε+γ), (6)\ngives the retarded Green’s function, which defines the\ndensity of states:\nN(ε,r) = Reg(ε,r). (7)\nHereγis the relaxation rate in the excitation spectrum\nof the superconductor.\nThe local density of states can be probed with the\nmethod of the tunnelling spectroscopy by measuring the\ntunnelling conductance G=dI/dVof the contact be-\ntween our superconducting structure and the normal\nmetal scanning tunnelling microscope’s (STM) tip. Atlow temperature the dependence of the conductance on\nthe bias voltage Vis given by the following relation [17]:\nG(eV) =GN/angbracketleftD(pF)N(eV,pF)/angbracketright, (8)\nwhereGNis the conductance in the normal state; D(pF)\nis the angle-dependent superconductor-insulator-normal\nmetal barrier transmission probability. The barrier can\nbe modelled e.g.as in Ref. [7] with the uniform proba-\nbility within the acceptance cone |ϑ|< ϑc, whereϑis the\npolar angle and the small value of ϑcdescribes the thick\ntunnelling barrier:\nD(ϑ) =1\n2ϑcθ(ϑ2\nc−ϑ2), (9)\nwhereθ(...) is the theta function.\nCONDUCTANCE CHARACTERISTICS OF THE\nHOMOGENEOUS CURRENT-CARRYING FILM\nBeforestudying the current-carryingweaklink we con-\nsider the homogeneous situation. We will consider the\nd-wave film as shown in the left inset in Fig. 1. The\nmotivation behind this study is twofold: first, to demon-\nstrate the application of the theory presented above, and\nsecond, to describe recent experimental results [14].\nFIG. 1: Normalized (divided by GN) conductance dI/dVfor\nthe homogeneous current-carrying state in the d-wave film\nfor different values of the transport current. The curves are\nplottedwith γ/∆00= 0.15andϑc= 0.1π[∆00= ∆0(vs= 0)].\nLeft and right insets show the schemes for probing the densit y\nof states in the current-carrying d-wave film and in the weak\nlink (see text for details).\nThe system considered consists of the d-wave film,\nin which the current is injected along the y-axis, and\nthe STM normal metal tip (another STM contact is not\nshown in the scheme for simplicity; for details see [14]).\nFollowing the experimental work [14], we consider the\nc-axis along the x-axis and the misorientation angle be-\ntweena-axis and the direction of current ( y-axis) to be3\nπ/4. Such problem can be described with the equations\npresented in the previous section as following [7, 15].\nConsider the specular reflection at the border, when\nthe boundary between the current-carrying d-wave su-\nperconductor and the insulator can be modelled as the\ncontact between two superconductors with the order pa-\nrameters given by ∆ L= ∆(ϑ) = ∆ 0cos2(ϑ−χ) and\n∆R= ∆(−ϑ)≡∆ and with φ= 0. Then from Eq. (3)\nwe have the following:\ng(ω) =/tildewideω/parenleftbig\nΩ+Ω/parenrightbig\nΩΩ+/tildewideω2+∆∆, (10)\nwhere Ω =√\n/tildewideω2+∆2andΩ =/radicalBig\n/tildewideω2+∆2. This expres-\nsion is valid for any relative angle χbetween the a-axis\nand the normal to the boundary; in particular,\ng(ω) =/tildewideω\nΩ, χ= 0 (∆( ϑ) = ∆0cos2ϑ),(11)\ng(ω) =Ω\n/tildewideω, χ=π\n4(∆(ϑ) = ∆0sin2ϑ).(12)\nThe accurate dependence of the gap function ∆ 0=\n∆0(vs,γ) can be obtained from Ref. [4] with introducing\nγas following: pFvs−→pFvs−iγ(which is analogous\nto Eq. (6)). The energy values in this paper are made\ndimensionless with the zero-temperature gap at zero cur-\nrent: ∆ 00= ∆0(vs= 0).\nAnd now with Eqs. (12) and (6-8) we plot the STM\nconductance for the current-carrying d-wave film in Fig.\n(1). We obtain the suppression of the zero-bias conduc-\ntance peak by the transport supercurrent, as was studied\nin much detail in Ref. [14]. Our results are in agreement\nwith their Fig. 1. Also the authors of Ref. [14] devel-\noped the model based on phase fluctuations in the BTK\nformalismtoexplainthe suppressionofthe zero-biascon-\nductance peak. However, their theoretical result, Fig. 2,\ndescribes the experimental one only qualitatively, leaving\nseveral distinctions. They are the following: (i) position\nof the minima ( eV/∆00∼0.5 and 1 for the experiment\nand the theory respectively); (ii) height of the zero-bias\npeak atzerotransportcurrent( ∼2.5and 4respectively);\n(iii) height of the peak at maximal transport current\n(∼1.3 and 2.5 respectively); (iv) presence/absence of\nthe minima for all curves. Our calculations, Fig. (1),\ndemonstrate agreement with the experiment in all these\nfeatures. The agreement we obtained with two fitting\nparameters, γandϑc.\nTo further demonstrate the impact of the two fitting\nparametersofourmodel, γandϑc, inFig. (2)weplotthe\nnormalized conductance fixing one of them and chang-\ning another. The figure clearly demonstrates how they\nchange the shape of the curves: the position of the min-\nima, splitting of the zero-bias peak etc. Note that the\nsplitting is suppressed at small ϑcand high γ. This ab-\nsence of the splitting was observed in the experiment [14]\nand studied in several articles, e.g. [18].\nFIG. 2: Normalized conductance dI/dVfor the homogeneous\ncurrent-carrying state in the d-wave film for different values\nofγandϑcatpFvs/∆00= 0.5.\nCONDUCTANCE CHARACTERISTICS OF THE\nCURRENT-CARRYING WEAK LINK\nConsider now the weak link between two d-wave\ncurrent-carrying banks. For studying the effect of both\nthe transport current and the phase difference on the\ndensity of states in the contact, we propose the scheme,\npresented in the right inset in Fig. 1. The supercurrent\nis injected along y-axis in the superconducting film, as it\nwas discussed in the previous section. Besides, the weak\nlink is created by the impenetrable for electrons parti-\ntion atx= 0. The small break in this partition ( a < ξ0)\nplays the role of the weak link in the form of the pinhole\nmodel [8, 15, 19]. The STM tip in the scheme is posi-\ntioned above the weak link to probe the density of states\nin it. Two more contacts along the x-axis provide the\norder parameter phase difference φalong the weak link.\nThis can be done, for example, by connecting the con-\ntacts with the inductance, as shown in the scheme, and\napplying magnetic flux Φ eto this inductance. Then one\nobtainsthephasecontrolofthe contactwith the relation:\nφ= Φe/Φ0.\nThe two half-plains (for x <0 andx >0) play the\nrole of the two banks of the contact, which we also call\nleft and right superconductors. In our scheme the banks\ncarry the transport current along the boundary, and the\nJosephson current along the contact is created due to\nthe phase difference. The banks we consider to be d-\nwave superconductors with c-axis along the z-axis and\nwith the misorientation angles χL= 0 and χR=π/4.\nNow we can apply the equations presented in Sec. II to\ndescribe the conductance characteristics of the contact\nbetween current-carrying d-wave superconductors. This\nis done in Fig. 3, where the normalized conductance is\nplotted for two values of the phase difference, for φ=\n±π/2 and with γ/∆00= 0.1. The two values of the\nphase difference, φ=±π/2, are particularly interesting\nfor the application since they correspond to the double-4\ndegenerate states [3, 4]. So, the density of states is the\nsameinthe absenceofthetransportsupercurrentin both\npanelsin Fig. 3with mid-gapstates(at eV <∆00)which\ncreate the spontaneous current along the boundary. The\ntransport supercurrent ( vs/negationslash= 0) removes the degeneracy\nby significantly changing the mid-gap states (Fig. 3),\nwhich explains different dependencies of the current in\nthe contact on the applied transport current (i.e. on vs),\nstudied in [4, 9].\nFIG. 3: Normalized conductance dI/dVat the contact be-\ntween two current-carrying d-wave superconductors for dif-\nferent values of the transport supercurrent ( vs) and for two\nvalues of the phase difference φ=±π/2.\nCONCLUSION\nWe have studied the density of states in the current-\ncarrying d-wave structures. Namely, we have considered,\nfirst, the homogeneous situation and, second, the super-\nconducting film with the weak link. The former case was\nrelated to recent experimental work, while the latter is\nthe proposition for the new one. The local density of\nstates was assumed to be probed with the scanning tun-\nnellingmicroscope. The densityofstatesatthe weaklink\nand the current ( i.e.its components through the contact\nand along the contact plane) are controlled by the values\nofφandvs. Thesystemisinterestingbecauseofpossible\napplications: in the Josephsontransistorwith controlling\nparameters φandvsgoverned by external magnetic flux\nand the transport supercurrent [11], and in solid-state\nqubits, based on a contact of d-wave superconductors [3].\nThe author is grateful to Yu.A. Kolesnichenko and\nA.N. Omelyanchouk for helpful discussions. This work\nwas supported in part by the Fundamental Researches\nState Fund (grant number F28.2/019).\n[1] C.C. Tsuei and J.R. Kirtley, Rev. Mod. Phys. 72, 969\n(2000).[2] Yu.A. Kolesnichenko, A.N. Omelyanchouk, and A.M.\nZagoskin, Fiz. Nizk. Temp. 30, 714 (2004) (Low Temp.\nPhys.30, 535 (2004)).\n[3] A.M. Zagoskin, J. Phys.: Condens. Matter 9, L419\n(1997); L.B. Ioffe et al., Nature 398, 679 (1999); A.M.\nZagoskin, Turk. J. Phys. 27, 491 (2003).\n[4] Yu.A. Kolesnichenko, A.N. Omelyanchouk, and S.N.\nShevchenko, Fiz. Nizk.Temp. 30, 288(2004) (LowTemp.\nPhys.30, 213 (2004)).\n[5] W. Braunisch, N. Knauf, V. Kateev, S. Neuhausen,\nA. Grutz, A. Koch, B. Roden, D. Khomskii, and D.\nWohlleben, Phys. Rev. Lett. 68, 1908 (1992); H. Wal-\nter, W. Prusseit, R. Semerad, H. Kinder, W. Assmann,\nH. Huber, H. Burkhardt, D. Rainer, and J. A. Sauls,\nPhys. Rev. Lett. 80, 3598 (1998); E. Il’ichev, F. Tafuri,\nM. Grajcar, R.P.J. IJsselsteijn, J. Weber, F. Lombardi,\nand J.R. Kirtley, Phys. Rev. B 68, 014510 (2003).\n[6] S. Higashitani, J. Phys. Soc. Jpn. 66, 2556 (1997).\n[7] M. Fogelstr¨ om, D. Rainer, and J.A. Sauls, Phys. Rev.\nLett.79, 281 (1997).\n[8] I.O. Kulik and A.N. Omelyanchouk, Fiz. Nizk. Temp. 4,\n296 (1978) (Sov. J. Low Temp. Phys. 4, 142 (1978)).\n[9] Yu.A. Kolesnichenko, A.N. Omelyanchouk, and S.N.\nShevchenko, Phys. Rev. B 67, 172504 (2003); S.N.\nShevchenko, in Realizing Controllable Quantum States–\nIn the Light of Quantum Computation, Proceedings of\nthe International Symposium on Mesoscopic Supercon-\nductivity and Spintronics, Atsugi, Japan, edited by H.\nTakayanagi and J. Nitta, World Scientific, Singapore, p.\n105 (2005); arXiv:cond-mat/0404738.\n[10] T.-Z.Qian, Y.-S. Yi, and Z.-B. Su, Phys. Rev. B 55,\n4065 (1997); V.A. Cherkassky, S.N. Shevchenko, A.S.\nRozhavsky, I.D. Vagner, Fiz. Nizk. Temp. 25, 725 (1999)\n(Low Temp. Phys. 25, 541 (1999)).\n[11] F.K. Wilhelm, G. Sh¨ on, and A.D. Zaikin, Phys. Rev.\nLett.81, 1682 (1998); E.V. Bezuglyi, V.S. Shumeiko,\nand G. Wendin, Phys. Rev. B 68, 134606 (2003).\n[12] G. Rashedi and Yu.A. Kolesnichenko, Phys. Rev. B 69,\n024516 (2004).\n[13] D. Zhang, C.S. Ting, and C.-R. Hu, Phys. Rev. B 70,\n172508 (2004).\n[14] J. Ngai, P. Morales, and J.Y.T. Wei, Phys. Rev. B 72,\n054513 (2005).\n[15] S.N. Shevchenko, Phys. Rev. B 74, 172502 (2006).\n[16] V. Lukic and E.J. Nicol, Phys. Rev. B 76, 144508 (2007).\n[17] C. Duke, Tunneling in solids , Academic Press, NY\n(1969).\n[18] H. Aubin, L.H. Greene, S. Jian, and D.G. Hinks, Phys.\nRev. Lett. 89, 177001 (2002).\n[19] M. Fogelstr¨ om, S. Yip, and J. Kurkij¨ arvi, Czech. J. Ph ys.\n46, 1057 (1996).\n[20] G. Rashedi, J. Phys.: Condens. Matter 21, 075704\n(2009).\n[21] A.N. Omelyanchouk, S.N. Shevchenko, and Yu.A.\nKolesnichenko, J. Low Temp. Phys. 139, 247 (2005)."    },    {        "title": "0909.1362v1.Theory_of_the_Magnetic_Field_Induced_Insulator_in_Neutral_Graphene.pdf",        "content": "arXiv:0909.1362v1  [cond-mat.mes-hall]  8 Sep 2009Theory oftheMagnetic-Field-Induced InsulatorinNeutral Graphene\nJ. Jung1and A. H. MacDonald1\n1Department of Physics, University of Texas at Austin, 78712 USA\n(Dated: November 14, 2018)\nRecent experiments have demonstrated that neutral graphen e sheets have an insulating ground state in the\npresence of an external magnetic field. We report on a π-band tight-binding-model Hartree-Fock calculation\nwhich examines the competition between distinct candidate insulating ground states. We conclude that for\ngraphenesheetsonsubstratesthegroundstateismostlikel yafield-inducedspin-density-wave,andthatacharge\ndensitywavestateispossible forsuspended samples. Neith erofthesedensity-wave statessupport gaplessedge\nexcitations.\nI. INTRODUCTION\nThe magnetic band energy quantization properties of\ngraphene sheets lead to quantum Hall effects1,2,3(QHEs)\nwithσxy=νe2/hat filling factors ν=4(n+1/2) =\n···,−6,−2,2,6,···for any integer value of n. The factor of\n4 in this expression accounts for a graphene sheet’s two-fol d\nvalley and spin degeneracies. When Zeeman spin-splittingo f\nLandau levels is included, quantum Hall effects are expecte d\nattheremainingevenintegervaluesof ν,includingtheneutral\ngraphene ν=0 case. The ν=0 quantum Hall effect of neu-\ntral graphenesystems is interesting from two-differentpo ints\nofview. Firstofall,thetransportphenomenologyofthequa n-\ntum Hall effect3is different at ν=0 because of the possi-\nble absence of edge states. Indeed the initial experimental\nindications3thataν=0quantumHalleffectoccursinneutral\ngraphenedidnotexhibiteithertheclearpleateauin ρxyorthe\ndeepminimumin ρxxwhicharenormallycharacteristicofthe\nQHE. Secondly, although a quantum Hall effect is expected\natν=0 even for non-interacting electrons, the large energy\ngaps identified experimentally suggest that interactions p lay\na substantial role in practice. Gaps due entirely to electro n-\nelectron interactions in ordered states are in fact common4,5\ninquantumHallsystemswhentwoormoreLandaulevelsare\ndegenerate. Partly for this reason, a number of different sc e-\nnarios have been proposed6,7,8,9,10,11,12,13,14,15,16in which the\ngapatν=0isassociatedwithdifferenttypesofbrokensym-\nmetrywithinthefourquasi-degenerateLandaulevelsneart he\nFermi level of a neutral graphene sheet. The prevailing view\nhas been that the ground state is spin-polarized, with parti al\nfilling factors νσequal to 1 and −1 for majority and minor-\nity spins respectively. This state has an interesting edge s tate\nstructureidenticaltothatofquantum-spin-Hallsystems,17and\ntransport propertiesin the quantum Hall regime that are con -\ntrolledbythepropertiesofcurrent-carryingspin-resolv edchi-\nraledge-states.8,18\nThe simplest picture of strong-field physics in nearly-\nneutral graphene sheets is obtained by using the Dirac-\nequationcontinuummodelandneglectinginteraction-indu ced\nmixing between Landau levels with different principal quan -\ntum number n. In this model, electron-electron interactions\nare valley and spin-dependent. When Zeeman interactions\nanddisorderareneglected,thebrokensymmetrygroundstat e\nconsists12of two-filled n=0 Landau levels with arbitrary\nspinors in the 4-dimensional spin-valley space. This famil yof states is favored by electron-electron interactions bec ause\nofFermistatisticswhichlowersCoulombinteractionenerg ies\nwhentheorbitalcontentofelectronsinthefermionsea ispo -\nlarized. When Zeeman coupling is included, it uniquely se-\nlects from this family the state in which both n=0 valley\norbitals are occupied for majority-spin states and empty fo r\nminority-spin states. The interacting system ground state is\nthen identical to the non-interacting system ground state, al-\nthoughinteractionsareexpected12todramaticallyincreasethe\nenergygapforchargedexcitations.\nThisargumentforthecharacterofthegroundstate appears\nto be compelling, but its conclusions are nevertheless unce r-\ntain. First of all, Landau-level mixing interactions are no r-\nmally stronger than Zeeman interactions, and could play a\nrole19. In addition, although corrections7,11,20to the contin-\nuum model for graphene are known to be small at experi-\nmentalfieldstrengths,theycouldstillbemoreimportantth an\nthe Zeeman interactions. Suspicions that the character of t he\ngroundstate couldbemisrepresentedbythe n=0continuum\nmodel theory have been heightened recently by the work of\nOngandcollaborators,whofoundasteepincreaseintheDira c\npointresistance21withmagneticfieldandevidenceforafield-\ninducedtransitiontoastronglyinsulatingstateatafinite mag-\nneticfieldstrength.22Somewhatlessdramaticincreasesinre-\nsistance at the Dirac point have also been reported by other\nresearchers.23,24,25\nIn this article we attempt to shed light on the ground\nstate of neutral graphene in a magnetic field by performing\nself-consistentHartree-Fockcalculationsfora π-orbitaltight-\nbinding model. In the continuum model, Hartree-Fock the-\nory is known12to yield the correct ground state. By us-\ning aπ-orbital tight-binding model we can at the same time\nconveniently account for Landau-level mixing effects and\nsystematically account for lattice corrections to the Dira c-\nequation continuum model. As we will discuss at length be-\nlow, it is essential to perform the mean-field-theory calcul a-\ntions with Coulombic electron-electron interactions, and not\ntheHubbard-likeinteractionscommonlyused10,26withlattice\nmodels. One disadvantage of our approach is that our cal-\nculations are feasible only at magnetic fields strengths whi ch\narestrongerthanthoseavailableexperimentally. Wethere fore\ncarefully examine the dependence on magnetic field strength\nandextrapolatetoweakerfields. Weconcludethatundertypi -\ncalexperimentalconditionsthemost-likelyfield-induced state\nof neutral graphene on a SiO 2substrate is an spin-density-2\nwave state, and that suspended samples might have a charge\ndensity wave state. Neither of these orderings support edge\nstates in the ν=0 gap. We also discuss the magnetic field\ndependence of different contributions to the total energy a nd\nestimate a critical value of perpendicular and tilted magne tic\nfield at which Zeemansplitting will bringabout a phase tran-\nsitiontoa solutionwith netspinpolarizationwhich doessup-\nportedgestates.\nAlthough our calculation captures some realistic features\nof graphene sheets that are neglected in continuum models,\nit is still not a complete all electron many-body theory. In\nparticularwe neglect the carbon σandσ∗orbitalswhose po-\nlarization is expected to screen the Coulomb interactions a t\nshort distances. Because of our uncertainty as to the streng th\nof this screening, our conclusions cannot be definitive. We\nnevertheless hope that our calculations, in combination wi th\nexperiment, will prove useful in identifying the character of\nthefield-inducedinsulatingstate in neutralgraphene.\nOurpaperisorganizedasfollows. InSection IIweexplain\nin detail the model which we study which has two parame-\nters, a relative dielectric constant εrwhich accounts for the\ndielectric environment of the graphene sheet, and on on-sit e\ninteraction Uwhich accountsfor short-distance screeningef-\nfects, for example by σ-band polarization. Our main results\non the competition between different ordered states are pre -\nsented in Section III. In Section IV we turn to a discussion\nof the electronic structure of neutral graphene ribbons, pa y-\ning particular attention to their edge states which play a ke y\nrole in most quantum Hall transport experiments. Finally in\nSection Vwe summarizeourmainconclusions.\nII. INTERACTING-ELECTRONLATTICEMODELFOR\nGRAPHENESHEETS\nA. Non-InteractingElectron π-bandmodel\nWefirstcommentbrieflyonthe π-bandtight-bindingmodel\nof graphene in the presence of a magnetic field.32,33,34,35,36\nEach carbonatom on graphene’shoneycomblattice hasthree\nnear-neighborswith π-orbitalhoppingparameter t=−2.6eV.\nMagnetic field effects are captured by a phase factor in the\nhopping amplitudes: t→t×e2πiφwhereφ= (e/ch)/integraltextAdl\ndependson line integral of the vector potential Aalong a tra-\njectory linking the two lattice sites. When the dimensionle ss\nmagneticfluxdensityis φ≡BShex/φ0=1/q,whereqisanin-\ntegerand φ0=ch/eis a magneticflux quantum,it ispossible\nto apply Bloch’s theorem in a unit cell which is enlarged by\na factor of qrelative to the honeycomblattice unit cell. (The\nhoneycomblatticeunitcellarea Shex=√\n3a2/2anda=2.46˚A\nfor a graphene sheet.) Lattice model Landau levels have a\nsmall width which increases with magnetic field strengthand\nreflects magnetic breakdown effects neglected in the contin -\nuummodel.\nThe ground state energy density differences discussed be-\nlow scale approximately as powers of the magnetic length\nℓB, defined by 2 πℓ2\nBB=φ0. (ℓBandqare related by ℓB=\n(Shexq/2π)1/2=0.371√qa=0.913√q˚A.) In a continuummodeldescriptionthedensitycontributedbyasinglefullL an-\ndau level is 1 /2πℓ2\nBand the energy of the nthLandau level is\ngiven by En=±2¯hvF/radicalbig\n|n|/ℓBwherevF=√\n3at/2¯his the\nFermi velocity of graphene. All energy levels evolve with\nmagneticfield except forthe n=0 level,E0=0.38When the\nnthLandaulevelisfullitcontributes En//parenleftbig\n2πℓ2\nB/parenrightbig\ntotheenergy\ndensity. From this we immediately see that in the weak-field\nlimitimportantenergiestendtoscaleas ℓ−3\nB∝B3/2. It iseasy\nto show, for example, that the magnetic-field dependence of\nthe total band energy of a neutral non-interacting graphene\nsheet is given by E(ℓB) =akin/ℓ3\nBwhereakin=2.65eV·˚A3.\nThis non-analytic field-dependence is responsible for the d i-\nvergent weak-field diamagnetic response ( (∂Etot/∂B)/B) of\ngraphene discussed some time ago by McClure39. We show\nbelow that when interactions are included, the energy diffe r-\nences between competing field-induced-insulator states al so\ntendto varyas ℓ−3\nB.\nB.π-bandModelEffective Interactions\nIt is clear from previous analysis of lattice-corrections t o\ncontinuum models7,11,13,20and from lattice-model calcula-\ntions based on extended Hubbard models10that conclusions\non the nature of the field-induced insulating ground state ar e\nvery dependenton the effective electron-electroninterac tions\nused in a π-band lattice model of graphene. In particular,\nit seems clear that the long-range 1 /rCoulomb interaction\ntail is essential. We approximate the interaction between π-\norbitals located at sites separated by a distance dbyV(d) =\n1/(εr/radicalbig\na2o+d2)whereao=a//parenleftbig\n2√\n3/parenrightbig\n, the bondingradiusof\nthe carbon atoms, accounts approximately for interaction r e-\nduction due to π-charge smearing on each lattice site,40and\nεraccounts for screening due to the dielectric environment\nof the graphene sheet. (Here energies are in Hartree ( e2/aB)\nunits and lengthsare in units of the Bohr radius aB.) The on-\nsite repulsiveinteractionparameter, U,isnotwell knownand\nwe take it to be a separate parameter. We motivate the range\nofvaluesconsideredforthisinteractionparameterbelow. The\nvalue chosen for εrcan also represent in part screening by σ\norbitals neglected in our approximation, or be understood a s\nanad-hoccorrection for overestimates of exchange interac-\ntions in Hartree-Fock theory. Although we study a range of\nvaluesforthisinteractionparametermodelinordertotest the\nrobustnessofourconclusions,webelievethatavalueof εr∼4\nisnormallyappropriateforgraphenesheetsplacedonadiel ec-\ntric substrate. For practical reasons we truncate the range of\nCoulomb interaction in real space at d=Lmax=6.5a. This\ntype of truncationis especially helpfulwhen treatingsyst ems\nwithout periodic boundary conditions and allows us to avoid\nproblemsduetoslowlyconvergingsumsinrealspacethatcan\notherwise be treated throughthe Ewald sum method.41Trun-\ncation of the Coulomb interaction at a reasonably large Lmax\nmust however be applied with utmost care in order to obtain\nsolutionsconsistent42,43withthelimit Lmax→∞.\nInconsideringappropriatevaluesfortheon-siteinteract ion\nUwe can start from the Coulomb interaction energy at the\ncarbon radius length scale which is ∼20eV, while this es-3\ntimate can be reduced if one considers a charge distribution\ncorrespondingtoa p-orbital. Infactanestimate fromthefirst\nionizationenergyandelectronaffinitygives U=9.6eV.27Itis\nknownthat the effectiveon-site interactionstrengthis gr eatly\nreducedfromthisbarevalueinthesolidstateenvironmentb e-\ncauseofscreeningbypolarizationofboundorbitalsonnear by\ncarbonatoms. Weconsidervaluesof Ubetween2eVand6eV,\nbracketingvaluesdeemedappropriatebyavarietyofdiffer ent\nresearchers7,28,29,30. A larger value of Uincreases the inter-\naction energy cost of any charge-density-wave (CDW) state\nwhichmightoccur. Thedirectinteractionenergyiszerowhe n\nall carbonsites stay neutral,butcan bepositiveornegativ ein\nCDW states. In the CDW states we discuss below electron\ndensityδnistransferredbetweenAandBhoneycombsublat-\ntices. Inthisstate thedirectinteractionenergypersite i s\nδEDI=(δn)2\n2/bracketleftBig\nU+∑\nj∈AV(dij)−∑\nj∈BV(dij)/bracketrightBig\n,(1)\nwheredijis the distance between lattice sites iandj,U=\nV(dii)andiis a fixed label belonging to sublattice A. The\nlargest terms in Eq.( 1) are the repulsive on-site interacti ons\nwhich are proportional in our model to Uand attractive ex-\ncitonicinteractionbetweenelectronsonneighboringoppo site\nsublattice sites which are inversely proportional to εr. Us-\ning an Ewald technique to sum over distant sites we find that\nδEDIispositivefor εr·U>13.05eV.(Thecorrespondingcri-\nterion for the truncated Coulomb interactions we use in our\nself-consitent-fieldcalculationsis εr·U>=12.23eV;thedif-\nferencebetween the right-hand-sideof these two equations is\none indicator of the inaccuracy introduced by truncating th e\nCoulombinteraction.) When εr·U<12.23eVtheCDWstate\nisstableunlessbandandexchangeenergiessupportaunifor m\ndensitystate43.\nGiven the band structure model and the interaction model,\nthe Hartree-Fock mean-field-theory calculations for bulk\ngraphene sheets with periodic boundary conditions and for\ngraphene ribbonsreported in the following sections are com -\npletelystandard.31Thebandquasiparticlesaredeterminedby\ndiagonalizing a single-particle Hamiltonian which includ es\ndirect and exchange interaction terms. The direct and ex-\nchange potentials are expressed in terms of the occupied\nquasiparticle states and must be determined self-consiste ntly.\n(We do not quote the detailed expressions for these terms\nhere.) Since the Hartree-Fock equations can be derived by\nminimizing the total energy for single Slater determinant\nwavefunctions, every solution we find corresponds to an ex-\ntremum of energy. The iteration procedure is stable only if\nthe extremum is a minimum so we can be certain that all the\nsolutions found below represent local energy minima among\nsingle Slater determinant wavefunctions with the same sym-\nmetryproperties.III. FIELD-INDUCEDINSULATINGGROUND STATES\nA. Identificationof CandidateStates\nAt zero-field band energy favors neutral graphene states\nwithout broken symmetries, and there is no compelling evi-\ndence from experiment that they are induced by interactions .\nIn a perpendicular magnetic field, however, the systems is\nparticularly susceptible to the formation of broken symme-\ntry ground states because of the presence of a half-filled set\nof four-fold spin (neglecting Zeeman) and valley degenerat e\nLandau levels with (essentially) perfectly quenched band e n-\nergy. Althoughthe final groundstate selection probablyres ts\nonconsiderationsthat it fails to capture,the n=0 continuum\nmodel captures the largest part of the interaction energy an d\nmostofthequalitativephysics. Thegroundstateisformedb y\noccupyingtwoofthefour n=0Landaulevels,selectedatran-\ndom from the four-dimensional orbital space, and producing\nagapforchargedexcitations.\nThreerepresentativebrokensymmetrystatesareillustrat ed\ninFig.1. Because n=0Landaulevelsorbitalsassociatedwith\ndifferentvalleys are completely localized on different ho ney-\ncomb sublattices, a charge density wave (CDW) solution re-\nsults when n=0 orbitals are occupied for both spins of one\nvalley. (WhenLandaulevelmixingisneglectedvalleyindic es\nand A or B sublattice indices are equivalent.) This state low -\ners the translational symmetry of the honeycomb lattice in a\nway which removes inversion symmetry. The other extreme\nis a spontaneouslyspin-polarizeduniformdensitystate (f erro\n- F) in which n=0 orbitals are occupied in both valleys but\nonlyforonespin-component. A thirdtypeof brokensymme-\ntry state, the spin-density-wave(SDW)state, hasbothbrok en\ninversionsymmetry and brokenspin-rotationalinvariance . In\nthe cartoon version of Fig. 1, n=0 electrons occupy states\nwith one spin-orientation on one sublattice and the opposit e\nspin-orientationontheothersublattice. Possible broken sym-\nmetrystates, some at other filling factors, had been discuss ed\npreviously by several authors.7,10These three states are all\ncontainedwithinthe n=0continuummodelfamilyofground\nstateswhosedegeneracyisliftedbybylatticendLandaulev el\nmixing effects. In the self-consistent mean-field-theory c al-\nculations described in detail below, the three states ident ified\naboveallappearasenergyextremainourcollinear-spinstu dy.\nB. EnergyComparisons\nInorderto examinethe physicsbehindthecompetitionbe-\ntweenthecandidategroundstateswedecomposethetotalen-\nergy for all three contributions into band, direct interact ion,\nand exchange interaction contributions. We have obtained\nself-consistentsolutionsforallthreestatesoverarange ofon-\nsite interaction Uand the dielectric screening εrvalues. Be-\ncause of kinetic-energy quenching in the (essentially dege n-\nerate)n=0 Landau level, the interaction strengths required\nto drive the system into an ordered state are essentially zer o.\nThe key question, then, is which state is favored. In Fig. ( 1)\nwe illustrate how the energy differences between the three4\nb.  SDW \nc.  F           a.  CDW Valley  A Valley  B a.b. c.\n0\n2 3 4 5 6 246810 \n-2E-3 -1E-3 -5E-4 03E-4 1E-3 2E-3 3E-3 \n0\n2 3 4 5 6 246810 \n-9E-4 -7E-4 -5E-4 -3E-4 -2E-4 02E-4 4E-4 6E-4 \n0\n2 3 4 5 6 246810 \n-2E-3 -2E-3 -1E-3 -7E-4 01E-4 6E-4 1E-3 \nCDW SDW F\nCDW SDW \nF/parenleftBig!!\n!(\"# ) !(\"# ) !(\"# )eV \nelec .eV \nelec .eV \nelec .\nFIG.1: (Coloronline) Upper panel: Schematic representationofvalleypolarizedchargedensi tywave (CDW),spindensitywave (SDW)and\nferromagnetic (F)spinpolarized broken symmetrysolution s thatcanbe obtained ina self-consistent meanfieldcalcula tionof graphene under\na perpendicular magnetic field at half filling. Each arrow rep resents the filling of one n=0 Landau level of a given spin and valley. Lower\npanel:From left to right ECDW−ESDW,ECDW−EFandESDW−EFtotal energy differences per electron in eVas a function of the onsite\nrepulsion Uandεrobtained from a data mesh of 9 ×10 points, calculated neglecting the Zeeman term and for a ma gnetic field of 792 Teslas\ncorresponding to 1 /100 of a flux quantum per honeycomb hexagon. For smaller value s ofUtheCDWsolutions are energetically favored\nwhereas for larger values of UtheSDWsolutions are favored in a wide range of εr. TheFsolutions are never the lowest in energy. The red\ncontour lines indicate degeneracy betweentwodifferent so lutions.\nF\nπ/L 0εr= 4U= 5eVSDW\nkπ/L 0CDWEband (eV)\nπ/L 01.2\n0.6\n0\n-0.6\n-1.2F\nπ/L 0εr= 2U= 5eVSDW\nkπ/L 0CDWEband (eV)\nπ/L 02\n1\n0\n-1\n-2\nFIG. 2: (Color online) One dimensional representation of th e dispersionless band structure of a graphene sheet under a s trong magnetic field\nB=440Trepresented inthe momentum coordinate kparallel tothe narrower directionof the unitcell. The band -gaps followthe B1/2scaling\nlawexpectedfromthecontinuummodel. Theredcolorisusedt orepresentupspinwhileblueisusedfordownspin. LeftPanel: Bandstructure\nfor CDW, SDW and F solutions obtained for U=5eVandεr=4. For these interaction parameters the SDW state has the low est energy and\nthe largest gap at the Fermi level. Right Panel: Band structures for U=5eVandεr=2. When the on site repulsion is sufficiently weak the\nenergeticallyfavoredsolution corresponds tothe CDWstat e andthis solution thenhas the largest gap.5\nstates depend on the model interaction parameters. The re-\nsultsinthisfigurewereobtainedfor q=100unitcellsperflux\nquantum, which corresponds to perpendicular field strength\nB=792 Tesla. The unit cells in which we can apply peri-\nodic boundary conditions in this case contain 100 ×2 lattice\nsites. The k-space integrations in the self-consistent Hartree-\nFockcalculationswereperformedusinga60 k-pointBrillouin\nzone sampling. The self-consistent field equations were ite r-\nateduntilthetotalenergieswereconvergedtoninesignific ant\nfigures. High accuracy is required because the three states\nare very similar in energy since the ordering occurs primar-\nily in the n=0 Landau level, involving only 1% or so of the\nelectrons for this value of q. This accuracy was sufficient to\nevaluate energy differences that typically have three sign ifi-\ncantfigures.\nThefirstpointtonoticein theseplotsofenergydifferences\nis that the two uniform charge density solutions, the F so-\nlution and the SDW solution, behave similarly. The largest\ncontrastthereforeisbetweentheCDWsolutionandtheSDW\nand F solutions. Focusing first on the CDW/SDW compari-\nson we notice that the SDW state is favored when Uis large\norεris large. The crossover occurs near εr·U∼12eV,\nvery close to the line along which δEDIchanges sign. The\nfact that the CDW/SDW phase boundaryoccursveryclose to\nthis line is expected because of kinetic energy quenching in\na magnetic field. When the non-uniform density CDW state\nis compared with the uniform-density spin-polarized F stat e\nthe phase boundarymovesvery close to a larger value of this\nproduct with εr·Uranging from ∼14 to∼18eV along the\nphase boundary. Evidently the competition between CDW\nandSDWstatesisbasedverycloselyonthedirectinteractio n\nenergy, with additional weaker elements of the competition\nenteringwhenthe F stateis considered.\nDirect comparison between the uniform density SDW and\nF solutions indicates that the latter is favored only at valu es\nofUandεrwhichareoutsidetherangeofmostlikelyvalues.\nAs discussed in more detail below, we find that the direct in-\nteraction energy in these two states is identical, and that t he\nmore negative exchange energy of the SDW state overcomes\nalargerbandenergy. Inthiscasethemaindifferencebetwee n\ntheenergiesofthetwostatesarisesfromLandaulevelmixin g\neffects. As we explain later, Landau level mixing leads to a\nlocal spin-polarization which is larger in the SDW state tha n\ninthe Fstate.\nInFig. (1)wehaveintroducedthemaintrendsin theener-\ngetic competition between CDW, SDW, and F states. How-\never, as we have explained, these calculations were under-\ntakenatfieldstrengthsthatexceedthoseavailableexperim en-\ntally. Inthefollowingsubsectionwedemonstratethatthefi eld\ndependence of the energy comparisons is extremely system-\naticsothatextrapolationsdowntophysicalfieldstrengths are\nreliable. So farwe havealso ignoredZeemancouplingwhich\nfavors F states. This coupling can be important and is also\naddressedinthe followingsubsection.−∆Ekin∆Etot∆EX\nU= 5eV, ε r= 4\nlB/aEF−EAF(meV)\n5 41\n0.4\n0.16−∆Ekin\nU= 5eV, ε r= 2\nlB/aEF−ECDW (meV)\n5 42.5\n1\n0.4∆Etot∆EX∆Ees\nU= 5eV, ε r= 2\nlB/aEF−ECDW (meV)\n5 42.5\n1\n0.4\nFIG. 3: (Color online) Total energy per site differences ΔEtot, sep-\narated into kinetic energy ΔEkin, electrostatic energy ΔEDIand ex-\nchange energy ΔEXIcontributions as a function of magnetic length\nℓBin lattice constant a=2.46˚Aunits. The total energy differences\nwere fitted to a C3/2B3/2+C2B2curve. The fitting parameters are\nlisted in Table I. Left panel: Energy differences between F and\nSDW solutions ΔEF/SDW=EF−ESDW. These results were ob-\ntained with interaction parameter values U=5eVandεr=4 for\nwhich SDW is the lowest energy configuration. The more negati ve\nvalues ofexchange energy intheSDWstatecompensates the ki netic\nenergy penalty related to the inhomogeneous accumulation o f the\nelectron wave functions at alternating lattice sites. The e lectrostatic\nenergy differences are zero thanks to the uniform electron d ensity\nfor both solutions. Right panel: Same as the previous figure but for\nΔEF/CDW=EF−ECDW. Theinteractionparameters inthiscaseare\nU=5eVandεr=2 for which CDW is the lowest energy configu-\nration. When the onsite repulsion Uis small enough that the elec-\ntrostatic energy penalty for the inhomogeneous charge dist ribution\nis small, exchange is the main contribution driving the CDW i nsta-\nbility. However, in the case illustrated here Uis so small that the\nelectrostatic part of the hamiltonian does play an importan t role in\nfavoringtheCDWstate. Theenergycontributionsfollowama gnetic\nfield decay law that deviates more from B3/2than in the previous\ncase because the on-site interaction Uplays anessential role.\nU= 5eV, ε r= 2U= 5eV, ε r= 4\nθBc(θ) ( Tesla )2000\n1000\n0\nπ/2 π/3 π/6 0200\n100\n0\nFIG. 4: Tilt angle θdependence of the critical magnetic field re-\nquiredtoinduce atransitiontotheFstate. Theblacksolidc urveand\ndashedblue curve represent the criticalmagnetic fields sta rtingfrom\nthe SDW and CDW states respectively. In the CDW curve we ob-\nserve a larger deviation from a simple cos (θ)law due to a stronger\ninfluence of lattice scale physics described by the C2coefficient in\nequation 3.6\nεr=2 εr=3 εr=4\nUCCDW\n3/2CCDW\n2CCDW\n3/2CCDW\n2CSDW\n3/2CSDW\n2CCDW\n3/2CCDW\n2CSDW\n3/2CSDW\n2\n2 NA NA 322 -1.89 — — 52.6 1.05 — —\n3 NA NA 130 0 — — 23.2 0.0314 — —\n4 1070 -13.6 54.2 -0.210 — — — — 51.6 0.0209\n5 417 -3.15 — — 111 -0.630 — — 92.8 0.839\n6 198 -1.67 — — 255 -2.10 — — 241 0\nU BCDW\nCBCDW\nCBSDW\nCBCDW\nCBSDW\nC\n2 NA 1200 — 70 —\n3 NA 300 — 10 —\n4 2600 52 — — 50\n5 1600 — 200 — 200\n6 490 — 740 — 1100\nTABLE I: In the upper table we list the values of C(SDW/CDW)\n3/2in units of 10−10eV/T3/2andC(SDW/CDW)\n2in units of 10−10eV/T2, obtained\nfrom fitstothe their energy difference withrespect toF stat esas given inequation (2) for twodifferent values of the int eractionparameters U\nandεr. Thecriticalfield Bcestimatesaredependent onthereliabilityofthesefits. Not ethatacrossoverbetweenSDWandCDWstatescanbe\ndriven by changes inthe dielectric screening environment c aptured by εr. The lower table lists Bcvalues at which the Fstates become critical\naccording toEq.(3).\nC. FieldStrengthandZeeman CouplingDependence\nWenowturnourattentiontothemagneticfielddependence\nof the solutions. For this purpose we found self-consistent\nsolutionsovera rangeofmagneticfieldsfortwo setsofinter -\naction parameters, U=5eVandεr=4 for which the SDW\nsolution has the lowest energy, and U=5eVandεr=2 for\nwhich the CDW solution has the lowest energy. The band\nstructures of the different possible solutions for these se t of\nparametersareshowninFig. (2)andthefielddependencesof\nthe three energy differences are plotted in Fig. (3). We see\nthat every contribution accurately follows a B3/2∝l−3\nBlaw\nwith small deviations that can be accounted for by allowing\na term proportional to B2. This is the same field-dependence\nlaw that we discussed earlier for the case ofa non-interacti ng\nelectron system. In the continuum model it is guaranteed in\nneutralgraphenewhenelectronsinteract via the Coulombin -\nteractions by the fact that both kinetic and interaction ene rgy\ndensities then scale as (length)−3; the magnetic field sim-\nply provides a scale for measuring density. The fact that we\nfindthisfielddependencesimplyshowsthatthecondensation\nenergies of all three ordered states are driven by continuum\nmodel physics. This is in agreement with the intuitive pic-\nture of the interactionenergyas the productof the numberof\nelectrons occupying a Landau level which is directly propor -\ntional to B, multiplied by the Coulomb interaction scale for\nelectrons in the n=0 Landau level which is proportional to\nB1/2. Thefactthatthedifferencesinenergybetweenthethree\nstatesfollowsthisrulesuggeststhatthemostimportantso urce\nof differences in energy between these states is Landau-lev el\nmixing, which should not violate the B3/2law. Small devia-\ntionsfromthislawareexpectedbecauseoflatticeeffects. The\ndeviations are stronger in CDW solutions than in the SDWsolutions because of the charge density inhomogeneityat th e\nlatticescale presentintheformer.\nWecandrawtwoimportantadditionalconclusionsfromthe\nB3/2behavior. First of all, lattice effects are not dominant\neffect at the field strengths for which we are able to perform\ncalculations,andshouldbelessimportantattheweakerfiel ds\nfor which experiments are performed because the magnetic\nlengthlBwillthenbeevenlongercomparedtothehoneycomb\nlattice constant. The difference in energy between the thre e\nstates should mainly vary as B3/2all the way down to zero\nfield, provided only that disorder is negligible. (We discus s\nthe role of disorder againin Sec. V.) Our calculationsshoul d\nthereforereliablypredicttheenergeticorderingofthest atesin\nthe experimental field range. The second conclusion we can\nmakeconcernsthe importanceofZeemancouplingwhichwe\nhave ignored to this point. First of all, Zeeman coupling wil l\nhaveanegligibleeffectontheenergiesoftheSDWandCDW\nstatessincetheyhaveavanishingspinmagneticsusceptibi lity.\nTheenergydifferencepersitebetweentheFstateandthetwo\ndensity-wavestates canbewrittenintheform\nΔE=E(SDW/CDW)−EF\n=B3/2C(SDW/CDW)\n3/2cos(θ)3/2\n+B2·/parenleftBig\nC(SDW/CDW)\n2cos(θ)2−CZcos(θ)/parenrightBig\n(2)\nwhereBis the total magnetic field strength and θis the field\ntilt angle relative to the graphene plane normal. Factors of\nBcos(θ)in this expression therefore account for the perpen-\ndicular field dependence. The second term in Eq. 2 contain\nthe contributions that scale with B2. The factor of Bcos(θ)\nwhichappearsintheZeemantermispresentbecausethespin-\npolarization of the F state is proportional to the Landau lev el\ndegeneracy. The coefficients C3/2andC2can be obtained7\nby fitting energy differences obtained from numerical solu-\ntions of the self-consistent field equations, like those plo t-\nted in Fig. 3, and depend on the interaction model parame-\nters as shown in Table I. The Zeeman coefficient in Eq. 2 is\nCZ=7.3·10−10eV/T2is independent of interaction param-\neters. From the above equation we find that the Fstate has\nlowerenergythanthespin-unpolarizedstatesfor\nB>Bc(θ)=C2\n3/2cos(θ)\n(Cz−C2cos(θ))2(3)\nThe fields required to achieve an energetic preference for th e\nspin-polarized state are smaller at larger tilt angles beca use\ntheorbitalenergyhasa stronger θdependence.\nIn table I we show the values of C3/2andC2for SDW and\nCDWconfigurationsfavoredwith respectto F fora set ofpa-\nrametersof Uandεr. We noticethatthecoefficientsdictating\nthe critical field transition to F solutions can be made rela-\ntivelysmall iftheparametersarenearthe crossoverbounda ry\nto F states. It is possible that a SDW or CDW to F transition\ncould be induced by varying magnetic field. If a transition\nwas observed, most likely by a change in transport proper-\nties asdiscussed in the nextsection,it couldprovidevalua ble\ninput on the effective interaction parameters of the π-orbital\ntight-bindingmodel.\nIV. QUANTUMHALL EDGESTATESIN GRAPHENE\nRIBBONS\nZigzag ribbon unit cell Armchair ribbon unit cell a=2.46 ˚A\nL=2a/√\n3Themagneticbandenergyquantizationprop-\nxy\nFIG. 5: Representation of the unit cell choices for armchair and\nzigzag edge terminated graphene nanoribbons.\nThe quantum Hall effect occurs when a two-dimensional\nelectron system has a chemical potential discontinuity (a g ap\nfor charged excitations) at a density which depends on mag-\nneticfield. Agapat afield-dependentdensitynecessarily50,51\nimpliesthepresenceofchiraledgestatesthatsupportaneq ui-\nlibrium circulating current. The current varies with chemi -\ncal potential at a rate defined by the field-dependence of thebulk gap density. Most quantum Hall measurements simply\nreflect the property51that separate local equilibria are estab-\nlished at opposite edges of a ribbon in systems with a bulk\nenergy gap or mobility gap. It is immediately clear therefor e\nthat theν=0 quantumHall effect is special since it is due to\nan energy gap at the neutrality point, i.e.at a density which\ndoes not depend on magnetic field. The issue of whether or\nnot theν=0 gap and associated phenomena should be re-\nferred to as an instance of the quantum Hall effect is perhaps\na delicate one. The ν=0 gap is intimately related to Landau\nquantization and in this sense is comfortably grouped with\nquantum Hall phenomena. This view supports the language\nwe use in referring to the ν=0 quantum Hall effect. On the\nother hand, since it occurs at a field-independent density, i ts\ntransportphenomenaaremorenaturallyviewedasthoseofan\nordinary insulator16which just happens to be induced by an\nexternalmagneticfield.\nAn exception occurs for the F state which does have edge\nstates8,18, and can be viewed as having ν=1 for majority\nspins and ν=−1 for minority spins. In the simplest case,\nit has two branches of edge state with opposite chirality for\nopposite spin, much like those of quantum-spin-Hall17sys-\ntems. In a Hall bar geometry most transport measurements\nare very stronglysensitive to the presence or absence of edg e\nstates. Inordertoaddressedgestate physicsdirectlyat ν=0\nwe have extended our study from the bulk graphene to the\ngraphenenanoribboncase. Tight-bindingmodelsolutionsf or\naribboninthepresenceofamagneticfieldcanbeobtainedin\nessentially the same way as for bulk graphene, with the sim-\nplification that any magnetic field strength preservesthe on e-\ndimensional ribbon wavevector kas a good quantum num-\nber when the gauge is chosen appropriately. This graphene\nribbon problem in a magnetic field was studied time ago by\nWakabayashi et. al.47andrevisitedrecentlywithinbothtight-\nbinding36,37and continuum8,48,49models in order to provide\na microscopic assessment of the relationship between Lan-\ndau levels and edge states. The general feature of the ribbon\nbandstructureinthepresenceofamagneticfieldisthatthos e\nstateslocalizedneartheedgeshavedispersivebands,wher eas\nthoseintheflat bandregionarelocatedmostlyinthe bulk. In\nthe case of zigzag edge termination, edge localized states a re\npresentevenintheabsenceofamagneticfield46. Inthequan-\ntum Hall regime these states are in the non-dispersive band\nregionlikethebulklocalizedstatesandtheydonotcontrib ute\nto edge currents, although they can interact with other edge\nlocalizedstates.\nFig. 6 explicitly illustrates how the character of the bulk\nbroken symmetry is manifested in ribbon edge state prop-\nerties. Because of practical limitations our calculations are\nrestricted to moderately narrow ribbons with widths of or-\nder 10nm. In order to properly reproduce bulk Landau level\nquantizationinthesenarrowsystemswehavetochoosemag-\nnetic field strengths strong enough to yield magnetic length s\nℓB∼25nm/(B[Tesla])1/2substantiallysmallerthantheribbon\nwidth,i.e.fieldsstrongerthantypicalexperimentalfields. On\ntheotherhandifthemagneticfieldsaretoostrong,say ℓB<a\nthelevelswillbestronglyaffectedbythelatticeandthepr op-\nertiesofthesolutionswillsubstantiallydepartfromtheb ehav-8\nCDW\nkEband (eV)\n2π/L π/L 01\n0\n-1εr= 4U= 4eVSDW\nk2π/L π/L 01\n0\n-1F\nk2π/L π/L 01\n0\n-1\nCDW\nx/an↑/↓(x)\n50 40 30 20 10 00.505\n0.5\n0.495SDW\nx/a50 40 30 20 10 00.51\n0.505\n0.5\n0.495\n0.49F\nx/a50 40 30 20 10 00.505\n0.5\n0.495\nFIG. 6: (Color online) Band structure and spin density distr ibutions in armchair ribbons calculated for U=4eVandεr=4 for a magnetic\nfieldcorresponding to ℓB=4.24a( B = 606 Tesla) using a 100 ×2 lattices in the unit cell and 80 k-point sampling in the Bril louinzone. The\nred color is used to represent up spin while blue is used for do wn spin. Upper row: Armchair ribbon band structures for the F, SDW and\nCDW solutions. Only the F solution is metallic, while the oth er two configurations are insulating. For this choice of para meters SDW is the\nenergetically favoredstate, but the energies of the other s tates are similarand their band gaps also have a similar size .Lower row: Spin-↑and\nspin-↓density distributions across one of the zigzag rows in the un it cell of an armchair ribbon. The influence of Landau-level m ixing on the\nthree solutions is apparent in the differences between the t hree sets of density distributions. In the case of AF solutio ns Landau-level mixing\nenhances the local spin-density, while in the case of the CDW solution the magnitude of the charge density oscillation is reduced by Landau\nlevel mixing. Thedashed horizontal lines represent the occ upation inabsence of Landau level mixing.\niorwe shouldexpectat weakermagneticfields, forwhichthe\ncontinuum model description is approximately correct. For\nfieldstrengthsintheappropriaterange,wefindthesamethre e\ntypes of self-consistent field solutions as in the bulk calcu la-\ntions, namely F, CDW, and SDW solutions. The band struc-\ntures and spin resolved densities presented in Fig. 6 confirm\nthat onlytheF solutionhasstates in thebroken-symmetryin -\nduced gap. Hall-bar transport properties for the F configura -\ntion have been discussed by Abanin et al.8and Fertig et al.18\nfrom a theoretical point of view. The CDW and SDW state\nelectronic structure is insulating, both at the edge and in t he\nbulk. The bandstructuresof these two-states are similar ev en\nthoughtheirspin-densityprofilesarequitedistinct.\nThe plots of spin- ↑and spin- ↓partial densities across the\nribbonshintatsomeofthephysicswhichselects betweenthe\nthreecandidateorderedstates. Inthetruncated n=0Landau-\nlevel continuum theory, the F state has one excess occupied\nLandau level for majority spins and one deficiency in occu-\npied Landau levels for minority spins. We see in Fig. 6, that\nthesizeofthesepolarizationsisnotstronglyinfluencedby the\nLandaulevelmixingeffectsincludedinourlatticecalcula tion.\nThen=0continuumtheorySDWstatehasthesamespinex-\ncesses and deficiencies, but they have opposite sign on oppo-\nsite sublattices. We see in Fig. 6, that these order paramete rs\nare actually enhanced by Landau level mixing effects; inter -Landaulevelexchangeeffectspolarizelower-energyoccup ied\nLandaulevelstatessothattheyenhancetheSDWpattern. For\nthe CDW state on the other hand, the n=0 Landau level ex-\ncess density on one sublattice is suppressed by Landau-leve l\nmixing. Inthis case the direct electrostaticinteractioni s non-\nzero so that the occupiedLandau levels away from the Fermi\nenergyarepolarizedbetweensublatticesintheoppositese nse\nof then=0 levels. The fact that the SDW state is enhanced\nby Landau level mixing explains why it is favored over the F\nstate fortheinteractionparametersusedto constructFig. 6.\nFinally, we comment on the microscopic electronic struc-\nture at the edges of an F state. As illustrated in Fig. 6,\nthe mean-field electronic structure at the edge contains a do -\nmainwall.18Becausetexturesinthisdomainwallcan18carry\ncharge, the F state edge is however not necessarily insulati ng\nintheabsenceofdisorder. It wasrecentlyarguedthatimpur i-\nties with magnetic momentsnear the sample edges can intro-\nducespin-flipbackscatteringpotentials18whoseeffectiveness\nisenhancedbythepresenceofadomainwall. Thehighresis-\ntancesseenexperimentallyat ν=0thereforedonotnecessar-\nily provethat the groundstate is not an Fstate. The analysis\nin Ref.18 was carriedout within a Luttingerliquid formalis m\nwhose parameters depend on the domain wall shape. As il-\nlustrated in Fig. 7, our microscopic domain walls exhibit an\ninteresting anisotropy in which inner and outer segments of9\nCL\n1Hyb, ↓CL\n1Hyb, ↑\ny(˚A)0 20 400.4\n0\n-0.4CL\n2,↓CL\n1,↑ 0.4\n0\n-0.4\nEHyb\nkEband (eV)0.4\n0.2\n0\n-0.2\n-0.4E↓\nkEband (eV)0.4\n0.2\n0\n-0.2\n-0.4E↑\nkEband (eV)0.4\n0.2\n0\n-0.2\n-0.4\nFIG. 7: (Color online) Quantum hall edge domain-wall wave fu nc-\ntions and energy bands corresponding to an Fsolution calculated\nfor a zigzag graphene nanoribbon with 20 carbon atom pairs in the\nunit cell ( W=42.6˚A) under a magnetic field of B=5660 Teslas.\nLeft panel: In the upper figure we represent the edge state wave-\nfunctions for the occupied and unoccupied ↑-spin and ↓-spin states\nwhen hybridization is not included. The lower figure represe nts\nthe coefficients of the hybridized domain wall state. All wav efunc-\ntions are plotted at the Bloch wavevector indicated by arrow s on\nthe right panel. Right panel: Quantum hall edge state bands with\ncollinear spin states represented by black and red symbols a nd the\nnon-collinear spin edge states formed by optimizing the hyb ridiza-\ntion of up and down spin orbitals represented by the blue symb ols.\nWe observe a clear single-particle energy gap related to the energy\ngained byformingthe non-collinear domain wall structure.\nthewall differ.\nV. SUMMARYAND DISCUSSIONS\nWehavecarriedoutamean-fieldstudyofgraphene’s ν=0\nground state which aims to shed light on the character of the\ninteraction-inducedgapthat appearsexperimentally. The fact\nthat the groundstate hasa chargegapat thisfilling factorca n\nbe established essentially rigorously12within the continuum\nmodel often used to describe graphene. When interaction in-\nducedmixingbetween n=0 and|n|/negationslash=0 Landaulevelsis ne-\nglected in the continuum model, the family of broken sym-\nmetrystatesisrelatedbyarbitraryspinandvalleypseudos pin\nrotations and includes both fully spin-polarized Fand spin\n(SDW) and charge (CDW) wave states. Our effort to deter-\nmine the ground state character when Landau-level mixing\nand lattice effects are included, is motivated by the observ a-\ntion of magnetic-field-inducedinsulating transport prope rties\nandbytheexpectationthattransportpropertiesinthequan tum\nHall regime should be very different for F, SDW, and CDW\nstates.\nThe lattice model we study has two phenomenologicalpa-\nrameters, a relative dielectric constant εrand an on-site in-\nteraction parameter U. The most appropriate values of both\nparameters are somewhat uncertain. Our two main findingsarethati)Landau-levelmixingeffectsfavorthedensity-w ave\nstates over the ferromagnetic state and ii) The competition\nbetween CDW and SDW states is sensitive to the relative\nstrengthofon-siteandinter-siteelectron-electroninte ractions\nand hence on the product εr·U. Large values of this prod-\nuct increase the direct mean-field energy cost of CDW order\nand favor the SDW state. Exchange energies are stronger for\ndensity wave states than for ferromagnetic states because o r-\nderwithinthe n=0level,inducesorderin thefullnegative n\nLandaulevelsintheformercase,butnotinthelattercase. F or\ngrapheneonSiO 2andothertypicalsubstrates,sensiblevalues\nforεrandUsuggest that the field-induced state at n=0 is a\nSDW state. CDW states could occur in suspended graphene\nsamples.\nThe atomic value of the on-site interaction term in carbon\nisU∼10eV, so graphene values should be smaller. Most\nof the illustrative calculations we have described use eith er\nU=4eVorU=5eV. The dielectric constant is εr∼1 for\nfree standing graphene. For a SiO 2substrate, with dielectric\nconstantεr∼4.5,theeffective2Ddielectricconstantatasub-\nstrate/vacuuminterfaceis εr∼2.5. BecausetheHartree-Fock\napproximation overestimates the strength of exchange inte r-\nactions, it can be argued that somewhat larger values of εr\nare appropriate - perhaps ∼2 for free standing graphene and\nεr∼4 or 5 for grapheneon a substrate. Because of these un-\ncertainties we view Uandεras effective parameters whose\nvalues are somewhat uncertain and have made energy com-\nparisonsovera widerangeofvalues.\nWe are able to complete our calculations only at magnetic\nfieldstrengthsmuchstrongerthanthoseavailableexperime n-\ntally. Partly to verify that the field-dependence is system-\natic, and partly in an effort to estimate the magnetic field\nstrengths necessary for Zeeman energies to drive the system\ninto a F state, we have fit the energy differences between F\nand density-wave states to the form ΔE=C3/2B3/2+C2B2.\nOnly the first term can be present for a continuum model\nwith Coulomb interactions, and we find that this term is in-\ndeeddominant. WefindthatZeemancouplingattypicalfields\ncanchangethenatureofthestate onlyinthe parameterrange\nwhere the crossover from CDW to SDW states occurs. Al-\nthough we assume collinear states in analyzing the SDW/F\ncompetition, we presume that the SDW to F crossover is ac-\ntually a continuousone in which the antiparallel spins on op -\nposite sublattices are smoothly rotated until they are alig ned.\nEvenifthebulkgapremainsopentherotationofthespinswil l\nbringaboutaprogressiveclosingofedgestatebandgaps,un til\nthegapiscompletelyclosedintheFconfiguration. Themod-\nulation of the edge charge gap due to Pauli paramagnetism\nmight be detected in edge transport experiments. Since a rel -\nativelystrongmagneticfield is requiredto inducethe insul at-\ning state in typical samples, we do not expect that it will be\neasytoproduceparallelfieldslargeenoughtoturnthesyste m\nmetallicwhile maintainingthisminimumperpendicularfiel d.\nNevertheless,astudyoftheparallel-fielddependenceoftr ans-\nportpropertiesis likely to hint at the natureof the underly ing\nbrokensymmetrystate ofthesystem.\nAlloftheseresultsignoretheinfluenceofdisorder. Experi -\nmentsappeartoshowthatthetransitiontotheinsulatingst ate10\noccurs at magnetic fields above some critical value that be-\ncomes smaller when the sample is cleaner. This is expected,\nsince disorder favors states without broken symmetries. Th e\nrelationshipbetweenthe minimumfield and themobilitywas\ncarefully examined some time ago,6but can be crudely de-\nscribed using the following simple argument. Assuming un-\ncorrelated scatterers and using the Fermi golden rule the mo -\nbility isµ∝1/V2\ndiswhereVdisis the typical energy scale of\ndisorder. The crossoveroccurswhen disorderstrengthequa ls\nthe interaction energy scale Vdis=Uint∝B3/2, therefore the\ncritical magnetic field between samples with different mobi l-\nity are relatedthrough B′\nc/Bc=(µ/µ′)1/3. One physicalpic-\ntureofthestrongdisorderlimitassertsthatcurrentflowsa long\ndomain walls which separate disorder-induced electron-ho le\npuddles16domain walls along which current carrying states\nwithν/negationslash=0 can dominate bulk transport and suppress the di-\nvergentresistivity. Ingrapheneonsubstratesthemobilit yval-\nuesrangebetween2000and25,000 cm2V−1s−1,andvaluesas\nhigh as 230,000 cm2V−1s−1have been achieved in annealed\nsuspendedsamples52. Typicalcrticalfieldsinsamplesonsub-\nstrates are in the 20 to 30 Tesla range. From the above argu-ment we can expect that the critical magnetic fields in sus-\npendedsamplesshouldberoughly2to5 timeslower.\nAn important goal of our work was to shed light on the\ncharacter of the ν=0 edge states. We examined the elec-\ntronic structure of armchair ribbons with CDW, SDW, and F\nstates, finding that edge states in the gap are absent for both\nCDW and SDW solutions. Given that the field-induced insu-\nlatingstate appearstohavean extremely largeresistanceonce\nestablished, it appears likely to us that the experimental s tate\ndoes not have edge states and that it thereforemust be a den-\nsitywave,assuggestedbythesecalculations. Ifso,astudy of\nthe influence of magnetic field tilting might be able to distin -\nquishbetweenSDWandCDWstates.\nAcknowledgments. We acknowledge helpful discussions\nwith R. Bistritzer, P. Cadden-Zimansky,Y. Zhao, K. Bolotin ,\nF. Ghahari, P. 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Tutuc1\n1Microelectronics Research Center, The University of Texas at Austin, Austin, TX 78758\n(Dated: November 8, 2018)\nWe study the magnetotransport properties of dual-gated gra phene bilayers, in which the total\ndensity and layer density imbalance are independently cont rolled. As the bilayer is imbalanced we\nobserve the emergence of a quantum Hall state (QHS) at filling factorν= 0 evinced by a plateau\nin the Hall conductivity, consistent with the opening of a ga p between the electron and hole bands.\nBy varying the layer density imbalance at fixed total density , we observe a suppression of the QHS\nat filling factors ν= 8 and ν= 12 when the layer densities are balanced, an observation at variance\nwith theoretical expectations in the absence of electron-e lectron interaction and disorder.\nPACS numbers: 73.43.-f, 71.35.-y, 73.22.Gk\nGraphene, a layerof carbon atoms in a honeycomb lat-\ntice, has emerged in recent years as a new test-bed for\nelectron physics in reduced dimensions [1, 2]. The lin-\near energy band dispersion, zero energy band-gap, and\nchiral quasi-particles set this material apart from con-\nventional two-dimensional electron systems (2DES) real-\nized in semiconductor heterostructures. Graphite bilay-\ners, consisting of two closely coupled graphene layers, are\nan equally interesting system with parabolic momentum-\nenergy dispersion and chiral quasi particles possessing 2 π\nBerryphase[3]. Mostinteresting, in dual-gatedgraphene\nbilayers the electron on-site energy and density in each\nlayer can be independently controlled, which in turn en-\nables the band-gap energy tuning. Here we investigate\nthe magnetotransport properties of dual-gated graphene\nbilayers with high-k dielectrics, a device geometry that\nallows an independent control of the total density and\nlayer density imbalance. As the bilayer is imbalanced we\nobserve the emergence of a quantum Hall state (QHS) at\nfilling factor ν= 0, evinced by a plateau in the Hall con-\nductivity, consistent with the opening of a gap between\nthe electron and hole bands. By varying the layer den-\nsity imbalance at fixed total density, we observe an un-\nexpected suppression of the QHSs at filling factors ν= 8\nandν= 12 when the layer densities are closely balanced.\nThis observation is at variance with the single particle\npicture, strongly suggesting that electron-electron inter-\naction or disorder play a role in this system.\nOur samples consist of natural graphite mechanically\nexfoliated [4] on a 300nm SiO 2dielectric layer, thermally\ngrown on a highly doped n-type Si substrate, with an As\ndoping concentration of ∼1020cm−3. Optical inspec-\ntion combined with Raman scattering are used to iden-\ntify graphene bilayer flakes for device fabrication. We\ndefine six metal contacts using electron beam (e-beam)\nlithography followed by 50 nm Ni deposition and lift-off\n[Fig. 1(a)]. A second e-beam lithography step followed\nby O2plasma etching are used to pattern a Hall bar\non the graphene bilayer flake. A key issue for graphene\ngated devices is the deposition of a top dielectric withoutdegrading the carrier mobility. Owing to the chemically\ninertnessofgraphite, attempts to growhigh- kdielectrics,\nsuchasAl 2O3orHfO 2, byatomiclayerdeposition(ALD)\noncleanhighly oriented pyrolytic graphite leads to se-\nlective growth on terraces, where broken carbon bonds\nserve as nucleation centers. To deposit an Al 2O3top di-\nelectric on our graphene bilayer samples, we first deposit\na∼20˚Athin Al layer, which serves as a nucleation layer\nfor the ALD of Al 2O3. The sample is then taken out in\nair and transferred to an ALD chamber. Based on X-ray\nphotoelectron spectroscopy and electrical measurements,\nthe Al layer is fully oxidized thanks to the presence of\nresidual O 2in the Al evaporation chamber and the expo-\nsure to ambient O 2[5]. Next a 15 nm-thick Al 2O3film is\ndeposited using trimethyl aluminum as the Al sourceand\nH2O as oxidizer. Lastly, a Ni top gate is deposited using\ne-beam lithography, metal deposition, and lift off [Fig.\n1(a)]. The dielectric deposition technique used here has\nbeen shown to produce top-gated monolayer graphene\nsamples with high, over 8,000 cm2/Vs, carrier mobility\n[6]. The mobility ofthe dual-gatedbilayergraphenesam-\nples investigated here is over ≃2200 cm2/Vs [7]. Lon-\ngitudinal ( ρxx) and Hall ( ρxy) resistivity measurements\nare performed down to a temperature of T= 0.3 K, and\nusing standard low-current, low-frequency lock-in tech-\nniques.\nHall measurementsallow us to determine the total car-\nrier density ( ntot) as a function of VTGandVBG, as well\nas the corresponding capacitance values. For the sample\ninvestigatedhere the top- and back-gatecapacitancesare\nCTG= 225 nF ·cm−2andCBG= 10 nF ·cm−2, respec-\ntively. A second parameter relevant for graphene bilay-\ners is the layer density imbalance ∆ n= (nB−nT)/2,\ndefined in terms of the difference between the bottom\n(nB) and top ( nT) layer densities. The use of top- and\nback- gate allows us to independently control ntotand\n∆n. Up to an additive constant, ntotand ∆nare related\ntoVTGandVBGbyntot= (CBG·VBG+CTG·VTG)/e,\nand ∆n= (CBG·VBG−CTG·VTG)/2e;eis the electron\ncharge. The definition of ∆ nrepresents the layer density2\n(b)(a)\n1 µm\nFIG. 1: (color online) (a) Scanningelectron micrograph sho w-\ning a dual gated graphene bilayer. The contact and top-gate\nterminals have been color coded in yellow and red, respec-\ntively. (b) Contour plot of the bilayer ρxxmeasured as a\nfunction of VTGandVBG. The right and top axis repre-\nsent the corresponding charge density change for the back-\nand top-gates, respectively. The top panel shows ρxxvsVTG\nmeasured at ntot= 0, which represents the cut along the ∆ n\ndiagonal of the contour plot. These data allow a one-to-one\ncorrespondence between VTG,VBGon one hand, and ntot,∆n\non the other.\nimbalancein the limit ofexcellent screeningin eachlayer,\nwhen the top and bottom gates control only the charge\ndensity in the top and bottom layers, respectively; the\nactual layer density imbalance is smaller than ∆ n[9].\nThe layer density imbalance translates into a transverse\nelectric field on the graphene bilayer, E=e∆n/ε0;ε0is\nthe vacuum dielectric permitivity.\nIn Fig. 1(b) we show the sample longitudinal resistiv-\nity (ρxx) measured as a function of top (V TG) and back\n(VBG) gate biases, at a temperature T= 0.3 K. The di-\nagonals of constant CBG·VBG+CTG·VTGrepresent the\nloci of constant ntotand varying ∆ n, while diagonals of\nconstant CBG·VBG−CTG·VTGdefine the lociofconstant\n∆nat varying ntot. The diagonal of ntot= 0 is defined\nby the points of maximum ρxxmeasured as a function\nofVTGat fixed VBGvalues. In order to determine the\nVTGandVBGvalues at which ntot= 0 and ∆ n= 0,\nwe consider ρxxmeasured as a function of VTGalong thediagonal ntot= 0 [Fig. 1(b) (top panel)]. As a function\nofVTG,ρxxdisplays a minimum and increases for both\nnegative and positive VTG. This trend can be explained\nby the opening of a band gap in the graphene bilayer,\nas a result of different electron on-site energies on the\ntwo layers [8, 9, 10]. The band-gap increases with the\ntransverse electric field and results in a reduced electri-\ncal conductivity [11, 12]. The ρxxdependence on VTG\nalong the diagonal ntot= 0 allows us to determine the\ngate biases at which ∆ n= 0. The ρxxminimum on the\nntot= 0 diagonal of Fig. 1(b)(top panel) defines the\n∆n= 0 point [13]. Having established a one-to-one cor-\nrespondencebetween VTGandVBGononehand, and ntot\nand ∆non the other, in the reminder of the manuscript\nwe will characterize the bilayer in terms of ntotand ∆n.\nThe dual-gated device of Fig. 1(a) allows indepen-\ndent control of ntotand ∆n. In Fig. 2(a) we show ρxx\nvsntot, measured at fixedvalues of ∆ n, at a magnetic\nfieldB= 18 T and at T= 0.3 K. These data are mea-\nsured by simultaneously sweeping VTGandVBG, with\nthe sweep rates adjusted such that ∆ nremains constant.\nThe data shows QHSs, marked by vanishing ρxxat in-\nteger filling factors that are multiples of four [14]. This\nobservation is explained by the four-fold degeneracy as-\nsociated with both spin and valley degrees of freedom\nof each Landau level [8]. Using the measured ρxxand\nρxy, we determine the Hall conductivity ( σxy) via a ten-\nsor inversion, σxy=ρxy/(ρ2\nxx+ρ2\nxy). Figure 2(b) data\nshows the Hall conductivity ( σxy) measured as a func-\ntion ofntot, atB= 18 T and T= 0.3 K, and for dif-\nferent, fixed values of ∆ n. Interestingly, the data of Fig.\n2(a,b) reveals an increasing ρxxatntot= 0 with increas-\ning ∆n, accompanied by the emergence of a plateau in\nthe Hall conductivity at σxy= 0, which in turn indicates\nan emerging QHS at ν= 0.\nThe emergence of a QHS at ν= 0 in graphene bilayers\ncan be explained by considering the electron energy lev-\nels in this system with and without an applied magnetic\nfield. At E= 0 and B= 0 the graphene band structure\nhas symmetric, parabolic electron and hole bands with a\nzero energy gap, and an effective mass m≈0.054m0;m0\nis the free electron mass [8]. The electron and hole band\nare four-fold degenerate, owing to the spin and valley\ndegrees of freedom. At finite E-field, or ∆ n, the elec-\ntron and hole bands are separated by an energy gap (∆)\nbecause of the different on-site electron energies on the\ntwo layers. In an applied perpendicular B-field the car-\nrier energy spectrum consists of the four-fold degenerate\nLandau levels (LLs). At E= 0 an eight fold degenerate\nLL exists at energy ǫ= 0, the electron-hole symmetry\npoint. As a consequence of this degeneracy QHSs emerge\nat fillings that are multiples of four, and for ∆ n= 0,σxy\nexhibits a double 8 e2/hstep across ntot= 0 as shown in\nFig. 2(b).\nInanappliedtransverse E-field, theeight-folddegener-\nate LL at ǫ= 0 splits into two, four-fold degenerate LLs.3\n051015202530-8.6 -5.7 -2.9 0.0 2.9 5.7 8 .6\n16128ν=4∆n (1012 cm-2)\n 0\n 0.94\n 1.87\n 3.75T=0.3K, B=18T(a)ntot (1012 cm-2) ρxx(kΩ//box3)\n-3 -2 -1 0 1 2 3σxy(kΩ-1) \nVTG(V)-0.62-0.310.000.310.62 (b)T=0.3K, B=18T\n∆n (1012 cm-2)\n 0\n 0.94\n 1.87\n 3.75 \n-4 -3 -2 -1 0 1 2 3 404080120160200 (c)\nT=0.3K, ntot=0ρxx(kΩ)\n∆n (1012 cm-2) B=0T\n B=18T//box3\nFIG. 2: (color online) (a) ρxxvs.ntotmeasured at B= 18\nT andT= 0.3 K, for different values of ∆ n. (b)σxyvs.\nntotmeasured at different ∆ nvalues, and at B= 18 T and\nT= 0.3 K. The bottom axis shows the corresponding VTG\nvalue measured with respect to the ntotand ∆nneutrality\npoint; the VBGvalues (not shown) can be inferred using the\nCTGandCBGvalues. The horizontal grid lines are spaced by\n4e2/h. (c)ρxxvs. ∆nmeasured at ntot= 0 and B= 0 (line),\nand atν= 0 and B= 18 T (symbols).\nThese two LLs, containing electron and holes states re-\nspectively, are symmetrically positioned with respect to\nǫ= 0, and areseparatedby the same energygap∆ which\nsplits the electron and hole bands at B= 0 [8]. In order\nto get further insight into the physics of the ν= 0 QHS,\nin Fig. 2(c) we show ρxxvs ∆n, measured at ntot= 0\nandB= 0 T, along with the ρxxvs ∆n, measured at\nν= 0 and at B= 18 T. These data show that as a func-\ntion of layer density imbalance, the ρxxvalues at ν= 0\nin highB-field are comparable to the ρxxmeasured at\nntot= 0 and B= 0, strongly suggesting that the ν= 0\nQHS and the energy band-gap opening at B= 0 as a\nresults of layer density imbalance have the same origin.0 5 10 15 20 25 30 3501234567\nρxy\nρxxν=4 ν=12ν=8ρ (kΩ)\nB(T)T=0.3 K, ntot=3.5×1012 cm-2\n ∆n=0 \n ∆n=1.1\n ∆n=2.45\nFIG. 3: (color online) ρxxandρxyvs.B, measured at T= 0.3\nK and at different ∆ nvalues. The total density is constant,\nntot= 3.5×1012cm−2.\nNext, we examine the resistivity values measured as\na function of the B-field, at fixed ntotand ∆n. Figure\n3 data shows ρxxandρxyvs.B, measured at T= 0.3\nK, at a total density ntot= 3.5×1012cm−2, and for\ndifferent ∆ nvalues. Consistent with the four-fold de-\ngeneracy of the LLs, QHSs emerge at integer fillings that\nare multiples of four, namely ν= 4,8,12,16etc... Exam-\nination of the Fig. 3 data reveals an interesting finding.\nTheρxxmeasured at ν= 8 and ν= 12 is maximum at\n∆n= 0, and decreases with increasing ∆ n. This obser-\nvation implies that the ν= 8 and ν= 12 are weakest at\nbalance (∆ n= 0), and become stronger as the bilayer is\nimbalanced. The dependence of the ν= 8 and ν= 12\nQHSasafunction ofchargeimbalanceisatvariancewith\nthe single-particle theoretical picture [8], in which the LL\nspacingis independent ofthe appliedtransversefield, and\nconsequently of ∆ n.\nTo further explore the unusual dependence on ∆ nof\nthe QHSs ν= 8 andν= 12, in Fig. 4 we show ρxxvs.B\nmeasured at ntot= 3.5×1012cm−2for different values\nand signs of ∆ n. Figure 4(a) shows the ρxxvs.Bdata\nmeasured for ∆ n≥0, when the carriers are transferred\nfrom the top to the bottom layer, while Fig. 4(b) shows\ntheρxxvs.Bdata measured for ∆ n≤0, when the car-\nriers are transferred from the bottom to the top layer.\nThe data of Fig. 4 substantiate our finding that ν= 8\nandν= 12 QHSs are weakest at balance and become\nstronger when the bilayer is imbalanced. A possible ex-\nplanation for this finding would be a disorder asymmetry\nbetween the two layers, which may suppress QHS when\nthe carriers reside predominantly in the more disordered\nlayer. This is however ruled out by Fig. 4 data, which\nshow that the ν= 8 and ν= 12 QHSs become stronger\nfor both positive andnegative ∆ n, roughly symmetric4\n0123410 12 14 16 18 20 22 24 B(T) \n(a) \n n = 1.1  n = 0.7  n = 0.35  n = 0 \n!=12 !=8 T = 0.3K, n tot  = 3.5 \"10 12  cm -2 #xx  (k $/ )\n10 12 14 16 18 20 22 24 01234 (b) \n n = -1.1  n = -0.7  n = -0.35  n = 0 \n!=12 !=8 T=0.3K, n tot =3.5 \"10 12  cm -2 #xx  (k $/ )\nB(T) \nFIG. 4: (color online) ρxxvs.Bmeasured at constant total\ndensityntot= 3.5×1012cm−2, and at different ∆ nvalues.\nPanels (a) and (b) show data that for ∆ n≥0 and ∆ n≤0,\nrespectively. These data show that the ν= 8 and ν= 12\nQHSs are suppressed when the bilayer is balanced. The ρxx\nvs.Bdata at ∆ n= 0 in panels (a) and (b) are separate\nmeasurements.\nwith respect to ∆ n= 0 [13]. Equally noteworthy is that\ntheρxxvs.Bmeasured at ∆ n= 0 display a small oscil-\nlation pattern, reminiscent of universal conductance fluc-\ntuations. The ρxxvs.Bdata at ∆ n= 0 in panels (a)\nand (b) of Fig. 4 represent two different measurements,\nand show that the ρxxoscillations are repeatable. These\noscillations indicate edge-to-edge carrier scattering, con-\nsistent with the presence of extended electron states at\nν= 8 and ν= 12 when these QHSs are weakest.\nWhile the emergence of the ν= 0 QHS as a function\nof charge imbalance can be explained by a single-particle\npicture as a signature of the opening of an energy gap\nas a function of an applied E-field [8], the dependence\nof theν= 8 and ν= 12 QHS on ∆ ncannot. Indeed,\nin this picture the LLs energy spacing is independent of\n∆n, which translates into QHSs that should be insensi-\ntive to an applied transverse E-field. We note howeverthat this picture does not consider the role of disorder or\nelectron-electroninteraction. DisorderleadstoLLbroad-\nening, which may also depend on the applied transverse\nE-field. ALLbroadeningwhichisminimum at∆ n= 0is\nconsistent with the observed trend of Fig. 3 and 4. How-\never, atheoretical study [15] consideringthe roleofdisor-\nder in graphene bilayers suggests that the QHSs strength\nin a disordered bilayer is independent of the applied E-\nfield. Another mechanism that can explain the ν= 8\nandν= 12 QHSs dependence of ∆ n, is the electron-\nelectron interaction in this system. While the role of\nelectron-electron interaction on the LLs energy spacings\nin graphenebilayershas not been examined theoretically,\nMinet al.[16] showed that at B= 0 T the exchange in-\nteraction favors the spontaneous charge transfer between\nthe layers, when a transverse E-field is applied.\nIn summary we report a magneto-transport study in\na dual-gated graphene bilayer, in which the total density\nandchargeimbalanceareindependently controlled. Con-\ncomitant with the energy band-gap opening at B= 0 in\nthe presence of an applied transverse E-field, we observe\nthe emerge of a QHS at filling factor ν= 0, in agreement\nwith single-particle theory [8]. Surprisingly, the ν= 8\nandν= 12 QHSs are suppressed when the bilayer is bal-\nanced, and become stronger when a transverse E-field\nis applied, rendering the bilayer imbalanced. This ob-\nservation is at variance with the single-particle picture,\nin which the QHS energy gaps are independent of the\nE-field, and strongly suggests that electron-electron in-\nteractionordisorderplayaroleinstabilizingtheseQHSs.\nWe thankA. H.MacDonald, B.Sahu, andS.K. Baner-\njee for discussions, and NRI-SWAN Research Center for\nsupport. This work was performed at the National High\nMagnetic Field Laboratory, which is supported by NSF\n(DMR-0654118), the State of Florida, and the DOE.\n[1] K. S. Novoselov et al., Science 306, 666 (2004).\n[2] K. S. Novoselov et al., Nature 438, 197 (2005); Y. Zhang\net al., Nature 438, 201 (2005); A. K. Geim and K. S.\nNovoselov, Nature Mat. 6, 183 (2007).\n[3] K. S. Novoselov et al., Nature Phys. 2, 177 (2006).\n[4] NGS Naturgraphit GmbH, Winner Strasse 9, D-91227\nLeinburg.\n[5] M. J. Dignam, W. R. Fawcett, H. Bohni, J. Electrochem\nSoc.113, 656 (1966).\n[6] S. Kim et al., Appl. Phys. Lett. 94, 062107 (2009).\n[7] The lower mobilities in bilayer vs monolayer graphene\ncan be explained by considering the Fermi surface prop-\nerties, which allow back-scattering in bilayers but not in\nmonolayers; see e.g. S. Adam and S. Das Sarma, Phys.\nRev. B77, 115436 (2008).\n[8] E. McCann and V. I. Fal’ko, Phys. Rev. Lett. 96, 086805\n(2006).\n[9] E. McCann, Phys. Rev. B 74, 161403(R) (2006).\n[10] H. Min et al., Phys. Rev. B 75, 155115 (2007).5\n[11] T. Ohta et al., Science 313, 951 (2006).\n[12] J. B. Oostinga et al., Nature Mat. 7, 151 (2008).\n[13] The balance (∆ n= 0) point is determined with an accu-\nracy of 0 .2×1012cm−2.\n[14] The filling factor νrepresents the number of occupiedLandau levels. The degeneracy of each Landau level is\neB/h, andν=ntot·h/eB;his Planck’s constant.\n[15] R. Ma et al., Europhys. Lett. 87, 17009 (2009).\n[16] H. Min et al., Phys. Rev. B 77, 041407(R) (2008)."    },    {        "title": "0910.0753v1.Correlation_density_matrices_for_1__dimensional_quantum_chains_based_on_the_density_matrix_renormalization_group.pdf",        "content": "Correlation density matrices for 1- dimensional\nquantum chains based on the density matrix\nrenormalization group\nW M under1, A Weichselbaum1, A Holzner1, Jan von Delft1and\nC L Henley2\n1Physics Department, Arnold Sommerfeld Center for Theoretical Physics and Center\nfor NanoScience, Ludwig-Maximilians-Universit at, 80333 Munich, Germany\n2Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New\nYork, 14853-2501\nE-mail: wolfgang.muender@physik.uni-muenchen.de\nAbstract. A useful concept for \fnding numerically the dominant correlations of\na given ground state in an interacting quantum lattice system in an unbiased way\nis the correlation density matrix. For two disjoint, separated clusters, it is de\fned\nto be the density matrix of their union minus the direct product of their individual\ndensity matrices and contains all correlations between the two clusters. We show how\nto extract from the correlation density matrix a general overview of the correlations\nas well as detailed information on the operators carrying long-range correlations and\nthe spatial dependence of their correlation functions. To determine the correlation\ndensity matrix, we calculate the ground state for a class of spinless extended Hubbard\nmodels using the density matrix renormalization group. This numerical method\nis based on matrix product states for which the correlation density matrix can be\nobtained straightforwardly. In an appendix, we give a detailed tutorial introduction\nto our variational matrix product state approach for ground state calculations for 1-\ndimensional quantum chain models. We show in detail how matrix product states\novercome the problem of large Hilbert space dimensions in these models and describe\nall techniques which are needed for handling them in practice.\nPACS numbers: 02.70.-c, 05.10.Cc, 03.65.Fd, 01.30.Rr, 71.10.Pm, 71.10.Hf\nSubmitted to: New J. Phys.arXiv:0910.0753v1  [cond-mat.str-el]  5 Oct 2009CONTENTS 2\nContents\n1 Introduction 3\n1.1 The correlation density matrix . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.2 Lessons from Luttinger liquid theory . . . . . . . . . . . . . . . . . . . . 4\n1.3 Operator basis and f-matrix . . . . . . . . . . . . . . . . . . . . . . . . . 5\n1.4 Ground state calculation with DMRG . . . . . . . . . . . . . . . . . . . . 6\n1.5 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n2 Model 6\n2.1 De\fnition of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n2.2 Expectations for simple limiting cases . . . . . . . . . . . . . . . . . . . . 7\n2.3 Smooth boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 8\n3 Calculation of the CDM 9\n3.1 De\fnition of the CDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n3.2 DMRG-calculation of the CDM . . . . . . . . . . . . . . . . . . . . . . . 9\n3.3 Symmetry sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n3.4 \\Restoration\" of numerically broken symmetries . . . . . . . . . . . . . . 10\n4 Finding a distance-independent operator basis 11\n4.1 Need for operator bases for clusters AandB. . . . . . . . . . . . . . . . 12\n4.2 Construction of operator bases . . . . . . . . . . . . . . . . . . . . . . . . 12\n4.3 De\fnition of f-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14\n4.4 Fourier-analysis and decay of f-matrix . . . . . . . . . . . . . . . . . . . . 14\n5 Numerical results: general remarks 15\n5.1 Speci\fcation of the clusters AandB. . . . . . . . . . . . . . . . . . . . 15\n5.2 Average site occupation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15\n5.3 r.m.s. net correlations w\u0001N(r) . . . . . . . . . . . . . . . . . . . . . . . 16\n6 Numerical results: symmetry sectors 17\n6.1 Charge-density correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n6.1.1 Operator basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n6.1.2 f-matrix elements: oscillations and decay . . . . . . . . . . . . . . 19\n6.2 One-particle correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 23\n6.3 Two-particle correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\n7 Comparison to previous results 28\n8 Conclusions 29Introduction 3\nA The variational matrix product state approach 30\nA.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31\nA.2 Matrix product states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32\nA.2.1 Construction of matrix product states . . . . . . . . . . . . . . . 32\nA.2.2 Global view and local view . . . . . . . . . . . . . . . . . . . . . . 33\nA.2.3 Details of the A-matrices . . . . . . . . . . . . . . . . . . . . . . . 34\nA.2.4 Orthonormalization of e\u000bective basis states . . . . . . . . . . . . . 35\nA.2.5 Hilbert space truncation . . . . . . . . . . . . . . . . . . . . . . . 37\nA.2.6 Scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\nA.2.7 Reduced density matrix . . . . . . . . . . . . . . . . . . . . . . . 40\nA.2.8 Operators in an e\u000bective basis . . . . . . . . . . . . . . . . . . . . 41\nA.2.9 Local operators acting on j i. . . . . . . . . . . . . . . . . . . . 42\nA.2.10 Expectation values . . . . . . . . . . . . . . . . . . . . . . . . . . 43\nA.3 Variational optimization scheme . . . . . . . . . . . . . . . . . . . . . . . 44\nA.3.1 Energy minimization of the current site . . . . . . . . . . . . . . . 44\nA.3.2 Sweeping details . . . . . . . . . . . . . . . . . . . . . . . . . . . 45\nA.4 Abelian symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47\nA.5 Additional details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48\nA.5.1 Derivation of the orthonormality condition . . . . . . . . . . . . . 48\nA.5.2 Singular value decomposition . . . . . . . . . . . . . . . . . . . . 49\nA.5.3 Numerical costs of index contractions . . . . . . . . . . . . . . . . 49\n1. Introduction\nIn an interacting quantum lattice model the ground state may have several kinds of\ncorrelations, such as long-range order, power-law, or exponentially decaying correlations.\nIn the numerical treatment of such a model it is not clear a priori what kind\nof correlation will be dominant and what kind of operators corresponds to these\ncorrelations. Before calculating correlation functions, one typically chooses in advance\nwhich operators to consider, using prior knowledge and making initial assumptions. The\nneed to make such choices introduces a certain bias into the investigation, which can be\nsomewhat unsatisfying, especially when hidden or exotic correlations are present.\n1.1. The correlation density matrix\nThe correlation density matrix (CDM) [1] has been proposed as an unbiased tool to\ndiscover the dominant kind of correlations between two separated clusters, given the\ndensity matrix for their union (obtained by tracing out the rest of the system). For two\ndisjoint, separated clusters AandBthe CDM is de\fned to be the density matrix of\ntheir union minus the direct product of their respective density matrices to get rid of\ntrivial correlations,\n^\u001aC\u0011^\u001aA[B\u0000^\u001aA\n^\u001aB; (1.1)Introduction 4\nwhich is completely unbiased except for the speci\fcation of the clusters. If the two\nclusters were not correlated at all, this would imply ^ \u001aAB= ^\u001aA\n^\u001aBand therefore\n^\u001aC= 0. The CDM encodes all possible correlations between the clusters AandB, as\ncan be seen from the fact that\ntr\u0010\n^\u001aC^OA\n^O0B\u0011\n= tr\u0010\n^\u001aA[B\u0010\n^OA\n^O0B\u0011\u0011\n\u0000tr\u0010\u0010\n^\u001aA^OA\u0011\n\n\u0010\n^\u001aB^O0B\u0011\u0011\n=h^OA^O0Bi\u0000h ^OAih^O0Bi\u0011C^O^O0; (1.2)\nwhere ^OAand ^O0Bare operators acting on clusters AandB, respectively.\n1.2. Lessons from Luttinger liquid theory\nTo extract useful information from the CDM, it will be helpful to develop some intuition\nfor its general structure. To this end, let us recall some fundamental facts from one-\ndimensional critical fermion systems. They are described by Luttinger liquid theory, in\nwhich one of the key parameters is the Fermi wave vector kF. The asymptotic behavior of\nany kind of correlation or Green's function is typically an oscillation inside a power-law\nenvelope,\nC(r)\u0018cos (mkFr+\u001e)=r\r; (1.3)\nfor some exponent \r, wheremis some integer. For the particular model to be used in\nthis study, a nontrivial mapping is known to a free fermion chain [2], a special case of\nLuttinger liquid.\nRenormalization group theory [6] quite generally implies the existence of scaling\noperators in any critical system such as a Luttinger liquid. They are eigenvectors of\nthe renormalization transformation and consequently their correlations are purely of a\nform like (1.3) for all r, not just asymptotically. The scaling operators usually have\ncomplicated forms. The correlation of a simple operator (e.g. fermion density n(x) at\npositionxalong a chain) has overlap with various scaling operators, and correspondingly\nthe correlation function of that simple operator is a linear combination of contributions\nlike (1.3) from those scaling operators.\nOur aim is to discover the leading scaling operators numerically. The leading scaling\noperator encodes all the local \ructuations that are correlated with faraway parts of the\nsystem. Intuitively, for a given cluster A, that operator does not depend signi\fcantly\non the exact position of the (distant) cluster B. That is particularly obvious in a one\ndimensional system: any correlation at distances r0> rmust be propagated through\nsome sort of correlation at r, so we expect the same operators from cluster Ato be\ninvolved in ^ \u001aC(r), irrespective of the distance r.\nThis suggests an ansatz for leading contributions in the CDM:\n^\u001aC(r) =X\ns^OA;s\n^OB;scseiksr\nr\rs: (1.4)\nHere ^OA;sand ^OB;sare a pair of (distance-independent) scaling operators acting on\nclustersAandB, respectively, ksis the characteristic wave vector for oscillations in theirIntroduction 5\ncorrelation, and \rsis the corresponding scaling exponent. When ks6= 0, the operator\npairs must themselves come in pairs, labelled, say, by sands+ 1, withks+1=\u0000ks,\ncs+1=c\u0003\ns, and\rs+1=\rs, so that ^\u001aCis hermitian. The scaling operators for each cluster\nform an orthonormal set. We expect that only a few terms in the sum in (1.4) capture\nmost of the weight. Correspondingly, it may be feasible to truncate the complete basis\nsets ^OA;sand ^OB;sto a smaller set of \\dominant operators\", whose correlators carry\nthe dominant correlations of the system. The ansatz (1.4) will guide our steps in the\nensuing analysis; at the end, we shall check how well it is satis\fed by the actual CDMs\ncalculated for the model studied in this paper (see section 6.1.2).\nNotice that although a particular correlation function may have nodes, see (1.3),\nfor a CDM of the form (1.4) the norm,\nk^\u001aC(r)k2=X\nsjcsj2\nr2\rs; (1.5)\nis monotonically decaying with r. This expresses the fact that information can only be\nlost with increasing distance, never restored, in a one-dimensional system.\n1.3. Operator basis and f-matrix\nIn [1] the operators entering the dominant correlation were found by a kind of\nsingular value decomposition (SVD), which was done independently for each separation.\nHowever, the operators obtained from the SVD will in general be di\u000berent for di\u000berent\nseparations r. This does not correspond to the form (1.4), where the operators are\ndistance-independent and only the coe\u000ecients are r-dependent. Therefore, we shall\nexplore in this paper a new scheme to decompose the CDMs for all separations in\nconcert, so as to obtain a small set of scaling operators characterizing the dominant\ncorrelations at any (su\u000eciently large) separation. We decompose ^ \u001aCin the form\n^\u001aC(r) =X\nSi X\n\u0016\u00160f\u0016;\u00160(r)^OA;\u0016\n^OB;\u00160!\nSi; (1.6)\nwhere the Sirepresent the symmetry-sectors of the discrete, Abelian symmetries of\nthe Hamiltonian (see section 3.3). The subscript of the brackets indicates that the\ndecomposition within the brackets is done for each symmetry-sector individually. This\ndecomposition is possible for any two complete, r-independent operator sets ^OA;\u0016\nand ^OB;\u00160acting on the part of the Hilbert space of clusters AandB, respectively,\nwhich correspond to the symmetry sector Si. The goal is to \fnd two operator sets\n^OA;\u0016and ^OB;\u00160such that these operator sets may be truncated to a small number of\noperators each, while still bearing the dominant correlations of the system. The distance\ndependence of the CDM is then only contained in the matrix f\u0016;\u00160(r). Then, all analysis\nconcerning the distance-dependence of correlations can be done in terms of this f-matrix.Model 6\n1.4. Ground state calculation with DMRG\nThe CDM in [1] was calculated using the full ground state obtained from exact\ndiagonalization. This limits the system size, so that the method was appropriate mainly\nin cases of rapidly decaying, or non-decaying correlations { not for critical or slowly\ndecaying ones. In the present work, we use the density matrix renormalization group\n(DMRG) [3] (see the excellent review by U. Schollw ock [4]) to compute the ground\nstate for a ladder system which is known to have algebraic correlations [2]. We use the\nmatrix product state (MPS) formulation of DMRG [5] in which an e\u000ecient variational\nprocedure is used to obtain the ground state.\n1.5. Structure of the paper\nThe structure of the main body of the paper is as follows: in section 2 we introduce the\nmodel to be considered for explicit calculations. In section 3 we show how the CDM is\nde\fned, how to calculate it, and explain how a \frst overview of the relative strengths\nof various types of correlations can be obtained. In section 4 we show how to analyze\nthe CDM and its distance dependence. Sections 5 to 7 present our numerical results,\nand section 8 our conclusions. In an extended appendix we o\u000ber a tutorial introduction\nto the MPS formulation of DMRG, and also explain how it can be used to e\u000eciently\ncalculate the CDM.\n2. Model\nTo be concrete in the following analysis of the CDM, we begin by introducing the model\nfor which we did our numerical calculations. This model contains rich physics and its\ntreatment below can readily be generalized to other models.\n2.1. De\fnition of the model\nWe analyze the CDM for a class of spinless extended Hubbard models for fermions, which\nwas intensely studied by Cheong and Henley [2]. They computed correlation functions\nup to separations of about r= 20, using nontrivial mappings to free fermions and\nhardcore bosons. The correlation functions are calculated with an intervening-particle\nexpansion [2], which expresses the correlation functions in terms of one-dimensional\nFermi-sea expectation values (an evaluation of the CDM for that model has also been\ndone by Cheong and Henley [1], using exact diagonalization, but the system sizes are\ntoo short to be conclusive). For spinless fermions on a two-leg ladder with length N,\nwe use the following Hamiltonian:\nH=\u0000tk2X\na=1N\u00001X\nx=1(^cy\na;x^ca;x+1+ h:c:)\u0000t?NX\nx=1(^cy\n1;x^c2;x+ h:c:)\n\u0000tcN\u00001X\nx=2(^cy\n1;x\u00001^n2;x^c1;x+1+ ^cy\n2;x\u00001^n1;x^c2;x+1+ h:c:)Model 7\nFigure 1. Ladder model with the terms of the Hamiltonian in (2.1). Fermions are\ndepicted by black circles and empty lattice positions by white circles. The ordering used\nfor our Jordan-Wigner transformation of fermionic creation and annihilation operators\nis depicted by the red line.\n+V2X\na=1N\u00001X\nx=1^na;x^na;x+1+VNX\nx=1^n1;x^n2;x; (2.1)\nwhere ^ca;xdestroys a spinless fermion on leg aand rungx, and ^na;x= ^cy\na;x^ca;xis the\ncorresponding number operator. E\u000bectively, the model corresponds to a one-dimensional\npseudo-spin chain, where the a= 1 leg is denoted by spin \"and thea= 2 leg by spin\n#. Hence, in the following sections which generally apply to quantum chain models we\nwill treat this model as a quantum chain consisting of Nsites and return to view the\nsystem as a ladder model in the sections where we discuss our results.\nWe will focus on in\fnite nearest-neighbour repulsion V! 1 , which we treat\ndi\u000berently along the legs and the rungs in our numerical calculations. In the pseudo-\nspin description we can enforce the nearest-neighbour exclusion along rungs by removing\ndouble occupancy from the local Hilbert space of the pseudo-spin sites. The nearest-\nneighbour exclusion along the legs cannot be implemented so easily and we mimic\nV! 1 by a value of Vwhich is much larger than all the other energies in the\nHamiltonian (typically V=tk= 104).\nFor fermionic systems, the fermionic sign due to the anti-commutation relations\nof the fermionic creation- and annihilation-operators needs to be taken into account.\nSpeci\fcally, we have to choose an order in which we pick the Fock basis, where we\nhave to keep in mind that this choice produces a so called Jordan-Wigner-string of the\nformPx0\u00001\nx00=x+1ei\u0019^nx00when evaluating correlators h^cx^cy\nx0iat distance r=jx\u0000x0j. In the\npresent system it is convenient to choose this order such that the operators of the two\nsites of a rung are succeeding each other (see \fgure 1), as this choice yields the shortest\nJordan-Wigner strings.\n2.2. Expectations for simple limiting cases\nSettingtk\u00111 as a reference scale, we are left with two parameters in the Hamiltonian:\nthe rung hopping t?and the correlated hopping tc. The physics of the system is governed\nby the competition of t?to localize the fermions on the rungs and tcto pair the fermions.\nThere are three limiting cases which have been studied in detail by Cheong and Henley\n[1, 2].\n(i) The paired limit, tc\u001dtk;t?(we usedtc=tk= 102andt?= 0 for our calculations).Model 8\nIn this limit the fermions form tight pairs which behave similar to hardcore bosons\n[2]. For two given rungs xandx+ 1, there are two possibilities to create a pair of\nfermions, due to in\fnite nearest-neighbour repulsion: ^ cy\n\"x^cy\n#x+1and ^cy\n#x^cy\n\"x+1. It has\nbeen shown in [2] that, based on these two bound pairs, one may classify the bound\npairs in two \ravours along the ladder and that the ground state has only one de\fnite\n\ravour, causing a twofold symmetry breaking in the ground state. This symmetry\nbreaking introduces complications that will be addressed below. The dominant\ncorrelations are expected to be charge-density correlations at short distances and\ntwo-particle at long distances. These charge-density and two-particle correlations\ndecay as power laws, oscillating with k= 2kF, where the Fermi wavelength kFis\nrelated to the \flling as kF= 2\u0017[2]. In this system, the one-particle correlations are\nsuppressed and are expected to decay exponentially, as a nonzero expectation value\ndepends on a local \ructuation completely \flling the rungs between the clusters (as\nelaborated in section 6.2).\n(ii) The two-leg limit, t?\u001ctk,tc= 0. In this limit the two legs are decoupled with\nrespect to hopping, but still the in\fnite nearest-neighbour repulsion introduces\ncorrelations between the two legs. At large distances, power-law charge-density\ncorrelations dominate, while two-particle correlations show much faster power-law\ndecay and one-particle correlations decay exponentially.\n(iii) The rung-fermion limit, t?\u001dtk,tc= 0. In this limit the particles are delocalized\nalong the rungs. For \fllings smaller than quarter-\flling, charge-density , one-\nparticle and two-particle correlations all decay as power laws where charge-density\ncorrelations dominate at large distances.\nOur analysis in this paper is limited to the case (i), where DMRG also showed best\nperformance.\n2.3. Smooth boundary conditions\nFor a ladder of length N(treated as a pseudo-spin chain), we have attempted to reduce\ne\u000bects from the boundaries by implementing smooth boundary conditions , adopting a\nstrategy proposed in [7] for a spin chain to our present fermionic system. (Alternatively,\nit is possible to use periodic boundary conditions [5]. However, this leads to some\ndi\u000eculties, since it is not possible to work with orthonormal basis sets describing\nthe left or right part of the chain with respect to a given site.) Smooth boundary\nconditions are open boundary conditions together with an arti\fcial decay of all terms\nof the Hamiltonian over the last Mrungs at each end of the chain. We shall calculate\nexpectation values only of operators located in the central part of the system (sites x,\nwithM <x\u0014N\u0000M), thus the system's e\u000bective length is N0=N\u00002M.\nFor both smooth and open boundary conditions the average site \flling strongly\ndecreases near the boundaries. To determine the average \flling \u0017, which in\ruences theCalculation 9\nsystem's correlations in an important manner, we thus use only the central N0sites:\n\u0017=N\u0000MX\nx=M+1(h^n\"xi+h^n#xi)=(2N0): (2.2)\nDue to the in\fnite nearest neighbour repulsion, this implies that \u00172[0;0:5].\n3. Calculation of the CDM\nThroughout the paper we will use the Frobenius inner product and norm for any matrices\nMijandM0\nijof matching dimension,\nhM;M0i\u0011X\nijM\u0003\nijM0\nij= tr\u0000\nMyM0\u0001\n(3.1)\nkMk \u0011hM;Mi1=2: (3.2)\n3.1. De\fnition of the CDM\nWe take two disjoint, separated clusters AandBof equal size from a one-dimensional\nquantum chain, i.e. two sets of adjacent sites xA\n1;:::;xA\nnandxB\n1;:::;xB\nnwherenis the\nsize of the clusters and all the indices xare distinct from each other. The local Hilbert\nspaces of clusters AandBwith dimension dnare described in terms of sets of basis\nstatesj\u000biandj\fi, which are product states of the local states of each site in the cluster.\nThe CDM of the two clusters, de\fned by (1.1), can be expanded in this basis as\n^\u001aC=\u001aC\n\u000b\f\u000b0\f0j\u000bij\fih\u000b0jh\f0j: (3.3)\nFor processing the CDM we fuse the two indices of each cluster [1]:\n~\u001aC\n~\u000b~\f\u0011~\u001aC\n(\u000b\u000b0)(\f\f0)j\u000bih\u000b0jj\fih\f0j (3.4)\nwith ~\u000b= (\u000b\u000b0) and ~\f= (\f\f0), and denote the reshaped object ~ \u001aCitself by an extra\ntilde. This corresponds to a partial transpose of the CDM (note that ~ \u001aCis no longer\na symmetric tensor). For the CDM expressed in the indices ~ \u000band ~\f, we may use the\nFrobenius inner product (3.1) and norm (3.2).\nTo study the distance dependence of the correlations, we vary the position of the\nclustersAandB, resulting in a position-dependent CDM ~ \u001aC\u0000\nxA\n1;xB\n1\u0001\n. If the system is\ntranslationally invariant, this object depends only on the distance r=jxA\n1\u0000xB\n1j(the\nminimal distance for two adjacent clusters is equal to the cluster size n). For a \fnite\nsystem, though, ~ \u001aCwill also depend on1\n2\u0000\nxA\n1+xB\n1\u0001\n, at best weakly if the system is long.\nStrategies for minimizing the dependence on1\n2\u0000\nxA\n1+xB\n1\u0001\nby taking suitable averages\nwill be discussed in section 3.4.\n3.2. DMRG-calculation of the CDM\nThe fact that the Hamiltonian in (2.1) is a one-dimensional pseudo-spin chain allows\nus to calculate ground state properties with the density matrix renormalization groupCalculation 10\n(DMRG) [3]. Using the variational matrix product state formulation of that method (see\nappendix for a detailed description), we calculated the ground state of the Hamiltonian\nin (2.1) for several values of t?andtc. The framework of MPS also allows the CDM\nto be calculated e\u000eciently (see section A.2.7 for details). Limiting ourselves to the case\nt?= 0 in this paper, we have calculated the CDM derived from the ground state for\ndistances up to 40 rungs, which is signi\fcantly larger than in previous approaches.\n3.3. Symmetry sectors\nAll the symmetries of the Hamiltonian are re\rected in the CDM, making the CDM\nblock-diagonal, where each block can be labeled uniquely by a set of quantum numbers\nthat are conserved by the Hamiltonian. This means for Abelian symmetries (which\nare the only ones we are considering in practice), that the CDM in the original form\n\u001aC\n\u000b\f;\u000b0\f0ful\fllsQ\u000b+Q\f=Q\u000b0+Q\f0, whereQ\u000bcorresponds to the quantum numbers\nof statej\u000bi, etc. The rearrangement of the CDM into ~ \u001aC\n~\u000b~\fthen implies \u0001 Q~\u000b=\u0000\u0001Q~\f\nwith \u0001Q~\u000b\u0011Q\u000b\u0000Q\u000b0and \u0001Q~\f\u0011Q\f\u0000Q\f0. Since ^\u001aABis hermitian, for every block\nof the CDM involving \u0001 Q~\u000b(\u0001Q~\f) there has to be a block involving \u0000\u0001Q~\u000b(\u0000\u0001Q~\f),\nrespectively. Therefore, it is convenient to sort the various parts of the CDM in terms of\ntheir change in quantum numbers \u0001 Q\u0011j\u0001Q~\u000bj=j\u0001Q~\fjand to analyze each symmetry\nsector individually.\nTo obtain a general classi\fcation of the CDM we sort the various contributions\nof the CDM according to the conserved quantum number(s) Q. In the case of the\nHamiltonian in (2.1), we consider particle conservation ( Q=^Ntot) which breaks the\nCDM into blocks with well-de\fned particle transfer \u0001 N\u0011j\u0001N~\u000bj=j\u0001N~\fjbetween\nclustersAandB. The following r.m.s. net correlations then is a measure for the\ncorrelations with transfer of \u0001 Nparticles between AandB(with \u0001N= 0;1;2):\nw2\n\u0001N(r) =X\n~\u000b~\f2S\u0001N\u0010\n~\u001aC\n~\u000b~\f(r)\u00112\n; (3.5)\nwhereP2\n\u0001N=0w2\n\u0001N(r) =k\u001aC(r)k2. Here the notation ~ \u000b\u0011(\u000b\u000b0)2 S \u0001Nindicates\nthat only pairs of states ( \u000b\u000b0) are considered which di\u000ber by \u0001 Nin particle number\n(similarly for ~\f\u0011(\f\f0)2S \u0001N). In the following we will call correlations involving\n\u0001N= 0;1;2 particles charge-density correlations (CD), one-particle correlations (1P),\nand two-particle correlations (2P), respectively. The following analysis is done for each\nsymmetry sector individually. Depending on the decay of the r.m.s. net correlations\n(3.5), some symmetry sectors may become irrelevant with increasing distance.\n3.4. \\Restoration\" of numerically broken symmetries\nAlthough we have tried to minimize the e\u000bect of boundaries, our numerical methods for\ncalculating the ground state and CDM do not produce strictly translationally invariant\nresults. (In contrast, analyses based on exact diagonalization start from a ground state\nwavefunction in which the symmetry (in a \fnite system) is restored, even if there isOperator basis 11\na symmetry breaking in the thermodynamic limit.) Therefore, we construct the CDM\n~\u001aC(r) for a given distance rfrom an average over several CDMs ~ \u001aC(x;x0) with constant\nr=jx\u0000x0j, wherexandx0give the position of the \frst site of clusters AandB,\nrespectively.\nMoreover, if the exact ground state is degenerate under a discrete symmetry, we\nexpect that DMRG breaks this symmetry unless it is implemented explicitly in the\ncode. As mentioned in section 2.2 for the speci\fc models of this paper we expect a\ndiscrete symmetry under interchange of legs for some parameter regimes. Since we\ndid not implement this symmetry explicitly in our code, we also average the CDM by\ninterchanging the legs of the ladder. Thus, all the data analysis presented in subsequent\nsections will be based on using the following \\symmetry-restored\" form of the CDM,\n~\u001aC(r) =1\nNX\nxx0;jx\u0000x0j=r\u0000\n~\u001aC(x;x0) + ~\u001a0C(x;x0)\u0001\n; (3.6)\nwhere ~\u001a0Cis obtained from ~ \u001aCby interchanging the legs of the ladder, and Nis some\nnormalization factor.\nOne might argue that it is not su\u000ecient to average over the broken symmetry w.r.t.\nleg-interchange on the level of the density matrix, but that instead the symmetry should\nbe restored on the level of the ground state wave function. Speci\fcally, for a ground\nstatej 1i(however it is calculated) which breaks this symmetry, we could restore the\nsymmetry in the following way,\n\f\f +\u000b\n=1p\n2(j 1i+j 2i); (3.7)\nwherej 2i=^Sj 1iand ^Sdescribes the action of interchanging the legs. This would\nlead to a total density matrix\n\f\f +\u000b\n +\f\f=1\n2(j 1ih 1j+j 2ih 2j+j 1ih 2j+j 2ih 1j): (3.8)\nNow, for two clusters AandB, the \frst two terms on the r.h.s. yield the CDM of (3.6),\nwhile the last two terms turn out to be negligible when traced out over all sites except\nfor the two local clustersAandB. This follows from j 1iandj 2ibeing orthogonal,\nhence tr(j 1ih 2j) =h 2j 1i= 0, implying that for a long chain with local clusters A\nandB, the reduced density matrix ^ \u001aAB;12\u0011trx=2A;B(j 1ih 2j) will be very close to zero\ndue to the orthogonality of the wave functions on the sites outside of clusters AandB.\nConsequently, it is su\u000ecient to retain only the \frst two terms of (3.8), i.e. to restore\nthe broken symmetry on the level of the density matrices only, as done in (3.6).\n4. Finding a distance-independent operator basis\nThe goal of this section is to extract a (likely) small set of operators from the CDM,\nwhich will describe the dominant correlations in the system as a function of distance.\nWe will assume in this section that the CDM does not include any broken symmetries\nas indicated in section 3.4.Operator basis 12\n4.1. Need for operator bases for clusters AandB\nAs already mentioned, the CDM (obtained from (3.6)) may be investigated by applying\na singular value decomposition (SVD) for each distance individually [1]:\n~\u001aC\n~\u000b~\f=X\nswsOA;s\n~\u000b\nOB;s\n~\f; (4.1)\nor, in operator notation:\n^\u001aC=X\nsws^OA;s\n^OB;s; (4.2)\nwhere ^OA;sand ^OB;sact on clusters AandB, respectively. Here the singular values ws\nare strictly positive real numbers. By construction, ^OA;sand ^OB;sform orthonormal sets\nin their corresponding Hilbert spaces, i.e. OA;s\n~\u000b=OA;s\n\u000b\u000b0andOB;s\n~\f=OB;s\n\f\f0form a complete\nset in the operator space of clusters AandB, respectively, using the inner product as\nin (3.1). The set includes operators with ws= 0, such as the identity operator, since\nthese will be produced by the SVD. The SVD (4.2) yields for each speci\fc distance ra\nset of operators ^OA;s(r) and ^OB;s(r) acting on clusters AandB, respectively.\nHowever, the dominant operators so obtained, i.e. the ones with large weight from\nthe SVD of ~ \u001aC(r), are likely not the same as each other for di\u000berent distances and hence\nnot convenient for characterizing the \\dominant correlations\" of the system. What is\nneeded, evidently, is a strategy for reducing the numerous sets of operators ^OA;s(r) and\n^OB;s(r) to two \\basis sets of operators\" for clusters AandB, respectively, say ^OA;\u0016and\n^OB;\u0016, which are r-independent and whose correlators yield the dominant correlations in\nthe system in the spirit of (1.4). (For a translationally invariant system the two sets\nhave to be equal for both clusters AandB, but we will treat them independently in the\nanalysis.) Following the ansatz (1.4) from the Luttinger liquid theory, these operators\nought to be distance-independent, carrying common correlation content for all distances.\nThus we seek an expansion of ~ \u001aC(r) of the form (1.6), in which only the coe\u000ecients,\nnot the operators, are r-dependent.\n4.2. Construction of operator bases\nWe have explored a number of di\u000berent strategies for extracting operators from the\nCDM which carry common information for all distances. We will discuss in detail only\none of these, which is rather simple to formulate and reliably yields operator sets with\nthe desired properties. (Several other strategies yielded equivalent results, but in a\nsomewhat more cumbersome fashion.)\nThe simplest possible strategy one may try is to average over all the CDMs at\ndi\u000berent distances and to singular-value decompose the resulting crude \\average CDM\".\nHowever, since the elements for the CDM are expected to be oscillating functions of r,\nsuch a crude average can cancel out important contributions of the CDM. Thus we\nneed a procedure that avoids such possible cancellations. To this end, we construct theOperator basis 13\nfollowing operators, bilinear in the CDM:\n^KA(r)\u0011trB\u0000\n^\u001aCy(r) ^\u001aC(r)\u0001\n=k^\u001aCk2(4.3a)\n^KB(r)\u0011trA\u0000\n^\u001aC(r) ^\u001aCy(r)\u0001\n=k^\u001aCk2; (4.3b)\nwith matrix elements\nKA\n~\u000b~\u000b0(r) =X\n~\f~\u001aC\n~\u000b~\f(r) ~\u001aC\n~\u000b0~\f\u0003(r)=k~\u001aC(r)k2(4.4a)\nKB\n~\f~\f0(r) =X\n~\u000b~\u001aC\n~\u000b~\f(r) ~\u001aC\n~\u000b~\f0\u0003(r)=k~\u001aC(r)k2: (4.4b)\nWe normalize by k~\u001aC(r)k2in order to treat the operator correlations of ~ \u001aC(r) for\ndi\u000berent distances on an equal footing. Note that the eigenvalue decomposition on the\nhermitian matrices KA(r) andKB(r) (in short K-matrices) yields the same operators\n^OA(r) and ^OB(r) as the SVD of ~ \u001aC(r), with eigenvalues being equal to singular values\nsquared, up to the additional normalization factor k~\u001aC(r)k2. (Reason: for a matrix of\nthe formM=usvywe haveMMy=us2uyandMyM=vs2vy.)\nThe object ^KX(forX=A;B) is positive-de\fnite and according to ansatz (1.4), it\nis expected to have the form\n^KX(r) =N\u00001\nKX\nsjcsj2\nr2\rs^OX^OXy: (4.5)\nIn particular, it no longer contains any oscillating parts (in contrast to (1.4)), and hence\nis suitable for being averaged over r.\nSumming up the KX-matrices over a range Rof distances ( r2R, whereR\nwill be speci\fed below) gives a mean \u0016KX-matrix for cluster X(=A;B), namely\n\u0016KX;R\u0011P\nr2R^KX(r). We do not divide the latter expression by the number of terms in\nthe sum (as would be required for a proper mean), as at this stage we are only interested\nin the operator eigendecomposition,\n\u0016KX;R=X\n\u0016wR;\u0016\u0010\n^OX;R;\u0016\n^OX;R;\u0016y\u0011\n; (4.6)\nwith the operators normalized such that k^OX;R;\u0016k= 1. The operator set ^OX;R;\u0016gives\nan orthonormal, r-independent basis for cluster X. In practice, however, many of the\nwR;\u0016(which turn out to be the same for X=AorB) will be very small. Thus, it will\nbe su\u000ecient to work with a truncated set of these operators having signi\fcant weight.\nTo explore the extent to which \u0016KXdepends on the summation range, we shall\nstudy several such ranges: Rallincludes all distances, Rshortshort distances (\frst third of\ndistances analyzed), Rintintermediate distances (second third) and Rlonglong distances\n(last third). The resulting (truncated) sets of operators can be compared via their\nmutual overlap matrix ORR0\n\u0016\u00160= tr( ^OR;X;\u0016 ^OR0;X;\u00160), or more simply, by the single number\nORR0=P\n\u0016\u00160(ORR0\n\u0016\u00160)2, which may be interpreted as the dimension of the common\nsubspace of the two operator sets. The value of ORR0ranges from 0 to dim(^OR;X;\u0016).\nBy comparing ORR0for the di\u000berent distance ranges, additional clues can be obtainedOperator basis 14\nabout how the relative weight of correlations evolves from short to long distances. (Such\na comparison is carried out in table 1 below.)\n4.3. De\fnition of f-Matrix\nOnce a convenient basis of operators ^OA;\u0016and ^OB;\u0016has been found, the correlation\ndensity matrix can be expanded in terms of this basis as in (1.6),\n~\u001aC\n~\u000b~\f(r) =X\n\u0016\u00160f\u0016;\u00160(r)OA;\u0016\n~\u000bOB;\u00160\n~\f; (4.7)\nwith matrix elements\nf\u0016;\u00160(r)\u0011X\n~\u000b~\f~\u001aC\n~\u000b~\f(r)OA;\u0016\n~\u000bOB;\u00160\n~\f: (4.8)\nFor complete operator spaces ^OA;\u0016and^OB;\u00160, by de\fnition, the set of amplitudes squared\nsum up to the norm of the CDM:\nX\n\u0016\u00160jf\u0016;\u00160(r)j2=k~\u001aC(r)k2: (4.9)\nHowever, as alluded to above, we expect that the dominant correlators can be expressed\nin terms of a truncated set of dominant operators. If the sum on the left hand side of\n(4.9) is restricted to this truncated set, its deviation from the right hand side gives an\nestimate of how well ~ \u001aCis represented by the truncated set of operators. It will turn\nout that only a handful of dominant operators (typically 4 or 6) are needed, implying\nvery signi\fcant simpli\fcations in the analysis. Thus, the data analysis will be done in\nterms of the matrices f\u0016;\u00160(r) (in short \\f-matrix\") for this truncated set of dominant\noperators.\n4.4. Fourier-analysis and decay of f-matrix\nAccording to the expectations expressed in (1.4), the elements of the f-matrix are\nexpected to be products of oscillating and decaying functions of r. The corresponding\ndominant wave vectors can be identi\fed via Fourier transform on each element of the\nf-matrix. For an oscillating function times a monotonically decaying envelope, the peaks\nof the Fourier spectrum of the oscillating function will be broadened by the presence\nof the envelope. To minimize this unwanted broadening, we introduce a rescaled f-\nmatrix (denoted by a tilde), ~f\u0016;\u00160(r) =u(r)f\u0016;\u00160(r), where the positive weighting-\nfunctionu(r) is chosen such that all values of j~f\u0016;\u00160(r)jare of the same order, and\nFourier decompose the rescaled ~f-matrix as ~f\u0016;\u00160(k) =P\nre\u0000ikr~f\u0016;\u00160(r). Its norm\nk~f(k)k2=P\n\u0016\u00160j~f\u0016;\u00160(r)j2, plotted as a function of k, will contain distinct peaks\nthat indicate which wave vectors characterize the dominant correlations. Subsequently,\nthe elements of the f-matrix, can be \ftted to the forms\nf\u0016;\u00160(r) =X\njA[j]\n\u0016;\u00160eikjrfj(r); (4.10)Numerical results 15\nwhereA[j]\n\u0016;\u00160are complex amplitudes, fj(r) describes the decay with distance (e.g.\nfj(r) =r\u0000\rjore\u0000r=rjfor power-law or exponential decay, respectively), and kjis a\nset of dominant wave vectors. The latter appear pairwise in combinations (+ k;\u0000k),\nsincef\u0016;\u001602 R, which implies A[i]\n\u0016;\u00160=A[j]\u0003\n\u0016;\u00160forki=\u0000kj. The results of such a \ft for\neach pair of dominant operators ^OA;\u0016and ^OB;\u00160, is the \fnal outcome of our analysis,\nsince it contains the information needed to check the applicability of ansatz (1.3).\n5. Numerical results: general remarks\nIn this section, we illustrate the analysis proposed above for the model introduced in\nsection 2. We will focus on the limiting case of large tc, which we expect to have the most\ncomplex behavior among all three limiting cases introduced in [1] and [2]. After some\npreliminary analysis, we will discuss in section 6 each of the three symmetry sectors\n(CD, 1P, and 2P) characterized by the operators' fermion number, and in section 7\ncompare our results to those found by [2] using a di\u000berent method.\n5.1. Speci\fcation of the clusters AandB\nFor the following analysis it is convenient to take the size of the clusters AandBto be\ntwo rungs, because clusters of at least that size allow for up to two particles in one cluster\n(due to in\fnite nearest-neighbour repulsion). Thus, correlations involving \u0001 N= 0;1;2\nare possible, i.e CD, 1P, and 2P correlations, respectively. Note that larger clusters\ncan be studied, but would signi\fcantly increase numerical costs. Taking into account\nthe in\fnite nearest-neighbour repulsion, clusters of size two have a seven-dimensional\nHilbert space spanned by the kets j00i,j0\"i,j0#i,j\"0i,j#0i,j\"#i,j#\"i, where the\n\frst (second) entry corresponds to the \frst (second) rung, 0 represents an empty rung\nand\"and#a fermion on the upper and lower leg in pseudo-spin notation (recall that\nwe are dealing with spinless fermions). The space of operators acting on a cluster has\ndimension 72= 49, where the subspaces for \u0001 N= 0, 1 or 2 have dimensions 21, 24 and\n4, respectively, as depicted schematically in \fgure 2.\n5.2. Average site occupation\nAs a \frst check of the in\ruence of the boundaries, we investigate the average site\noccupation on the ladder. It is expected to be uniform in a translationally invariant\nsystem. However, there are two ways in which our calculation breaks translational\nsymmetry, which cause residual oscillations in the density of particles along the ladders.\nFirstly, there is the spontaneous breaking of the pair \ravor symmetry described in\nsection 2.2. In the ground state produced by DMRG, all pairs have the same \ravor, so\nonly one of the two sublattices actually has any fermions on it. Thus a strong alternation\nin the density is observed between one leg for even rungs and the other leg for odd rungs;\nthis can be taken care of by the symmetrization with respect to legs (as in (3.6)).Numerical results 16\nFigure 2. The symmetry sectors of an operator acting on a cluster of two rungs in\nthe basisj00i,j0\"i,j0#i,j\"0i,j#0i,j\"#i,j#\"iin pseudo-spin notation.\nSecondly, translational symmetry is broken due to \fnite size in the DMRG\ncalculation. This induces oscillations in the average occupation as a function of x(see\n\fgure 3), whose period is clearly dependent on the \flling. In fact, their period is 2 kF, so\nthey may be interpreted as Friedel-like oscillations caused by the boundaries. Although\nthe amplitude of density oscillation appears rather \rat in the central portion of the\nsystem, it does have a minimum there; so we expect that the amplitude in the center of\nthe system would vanish in a su\u000eciently large system.\nAlthough the intent of the smooth boundary conditions is to minimize e\u000bects such\nas these oscillations, in fact, their amplitude appeared to be of about the same strength\nindependent of whether we used smooth or plain open boundary conditions. We suspect,\nhowever, that the amplitude could be reduced by further careful optimization (not\nattempted here) of the parameters of the smooth boundary conditions.\n5.3. r.m.s. net correlations w\u0001N(r)\nThe next basic step is to identify the leading correlations in terms of the r.m.s. net\ncorrelations w\u0001Nde\fned in (3.5). These reveal which sectors of correlations dominate\nat large distances. The results (see \fgure 4) show that the r.m.s. net correlations\ndecay exponentially in the 1P sector, whereas they decay algebraically in both the CD\nand 2P sectors, consistent with [2]. The latter two correlations are comparable in size\nover a signi\fcant range of distances, but for the \fllings we investigated, 2P correlations\nultimately dominate over CD correlations at the largest distances. Both the CD and\n2Pr.m.s. net correlations can be \ftted to power laws, with the exponent dependent on\nthe \flling. The r.m.s. net correlations in each sector are monotonic and only weakly\nmodulated, even though the dominant correlation functions and the dominant parts of\nthe CDM itself are oscillating (as will be discussed in more detail in section 6.1, see,\ne.g., \fgure 7). This implies that the correlations in each sector can be represented\nby a linear combination of correlation functions (associated with di\u000berent operators)\nwhich oscillate out of phase, in such a way that in the sum of their squared moduli theNumerical results 17\n20 40 60 8000.51\nxˆnx\n  \nˆn↑\nˆn↓\n  (a) ¯n\n20 40 60 8000.51\nxˆnx\n  \nˆn↑\nˆn↓\n  (b) ¯n\nk/π¯n(k)(c)\n−1 0 110−5100\nk/π¯n(k)(d)\n−1 0 110−5100\nFigure 3. The average occupation along the legs of the ladder for a \flling of \u0017= 0:248\n(panels a,c) and a \flling of \u0017= 0:286 (panels b,d). Panels (a) and (b) show the average\noccupation ^ n\"on the upper (red) and ^ n#on the lower (green) leg, with every second\nvalue being zero. The end regions i= 1:::20 andi= 81:::100 were skipped in the\n\fgures and also in the analysis as these are a\u000bected by the smooth open boundary\ncondition. The leg symmetrized occupation \u0016 n=1\n2(^n\"+ ^n#) (blue, same for upper and\nlower leg) eliminates this strong even odd alternation but still shows small modulations.\nThis can be seen in detail in the Fourier transform of the symmetrized occupation in\npanels (c) and (d). There is a clear peak at k=\u00062kF(dashed vertical lines).\noscillations more or less average out, resulting in an essentially monotonic decay with\nr, as expected according to (1.5).\nWe will next apply the analysis proposed in section 4.2 to the respective symmetry\nsectors (which will provide more exact \fts of the exponents of the power-law decays).\nThe analysis in any sector consists of two stages. First, following section 4.2, we try to\n\fnd an optimal truncated basis which describes best the dominant correlations. Second,\nwe examine the f-matrix of section 4.3 (i.e. represent the CDM in the truncated basis)\nto see the nature of its rdependence, and to \ft this to an appropriate form, following\nsection 4.4.\n6. Numerical results: symmetry sectors\n6.1. Charge-density correlations\n6.1.1. Operator basis First we calculated the mean K-matrices \u0016KA;Rand \u0016KB;Rfrom\n\u0016\u001aC\nRde\fned in (4.3 a) and (4.3 b), and obtained operator sets from their eigenvalueNumerical results 18\n2 10 3010−310−210−1100\nrw∆N(r)\n  (a) ∆N= 0 , γ0= 1.45\n∆N= 1 , r1= 0.46\n∆N= 2 , γ2= 0.95\n2 3010−20100\n2 10 3010−310−210−1100\nrw∆N(r)\n  (b) ∆N= 0 , γ0= 1.33\n∆N= 1 , r1= 0.53\n∆N= 2 , γ2= 1.11\n2 3010−20100\nFigure 4. Ther.m.s. net correlations of (3.5), plotted as a function of distance for (a)\na \flling of\u0017= 0:248 and (b) a \flling of \u0017= 0:286. The symmetry sectors are \u0001 N= 0\n(blue, no particle transfer, CD), \u0001 N= 1 (green, transfer of one particle, 1P) and\n\u0001N= 2 (red, transfer of two particles, 2P). We see that CD and 2P correlations decay\nas power-laws ( r\u0000\r, blue and red solid lines) with small residual oscillations at k= 2kF,\nwhile the 1P correlations show exponential decay ( e\u0000r=r1, see semi-logarithmic plot in\nthe inset). The value r1'0:5 for both \fllings is reasonable as we would expect a value\nof the order of one, which is the size of the bound pairs.\ndecomposition, using various distance ranges.\nIn order to decide how many operators to include in the truncated basis, we used\nthe diagnostic described in section 4.2. In presenting the results, we limit ourselves\nto clusterAas the results for cluster Bare completely analogous. The operator set\n^OA;Rall;\u0016corresponding to the full range of distances Rall(speci\fed in section section 4.2)\nis used as a reference set to be compared with the operator sets obtained from Rshort,\nRintandRlong. The results are given in table 1. We see that, for intermediate or long\ndistances, the e\u000bective dimension ( ORallRintandORallRlong) of the common operator space\nshared between the operator set ^OA;Rall;\u0016and the operator sets ^OA;Rint;\u0016and ^OA;Rlong;\u0016,\nrespectively, saturates at six even if a larger operator space is allowed. Similarly, also\nthe short-distance operator set ^OA;Rshort;\u0016agrees best with the other three operator sets\nat dimension six: a further increase of the number of operators, however, adds only\noperators in the short range sector of the CDM. Hence we truncate to a six-dimensional\noperator basis. Within this reduced operator space, all dominant correlations are well-\ncaptured, as can be seen from the relative weights of table 1. For the resulting truncated\nbasis set equation (4.9) holds up to a relative deviation of the order of O(10\u00005).\nInvestigating the six-dimensional set of operators in more detail reveals that they\ncan be classi\fed with respect to their symmetry with respect to interchanging the legs\nof the ladder, i.e. they obey ^S^OA;Rall;\u0016=\u0006^OA;Rall;\u0016, with ^Sdescribing the action of\ninterchanging legs. The set breaks into two subsets of three operators each, which have\npositive or negative parity with respect to ^S, respectively. It turns out that all six\noperators are linear combinations of operators having matrix elements on the diagonal\nonly, in the representation of \fgure 2. Moreover, together with the unit matrix they spanNumerical results 19\nTable 1. Comparison of the operator sets on cluster Afor a \flling of \u0017= 0:286. (The\nresults for\u0017= 0:248 are similar, with only minor di\u000berences.) The \frst and second\ncolumn of the table give the number of operators kept and the corresponding smallest\nsingular value of the set of operators ^OA;R all;\u0016obtained from the full range of distances\nRall. The other three columns show ORallRshort,ORallRintandORallRlongfor the given\nnumber of operators.\nnumber of wRall;\u0016=wRall;1ORallRshortORallRintORallRlong\noperators (short) (intermediate) (long)\n1 1 1 0.99 1\n2 0.784122 1.99 2 2\n3 0.579242 2.99 3 3\n4 0.176043 3.99 4 4\n5 0.011250 5 5 4.99\n6 0.003040 6 6 5.99\n7 0.000004 7 6 6\n8 0.000001 8 6 6\n9 0.000001 9 6 6\n10 0.000001 10 6 6\nthe full space of diagonal operators (therefore the dimension of 6 = 7 \u00001). Explicitly,\nthe symmetric operators are given by\n^O1=1p\n12(\u0000^n0;x^n\";x+1\u0000^n\";x^n0;x+1+ 2^n\";x^n#;x+1+ leg symmetrized) (6.1 a)\n^O2=1\n2(^n0;x^n\";x+1\u0000^n\";x^n0;x+1+ leg symmetrized) (6.1 b)\n^O3=1p\n42[\u00006^n0;x^n0;x+1+ (^n0;x^n\";x+1+ ^n\";x^n0;x+1+ ^n\";x^n#;x+1+ leg symmetrized)](6.1 c)\nand the antisymmetric operators by\n^O4=1p\n2^n0;x(^n\";x+1\u0000^n#;x+1) (6.2 a)\n^O5=1p\n2(^n\";x\u0000^n#;x) ^n0;x+1 (6.2b)\n^O6=1p\n2(^n\";x^n#;x+1\u0000^n#;x^n\";x+1) (6.2 c)\nwhere ^n0= (1\u0000^n\"\u0000^n#). We use this operator basis for both cluster Aand cluster B.\nIf we calculate the f-matrix (4.7) based on these operators we see that it breaks into two\nblocks corresponding to their symmetry with respect to leg interchange.\n6.1.2. f-matrix elements: oscillations and decay We now turn to extracting the\ndistance-dependence of the dominant correlation in this symmetry sector, which is\nnow visualizable since we drastically reduced the operator space to six dimensions.\nAll relevant information is contained in the f-matrix and its Fourier transform. The\n\frst step is to identify the oscillation wave vector(s) kto be used as initial guesses in\nthe \ft. A general method is to plot the Fourier spectrum k~f(k)kof the rescaled f-\nmatrix (\fgure 5). When using a logarithmic scale for the vertical axis, even sub-leading\ncontributions show up clearly. We \fnd that the spectra belonging to the symmetric\nand anti-symmetric operators are shifted against each other by \u0019. This relative phaseNumerical results 20\nk/π/bardbl˜f(k)/bardbl2\n  (a)\n−1 0 110−2100102\n˜f+\n˜f−\nk/π/bardbl˜f(k)/bardbl2\n  (b)\n−1 0 110−2100102\n˜f+\n˜f−\nFigure 5. Fourier transform of the rescaled f-matrix ~ffor CD correlations based\non operators chosen from a reduced six-dimensional operator space, for a \flling of\n(a)\u0017= 0:248 and (b) \u0017= 0:286. We obtain these Fourier spectra from the\nrescaled f-matrix ~f\u0016;\u00160(r) =r\r00f\u0016;\u00160(r), with\r00extracted from a power-law \ft\nonjf\u0016;\u00160(r)j. The Fourier spectrum breaks up into a contribution coming from the\noperators symmetric or antisymmetric under leg-interchange, labelled ~f+(blue) and\n~f\u0000(red), respectively. The spectrum of ~f+shows strong peaks at k=\u00062kF(dashed\nlines) and a smaller peak at k= 0 withkF=\u0019=\u0017. The spectrum of ~f\u0000, having peaks\natk=\u00062kF+\u0019(dashed lines) and k=\u0019, is shifted w.r.t. ~f+by\u0019. For a \flling close\nto1\n4the dominant peaks of ~f\u0006, atk=\u00062kFandk=\u00062kF+\u0019. are nearly at the\nsame position.\nshift implies a trivial additional distance dependence of ei\u0019roff\u0000(r) with respect to\nf+(r), re\recting the di\u000berent parity under leg interchange of the two operator sets. We\nhave found it convenient to undo this shift by rede\fning f\u0000(r), the part of the f-matrix\nbelonging to the anti-symmetric operators, to ei\u0019rf\u0000(r). The resulting combined Fourier\nspectrum for f+andei\u0019rf\u0000has strong peaks at k= 2kFand a smaller peak at k= 0,\nin agreement with the result from [2].\nBased on the Fourier spectrum, we rewrite the \ftting form (4.10) as\nf\u0016;\u00160(r) =A\u0016\u00160r\u0000\rcos (kr+\u001e\u0016\u00160) +B\u0016\u00160r\u0000\r0; (6.3)\nwith real numbers A\u0016\u00160>0 andB\u0016\u00160, where we expect \r0> \r, due to the relative\nsharpness of the peaks in the Fourier spectrum. The non-linear \ftting over the full\nrange of distances is done in several steps to also include the decaying part at long\ndistances on an equal footing. First, the data is rescaled by r+\r00, where we obtained \r00\nfrom a simple power-law \ft, in order to be able to \ft the oscillations for all distances\nwith comparable accuracy. Then we \ft the rescaled data to (6.3), where initially we use\nthe information from the Fourier spectrum in keeping k\fxed tok= 2kF, but \fnally\nalso release the constraint on k. This procedure showed best results, with relative error\nbounds up to 2%. The uncertainties are largest for the second term in (6.3) as it acts\nmainly on short distances, having \r0>\r.\nThe results of this \ftting procedure are depicted in \fgure 6, for all 18 nonzero\nelements of the f-matrix. We see that the leading power-law exponents deviate from theNumerical results 21\nγ(a)\n1.251.31.35 Aµµ/prime(b)\n00.10.2φµµ/prime/π(c)\n−0.500.5\nµµ/primerel. error(d)\n33133154664455221145325646122364652100.010.02\nFigure 6. The results of the \ft in (6.3) to the 18 independent elements f\u0016;\u00160of the\nf-matrix, labeled along the horizontal axis by the index pair \u0016\u00160, for \u0001N= 0 at \flling\n\u0017= 0:286. The results for \u0017= 0:248 are qualitatively the same. Panel (a) shows\n\r, panel (b) A\u0016\u00160, panel (c) \u001e\u0016\u00160and panel (d) the error of the \ftting \u000f, de\fned by\n\u000f2=P\nr(f\u0016\u00160(r)\u0000f\ft(r))2=r\u00002\r00, wherer\u0000\r00is the power-law we used to rescale the\ndata before Fourier-transforming. The red, dashed line in the \frst panel shows the\npower-law exponent obtained from the r.m.s. net correlations, \r0= 1:33. The phase\n\u001e\u0016\u00160is de\fned such that it is in the interval [ \u0000\u0019;\u0019]. The matrix elements have been\ngrouped according to their relative phases \u001e\u0016\u00160(separated by the black, dashed line),\nwhich clearly indicate cos and sin behaviour for \u001e\u0016\u00160= 0 and\u001e\u0016\u00160=\u0006\u0019\n2, respectively.\nThe solid red lines in panels (a) and (b) show the exponent \r0and the amplitude A,\nrespectively, from the single \ft (6.4).\n\ft to the r.m.s. net correlations in (3.5) (compare \fgure 4) by about 5%. The k-vectors\nfrom the non-linear \ft are close to k= 2kFand deviate by less than 1%. The \ft to\nthe sub-leading second term in (6.3) is not reliable, so we do not show the results for \r0\nhere, but note that every \ft satis\fed \r0>\r.\nSince most of the exponents \rand amplitudes A\u0016\u00160are of comparable size, we \ft\nthe f-matrix elements to a single\r0andA(as well as a single \r0\n0andBfor the second\nterm) for all the f-matrix elements, using the Ansatz:\nf(r) =Ar\u0000\r00\nBB@0\n@cos(kr) sin(kr) cos(kr)\n\u0000sin(kr) cos(kr)\u0000sin(kr)\ncos(kr) sin(kr) cos(kr)1\nA 0\n0 ei\u0019r0\n@cos(kr) cos(kr) sin(kr)\ncos(kr) cos(kr) sin(kr)\n\u0000sin(kr)\u0000sin(kr)cos(kr)1\nA1\nCCA+Br\u0000\r0\n0:(6.4)\nThe form of the matrices in the two blocks was obtained by inserting into (6.3) the\nexplicit values of the phases \u001e\u0016\u00160determined from the previous \ft and summarized inNumerical results 22\n2102030−0.0100.01\nrfµ , µ/prime(r)\n  \n(µ , µ/prime) = ( 1 ,3)\n(µ , µ/prime) = ( 4 ,6)\nr. m. s.\n  (a) (µ , µ/prime) = ( 1 ,3)\n(µ , µ/prime) = ( 4 ,6)\nr. m. s.\n2102030−0.0100.01\nrfµ , µ/prime(r)\n  \n(µ , µ/prime) = ( 1 ,3)\n(µ , µ/prime) = ( 4 ,6)\nr. m. s.\n  (b) (µ , µ/prime) = ( 1 ,3)\n(µ , µ/prime) = ( 4 ,6)\nr. m. s.\nFigure 7. Two entries of the f-matrix for (a) \u0017= 0:248 and (b) \u0017= 0:286 \ftted to\nthe form in (6.3). The single points (blue circles and squares) are data points from\nthe f-matrix and the lines (red and green) are the result of the \ftting. They evidently\noscillate with a relative phase of \u0001 \u001e=\u0019=2. As a result, their contribution to the\nr.m.s. net correlations , ( jf1;3j2+jf4;6j2)1\n2, shown by the thick orange curve, has only\nsmall oscillations at large distances.\n\fgure 6. Fitting to (6.4) gives an error of about 10%, with largest errors arising for\nthe f-matrix elements where A\u0016\u00160deviates strongly from A(see \fgure 6). For the \flling\n\u0017= 0:286 we \fnd \r0= 1:26 andA= 0:06. The values of \r0\n0andBare unreliable in\nthat the results from several \fttings di\u000ber by about 30%, but still it holds that \r0\n0>\r 0.\nThe form of (6.4) allows us to understand why the r.m.s. net correlations displayed\nin \fgure 4 show some residual oscillations, instead of decaying completely smoothly, as\nanticipated in section 1.2. The reason is that (6.4) contains 10 cos( kr) terms but only\n8 sin(kr) terms. Although any two such terms oscillate out of phase, as illustrated in\n\fgure 7, the cancellation of oscillations will thus not be complete. Instead, the r.m.s.\nnet correlations contain a factor [8 + 2 cos2(kr)]1\n2(compare to (3.5)), which produces\nrelative oscillations of about 10%, in accord with \fgure 4. (The fact that the total\nnumber of cos( kr) and sin(kr) terms is not equal is to be expected: the total operator\nHilbert space per cluster is limited, and its symmetry subspaces might have dimensions\nnot a multiple of 4.)\nFor each pair of wave vectors \u0006kin each parity sector, the e\u000bective operator basis\nper cluster can be reduced even further, from 3 operators to one conjugate pair of\noperators. This can be seen by rewriting (6.4) as follows:\nf(r) =Ar\u0000\r0\"\neikr \nf+ 0\n0ei\u0019rf\u0000!\n+ c:c:#\n+Br\u0000\r0\n0; (6.5)\nwith the matrices f+andf\u0000de\fned as\nf+=1\n20\nB@1\u0000i1\ni1i\n1\u0000i11\nCA; f\u0000=1\n20\nB@1 1\u0000i\n1 1\u0000i\ni i 11\nCA: (6.6)Numerical results 23\nTable 2. Comparison of the 1P operator sets on cluster Afor a \flling of \u0017= 0:286,\nusing the same conventions as for table 1.\nnumber of wRall;\u0016=wRall;1ORallRshortORallRintORallRlong\noperators (short) (intermediate) (long)\n4 1 4 4 4\n8 0.297162 8 8 8\n12 0.014661 12 12 12\n16 0.000402 16 16 16\n20 0.000001 19.97 19.95 19.31\nNote that both f+andf\u0000are matrices of rank one with eigenvalues3\n2, 0 and 0. The\neigenvectors with eigenvalue3\n2are1p\n3(1;i;1) and1p\n3(1;1;i), respectively. Thus, by\ntransforming to an operator basis in which f\u0006is diagonal, one \fnds that in both the\neven and the odd sector, the dominant correlations are actually carried by only a pair of\noperators, namely1p\n3(^O1+i^O2+^O3) and its hermitian conjugate, and1p\n3(^O4+^O5+i^O6)\nand its hermitian conjugate, respectively. This result, whose precise form could hardly\nhave been anticipated a priori, is a pleasing illustration of the power of a CDM analysis\nto uncover nontrivial correlations.\n6.2. One-particle correlations\nThe correlations in the 1P sector are exponentially decaying, as already mentioned in\nsection 5.3. The reason for this was given in [1] and is the key to understanding the\noperators and correlations in this sector. In the limit where the fermions are all paired,\nthe only possible way to annihilate one at xand create one at x0> x , such that the\ninitial and \fnal states are both paired, is that every rung in the interval ( x;x0) has a\nfermion (necessarily on alternating legs). These fermions can be grouped as pairs in\ntwo di\u000berent ways: ( x;x+ 1), (x+ 2;x+ 3), . . . , ( x0\u00002;x0\u00001) in the initial state,\nbut (x+ 1;x+ 2), . . . , (x0\u00001;x0) in the \fnal state. (Notice this requires that xandx0\nhave the same parity.) [1] showed that the probability of such a run of \flled sites decays\nexponentially with its length.\nApplying the operator analysis in this sector using the eigenvalue decomposition in\n(4.6) gives a series of fourfold degenerate eigenvalues for both clusters, see table 2 for\nclusterA. The table for cluster Bis exactly the same. For a speci\fc eigenvalue, also\nthe operators for cluster B(residing at rungs ( x0;x0+ 1)) are the same as for cluster A\n(residing at rungs ( x;x+ 1)), but with mirrored rungs, i.e. an operator acting on rungs\n(x;x+ 1) acts in the same fashion on rungs ( x0+ 1;x0).\nLooking more closely, the \frst four operators annihilate or create a particle on\nrungsx+ 1 orx0, respectively, thereby breaking or regrouping bound pairs residing on\n(x+ 1;x+ 2) or (x0\u00001;x0), respectively. The second set of four operators annihilates\nor creates a particle on rungs xorx0+ 1, thereby breaking or regrouping bound pairs\nresiding on rungs ( x;x+ 1) or (x0;x0+ 1). For a given odd separation x0\u0000x, theNumerical results 24\n(a)\n(b)\n(c)\nFigure 8. Three con\fgurations of bound pairs contributing to 1P correlations for a\ndistance (a) r= 2 and (b),(c) r= 3. Clusters AandBare depicted by the green\nand red squares, respectively. Fermions are depicted by black circles, empty lattice\npositions by white circles and the position where a fermion will be created is depicted\nby concentric circles. The crosses show the center of mass of the bound pairs. In\ncon\fguration (a) we have a correlation between an operator corresponding to the\n\frst four eigenvalues and an operator corresponding to the second four eigenvalues\nin clusters AandB, respectively. In contrast, con\fguration (b) shows a correlation\nbetween operators corresponding to the largest eigenvalue only and con\fguration (c)\na correlation between operators corresponding to the second eigenvalue only.\ncombination of x+ 1 withx0requires the smallest number of pairs to be present in\nbetween the two clusters. The alternative combination is xwithx0+ 1, which requires\nan additional pair in between (see \fgure 8). We could estimate their weights since\nthe relative probability of an extra pair is the factor associated with increasing the\nseparation by two. Since the correlations decay roughly as \u001810\u0000r(see \fgure 10), we\npredict two orders of magnitude. Similarly, when x0\u0000xis even, we get at mixture of\nthe \frst and second four operators (see \fgure 8). This explains the di\u000berence in the\nweights of the two operator sets.\nThus, it turns out that for the 1P correlations a cluster size of one rung would\nalready have been large enough to reveal the dominant correlations. We will hence use\nas operator basis\n^OA;\u0006= 1x\n1p\n2(^c\";x+1\u0006^c#;x+1) (6.7 a)\n^OB;\u0006=1p\n2(^c\";x\u0006^c#;x)\n 1x+1; (6.7b)\ntogether with their hermitian conjugates. (The fact that our operator basis consists only\nof operators acting on a single rung implies that it would have been su\u000ecient to use\nsingle-rung clusters. However, for the sake of consistency with the rest of our analysis,\nwe retain two-rung clusters here, too.)\nThe f-matrix based on these four operators (per cluster) is diagonal with equal\nentries for a given distance r. Its Fourier transform (see \fgure 9) gives a result distinct\nfrom the Fourier transform for CD and 2P correlations. The dominant wave vectors are\nk=\u0006kFandk=\u0019\u0006kF, where the latter is the product of an oscillation with k=\u0019Numerical results 25\nk/π/bardbl˜f(k)/bardbl2(a)\n−1 0 1102104106\nk/π/bardbl˜f(k)/bardbl2(b)\n−1 0 1102104106\nFigure 9. Fourier transform of the f-matrix obtained similarly as \fgure 5, for 1P\ncorrelations based on the four operators per cluster for (a) a \flling of \u0017= 0:248 and\n(b) a \flling of \u0017= 0:286. We \fnd peaks at about k=\u0006kFandk=\u0006kF+\u0019(dashed\nblack lines).\nand an oscillation with k=\u0006kF. In total we have an oscillation in the correlations of\nthe form (1 + (\u00001)r)e\u0006ikFr, i.e. an oscillation with k=\u0006kF, and every second term\nbeing close to zero. The dominant wave vector k=\u0006kFi s consistent with the usual\nbehaviour of 1P Green's functions.\nThe reason for every second term being essentially zero is that the dominant hopping\nin the system, the correlated hopping, always changes the position of a particle by two\nrungs, so every second position is omitted. The small but \fnite value for hopping onto\nintermediate rungs is related to the \fnite tk=tc= 10\u00002that we use. It results in a second\noscillation at k=\u0006kFlocated at intermediate rungs, whose relative strength compared\nto the dominant one is about 10\u00002, which is consistent with the ratio tk=tcthat we used\n(see \fgure 10).\nWe \ft the one independent f-matrix element f\u0016;\u0016to an exponential decay of the form\nAe\u0000r=r1(see \fgure 10), but apart from this we were not able to \ft the exact functional\ndependence on r, especially the oscillations with k=\u0006kF. The reason for this is the\nexistence of two oscillations where one is zero on every second rung, and that the data\nrange for which reasonable 1P correlations are still present is too small and thus makes\nit susceptible to numerical noise. This can be seen already in the Fourier spectrum,\nwhere we \fnd relatively broad peaks, as a result of the in\ruence of the exponential\nenvelope and the relatively short distance range available.\n6.3. Two-particle correlations\nThe operator subspace for 2P (\u0001 N= 2), in a cluster including two rungs has the\ncomparatively small dimension of four due to the in\fnite nearest-neighbour repulsion\n(see \fgure 2). These are ^ c\";x^c#;x+1, ^c#;x^c\";x+1and their hermitian conjugates. In the\npresent case of dominating tc, these operators represent the creation- and annihilation-\noperators of bound pairs [2]. The operator analysis yields exactly the same fourNumerical results 26\n21020304010−4010−20100\nr1= 0.46\nA= 24 .0\nrfµ , µ(r)(a)\n21020304010−4010−20100\nr1= 0.54\nA= 10 .1\nrfµ , µ(r)(b)\n5101520−20020\nrfµ , µ(r)/Ae−r /r 1\n(c)\n5101520−25025\nrfµ , µ(r)/Ae−r /r 1\n(d)\nFigure 10. The 1P correlations for a \flling of (a),(c) \u0017= 0:248 and (b),(d)\n\u0017= 0:286. Panels (a) and (b) show the 1P correlations (blue symbols) together with\na \ft of the form Ae\u0000r=r1(red line). Panels (b) and (d) show the rescaled correlator\nf\u0016;\u0016(r)=Ae\u0000r=r1(blue symbols) for distances up to r= 20. (Larger distances are\nomitted, because for these f\u0016;\u0016(r)<10\u000016, which is the maximal computer precision.)\nWe see a strong oscillation (green curve) and a weak oscillation (red curve).\noperators with degenerate weight for all distance regimes for both cluster AandB. The\nfour operators are 1 =p\n2 (^c\";x^c#;x+1\u0006^c#;x^c\";x+1) together with their hermitian conjugates,\nand they already represent the symmetric and antisymmetric combinations of the\noperators mentioned above.\nThe f-matrix (4.7) is diagonal in the basis of the four operators, with equal strength\nof correlations for a \fxed distance apart from a possible sign. This may be expected,\ngiven the similar structure of the operators.\nAs for the CD correlations (\u0001 N= 0), we apply a Fourier transform on the f-matrix\n(see \fgure 11) to identify the dominant wave vectors. Again, we \fnd two spectra of\nsimilar form but shifted by \u0019with respect to each other. Consequently we rede\fne f+\ntoei\u0019rf+, the part of the f-matrix belonging to the symmetric operators. Thus, we\nobtain one leading peak at k= 0 and sub-leading peaks at k= 2kF. Given the similar\nstructure of the Fourier spectrum to that of the CD correlations, we \ft the elements of\nthe f-matrix to the form (6.3), but now expect \r0<\rfrom the relative sharpness of the\npeaks. Already at the level of the f-matrix elements we \fnd an overall leading decay\nwith residual oscillations, whose relative magnitude becomes smaller at large distances\n(since\r0<\r). Since all matrix elements are the same after rede\fning f+, it is su\u000ecient\nto \ftjf\u0016;\u0016jfor a given\u0016, which will have dominant k-vectorsk= 0 andk=\u00062kF. The\n\ft has errors of less than 5% throughout, with results as shown in \fgure 12. The overall\nbehaviour is very similar to the one already found from the r.m.s. net correlations ofNumerical results 27\nk/π/bardbl˜f(k)/bardbl2\n  (a)\n−1 0 110−2100102\n˜f+\n˜f−\nk/π/bardbl˜f(k)/bardbl2\n  (b)\n−1 0 110−410−2100102\n˜f+\n˜f−\nFigure 11. Fourier transform of the f-matrix for 2P correlations based on the operators\nchosen from the four-dimensional operator space for (a) a \flling of \u0017= 0:248 and (b)\na \flling of\u0017= 0:286. For a detailed description see \fgure 5.\n2 10 3010−310−210−1\nrfµ , µ(r)\n  \nν= 0 .248 , γ/prime= 0 .92 , γ= 2 .03\nν= 0 .286 , γ/prime= 1 .10 , γ= 2 .03\n  \nν= 0 .248 , γ/prime= 0 .92 , γ= 2 .03\nν= 0 .286 , γ/prime= 1 .10 , γ= 2 .03\nFigure 12. Fitting the 2P correlations to the form in (6.3) for a \flling of \u0017= 0:248\nand\u0017= 0:286. The single points (blue circles and squares) are data points from the\nf-matrix and the lines (red and green) are the result of the \ftting.\nthis sector (see \fgure 4), up to the oscillatory part from the second term in (6.3). We\nsee that the oscillations clearly decay more strongly than the actual strength jf\u0016;\u0016j, in\naccord with \r0<\r.\nIn contrast to the CD correlations (see \fgure 6.1.2), for the 2P correlations we do\nnot \fnd correlations which oscillate with phases shifted by \u0001 \u001e=\u0006\u0019=2 . This may come\nfrom the fact that clusters with the size of two rungs have the minimal possible size to\ncapture 2P correlations. The corresponding operator space has dimension four and the\nfour possible operators are very similar in structure. We expect that for larger clusters\nand hence a larger operator space, we would \fnd correlations which also oscillate out of\nphase such that their oscillations cancel in the r.m.s. net correlations , in accord with\n(1.4).Comparison 28\nTable 3. Comparison of the power-law exponents, which we extracted from our\nnumerical data, with those predicted in [2].\nCD 2P\n\flling [2] \r0 [2]\r2\n0.248 1.13 1.45 0.5 0.95\n0.286 1.04 1.33 0.5 1.11\n7. Comparison to previous results\nWe are now ready to compare our CDM-based results with those obtained in [2] by\nCheong and Henley (CH) from \ftting simple correlation functions. The latter were\ncomputed exactly in [2] for accessible separations after mapping the large tcmodel onto\na hard-core bosonic system, but the functional forms of the rdependencies were inferred\nfrom a purely numerical \ftting procedure.\nOverall, our results for the Hamiltonian (2.1) in the strongly correlated hopping\nregime agree with [2], in that (i) 2P correlations and CD correlations show power-law\nbehaviour, (ii) the 2P correlations dominate at large distances for the \fllings we were\ninvestigating, (iii) 1P correlations are exponentially decaying and are negligible over all\nbut very short distances, and (iv) the dominating k-vector, for either 2P or CD sectors,\nis 2kF.\nHowever, the power-law exponents obtained from \ftting f-matrix elements to (4.10)\nand summarized in table 3, clearly deviate from the results in [2] by CH. For the CD\ncorrelations, in [2] the dependence of \r0on the \flling \u0017was given by the exponent\n\rCH\n0=1\n2+5\n2\u00001\n2\u0000\u0017\u0001\n, from which our results deviate (see \fgure 4 a,b) by about 25%.\nNevertheless, our results for \r0agree qualitatively with this prediction, in that we also\n\fnd\r0to decrease linearly with increasing \flling.\nThe 2P correlations deviate more strongly. For the dominant 2P correlations, CH\npredicted a constant power-law exponent of \rCH\n2=1\n2independent of \flling, coming\nfrom a universal correlation exponent for a chain of tightly-bound spinless fermion pairs\n[8]. In contrast, we obtain a larger exponent (see \fgure 4 a,b) for given \fllings. Our\nresult for\r2linearly decreases as the \flling gets smaller and appears to approach1\n2\nonly in the limit \u0017!0. We also explicitly calculated the same correlation function\nas investigated in [2] but found a stronger decay than the r\u00001\n2suggested there. We do\nnot know whether the deviation is an artifact of the boundaries of our \fnite system,\nor whether the mapping used in [2] to a set of hardcore bosons might have omitted an\nimportant contribution.\nMoreover, it may be noted that by extrapolating the exponents in a linear fashion\ntowards large \fllings ( \u0017!1\n2), it appears that for \fllings larger than \u00180:35 eventually\nthe CD correlations dominate over 2P correlations (see \fgure 13). This conclusion\nhas also been found in [9] which similarly addresses diatomic real space pairing in the\ncontext of superconductivity. Their discussion, however, is not speci\fcally constrainedConclusions 29\n00.1 0.2 0.3 0.4 0.50.511.522.533.5\nνγ\n  γ0(N= 100)\nγ0(N= 150)\nγ0(N= 200)\nγ2(N= 100)\nγ2(N= 150)\nγ2(N= 200)\n  −2.96ν+ 2.23\n2ν+ 0.5\nγCH\n0=1\n2+5\n2(1\n2−ν)\nγCH\n2=1\n2\nFigure 13. The power-law exponents for CD correlations ( \r0, blue symbols) and 2P\ncorrelations ( \r2, red symbols) obtained from the r.m.s. net correlations for several\n\fllings\u0017. We used chain lengths of N= 100 (circles), N= 150 (crosses), and\nN= 200 (triangles). The dashed blue and red lines are linear \fts to our numerical\ndata for\r0and\r2, respectively. The solid blue and red lines show the corresponding\npredictions of Cheong and Henley [2]. For the 2P correlations, our data implies a\nlinear\u0017-dependence going from1\n2for\u0017= 0 to3\n2for\u0017=1\n2. This crossover from1\n2to\n3\n2is predicted by Cheong and Henley as a sub-leading contribution, without giving an\nexplicit functional dependence on \u0017. The two linear \u0017-dependencies imply that for large\n\fllings CD correlations should become dominant over 2P correlations. Unfortunately,\nwe do not have been able to obtain reliable data in that regime, because the r.m.s.\nnet correlations showed strong oscillations here, contrary to our expectations from\nsection 1.2.\nto one-dimensional systems, and one may wonder how the speci\fc choice of parameters\ncompare.\nAs the \flling approaches 0 :5 in an excluded-fermion chain, it is appropriate to think\nabout the degrees of freedom as impurity states or holes in the crystalline matrix of pairs\n[9]. Then the natural length scale is the spacing between holes. The longer that spacing\ngets (it diverges as \u0017!0:5), the larger also the system under investigation must be in\norder to reach the asymptotic limit. In other words, to see proper scaling behavior in a\nuniform way, the system size should increase proportional to 1 =(0:5\u0000\u0017). In our case the\ndata became unreliable for \u0017&0:4 (see \fgure 13). On the other hand, for certain \fllings\n\u0017.0:4, we calculated the power-law exponents for CD and 2P correlations for ladders\nof lengthN= 150 and N= 200 (this data is also included in \fgure 13) and did not\n\fnd di\u000berent behaviour compared to out original data for ladders of length N= 100.\n8. Conclusions\nSummarizing, we found that the CDM is a useful tool to detect dominant correlations\nin a quantum lattice system. Starting from a ground state calculated with DMRG, weVariational matrix product state approach 30\nextracted all the important correlations present in our model system. We developed\na method which, \frst, determines the distance-independent operators on each cluster\nthat carry the dominant correlations of the system, and second, encodes the distance-\ndependence of the correlations in the f-matrix. The latter is then analyzed in terms of\ndecaying and oscillatory terms to extract the long-range behaviour of the correlations.\nWe saw that the size of the clusters AandBis a limitation of the method as it\nconstrains the analysis to local operators. For some kind of correlations, however, larger\nclusters are needed to capture the relevant physics. This is not too easily implemented\nas it requires signi\fcantly more resources. As a possible alternative and as an outlook\nfor possible future work, one may think of using a di\u000berent cluster structure: one cluster\nas before and one \\super-cluster\" representing a larger continuous part of the system\nincluding one boundary. As MPS introduces, for each site, e\u000bective left and right\nHilbert spaces describing the part of the chain to the left and to the right of that site,\nthe description of such a super-cluster should be straightforward. The resulting e\u000bective\ndensity matrix describing a large part of the system can be calculated accordingly.\nOverall, DMRG is a suitable method to calculate the CDM. The latter is easily\nand e\u000eciently calculated within the framework of the MPS. The explicit breaking of (i)\ntranslational invariance by using \fnite system DMRG and (ii) a discrete symmetry of\nthe model, lead us to develop certain strategies to restore these broken symmetries. The\nsmoothing of the boundaries can still be further optimized, or be replaced by periodic\nboundary conditions. However, we do not expect that this will have signi\fcant in\ruence\non the conclusions drawn.\nAcknowledgments\nWe would like to thank S.-A. Cheong and A. L auchli for discussions and comments on\nthe manuscript. This work was supported by DFG (SFB 631, SFB-TR 12, De-730/4-\n1 and De-730/4-2), CENS (Center for NanoScience, LMU) and NIM (Nanosystems\nInitiative Munich). C. L. Henley acknowledges NSF grant DMR-0552461 for support.\nThis research was supported in part by the National Science Foundation under Grant\nNo. NSF PHY05-51164. J. von Delft acknowledges the hospitality of the Kavli Institute\nfor Theoretical Physics, UCSB, and of the Institute for Nuclear Theory, University of\nWashington, Seattle.\nA. The variational matrix product state approach\nThis appendix o\u000bers a tutorial introduction to the variational formulation of DMRG for\n\fnding the ground state of a one-dimensional quantum lattice model, , based on matrix\nproduct states (MPS). It also explains how this approach can be used to e\u000eciently\ncalculate the CDM. We point out all the important properties of the MPS and explain\nhow to perform basic quantum calculations such as evaluating scalar products and\nexpectation values, as well as determining the action of local operators on the MPS andVariational matrix product state approach 31\nconstructing a reduced density matrix. We explain how a given MPS can be optimized\nin an iterative fashion to \fnd an excellent approximation for the global ground state.\nWe also indicate brie\ry how the e\u000eciency of the method can be enhanced by using\nAbelian symmetries.\nWe would like to emphasize that we make no attempt below at a historical overview\nof the DMRG approach, or at a complete set of references, since numerous detailed\nexpositions of this approach already exist in the literature (see the excellent review by\nU. Schollw ock [4]). Our aim is much more modest, namely to describe the strategy\nimplemented in our code in enough detail to be understandable for interested non-\nexperts.\nA.1. Introduction\nQuantum many-body systems deal with very large Hilbert spaces even for relatively\nsmall system sizes. For example, a one- dimensional quantum chain of Nspin1\n2\nparticles forms a Hilbert space of dimension 2N, which is exponential in system size. For\nquantum lattice models in 1D a very e\u000ecient numerical method is the density matrix\nrenormalization group (DMRG), introduced by Steven R. White [3]. The problem of\nlarge Hilbert space dimension is avoided by an e\u000ecient description of the ground state,\nwhich discards those parts of the Hilbert space which have negligible weight in the\nground state. In this manner the state space dimension of the e\u000bective description\nbecomes tractable, and it has been shown that this produces excellent results in many\nquasi one-dimensional systems.\nThe algebraic structure of the ground state for one-dimensional systems calculated\nwith DMRG is described in terms of matrix product states (MPS) [10, 11, 12, 5, 13]. The\norigin of this MPS structure can be understood as follows (a detailed description will\nfollow later): pick any speci\fc site of the quantum lattice model, say site k, representing\na local degree of freedom whose possible values are labeled by an index \u001bk(e.g., for a\nchain of spinless fermions, \u001bk= 0 or 1 would represent an empty or occupied site). Any\nmany-body state j iof the full chain can be expressed in the form\nj i=X\nlkrk\u001bkA[\u001bk]\nlkrkjlkij\u001bkijrki; (A.1)\nwherejlkiandjrkiare sets of states (say NlandNrin number) describing the parts of\nthe chain to the left and right of current site k, respectively, and for each \u001bk,A[\u001bk]is a\nmatrix with matrix elements A[\u001bk]\nlkrkand dimension Nl\u0002Nr. Since such a description is\npossible for any site k, the statej ican be speci\fed in terms of the set of all matrices\nA[\u001bk], resulting in a matrix product state of the form\nj i=X\n\u001b1:::\u001bN\u0000\nA[\u001b1]:::A[\u001bN]\u0001\nl1rNj\u001b1i:::j\u001bNi: (A.2)\nOne may now seek to minimize the ground state energy within the space of all MPS,\ntreating the matrix elements of the A-matrices as variational parameters to minimize the\nexpectation value h jHj i. If this is done by sequentially stepping through all matricesVariational matrix product state approach 32\nin the MPS and optimizing one matrix at a time (while keeping the other matrices\n\fxed), the resulting procedure is equivalent to a strictly variational minimization of the\nground state energy within the space of all MPS of the form (A.2) [5, 10, 11, 12, 13]. If\ninstead the optimization is performed for two adjacent matrices at a time, the resulting\n(quasi-variational) procedure is equivalent to White's original formulation of DMRG\n[5, 10, 11, 12, 13]. The MPS based formulation of this strategy has proven to be very\nenlightening and fruitful, in particular also in conjunction with concepts from quantum\ninformation theory [5].\nIn general, such an approach works for both bosonic and fermionic systems.\nHowever, to be e\u000ecient the method needs a local Hilbert space with \fnite and small\ndimension, limiting its applicability to cases where the local Hilbert space is \fnite\ndimensional a priori (e.g. fermions or hard-core bosons) or e\u000bectively reduced to a \fnite\ndimension, e.g. by interactions. For example, such a reduction is possible if there is a\nlarge repulsion between bosons on the same site such that only a few states with small\noccupation number will actually take part in the ground state. For fermions, on the\nother hand, the fermionic sign must be properly taken care of. The anti-commutation\nrules of fermionic creation and annihilation operators causes the action of an operator\non a single site to be non-local because the occupations of the other sites have to be\naccounted for. To simplify the problem, a Jordan-Wigner transformation [14] can be\nused to transform fermionic creation and annihilation operators to new operators that\nobey bosonic commutation relations for any two operators referring to di\u000berent sites.\nThis greatly simpli\fes the numerical treatment of these operators as fermionic signs can\nbe (almost) ignored.\nBefore outlining in more detail the above-mentioned optimization scheme for\ndetermining the ground state (see section A.3), we present in section A.2 various\ntechnical ingredients needed when working with MPS.\nA.2. Matrix product states\nA.2.1. Construction of matrix product states We consider a chain with open boundary\nconditions consisting of Nequal sites with a local Hilbert space dimension of d. A state\nj iis described by\nj i=X\n\u001b1:::\u001bN \u001b1;:::;\u001bNj\u001b1i:::j\u001bNi; (A.3)\nwhere\u001bi= 1;:::;d labels the local basis states of site i. In general, the size of the\ncoe\u000ecient space  scales withO(dN). This can be rewritten in a matrix decomposition\nof the form (A.2) with a set of NtimesdmatricesA[\u001bk](see section A.2.3 for details).\nFormally, this decomposition has two open indices, namely the \frst index of A[\u001b1]and\nthe second index of A[\u001bN], asA[\u001b1]andA[\u001bN]are not multiplied onto a matrix to the left\nand to the right, respectively. For periodic boundary conditions these two indices would\nbe connected by a trace over the matrix decomposition, giving a scalar. In the case ofVariational matrix product state approach 33\nopen boundary conditions, the two indices range only over one value (see section A.2.3),\ni.e. the matrix decomposition is a 1 \u00021 matrix which is a scalar.\nIf theseA-matrices are su\u000eciently large this decomposition is formally exact, but\nsince that would require A-matrices of exponentially large size, such an exact description\nis of academic interest only. The reason why the A-matrices are introduced is that they\no\u000ber a very intuitive strategy for reducing the numerical resources needed to describe a\ngiven quantum state. This strategy involves limiting the dimensions of these matrices by\nsystematically using singular-value decomposition and retaining only the set of largest\nsingular values. The A-matrices can be chosen much smaller while still giving a very\ngood approximation of the state j i.\nSelecting a certain site k, the state can be rewritten in the form (A.1). The\ne\u000bective 'left' basis jlki=P\n\u001b1:::\u001bk\u00001A[\u001b1]:::A[\u001bk\u00001]j\u001b1i:::j\u001bk\u00001idescribes the sites j=\n1;:::;k\u00001, the e\u000bective 'right' basis jrkisimilarly describes the sites j=k+ 1;:::;N .\nSitekis called the current site, as the description of the state makes explicit only the\nA-matrix of this site (see \fgure A1).\nFigure A1. Current site with e\u000bective basis sets.\nSo far (A.3) and (A.1) are equivalent, but now we have a representation of the state\nwhich allows a convenient truncation of the total Hilbert space, used for the description\nof a MPS. For example, if we introduce a parameter Dand truncate all e\u000bective Hilbert\nspaces of all sites to the dimension D, eachA[\u001bk]-matrix has at most the dimension D\u0002D.\nThis reduces the resources used to describe a state from O(dN) for the full many-body\nHilbert space down to O(ND2d). This is linear in the system size, assuming that\nthe size required for Dto accurately describe the state grows signi\fcantly slower than\nlinearly inN. This, in fact, turns out to be the case for ground state calculations [15].\nDetails of this truncation procedure and estimates of the resulting error are described\nin section A.2.5.\nA.2.2. Global view and local view Matrix product states can be viewed in two\nalternative ways: a global view and a local view. Both views are equivalent and both\nhave their applications. In the global view the state is expressed as in (A.2), i.e. the\ne\u000bective Hilbert spaces have been used 'only' to reduce resources. The state is stored\nin theA-matrices, but the e\u000bective basis sets will be contracted out. This perception\nhas to be handled very careful, because contracting out the e\u000bective basis sets leads to\nhigher costs in resources! In the local view the state is expressed as in (A.1). It is called\nlocal because there is one special site, the current site, and all other sites are combined\nin e\u000bective orthonormalized basis sets. Usually, the local view is used iteratively forVariational matrix product state approach 34\nevery site. In this perception, we need e\u000bective descriptions of operators contributing\nto the Hamiltonian acting on other sites than the current site (see section A.2.8).\nA.2.3. Details of the A-matrices TheA-matrices have some useful properties that\nhold independently of the truncation scheme used to limit the e\u000bective Hilbert spaces.\nFirst of all, we notice that by construction dim(Hrk\u00001)\u0011dim(Hlk), otherwise the matrix\nproducts in (A.2) would be ill de\fned. Based on this, we can \fnd another interpretation\nof theA-matrices in the local view. The part of the chain to the left of site k(wherek\nis far from the ends for simplicity) is described by the e\u000bective basis jlki, which is built\nof truncated A-matrices:\njlki=X\n\u001b1;:::;\u001bk\u00001\u0000\nA[\u001b1]:::A[\u001bk\u00001]\u0001\n1lkj\u001b1i:::j\u001bk\u00001i\n=X\n\u001bk\u00001X\nlk\u00001X\n\u001b1;:::;\u001bk\u00002\u0000\nA[\u001b1]:::A[\u001bk\u00002]\u0001\n1lk\u00001j\u001b1i:::j\u001bk\u00002i\n| {z }\njlk\u00001iA[\u001bk\u00001]\nlk\u00001;lkj\u001bk\u00001i\n=X\n\u001bk\u00001;lk\u00001A[\u001bk\u00001]\nlk\u00001lkjlk\u00001ij\u001bk\u00001i: (A.4)\nTheA[\u001bk\u00001]-matrix maps the e\u000bective left basis jlk\u00001itogether with the local j\u001bk\u00001ibasis\nonto the e\u000bective left basis jlki! The same argument applied on the e\u000bective right basis\nof sitekleads to the transformation of jrk+1iandj\u001bk+1iontojrkivia theA[\u001bk+1]-matrix:\njrki=X\n\u001bk+1;rk+1A[\u001bk+1]\nrkrk+1j\u001bk+1ijrk+1i: (A.5)\nSo far, this may be any transformation, but in order to deal with properly orthonormal\nbasis sets, we may impose unitarity on the transformation (see below).\nTheA-matrices towards the ends of the chain have to be discussed separately. The\nuse of open boundary conditions implies that we have a 1-dimensional e\u000bective state\nspace to the left of site one and the right of site N, respectively, both representing the\nempty state. This implies that dim( Hl1) = 1 = dim(HrN). Moving inwards from the\nends of the chain, the e\u000bective Hilbert spaces acquire dimension d1;d2;:::until they\nbecome larger than Dand need to be truncated. Correspondingly, the dimension of\nmatrixA[\u001bk]isDk\u00001\u0002Dk, whereDk= min(dk;dN\u0000k;D). There is no truncation needed\nif dim(Hlk)\u0003d= dim(Hrk) or dim(Hrk)\u0003d= dim(Hlk). In these cases we simply choose\nA(lk\u001bk)rk= 1andAlk(rk\u001bk)= 1, respectively.\nSummarizing, the A-matrices have two functions. If site iis the current site in (A.1),\ntheA[\u001bi]-matrices represent the state, i.e. its coe\u000ecients specify the linear combination\nof basis statesjlki,j\u001bkiandjrki. On the other hand, if not the current site, the A-\nmatrices are used as a mapping to build the e\u000bective orthonormal basis for the current\nsite, as we describe next:\nOrthonormal basis sets In the local view, the whole system is described by the A-\nmatrices of the current site kin the e\u000bective left basis, the e\u000bective right basis, and theVariational matrix product state approach 35\nlocal basis of site k. A priori, the basis states form an orthonormal set only for the local\nbasis set, but we may ask for the e\u000bective basis sets jliandjrizto be orthonormal,\ntoo, i.e. require them to obey:\nhl0jli=\u000el0l;\nhr0jri=\u000er0r: (A.6)\nThis immediately implies the following condition on the A[\u001bj]-matrices, using (A.4) and\n(A.5) (for a derivation, see section A.5.1):\nX\n\u001bjA[\u001bj]yA[\u001bj]= 1 forj <k;\nX\n\u001bjA[\u001bj]A[\u001bj]y= 1 forj >k: (A.7)\nThe orthonormality (A.6) for both the left- and right basis states holds only for the\ncurrent site. For the other sites there is always only one orthonormal e\u000bective basis.\nGraphical representation Matrix product states can be depicted in a convenient\ngraphical representation (see \fgure A2). In this representation, A-matrices are displayed\nas boxes and A[\u001bk]is replaced by Akfor brevity. Indices correspond to links from the\nboxes. The left link connects to the e\u000bective left basis, the right link to the right one,\nand the link at the bottom to the local basis. Sometimes indices are explicitly written on\nthe links to emphasize the structure of the sketch. Connected links denote a summation\nover the indices (also called contraction) of the corresponding A[\u001b]-matrices. At the\nboundaries of the chain, a cross is used to indicate the vacuum state.\n(a)\n(b)\nFigure A2. Graphical representation of a matrix product state in the (a) global view\nand (b) local view.\nA.2.4. Orthonormalization of e\u000bective basis states We now describe how an arbitrary\nMPS state can be rewritten into a form where its local view with respect to a given site\nhas orthonormal left- and right basis states. It should be emphasized that this really just\namounts to a reshu\u000fing of information among the state's A-matrices without changing\nthe state itself, by exploiting the freedom that we always can insert any X\u00001X= 1at\nany position in the matrix product state without altering it.\nzFrom now on the index kis only displayed when several sites are involved. For the current site or in\nthe case when only one A-matrix is considered the index will be dropped.Variational matrix product state approach 36\nAssume site kto be the current site and assume that it has an orthonormal left basis\n(the latter is automatically ful\flled for k= 1). We need a procedure to ensure that, when\nthe current site is switched to site k+1, this site, too, will have an orthonormal left basis.\n(This is required for the orthonormality properties used in the proof in section A.5.1. A\nsimilar procedure can be used to ensure that site k\u00001 has an orthonormal right basis\nprovidedkhas such a basis.) For this purpose we use the singular value decomposition\n(SVD, see section A.5.2) for which we have to rewrite A[\u001bk]\nlkrkbyfusing the indices lkand\n\u001bk:\nA[\u001bk]\nlkrkb=A(lk\u001bk)rk=X\nm;nu(lk\u001bk)msmn\u0000\nvy\u0001\nnrkb=X\nmu[\u001bk]\nlkm\u0000\nsvy\u0001\nmrk; (A.8)\nwherem,nandrkhave the same index range (see \fgure A3). Speci\fcally, uful\flls\n1=uyu=X\n(lk\u001bk)u\u0003\n(lk\u001bk);m0u(lk\u001bk);m; (A.9)\nwhich is equivalent to the orthonormality condition (A.7) for the A[\u001bk]-matrices.\nSVD\nFigure A3. Singular value decomposition of the A-matrices\nAsureplacesA[\u001bk]andsvyis contracted onto A[\u001bk+1], this leaves the overall state\nunchanged (for a graphical depiction see \fgure A4):\nA[\u001bk]A[\u001bk+1]=X\n(rk=lk+1)A[\u001bk]\nlkrkA[\u001bk+1]\nlk+1rk+1=X\n(rk=lk+1)X\nmu[\u001bk]\nlkm\u0000\nsvy\u0001\nmrkA[\u001bk+1]\nlk+1rk+1\n=u[\u001bk]\u0000\nsvyAk+1\u0001[\u001bk+1]\u0011~A[\u001bk]~A[\u001bk+1]: (A.10)\nSVD\nFigure A4. Rearrangement of the A-matrices to switch the current site from site k\ntok+ 1.\nSitek+1 now has an orthonormal e\u000bective left basis. A similar procedure works for\nthe e\u000bective right basis, see \fgure A5. To obtain an orthonormal e\u000bective left basis for\nFigure A5. Orthonormal e\u000bective right basis for site k\u00001.\nthe current site k, we start with the \frst site, update A[\u001b1]andA[\u001b2], move to the nextVariational matrix product state approach 37\nsite, update A[\u001b2]andA[\u001b3], and so on until site k\u00001. For an orthonormal e\u000bective right\nbasis, we start from site Nand apply an analogous procedure in the other direction.\nIf the statej iis in the local description of site kwith orthonormal basis sets\njlki,j\u001bkiandjrki, it is now very easy to change the current site to site k\u00061, with\ncorresponding new orthonormal basis sets jlk\u00061i,j\u001bk\u00061i,jrk\u00061i. Suppose we want to\nchange the current site from site kto sitek+ 1. Following the procedure described\nabove, site k+ 1 already has an orthonormal right basis and all sites left of site kful\fll\nthe orthonormality condition. All that is left to do, is to update site kandk+ 1 to\nobtain an orthonormal left basis for site k+ 1. This is called a switch of the current site\nfrom sitektok+ 1. The switch from site kto sitek\u00001 is done analogously.\nA.2.5. Hilbert space truncation A central ingredient in the variational optimization of\nthe ground state (see section A.3.1 below) is the truncation of the e\u000bective Hilbert spaces\nassociated with a given A-matrix. The strategy for truncating the e\u000bective Hilbert\nspaces is completely analogous to the original DMRG formulation [11]. The DMRG\ntruncation scheme is based on discarding that part of the Hilbert space on which a\ncertain density matrix has su\u000eciently small weight. There are two ways to obtain an\nappropriate reduced density matrix: two-site DMRG [3, 4] and one-site DMRG [4]. The\ncrucial di\u000berence between the two is that one-site DMRG is strictly variational in the\nsense that the energy is monotonically decreasing with each step,, whereas in two-site\nDMRG the energy may (slightly) increase in some steps, but with the advantage that\nthe cuto\u000b dimension can be chosen dynamically in each step.\nTwo-site DMRG Two-site DMRG arises when variationally optimizing two sites at a\ntime. We consider two current sites, say kandk+ 1, and we may choose the cuto\u000b\ndimension site-dependent: D!Dk\u0011dim(Hlk). Following section A.2.4, we assume\nsitekto have an orthonormal left basis and site k+ 1 to have an orthonormal right\nbasis. After contracting the indices connecting A[\u001bk]andA[\u001bk+1](see \fgure A6), the\nstate is described by A[\u001bk\u001bk+1]\nlkrk+1. In this description we may optimize the ground state\nlocally by variationally minimizing the ground state energy with respect to A[\u001bk\u001bk+1]\nlkrk+1\n(see section A.3.1). Afterwards, we need to decompose A[\u001bk\u001bk+1]\nlkrk+1intoA[\u001bk]andA[\u001bk+1]\nagain. This can be accomplished via singular value decomposition (see section A.5.2) by\nfusing the indices lk;\u001bk!(lk\u001bk) andrk+1;\u001bk+1!(rk+1\u001bk+1) (see \fgure A6) to obtain\nA[\u001bk\u001bk+1]\nlkrk+1=P\niu[\u001bk]\nlkisi\u0000\nvy\u0001[\u001bk+1]\nirk+1, wherei= 1:::min(dDk;dDk+2). Using the column\nunitarity of uand the row unitarity of vy(see section A.5.2), we rewrite the state as\nj i=X\nlkrk+1\u001bk\u001bk+1 X\niu[\u001bk]\nlkisi\u0000\nvy\u0001[\u001bk+1]\nirk+1!\njlkij\u001bkij\u001bk+1ijrk+1i\n=X\nisi X\nlk\u001bku[\u001bk]\nlkijlkij\u001bki!\n|{z}\nj~lii0\n@X\nrk+1\u001bk+1\u0000\nvy\u0001[\u001bk+1]\nirk+1j\u001bk+1ijrk+1i1\nA\n| {z }\nj~riiVariational matrix product state approach 38\n=X\nisij~liij~rii; (A.11)\nwhere the new set of basis states j~liiandj~riiis orthonormal with h~li0j~lii=\u000ei0iand\nh~ri0j~rii=\u000ei0i. This representation of the state may be seen as residing on the bond\nbetweenkandk+ 1, with e\u000bective orthonormal basis sets for the parts of the system to\nthe left and right of the bond. Reduced density matrices for these parts of the system,\nobtained by tracing out the respective complementary part, have the form:\n\u001a[L]=X\nis2\nij~liih~lij; \u001a[R]=X\nis2\nij~riih~rij: (A.12)\nThe standard DMRG truncation scheme amounts to truncating \u001a[L]and\u001a[R]according\nto their singular values si. We could either keep all singular values greater than a\ncertain cuto\u000b, thereby specifying a value for Dk+1between 1 and min ( dDk;dDk+2), or\nalternatively choose Dk=Dto be site-independent for simplicity. This step makes the\nmethod not strictly variational, since we discard some part of the Hilbert space which\ncould increase the energy. It turns out that this potential increase of energy is negligible\nin practice. We can obtain a measure for the information lost due to truncation by using\nthe von Neumann entropy S=\u0000tr (\u001aln\u001a), given by\n\"\u0011\u0000X\ni>Ds2\niln\u0000\ns2\ni\u0001\n; (A.13)\nwherePs2\ni= 1 due to the normalization of j i.\nSVD\nFigure A6. Procedure for site update within two-site DMRG. The grey line under\nthesindicates that sis the diagonal matrix of singular values.\nOne-site DMRG One-site DMRG arises when variationally optimizing one site at a\ntime. In contrast to two-site DMRG, one-site DMRG does not easily allow for dynamical\ntruncation during the calculation. (It is possible in principle to implement the latter,\nbut if one decides to use dynamical truncation, it would be advisable to do so using two-\nsite DMRG.) The truncation is \fxed by the initial choice of D, but it is still possible to\ndetermine an estimate on the error of this truncation by analyzing the reduced density\nmatrix. Starting from an expression for the full density matrix in the local view (current\nsitekwith orthonormal e\u000bective basis sets)\n\u001a=j ih j= X\nlr\u001bA[\u001b]\nlrjlij\u001bijri! X\nl0r0\u001b0A[\u001b0]\nl0r0\u0003hl0jh\u001b0jhr0j!\n=X\nlr\u001bl0r0\u001b0A[\u001b]\nlrA[\u001b0]\nl0r0\u0003jlihl0jj\u001bih\u001b0jjrihr0j; (A.14)Variational matrix product state approach 39\nwe trace out the e\u000bective right basis and obtain a reduced density matrix for the current\nsite and the left part of the system:\n\u001a[lk+1]=X\nlr\u001bl0\u001b0A[\u001b]\nlrA[\u001b0]\nl0r\u0003jlihl0jj\u001bih\u001b0j: (A.15)\nThis reduced density matrix carries the label lk+1because it corresponds precisely to\nthe density matrix jlk+1i\nl0\nk+1\f\f. So if we switch the current site from site kto sitek+ 1,\nwe can check the error of the truncation of Hlk+1. Fusing the indices land\u001b, we obtain\n\u001a[lk+1]=X\nlr\u001bl0\u001b0A(l\u001b)rA\u0003\n(l0\u001b0)rj(l\u001b)ih(l0\u001b0)j=X\nlr\u001bl0\u001b0A(l\u001b)r\u0000\nAy\u0001\nr(l0\u001b0)j(l\u001b)ih(l0\u001b0)j\n=X\nl\u001bl0\u001b0\u0000\nAAy\u0001\n(l\u001b)(l0\u001b0)j(l\u001b)ih(l0\u001b0)j: (A.16)\nWe do not need to diagonalize the coe\u000ecient matrix AAyto obtain the largest weights\nin the density matrix, because we get its eigenvalues as a byproduct of the following\nmanipulations anyway [4]. To switch the current site we need to apply a singular\nvalue decomposition (see section A.2.4) and obtain A=usvy(this is not the usual A-\nmatrix, but the index- fused form). This directly yields AAy=usvyvsuy=us2uy, which\ncorresponds to the diagonalization of \u001a[lk+1], implying that the weights of the density\nmatrix are equal to s2. Of course this works also for the right e\u000bective basis. With such\nan expression, we can check whether the e\u000bective Hilbert space dimension DofHlk+1is\ntoo small or not. For example, we could ask for the smallest singular value sDto be at\nleastnorders of magnitude smaller than the largest one s1, i.e. the respective weights\nin the density matrix would be 2 norders of magnitude apart. If the singular values do\nnot decrease that rapidly, we have to choose a greater D.\nA.2.6. Scalar product The scalar product of two states j iandj 0iis one of the\nsimplest operations we can perform with matrix product states. It is calculated\nmost conveniently in the global view because then we do not need to care about\northonormalization of the A-matrices:\nh 0j i=h\u001b0\n1j:::h\u001b0\nNjX\n\u001b0\n1:::\u001b0\nN\u0010\nA0[\u001b0\n1]:::A0[\u001b0\nN]\u0011\u0003X\n\u001b1:::\u001bN\u0000\nA[\u001b1]:::A[\u001bN]\u0001\nj\u001b1i:::j\u001bNi\n=X\n\u001b1:::\u001bN\u0000\nA0[\u001b1]:::A0[\u001bN]\u0001\u0003\u0000\nA[\u001b1]:::A[\u001bN]\u0001\n; (A.17)\nusing the orthonormality of the local basis h\u001b0\nkj\u001bli=\u000ekl\u000e\u001b0\nk\u001bk. In principle the order\nin which these contractions are carried out is irrelevant, but in practice it is possible\nto choose an order in which this summation over the full Hilbert space is carried out\nvery e\u000eciently by exploiting the one-dimensional structure of the matrix product state\n(see \fgure A7 for a graphical explanation). For details on the numerical costs, see\nsection A.5.3. In method (a), after contracting all A-matrices ofj iandj 0i, we have\nto perform a contraction over the full Hilbert space, i.e. a 1 \u0002dNmatrix is multiplied\nwith adN\u00021 matrix. This contraction is of order O\u0000\ndN\u0001\n, which is completely unfeasible\nfor practical purposes. In method (b) the most 'expensive' contraction is in the middleVariational matrix product state approach 40\nof the chain, say at site k, and it is of order O(dD3). Here the A-matrices are viewed\nas three-index objects Alkrk\u001bkwith dimension D\u0002D\u0002d. All sites left of site kare\nrepresented by a D\u0002Dmatrix, say Llk\nl0\nk. Contracting this with the matrix at site k\nyields the objectP\nlkLlk\nl0\nkAlkrk\u001bk, which has dimensions D\u0002D\u0002d, and since the sum\ncontainsDterms, the overall cost is O(dD3). Thus, in practice, method (b) is rather\ne\u000ecient and renders such calculations feasible in practice.\n1 2\n3(a) (b)\nFigure A7. Scalar product, computed in two di\u000berent orders. (a) First all A-matrices\nofj iandj 0iare contracted and then contraction over the local indices is carried\nout. b) First, for site one, we contract over the local indices of A1andA0\n1. Then we\ncontract over the e\u000bective index between A1andA2and afterwards over the indices\nbetween the resulting object and ( A0\n2)\u0003. Proceeding over the whole chain yields the\nscalar product.\nPartial product Sometimes it is required to calculate a product over only a part of the\nmatrix product state. This is done the same way as the scalar product\n\u0000\nP[Lk]\u0001\nlkl0\nk\u0011X\n\u001b1:::\u001bk\u00001\u0000\nA[\u001b1]:::A[\u001bk\u00001]\u0001\u0003\nl0\nk\u0000\nA[\u001b1]:::A[\u001bk\u00001]\u0001\nlk; (A.18)\n\u0000\nP[Rk]\u0001\nrkr0\nk\u0011X\n\u001bk+1:::\u001bN\u0000\nA[\u001bk+1]:::A[\u001bN]\u0001\u0003\nr0\nk\u0000\nA[\u001bk+1]:::A[\u001bN]\u0001\nrk; (A.19)\n\u0010\nP[kk0]\u0011\nrkr0\nk;lk0l0\nk0\u0011X\n\u001bk+1:::\u001bk0\u00001\u0000\nA[\u001bk+1]:::A[\u001bk0\u00001]\u0001\u0003\nr0\nkl0\nk0\u0000\nA[\u001bk+1]:::A[\u001bk0\u00001]\u0001\nrklk0:(A.20)\nNotice that P[Lk]andP[Rk]are matrices in the indices lkandrk, respectively (see\n\fgure A8). In fact, they correspond to the overlap matrices hl0\nkjlkiandhr0\nkjrki,\nrespectively.\nFigure A8. Partial products associated with site k.\nA.2.7. Reduced density matrix The pure density matrix given by the matrix product\nstatej iis de\fned as \u001a=j ih j. To describe only a part of the system, we need to\ncalculate the reduced density matrix. Let Ibe a set of sites and \u001bs=f\u001bk2Iga fusedVariational matrix product state approach 41\nindex for their local states. Tracing out all other sites with combined index \u001bb=f\u001bk=2Ig\nwe obtain\n\u001aI=X\n\u001b1:::\u001bN\u001b0\n1:::\u001b0\nN\u000e\u001bb\u001b0\nb\u0010\nA[\u001b0\n1]:::A[\u001b0\nN]\u0011\u0003\u0000\nA[\u001b1]:::A[\u001bN]\u0001\nj\u001bsih\u001b0\nsj:(A.21)\nThis is a completely general expression, but in the cases where I=fkgorI=fk;k0g\nit reduces to (see \fgure A9)\n\u001afkg=P[Lk]\u0010\nA[\u001bk]\nA[\u001b0\nk]\u0003\u0011\nP[Rk]j\u001bkih\u001b0\nkj; (A.22)\n\u001afkk0g=P[Lk]\u0010\nA[\u001bk]\nA[\u001b0\nk]\u0003\u0011\nP[kk0]\u0010\nA[\u001bk0]\nA[\u001b0\nk0]\u0003\u0011\nP[Rk0]j\u001bkij\u001bk0ih\u001b0\nkjh\u001b0\nk0j:(A.23)\nA similar strategy can be used to calculate the density matrices needed for the main\ntext, by contracting out the \u001bk's for all sites except those involved in the clusters A,B\norA[B. In fact, (A.23) gives ^ \u001aA[Bfor two clusters of size one at sites kandk0.\n(a) (b)\nFigure A9. Reduced density matrix (a) \u001afkgfor sitekand (b)\u001afkk0gfor siteskand\nk0, wherek<l<k0.\nA.2.8. Operators in an e\u000bective basis Letkbe the current site with orthonormal\ne\u000bective basis sets jlkiandjrki. Consider an operator B, which acts on the local basis\nof sitek\u00001 only, with matrix elements B\u001b0\nk\u00001\u001bk\u00001=\n\u001b0\nk\u00001\f\fBj\u001bk\u00001i. We call this the\n(k\u00001)-local-representation ofB. To represent Bin the e\u000bective left basis of site k,\ncalled thek-left-representation ofB, we use the transformation properties of A[\u001bk\u00001]\n(see \fgure A10),\nhl0\nkjBjlki=0\n@\nl0\nk\u00001\f\f\n\u001b0\nk\u00001\f\fX\nl0\nk\u00001\u001b0\nk\u00001A[\u001b0\nk\u00001]\nl0\nk\u00001l0\nk\u00031\nAB\u001b0\nk\u00001\u001bk\u000010\n@X\nlk\u00001\u001bk\u00001A[\u001bk\u00001]\nlk\u00001lkjlk\u00001ij\u001bk\u00001i1\nA\n=X\nlk\u00001\u001b0\nk\u00001\u001bk\u00001A[\u001b0\nk\u00001]\nlk\u00001l0\nk\u0003\nA[\u001bk\u00001]\nlk\u00001lkB\u001b0\nk\u00001\u001bk\u00001; (A.24)\nwhere the only condition to derive these results, was that site k\u00001 has an orthonormal\ne\u000bective left basis. Similarly, if the ( k\u00001)-left-representation of an operator Cis known,\nitsk-left-representation can be obtained via (see \fgure A10)\nhl0\nkjCjlki=X\nlk\u00001l0\nk\u00001\u001bk\u00001A[\u001bk\u00001]\nl0\nk\u00001l0\nk\u0003A[\u001bk\u00001]\nlk\u00001lkCl0\nk\u00001lk\u00001: (A.25)\nEquation (A.24) and (A.25) can be used iteratively to transcribe the i-local-\nrepresentation of Binto itsk-left-representation for any k > i (see \fgure A11). ThisVariational matrix product state approach 42\n(a)\n(b)\nFigure A10. Thek-left-representation of (a) the operator B, obtained from its\n(k\u00001)-local-representation and (b) the operator C, obtained from its ( k\u00001)-left-\nrepresentation.\nFigure A11. Iterative calculation of the k-left-description of an operator B, given in\nthei-local-description, by (A.24) and (A.25) for any k>i .\nreasoning also applies to the right site of site kand so it is possible to obtain a description\nof any local operator on any site.\nTo obtain a description of a pair of local operators acting on di\u000berent sites, we\nhave to transcribe them step by step. Let site kbe the current site with orthonormal\ne\u000bective basis sets and B;C two operators acting locally on site iandjrespectively\n(i < j < k ). First we obtain the j-left-representation of B, namelyBl0\njlj, as described\nabove. Then both operators are transformed together into the ( j+ 1)-left-representation\n(see \fgure A12),\n\nl0\nj+1\f\f(BC)jlj+1i=X\nljl0\nj\u001bj\u001b0\njA[\u001b0\nj]\nl0\njl0\nj+1\u0003\nA[\u001bj]\nljlj+1Bl0\njljC\u001b0\nj\u001bj; (A.26)\nwhich in turn can be transformed iteratively into the desired k-left-representation of the\noperatorsBandC.\nFigure A12. The (j+ 1)-left-representation of the operators C, given in the j-local-\nrepresentation, and B, given in the j-left-representation.\nA.2.9. Local operators acting on j iAny combination of operators can be calculated\ndirectly in the global view or in the local view via the e\u000bective descriptions introduced\nin the previous section.Variational matrix product state approach 43\nGlobal view The operators, known in the local basis of the site they are acting on, are\ncontracted directly with the corresponding A-matrix. For example, the formula for a\nnearest neighbour hopping term cy\nkck+1(see \fgure A13) reads as\ncy\nkck+1j i =X\n\u001b1:::\u001bN0\n@X\n\u001b0\nk\u0010\ncy\nk\u0011\n\u001b0\nk\u001bk1\nA0\n@X\n\u001b0\nk+1(ck+1)\u001b0\nk+1\u001bk+11\nA\u0000\nA[\u001b1]:::A[\u001bN]\u0001\nj\u001b1i:::j\u001bk\u00001ij\u001b0\nki\f\f\u001b0\nk+1\u000b\nj\u001bk+2i:::j\u001bNi: (A.27)\nLocal view Letkbe the current site with orthonormal e\u000bective basis sets. If we want\nto evaluate operators acting on other sites than the current site k, we need an e\u000bective\ndescription of these operators in one of the e\u000bective basis sets of site kto contract these\noperators with the A-matrix of the current site. For example, to calculate the action of\nthe nearest neighbour hopping term cy\nkck+1onj i=A[\u001bk]\nlrjlij\u001bkijri, we need ( cy\nk)\u001b0\nk\u001bk\nand (ck+1)r0rto obtain (see \fgure A13)\ncy\nkck+1j i=X\nr\u001bk0\n@X\n\u001b0\nk\u0010\ncy\nk\u0011\n\u001b0\nk\u001bk1\nA X\nr0(ck+1)r0r!\nA[\u001bk]\nlrjlij\u001b0\nkijr0i:(A.28)\n(a)\n(b)\nFigure A13. The nearest neighbour hopping term cy\nkck+1acting onj iin (a) the\nglobal view and (b) the local view.\nA.2.10. Expectation values Expectation values are merely the scalar product between\nthe state with itself including the action of an operator and can be easily worked out in\nboth the global and the local view (see \fgure A14). Since both methods are equivalent,\nthe local variant is much more e\u000ecient as it involves much less matrix multiplications.\nHowever, it requires careful orthonormalization of the remainder of the A-matrices. The\niterative scheme, introduced in section A.3, allows for that and works in the local picture.\n(a) (b)\nFigure A14. The expectation value of the nearest neighbour hopping cy\nkck+1in (a)\nthe global view and (b) the local view.Variational matrix product state approach 44\nA.3. Variational optimization scheme\nThe basic techniques introduced in the previous sections are the building blocks for\nDMRG sweeps, an iterative scheme to determine the ground state in the usual DMRG\nsense. This scheme starts at some site as current site, for example the \frst site where\ntruncation occurs, and minimizes the energy of j iwith respect to that site. Afterwards\nthe current site is shifted to the next site, and the energy of j iwith respect to that site\nis minimized. This is repeated until the last site where truncation occurs is reached and\nthe direction of the switches is reversed. When the starting site is reached again, one\nsweep has been \fnished (see \fgure A15). These sweeps are repeated until j iconverges.\nFigure A15. One complete sweep.\nA.3.1. Energy minimization of the current site In order to \fnd the ground state of the\nsystem we have to minimize the energy E=h jHj iof the matrix product state j i\nwith the constraint that the norm of j imust not change. Introducing \u0015as Lagrange\nmultiplier to ensure proper normalization, we arrive at the problem of determining\nmin\nj i(h jHj i\u0000\u0015h j i): (A.29)\nIn the sweeping procedure introduced above, the current site is changed from one site\nto the next and the energy is minimized in each local description. Thus, we need (A.29)\nin terms of the parameters of the current site. Let us describe how to do this for the\ncase of one-site DMRG, where the A-matrices are optimized one site at a time. (The\nprocedure for two-site DMRG is entirely analogous, except that it involves combining\nA-matrices of two neighboring sites by fusing their indices to obtain a combined two-site\nA-matrix, see section A.2.5.) Inserting (A.1) into (A.29) yields (see \fgure A16)\nmin\nA[\u001b] X\nlr\u001bl0r0\u001b0A[\u001b0]\nl0r0\u0003Hl0r0\u001b0lr\u001bA[\u001b]\nlr\u0000\u0015X\nlr\u001bA[\u001b]\nlr\u0003A[\u001b]\nlr!\n; (A.30)\nwhereHl0r0\u001b0lr\u001b=hl0jh\u001b0jhr0jHjlij\u001bijriis the Hamiltonian expressed in the two\northonormal e\u000bective basis sets and the local basis of the current site.\nThe multidimensional minimization problem (A.29) has been transformed to a local\nminimization problem where one A-matrix (or two) is optimized at a time and all others\nare kept constant. Such a procedure could, in principle, cause the system to get stuck\nin a local minimum in energy, but experience shows that the procedure works well [4],\nespecially in the presence of a gap.Variational matrix product state approach 45\nFigure A16. The minimization problem expressed in the current site.\nTo obtain a solution for (A.30), we di\u000berentiate the equation with respect to A[\u001b0]\nl0r0\u0003\n(this is possible because the Hilbert space has an hermitian scalar product) and obtain\n0 =X\nl0r0\u001b0Hl0r0\u001b0lr\u001bA[\u001b]\nlr\u0000\u0015A[\u001b0]\nl0r0: (A.31)\nThe matrix elements Hl0r0\u001b0lr\u001bmay be calculated easily using the techniques introduced\nin section A.2 (see section A.3.2 for details). Changing to matrix notation and replacing\n\u0015withE0in anticipation of its interpretation as an energy, we obtain an eigenvalue\nequation:\nHA[\u001b]\nlrjlij\u001bijri=E0A[\u001b]\nlrjlij\u001bijri: (A.32)\nThe minimization problem reduces to a local eigenvalue problem, which can be solved\nby standard techniques. The full Hilbert space of the current site has dimension dD2\nand may become large, but it is not necessary to determine the full spectrum of H,\nsince we are interested only in the ground state. The Lanczos algorithm is an e\u000bective\nalgorithm to achieve exactly that. The advantage of this algorithm is that we only have\nto compute Hj i, which saves much e\u000bort. The Lanczos algorithm produces as output\nthe ground state eigenvalue and eigenvector. The latter gives the desired optimized\nversion of the matrix A\u001b\nlr, which then has to be rewritten (with or without Hilbert space\ntruncation, as needed) into a form that satis\fes the orthonormality requirements of the\nleft and right basis sets, as described in section A.2.4.\nA.3.2. Sweeping details Before the actual sweeping may be started we have to set\nup an initial state, prepare a current site with orthonormal e\u000bective basis sets and\ncalculate e\u000bective descriptions of operators which are part of the Hamiltonian. After\nthis initialization we may determine the ground state with respect to this current site\nand shift the current site to the next site. That current site again has orthonormal\ne\u000bective basis sets due to the switching procedure introduced in section A.2.4, but we\nalso need e\u000bective representations of the operators acting in the Hamiltonian. At this\nstep the structure of the matrix product state saves much e\u000bort, as most of the needed\nrepresentations are already calculated.\nStructure of the Hamiltonian terms The Hamiltonian Hl0r0\u001b0lr\u001b, acting in the space\nspanned by the states jli,j\u001bi,jri, breaks up into several terms:\nHl0r0\u001b0lr\u001b = 1l0l\n(H\u000f)\u001b0\u001b\n 1r0r+ (HL)l0l\n 1\u001b0\u001b\n 1r0r+ 1l0l\n 1\u001b0\u001b\n(HR)r0rVariational matrix product state approach 46\n+ (HL\u000f)l0l\u001b0\u001b\n 1r0r+ 1l0l\n(H\u000fR)r0r\u001b0\u001b+ (HL\u000fR)l0lr0r\u001b0\u001b; (A.33)\nwhere the indices denote on which parts of the system the respective term acts on ( L\nandRindicate left and right of the current site, respectively, \u000findicates action on the\ncurrent site). Of course, the six terms of (A.33) depend on the current site k:H(k)\n\u000f,\nH(k)\nL,H(k)\nR,H(k)\nL\u000f,H(k)\n\u000fRandH(k)\nL\u000fR. The terms ( HL)l0land (HR)r0rcontain all terms which\ninvolve only sites k0<kandk0>k, respectively. The iterative structure of the method\ndirectly yields the following equalities:\nH(k+1)\nL =H(k)\nL+H(k)\nL\u000f+H(k)\n\u000f; (A.34)\nH(k\u00001)\nR =H(k)\n\u000f+H(k)\n\u000fR+H(k)\nR; (A.35)\nwhere the terms on the right hand side are meant to be expressed in the e\u000bective basis\nof the operator on the left hand site (see \fgure A17).\nFigure A17. Iterative calculation of the operator H(k+1)\nL . The sum over iindicates\nthatH(k)\nL\u000fhas the formP\niH(k)\nL;i\nH(k)\n\u000f;i, whereH(k)\nL;iacts only on sites k0<kandH(k)\ni;\u000f\nonly on site k. The calculation of H(k\u00001)\nR works analogously.\nInitialization First of all we need an initial matrix product state, which is most\nconveniently chosen to consist of identity transformations at the ends of the chain\n(see section A.2.3) and random A-matrices for the rest of the chain. We take the\n\frst site where Hilbert space truncation is applied as current site kand obtain an\northonormal e\u000bective right basis (the e\u000bective left basis is already orthonormal) using\nthe orthonormalization procedure introduced in section A.2.4 starting from site N.\nAdditionally it is convenient, while dealing with site N, to calculate and store the\noperatorH(N\u00001)\nR (see (A.35)) and the e\u000bective description of all operators of site N\nwhich contribute to H(k)\n\u000fRandH(k)\nL\u000fRin the e\u000bective right basis of site N\u00001 (see\nsection A.2.8). This ensures, when the sweeping procedure reaches site N\u00001, that\nall necessary operators are already calculated. This is repeated from site Ndown to\nsitek+ 1, and similarly for the sites k0<kin the other direction. The result of these\ninitialization steps is that we have a current site kwith orthonormal e\u000bective basis sets,\ne\u000bective descriptions of the Hamiltonian terms H(k)\nLandH(k)\nRand e\u000bective descriptions\nof all operators contributing to H(k)\nL\u000f,H(k)\n\u000fRandH(k)\nL\u000fR. Moreover, with an appropriate\nextension to the switching procedure of section A.2.4, all e\u000bective descriptions for other\ncurrent sites are available for use when needed in future sweeping steps.Variational matrix product state approach 47\nExtended switching procedure The switching procedure of section A.2.4 is applied as\nbefore. Additionally, depending on the direction of the switch, H(k+1)\nL orH(k\u00001)\nR are\ncalculated and stored as well as the operators needed for the Hamiltonian (A.33). This\nextended switching ensures that for the new current site all required operators are\ncalculated, if they had been for the old current site.\nComplete ground state calculation The methods introduced above make the procedure\nto determine the ground state very e\u000ecient as the global problem is mapped onto\nmany local problems involving only a few terms to calculate. The iterative structure\nof the matrix product states and the e\u000bective Hamiltonian terms strongly increase the\ne\u000eciency. A full ground state calculation consists of:\n(i) Initialization as described above\n(ii) Full sweeps from site Kto siteK0and back to site K, with sites KandK0the \frst\nand last site where the e\u000bective Hilbert spaces are truncated.\n(iii) After each sweep ithe overlaph i\u00001j iibetween the state before and after the\nsweep is calculated. If the matrix product state does not change any more, stop\nthe sweeping. A criterion, for example, for when to stop would be to require that\njh i\u00001j ii\u0000h i\u00002j i\u00001ij\njh i\u00001j iij\u0014\u000f; (A.36)\nwhere\u000fis a small control parameter, typically of order 10\u000010.\nNumerical costs The step with the most impact on the numerical costs of the algorithm\nis the calculation of Hj iin the Lanczos method. This method is an iterative scheme\nusing several Lanczos steps , of which usually less than 100 are needed for one ground\nstate calculation. Each Lanczos step calculates Hj iexactly once. This calculation\nbasically consists of elementary matrix multiplications, see section A.5.3 for details on\nthe numerical costs of such calculations. The six terms introduced in (A.33) are not all\nequally time consuming. Most of them contain identity maps which do not need to be\ncarried out and thus the term HL\u000fRis the most time consuming, requiring operations\nof orderO(dD2(2D+d)). The total numerical cost for the minimization process is\nC=NSweep\u00022N\u0002NLanczos\u0002\u0000\ndD2(2D+d)\u0001\n; (A.37)\nwhereNSweep is the number of sweeps, Nthe chain length and NLanczos the number of\nLanczos steps. In practice the cuto\u000b dimension is signi\fcantly higher than the local\nHilbert space dimension dand thus (A.37) is nearly linear in d.\nA.4. Abelian symmetries\nMatrix product states can be easily adapted to properly account for conserved quantum\nnumbers, representing the global symmetries of the Hamiltonian. We will limit ourselves\nto Abelian symmetries, meaning that the irreducible representation of the symmetryVariational matrix product state approach 48\ngroup is Abelian, as these are easily implemented, which is not necessarily the case for\nnon-Abelian symmetries [16].\nAn Abelian symmetry allows a quantum number Qto be attached to every state.\nThe property that the symmetry is Abelian manifests itself in that this quantum number\nis strictly additive. For two states jQ1iandjQ2i, the quantum number of the direct\nproduct of these two states is given by jQ1i\njQ2i=jQ1+Q2i. For example, if the\nHamiltonian commutes with the number operator for the full system, the quantum\nnumberQcould represent particle number.\nFor matrix product states, the introduction of Abelian symmetries has the\nconsequence that the A-matrixA[\u001b]\nlrmay be written as ( AQ\u001b\nQlQr)\r\u001b\n\u000bl\fr. HereQ\u001b,Ql,Qr\nare the quantum numbers attached to the local, left e\u000bective and right e\u000bective basis,\nrespectively. The index \u000bldistinguishes di\u000berent states jQl;\u000blicharacterized by the\nsame quantum number Ql, and similarly for jQr;\friandjQ\u001b;\r\u001bi. IfAdescribes, for\nexample, the mapping of the jli-basis of the left block together with the local basis to\na combined (truncated) jri-basis, then the only non-zero blocks of the A-matrix are\nthose for which Q\u001b+Ql=Qr. For the current site, the total symmetry Qtotof the full\nquantum many-body state manifests itself in that the corresponding A-matrix ful\flls\nQl+Qr+Q\u001b=Qtot.\nFor the handling of matrix product states quantum numbers imply a signi\fcant\namount of bookkeeping, i.e. for every coe\u000ecient block we have to store its quantum\nnumber. The bene\ft is that we can deal with large e\u000bective state spaces at reasonable\nnumerical cost. The Lanczos algorithm, in particular, takes advantage of the block\nstructure.\nOf course, the treatment of Abelian symmetries is generic and not limited to\nonly one symmetry. We may incorporate as many symmetries as exist for a given\nHamiltonian, by writing Qas a vector of the corresponding quantum numbers.\nA.5. Additional details\nA.5.1. Derivation of the orthonormality condition The orthonormality condition (A.7)\nis easily derived by induction. The starting point is condition (A.6) and we limit to the\nderivation for the left basis. The derivation for the right basis is analogous.\nThe induction argument can be initialized with site k= 1 because its e\u000bective left\nbasis is already orthonormal as it consists only of the vacuum state. Now, consider the\ncase that site khas an orthonormal e\u000bective left basis and construct the condition for\nsitek+ 1 to have an orthonormal e\u000bective left basis:\n\nl0\nk+1\f\flk+1\u000b\n=0\n@X\nl0\nk\u001b0\nkhl0\nkjh\u001b0\nkjA[\u001b0\nk]\nl0\nkl0\nk+1\u00031\nA X\nlk\u001bkA[\u001bk]\nlklk+1jlkij\u001bki!\n=X\nl0\nklk\u001b0\nk\u001bkA[\u001b0\nk]\nl0\nkl0\nk+1\u0003\nA[\u001bk]\nlklk+1hl0\nkjlki|{z}\n\u000el0\nklkh\u001b0\nkj\u001bki|{z}\n\u000e\u001b0\nk\u001bk=X\nlk\u001bkA[\u001bk]\nlkl0\nk+1\u0003A[\u001bk]\nlklk+1Variational matrix product state approach 49\n= X\n\u001bkA[\u001bk]yA[\u001bk]!\nl0\nk+1lk+1: (A.38)\nCondition (A.7) follows with\nl0\nk+1\f\flk+1\u000b!=\u000el0\nk+1lk+1.\nA.5.2. Singular value decomposition The singular value decomposition can be seen as\na generalization of the spectral theorem, i.e. of the eigenvalue decomposition. It is valid\nfor any real or complex m\u0002nrectangular matrix. Let Mbe such a matrix, then it can\nbe written in a singular value decomposition\nM=USVy; (A.39)\nwhereUis am\u0002munitary matrix, Sam\u0002nmatrix with real, nonnegative entries on\nthe diagonal and zeros o\u000b the diagonal, and V a n\u0002nunitary matrix. The numbers on\nthe diagonal of Sare called singular values , and there are p=min(n;m) of them. The\nsingular values are unique, but UandVare not, in general. It is convenient to truncate\nand reorder these matrices in such a fashion that their dimension are m\u0002pforU,p\u0002p\nforS(with the singular values ordered in a non-increasing fashion) and n\u0002pforV(i.e.\np\u0002nforVy). A consequence of this truncation is that UorVis no longer quadratic and\nunitarity is not de\fned for such matrices. This property is replaced by column unitarity\n(orthonormal columns) of Uand row unitarity (orthonormal rows) for Vy- no matter\nwhich one is no longer quadratic. In this article all singular value decompositions are\nunderstood to be ordered in this fashion.\nA.5.3. Numerical costs of index contractions The numerical costs of matrix\nmultiplications and index contractions of multi-index objects depend on the dimension\nof both the resulting object and of the contracted indices. In the case of matrix\nmultiplications this is quite simple. Consider a n\u0002mmatrixM1multiplied by a m\u0002p\nmatrixM2. The result is a n\u0002pmatrixM:\nMij=mX\nk=1(M1)ik(M2)kj: (A.40)\nEvidently, each of the n\u0003pmatrix elements Mijrequires a sum over mproducts of the\nform (M1)ik(M2)kj. Thus the process for calculating M1M2is of orderO(nmp).\nThe numerical costs of multi-index objects are obtained analogously. Consider two\nmulti-index objects, M1with indices i1;:::;inand dimensions p1\u0002:::\u0002pnandM2with\nindicesj1;:::;jmand dimensions q1\u0002:::\u0002qm. If we contract the indices i1andi2of\nM1with the indices j1andj2ofM2(assuming that p1=q1andp2=q2), we obtain the\nmulti-index object M:\nMi3:::inj3:::jm=p1X\nk=1p2X\nl=1(M1)kli3:::in(M2)klj3:::jm: (A.41)\nThus for every entry of M,p1timesp2multiplications have to be done, so that the\nprocess is of order O((p3:::pn) (p1p2) (q3:::qm)).Variational matrix product state approach 50\nReferences\n[1] Cheong S A and Henley C L 2009 Phys. Rev. B79212402\n[2] Cheong S A and Henley C L 2009 Exact ground states and correlation functions of chain and\nladder models of interacting hardcore bosons or spinless fermions Preprint cond-mat/0907.4228\n[3] White S R 1992 Phys. Rev. Lett. 692863 { 66\n[4] Schollw ock U 2005 Rev. Mod. Phys. 77259 { 315\n[5] Verstraete F, Porras D and Cirac J I 2004 Phys. Rev. Lett. 93227205\n[6] Wilson K G 1975 Rev. Mod. Phys. 47773 { 840\n[7] Veki\u0013 c M and White S R 1993 Phys. Rev. Lett. 714283 { 86\n[8] Efetov K B and Larkin A I 1976 Sov. Phys. JETP 42390 { 96\n[9] Micnas R, Ranninger J and Robaszkiewicz S 1988 J. Phys. Colloques 49C8-2221 { 26\n[10] Ostlund S and Rommer S 1995 Phys. Rev. Lett. 753537 { 40\n[11] Dukelsky J, Martin-Delgado M A, Nishino T and Sierra G 1998 Europhys. Lett. 43457 { 62\n[12] Fannes M, Nachtergaele B and Werner R F 1992 Comm. Math. Phys. 144443 { 90\n[13] Takasaki H, Hikihara T and Nishino T 1999 J. Phys. Soc. Jpn. 681537 { 40\n[14] Jordan P and Wigner E 1928 Zeitschrift f ur Physik 47631 { 51\n[15] Verstraete F and Cirac J I 2006 Phys. Rev. B73094423\n[16] McCulloch I P and Gul\u0013 acsi M 2002 Europhys. Lett. 57852 { 8"    },    {        "title": "0910.1430v1.State_price_density_estimation_via_nonparametric_mixtures.pdf",        "content": "arXiv:0910.1430v1  [q-fin.CP]  8 Oct 2009The Annals of Applied Statistics\n2009, Vol. 3, No. 3, 963–984\nDOI:10.1214/09-AOAS246\nc/circlecopyrtInstitute of Mathematical Statistics , 2009\nSTATE PRICE DENSITY ESTIMATION VIA\nNONPARAMETRIC MIXTURES1\nBy Ming Yuan\nGeorgia Institute of Technology\nWe consider nonparametric estimation of the state price den sity\nencapsulated in option prices. Unlike usual density estima tion prob-\nlems, we only observe option prices and their corresponding strike\nprices rather than samples from the state price density. We p ropose\nto model the state price density directly with a nonparametr ic mix-\nture and estimate it using least squares. We show that althou gh the\nminimization is taken over an infinitely dimensional functi on space,\nthe minimizer always admits a finite dimensional representa tion and\ncan be computed efficiently. We also prove that the proposed es ti-\nmate of the state price density function converges to the tru th at a\n“nearly parametric” rate.\n1. Introduction. In this paper we consider estimating the risk-neutral\ndistribution encapsulated in option prices. Risk-neutral distributions, often\ncharacterized by state price densities, recovered from opt ion prices reflect\ninvestors’ expectation toward the future returns of the und erlying assets. It\nmanifests the preferences and risk aversion of a representa tive agent [A¨ ıt-\nSahalia and Lo ( 2000); Jackwerth ( 2000); Rosenberg and Engle ( 2002)].\nConsider, for example, a European call option with maturity dateTand\nstrike price X. Under the no-arbitrage principle, its price at tcan be given\nas\nC(X,St,rt,τ,τ)=e−rt,ττ/integraldisplay∞\n0ψ(ST)f(ST)dST, (1.1)\nwhereτ=T−tis the time to maturity, rt,τis the interest rate, ψ(ST)=\nmax{ST−X,0}is the payoff function, and fis the state price density. For\nbrevity, we leave implicit the dependence of fon the horizon as well as other\nReceived May 2008; revised December 2008.\n1Supported in part by NSF Grants DMS-0624841 and DMS-0706724 .\nKey words and phrases. Black–Scholes equation, European call options, nonparame tric\nmixture, state price density.\nThis is an electronic reprint of the original article published by the\nInstitute of Mathematical Statistics inThe Annals of Applied Statistics ,\n2009, Vol. 3, No. 3, 963–984 . This reprint differs from the original in pagination\nand typographic detail.\n12 M. YUAN\neconomic variables such as the current asset price St, the interest rate and\nthe dividend yield over the period.\nThe renowned Black–Scholes model assumes that the underlyi ng asset\nprice process {St}follows a geometric Brownian motion and, therefore, the\nrisk-neutral distribution is a log-normal distribution. D espite its elegance\nand popularity, it is now well understood that the log-norma l assumption\nmade by the Black–Scholes model can be problematic in practi ce and may\nresult in severe bias of option prices. A number of econometr ic models have\nbeen developed to address this issue. Most notable examples include the\nstochastic volatility model and the GARCH model. The reader s are referred\nto Garcia, Ghysels and Renault ( 2009) for a survey of recent developments\nin this direction. Although useful in a variety of contexts, these parametric\nmodels are still susceptible to model misspecification.\nVarious nonparametric methods have been employed to overco me this\nproblem. Derman and Kani ( 1994), Dupire ( 1994) and Rubinstein ( 1994)\npropose implied binomial tree techniques to recover the sta te price density\nfrom a set of option prices without assuming the log-normali ty. Buchen\nand Kelly ( 1996) and Stutzer ( 1996) reconstruct the state price density\nunder the maximum entropy principle. Jackwerth and Rubinst ein (1996)\nintroduce a smoothness penalized estimate. However, littl e is known about\nthe econometric properties of these methods.\nThe state price density estimation is closely related to the recovery of the\noption pricing function Citself. As observed by Banz and Miller ( 1978) and\nBreeden and Litzenberger ( 1978),\nf(ST)=ert,ττ∂2C\n∂X2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nX=ST. (1.2)\nTaking advantage of this relationship, the state price dens ity can be derived\nas the second derivative of an estimate of the pricing functi onC. In the\npresence of pricing error, the estimation of the pricing fun ctionCcan be\ncast as a regression problem:\nC=C(X)+ε, (1.3)\nwhere, with slight abuse of notation, we use Cto denote the observed option\nprice andC(·) to denote the correct pricing as a function of the strike pri ce,\nandεrepresents the pricing error. Various nonparametric regre ssion tech-\nniques have been applied to estimate C(·). In one of the pioneering papers,\nHutchinson, Lo and Poggio ( 1994) consider estimating Cnonparametrically\nusing various learning networks. More recently, A¨ ıt-Saha lia and Lo ( 1998)\nintroduce a semiparametric alternative where the volatili ty of the Black–\nScholes formulation is modeled nonparametrically. The rea ders are referredSTATE PRICE DENSITY ESTIMATION 3\nto Ghysels et al. ( 1997) and Fan ( 2005) for recent reviews of other nonpara-\nmetric methods for estimating the option pricing function o r the state price\ndensity.\nFrom a statistical point of view, estimating the state price density now\nbecomes estimating the second derivative of a regression fu nction. But un-\nlike other regression problems, the state price density nee ds to be a proper\ndensity function that is non-negative and integrates to the unity. This dic-\ntates, for example, that the price function C(·) is monotonically decreasing\nand convex in terms of the strike price X. More precisely,\n−ert,ττ≤C′(X)≤0,\nC′′(X)≥0.\nHow to impose these constraints presents the main difficultie s in nonpara-\nmetric regression ( 1.3). A¨ ıt-Sahalia and Duarte ( 2003) and Yatchew and\nH¨ ardle (2005) stress the importance of enforcing such shape constraints in\nestimatingtheoptionpricingfunctionandproposeanonpar ametricestimate\nofCthat respects these constraints. It is shown that both appro aches lead\nto improved accuracy in recovering the pricing function and can guarantee\nthe non-negativity of the state price density estimate. Nei ther state price\ndensity estimate, however, is guaranteed to integrate to on e as required by\na proper density. A post-estimate normalization is only nec essary to ensure\nsuch constraint.\nIn this paper we develop a new approach to nonparametric esti mation\nof the state price density. We propose to estimate the regres sion functions\nby minimizing the (weighted) least squares over a set of admi ssible pric-\ning functions. The admissible pricing function is deducted directly from\nthe very existence of a state price density. We consider a par ticular ad-\nmissible set of pricing functions whose corresponding stat e price density is\na nonparametric mixture of log-normals. We show that even th ough the\nminimization is taken over a infinite dimensional space, the minimizer ac-\ntually admits a finite dimensional representation. In parti cular, all solutions\ncan be expressed as a convex combination of at most n+1 Black–Schole\ntype of pricing functions. In addition, we prove that, by foc using on the\nset of admissible pricing functions, not only the estimated state price den-\nsity can be ensured to be a legitimate density function, but a lso the esti-\nmation accuracy can be drastically improved. More specifica lly, we show\nthat as the sample size nincreases, the pricing function can be recov-\nered with squared error converging to zero at the rate of ln2n/n, which\nis very close to the 1 /nconvergence rate that is typically achieved only with\nmuch more restrictive parametric assumptions such as the lo g-normality.\nFurther, we show that integrated squared error of the estima te of the state\nprice density converges to zero at the rate of ln4n/n, which again differs4 M. YUAN\nfrom the usual parametric rate only by a factor of the power of log sample\nsize.\nThe rest of the paper is organized as follows. We describe the methodol-\nogy in the next section. Section 3discusses the asymptotic properties of the\nproposed estimate of both the call option prices and the stat e price density.\nThe proposed estimating scheme is illustrated through an em pirical study\nin Section 4. We close with some conclusions in the last section. All proo fs\nare relegated to the Appendix .\n2. Method. In the Black–Scholes paradigm, the state price density f\ncorresponds to a log-normal distribution. More precisely, the log return\nln(ST/St) follows a normal distribution with mean ( rt,τ−δt,τ−σ2/2)τand\nvarianceσ2τ,whereδt,τis the dividend yield in this period. Under this\npremise, ( 1.1) yields\nC(X,St,rt,τ,τ)=Ste−δt,ττΦ(d1)−Xe−rt,ττΦ(d2), (2.1)\nwhere Φ( ·) is the cumulative distribution function of the standard no rmal\ndistribution and\nd1=ln(St/X)+(rt,τ−δt,τ+σ2/2)τ\nσ√τ,\nd2=ln(St/X)+(rt,τ−δt,τ−σ2/2)τ\nσ√τ.\nThe Black–Scholes formula ( 2.1) prices a European call option with only\none parameter, σ, often referred to as the implied volatility. The Black–\nSchole model works remarkably well in the early years of opti on markets.\nIt, however, becomes increasingly conspicuous that it fail s to explain the\noption prices observed in the post-1987 crash market [Rubin stein (1994)].\nTo illustrate, in Figure 1, we plot a cross section of S&P 500 index option\nprices versus the strike price during a three week span in Dec ember 2002.\nThe options expired on March 2003. We shall explain the main d ata char-\nacteristics in more detail in Section 4. Along with the observed data, we\nalso plot the best fit given by the Black–Scholes model. It can be observed\nthat the Black–Scholes model tends to underprice the deep in -the-money\noptions. The discrepancy can be as much as 25% or about $20, wh ich is\nrather significant.\nVarious approaches have been developed to improve the origi nal Black–\nScholes model. In the Black–Scholes paradigm, the underlyi ng asset price is\nassumed to follow a geometric Brownian motion:\ndS=ηSdt+σSdB, (2.2)\nwhereηis the growth rate of the stock price and Btis a standard Brownian\nmotion. One popular alternative to the original Black–Scho les model is theSTATE PRICE DENSITY ESTIMATION 5Fig. 1. S&P 500 index option prices together with the best Black–Sch ole model fit.6 M. YUAN\nstochastic volatility model where V=σ2is further modeled as a stochastic\nprocess:\ndV=ζV dt+ξV dW, (2.3)\nwhereWis another standard Brownian motion. In the risk-neutral wo rld,\nthe asset price and its instantaneous variance σ2follow similar processes:\ndS=(r−δ)Sdt+σSd˜B,\ndσ2=ασ2dt+ξσ2d˜W.\nHull and White ( 1987) showed that if ˜Btand˜Wtare two independent Brow-\nnian motions, then the conditional distribution of ln( ST/St) given “average”\nvolatility\n¯V=1\nT−t/integraldisplayT\ntσ2(u)du (2.4)\nis normal with mean ( rt,τ−δt,τ−¯V/2)τand variance ¯Vτ. More generally,\nusing the same argument as Hull and White ( 1987), it can beshown that the\nstatement holds true as long as ˜Btis independent of the volatility process\nV(t).\nIn other words, under the stochastic volatility model, ln( ST/St) follows a\nmixture of normal distribution\nf(ln(ST/St))=/integraldisplay\nφ(ln(ST/St)|(rt,τ−δt,τ−¯V/2)τ,¯Vτ)dF(¯V), (2.5)\nwhereφ(·|µ,σ2) is the normal density function with parameters µandσ2\nandFis the distribution function of ¯V. Clearly, this reduces to the Black–\nScholes model when Fis a degenerated distribution. Motivated by this and\nto allow for more flexibility, we consider in this paper state price densities\nsuch that ln STfollows a nonparametric mixture of normal densities:\nh(lnST)=/integraldisplay\nφ(lnST|µ,σ2)dG(µ,σ), (2.6)\nwhereG, referred to as mixing distribution, is an unknown bivariat e distri-\nbution function. The corresponding state price density can be written as\nf(ST)=/integraldisplay\nυ(ST|µ,σ2)dG(µ,σ), (2.7)\nwhereυ(·|µ,σ2) is the density function of log normal distribution with loc a-\ntion parameter µand scale parameter σ. It is evident that when Gassigns\nprobability one to (ln( St)+(rt,τ−δt,τ−σ2/2)τ,σ√τ), the proposed mixture\nmodel reduces to the Black–Scholes model. The stochastic vo latility model\ndescribed above is also the special case of our nonparametri c mixture model.STATE PRICE DENSITY ESTIMATION 7\nDifferent from the Black–Scholes models, our model for the sta te price den-\nsity is nonparametric in that we do not impose any parametric assumption\nto the mixing distribution G. Mixtures of form ( 2.6) are known to be a\nrich family of distributions and can approximate any differen tiable density\nfunction to an arbitrary precision [Silverman ( 1986)].\nOur goal is to extract the state price density function fas well as the\npricingfunction Cgiven aset of observations onstrike priceandoption price\npairs, (X1,C1),(X2,C2),...,(Xn,Cn), that follow a regression relationship\nCi=C(Xi)+εi, i=1,2,...,n. (2.8)\nIn particular, we assume that the state price density functi on lies in the\nfollowing class:\nF=/braceleftbigg\nf(·):f(S)=/integraldisplay\nυ(S|µ,σ2)dG(µ,σ),\n(2.9)\nsupp(G)⊆[−M,M]×[σ,¯σ]/bracerightbigg\n,\nfor some constants M<∞and 0<σ≤¯σ<∞. Correspondingly, the pricing\nfunction belongs to the following function class:\nC=/braceleftbigg\nC(·):C(X)=e−rt,ττ/integraldisplay\nψ(S)f(S)dS,f∈F/bracerightbigg\n. (2.10)\nFollowing A¨ ıt-Sahalia and Duarte ( 2003), we consider estimating the pricing\nfunction by minimizing the weighted least squares:\nˆC(·)=arg min\nC(·)∈C1\nnn/summationdisplay\ni=1wi(Ci−C(Xi))2. (2.11)\nAs argued by A¨ ıt-Sahalia and Duarte ( 2003), the weights w′\nis can be cho-\nsen to reflect the relative liquidity of different options. Mor e actively traded\noptions would receive a higher weight than those less active ly traded ones.\nThey also suggest that the actual weights be determined on th e basis of the\nsize and time of the most recent transaction and the bid-ask s pread, which\nare readily available in practice. For brevity, in the follo wing discussion, we\nshall assume equal weights w1=w2=···=wn= 1. Our results, however,\nalso apply to the more general and realistic weighting schem es.\nNote that when the state price density is given by ( 2.7), the pricing func-\ntion is also determined by the mixing distribution G:\nC(X;G)=e−rt,ττ/integraldisplay\nψ(ST)f(ST)dST\n=e−rt,ττ/integraldisplay\nψ(ST)/integraldisplay\nυ(ST|µ,σ2)dG(µ,σ)dST8 M. YUAN\n=e−rt,ττ/integraldisplay/bracketleftbigg/integraldisplay\nψ(ST)υ(ST|µ,σ2)dST/bracketrightbigg\ndG(µ,σ)\n≡/integraldisplay\nC(X;µ,σ2)dG(µ,σ),\nwhere\nC(X;µ,σ2)=e−rt,ττ/integraldisplay∞\n0ψ(ST)υ(ST|µ,σ2)dST\n=e−rt,ττ/integraldisplay∞\n−∞(es−X)+φ(s|µ,σ2)ds\n=e−rt,ττ/parenleftbigg/integraldisplay∞\nlnXesφσ(s−µ)ds−X/integraldisplay∞\nlnXφσ(s−µ)ds/parenrightbigg\n=e−rt,ττ+σ2/2+µ/braceleftbigg\n1−Φ/parenleftbigglnX−(µ+σ2)\nσ/parenrightbigg/bracerightbigg\n−e−rt,ττX/braceleftbigg\n1−Φ/parenleftbigglnX−µ\nσ/parenrightbigg/bracerightbigg\n=e−rt,ττ+σ2/2+µ¯Φ/parenleftbigglnX−(µ+σ2)\nσ/parenrightbigg\n−e−rt,ττX¯Φ/parenleftbigglnX−µ\nσ/parenrightbigg\n,\nand¯Φ(·)=1−Φ(·).\nThe least squares estimate of the pricing function can be equ ivalently\nwritten as ˆC(·)=C(·;ˆG) with\nˆG(·)=argmin\nG∈G1\nnn/summationdisplay\ni=1(Ci−C(Xi;G))2, (2.12)\nwhereGis the collection of all probability measures on µandσ2. Note\nthat the minimization is taken over a function space of infini te dimension,\nwhichisnotdirectlycomputableatthefirstglance.However , asthefollowing\ntheoremshows,thesolutioncanalways berepresentedinafin itedimensional\nspace and therefore make the minimization possible.\nTheorem 2.1. The minimum of ( 2.12) exists and there is a distribution\nwhose support contains no more than n+1points achieves the minimum.\nFurthermore, at each support point, σ2=σ2.\nTheorem 2.1is of great practical importance since it now suffices to find a\nminimizer of ( 2.12) that has a support of n+1 points or fewer, which can be\nsolved numerically. The theorem is similar to theorems for o ptimal design\n[Silvey (1980)] and the the famous result for mixture likelihood [Lindsay\n(1983)].STATE PRICE DENSITY ESTIMATION 9\nIn practice, it is common to ensure that the expected value of the price\nof the underlying security under the risk neutral measure is equal to the\nforward price of the underlying. This constraint can be easi ly incorporated\nin our framework. Note that when the state price density fcomes from F,\nthe expected value of STcan be conveniently expressed as\nEf(S)=/integraldisplay\neµ+σ2/2dG(µ,σ2). (2.13)\nDenote by Ft,Tthe forward price of the underlying security. Enforcing the\naforementioned constraint means that instead of F, we restrict our attention\nto the following family of densities:\nF∗=/braceleftbigg\nf(·):f∈F,/integraldisplay\neµ+σ2/2dG(µ,σ2)=Ft,T/bracerightbigg\n. (2.14)\nNote that F∗is a convex subset of F. Theorem 2.1remains true. For the\nsame reason, in the subsequent theoretical development, we shall neglect\nsuch constraint for brevity. But it is noteworthy that all ou r discussion also\napplies to the situation when this constraint is in place wit h little notational\nchanges.\n3. Theoretical properties. Before stating the main theoretical results,\nwe first describe a set of conditions for the pricing errors. A ssume that the\npricing errors are independent and satisfy the following:\n(a)E(εi)=0, fori=1,...,n;\n(b) for some β>0, Γ>0,\nsup\nnmax\n1≤i≤nE(exp(βε2\ni))≤Γ≤+∞. (3.1)\nBoth conditions are rathermild. Condition(a) indicates th at theobserved\nprice is unbiased, which provides the basis for estimating t he pricing func-\ntion. Condition (b) concerns how fast the tail of error distr ibutions decays.\nDistributionsthat satisfy Condition(b) areoftencalled s ub-Gaussian.When\nthe pricing errors follow normal distributions, this condi tion is satisfied. In\nthe most realistic situations, the pricing error is bounded and this condition\nis also trivially satisfied.\nWe now consider the property of the estimated price function .\nTheorem 3.1. LetˆCnbe the minimizer of ( 2.11) overF. Then under\nConditions (a)and(b)there exist constants L0,C0>0such that, for any\nn≥n0andL≥L0,\nP/parenleftbigg√n\nlnn/bardblˆCn−C/bardbln>L/parenrightbigg\n≤n−C0L2, (3.2)10 M. YUAN\nwhere\n/bardblˆCn−C/bardbl2\nn=1\nnn/summationdisplay\ni=1{ˆC(Xi)n−C(Xi)}2. (3.3)\nConsequently,\n/bardblˆCn−C/bardbl2\nn=Op/parenleftbiggln2n\nn/parenrightbigg\n.\nThe convergence rate obtained in Theorem 3.1is to becompared with the\nusualparametric situation such as theBlack–Scholes model . Whenassuming\nthat the state price density follows a log-normal distribut ion, the pricing\nfunction can be given as ( 2.1). The volatility can also be estimated, for\nexample, by means of the least squares. Such procedure would lead to the\nusual parametric convergence rate that /bardblˆCn−C/bardbl2\nn=Op(1/n). Note that,\nassuming the state price density resides in a much more gener al family F,\nthe convergence rate we obtained here only differs from the par ametric rate\nby ln2n.\nNext, we study the properties of the estimated state price de nsities.\nTheorem 3.2. DenoteρXthe sampling density of strike prices X1,...,X n.\nLetΩbe an open set such that\nmin\nx∈ΩρX(x)≥L0>0 (3.4)\nfor some constant L0. Then under Conditions (a)and(b)\n/integraldisplay\nΩ(ˆfn−f)2=Op/parenleftbiggln4n\nn/parenrightbigg\n. (3.5)\nSimilar to the price function, the state price density estim ate converges at\na“nearly”parametricrate,nowwithanextratermln4n.Comparedwiththe\npricefunction,therateisslightlyslower. Itistypical in nonparametricstatis-\ntics that differentiation results in slower convergence rate . Different from\nthe usual nonparametric setting, however, the convergence rate deteriorates\nonly by ln2n. In contrast, for both approaches from A¨ ıt-Sahalia and Dua rte\n(2003) and Yatchew and H¨ ardle ( 2005), the price function can be estimated\nat convergence rate n−2q/(1+2q)and state price density at n−2(q−2)/(1+2q),\nboth in the integrated squared error sense when assuming tha t the price\nfunction is qtimes differentiable.\nIn characterizing the performance of the state price densit y, confining to\na set such as Ω is often necessary. The state price density is e stimated based\non observed pairs of strike and call price. Because we rarely observe strikes\nfrom regions where ρXis close to zero, it is impossible to estimate it wellSTATE PRICE DENSITY ESTIMATION 11\nin these regions without further restrictions. In practice , this is irrelevant\nbecause the strikes are most often evenly distributed in a co mpact region\naround the asset price. Putting it in our notation, this amou nts toρXbeing\na uniform distribution and set Ω of Theorem 3.2can be taken as the whole\nsupport of the distribution.\n4. Numerical studies.\n4.1.Implementation. Theorem 2.1showsthat,withoutlossofgenerality,\nthe estimated state price density admits the following expr ession:\nˆf(ln(ST/St))=π1φ(ln(ST/St);µ1,σ2)+π2φ(ln(ST/St);µ2,σ2)\n+···+πn+1φ(ln(ST/St);µn+1,σ2),\nand it suffices to estimate the mixing proportions πjs and means µjs in\nminimizing the least squares. We propose to iteratively com pute the mixing\nproportions and the means for a given σ2. Givenµ1,...,µn+1, updating the\nmixingproportionscanthenbecastasaquadraticprograman deasilysolved\nusing the standard quadratic program solvers. Once the mixi ng proportions\nare available, we update the means by Newton Ralphson iterat ions.\nThe constraint that the expected value of the price of the und erlying\nsecurity underthestatepricedensityisequal totheforwar dpricecanalsobe\neasily incorporatedinthisalgorithm,asitcannowbeconve niently expressed\nas\nFt,T=Steσ2/2(π1eµ1+···+πn+1eµn+1). (4.1)\nIt is of great importance to choose a good initial value. A car eful exami-\nnation of the proofs to Theorems 3.1and3.2suggests that any density from\nFcan be approximated well by a member of Fbut with the means µto\nbe equally spaced between [ −M,M]. Motivated by this fact, we can take\nthe means to be equally spaced as the initial value. The algor ithm there-\nfore starts with a natural initial solution, which is alread y a good estimate.\nA limited number of iterations are usually sufficient to achie ve good per-\nformance in practical applications. We observe empiricall y that the least\nsquares objective function decreases quickly in the first it eration, and the\nobjective function after the first iteration is already very close to the objec-\ntive function at convergence, as the magnitude of the decrea se in the first\niteration dominates the decreases in subsequent iteration s. This motivates\nus to use a one-step iteration in our implementation.\n4.2.Simulation. To gain insights to the finite sample performance of the\nproposedmethod, we firstconducted a set of simulation studi es. We adopted\ntheexperimentsettingofA¨ ıt-Sahalia andDuarte( 2003),whichwasdesigned12 M. YUAN\nFig. 2. Estimated call option price function and state price densit y summarized over\n5000 simulation runs.\nto mimic S&P 500 index options. In particular, the current in dex price was\nset at 1365, the short term interest rate at 4 .5%, the dividend yield at 2 .5%,\nand the time to maturity at 30 days. The volatility smile was a ssumed to be\na linear function of the strike with volatility equal to 40% a t the strike price\n1000 and 20% at the strike price 1700. We assume that we observ en=25\noption prices with strike prices equally spaced between 100 0 and 1700. The\noption prices were simulated by adding uniformly distribut ed random noise\nto the theoretical option prices. Following A¨ ıt-Sahalia a nd Duarte ( 2003),\nthe range of the noise varies linearly from 3% of the option va lue for deep\nin the money options to 18% for deep out of the money options.\nFor each run, the call function and the corresponding state p rice density\nare estimated by the proposed method with σchosen by leave one out cross\nvalidation [Wahba ( 1990)]. Figure 2displays the average estimates and 95%\npointwise confidence intervals for the call price function a nd the state price\ndensity based on 5000 simulations. It is evident that the est imate works very\nwell.\n4.3.Real data analysis. To illustrate the proposed methodology, we now\ngoback tothehistorical option databrieflymentioned inSec tion2.ThedataSTATE PRICE DENSITY ESTIMATION 13\nFig. 3. Historical S&P 500 index price and Eurodollar deposit rates from December 2,\n2002 to December 20, 2002.\nconsist of a cross section of European call option prices wri tten on the S&P\n500 index duringthe firstthree weeks of December, 2002. Figu re3shows the\nclosing price of the index itself and the Eurodollar deposit rates (London)\nin the same period. The deposit rate is used as the risk-free i nterest rate.\nBecause the maturity ranges from 3 months to about 4 months, w e linearly\ninterpolated the 3 month rate and 6 month rate to yield the dai ly risk-free\nrate.\nThe cross-section of the option prices are given in the leftm ost panel of\nFigure4. Following convention, we use the average of the end-of-day bid and\nask price as the option price. Different lines correspond to di fferent dates.\nIt is clear that the option price can be modeled as a smooth fun ction of the\nstrike. This leads to the misperception that we can always es timate the state\npricedensitybydirectlydifferentiatinganinterpolation o ftheoptionsprices.\nSuch na¨ ıve strategy does not work in practice, however. To e laborate, the\nmiddle and right panels of Figure 4show∂C/∂Xand∂2C/∂X2respectively\nestimated by straightforward differentiation. It can be obse rved that the\nderivatives aremuch morewiggly as functions of thestrikea nd, furthermore,\nthere is no guarantee that the resulting estimate of the stat e price density\nis positive as required by a legitimate density.14 M. YUAN\nFig. 4. Historical European call option prices for S&P 500 index fro m December 2,\n2002 to December 20, 2002 and its derivatives with respect to the strike. Different lines\ncorrespond to different dates.\nWe now apply the proposed method to the option prices on a dail y basis.\nAs in A¨ ıt-Sahalia and Duarte ( 2003), we set the weights to be the inverse of\nthe option price since the prices fluctuate considerably mor e when the price\nitself is high. The Black–Scholes model fit reported in Secti on2is produced\nin the same fashion. We also reconstruct the dividend rate th rough the put-\ncall parity:\nPt+Ste−δτ=Ct+Xe−rτ(4.2)\nusing the put-call pair at the money. We choose σusing leave one out cross\nvalidation. Our experience suggests, however, usually thr ee quarters of the\nvolatility obtained from the Back–Scholes model fit works fa irly well in prac-\ntice. The estimated pricing functions and state price densi ties are given in\nFigures5and6respectively. In contrast to theBlack–Schole model fit show n\nin Figure 1, our nonparametric estimate fits the historical option pric es very\nwell. Thedepartureoftheunderlyingstatepricedensities fromlognormality\nis also evident from Figure 6.\nThese nonparametric state price density estimates can have many uses.\nFor example, as pointed out in A¨ ıt-Sahalia and Duarte ( 2003), they canSTATE PRICE DENSITY ESTIMATION 15 Fig. 5. Estimated call prices versus Moneyness.16 M. YUAN\nFig. 6. Estimated state price density versus the excess log return ln(ST/St)−rt,τ.STATE PRICE DENSITY ESTIMATION 17\nbe employed to price new and more complex or less liquid optio ns in an\narbitrary-free fashion. With the knowledge of the entire st ate price density\navailable, it is also straightforward to derive interestin g quantities such as\nvalue-at-risk. Our nonparametric estimate can provide mor e reliable infor-\nmation than its parametric counterpart from these perspect ives. For exam-\nple, as evidenced in Figure 6, the Black–Scholes paradigm may significantly\nunder-evaluate investment risk.\n5. Conclusions. Inthispaperweintroducedanewnonparametricoption\npricing technique. We consider state price densities that c an be represented\nas a nonparametric mixture of log normals. Our nonparametri c model is\ninspired by the stochastic volatility model of Hull and Whit e (1987) and\nextends the original Black–Scholes model. Both the option p rice function\nand the state price density can be estimated through the leas t squares. We\nshowed that such estimates enjoy nice asymptotic propertie s. An applica-\ntion to the historical data also demonstrates the merits of t he proposed\nmethodology in finite samples.\nAPPENDIX\nProof of Theorem 2.1.The existence of a minimum comes from the\nfact that both the objective function and the feasible regio n are convex.\nDenote\nA={C(·;µ,σ2):µ∈R}. (5.1)\nIt is not hard to see that Cis a subset of the convex hull of A. LetˆG\nbe a minimizer of ( 2.12). Clearly, ( C(X1,ˆG),C(X2,ˆG),...,C(Xn,ˆG))′is\nan element of the convex hull spanned by A. By Carath´ eodory’s theorem,\nthere exits a subset BofAconsisting of n+ 1 points or fewer so that\n(C(X1,ˆG),C(X2,ˆG),...,C(Xn,ˆG))′is in the convex hull spanned by B. In\nother words, we can find µ1,µ2,...,µn+1∈Rso that\nC(Xi,ˆG)=n+1/summationdisplay\nj=1πjC(Xi;µj,σ2)∀i=1,...,n (5.2)\nfor someπj≥0 andπ1+···+πn+1=1. Therefore,\nG(·)=n+1/summationdisplay\nj=1πjΦ(·;µj,σ2) (5.3)\nminimizes ( 2.12)./square\nProof of Theorem 3.1.Without loss of generality, we assume that\nrt,τis zero throughout the proof.18 M. YUAN\nForε>0, theε-covering number of C,N(ε,C,/bardbl·/bardbl∞), is defined as the\nnumber of balls with radius εnecessary to cover C. Denote\nH(ε,C,/bardbl·/bardbl∞)=lnN(ε,C,/bardbl·/bardbl∞) (5.4)\ntheε-entropy of C. In the light of Theorem 4.1 from van de Geer ( 1990), it\nsuffices to show that, for any δ>0 small enough,\n/integraldisplayδ\n0H1/2(u,C,/bardbl·/bardbl∞)du≤Cδln1\nδ. (5.5)\nWe now set out to establish this inequality. We proceed by exp licitly con-\nstructing an ε-covering set for C.\nAn application of the Taylor expansion yields\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ(u)−1√\n2πk−1/summationdisplay\nj=0(−1)ju2j\n2jj!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1√\n2πu2k\n2kk!≤1√\n2π/parenleftbiggeu2\n2k/parenrightbiggk\n. (5.6)\nFor anyU>|u|,\n¯Φ(u)=/integraldisplay∞\nUφ(z)dz+/integraldisplayU\nuφ(z)dz\n=/integraldisplay∞\nUφ(z)dz+/integraldisplayU\nu/parenleftBigg\n1√\n2πk−1/summationdisplay\nj=0(−1)jz2j\n2jj!/parenrightBigg\ndz\n+/integraldisplayU\nu/parenleftBigg\nφ(u)−1√\n2πk−1/summationdisplay\nj=0(−1)jz2j\n2jj!/parenrightBigg\ndz.\nTherefore,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle¯Φ(u)−/integraldisplayU\nu/parenleftBigg\n1√\n2πk−1/summationdisplay\nj=0(−1)jz2j\n2jj!/parenrightBigg\ndz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay∞\nUφ(z)dz+/integraldisplayU\nu1√\n2π/parenleftbiggez2\n2k/parenrightbiggk\ndz\n≤Cφ(U)/U+U2k+1−u2k+1\n√\n2π(2k+1)/parenleftbigge\n2k/parenrightbiggk\n≤C\nU/braceleftbigg\nφ(U)+/parenleftbiggeU2\n2k/parenrightbiggk+1/bracerightbigg\n.\nHereafter, we use C>0 as a generic constant.\nNow consider distribution functions FandGsuch that\n/integraldisplay\nujdF(u)=/integraldisplay\nujdG(u), j=1,...,2k−1. (5.7)STATE PRICE DENSITY ESTIMATION 19\nThen,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nC(x;µ,σ)dF(µ)−/integraldisplay\nC(x;µ,σ)dG(µ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nC(x;µ,σ)d{F(µ)−G(µ)}/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\neσ2/2+µ¯Φ/parenleftbiggx−(µ+σ2)\nσ/parenrightbigg\nd{F(µ)−G(µ)}/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingleex/integraldisplay\n¯Φ/parenleftbiggx−µ\nσ/parenrightbigg\nd{F(µ)−G(µ)}/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤eσ2/2+M/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n¯Φ/parenleftbiggx−(µ+σ2)\nσ/parenrightbigg\nd{F(µ)−G(µ)}/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+ex/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n¯Φ/parenleftbiggx−µ\nσ/parenrightbigg\nd{F(µ)−G(µ)}/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤eσ2/2+M/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay/parenleftBigg/integraldisplayU\n(x−µ−σ2)/σ/parenleftBigg\n1√\n2πk−1/summationdisplay\nj=0(−1)jz2j\n2jj!/parenrightBigg\ndz/parenrightBigg\nd{F(µ)−G(µ)}/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+ex/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay/parenleftBigg/integraldisplayU\n(x−µ)/σ/parenleftBigg\n1√\n2πk−1/summationdisplay\nj=0(−1)jz2j\n2jj!/parenrightBigg\ndz/parenrightBigg\nd{F(µ)−G(µ)}/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+(eσ2/2+M+ex)C\nU/braceleftbigg\nφ(U)+/parenleftbiggeU2\n2k/parenrightbiggk+1/bracerightbigg\n=(eσ2/2+M+ex)C\nU/braceleftbigg\nφ(U)+/parenleftbiggeU2\n2k/parenrightbiggk+1/bracerightbigg\n.\nChoosingU=/radicalbig\nk/2 yields\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nC(x;µ,σ)dF(µ)−/integraldisplay\nC(x;µ,σ)dG(µ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(1+r)−k/√\nk (5.8)\nfor some 0<r<exp(1/2σ2)−1 and allx≤/radicalbig\nk/2.\nOn the other hand, when x>/radicalbig\nk/2,\n/integraldisplay\nC(x;µ,σ)dF(µ)≤eσ2/2+M/integraldisplay\n¯Φ/parenleftbiggx−(µ+σ2)\nσ/parenrightbigg\ndF(µ)\n≤eσ2/2+M¯Φ/parenleftbiggx−M−σ2\nσ/parenrightbigg\n≤Ceσ2/2+Mφ/parenleftbiggx−M−σ2\nσ/parenrightbigg/slashBig/parenleftbiggx−M−σ2\nσ/parenrightbigg\n≤Ceσ2/2+Me−k/2σ2/√\nk20 M. YUAN\n≤C(1+r)−k/√\nk.\nSubsequently, for any x>/radicalbig\nk/2,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nC(x;µ,σ)dF(µ)−/integraldisplay\nC(x;µ,σ)dG(µ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/integraldisplay\nC(x;µ,σ)dF(µ)+/integraldisplay\nC(x;µ,σ)dG(µ)\n≤C(1+r)−k/√\nk.\nIn summary, we conclude that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\nC(·;µ,σ)dF(µ)−/integraldisplay\nC(·;µ,σ)dG(µ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞≤C(1+r)−k/√\nk. (5.9)\nLemma A.1 of Ghosal and van de Vaart ( 2001) shows that, for any dis-\ntribution function F, there exists a probability measure Gsupported on\n2kpoints so that ( 5.7) is satisfied. In other words, we can always find a\nprobability measure Gwith support on at most O(ln(1/ε)) points such that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\nC(·;µ,σ)dF(µ)−/integraldisplay\nC(·;µ,σ)dG(µ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞≤ε/2. (5.10)\nNow note that, for ε(>0) small enough,\n|C(X;µ,σ)−C(X;µ+ε,σ)|\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞\n−∞{(es−X)+−(es+ε−X)+}dΦ(s;µ,σ2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/integraldisplay+∞\nlnX(es+ε−es)dΦ(s;µ,σ2)+/integraldisplaylnX\nlnX−ε(es+ε−X)dΦ(s;µ,σ2)\n=(eε−1)eσ2/2+µ/braceleftbigg\n1−Φ/parenleftbigglnX−(µ+σ2)\nσ/parenrightbigg/bracerightbigg\n+eσ2/2+µ+ε/braceleftbigg\nΦ/parenleftbigglnX−(µ+σ2)\nσ/parenrightbigg\n−Φ/parenleftbigglnX−(µ+σ2+ε)\nσ/parenrightbigg/bracerightbigg\n−X/braceleftbigg\nΦ/parenleftbigglnX−µ\nσ/parenrightbigg\n−Φ/parenleftbigglnX−(µ+ε)\nσ/parenrightbigg/bracerightbigg\n≤(eε−1)eσ2/2+µ+εeσ2/2+µ+ε/√\n2πσ2\n≤{(2+1/σ)e¯σ2/2+M}ε≡Kε.\nIn conclusion, for any F∈F, we can always find a Gwho is supported only\non 0,±ε/K,±2ε/K,..., ±[KM]ε/Ksuch that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\nC(·;µ,σ)dF(µ)−/integraldisplay\nC(·;µ,σ)dG(µ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞≤ε. (5.11)STATE PRICE DENSITY ESTIMATION 21\nTogether with the fact that there exists a generic constant D>0 such that\n|C(X;µ,σ)−C(X;µ,σ+ε)|≤Dε, (5.12)\nthis implies the following bound on the covering number for C:\nN(ε,C,/bardbl·/bardbl∞)≤C/parenleftbigg1\nε/parenrightbiggCln(1/ε)\n. (5.13)\nTherefore, the entropy can be bounded as well:\nH(ε,C,/bardbl·/bardbl∞)=lnN(ε,C,/bardbl·/bardbl∞)≤C/parenleftbigg\nln1\nε/parenrightbigg2\n. (5.14)\nHence, forδsmall enough,\n/integraldisplayδ\n0H1/2(u,C,/bardbl·/bardbl∞)du≤Cδln1\nδ. (5.15)\nThe proof is now completed. /square\nProof of Theorem 3.2.Denoteh=ˆCn−C. From Theorem 3.1,\n/integraldisplay\nΩh2≤1\nL0/integraldisplay\nΩh2ρX≤1\nL0/integraldisplay\nh2ρX=Op/parenleftbiggln2n\nn/parenrightbigg\n. (5.16)\nIt is not hard to see that h′′=ˆfn−f. By Parseval’s theorem,\n/integraldisplay\n(h[k])2=/integraldisplay\n/bardblF{h[k]}/bardbl2, (5.17)\nwhereF{h[k]}is the continuous Frourier transform of h[k]. Furthermore,\nnote that\nF{h[k]}(s)=sk−2F{h′′}(s)=n+1/summationdisplay\nj=1πjsk−2exp/parenleftbigg\n−iµjs−σ2\njs2\n2/parenrightbigg\n. (5.18)\nBy the triangular inequality,\n/bardblF{h[k]}(s)/bardbl≤n+1/summationdisplay\nj=1πj|s|k−2exp/parenleftbigg\n−σ2\njs2\n2/parenrightbigg\n≤|s|k−2exp/parenleftbigg\n−σ2s2\n2/parenrightbigg\n. (5.19)\nTogether with ( 5.17), we have\n/integraldisplay\n(h[k])2≤/integraldisplay\ns2(k−2)exp(−σ2s2)ds\n(5.20)\n=/radicalbiggπ\nσ2(2k−4)!\n2k−2(k−2)!(2σ2)−(k−2).22 M. YUAN\nAn application of the Kolmogorov interpolation inequality yields\n/integraldisplay\nΩ(ˆfn−f)2=/integraldisplay\nΩ(h′′)2\n≤C/parenleftbigg/integraldisplay\nΩh2/parenrightbigg1−2/k/parenleftbigg/integraldisplay\nΩ(h[k])2/parenrightbigg2/k\n≤C/parenleftbigg/integraldisplay\nΩh2/parenrightbigg1−2/k/parenleftbigg/integraldisplay\n(h[k])2/parenrightbigg2/k\n=Op/parenleftbigg/parenleftbiggln2n\nn/parenrightbigg1−2/k/parenleftbiggk\n2σ2/parenrightbigg2/parenrightbigg\n.\nThe proof can now be completed by setting k=lnn./square\nAcknowledgments. The author thanks the editor, the associate editor\nandtwoanonymousrefereesforcommentsthatgreatlyimprov edthemanuscript.\nThis research was supported in part by grants from the Nation al Science\nFoundation.\nREFERENCES\nA¨ıt-Sahalia, Y. andDuarte, J. (2003). Nonparametric option pricing under shape\nrestrictions. J. Econometrics 1169–47.MR2002521\nA¨ıt-Sahalia, Y. andLo, A. W. (1998). 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SIAM, Philadelphia.\nMR1045442\nYatchew, A. andH¨ardle, W. (2005). Dynamic nonparametric state price density esti-\nmation usingconstrained least squares andthebootstrap. J. Econometrics 133579–599.\nMR2252910\nSchool of Industrial and Systems\nEngineering\nGeorgia Institute of Technology\n755 Ferst Drive NW\nAtlanta, Georgia 30332\nUSA\nE-mail: myuan@isye.gatech.edu"    },    {        "title": "0912.3154v2.Reduced_density_matrix_functional_theory_in_quantum_Hall_systems.pdf",        "content": "arXiv:0912.3154v2  [cond-mat.mes-hall]  3 Feb 2010Reduced density-matrix functional theory in quantum Hall s ystems\nE. T¨ ol¨ o and A. Harju\nHelsinki Institute of Physics and Department of Applied Phy sics, Aalto University, FI-02150 Espoo, Finland\nWe apply reduced density-matrix functional theory to the pa rabolically confined quantum Hall\ndroplet in the spin-frozen strong magnetic field regime. One -body reduced density matrix functional\nmethod performs remarkably well in obtaining ground states , energies, and observables derivable\nfromtheone-bodyreduceddensitymatrixfor awiderange ofs ystemsizes. Atthestronglycorrelated\nregime, theresultsgowell beyondwhatcan beobtainedwith t hedensityfunctional theory. However,\nsome of the detailed properties of the system, such as the edg e Green’s function, are not produced\ncorrectly unless we use the much heavier two-body reduced de nsity matrix method.\nI. INTRODUCTION\nQuantum Hall fluids occur at low temperature in clean\ntwo-dimensional electron systems exposed to a perpen-\ndicular magnetic field.1,2Different phases are character-\nized by the number νthat tells the ratio of the num-\nber of electrons to the number of single-particle states\nin a highly degenerate Landau level (the number of flux\nquanta piercing the sample). Near certain fractional fill-\ning factors ν, the interactions between electrons induce\nan energy gap and lead to a ground state with topologi-\ncal order manifest in exotic quasiparticles and non-Fermi\nliquid edge modes.3–11\nStraight from the outset, numerical simulations have\nbeen an indispensable guide in development of the\ntheory.4While majority of the numerics are exact diag-\nonalization studies only viable with small electron num-\nbers, variational Monte Carlo12and density functional\ntheory13,14(DFT) have also been applied to larger sys-\ntems. For example, accurate wave functions incorporat-\ningthecomplexnon-perturbativeeffectsofelectroninter-\nactions have been theoretically devised and later backed\nup by the high overlap with exact numerical results for\nsmall systems. Due to such developements, reason be-\nhind the energy gap of simplest of the many fractions is\nnow well understood within the framework of composite-\nfermion theorythat allowsforexplicit constructionofthe\nmany-body wave function and calculation of topologi-\ncal quantum numbers.8However, in going beoynd the\ncomposite-fermion theory to more complex phases, the\nMonte Carlo method is crippled since a trustworthy trial\nwave function for the phases we would be interested to\nknow more about is not known. In addition, owing to the\nstrong correlations, the DFT is inaccurate at, for exam-\nple, the paradigm fractional quantum Hall state at filling\nfractionν= 1/3 where the vortices supposed to form\na bound state with the electrons localize at fixed posi-\ntions instead.15Consequently, the exact diagonalization\nis frequently the only viable alternative, leaving the large\nelectron numbers beyond the reach of direct calculations,\nthoughthe densitymatrixrenormalizationgroupmethod\nhas brought some progress making a bit larger systems\ncomputationally feasible.16,17\nDuring the past 10 years, reduced density matrix func-\ntional theory has been revived and applied successfullyin the chemical physics community.18–20The method is\nknown to handle e.g. molecular dissociation limits better\nthan standardDFT,21,22and it has recently been applied\nto homogeneous electron gas.23,24In this manuscript,\nwe aim to probe the potential of the one-body reduced\ndensity-matrix functional theory (1-RDMFT) in a frac-\ntional quantum Hall system, specifically a parabolically\nconfined quantum dot in the spin-frozen strong magnetic\nfield regime. In contrast to many molecular and atomic\nsystemswherethedominantoccupationnumbersaretyp-\nically close to one, this is a highly demanding application\nfor any numerical method as fractional quantum Hall\nstates have long-range quantum entanglement with all\nthe occupation numbers small for example near 1/3 in\ntheν= 1/3 state.\nThe performance of various functionals in predicting\nground states, energies, and observables attainable by\nthe reduced density matrix is compared for small system\nto the exact diagonalization and Hartree-Fock with and\nwithout the Brillouin-Wigner perturbation theory. For\nlarger systems with tens of particles, the comparison is\ndone utilizing the accurate Laughlin wave function4for\nfilling fraction ν= 1/3 state and Monte Carlo methods.\nEnergies are produced quite well in all systems at the\nstrong-correlation regime ν≪1. Even the bulk densi-\nties appear reasonably good and reproduce the predicted\nedge stripe phase.25However, we are still dealing with an\napproximate method that has its limitations. Detailed\nproperties of the edge of the electron droplet, such as the\nedge tunneling exponent,26,27are not produced correctly\nby the present functionals.\nFor this reason, we also perform a small comparison\nwith the heavier two-body reduced density matrix func-\ntional theory28(2-RDMFT) (see the closely related work\nin Ref. 29). Including the exact electron interaction by\nthe two-body reduced density-matrix functional appears\ntofacilitate a moreaccuratedescriptionofthe edge, how-\never with a computational cost in large systems beyond\nthe reach of present day computers.\nThe rest of the manuscript is organized as follows. In\nthe next section, we briefly introduce the idea of reduced\ndensity matrix functional methods. In Section III, we\ndescribe the model system and derive an exact formula\nfor the energy contribution due to one-body operators\npresent in our Hamiltonian such that only the interac-2\ntion energy remains to be solved. Section IVA and B\npresent the details of our 1-RDMFT and 2-RDMFT im-\nplementations, respectively. The 1-RDMFT results are\ndivided according to the small or large size of the system\ninto Section VAand B, followedby the 2-RDMFT calcu-\nlation in Section VC. Conclusions and future prospects\nare found in Section VI.\nII. REDUCED DENSITY MATRIX\nFUNCTIONAL THEORIES\nThe 1-RDMFT is based on the Gilbert’s theorem,\nwhichguaranteesthattheground-stateexpectationvalue\nof any observable is a unique functional of the 1-RDM\nγ.30It follows that the ground state energy can be writ-\nten as\nF[γ] =/integraldisplay\nR2ddrdr′δ(r−r′)(T(r)+U(r))γ(r,r′)+Vee[γ]\n(1)\nwhereTandUare the standard operators for the kinetic\nenergy and an external potential while the functional for\nthe interaction energy Vee[γ] is unknown. It is simple to\nshow that this functional yields the exact ground state\nenergy for the exact 1-RDM\nγ(r,r′) =N/integraldisplay\nΨ∗(r,r2,r3,...,rN)\n×Ψ(r′,r2,r3,...,rN)dr2dr3...drN(2)\nifVeeis replaced by half the Coulomb energy of the exact\npair-density\nEee=e2\n2ǫ/integraldisplay\nR2ddrdr′ρ2(r,r′)\n|r−r′|,\nρ2(r,r′) =N(N−1)/integraldisplay\nΨ∗(r,r′,r3,...,rN)\n×Ψ(r,r′,r3,...,rN)dr3...drN.(3)\nThe 2-RDMFT minimizes the resulting exact functional\nFsubject to asubset ofcomplete N-representabilitycon-\nditions, known as the 2-representability conditions, and\nother possible constraints due to additional symmetries\n(see Sec. IVB). On the other hand, the crux of the 1-\nRDMFT is to approximate the pair-density by a func-\ntional of the 1-RDM.31The customary way to do the\napproximation, which we will also employ in this paper,\nis to replace the pair-density above by\nρ(r)ρ(r′)−/summationdisplay\ni,jf(ni,nj)φ∗\ni(r)φ∗\nj(r′)φj(r)φi(r′) (4)\nwhereρ(r) =γ(r,r) is the density at r,φiarethe natural\norbitals (eigenvectors of γ), andfis a function solely of\nthe natural occupation numbers ni∈[0,1] (eigenvalues\nofγ). The functional forthe interactionenergyin Eq.(1)then reads\nVee[γ] =e2\n2ǫ/bracketleftbigg/integraldisplay\nR2ddrdr′ρ(r)ρ(r′)\n|r−r′|−/summationdisplay\ni,jf(ni,nj)\n×/integraldisplay\nR2ddrdr′φ∗\ni(r)φ∗\nj(r′)φj(r)φi(r′)\n|r−r′|/bracketrightbigg\n.(5)\nThe form of fcould vary in different type of systems,\nwhereasthose used in this study are enlisted in Table I in\nSec. IVA. In analogy with the density functional theory,\nthe first term is referred to as the Hartree term while the\nlatter is the exchange-correlation term. However, com-\nparedtothe DFT, 1-RDMFT hasacoupleofadvantages.\nFirstly, the method obtains not only the density but the\nwhole 1-RDM so that the ground-state expectation value\nof any one-body observable can be readily computed.\nThus, for example kinetic and interaction energies can\nbe readily separated and Green’s functions calculated.\nSecondly, although both methods are in principle exact\nfor an exact functional, due to the variable γ(vs.ρ),\nit is easier to develop a good 1-RDM functional than a\ngood density functional. This is the reason why the DFT\ndoesnotworkin thestronglycorrelatedregimewherethe\nproper treatment of electronic correlations is important.\nHowever, there are fresh ideas of how to treat strongly\ncorrelated electrons with DFT.32,33\nIII. QUANTUM DOT MODEL\nThe quantum Hall droplet is modeled by the two-\ndimensional effective-mass Hamiltonian\nH=N/summationdisplay\ni=1/bracketleftBigg/parenleftbig\npi+e\ncAi/parenrightbig2\n2m∗+m∗ω2\n0r2\n2/bracketrightBigg\n+/summationdisplay\ni<je2\nǫrij,(6)\nwhereNis the number of electrons, Ais the planar\nvector potential of the homogeneous magnetic field B\nperpendicular to the sample plane, and the energy scale\nof the external confinement potential /planckover2pi1ω0is typically a\nfew meVs. The material parameters are the effective\nmass of the electrons m∗= 0.067meand the dielectric\nconstant of the GaAs semiconductor medium ǫ= 12.7.\nCoulomb interactions tend to spontaneously polarize the\nelectron spins in an effect known as quantum Hall ferro-\nmagnetism. For this reason, the relatively weak Zeeman\nterm has been omitted and the electrons are assumed\nspin-polarized. From here on, we use oscillator units so\nthat the energies are expressed in units of /planckover2pi1ωand lengths\nin units of/radicalbig\n/planckover2pi1/m∗ωwhereω2=ω2\n0+ (ωc/2)2with cy-\nclotron frequency ωc=eB/m∗c.\nForN= 1, the energy states are written as\nψm\nn(z) =/radicalBigg\nn!\nπ(n+m)!zmLm\nn(z¯z)e−z¯z/2, m/greaterorequalslant−n, n/greaterorequalslant0\n(7)3\nwherez=x+iyandLm\nnare the generalized Laguerre\npolynomials. The correspondingeigenvalues are given by\nEm\nn= 2n+1+/parenleftbigg\n1−/planckover2pi1ωc\n2/parenrightbigg\nm . (8)\nWe take on interest in developing a computational\nmethod for the fractional quantum Hall states at the\nstrong magnetic field regime, so we may assume that\nω0≪ωc. Then/planckover2pi1ωc/2 is close to unity, such that the\nterm in parentheses in Eq. (8) becomes small and values\nof quantum number nother than 0 become irrelevant to\nthe low energy physics. This is the Landau level projec-\ntion to the band with n= 0. It should not be difficult\nto generalize the 1-RDMFT method to include spin and\nseveral Landau levels and study the region ν >1 as well,\nhowever, forsimplicity we stickto the spin-polarizedlow-\nest Landau level ν/lessorequalslant1 in this study.\nNote that in our two dimensional model, min Eq. (7)\nistheoneandonlyangularmomentumquantumnumber.\nWe can write the contribution to the total energy due to\nterms other than the Coulomb interaction for a system\nofNelectrons in the lowest Landau level n= 0 in terms\nof the total angular momentum (quantum number) M\nexactly as\nT+U=N/summationdisplay\ni=1/bracketleftbigg\n1+/parenleftbigg\n1−/planckover2pi1ωc\n2/parenrightbigg\nmi/bracketrightbigg\n=N+/parenleftbigg\n1−/planckover2pi1ωc\n2/parenrightbigg\nM .\n(9)\nAsthetotalangularmomentumoperator ˆMcommutes\nwith the total Coulomb interaction energy operator Vee,\ntosolvetheLandaulevelprojectedspectrum, theremain-\ning task is to find the common eigenstates of the Landau\nlevel projected VeeandˆM.\nForNandMsmall enough, these are solved exactly\nwith the configuration interaction method (exact diag-\nonalization) since the number of possible single-particle\nstates is finite. We may go to a bit larger NandM\nby taking interest in only the ground state and finding\nit by the Lanczos algorithm (cf. Refs. 36,37). However,\nthe exponential growthof the many-body basis limits the\nparticle number to around 10 depending on M, and some\nkind of an approximate calculation method becomes nec-\nessary. The Kohn-Sham DFT is still a good method for\nthe weakly correlated regime34but for larger Mthe nat-\nural occupations tend far from 1 and the DFT no longer\ngives us good results. Absence of a reliable trial wave\nfunction forgeneric Mmakesthe implementation ofvari-\national Monte Carlo rather uncertain.12,35In this paper\nwe are going to see, if the reduced density matrix func-\ntional theory can make itself useful. The expectation is\nthatitwillworkbetterthanDFTatleastwhentheeigen-\nvalues of the density matrix are small meaning ν≪1.\nWith several Landau levels, this would correspond to the\nsituation where the highest occupied Landau level has\nlow filling fraction.IV. COMPUTATIONAL METHODS\nA. 1-RDMFT\nDue to the exact formula for the one-body operators’\nenergy contribution (Eq. (9)), the energy functional of\nEq. (1) simplifies to\nF[γ] =N+/parenleftbigg\n1−/planckover2pi1ωc\n2/parenrightbigg\nM+Vee[γ],(10)\nwith the constraints that the particle number and the to-\ntalangularmomentumare NandM, respectively. More-\nover, the natural orbitals for energy state |Ψ∝an}bracketri}htin the low-\nest Landau level are directly the single-particle energy\nstates since the angular momentum conservation yields a\ndiagonal density matrix in this basis\nγmm′=∝an}bracketle{tΨ|a†\nmam′|Ψ∝an}bracketri}ht=δmm′nm,(11)\nwhereamand its adjoint annihilate and create a particle\nwith quantum numbers n= 0 andm(see Eq. (7)). As\nN−1 particles have at least a total angular momentum\n(N−1)(N−2)/2, the maximum single-particle angular\nmomentum becomes k=M−(N−1)(N−2)/2. There-\nfore, the minimization of the interaction energy reduces\nto the constrainedminimization of a function whose vari-\nables arek+1 occupation numbers\nVee({ni}k\ni=0) =1\n2/summationdisplay\ni,j(ninjVijij−f(ni,nj)Vijji) (12)\nwhereVijklare the interaction matrix elements of the\nlowest Landau level orbitals computed in Ref. 38, and\nthe constraints are explicitly written as\nk/summationdisplay\nm=0nm=Nandk/summationdisplay\nm=0nmm=M . (13)\nTo optimize the occupation numbers, we first express\nthem in terms of variables θisuch thatni= sin2θi.\nThe minimization of Veewith the above two remaining\nconstraints is then performed with the interior point or\nNelder-Mead algorithm of Mathematica.39\nA vast number of functions f(ni,nj) have been pro-\nposed in the literature in treatment of simple atoms and\nmolecules, and it is not immediately clear, which of them\nwould work well in the current system. Table I summa-\nrizes those used in this work for small systems to find\nout the optimal ones. For large systems, we only use\nthe density-matrixpowerfunctional f(ni,nj) = (ni,nj)α\n(P-α).24Each form of the off-diagonal f(ni,nj/ne}ationslash=i) corre-\nsponds to two different functions, first of which has diag-\nonalf(ni,ni) =niwhile the second has f(ni,ni) =n2\ni.\nSince the natural orbitals are orthonormal, integration\nof the approximate pair-density in Eq. (4) over the co-\nordinates yields the proper normalization N(N−1) for\nf(ni,ni) =ni. The form n2\ni, on the other hand, is jus-\ntified as it cancels the self-interaction terms Viiiiarising\nfrom the Hartree term.4\nA few words about different forms of the off-diagonal\nterms are in order (see last column in Table I). The form\nnα\nin1−α\njforfwas first introduced by M¨ uller, who found\nα= 1/2 (MU) to be the optimal value.18Goedecker and\nUmrigar(GU) used the modification to the diagonalthat\nremoves the self-interaction terms.40Much similar to the\ndensity-matrix power functional, we generalize these to\nMU-αand GU-αwhere the square root is replaced by an\narbitrary power, presumably lying between half and one.\nAlternatively, it is often physically motivated to reduce\ntheovercorrelationofMUbyswitchingthesignoftheoff-\ndiagonal terms between weakly occupied natural orbitals\nW(BBC1) or additionally banishing the square root for\npairs of strongly occupied orbitals S(BBC2).22,23For\nsimplicity, we define the strongly and weakly occupied\norbitals directly from the Hartree-Fock solution where\nthe occupations are 0 or 1.\nTABLE I: Functions f(ni,nj).SandWrefer to the strongly\nand weakly occupied natural orbitals, respectively.\nf(ni,ni)f(ni,nj/negationslash=i)\nMU\nGUni\nn2\ni√ninj\nMU-α\nGU-α\nP-αni\nn2\ni\nn2α\ni(ninj)α\nBBC1\nBBC1Sni\nn2\ni/braceleftbigg\n−√ninji,j∈W√ninji,j /∈W\nBBC2\nBBC2Sni\nn2\ni\n\n−√ninji,j∈W\nninji,j∈S√ninjelse\nFinally, we would like to point out an issue with the\nphysicality of the obtained solution, which is generally\nmore of a problem in the higher order RDM methods.\nSpecifically, Coleman has shown that necessary and suf-\nficient condition for the 1-RDM to be N-representable,\nmeaning that there exists |Ψ∝an}bracketri}htin the Hilbert space of the\nsystem such that γij=∝an}bracketle{tΨ|a†\niaj|Ψ∝an}bracketri}ht, is that its eigenval-\nues are between 0 and 1 and their sum is N.41However,\nif we additionally demand a symmetry, it may be that\nthe obtained solution is not representable in the symme-\ntry restricted part of the Hilbert space. To be explicit,\nif in our model we demand total angular momentum of\na two electron system to be 2, the symmetry restricted\nHilbert space of states with m= 0,1, and 2 consists\nonly of one state with occupations (1 ,0,1) while 1-RDM\nmethod could give us unphysical occupations (0 .5,1,0.5)\nboth having the same particle number and angular mo-\nmentum (0.5×0+1×1+0.5×2 = 2). While this could\nbe avoidedby imposing additional constraints, we refrain\nfrom doing it as that would be exponentially unfeasible\nfor larger systems. Additionally, it is plausible that the\nunphysical solutions have less weight when the size of the\nphysical Hilbert space increases.B. 2-RDMFT\nIn analogy with the 1-RDMFT, we now minimize a\nfunctional of the 2-RDM\nΓij\nkl=∝an}bracketle{tΨ|a†\nia†\njalak|Ψ∝an}bracketri}ht. (14)\nThe markeddifference is that we now havean exact func-\ntional for the interaction energy\nVee(Γ) =1\n2/summationdisplay\ni,j,k,lΓij\nklVijkl, (15)\nhowever, with the cost of large number of additional pa-\nrameters to optimize with similarly large number of ad-\nditional constraints. Furthermore, the set of constraints\nto be listed below form a relatively stringent set of nec-\nessary conditions that only in special cases is sufficient\nfor the obtained solution to be exactly N-representable,\nmeaning a physical |Ψ∝an}bracketri}htto exist such that Eq. (14) holds.\nThe minimization of the functional (15) is performed\nby forming an augmented Lagrangian function and min-\nimizing it following the algorithm in Ref. 42. The\nminimization is performed with limited memory quasi-\nNewton algorithm of Mathematica, and the form of the\naugmented Lagrangian is\nL=F[γ]−/summationdisplay\niλici+/summationdisplay\nic2\ni/µ (16)\nwhereλiare the Lagrange multipliers and µ >0 is the\naugmentation parameter used to enforce the convergence\nof the constraints ci= 0. In the following, we first intro-\nduce the subset of applied N-representability conditions,\nand after that, impose the further constraints due to the\nfixed total angular momentum, M-representability.\n1.N-representability\nThe trace condition\n/summationdisplay\ni<jΓij\nij=N(N−1)\n2(17)\nis used to fix the particle number to N.\nPositivity conditions form the major part of the N-\nrepresentability constraints. Consider an operator of\nthe formA=/summationtext\ni1i2...ikti1i2...ikai1ai2...aik. Since\n∝an}bracketle{tΨ|A†A|Ψ∝an}bracketri}ht/greaterorequalslant0, it follows that\n/summationdisplay\ni1i2...ikj1j2...jkt∗\ni1i2...iktj1j2...jkΓi1i2...ik\nj1j2...jk/greaterorequalslant0.(18)\nFork= 2 we obtain the 2-positivity condition for the\n2-RDM\n/summationdisplay\nij\nklt∗\nijtklΓij\nkl/greaterorequalslant0. (19)5\nBy a different choice of A, positivity conditions of the\nexact same form can be derived for the two other repre-\nsentations of the 2-RDM\nQij\nkl=∝an}bracketle{tΨ|aiaja†\nla†\nk|Ψ∝an}bracketri}htand\nGij\nkl=∝an}bracketle{tΨ|a†\niaja†\nlak|Ψ∝an}bracketri}ht.(20)\nWhile the representations Γ, Q, andGare all equiva-\nlentastheyarerelatedbythefermionicanticommutation\nrules, the positivity conditions are inequivalent and must\nbe taken into account simultaneously. We use the anti-\nsymmetric basis |(ij)∝an}bracketri}ht= (|ij∝an}bracketri}ht−|ji∝an}bracketri}ht)/2 for Γ and Qma-\ntrices since Γij\nkl=−Γji\nkl=−Γij\nlkandQij\nkl=−Qji\nkl=−Qij\nlk.\nFurthermore following Ref. 42, since all the three matri-\nces are real and symmetric under ij↔kl, the positive-\ndefinite condition can be accounted for simply by writing\nthe matrices as square of symmetric matrices Γ = R2,\nQ=S2, andG=T2(meaning Γ(ij)\n(kl)=/summationtext\n(pq)R(ij)\n(pq)R(pq)\n(kl)\netc.) and optimizing the upper diagonal of matrices R,\nS, andT. The linear relations linking Γ and Qand Γ\nandGbecome the relevant constraint equations\nQij\nkl=Γij\nkl−δikγjl−δjlγik+δilγjk+\nδjkγil+δikδjl−δilδjk,\nGij\nkl=δjlγik−Γil\nkj,(21)\nwhere the 1-RDM γijmay be obtained through\n/summationdisplay\nkGij\nkk=Nγijor/summationdisplay\nkΓik\njk= (N−1)γij.(22)\n2.M-representability\nSince/summationtext\nimia†\niai|Ψ∝an}bracketri}ht=M|Ψ∝an}bracketri}ht, we can form one inde-\npendentnontrivialequationinvolving2-RDM(equivalent\nto the contracted Schr¨ odinger equation in Ref. 29)\n/summationdisplay\nkmkGij\nkk=Mγij. (23)\nThe trace of this is already fixed by Eqs. (17) and (22)\n/summationdisplay\nijmjGii\njj=MN . (24)\nSince in our system the angular momentum quantum\nnumbersmj=jare always non-negative, the maxi-\nmum angularmomentum for a pair ofelectrons to have is\nk2=M−(N−2)(N−3)/2 where the subtracted term is\nthe minimum angularmomentum of N−2electrons. Ad-\nditionally, some of the matrix elements of the 2-RDM are\nzero because states with different total angular momen-\ntum are orthogonal. The independent constraints due to\nthese considerations read\nΓij\nkl= 0 ,i+j∝ne}ationslash=k+lori+j >k2,(25)while Eq. (21) communicates them to QandG. Though\nwe can just drop the corresponding terms from our equa-\ntions, we still need to take into account the ensuing con-\nstraints on the actual variables R,S, andT.\nC. Monte Carlo\nThe calculation of the natural orbital occupations is\nrelatively simple using the Monte Carlo technique. Un-\nlike in the previous Monte Carlo study in Ref. 43, we\nknow from the start the natural orbitals that diagonalize\nthe density matrix, and thus we only need to calculate\nthe occupations.\nStarting from the definition (2), the orthogonality of\nthe natural orbitals, and the expansion of 1-RDM using\nthe natural orbitals\nγ(r,r′) =/summationdisplay\nmnmφ∗\nm(r)φm(r′), (26)\nthe occupations are integrated as\nnm=N/integraldisplay\nΨ∗(r1,r2,...)Ψ(r′,r2,...)\n×φm(r1)φ∗\nm(r′)dr′dr1...drN.(27)\nThis is further reformulated as\nnm=N/integraldisplay\n|Ψ(r1,r2,...)|2Ψ(r′,r2,...)\nΨ(r1,r2,...)\n×φm(r1)φ∗\nm(r′)dr′dr1...drN,(28)\nwhich can be symmetrized and rewritten in Monte Carlo\nexpectation value as\nnm=/angbracketleftBigg/summationdisplay\ni/integraldisplayΨ(r′,r2,...)\nΨ(r1,r2,...)φm(ri)φ∗\nm(r′)dr′/angbracketrightBigg\n{ri}∈|Ψ|2\nwhere the summation is over the coordinates {ri}N\ni=1.\nThe first strategy for Monte Carlo evaluation of the oc-\ncupation is to sample these coordinates from |Ψ|2and to\nintegrate over r′on a grid.43\nFor a better option in our case, we first rewrite\nφm(ri)φ∗\nm(r′) =|φm(r′)|2φm(ri)\nφm(r′)and then do a Monte\nCarlo integration also over r′as\nnm=/angbracketleftBigg/summationdisplay\niΨ(r′,r2,...)\nΨ(r1,r2,...)φm(ri)\nφm(r′)/angbracketrightBigg\n{ri}∈|Ψ|2,r′∈|φm|2\n(29)\nwhere{ri}N\ni=1is again sampled from |Ψ|2andr′from\n|φm|2. This option can be made more stable by noting\nthat the natural orbitals have rotation symmetry and\n|φm(r′)|2depends only on the radial coordinate r′and\nnot on the angle θ′. Now, the radial integral over r′can\nbe done using Monte Carlo integration and the angular\nintegral by averaging over a uniform grid {θ′\nj}Nθ′\nj=1as\nnm=/angbracketleftBigg\n1\nNθ′/summationdisplay\ni,jΨ(r′,r2,...)\nΨ(r1,r2,...)φm(ri,θi)\nφm(r′,θ′\nj)/angbracketrightBigg\n{ri}∈|Ψ|2, r′∈|φm|2.6\nNotice that r′is generated separately for each φm.\nV. RESULTS\nIn the following, we will first compare the performance\nof various 1-RDM functionals in calculating the interac-\ntion energies in different ( N,M) sectors in exactly solv-\nable small quantum Hall droplets. This analysis is deep-\nened by analyzing the occupation numbers obtained for\nthegroundstates. Oncewehaveestablishedthatwehave\na decent functional, we proceed to test its performance\nin larger systems. Since no exact results are available for\nlarger systems at the strongly correlated regime ν≪1,\nwe employ the next best gripping handle, which is the\nLaughlin’s variational wave function for filling fraction\nν= 1/3. Moreover, we compare the energies and occu-\npations obtained with density-matrix power functionals\n(P-α) to the results extracted from the Laughlin’s wave\nfunction with Monte Carlo techniques. From the occupa-\ntion numbers we calculate the edge Green’s function GE,\nwhich gives information about the topological order of\ndifferent quantum Hall phases. It is interesting to see if\nthe newly applied methods (1-RDMFT and 2-RDMFT)\nreproduce the correct decay properties of GE.\nA. Towards a good 1-RDM functional in small\nquantum Hall droplets\nThe energy states of the quantum dot may be written\nas\nEn\nMN=N/planckover2pi1ω+(/planckover2pi1ω−/planckover2pi1ωc/2)M+Vn\nee(M,N),(30)\nwhereVn\nee(M,N) is thenth eigenvalue of the total inter-\naction energyin sector ( M,N), whose dependence on the\nphysical parameters is scaling by a factor e2/ǫl. Because\nof the second term, it follows that all possible ground\nstates forNparticles are found at the intersections of\ntheM-Vee-curve and its convex lower envelope.\nFigure 1(a) shows the M-Vee-curves for 4 elec-\ntrons computed with the configuration interaction (CI),\nHartree-Fock method (HF), and the Brillouin-Wigner\n2nd order perturbation theory (BW) to the HF state.\nThe exact diagonalization (CI) ground states, detected\nby the convex envelope, are marked by circles. While\nboth HF and BW predict the correct ground states, the\nperturbation theory leads to a significant improvement\nto the HF energy. The 2nd order perturbation theory\nis very accurate and close to the CI result for M <14,\nand it is roughly half-way between HF and CI energy for\nM >14.\nPhysically the cusp structure seen in the results fol-\nlows from the energetic advantage of a configuration,\nin which the 4 electrons are located at vertices of a\nsquare. This configuration has non-zero weight only if\nthe angular momentum attains a special value such thatN(N−1)/2≡mod(M,N). The difference between sub-\nsequent magic angular momentum states is the number\nof vortices found at the center of the system. As mag-\nnetic field is increased, vortices that carry quantized an-\ngularmomentum, andinasensequantizedmagneticflux,\nemerge at the center. As the particle number is taken\nvery large, the number of cusps increases while all cusps\nno longer correspond to a ground-state at certain sys-\ntem parameters. Instead, a few of them are more special\nthan others forming the incompressible vacuum state of\ncertain fractional quantum Hall phase as the remaining\ncusps are related to the quasiparticle/vortex excitations\nthat are eventually responsible for the finite extension of\nthe quantum Hall plateaus.\nThe corresponding energies obtained with 1-RDMFT\nare shown in Figs. 1(b) and (c). At this point the choice\nof parameters α= 0.75,α= 0.7, andα= 0.65 for MU-\nα, P-α, and GU-αis an educated guess, whereas the\neffect of the parameter αbecomes apparent in the next\nsection (basically, it tunes the strength of electron cor-\nrelations). Typical to 1-RDMFT calculations in general,\nthe energies are below the CI result. The dashed lines\nsystematically lie below the solid lines of the same tone,\ndue to the self-energy cancellation present in the latter.\nBasic MU and GU functionals clearly overestimate the\ncorrelation energy and behave even qualitatively wrong\nas they fall too fast with increasing M. For the rest of\nthe functionals, Veeseems to decline at about the correct\nrate as a function of M. However, a nice cusp structure\nis only seen with the BBC1S and BBC2S functionals,\nwhich inherit the cusps from the HF state used in the\nselection of the strongly and weakly occupied orbitals.\nOverall the energetically best of these 1-RDM function-\nals(P-0.7, GU-0.65, BBC1S,andBBC2S) performbetter\nthan the 2nd order perturbation theory when M >14,\nalthough only BBC2S produces the correct ground state\nstructure.\nThe equivalent curves for 6 electrons are shown in\nFigs. 1 (d)-(f). As the 6-electron results are quantita-\ntively like the 4-electron results, the performance of dif-\nferent functionals seems to be rather insensitive to the\nparticle number. For six electrons, the cusps should oc-\ncur atN(N−1)/2≡mod(M,N) orN(N−1)/2≡\nmod (M,N−1) corresponding to a hexagonal configura-\ntion or a pentagonal configuration with one electron at\nthe center. All the cusps are not correctly reproduced\nwith any 1-RDM functional, though BBC2S result fol-\nlows the cusp structure quite well at M >27.\nThe quantumHall dropletmodel hasthe propertythat\nanincreasein Mincreasestheareaofthedropletandalso\nthe electronic correlations quantified in reduction of the\noccupation numbers of the natural Landau level orbitals.\nThe average occupation number of the relevant orbitals\nis close to the corresponding macroscopic quantum Hall\nfilling fraction νand becomes exact as Nis taken to\ninfinity. For example, the ν= 1 state occurs at the\nminimum angular momentum M=N(N−1)/2 where\nthe firstNorbitals haveoccupation1. Second exampleis7\n610141822263011.522.53\nMVee (e2/εl)\n  \nHF\nCI\nBW\n610141822263011.522.53\nMVee (e2/εl)\n  \nCI\nMU\nGU\nMU−0.75\nP−0.7\nGU−0.65\n610141822263011.522.53\nMVee (e2/εl)\n  \nCI\nBBC1\nBBC1S\nBBC2\nBBC2S\n(a) (b) (c)\n15212733394544.555.566.57\nMVee (e2/εl)\n  \nHF\nCI\nBW\n15212733394544.555.566.57\nMVee (e2/εl)\n  \nCI\nMU\nGU\nMU−0.75\nP−0.7\nGU−0.65\n15212733394544.555.566.57\nMVee (e2/εl)\n  \nCI\nBBC1\nBBC1S\nBBC2\nBBC2S\n(d) (e) (f)\nFIG. 1: (Color online) The minimum interaction energy Veeat each angular momentum Mfor 4 (a-c) and 6 electrons (d-f).\nThe exact diagonalization ground states detected by the con vex envelope are marked with circles in (a) and (d).\nM= 10 M= 14 M= 18 M= 22 M= 26 M= 30\nHF\nCI\nBW\nMU\nGU\nMU−0.75\nP−0.7\nGU−0.65\nBBC1\nBBC1S\nBBC2\nBBC2S\nM= 21 M= 25 M= 30 M= 35 M= 39 M= 45\nHF\nCI\nBW\nMU\nGU\nMU−0.75\nP−0.7\nGU−0.65\nBBC1\nBBC1S\nBBC2\nBBC2S\nFIG. 2: (Color online) The occupation numbers for selection of the 4-electron (upper panel) and 6-electron (lower panel ) ground\nstates indicated in Fig. 1 (a) and (d), respectively. A bar th at fills the assigned height corresponds to occupation numbe r 1.\nOccupations are ordered according to the increasing single -particle angular momentum mof the lowest Landau level orbitals\nstarting with 0 at the left. Because most likely location of e lectron at orbital mis at distance r=√mfrom the center, one\ncan think the set of orbitals as a radially discretized disk.8\nthe Laughlin’s wave function4for filling fraction ν= 1/3\nΨ1/3\nL({zi}) =/productdisplay\ni<j(zi−zj)3e−1\n2/summationtext\nizi¯zi.(31)\nIt has angular momentum M= 3N(N−1)/2 and 3(N−\n1)+1nonemptyorbitalssuchthatonaveragethefraction\nν=N/(3(N−1)+1)≈1/3 is filled. The Laughlin state\nhas about 0.98 overlap with the exact ground state for 4\nand 6 electrons, and it is the lowest angular momentum\nzero-energy state of the short-range model interaction44\nV1(zij) =∂zi∂¯ziδ(zi−zj)+i↔j . (32)\nNote, howeverthatwhileforfourelectronsthehighlycor-\nrelated Laughlin state occurs at M= 18 near the center\nof theM-windowin the M-Vee-curves, the corresponding\nangular momentum for six electrons is M= 45, and it\nis the highest Mincluded in the corresponding figures.\nOverall, it seems that the perturbation theory works en-\nergetically well near ν= 1 where the correlations are\nweak while the 1-RDM functionals perform better at the\nstronglycorrelatedregime ν≪1, whichraisessomehope\nfor the 1-RDMFT to provevaluable in quantum Hall sys-\ntems.\nBut how close are the obtained minimizing 1-RDMs\nactually to the exact results? Recall that the natural\norbitals in the lowest Landau level are fixed and their oc-\ncupations completely specify the 1-RDM. Figure 2 shows\nthe occupation numbers corresponding to the ground\nstates indicated in Fig. 1 (a) and (d).\nLooking at the first row of occupation numbers for 4\nelectrons, the next HF ground state is obtained from the\nprevious by adding a hole to the center leading to angu-\nlar momentum increase N. The exact CI result below\nis similar but, in addition, the correlations spread the\noccupations at each step. On the third row, the sec-\nond order perturbation theory BW has a small spread\nof occupations in accordance with the energy curves in\nFig. 1(a). The 1-RDM functional results on the follow-\ning 9 rows are varied in nature. In accord with the poor\nenergies, MU functional leads to a way too large spread\nof occupation and so does the GU, although the latter\nalso pins some occupations to one. The inclination to-\nwardspinning is due to the self-energycancellation, since\nwithout the cancellation non-pinned occupations lead to\nnegative self-energy contribution lowering the total en-\nergy (n2\ni−ni<0 for 0<ni<1). This is the reason why\nMU-0.75 is more spread out than GU-0.65, and BBC oc-\ncupations are a bit less pinned than BBCS occupations.\nDespite the better energetics, the GU-0.65, BBC1S, and\nBBC2s occupations seem not to be much better than the\nsecond order perturbation theory. On the contrary, P-\n0.7, on the other hand, has both quite good energy and\noccupations numbers only slightly less spread than the\nexact result. Note however, the nonzero first occupation\natM= 26 andM= 30 in contrast to the exact result.\nFor six electrons (Fig. 1 (lower)), the occupations are\nsimilar. Note the high probability for one electron to beat the center in some of the HF and CI ground states,\ncorrectly reproduced by many of the functionals. The\nsix-electron occupation numbers at M= 30 andM= 35\ncan be directly compared to the occupations obtained\nwith DFT in Ref. 15, and they are found to be a bit\nsimilar to our BBC1 or BBC2 results.\nOn the whole, the P-0.7 power functional seems like a\ngood candidate functional for systems with large number\nof electrons. GU- αwithα <0.65 and MU- αwithα >\n0.75 could also work well, however, since the diagonal\npart of the P- αfunctional is somewhat a compromise\nbetween these two, we employ the P- αfunctional in the\nremainder of the manuscript.\nB. 1-RDM at large N\nThe results of the previous section suggest that the\ndensity matrix power functional (P- α) could be a good\nfunctional in quantum Hall systems, and thus we ap-\nply it to large systems for a few parameters α. More-\nover, we concentrate on the ν= 1/3 state, whereby close\nto exact nonperturbative results can be computed with\nthe Laughlin’s variational wave function (Eq. (31)) using\nMonte Carlo. It is natural to limit the number of natural\norbitalstothatofthe Laughlinwavefunction3( N−1)+1\nalthough the realistic Coulomb ground state would actu-\nally extend, weakly though, to a few more orbitals.\nThe occupation numbers obtained in such way for\nN= 10,N= 20, andN= 30 are presented in Fig. 3.\nAs seen in the exact result (first row in Fig. 3(a)) the\nlong-range Coulomb correlations cause oscillations in the\noccupations around 1 /3 (CI), apart from the edge den-\nsity modulation not present in the Laughlin’s occupa-\ntion numbers (MC). Sliding αfrom 0.6 to 0.75 gradu-\nally strengthens these oscillations. P-0.65 is close to the\nLaughlin’soccupations while P-0.675is closeto the exact\noccupations (CI). Similar behavior is seen at larger par-\nticle numbers in Figs. 3 (b) and (c), where P-0.675yields\nagainoccupations plausibly closest to the exact unknown\nresult.\nThe oscillations in the occupation numbers reflect the\nformation of an edge striped phase. An extrapolated\nphenomenological formula for the slow-decaying charge\ndensity oscillations at the ν= 1/3 edge at the thermo-\ndynamic limit is given in Ref. 25\nρ(s) =1\n6(Erf(s)+1)/bracketleftbigg\n1+1\n2J0/parenleftbigπ\n2(s−1)/parenrightbig/bracketrightbigg\n(33)\nwheres/√\n2 is the distance from the edge located at/radicalbig\n3(N−1), Erfis the Gauss errorfunction, and J0is the\nBessel function of the first kind. Fig. 4(a) shows the ra-\ndialchargedensities ∝an}bracketle{tΨ|ψ†(r)ψ(r)|Ψ∝an}bracketri}htcalculatedfromthe\n30 electron occupation numbers compared to the extrap-\nolated formula (red dashed line). The latter has slightly\nlongeroscillationwavelengthcomparedtotheP- αresults\nwhile the amplitude of oscillationssuggeststhat the opti-\nmalvalueof αissomewherebetween0.675and0.7. Thus,9\nCI    \nMC\nP−0.6  \nP−0.65 \nP−0.675\nP−0.7  \nP−0.75 MC\nP−0.6  \nP−0.65 \nP−0.675\nP−0.7  \nP−0.75 MC\nP−0.6  \nP−0.65 \nP−0.675\nP−0.7  \nP−0.75 \n(a) (b) (c)\nFIG. 3: (Color online) The occupation numbers at the angular momentum of the 1 /3 Laughlin state for (a) N= 10, (b)\nN= 20, and (c) N= 30 electrons with 3( N−1)+1 natural orbitals. MC is the exact Laughlin wave functio n result extracted\nwith Monte Carlo. The dashed line marks 1 /3 occupation.\nr(l)ρ (1/π l2)\n0 5 1000.511.522.53\nsin(θ/2)|G| (1/π l2)\n  \n0.1 0.5 110−510−410−310−2\nCI\nMC\nP−0.6\nP−0.65\nP−0.675\nP−0.7\nP−0.75\nsin(θ/2)|G| (1/π l2)\n  \n0.1 0.5 110−510−410−310−2\nMC\nP−0.6\nP−0.65\nP−0.675\nP−0.7\nP−0.75\n(a) (b) (c)\nFIG. 4: (Color online) (a) 30 electron density profiles corre sponding to Fig. 3(c) shifted vertically by 0.5 unit. Red das hed line\nis the phenomenological estimate for the density oscillati ons at the thermodynamic limit. (b),(c) The decay of the edge Green’s\nfunction calculated at r=/radicalbig\n3(N−1)+1 from the occupations in Fig. 3(a) and (b), respectively . The short red lines illustrate\nslopes−3 (theoretical prediction for ν= 1/3) and−1 (Fermi liquid).\nalthough Eq. (33) has zero free parameters, it matches\nthe 1-RDM results reasonably well.\nEdge Green’s function GEis the amplitude for electron\nto propagate a distance along the edge. In the quantum\nHall droplet, the distance is related to the angle θbe-\ntween the two points, and GE=∝an}bracketle{tΨ|ψ†(z0eiθ)ψ(z0)|Ψ∝an}bracketri}ht,\nwherez0is a point of the edge. Chiral Luttinger liquid\ntheory of the fractional quantum Hall edge predicts theuniversal asymptotic behavior9,10\n|GE| ∝ |z0eiθ−z0|−g∝ |sin(θ/2)|−g,(34)\nwhereg= 3 forν= 1/3. Values of g∝ne}ationslash= 1 lead to non-\nOhmic current-voltagedependence I∝Vgin the tunnel-\ning experiments. However, thus observed experimental\nvalue,g≈2.2−2.8, is contrary to the theory possibly\nsample dependent.26,27\nThe decay of |GE|calculated with the density-matrix\npower functionals is compared to the exact (CI) and the10\nLaughlin wave function’s result (MC) in Figs. 4(b) and\n(c) forN= 10 andN= 20, respectively (not shown case\nN= 30 looks similar). The short lines signify power-law\nexponentsg= 1 andg= 3. Except for the curve cor-\nresponding to P-0.75, which oscillates heavily, the P- α\ncurves follow closely the theoretical dashed black line un-\ntil sin(θ/2)≈0.3, after which they sheer off the course to\nyield an exponent 1. This is due to the incorrect weights\nof the occupation numbers near the edge of the system\nand investigated further in the next subsection where we\napply 2-RDMFT to a smaller system.\nThe interaction energies are shown in Table II. The\nexactness of the Laughlin trial wave function’s energy\n(MC) up to 0.1% for N= 10 is expected to carry on to\nthe larger electron numbers. The interaction energy is\nseen to increase as a function of αin P-αand is optimal\nwith the functional P-0.7, which attains 99.9% of the\ninteraction energy for N= 20 and 30. Typical to 1-\nRDMFT calculations in general, the energies are mostly\nbelow the assumed nearly exact Monte-Carlo energy.\nWhile knowledge of the ground state energy may be\nuseful when comparing different methods, only energy\ndifferences are physically meaningful. In the RDM meth-\nods, we can calculate the energy differences between\nlowest energy states of different ( M,N) sectors such as\naddition energy (change in N) as well as quasiparticle\nand some edge excitations ( Mchanges). If the excited\nstate becomes the ground state for some parameters, the\nGilbert’s theorem guarantees the existence of a 1-RDM\nfunctional minimized by the exact 1-RDM but even if\nthis is not the case, a good functional might still exist.\nAs mentioned previously, some of the cusps in M-Vee-\ncurves correspond to the quasiparticle excitations of sta-\nTABLE II: Interaction energy in units of e2/ǫlat angular\nmomentum M= 3N(N−1)/2 and the percentage captured\nof the energy of theLaughlin wave function for density-matr ix\npower functionals P- α.\nN= 10N= 20N= 30\nCI 10.14\nMC 10.15 32.92 64.00\nP-0.6 8.9330.36 60.08\nP-0.65 9.5831.72 62.15\nP-0.675 9.8732.34 63.10\nP-0.7 10.12 32.88 63.93\nP-0.75 10.44 33.44 64.80N= 10N= 20N= 30\n99.9%\n100.0% 100.0% 100.0%\n88.0% 92.2% 93.9%\n94.4% 96.4% 97.1%\n97.2% 98.2% 98.6%\n99.7% 99.9% 99.9%\n102.9% 101.6% 101.2%\nTABLE III: Interaction energy of one elementary quasihole\nexcitation in units of e2/ǫland the percentage captured of\nthe interaction energy of the wave functional quasihole mod el\n(MC).\nN= 10N= 20N= 30\nMC -0.314 -0.497 -0.635\nP-0.6 -0.396 -0.601 -0.758\nP-0.65 -0.381 -0.588 -0.736\nP-0.675 -0.376 -0.583 -0.733\nP-0.7 -0.376 -0.584 -0.689\nP-0.75 -0.378 -0.585 -0.326N= 10N= 20N= 30\n100% 100% 100%\n126% 121% 119%\n121% 118% 116%\n120% 117% 115%\n120% 118% 109%\n120% 118% 51%blequantum Hallphases. Next, weconsidersuchaquasi-\nhole excitation above the ν= 1/3 state. The quasihole is\na charged vortex carrying fractional charge q=e/3 and\nobeyinganyonicstatistics.45,46Toobtaininteractionpart\nof the quasihole excitation energy at ν= 1/3, we need to\ncalculate the difference Vee(M1/3+N,N)−Vee(M1/3,N).\nThe angular momentum M1/3+Nfollows from the\nLaughlin’s quasihole wave function, which is also used to\ncompute an estimate for Vee(M1/3+N,N) with Monte\nCarlo. Due to the fact that Laughlin’s wave function is\nmore accurate than the quasihole wave function, varia-\ntional principle implies that the Monte Carlo estimate\nto the (negative) contribution to the excitation energy\nis likely an upper bound to the exact result. Neverthe-\nless, for 8 particles the difference to exact CI result is\nless than 0.2% so we expect the estimates to be quite\naccurate. The quasihole wave function reads\nΨ1/3\nL({zi}) =/productdisplay\ni(zi−zCM)/productdisplay\ni<j(zi−zj)3e−1\n2/summationtext\nizi¯zi,(35)\nwherezCM= (1/N)/summationtext\nizi. The interaction contribution\nto the excitation energy is shown in Table III. The power\nfunctionals appear to overestimate the energy gap by 10\nto 20 percent compared to the trial wave function though\nthe results seem to get more accurate with increasing N.\nThis preliminary result indicates that the method could\nprove useful in assessing the stability of different models\nfor quantum Hall phases characterized by certain angu-\nlar momentum and spin. Additionally, 1-RDM method\noffers a simple framework to include the higher Landau\nlevels, however, instead of the bare eigenvalues of the re-\nduced density matrix, one must then optimize the eigen-\nvectors also.\nC. 2-RDMFT results for three electrons\nInthisfinalpart, weapplythe exact2-RDMfunctional\n(Eq.(3)) toathreeelectrondropletinthe 1/3stateagain\nwiththemaximumsingle-particleangularmomentumset\nto 3(N−1). We will see that the resulting 2-RDM,\nthough not strictly physical, is close to the exact solution\nand yields better results than our 1-RDM functionals.\nCompared to the 1-RDM calculations seen above, the\ncomputational cost of the problem in the 2-RDM opti-\nmization is considerably larger and scales at higher order\np6(versusp4) with the number of single-particle orbitals\np. The 1/3 Laughlin state for 3 electrons has angular\nmomentum M= 9 and requires only 7 single-particle\nstates. However, the number of optimization variables in\nΓ,Q, andGare 276, 378, and 1225, respectively, and the\nconstraint equations (17,21,23,25) together lead to 2798\nconstraints each facilitated by a Lagrange multiplier. In\npractice, this means that this method takes more time\nthan exact diagonalization in any system that could be\nsolved in a reasonable time. However, owing to the ex-\nponential scaling of the exact diagonalization problem,\nthe situation could change in future with development11\nof faster computers and more efficient semi-definite pro-\ngramming and optimization algorithms.\nFigure 5 shows the occupation numbers and the decay\noftheedgeGreen’sfunctionforthreeelectronsat ν= 1/3\n(M= 3N(N−1)/2 = 9). Due to a finite size effect, the\nexact diagonalization Green’s function has a downward\ncusp at sin( θ/2)≈0.9. The 2-RDMFT result has a sim-\nilar cusp while the P-0.675 result, which turns smoothly\nto a lower slope decay, does not have one. We verified\nthat this is due to the difference in the weights of the last\nthree occupation numbers corresponding to the edge of\nthe system. Since the HF solution would have exponent\ng= 1, the correct decay property of the Green’s function\nfollows from the off-diagonal terms of the interaction op-\nsin(θ/2)|G| (1/π l2)\n  \n0.1 0.5 110−310−2\nCI\n2RDMFT\nP−0.675m\nnm\nFIG. 5: (Color online) The decay of the edge Green’s function\ncorresponding to the inset 3-electron occupation numbers. As\npreviously, the short red lines illustrate slopes −g=−3 and\n−1, andGEis evaluated at r=/radicalbig\n3(N−1)+1.\nx (l)y (l)\n−2 0 2−202\nx (l)−2 0 2\nx (l)−2 0 2\n(a) (b) (c)\nFIG. 6: The pair-correlation function g(z1,z2) =\nρ2(z1,z2)/ρ(z1) with the first coordinate placed at the den-\nsity maximum of the negative y-axis z1=−i√\n3 for (a) exact\ndiagonalization, (b) 2-RDMFT, and (c) 1-RDMFT with P-\n0.675. Contours are separated by 0.05 1 /πl2and start from\n0.05 1/πl2in (a) and (b) and from 0 in (c).erator. The 1-RDMFT that only uses the diagonal Vijij\nandVijjiterms of the interaction matrix can not yield\nthe correct behavior unless we have a very good density\nmatrix functional.\nRecall that the backbone of the 1-RDMFT was the\napproximation of the pair-density. Figure 6 shows the\npair-correlation functions corresponding again to exact\ndiagonalization, 2-RDMFT, and P-0.675, where the lat-\nter is reconstructed from the 1-RDM using Eq. (4). Al-\nthough the 1-RDM reconstructed pair-correlation func-\ntion is reasonable vanishing at the position of the fixed\nelectron, only the 2-RDMFT is able to produce the two-\npeak structure of the exact result with reduced density\nalong they-axis.\nGranted that the interaction energy functional in the\n2-RDMFT is exact, the results still do not coincide\nwith the exact diagonalization results because the 3-\nrepresentability conditions that ensure the physicality in\nthis3-electronsystemcannotbetakenintoaccountwith-\nout invoking the generalization of 2-RDMFT to include\nhigher order RDMs. Figure 7(a) shows the exact non-\nzero matrix elements of the 2-RDM Γ(ij)\n(kl)in the antisym-\nmetric basis |(ij)∝an}bracketri}ht= (|ij∝an}bracketri}ht − |ji∝an}bracketri}ht)/2 and (b) calculated\nwith the 2-RDMFT.The largestdiscrepancybetween the\nresults is the vanishing of the matrix elements involving\nstates|(14)∝an}bracketri}htor|(25)∝an}bracketri}htfor the exact result, while for ex-\nample Γ(05)\n(14)∝ne}ationslash= 0 in the 2-RDMFT result. The matrix\nelements should vanish, as they are related to the ficti-\ntiousmany-bodybasisstatesthathavedoubleoccupancy\nof orbital 4 or 2 (1+4+4= 2+2+5= 9, the total angu-\nlarmomentum). Consequently, the absolutevaluesofthe\nmatrix elements also differ. However, the computed pair-\ncorrelation function and edge Green’s function suggest\nthat many of the physical quantities are not significantly\naffected by the absence of exact N-representability, and\nthat the lack of computer power might be the only real\nstumbling stone in the way of the 2-RDMFT.\nVI. CONCLUSIONS\nIn summary, we have applied the one-body reduced\ndensity-matrixfunctionaltheorytosmallandlargequan-\ntum Hall droplets at the spin-polarized strong magnetic\nfield regime. The density-matrix power functional seems\nto work reasonably well at the strongly correlated ν≪1\nregime where the occupation numbers of the natural or-\nbitals are small. The newly applied method yields pre-\nviously inaccessible valuable information with large par-\nticle numbers about the energetics and quantities that\nderive from the one-body reduced density matrix. The\ndensity-matrix power functional yields reasonable bulk\ndensities with the power parameter in the range 0.65-\n0.7. However, the detailed properties of the edge are not\nproduced accurately with this functional. Moreover, it\nis not known if a good functional for a specific quan-\ntum Hall state would work universally at different filling12\n(03)(04)(05)(06)(12)(13)(14)(15)(16)(23)(24)(25)(26)(34)(35)(36)(45)(03)(04)(05)(06)(12)(13)(14)(15)(16)(23)(24)(25)(26)(34)(35)(36)(45)\n  \n(03)(04)(05)(06)(12)(13)(14)(15)(16)(23)(24)(25)(26)(34)(35)(36)(45)(03)(04)(05)(06)(12)(13)(14)(15)(16)(23)(24)(25)(26)(34)(35)(36)(45)\n−0.5−0.4−0.3−0.2−0.100.10.20.30.40.5\n(a) (b)\nFIG. 7: (Color online) Illustration of the 2-RDM Γ(ij)\n(kl)for three electrons in the ν= 1/3 state computed with (a) exact\ndiagonalization and (b) 2-RDMFT. The same color bar applies to both figures while zero values are left white. The negative\nmatrix elements are indicated by black edge.\nfractions. Nevertheless,thecomputationallyexpensive2-\nbody reduced density matrix method seems to facilitate\nthe properties of the edge, though this should be verified\nwith a larger electron number in future.\nProspects of the 1-RDMFT in quantum Hall systems\ninclude generalizations to spin and multiple Landau lev-\nels. Studies with systems without edge (sphere) and non-\ntrivial topology (torus) are also encouraged while new\nstate of the art energy functionals are of course very wel-\ncome.Acknowledgements\nThis study hasbeen supported bythe AcademyofFin-\nland through its Centres of Excellence Program (2006-\n2011). ET acknowledges financial support from the\nVilho, Yrj¨ o, and Kalle V¨ ais¨ al¨ a Foundation of the Finnish\nAcademy of Science and Letters. We also thank I.\nMakkonen for careful reading of the manuscript and E.\nR¨ as¨ anen and R. van Leeuwen for useful discussions.\n1K. v.Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.\n45, 494 (1980).\n2D. C. Tsui, H. L. 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The obtained probability distribution s are broad and the vacuum expectation\nvalue of the Hamiltonian density is not fully representativ e of the vacuum energy density.\nPACS numbers: 11.10.-z; 11.10.Ef; 03.70.+k; 98.80.Cq\nQuantum field theory (QFT) is a theoretical frame-\nwork that has enabled, via its phenomenologically suc-\ncessful models such as the Standard Model, a detailed\nquantitative understanding of a vast number of physical\nphenomena. In numerous areas of its application QFT\npredictions are confirmed at ever greater accuracy with\nevery new experiment. Yet, these QFT models with such\na remarkable performance in precise determination of\nscattering cross-sections and decay rates also give a defi-\nnite prediction for the vacuum expectation value (VEV)\nofthe Hamiltonian density. As longaswe disregardgrav-\nitationalinteraction, the sizeofthis VEVis irrelevant[1].\nWhen the gravitational interaction, described by general\nrelativity, is included the VEV of the Hamiltonian den-\nsity can no longer be ignored. Moreover, any estimate of\nthis quantity within QFT, also referred to as zero point\nenergy density , showsthat this contribution to the source\nterm of the Einstein equation is crucial. The zero point\nenergy density represents a major contribution to the to-\ntal value of the cosmological constant (CC) [2]. The fact\nthat the size of this contribution, as well as the size of\nother identified contributions to the cosmological con-\nstant, are many orders of magnitude larger than the ob-\nserved value of the CC is at the core of the notorious old\ncosmological constant problem . A deeper understanding\nofallcontributionsto the cosmologicalconstantis clearly\nneededto resolvethe oldCCproblem, perceivedasoneof\nthemostseriousproblemsintheoreticalphysics. Another\nimportant physical phenomenon where the understand-\ning of the energy density in the vacuum is paramount is\ninflation [3].\nIn this paper our primary goal is understanding of the\nenergydensitythatquantumfieldsdevelopinthevacuum\nstate. Hereafter we call this energy density the vacuum\nenergy density and investigate how well it is represented\nby a VEV of the Hamiltonian density. The central result\nof this paper is the calculation of the probability density\nfunction (p.d.f.) of the vacuum energy density for both\n∗Electronic address: gorand@thphys.irb.hr\n†Electronic address: dglavan@dominis.phy.hr\n‡Electronic address: shrvoje@thphys.irb.hrreal and complex free massless scalar fields in Minkowski\nspace. We present a fully analytically tractable treat-\nment of the problem and finally provide the analytical\nexpressions for the probability density functions of the\nvacuum energy density. As our motivation comes from\ncosmology we do not assume normal ordering during cal-\nculation. However, since normal ordering is a widely ac-\ncepted regularizationprocedure in QFT we present “nor-\nmally ordered” results in the discussion. Throughout the\npaper we adopt the notation and conventions of [1].\nWe start with a direct observation that, for both real\nand complex scalar fields, the Hamiltonian Hand the\nHamiltonian density Hoperators do not commute. The\nvacuum state |0∝an}bracketri}htis not an eigenstate of H. The mea-\nsurement of the Hamiltonian density in vacuum can re-\nsult in different values of vacuum energy density with\ndifferent probabilities. A quick calculation of the vari-\nanceσ2\nH=∝an}bracketle{tH2∝an}bracketri}ht − ∝an}bracketle{tH∝an}bracketri}ht2shows that σ2\nH= 2/3∝an}bracketle{tH∝an}bracketri}ht2for\nreal and σ2\nH= 1/3∝an}bracketle{tH∝an}bracketri}ht2for complex massless scalar field.\nThe result that σHis of the same order as ∝an}bracketle{tH∝an}bracketri}htreveals\nthe existence of a broad p.d.f. for the vacuum energy\ndensity. At this point, one can question the relevance of\nHas an observable. Namely, from the practical point\nof view, it is more natural (i.e. physical) to observe a\nsmoothed energy density in a finite space-time region. In\nthat case, the relevant region depends on the size of the\nsystem under investigation. It can be as big as the hori-\nzon in the case of cosmology or as small as experimental\nsetup in the case of Casimir effect. The upper limit cor-\nresponding to the infinite region gives an uninteresting\nresult (p.d.f. is a delta function because H ∝H) and\nthe lower limit corresponds to taking Has an observ-\nable. Smoothing of energy density in a finite space-time\nregion considerably complicates the calculation. Here we\nassume that His a legitimate observable and our result\nis directly useful as a limiting value for making measure-\nments in smaller and smaller volumes. It is important to\nnote that treating Has an observable can be crucial in\nunderstanding cosmological inflation where our patch of\nthe Universeoriginatesfrom a virtually pointlike element\nof space.\nSince the spectrum of the Hamiltonian density opera-\ntor is positively defined, the p.d.f. of the vacuum energy\ndensity isto be determined byusing aLaplacetransform.2\nIfp(ǫ) is the p.d.f. of interest, its Laplace transform is\ngiven by\nL{p(ǫ)}=/integraldisplay∞\n0e−sǫp(ǫ)dǫ=∞/summationdisplay\nn=0(−s)n\nn!/integraldisplay∞\n0ǫnp(ǫ)dǫ\n=∞/summationdisplay\nn=0(−s)n\nn!∝an}bracketle{tHn∝an}bracketri}ht, (1)\nwhere∝an}bracketle{tHn∝an}bracketri}htis the VEV of n-th power of the Hamilto-\nnian density operator. Dimensions of sandǫare [E]−4\nand [E]4, respectively. Since the original function is\nuniquelydefinedbyitsimage,oncethetransform L{p(ǫ)}\nis known, the p.d.f. p(ǫ) can be determined by using var-\nious properties of the Laplace transform [4].\nTo obtain the transform L{p(ǫ)}one has to calculate\n∝an}bracketle{tHn∝an}bracketri}ht,n= 1,...,∞. Hamiltonian densities of real ( HR)\nand complex ( HC) massless scalar fields are given by\nHR(x,t) =−/integraldisplay\nd3kd3qωkωq+k·q\n4(2π)3√ωkωqrk(x,t)rq(x,t),\n(2)\nHC(x,t) =/integraldisplay\nd3kd3qωkωq+k·q\n2(2π)3√ωkωqc†\nk(x,t)cq(x,t),\n(3)\nwhereωp=|p|,\nrp(x,t) =ape−i(ωpt−p·x)−a†\npei(ωpt−p·x),(4)\ncp(x,t) =ape−i(ωpt−p·x)−b†\npei(ωpt−p·x),(5)\nanda†\np,b†\np(ap,bp) are the usual creation (annihilation)\noperators of scalar fields. For further considerations it is\nconvenient to know contractions and commutators of the\nabove mentioned operators. The relevant contractions\nand commutators are\nrkrq=−δ3(k−q)e−i(ωk−ωq)t+i(k−q)·x,(6)\nckc†\nq=c†\nkcq=δ3(k−q)e−i(ωk−ωq)t+i(k−q)·x,(7)\nckcq=c†\nkc†\nq= 0, (8)\n[rk,rq] = [ck,cq] = [c†\nk,c†\nq] = [ck,c†\nq] = 0,(9)\nwhere, from now on, arguments ( x,t) of the operators r\nandcare omitted for the sake of notation simplicity.\nFrom Eqs. (2) and (3), it follows that vacuum expec-\ntation value of Hncan be written as\n∝an}bracketle{tHn\nR∝an}bracketri}ht=/integraldisplayd3k1...d3k2n\n4n(2π)3nn/productdisplay\ni=1ωkiωkn+i+ki·kn+i√ωkiωkn+i\n×(−)n∝an}bracketle{t0|n/productdisplay\nj=1rkjrkn+j|0∝an}bracketri}ht, (10)\n∝an}bracketle{tHn\nC∝an}bracketri}ht=/integraldisplayd3k1...d3k2n\n2n(2π)3nn/productdisplay\ni=1ωkiωkn+i+ki·kn+i√ωkiωkn+i\n×∝an}bracketle{t0|n/productdisplay\nj=1c†\nkjckn+j|0∝an}bracketri}ht, (11)where the notation for momenta is renamed for later con-\nvenience.\nThe expectation values in integrands of Eqs. (10) and\n(11) can be calculated by using Wick’s theorem [5] and\ncontractions from Eqs. (6) – (8). It follows that\n∝an}bracketle{t0|n/productdisplay\ni=1rkirkn+i|0∝an}bracketri}ht=(−)n\n2nn!(12)\n/summationdisplay\n{i1,...,i2n}=P{1,...,2n}δ(ki1−ki2)...δ(ki2n−1−ki2n),\n∝an}bracketle{t0|n/productdisplay\ni=1c†\nkickn+i|0∝an}bracketri}ht= (13)\n/summationdisplay\n{j1,...,jn}=P{n+1,...,2n}δ(k1−kj1)...δ(kn−kjn),\nwhereParepermutations. Thetotalnumbersoftermsin\nEqs. (12) and (13) are (2 n)!/(2nn!) andn!, respectively.\nBecause of the symmetry on the exchange of dummy\nvariables ki, it is possible to further simplify Eqs. (10)\nand (11) using binomial coefficients\n∝an}bracketle{tHn\nR∝an}bracketri}ht=n/summationdisplay\nl=0/parenleftBign\nl/parenrightBig/integraldisplayd3k1...d3k2n\n4n(2π)3n/parenleftBiggl/productdisplay\ni=1√ωkiωkn+i/parenrightBigg\n×/parenleftBiggn/productdisplay\ns=l+1ks·kn+s√ωksωkn+s/parenrightBigg\n(−)n∝an}bracketle{t0|n/productdisplay\nj=1rkjrkn+j|0∝an}bracketri}ht,(14)\n∝an}bracketle{tHn\nC∝an}bracketri}ht=n/summationdisplay\nl=0/parenleftBign\nl/parenrightBig/integraldisplayd3k1...d3k2n\n2n(2π)3n/parenleftBiggl/productdisplay\ni=1√ωkiωkn+i/parenrightBigg\n×/parenleftBiggn/productdisplay\ns=l+1ks·kn+s√ωksωkn+s/parenrightBigg\n∝an}bracketle{t0|n/productdisplay\nj=1c†\nkjckn+j|0∝an}bracketri}ht,(15)\nwhere we assume that in the case of an ill-defined prod-\nuct (/producttext0\ni=1and/producttextn\ni=n+1) the whole contribution in corre-\nsponding brackets is equal to 1.\nSince the factors ki·kjare odd functions of momenta\nand taking into account Eqs. (12) and (13), the nonva-\nnishing contributions in Eqs. (14) and (15) can be fac-\ntorized. In factorized form Eqs. (14) and (15) are given\nby\n∝an}bracketle{tHn\nR∝an}bracketri}ht= (16)\n1\n2nn/summationdisplay\nl=0/parenleftBign\nl/parenrightBig/bracketleftBigg/integraldisplayd3k1...d3kld3kn+1...d3kn+l\n2l(2π)3l\n×/parenleftBiggl/productdisplay\ni=1√ωkiωkn+i/parenrightBigg\n(−)l∝an}bracketle{t0|l/productdisplay\nj=1rkjrkn+j|0∝an}bracketri}ht/bracketrightBigg\n×/bracketleftBigg/integraldisplayd3kl+1...d3knd3kn+l+1...d3k2n\n2n−l(2π)3(n−l)\n×/parenleftBiggn/productdisplay\ns=l+1ks·kn+s√ωksωkn+s/parenrightBigg\n(−)n−l∝an}bracketle{t0|n/productdisplay\np=l+1rkprkn+p|0∝an}bracketri}ht/bracketrightBigg\n,3\n∝an}bracketle{tHn\nC∝an}bracketri}ht= (17)\nn/summationdisplay\nl=0/parenleftBign\nl/parenrightBig/bracketleftBigg/integraldisplayd3k1...d3kld3kn+1...d3kn+l\n2l(2π)3l\n×/parenleftBiggl/productdisplay\ni=1√ωkiωkn+i/parenrightBigg\n∝an}bracketle{t0|l/productdisplay\nj=1c†\nkjckn+j|0∝an}bracketri}ht/bracketrightBigg\n×/bracketleftBigg/integraldisplayd3kl+1...d3knd3kn+l+1...d3k2n\n2n−l(2π)3(n−l)\n×/parenleftBiggn/productdisplay\ns=l+1ks·kn+s√ωksωkn+s/parenrightBigg\n∝an}bracketle{t0|n/productdisplay\np=l+1c†\nkpckn+p|0∝an}bracketri}ht/bracketrightBigg\n,\nwhere, as before, in the case of ill-defined integrals (cases\nl= 0 and l=n) we assume that the contribution of the\nintegral is equal to 1.\nBecause of the delta functions emerging from matrix\nelements, integrals in the square brackets of Eqs. (16)\nand (17) reduce to the combinations of the integrals\n/integraldisplayd3k\n2(2π)3ωk=κ, (18)\n/integraldisplayd3k\n2(2π)3kikj\nωk=κ\n3δij, (19)\nwhere, for further convenience, a short notation ( κ) is\nintroduced for the first integral and the second integral\nis represented in terms of the first integral. It is im-\nportant to note that the above integrals are divergent,\nso is parameter κ, and strictly speaking one has to reg-\nularize integrals (18) and (19). Clearly the physically\nrelevant conclusions should not depend on the choice of\nregularization. However, it is important to discuss var-\nious regularizations since, if blindly used, they can give\nmisleading results. That is especially true for the case of\ndimensional regularization. Namely, if dimensional reg-\nularization is applied in a usual way the parameter κ\nis put to zero by convention using the argument that\nit has no scale dependence. Consequently, that puts all\nthe moments (except the zeroth one) of the distribution\nto zero and the p.d.f. would equal the delta functional.\nHowever that would imply that the Hamiltonian den-\nsity and the Hamiltonian commute (at least when their\ncommutator acts on the vacuum state) which is not the\ncase, as one can check by explicit calculation. To re-\nsolve this paradox, one has to remember that rigorously\nspeaking the integrals (18) and (19) are not zero in di-\nmensional regularization. They are either UV divergent\nor IR divergent depending on the concrete dimension\nused for regularization. That is because relevant inte-\ngral in dimensional regularization can be written in the\nform/integraltext∞\n0dkkα(wherekis magnitude of momenta in D-\ndimensional space) which is obviously divergent for any\nα. However, for practical purposes (calculation of cross-\nsections and decay rates) these integrals are convention-\nally put to zero because their contributions are always\nfully absorbed into renormalized quantities and we havecomplete control over them. There is not any justifica-\ntion to use such an approach to calculate the p.d.f. of\nthe vacuum energy density. In a sense, the standard di-\nmensional regularization procedure is not an adequate\nregularization procedure here. However, since relevant\nconclusions should not depend on the choice of the reg-\nularization, for our purposes, we can adopt more conve-\nnient regularization. For example, in three-dimensional\ncutoff regularization where the momentum cut-off is Λ,\nthe parameter κis equal to Λ4/(16π2). Anyway, let us\nnote that, fortunately, to obtain the functional forms of\nthe probability density functions it is not necessary to\nexactly calculate the above integrals i.e. to know the\nvalue of κ. Namely, when normalized and expressed in\nterms of expectation value the p.d.f.s do not depend on\nκexplicitly (see Eqs. (36) and (37) later in the text).\nAfter integration and counting of identical terms, it\ncan be shown that VEVs ∝an}bracketle{tHn\nR∝an}bracketri}htand∝an}bracketle{tHn\nC∝an}bracketri}htare equal to\n∝an}bracketle{tHn\nR∝an}bracketri}ht=1\n2nn/summationdisplay\nl=0/parenleftBign\nl/parenrightBig/bracketleftBigg\n(2l)!\n2ll!κl/bracketrightBigg\n×/bracketleftBigg/parenleftBigκ\n3/parenrightBign−ln−l/summationdisplay\nk=03k2n−l−k|S(k)\nn−l|/bracketrightBigg\n,(20)\n∝an}bracketle{tHn\nC∝an}bracketri}ht=n/summationdisplay\nl=0/parenleftBign\nl/parenrightBig/bracketleftBigg\nl!κl/bracketrightBigg/bracketleftBigg/parenleftBigκ\n3/parenrightBign−ln−l/summationdisplay\nk=03k|S(k)\nn−l|/bracketrightBigg\n,(21)\nwhere|S(j)\ni|are unsigned Stirling numbers of the first\nkind. The number |S(j)\ni|gives a number of permutations\nofielements which haveexactly jcycles[4]. In the above\nequations, terms are still grouped in square brackets to\nindicate the origin of each contribution if compared to\nEqs. (16) and (17).\nNow it is possible to calculate Laplace transforms of\nthe probability density functions of the vacuum energy\nfor real and complex massless scalar fields. After putting\nEqs. (20) and (21) in Eq. (1) and rearrangement of\nterms, it follows\nL{pR,C(ǫ)}=∞/summationdisplay\nn=01\nn!/parenleftBig\n−sκ\n3/parenrightBign\nL(n)\nR,C,(22)\nwhere\nL(n)\nR=n/summationdisplay\nl=0/parenleftBign\nl/parenrightBig(2l)!\nl!/parenleftbigg3\n4/parenrightbiggln−l/summationdisplay\nk=0|S(k)\nn−l|/parenleftbigg3\n2/parenrightbiggk\n,(23)\nL(n)\nC=n/summationdisplay\nl=0/parenleftBign\nl/parenrightBig\nl!3ln−l/summationdisplay\nk=0|S(k)\nn−l|3k. (24)\nSince\ndn\ndxn(1−x)−u/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0=n/summationdisplay\nk=0|S(k)\nn|uk,\ndn\ndxn(1−3x)−1/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0=(2n)!\nn!/parenleftbigg3\n4/parenrightbiggn\n,4\ndn\ndxn(1−3x)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0=n!3n,\nit can be seen that functions L(n)\nR,Cequal\nL(n)\nR=dn\ndxn1\n(1−x)3/2(1−3x)1/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0,(25)\nL(n)\nC=dn\ndxn1\n(1−x)3(1−3x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=0. (26)\nFinally, by taking into account identities (25), (26), and\n(22), the relevant Laplace transforms are\nL{pR(ǫ)}= (1+κs/3)−3/2(1+κs)−1/2,(27)\nL{pC(ǫ)}= (1+κs/3)−3(1+κs)−1.(28)\nBy using the differentiation theorem of Laplace trans-\nform, it further follows\nL{pR(ǫ)}=6\nκ/parenleftbigg\n−d\nds(1+κs/3)−1/2/parenrightbigg\n(1+κs)−1/2\n=6\nκL{ǫf(ǫ,κ/3)}L{f(ǫ,κ)}, (29)\nwhereL{f(ǫ,a)}= (1+as)−1/2. Since\nL{e−t/a\n√\nπat}=1√1+as,\nit can be written\nL{pR(ǫ)}=6\nκL{√\n3ǫe−3ǫ/κ\n√πκ}L{e−ǫ/κ\n√πκǫ} (30)\n=L{6√\n3ǫ\nπκ2e−3ǫ/κ/integraldisplay1\n0dx/radicalbigg\n1−x\nxe2ǫx/κ},\nwhere theorems of linearity and convolution are used.\nAfter calculating the remaining integral we obtain\npR(ǫ) =3√\n3ǫ\nκ2e−2ǫ/κ/bracketleftBig\nI0/parenleftBigǫ\nκ/parenrightBig\n−I1/parenleftBigǫ\nκ/parenrightBig/bracketrightBig\n,(31)\nwhereIn(x) isamodifiedBesselfunction ofthefirstkind.\nSimilarly, for the complex scalar field\nL{pC(ǫ)}=L{27ǫ2\n2κ3e−3ǫ/κ}L{1\nκe−ǫ/κ},(32)\nwhere the following identity is used,\nL{tn−1\nan(n−1)!e−t/a}=1\n(1+as)n.\nApplying the convolution theorem to Eq.(32) gives\nL{pC(ǫ)}=L{27ǫ3\n2κ4e−ǫ/κ/integraldisplay1\n0dxx2e2ǫx/κ},(33)0 1 2 3 4 5\nx00.20.40.60.8P(x)real massless scalar field\ncomplex massless scalar field\nFIG. 1: The probability density functions of the vacuum en-\nergy density for real (solid) and complex (dashed) massless\nscalar fields. Variable xis equal to the ratio ǫ/¯ǫwhereǫis\nthe energy density and ¯ ǫis the expectation value which is\ndifferent for the real and the complex cases. Mean values for\nboth curves are ¯ x= 1 and the standard deviation for the\nreal case is σ=/radicalbig\n2/3 = 0.816 and for the complex case is\nσ=/radicalbig\n1/3 = 0.577. For the real field maximal probability is\natx= 0.419 and for the complex field at x= 0.664.\nand finally after integration we get\npC(ǫ) =27\n8κ/braceleftbigg\ne−ǫ/κ−e−3ǫ/κ/bracketleftbigg\n2/parenleftBigǫ\nκ/parenrightBig2\n+2ǫ\nκ+1/bracketrightbigg/bracerightbigg\n.\n(34)\nIt is convenient to express the probabilities pR,C(ǫ)dǫin\nterms of vacuum expectation values ¯ ǫR,C=∝an}bracketle{tHR,C∝an}bracketri}htas\npR,C(ǫ)dǫ=PR,C/parenleftbiggǫ\n¯ǫR,C/parenrightbigg\nd/parenleftbiggǫ\n¯ǫR,C/parenrightbigg\n,(35)\nwhere ¯ǫR=κ, ¯ǫC= 2κ, and\nPR(x) = 3√\n3xe−2x[I0(x)−I1(x)], (36)\nPC(x) = 27/4/bracketleftbig\ne−2x−e−6x(1+4x+8x2)/bracketrightbig\n.(37)\nAs can be seen from Fig. 1, probability density func-\ntionsPR,C(x) are broad.\nThe distributions given in (31) and (34) depend on\nformally divergent quantity κand a regularization fol-\nlowed by renormalization must be employed. The only\nrequired renormalization on free field theories within the\nstandard approach in QFT is given by normal ordering.\nAs normal ordering corresponds to subtraction of ∝an}bracketle{tH∝an}bracketri}ht\nfromH, the effect of this renormalization on probability\ndistributions is their shift so that their expectation val-\nues vanish as shown in Fig. 2. This can be checked by\ndirect calculation if : H:=H − ∝an}bracketle{tH∝an}bracketri}ht is used instead of\nH, which requires calculation of ∝an}bracketle{t:H:n∝an}bracketri}htin intermediate\nsteps. However, the form of probability distributions for\n:H: still depends on formally divergent κ. In a sense,\nthis indicates that the normal ordering is insufficient to\nremove all divergences from the distributions. However,\nit is not realistic that κis really divergent. Any plausi-\nble energy cut-off makes it big but finite. All moments5\n-1 0 1 2 3 4\nx00.20.40.60.8P(x)real massless scalar field\ncomplex massless scalar field\nFIG. 2: The probability density functions of the normally\nordered vacuum energy density for real (solid) and complex\n(dashed) massless scalar fields. Variable xis equal tothe ratio\nǫ/κfor the real and ǫ/2κfor the complex field. Parameter ǫis\nthe energy density and κ= Λ4/(16π2) in three-dimensional\ncutoff regularization, where the momentum cutoff is Λ.\ncan be naively renormalized if moments are defined as\n∝an}bracketle{t:Hn:∝an}bracketri}htinstead of ∝an}bracketle{tHn∝an}bracketri}htor∝an}bracketle{t:H:n∝an}bracketri}ht. Here, it is impor-\ntant to note that suchprocedure correspondsto infinitely\nindependent renormalizationsof moments since each mo-\nment has a different divergency (beside the fact they all\ndepend on the same formally divergent integral κ). With\nsuch an approach, all the moments (except the zeroth\none) would be equal zero and the p.d.f. would equal the\ndelta functional, which leads us into contradiction with\nthe fact that the Hamiltonian density and Hamiltoniando not commute as mentioned above.\nIn conclusion, the calculations presented in this paper\nshow that the vacuum energy density of a massless scalar\nfield is a random variable with a broad p.d.f. Both the\nexpectation of the vacuum energy density and the most\nprobable value of the vacuum energy density are not rep-\nresentative for the distribution as a whole. There exists\na nonnegligible probability for a value of vacuum energy\nthat differs a lot from ǫ. These results, obtained analyti-\ncally and in a closed form, imply that the vacuum energy\ndensity is a more complex object than just the VEV of\nH.\nThe central results of this work have implications be-\nyond the scope of this paper. A thorough understanding\nof the vacuum energy density is essential for the expla-\nnation of the epochs of accelerated expansion such as the\ninflationaryphaseorthe late-time acceleratedexpansion.\nAn especially interesting prospect is that a multiverse\nscenario could be realized in the framework of QFT. To\nthis end, the present analysis needs to be generalized to\ncurved space-times to study the zero point energy den-\nsity in the presence of gravity. Natural extensions of the\npresent analysis comprise the study of effects of mass,\nother types ofquantum fields, and field interactions. The\nanalytical results presented here represent a sound basis\nfor the exploration of these directions of research.\nThis work was supported by the Ministry of Educa-\ntion, Science and Sports of the Republic of Croatiaunder\nContract No. 098-0982930-2864.\n[1] J.D. Bjorken and S.D. Drell, Relativistic Quantum Field s,\nMcGraw-Hill Publishing Company, New York (1965).\n[2] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989); P. J. E. Pee-\nbles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003);\nT. Padmanabhan, Phys. Rept. 380, 235 (2003); V. Sahni,\nA. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000);\nS.M. Carroll, Living Rev. Rel. 4, 1 (2001).[3] A. H. Guth, Phys. Rev. D 23,347 (1981); A. D. Linde,\nPhys. Lett. B 108,389 (1982); A. Albrecht and P. J. Stein-\nhardt, Phys. Rev. Lett. 48, 1220 (1982).\n[4] M. Abramowitz and I. A. Stegun, Handbook of Mathe-\nmatical Functions, Dover Publications, New York (1968).\n[5] G. C. Wick, Phys. Rev. 80, 268 (1950)."    },    {        "title": "1003.0246v1.Density_matrix_formalism_with_three_body_ground_state_correlations.pdf",        "content": "arXiv:1003.0246v1  [nucl-th]  1 Mar 2010Density-matrix formalism with three-body ground-state co rrelations\nMitsuru Tohyama\nKyorin University School of Medicine, Mitaka, Tokyo 181-86 11, Japan\nPeter Schuck\nInstitut de Physique Nucl ´ eaire, IN2P3-CNRS, Universit ´ eParis-Sud, F-91406 Orsay Cedex, France\n(Dated: November 1, 2018)\nA density-matrix formalism which includes the effects of thr ee-body ground- state correlations is\napplied to the standard Lipkin model. The reason to consider the complicated three-body correla-\ntions is that the truncation scheme of reduced density matri ces up to the two-body level does not\ngive satisfactory results to the standard Lipkin model. It i s shown that inclusion of the three-body\ncorrelations drastically improves the properties of the gr ound states and excited states. It is pointed\nout that lack of mean-field effects in the standard Lipkin mode l enhances the relative importance\nof the three-body ground-state correlations. Formal aspec ts of the density-matrix formalism such\nas a relation to the variational principle and the stability condition of the ground state are also\ndiscussed. It is pointed out that the three-body ground-sta te correlations are necessary to satisfy\nthe stability condition.\nPACS numbers: 21.60.Jz\nI. INTRODUCTION\nMean-field theories such as the Hartree-Fock theory (HF), the H aree-Fock Bogoliubov theory, the random-phase\napproximation(RPA), andthe quasi-particleRPAhaveextensivelyb een usedto studythe groundstatesandcollective\nexcitations of atomic nuclei [1]. For more realistic theoretical treat ment of nuclei such as inclusion of the ground-state\ncorrelations other than pairing correlations and the damping effect s of collective excitations, however, we must go\nbeyond the mean-field theories. The time-dependent density-mat rix theory (TDDM) [2–4] which has been formulated\nby truncating the chain of the equations of motion for reduced den sity matrices up to the two-body level is one\nof such extended mean-field theories. It has been pointed out tha t a stationary solution of the TDDM equations\ngives a correlated ground state and that the small amplitude limit of T DDM based on the correlated ground state\ncorresponds to an extended RPA (ERPA) including two-body trans ition amplitudes [5]. To test the reliability of the\ndensity-matrix formalism, we have applied it to solvable models [6–8] an d found that the obtained results improve\non deficiencies of the mean-field theories [9, 10]. In the case of the s tandard Lipkin model [6] where the interaction\nterm contains only two particle - two hole excitations, however, we f ound that the ground states in TDDM become\nslightly overbound as compared with the exact solutions. That the a pproximate ground states have lower energy than\nthe exact ones contradicts the variational principle [1] for the to tal wavefunction and should possibly be avoided,\nalthough TDDM is not based on a Raleigh-Ritz variational principle. The aim of this paper is to clarify the origin\nof such an unsatisfactory feature of TDDM in the standard Lipkin m odel. We will show that inclusion of three-body\nground-state correlations, though it is complicated, drastically imp roves the results for the standard Lipkin model.\nThe paper is organized as follows: The formulation of TDDM with the th ree-body ground-state correlations [11, 12]\nand ERPA built on the TDDM ground state are given in sect.2. Some for mal aspects of TDDM and ERPA, a relation\nof the TDDM equations to the variational principle, the stability cond ition of the ground-state, and a comparison of\nERPA with the self-consistent RPA (SCRPA) [13, 14], which have not b een pointed out in our earlier publications,\nare also discussed in sect. 2. The results for the standard Lipkin mo del are presented in sect.3. Section 4 is devoted\nto the summary.\nII. FORMULATION\nAlthough the numerical calculations are performed for the Lipkin mo del, the formulation is presented using the\nfollowing general hamiltonian\nˆH=/summationdisplay\nλλ′/an}bracketle{tλ|t|λ′/an}bracketri}hta†\nλaλ′+1\n2/summationdisplay\nλ1λ2λ′\n1λ′\n2/an}bracketle{tλ1λ2|v|λ′\n1λ′\n2/an}bracketri}hta†\nλ1a†\nλ2aλ′\n2aλ′\n1, (1)\nwheretis the kinetic energy operator, vis a two-body interaction and a†\nλ(aλ) the creation (annihilation) operator of\na nucleon in a single-particle state λ.2\nA. TDDM\nIn TDDM the ground state |0/an}bracketri}htis defined by the occupation matrix nαα′, the two-body correlation matrix Cαβα′β′\nand the three-body correlation matrix Cαβγα′β′γ′given by\nnαα′=/an}bracketle{t0|a†\nα′aα|0/an}bracketri}ht, (2)\nCαβα′β′=/an}bracketle{t0|a†\nα′a†\nβ′aβaα|0/an}bracketri}ht−A(nαα′nββ′), (3)\nCαβγα′β′γ′=/an}bracketle{t0|a†\nα′a†\nβ′a†\nγ′aγaβaα|0/an}bracketri}ht−A(nαα′nββ′nγγ′+S(nαα′Cβγβ′γ′)), (4)\nwhereAandSmean that the products in the parentheses are properly antisymm etrized and symmetrized under\nthe exchange of single-particle indices [2]. Equations of motion for nαα′,Cαβα′β′andCαβγα′β′γ′can be obtained by\ntruncating a coupled chain of equations of motion for reduced dens ity matrices (the so-called Bogoliubov-Born-Green-\nKirkwood-Yvon hierarchy) up to the three-body level [2] and can be written as\ni¯hd\ndtnαα′=/an}bracketle{t0|[a†\nα′aα,ˆH]|0/an}bracketri}ht=F1(αα′), (5)\ni¯hd\ndtCαβα′β′= :/an}bracketle{t0|[a†\nα′a†\nβ′aβaα,ˆH]|0/an}bracketri}ht:=F2(αβα′β′), (6)\ni¯hd\ndtCαβγα′β′γ′= :/an}bracketle{t0|[a†\nα′a†\nβ′a†\nγ′aγaβaα,ˆH]|0/an}bracketri}ht:=F3(αβγα′β′γ′), (7)\nwhere : : means that the time derivatives of the second terms on th e right-hand sides of eqs. (3) and (4) are\nsubtracted. Since a four-body correlation matrix is neglected, th e expectation values of four-body operators in eq.(7)\nare approximated by the products of nαα′,Cαβα′β′andCαβγα′β′γ′. The expressions for F1,F2andF3are given in\nthe Appendix, where the single-particle states which satisfy the HF -like mean-field equation\nh(ρ)φα(1) =ǫαφα(1) (8)\nare used. Here, ρis the one-body density matrix given by ρ(1,1′) =/summationtext\nαα′nαα′φα(1)φ∗\nα′(1′) and numbers indicate\nspatial, spin and isospin coordinates.\nTo obtain the ground state implies that all quantities nαα′,Cαβα′β′,Cαβγα′β′γ′andφαare determined under the\nstationary conditions\ni¯hd\ndtnαα′=F1(αα′) = 0, (9)\ni¯hd\ndtCαβα′β′=F2(αβα′β′) = 0, (10)\ni¯hd\ndtCαβγα′β′γ′=F3(αβγα′β′γ′) = 0. (11)\nand eq. (8). This task can be achieved using the gradient method [1 5]. The functional derivatives of eqs. (9)-(11) are\nwritten as\n\nδF1\nδnδF1\nδC20\nδF2\nδnδF2\nδC2δF2\nδC3δF3\nδnδF3\nδC2δF3\nδC3\n\n∆n\n∆C2\n∆C3\n=\n∆F1\n∆F2\n∆F3\n=−\nF1\nF2\nF3\n. (12)\nThe inversion of the above matrix gives the following equation to be us ed in the gradient method\n\nn(N+1)\nC2(N+1)\nC3(N+1)\n=\nn(N)\nC2(N)\nC3(N)\n−α\na c0\nb d e\nf g h\n−1\nF1(N)\nF2(N)\nF3(N)\n, (13)\nwheren(N),C2(N) andC3(N) imply nαα′,Cαβα′β′andCαβγα′β′γ′at theNth iteration step, respectively, and\na=δF1/δn,b=δF2/δn,c=δF1/δC2,d=δF2/δC2,e=δF2/δC3,f=δF3/δn,g=δF3/δC2, andh=δF3/δC3.\nThe expressions for these matrices are given in ref. [12]. The matric esa,b,d,f,g, andhdepend on the iteration step\nNthrough n(N),C2(N) andC3(N). To solve eq. (13), we start from a simple ground state such as th e HF ground\nstate where nαα′,Cαβα′β′andCαβγα′β′γ′can be easily evaluated and iterate eq. (13) until convergence is ac hieved.3\nA small parameter αis introduced to regulate the convergence process. When the mea n-field potential is present, eq.\n(13) couples to eq. (8) through nαα′. As was discussed in ref. [12], the matrix consisting of the functiona l derivatives\nofF1, F2andF3on the right-hand side of eq. (13) can be derived as the small amplitu de limit of TDDM and has\na close relation with the hamiltonian matrix of ERPA given in the next sub section [11, 16]. This indicates that the\nground state is not independent of the excited states in ERPA. Let us discuss this point in some more detail using\nthe eigenstates of the following matrix equation, which correspond s to the small amplitude limit of eqs. (5)-(7) [12],\n\na c0\nb d e\nf g h\n\nxµ\nyµ\nzµ\n= Ωµ\nxµ\nyµ\nzµ\n. (14)\nSince the closure relation is given as [16],\n/summationdisplay\nµ\nxµ\nyµ\nzµ\n/parenleftbig\n˜x∗\nµ˜y∗\nµ˜z∗\nµ/parenrightbig\n=I, (15)\nwhere (˜x∗\nµ˜y∗\nµ˜z∗\nµ) is the left-hand eigenvector of eq. (14) and Iis unit matrix, eq. (13) is written as\n\nn(N+1)\nC2(N+1)\nC3(N+1)\n=\nn(N)\nC2(N)\nC3(N)\n−α/summationdisplay\nΩµ/negationslash=01\nΩµ\nxµ\nyµ\nzµ\n/parenleftbig\n˜x∗\nµ˜y∗\nµ˜z∗\nµ/parenrightbig\nF1(N)\nF2(N)\nF3(N)\n. (16)\nIn eq. (16) only the eigenstates with Ω µ/ne}ationslash= 0 can contribute as is understood by multiplying eq. (12) with/parenleftbig\n˜x∗\nµ˜y∗\nµ˜z∗\nµ/parenrightbig\n.\nThe above equation indicates that the occupation matrix and the co rrelation matrices are constructed from the\neigenvectors of eq. (14) at each iteration step which have the sam e quantum numbers as the ground state: In the\nstandard Lipkin model these states are two-phonon states.\nFor the numerical calculations shown below we use not eq. (16) but e q. (13) because a considerable dimension\nsize of the three-body part of eq. (14) makes it impracticable to ad opt eq. (16) which requires the eigenvalues and\neigenvectors at each iteration step. In the matrix inversion of eq. (13), however, mixing of unphysical states with\nΩµ≈0 is unavoidable. This problem can be removed by slightly shifting the un perturbed energies in a,d, andhby\n∆ǫso that the inverse can be taken. In the case of the Lipkin model th e typical value of ∆ ǫused is 10−5ǫwhereǫis\nthe level spacing of the Lipkin model hamiltonian. The obtained result s are independent of ∆ ǫbecause the product\n/parenleftbig\n˜x∗\nµ˜y∗\nµ˜z∗\nµ/parenrightbig\nF1(N)\nF2(N)\nF3(N)\n (17)\nvanishes for the states with Ω µ= 0. Since the inversion of the matrix on the right-hand side of eq. (1 3) is time\nconsuming, the gradient method is supplemented with a time-depend ent approach as will be explained below. We\nmakeacomparisonwithasimplifiedversionofTDDM wherethethree-b odycorrelationmatrix Cαβγα′β′γ′isneglected.\nWe refer this TDDM as to TDDM1.\nB. Extended RPA\nIn this section we present our extended version of RPA (ERPA) [12]. We also discuss its close connection to TDDM.\nERPA is formulated for the following excitation operator consisting o f one-body and two-body operators\nˆQ†\nµ=/summationdisplay\nλλ′xµ\nλλ′:a†\nλaλ′:+/summationdisplay\nλ1λ2λ′\n1λ′\n2Xµ\nλ1λ2λ′\n1λ′\n2:a†\nλ1a†\nλ2aλ′\n2aλ′\n1:, (18)\nwhere : : implies that uncorrelated parts consisting of lower-level o perators are to be subtracted; for example,\n:a†\nα′aα: =a†\nα′aα−nαα′, (19)\n:a†\nα′a†\nβ′aβaα: =a†\nα′a†\nβ′aβaα−AS(nαα′:a†\nβ′aβ:)\n−[A(nαα′nββ′)+Cαβα′β′]. (20)4\nIt is assumed that the operator ˆQ†\nµsatisfies\nˆQ†\nµ|Ψ0/an}bracketri}ht=|Ψµ/an}bracketri}ht, (21)\nˆQµ|Ψ0/an}bracketri}ht= 0. (22)\nHere,|Ψ0/an}bracketri}htis the ground state in ERPA and |Ψµ/an}bracketri}htis an excited state. The ERPA equations are obtained from the\nequations-of-motion method [13]\n/an}bracketle{tΨ0|[[:a†\nα′aα:,ˆH],ˆQ†\nµ]|Ψ0/an}bracketri}ht=ωµ/an}bracketle{tΨ0|[:a†\nα′aα:,ˆQ†\nµ]|Ψ0/an}bracketri}ht, (23)\n/an}bracketle{tΨ0|[[:a†\nα′a†\nβ′aβaα:,ˆH],ˆQ†\nµ]|Ψ0/an}bracketri}ht=ωµ/an}bracketle{tΨ0|[:a†\nα′a†\nβ′aβaα:,ˆQ†\nµ]|Ψ0/an}bracketri}ht, (24)\nwhereωµis the excitation energy of |Ψµ/an}bracketri}ht. In the evaluation of the matrix elements we approximate the ERPA g round\nstate|Ψ0/an}bracketri}htby|0/an}bracketri}htwhich satisfies eqs. (8)-(11). The ERPA equations can then be writ ten in matrix form\n/parenleftbigg\nA C\nB D/parenrightbigg/parenleftbigg\nxµ\nXµ/parenrightbigg\n=ωµ/parenleftbigg\nS1T1\nT2S2/parenrightbigg/parenleftbigg\nxµ\nXµ/parenrightbigg\n, (25)\nwhere the matrix elements are given by\nA(αα′:λλ′) =/an}bracketle{t0|[[:a†\nα′aα:,ˆH],:a†\nλaλ′:]|0/an}bracketri}ht, (26)\nB(αβα′β′:λλ′) =/an}bracketle{t0|[[:a†\nα′a†\nβ′aβaα:,ˆH],:a†\nλaλ′:]|0/an}bracketri}ht, (27)\nC(αα′:λ1λ2λ′\n1λ′\n2) =/an}bracketle{t0|[[:a†\nα′aα:,ˆH],:a†\nλ1a†\nλ2aλ′\n2aλ′\n1:]|0/an}bracketri}ht, (28)\nD(αβα′β′:λ1λ2λ′\n1λ′\n2) =/an}bracketle{t0|[[:a†\nα′a†\nβ′aβaα:,ˆH],:a†\nλ1a†\nλ2aλ′\n2aλ′\n1:]|0/an}bracketri}ht, (29)\nS1(αα′:λλ′) =/an}bracketle{t0|[:a†\nα′aα:,:a†\nλaλ′:]|0/an}bracketri}ht, (30)\nT1(αα′:λ1λ2λ′\n1λ′\n2) =/an}bracketle{t0|[:a†\nα′aα:,:a†\nλ1a†\nλ2aλ′\n2aλ′\n1:]|0/an}bracketri}ht, (31)\nT2(αβα′β′:λλ′) =/an}bracketle{t0|[:a†\nα′a†\nβ′aβaα:,:a†\nλaλ′:]|0/an}bracketri}ht, (32)\nS2(αβα′β′:λ1λ2λ′\n1λ′\n2) =/an}bracketle{t0|[:a†\nα′a†\nβ′aβaα:,:a†\nλ1a†\nλ2aλ′\n2aλ′\n1:]|0/an}bracketri}ht. (33)\nAt this point we want to insist on the fact that in eqs.(9)-(11) antisy mmetrisation is fully respected and, thus, the\nmatrix elements entering eq. (25) also respect antisymmetrisation fully. This is at variance of most other extensions\nof the RPA approach.\nIn the following we show that eq. (25) can also be derived from the sm all amplitude limit of TDDM (eq. (14)). We\ntransform the eigenvector in eq. (14) using an extended norm mat rix including three-body components as\n\nxµ\nXµ\nYµ\n=\nS1T1T13\nT2S2T23\nT31T32T33\n\nxµ\nyµ\nzµ\n, (34)\nwhere\nT31=/an}bracketle{t0|[:a†\nα′a†\nβ′a†\nγ′aγaβaα:,:a†\nλaλ′:]|0/an}bracketri}ht, (35)\nT32=/an}bracketle{t0|[:a†\nα′a†\nβ′a†\nγ′aγaβaα:,:a†\nλ1a†\nλ2aλ′\n2aλ′\n1:]|0/an}bracketri}ht (36)\nT13=/an}bracketle{t0|[:a†\nα′aα:,:a†\nλ1a†\nλ2a†\nλ3aλ′\n3aλ′\n2aλ′\n1:]|0/an}bracketri}ht, (37)\nT23=/an}bracketle{t0|[:a†\nα′a†\nβ′aβaα:,:a†\nλ1a†\nλ2a†\nλ3aλ′\n3aλ′\n2aλ′\n1:]|0/an}bracketri}ht, (38)\nT33=/an}bracketle{t0|[:a†\nα′a†\nβ′a†\nγ′aγaβaα:,:a†\nλ1a†\nλ2a†\nλ3aλ′\n3aλ′\n2aλ′\n1:]|0/an}bracketri}ht. (39)\nThen eq. (14) is written as\n\naS1+cT2 aT1+cS2 aT13+cT23\nbS1+dT2+eT31bT1+dS2+eT32bT13+dT23+eT33\nfS1+gT2+hT31fT1+gS2+hT32fT13+gT23+hT33\n\nxµ\nXµ\nYµ\n= Ωµ\nS1T1T13\nT2S2T23\nT31T32T33\n\nxµ\nXµ\nYµ\n.(40)\nThe matrix elements on the left-hand side of the above equation can be written in the form of the double commutators\nwiththe hamiltonian[12] exceptforthosewith f,gandh: Since[: a†\nα′a†\nβ′a†\nγ′aγaβaα:,ˆH]containsfour-bodyoperators,5\nthe matrices fS1+gT2+hT31,fT1+gS2+hT32, andfT13+gT23+hT33cannot be expressed using the double\ncommutator with the hamiltonian. The matrices A,B,C, andDin eq. (25) have the following relations with a,b,\nc,d, ande[12],A=aS1+cT2,B=bS1+dT2+eT31,C=aT1+cS2, andD=bT1+dS2+eT32. The expressions\nforS1,T1,T2,S2,T31, andT32are given in ref. [12]. Thus it is shown that eq. (25) is obtained from eq . (40) by\nneglecting the three-body sections.\nAs has been discussed in detail in ref. [11], the hamiltonian matrix on t he left hand side of eq. (25) is hermitian\ndue to the ground-state conditions eqs. (9)-(11) which guarant ee the Jacobi’s identity\n/an}bracketle{t0|[[ˆQ,ˆH],ˆP]|0/an}bracketri}ht−/an}bracketle{t0|[[ˆP,ˆH],ˆQ]|0/an}bracketri}ht=/an}bracketle{t0|[ˆH,[ˆP,ˆQ]]|0/an}bracketri}ht= 0, (41)\nwhereˆPandˆQare either one-body or two-body operators. It is interesting to n ote that reversing the order of\narguments and imposing hermiticity of the matrix on the left-hand sid e of eq.(25) one can arrive at eqs. (9)-(11). We\nremind in this context that Rowe [13] achieved hermiticity of his equat ion of motion method taking the arithmetic\nmean of the off diagonal elements. This somewhat artificial procedu re finds a natural solution with the considerations\ngiven in our present procedure. It, therefore, can be stated th at in the equation of motion method the non hermitian\npart of the matrices has to be put to zero what imposes the fulfillmen t of extra equations.\nAs mentioned above, the hamiltonian matrix of eq. (40) is not hermitia n because of the asymmetry in the three-\nbody parts: It can be stated that eq. (25) is obtained omitting the non-hermitian parts of eq. (40). The commutator\n[ˆP,ˆQ] for three-body operators ˆPandˆQcontains three-body, four-body and five-body operators. In o rder to fulfill\nthe condition eq. (41) and make an extended RPA with the three-bo dy amplitude Yµhermitian, therefore, we need\nto consider two additional ground-state conditions for four-bod y and five-body operators. The problem of a further\nextended RPA with the three-body amplitude is beyond the scope of this paper.\nThe ortho-normal condition of ERPA (eq. (25)) is given as [19]\n(xµ∗Xµ∗)/parenleftbigg\nS1T1\nT2S2/parenrightbigg/parenleftbigg\nxµ′\nXµ′/parenrightbigg\n=±δµµ′, (42)\nwhere the negative sign is for a negative-energy solution. Equation (25) is equivalent to the second RPA [17, 18]\nwhen the ground-state correlations are neglected: Neglect of th e ground-state correlations means that nαα′=δαα′(0)\nfor occupied (unoccupied) states, Cαβα′β′= 0 and Cαβγα′β′γ= 0. In this limit, the one-body section of eq. (25),\nAxµ=ωµS1xµ, is equivalent to standard RPA. We make a comparison with a simplified v ersion of ERPA where the\nthree-body correlation matrix Cαβγα′β′γ′is neglected in the calculation of the ground state. We refer this ERP A as to\nERPA1. When eq. (25) is solved numerically, first the norm matrix on t he right-hand side of eq. (25) is diagonalized.\nThen eq. (25) is solved in the truncated space consisting of the eige nstates of the norm matrix with nonvanishing\neigenvalues.\nC. Stability condition\nWe discuss the relation of the ground state conditions eqs. (9) and (10) to the variational principle for the energy.\nWe also discuss the stability condition of the ground state. Suppose the following variational energy\nE=/an}bracketle{t0|e−iˆFˆHeiˆF|0/an}bracketri}ht, (43)\nwhereˆFis an arbitrary hermitian operator given as\nˆF=/summationdisplay\nαα′fαα′:a†\nαaα′: +/summationdisplay\nαβα′β′Fαβα′β′:a†\nαa†\nβaβ′aα′:. (44)\nThe energy Ecan be expressed as\nE=/an}bracketle{t0|ˆH|0/an}bracketri}ht+i/an}bracketle{t0|[ˆH,ˆF]|0/an}bracketri}ht+1\n2/an}bracketle{t0|[[ˆF,ˆH],ˆF]|0/an}bracketri}ht+····\n=/an}bracketle{t0|ˆH|0/an}bracketri}ht+i/summationdisplay\nαα′/an}bracketle{t0|[ˆH,:a†\nαaα′:]|0/an}bracketri}htfαα′\n+i/summationdisplay\nαβα′β′/an}bracketle{t0|[ˆH,:a†\nαa†\nβaβ′aα′:]|0/an}bracketri}htFαβα′β′\n+1\n2(f∗F∗)/parenleftbigg\nA C\nB D/parenrightbigg/parenleftbigg\nf\nF/parenrightbigg\n+····. (45)6\nThe stationary conditions eqs. (9) and (10) correspond to δE/δf αα′= 0 and δE/δF αβα′β′= 0, respectively, and\nthe single-particle hamiltonian of eq. (8) is obtained from the variatio nδ/an}bracketle{t0|ˆH|0/an}bracketri}ht/δnαα′=hαα′. The condition eq.\n(11) can also be obtained by adding a three-body operator to eq. ( 44). If the total wavefunction |0/an}bracketri}htwere used to\ncalculate nαα′,Cαβα′β′,Cαβγα′β′γ′, and also the four-body correlation matrix, eqs.(9)-(11) would co rrespond to the\nvariational principle for the total wavefuntion. However, it is impra cticable to find |0/an}bracketri}htitself. In TDDM, eqs.(9)-(11)\nare used to obtain not the total wavefunction but nαα′,Cαβα′β′, andCαβγα′β′γ′by approximating a four-body density\nmatrix with lower level density matrices. In this sense, TDDM deviate s from the variational principle for the total\nwavefunction, although eqs.(9)-(11) can be formally derived from the variational principle.\nThe stability matrix in the last line on the right-hand side of eq. (45) is n othing but the hamiltonian matrix of\neq. (25). As was mentioned above, the condition for the three-bo dy matrix eq. (11) in addition to eqs. (9) and (10)\nis necessary to make the stability matrix hermitian although the oper ators eqs. (18) and (44) are at most two-body\nones. The stability condition that |0/an}bracketri}htcorresponds to a minimum in the energy surface is given by\n(f∗F∗)/parenleftbigg\nA C\nB D/parenrightbigg/parenleftbigg\nf\nF/parenrightbigg\n≥0 (46)\nfor arbitrary fαα′andFαβα′β′. This condition is satisfied when the stability matrix is positive definite. In this case,\neq. (25) has only real eigenvalues since the norm matrix in eq. (25) is hermitian [1].\nD. Relation to self-consistent RPA\nIn this subsection we discuss some relation of the one-body part of eq. (25)\nAxµ=ωµS1xµ(47)\nto SCRPA [14]. The matrix Ais explicitly written as\nA(αα���:λλ′) = (ǫα−ǫα′)(nλ′α′δαλ−nαλδα′λ′)\n+/summationdisplay\nγγ′[(/an}bracketle{tαγ|v|γ′λ/an}bracketri}htAnλ′γ−/an}bracketle{tαλ′|v|γ′γ/an}bracketri}htAnγλ)nγ′α′\n−(/an}bracketle{tγ′γ|v|α′λ/an}bracketri}htAnλ′γ−/an}bracketle{tγ′λ′|v|α′γ/an}bracketri}htAnγλ)nαγ′]\n−/summationdisplay\nγγ′γ′′(/an}bracketle{tαγ|v|γ′γ′′/an}bracketri}htCγ′γ′′λγδα′λ′+/an}bracketle{tγγ′|v|α′γ′′/an}bracketri}htCλ′γ′′γγ′δαλ)\n+/summationdisplay\nγγ′(/an}bracketle{tαγ|v|λγ′/an}bracketri}htACλ′γ′α′γ+/an}bracketle{tλ′γ|v|α′γ′/an}bracketri}htACαγ′λγ)\n−/summationdisplay\nγγ′(/an}bracketle{tαλ′|v|γγ′/an}bracketri}htCγγ′α′λ+/an}bracketle{tγγ′|v|α′λ/an}bracketri}htCαλ′γγ′), (48)\nwhere the subscript Ameans that the corresponding matrix is antisymmetrized. The norm matrixS1is given by\nS1(αα′:λλ′) =nλ′α′δαλ−nαλδα′λ′. (49)\nThe first two terms with Cαβα′β′on the right-hand side of eq. (48) describe the self-energy of the particle - hole state\ndue to ground-state correlations [20]. The last four terms with Cαβα′β′may be interpreted as the modification of the\nparticle - hole interaction caused by ground-state correlations [20 ]. Equation (47) is formally the same as the SCRPA\nequation. Both equations include the effects of ground-state cor relations through nαα′andCαβα′β′. The difference\nbetween eq. (47) and the SCRPA equation lies in the fact that nαα′andCαβα′β′are self-consistently generated from\nxµ\nαα′in SCRPA, while they are given by eqs. (9) and (10), independently of eq. (47). Another difference stems from\nthe fact that eq. (9) is used in SCRPA to determine the optimal single particle basis whereas the occupation numbers\nare obtained from a separate relation [14].\nIII. APPLICATION TO LIPKIN MODEL\nA. Lipkin model\nThe Lipkin model [6] describes an N-fermions system with two N-fold d egenerate levels with energies ǫ/2 and−ǫ/2,\nrespectively. The upper and lower levels are labeled by quantum numb erpand−p, respectively, with p= 1,2,...,N.7\nWe consider the standard hamiltonian\nˆH=ǫˆJz+V\n2(ˆJ2\n++ˆJ2\n−), (50)\nwhere the operators are given as\nˆJz=1\n2N/summationdisplay\np=1(a†\npap−a−p†a−p), (51)\nˆJ+=ˆJ†\n−=N/summationdisplay\np=1a†\npa−p. (52)\nSince eq. (9) keeps the property np−p=n−pp= 0, the single-particle state in eq. (8) is equivalent to the original\nsingle-particle basis labeled by −pandp. When the matrix inversion in eq. (13) is taken, it is essential to pres erve\nsymmetry properties of all the matrices. Therefore, we use the s o-called m scheme to define each matrix throughout\nour numerical calculations: For example, n−p−p,n−p′−p′,C−p−p′pp′,C−p′−ppp′,C−p−p′p′pandC−p′−pp′pwithpand\np′= 1,2,...,Nare treated as independent quantities. Of course, eqs. (9) and ( 10) guarantee n−p−p=n−p′−p′and\nC−p−p′pp′=−C−p′−ppp′=−C−p−p′p′p=C−p′−pp′p. Let us mention that due to the use of the m-scheme the\ndimension of the matrices can become considerable, even for the ca se of the Lipkin model. One is forced to do that,\notherwise the full antisymmetry can not be maintained.\nB. Ground state\nSince eq. (13) involves the time-consuming inversion of a large matrix , we first find an approximate solution for\neqs. (9) - (11) to be used in the gradient method using the following t ime-dependent approach [15, 21]: Starting\nwith the HF ground state, we solve eqs. (5)-(7) using the time-dep endent interaction V(t) =V×t/τwith large τ\n[21, 22]. We tested the reliability of this time-dependent method for a nN= 2 system, to which the truncation of\nthe reduced density matrices up to the two-body level should give t he exact solution. Since there are no three-body\ndensity matrices, Eq. (6) for the N= 2 system is modified to\ni¯hd\ndtραβα′β′= (ǫα+ǫβ−ǫα′−ǫβ′)ραβα′β′\n+/summationdisplay\nλ1λ2[/an}bracketle{tαβ|v|λ1λ2/an}bracketri}htρλ1λ2α′β′−/an}bracketle{tλ1λ2|v|α′β′/an}bracketri}htραβλ1λ2], (53)\nwhereραβα′β′=A(nαα′nββ′)+Cαβα′β′. We found that τ= 2π¯h/ǫ×10 is sufficiently large to suppress the spurious\nmixing of excited states and the obtained ground-state energy is e quivalent to the exact one −√\nǫ2+V2within\nnumerical accuracy. After the time-dependent calculation eq. (1 3) is then solved for a few hundred iterations to\nachieve the conditions eqs. (9) - (11) for each matrix element, whic h guarantee the hermiticity of the hamiltonian\nmatrix of eq. (25). Since the time dependent approach already give s a good approximate solution for eqs. (9) - (11),\nthe change in the total energy during the iteration process of eq. (13) is negligible.\nThe ground-state energy in TDDM (solid line) is shown in fig. 1 as a func tion ofχ= (N−1)V/ǫforN= 4. The\nresults with TDDM1 and the exact ground-state energies are show n in fig. 1 with the dotted and dot-dashed lines,\nrespectively. As seen in fig. 1, the ground-state energies in TDDM1 are overestimated. This unpleasant feature of\nTDDM1 is removed by the inclusion of the three-body correlation mat rix and the agreement with the exact solution\nis much improved. In order to see the ground-state properties in m ore detail, we show the occupation probability\nof the upper level nppand the two-body correlation matrix Cpp′−p−p′in figs. 2 and 3, respectively. The solid and\ndotted lines denote the results in TDDM and TDDM1, respectively. Th e exact values are shown with the dot-dashed\nlines. Figures 2 and 3 show that the agreement of these matrices wit h the exact ones is also improved by the inclusion\nof the three-body correlation matrices. As is understood from eq s. (A2) and (A7) in the Appendix, the three-body\ncorrelation matrix interferes in particle - particle and particle - hole c orrelations and presumably plays a role in\nscreening these two-body correlations. In order to investigate t he effects of the three-body correlations in larger N\nsystems, we calculate the ground-state energies in TDDM for N= 8. As shown in fig. 4, an improvement similar to\ntheN= 4 case is achieved.\nThe importance of the three-body correlation matrix in the standa rd Lipkin model is strongly in contrast to the\ncase of an extended Lipkin-model hamiltonian which we have previous ly studied [10, 12]. The extended Lipkin model8\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s51/s46/s53/s45/s51/s46/s48/s45/s50/s46/s53/s45/s50/s46/s48/s69/s48/s32/s47/s32\nFIG. 1: Ground-state energy in TDDM (solid line) as a functio n ofχ= (N−1)|V|/ǫforN= 4. The dotted line depicts the\nresults in TDDM1 where the three-body correlation matrix is neglected, and the dot-dashed line the exact values.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s110/s112/s112\nFIG. 2: Occupation probability of the upper state as a functi on ofχforN= 4. The meaning of the three lines is the same as\nin fig. 1.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s67/s112/s112/s39/s45/s112/s45/s112/s39\nFIG. 3: Two-body correlation matrix Cpp′−p−p′as a function of χforN= 4. The meaning of the three lines is the same as in\nfig. 1.9\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s54/s46/s48/s45/s53/s46/s53/s45/s53/s46/s48/s45/s52/s46/s53/s45/s52/s46/s48/s69/s48/s32/s47/s32\nFIG. 4: Ground-state energy as a function of χforN= 8. The meaning of the three lines is the same as in fig. 1.\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s83/s49/s32\n/s69/s32/s47/s32\nFIG. 5: Strength function of the one-phonon state in ERPA (so lid line) for χ= 1 and N= 4. The dotted and dot-dashed lines\ndepict the result in ERPA1 and the exact solution, respectiv ely.\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s83/s49/s32\n/s69/s32/s47/s32\nFIG. 6: Strength function of the one-phonon state for χ= 1.5 andN= 4. The meaning of the three lines is the same as in fig.\n5.10\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s49/s50/s51/s52/s69/s32/s47/s32\nFIG. 7: Excitation energies of the one-phonon state (lower p art) and the two-phonon state (upper part) as a function of χfor\nN= 4. The filled and open squares depict the results in ERPA and E RPA1, respectively. The filled and open triangles indicate\nthe results in IRPA and IRPA1, respectively. The dashed line depicts the results in RPA. The exact solutions are given by t he\nsolid lines.\nhas the additional particle scattering term ( Uterm) [7, 8]\n1\n2U[ˆJz(ˆJ++ˆJ−)+(ˆJ++ˆJ−)ˆJz]. (54)\nThisUterm generates a mean-field potential [8] and gives nonzero off-dia gonal elements of the occupation matrix,\nthat is,np−p/ne}ationslash= 0 and n−pp/ne}ationslash= 0. Therefore, a significant portion of the interaction energy inclu ding the Vterm in eq.\n(50) is carried as the mean-field energy [10], almost independently of the strength of the Uterm. As a consequence,\nthe two-body correlation matrix and the correlation energy becom e quite small in the extended Lipkin model. The\neffect of the three-body correlationson the ground-state ener gy was found even smaller when the Uterm was included\n[12]. In the case of the extended Lipkin model the ground-state en ergies calculated in TDDM1 are always above the\nexact values and such an overbound problem as in the standard Lipk in model does not occur. Now the importance of\nthe three-body ground-state correlations in the standard Lipkin model is understood as a consequence of lack of the\nmean-field energy, which makes the truncation of the reduced den sity matrices up to the two-body level unreliable\nand enhances the relative importance of the three-body correlat ions.\nThe standard Lipkin model has deformed HF solutions with n−pp/ne}ationslash= 0 for χ >1 [1], which consequently carry\nthe mean-field energy which sums up already a lot of correlations. On e may then think that the truncation up to\nthe two-body level be reliable when the two-body correlation matrix is defined using the deformed HF single-particle\nstates. To answer this question, we performed a simplified gradient -method calculation where the three-body parts in\neq. (13) are neglected and a deformed HF state at χ= 1.5 is used as the initial state. We found that the converged\nresults are the same as those shown in figs. 1-3: The residual inter action plays a role in restoring the symmetry of eq.\n(50) which is violated by the deformed HF solution. In other words st arting with a ’deformed’ solution, at the end of\nthe iteration cycle the solution is back to ’sphericity’. Thus the use of the deformed HF basis does not improve the\ntwo-body level approximation for the standard Lipkin model. This so mewhat surprising feature may be due to the\nsmall number of particles considered. It is known that standard de formed RPA becomes exact in the macroscopic\nlimit. Therefore, we suppose that considering higher particle numbe rs, the deformation will catch on when we start\nwith the deformed HF state.\nC. Excited states\nFor the evaluation of the excited states, we use the ERPA equation (25). First we present the results for the\none-phonon states in the N= 4 system excited by the operator ˆQ=ˆJ++ˆJ−. Figures 5 and 6 show the strength\nfunctions of the one-phonon states at χ= 1 and χ= 1.5, respectively. To facilitate easy comparison among various\ncalculations, we smooth the strength functions with an artificial wid th ΓFWHM/ǫ= 0.025. The solid and dotted lines\ndenote the results in ERPA and ERPA1, respectively. The exact res ults are shown with the dot-dashed lines. The\nresults in ERPA almost coincide with the exact ones, while those in ERPA 1 are located above the exact solutions.11\n/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48/s50/s53/s48/s48/s83/s50/s32\n/s69/s32/s47/s32\nFIG. 8: Strength function of the two-phonon state in ERPA (so lid line) for χ= 1 and N= 4. The dotted and dot-dashed lines\ndepict the result in ERPA1 and the exact solution, respectiv ely.\n/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53/s48/s53/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48/s83/s50/s32\n/s69/s32/s47/s32\nFIG. 9: Strength function of the two-phonon state for χ= 1.5 andN= 4. The meaning of the three lines is the same as in fig.\n8.\nNote that RPA breaks down at χ= 1 [1] as shown in fig. 7. At χ= 1.5 the effect of the three-body correlations on the\none-phonon state is drastic because it significantly improves the oc cupation probabilities and the two-body correlation\nmatrices as shown in figs 2 and 3. Thus it is found that the inclusion of t he three-body ground-state correlations also\ngives a better description of the one-phonon state in the standar d Lipkin model. The χdependence of the excitation\nenergy of the one-phonon state is shown in fig. 7 for various appro ximations. The filled triangles indicate the results\nobtained from eq. (47) using the same nαα′andCαβα′β′as those used in ERPA. This approximation is referred to\nas the improved RPA (IRPA). Similarly, the results obtained from eq. (47) with the same nαα′andCαβα′β′as those\nused in ERPA1 are referred to as the IRPA1 results (open triangles ). The excitation energies of the one-phonon state\nin IRPA and IRPA1 are larger than the exact values and increase with increasing χ. This is explained by the increase\nin the self-energy terms in eq. (48). Since the deviation of the occu pation matrix and correlation matrix from the\nexact values is larger in TDDM1 than in TDDM, the excitation energies in IRPA1 are larger than those in IRPA. The\ndifference between the results in IRPA and ERPA and also between IR PA1 and ERPA1 is due to the coupling of the\none-body amplitudes to the two-body ones. Since the parity of the number of particle - hole pairs is conserved in the\nstandard Lipkin model, the one-body amplitudes can couple to the tw o-body amplitudes that have the same parity:\nFor example, xµ\np−pcouples to Xµ\npp′−pp′, which expresses excitations built on the two particle - two hole confi gurations\nin the ground state. As shown in fig. 7, the coupling to the two-body amplitudes is essential to obtain the one-phonon\nstates with accurate excitation energies.\nNow we discuss the two-phonon states excited by ˆQ2= (ˆJ++ˆJ−)2. The results at χ= 1 and 1.5 are shown in12\nfigs. 8 and 9, respectively. Due to the parity conservation, the tw o-phonon state excited by ˆQ2does not couple to\nthe one-phonon state in the standard Lipkin model. The excitation e nergies both in ERPA and ERPA1 are higher\nthan the exact value and the improvement of the two-phonon stat e due to the inclusion of the three-body correlations\nis not so large as in the case of the one-phonon state. This indicates that the correlations among the two-body\namplitudes are not sufficient to lower the energy of the two-phonon state, suggesting the importance of the coupling\nto the three-body amplitudes which are neglected in ERPA and ERPA1 . Theχdependence of the excitation energy\nof the two-phonon state is also shown in fig. 7.\nWe also studied the stability condition by diagonalizing the stability matr ix eq. (46). We found that all the\neigenvalues associated with the physical operators such as J+,J−andJzwhich consist of the hamiltonian eq. (50)\nare positive definite. This corresponds to the real eigenvalues of e q. (25) for the one-phonon and two-phonon states\nexcited by Q=J++J−andQ2. However, the stability matrix has some negative eigenvalues for ot her degrees of\nfreedom, though their absolute values are small; They are at most 1 % of the maximum positive eigenvalue for the\nphysical operators. This means that the stability condition is not alw ays satisfied for unphysical states. These states\ncorrespond to non-collective phonons as discussed in ref. [20].\nIV. SUMMARY\nThe density-matrix formalism which includes the effects of three-bo dy correlations on the ground state was applied\nto the standard Lipkin model. It was found that the inclusion of the t hree-body correlations removes unpleasant\nfeatures of the two-body level approximation and drastically impro ves the ground-state properties. It was discussed\nthat the importance of the three-body correlations in the standa rd Lipkin model is attributed to the fact that it\ndoes not have the mean-field contributions when the conservation of the number of particle - hole pairs is respected.\nThe extended RPA built on the ground state with three-body corre lations was applied to the one-phonon and two-\nphonon states. It was found that the spectrum of the one-phon on state is drastically improved by the inclusion of\nthe three-body correlations. In the case of the two-phonon sta tes, however, the agreement with the exact solutions is\nnot so good as in the case of the one-phonon states, which sugges ts the importance of the coupling to the three-body\namplitudes. Inclusion of the three-body correlations for realistic n uclei may be impracticable. Fortunately, mean-\nfield effects are quite important in nuclei and the mean-field theories give good first description for ground states\nand collective excitations. Therefore, the truncation up to the tw o-body level may be justified as in the case of the\nextended Lipkin model. When the three-body correlation matrix is ne glected, the hermiticity of the hamiltonian\nmatrix in ERPA is not guaranteed. Our applications so far performed for realistic cases indicate that this does not\ncause any serious problems.\nAppendix A\nF1,F2andF3in eqs. (9)-(11) are shown. The single-particle states which satisf y the HF-like equation hφα=ǫφα\nare used.\nF1(αα′) = (ǫα−ǫα′)nαα′+/summationdisplay\nλ1λ2λ3(Cλ1λ2α′λ3/an}bracketle{tαλ3|v|λ1λ2/an}bracketri}ht−Cαλ3λ1λ2/an}bracketle{tλ1λ2|v|α′λ3/an}bracketri}ht), (A1)\nF2(αβα′β′) = (ǫα+ǫβ−ǫα′−ǫβ′)Cαβα′β′+Bαβα′β′+Pαβα′β′+Hαβα′β′+Tαβα′β′, (A2)\nF3(αβγα′β′γ′) = (ǫα+ǫβ+ǫγ−ǫα′−ǫβ′−ǫγ′)Cαβγα′β′γ′\n+I(αβγα′β′γ′)−I(βαγα′β′γ′)−I(γβαα′β′γ′)\n−I∗(α′β′γ′αβγ)+I∗(β′α′γ′αβγ)+I∗(γ′β′α′αβγ)\n+J(αβγα′β′γ′)+J(βγαα′β′γ′)−J(αγβα′β′γ′)\n−J∗(α′β′γ′αβγ)−J∗(β′γ′α′αβγ)+J∗(α′γ′β′αβγ), (A3)\nwhere\nBαβα′β′=/summationdisplay\nλ1λ2λ3λ4/an}bracketle{tλ1λ2|v|λ3λ4/an}bracketri}htA[(δαλ1−nαλ1)(δβλ2−nβλ2)nλ3α′nλ4β′\n−nαλ1nβλ2(δλ3α′−nλ3α′)(δλ4β′−nλ4β′)], (A4)\nPαβα′β′=/summationdisplay\nλ1λ2λ3λ4/an}bracketle{tλ1λ2|v|λ3λ4/an}bracketri}ht[(δαλ1δβλ2−δαλ1nβλ2−nαλ1δβλ2)Cλ3λ4α′β′13\n−(δλ3α′δλ4β′−δλ3α′nλ4β′−nλ3α′δλ4β′)Cαβλ1λ2], (A5)\nHαβα′β′=/summationdisplay\nλ1λ2λ3λ4/an}bracketle{tλ1λ2|v|λ3λ4/an}bracketri}htA[δαλ1(nλ3α′Cλ4βλ2β′−nλ3β′Cλ4βλ2α′)\n+δβλ2(nλ4β′Cλ3αλ1α′−nλ4α′Cλ3αλ2β′)\n−δα′λ3(nαλ1Cλ4βλ2β′−nβλ1Cλ4αλ2β′)\n−δβ′λ4(nβλ2Cλ3αλ1α′−nαλ2Cλ3βλ1α′)], (A6)\nTαβα′β′=/summationdisplay\nλ1λ2λ3λ4/an}bracketle{tλ1λ2|v|λ3λ4/an}bracketri}ht[δαλ1Cλ3λ4βα′λ2β′+δβλ2Cλ4λ3αβ′λ1α′\n−δα′λ3Cαλ4βλ1λ2β′−δβ′λ4Cβλ3αλ2λ1α′], (A7)\nI(αβγα′β′γ′) =/summationdisplay\nλ1λ2λ3{/an}bracketle{tαλ3|v|λ1λ2/an}bracketri}ht(nγλ3Cλ1λ2βα′β′γ′−nβλ3Cλ1λ2γα′β′γ′+Cλ1λ2α′β′Cβγγ′λ3\n−Cλ1λ2α′γ′Cβγβ′λ3+Cλ1λ2β′γ′Cβγα���λ3)\n+/an}bracketle{tαλ3|v|λ1λ2/an}bracketri}htA[nλ1α′Cλ2βγβ′γ′λ3−nλ1β′Cλ2βγα′γ′λ3+nλ1γ′Cλ2βγα′β′λ3\n+Cλ2βγ′λ3Cλ1γα′β′+Cλ2γα′λ3Cλ1βγ′β′+Cλ2γβ′λ3Cλ1βα′γ′−Cλ2γγ′λ3Cλ1βα′β′\n−Cλ2βα′λ3Cλ1γγ′β′−Cλ2ββ′λ3Cλ1γα′γ′\n+nγλ3(nλ1α′Cλ2ββ′γ′−nλ1β′Cλ2βα′γ′+nλ1γ′Cλ2βα′β′)\n−nβλ3(nλ1α′Cλ2γβ′γ′−nλ1β′Cλ2γα′γ′+nλ1γ′Cλ2γα′β′)\n+nλ1α′(nλ2β′Cβγγ′λ3−nλ2γ′Cβγβ′λ3)+nλ1β′nλ2γ′Cβγα′λ3]}, (A8)\nJ(αβγα′β′γ′) =/summationdisplay\nλ1λ2[/an}bracketle{tαβ|v|λ1λ2/an}bracketri}htA(nλ1α′Cλ2γβ′γ′−nλ1β′Cλ2γα′γ′+nλ1γ′Cλ2γα′β′)\n+/an}bracketle{tαβ|v|λ1λ2/an}bracketri}htCλ1λ2γα′β′γ′]. (A9)\nThe matrix Bαβα′β′does not contain Cαβα′β′and may be called the Born term, whereas Pαβα′β′andHαβα′β′contain\nCαβα′β′and describe particle-particle (and hole-hole) and particle-hole cor relations to infinite order, respectively.\nThe last term on the right-hand side of eq. (A2) expresses the con tribution of the three-body correlation matrix.\nThe matrix J(αβγα′β′γ′) includes particle-particle (and hole-hole) correlations, while I(αβγα′β′γ′) contains both\nparticle-particle and particle-hole correlations.\n[1] P. Ring and P. Schuck, The Nuclear Many-Body Problem , (Springer-Verlag, Berlin, 1980).\n[2] S. J. Wang and W. Cassing, Ann. Phys. 159, 328 (1985); W. Cassing and S. J. Wang, Z. Phys. A 328, 423 (1987).\n[3] M. Gong and M. Tohyama, Z. Phys. A 335, 153 (1989).\n[4] A. Peter, W. Cassing, J. M. H¨ auser, A. Pfitzner, Nucl. Phy s. A573, 93 (1994).\n[5] M. Tohyama and M. Gong, Z. Phys. A 332, 269 (1989).\n[6] H. J. Lipkin, N. Meshkov and A. J. Glick, Nucl. Phys. 62, 188 (1965).\n[7] S. T. Yang, J. Heyer and T. T. S. Kuo, Nucl. Phys. A448, 420 (1986).\n[8] D. Shindo and K. Takayanagi, Phys. Rev. C 68, 014312 (2003).\n[9] S. Takahara, M. Tohyama and P. Schuck, Phys. Rev. C 70, 057307 (2004).\n[10] M. Tohyama, Phys. Rev. C 75, 044310 (2007).\n[11] M. Tohyama and P. Schuck, Eur. Phys. J. A 32, 139 (2007).\n[12] M. Tohyama and P. Schuck, Eur. Phys. J. A 36, 349 (2008).\n[13] D. J. Rowe, Rev. Mod. Phys. 40, 153 (1968).\n[14] J. Dukelsky and P. Schuck, Nucl. Phys. A 512, 466 (1990); J. Dukelsky, G. R¨ opke, P. Schuck, Nucl. Phys. A 628, 17(1998);\nD.S. Delion, P. Schuck, J. Dukelsky, Phys. Rev. C 72, 064305 (2005).\n[15] M. Tohyama, S. Takahara and P. Schuck, Eur. Phys. J. A 21, 217 (2004).\n[16] M. Tohyama and P. Schuck, Eur. Phys. J. A 19, 203 (2004).\n[17] J. Sawicki, Phys. Rev. 126, 2231 (1962).\n[18] S. Dro˙ zd˙ z, S. Nishizaki, J. Speth and J. Wambach, Phys . Rep.197, 1 (1990).\n[19] K. Takayanagi, K. Shimizu and A. Arima, Nucl. Phys. A 477, 205 (1988).\n[20] D. Janssen and P. Schuck, Z. Phys. A 339, 43 (1991).\n[21] A. Pfitzner, W. Cassing, and A. Peter, Nucl. Phys. A 577, 753 (1994).\n[22] M. Tohyama, J. Phys. Soc. Jpn. 78, 104003 (2009)."    },    {        "title": "1003.0378v1.Reentrant_fractional_quantum_Hall_states_in_bilayer_graphene__Controllable__driven_phase_transitions.pdf",        "content": "arXiv:1003.0378v1  [cond-mat.mtrl-sci]  1 Mar 2010Reentrant fractional quantum Hall states in bilayer graphe ne: Controllable, driven\nphase transitions\nVadim M. Apalkov\nDepartment of Physics and Astronomy, Georgia State Univers ity, Atlanta, Georgia 30303, USA\nTapash Chakraborty‡\nDepartment of Physics and Astronomy, University of Manitob a, Winnipeg, Canada R3T 2N2\n(Dated: November 20, 2018)\nHere we report from our theoretical studies that in biased bi layer graphene, one can induce phase\ntransitions from an incompressible state to a compressible state by tuning the bandgap at a given\nelectron density. Likewise, variation of the density with a fixed bandgap results in a transition\nfrom the FQHE states at lower Landau levels to compressible s tates at intermediate Landau levels\nand finally to FQHE states at higher Landau levels. This intri guing scenario of tunable phase\ntransitions in the fractional quantum Hall states is unique to bilayer graphene and never before\nexisted in conventional semiconductor systems.\nThe unconventional quantum Hall effect in monolayer\ngraphene, whose experimental observation [2] unleashed\nquite unprecedented interest in this system [3], reflects\nthe unique behavior of massless Dirac fermions in a mag-\nnetic field [4, 5]. In bilayer graphene this effect confirms\nthe presence of massive chiral quasiparticles [6]. An im-\nportant characteristic of bilayer graphene is that it is a\nsemiconductor with a tunable bandgap between the va-\nlence and conduction bands [7]. This modifies the Lan-\ndau level spectrum and influences the role of long-range\nCoulomb interactions [8]. Here we report that the frac-\ntional quantum Hall effect (FQHE), a distinct signature\nof interacting electrons in the system [9, 10] is very sensi-\ntive to the interlayer coupling strength and the bias volt-\nage. We propose that by tuning the bias voltage, one can\ninduce phase transitions from an incompressible state to\na compressible state at a given gate voltage. In the same\nvein, variation of the gate voltage at a fixed bias poten-\ntial results in a transition from the FQHE states in lower\nLandaulevelstocompressiblestatesin intermediateLan-\ndau levels and finally to FQHE states in higher Landau\nlevels. This interesting scenario of tunable phase tran-\nsitions in the FQH states is unique to bilayer graphene\nand never before existed in conventional semiconductor\nsystems. The FQHE in monolayer graphene was in fact,\nstudied theoretically by us [11] and subsequent experi-\nments confirmed the existence of that effect in suspended\nmonolayer graphene samples [12]. No such studies have\nbeen reported on bilayer graphene.\nWe assume that the bilayer graphene consists of two\ncoupled graphene layers with the Bernal stacking ar-\nrangement. In that case, we are mainly concerned with\nthe coupling between the atoms of sublattice A of the\nlower layer and atoms of sublattice B′of the upper\nlayer. The single-particle levels in bilayer graphene have\ntwo-fold spin degeneracy and two-fold valley degeneracy,\nwhich can be lifted in many-particle systems at relatively\nlarge magnetic fields [13]. Considering only one valley\n(say valley K) and one direction of spin, we describe the\nstateofthebilayersystemintermsofthefour-componentspinor(ψA,ψB,ψB′,ψA′)T. Here subindicesA, B and A′,\nB′correspondtolowerandupperlayersrespectively. The\nstrength of inter-layer coupling is described in terms of\nthe inter-layer hopping integral, t. In a biased bilayer\ngraphene the bias potential is introduced as the poten-\ntial difference, ∆ U, between the upper and lower layers.\nThe Hamiltonian of the biased bilayer system in a per-\npendicular magnetic field then takes the following form\nH=\n∆U/2vFπ+t 0\nvFπ−∆U/2 0 0\nt0−∆U/2vFπ−\n0 0 vFπ+−∆U/2\n,(1)\nwhereπ±=πx±iπy,/vector π=/vector p+e/vectorA/c,/vector pis an electron\ntwo-dimensional momentum, /vectorAis the vector potential,\nandvF≈106m/s is the fermi velocity.\nIn a perpendicular magnetic field the Hamiltonian\n(1) generates a discrete Landau level energy spectrum.\nThe corresponding eigenfunctions can be expressed in\nterms of the conventional nonrelativistic Landau func-\ntions. The electron states at sublattices A and A′are\nexpressed in terms of the n-th ‘nonrelativistic’ Landau\nfunctions, while the electron states at sublattices B and\nB′are described by the |n−1|andn+ 1 Landau func-\ntions, respectively. Therefore the Landau states in bi-\nlayer graphene can be described as a mixture of the n,\nn+ 1, andn−1 nonrelativistic Landau functions be-\nlonging to different sublattices [7]. This mixture, for a\ngiven value of n, results in four different Landau levels\nin bilayer graphene. The energies, ε, of the four Landau\nlevels corresponding to the index ncan be found from\nthe following equation [7]\n/bracketleftig\n(ε+δ)2−2(n+1)/bracketrightig/bracketleftbig\n(ε−δ)2−2n/bracketrightbig\n= (ε2−δ2)t2,(2)\nwhereδ= ∆U/2 and all energies are expressed in units\nof/planckover2pi1vF/lB. HerelB= (/planckover2pi1/eB)1\n2is the magnetic length.\nItisconvenienttointroducespeciallabelingoftheLan-\ndau levels in bilayer graphene. From Eq. (2), we can see2\nthat for each value of n,n= 0,1,2,..., there are four\nsolutions, i.e., four Landau levels. Then each of these\nLandau levels can be labeled as n(i), wherei= 1,...,4\nis the number of the Landau level corresponding to the\nsolution of Eq. (2) for a given value of n.\nFor a partially occupied Landau level, the properties\nof the system, e.g., the ground state and excitations, are\ncompletely determined by the inter-electron interactions,\nwhich can be expressed by Haldane’s pseudopotentials,\nVm, [14] which are the energies of two electrons with rel-\native angular momentum m. In a graphene bilayer the\nHaldane pseudopotentials in a Landau level with index\nnand the energy εhave the form\nV(n)\nm=/integraldisplay∞\n0dq\n2πqV(q)[Fn,ε(q)]2Lm(q2)e−q2,(3)\nwhereLm(x) are the Laguerre polynomials, V(q) =\n2πe2/(κlBq) is the Coulomb interaction in the momen-\ntum space, κis the dielectric constant, and Fn,ε(q) are\nthe corresponding form factors\nFn,ε(q) =d2\nn/bracketleftbigg/parenleftbig\n1+f2\nn/parenrightbig\nLn/parenleftig\nq2\n2/parenrightig\n+2n\n(ε−δ)2Ln−1/parenleftig\nq2\n2/parenrightig\n+2(n+1)\n(ε+δ)2f2\nnLn+1/parenleftig\nq2\n2/parenrightig/bracketrightbigg\n, (4)\nwherefn= [(ε−δ)2−2n]/[t(ε−δ)] and\ndn=/bracketleftbigg\n1+f2\nn+2n\n(ε−δ)2+2(n+1)\n(ε+δ)2f2\nn/bracketrightbigg−1\n2\n.(5)\nThe form factors of bilayer graphene [Eq. (4)] are clearly\ndifferent from the corresponding form factors of a sin-\ngle layer of graphene. For a single layer of graphene,\nthe form factor of the n= 0 Landau level is the same\nas that of conventional non-relativistic electrons [11, 15],\nFn=0(q) =L0/parenleftbig\nq2/2/parenrightbig\n. The form factors of higher Landau\nlevels are determined by the mixture of LnandLn+1\nterms. For bilayer graphene the form factors of the Lan-\ndau level with index n= 0 are the mixtures of the L0\nandL1terms and are different from that in the non-\nrelativistic case. There is of course, one special Landau\nlevel in bilayer graphenewith index n= 0, whose proper-\ntiesarecompletelyidenticaltothatofthenon-relativistic\nn= 0 Landau level. Indeed, it is clear from Eq. (2) that\natn= 0 there is a Landau level with energy ǫ=δ. This\nenergy does not depend on the coupling between the lay-\ners, i.e., the inter-layer hopping integral, t. The form\nfactor of this Landau level is exactly equal to the form\nfactor of a non-relativistic system of the n= 0 Landau\nlevel,Fn=0,ǫ=δ=L0. Therefore, all many-body proper-\nties of a bilayer system in the n= 0,ǫ=δLandau level\narecompletely identical tothose ofanon-relativisticcon-\nventional system in the n= 0 Landau level.\nFor Landau levels with higher indices the form fac-\ntor is the mixture of three different functions, Ln,Ln−1,\nandLn+1. Therefore, in general, the strength of inter-\nelectron interactions in bilayer graphene is strongly mod-\nified as compared to its value in monolayer graphene. Toaddress the effects of these modifications on the proper-\nties of the many-electronsystem in a graphene bilayer we\nstudy below the partially occupied Landau levels with\nfractional filling factor corresponding to the FQHE [9].\nIn what follows, we study the many-electron system at\nvarious fractional filling factors numerically within the\nspherical geometry [11, 14]. The radius of our spehere is\nR=√\nSlB, where 2Sis the number of magnetic fluxes\nthrough the sphere in units of the flux quanta. The\nsingle-electron states are characterized by the angular\nmomentum, S, and itszcomponent, Sz. For the many-\nelectron system the corresponding states are classified\nby the total angular momentum Land itszcomponent,\nwhile the energy of the state depends only on L[16]. A\ngiven fractional filling of the Landau level is determined\nby a special relation between the number of electrons N\nand the radius of the sphere R. For example, the1\n3-\nFQHE state is realized at S= (3\n2)(N−1), while the\n2\n5-FQHE state corresponds to the relation S= (5\n4)N−2.\nWith the Haldane pseudopotentials [Eq. (3)] we deter-\nmine the interaction Hamiltonian matrix [16] and then\ncalculateafew lowesteigenvaluesandeigenvectorsofthis\nmatrix. The FQHE states are obtained when the ground\nstateofthe systemisanincompressibleliquid, theenergy\nspectrum of which has a finite many-body gap [9, 10].\nWe begin our study with the celebrated1\n3-FQHE [10]\ncorresponding to the filling factor ν=1\n3. In Fig. 1 we\nshow the behavior of the Landau level spectra as a func-\ntion of the bias voltage and for different values of the\ninter-layer hopping integral, t. Only the Landau levels\nwith positive energies (corresponding to the conduction\nband) are shown in the figure. A similar behavior is valid\nfor other FQHE filling factors, e.g., for ν=2\n5. Figure 1\nclearly illustrates that the FQHE can be observed in all\nn= 0 Landau levels with the strongest FQHE being in\nthe first and the third n= 0 Landau levels, i.e., 0 (1)and\n0(3). An interesting behavior was also observed for the\nn= 1 Landau levels. For ∆ U≈200 meV the n= 1 Lan-\ndau levels show an anti-crossing , which is accompanied\nby strong changes in the properties of the FQHE. More\nspecifically, at a small ∆ Uthe FQHE can be observed\nonly in the lower n= 1 Landau level, while for larger\n∆Uthe FQHE is possible only in higher n= 1 Lan-\ndau level. These have important implications for pos-\nsible experimental observations of this unique behavior\n(see Fig. 1(b)):\n(i) By applying a gate voltage the electron density can\nbe tuned so that the first three Landau levels in bilayer\ngraphene are completely occupied and the next Landau\nlevel is partially occupied with the FQHE filling factor,\nfor example, ν=1\n3. According to the notations of Fig. 1,\nthis means that the 0 (1), 0(2), 0(3)Landau levels are fully\noccupied, while the 1 (1)Landau level has a filling factor\n1\n3. Then, by varying ∆ Ufrom a small value, e.g., 100\nmeV, to a larger value, e.g., 400 meV, one can observe\nthe disappearanceof the FQHE (see line (i) in Fig. 1(b)).\n(ii) The bias voltage is kept fixed at a large value, e.g.,3\nFIG. 1: A few lowest Landau levels of the conduction band\nare shown as a function of the bias potential, ∆ U, for dif-\nferent values of inter-layer coupling: (a) t= 100 meV (b)\nt= 200 meV and (c) t= 400 meV and the magnetic field is\n15 Tesla. The numbers next to the curves denote the corre-\nsponding Landau levels. The Landau levels where the FQHE\ncan be observed are drawn as blue and green dots. The green\ndots correspond to the Landau levels where the FQHE states\nare identical to that of a monolayer of graphene or a non-\nrelativistic conventional system. The red dots represent L an-\ndau levels with weak FQHE. The open dots correspond to\nthe Landau levels where the FQHE cannot be observed. In\n(b), the dashed lines labeled by (i) and (ii) illustrate two s it-\nuations: (i) under a constant gate voltage and variable bias\npotential; (ii) under a constant bias potential and variabl e\ngate voltage.\n∆U= 400 meV. Then by varying the gate voltage and\nthus increasing the electron density, one can observe the\ndisappearance and reappearance of the FQHE at large\nLandau levels (see line (ii) in Fig. 1(b)).\nThe appearance or suppression of the FQHE in differ-\nent Landau levels is correlated with the unique behavior\nof Haldane pseudopotentials in bilayer graphene. The\nFQHE is observed in Landau levels with a rapid decrease\nofthecorrespondingpseudopotentialswithincreasingan-\ngular momentum, m. For the first and second n= 0\nLandau levels (0 (1)and 0(2)levels), between m= 1 and\nFIG. 2: Low energy excitation spectra of the FQHE states\nshown for different values of the bias potential. The numbers\nnext to the lines are the values of the bias potential in meV.\n(a)ν=1\n3-FQHE (eight electrons) system in the Landau level\n1(2), (b)ν=1\n3-FQHE (eight electrons) system in the Lan-\ndau level 1 (1), (c)ν=2\n5-FQHE (ten electrons) system in the\nLandau level 1 (2), (d)ν=2\n5-FQHE (ten electrons) system in\nthe Landau level 1 (1). The systems are fully spin-polarized.\nThe flux quanta is 2 S= 21. The magnetic field is 15 T. The\nground state is shown as a solid dot at L= 0.\nm= 3, the pseudopotential has the fastest decay for the\nfirst level, which results in a higher FQHE gap in this\nLandau level. The pseudopotentials in the 0 (2)Landau\nlevel exactly coincide with the pseudopotentials of the\nn= 0 Landau level in a single layer of graphene. For\nthen= 1 Landau levels, the pseudopotentials in one of\nthen= 1 levels are close to those in the n= 1 Landau\nlevel of the monolayer graphene, resulting in a well pro-\nnounced FQHE. While in the other n= 1 Landau level\nof the bilayer graphene the pseudopotentials display a\nslow decrease with mand no FQHE is expected in that\nLandau level.\nThe collapse of the FQHE gap, corresponding to the\nappearence of anticrossing of the n= 1 Landau levels,\nis illustrated in Fig. 2. The FQHE gap has a monotonic\ndependence on the bias voltage. In the anticrossing re-\ngion the gap disappears for the lower n= 1 Landau level\n(seeFig. 2a,c)and reappearsforthe higher n= 1Landau\nlevel(seeFig. 2b,d). The evolutionsofthe energyspectra\nof the incompressible liquid with different filling factors\nare similar, which is illustrated in Fig. 2a,b (for ν=1\n3)\nandFig. 2c,d(for ν=2\n5). This behaviorwasneverbefore\nobserved in the FQHE of conventional two-dimensional\nelectron systems.\nThe strength of the FQHE effect, i.e., the magnitude\nof the excitation gap, depends on the parameters of the\nbilayer graphene, i.e., on the bias voltage and the inter-\nlayer hopping integral, t. In Fig. 3 such dependence is\nshown for1\n3- and2\n5-FQHE in different Landau levels as\na function of the hopping integral, t. In accordance with\nthe properties of Haldane pseudopotentials, the excita-\ntion gap of the 0 (2)Landau levels does not depend on4\nFIG. 3: The FQHE gaps are shown for different Landau lev-\nels. The labels next to the lines correspond to the labeling\nof Landau levels shown in Fig. 1. (a) ν=1\n3-FQHE (eight\nelectron) system at ∆ U= 10 meV, (b) ν=1\n3-FQHE (eight\nelectron) system at ∆ U= 400 meV, (c) ν=2\n5-FQHE (ten\nelectron) system at ∆ U= 10 meV, (d) ν=2\n5-FQHE (ten\nelectron) system at ∆ U= 400 meV. All systems are fully\nspin polarized. The flux quanta is 2 S= 21. The magnetic\nfield is 15 Tesla.\nthe bias voltage and on the inter-layer hopping integral.\nThe corresponding gap remains constant and is equal to\ngapoftheFQHEinasinglelayerofgrapheneinthe n= 0\nLandau level. This gap has the smallest value compared\nto that ofthe other Landau levels. For zerohopping inte-\ngralthetwolayersofgraphenebecomedecoupledandthe\nbilayersystembecomesidenticaltoasinglelayerwithad-\nditional double degeneracy. This property is clearly seen\nin Fig. 3, where at t= 0 there are only two doubly de-generate FQHE gaps, corresponding to n= 0 andn= 1\nsingle layer Landau levels.\nAt a small bias voltage (∆ U= 10 meV, see Fig. 3a,c),\nthe FQHE gaps have a monotonic dependence on the\nhopping integral. The values of the gaps increases for\nthen= 0 Landau levels, while for the n= 1 Landau\nlevel the gap decreases. At large values of the bias volt-\nage, ∆U= 400 meV (see Fig. 3b,d), the FQHE gap in\nthe 03Landau level has a nonmonotonic dependence on\nthe hopping integral with a maximum around t= 100\nmeV. These results also illustrate that the FQHE in bi-\nlayergraphenecanbe morestable, i.e., thecorresponding\nFQHE gap is much larger than in the case of FQHE in a\nsingle graphene layer. As an illustration, the FQHE gaps\nin the 0 (1)and 0(3)Landau levels at finite values of tare\nlarger than the FQHE gaps in a single layer of graphene,\ni.e., att= 0.\nOur present work suggests an unique opportunity to\ntune the reentrant FQHE states. The transition from\nthe FQHE state to a gapless compressible state is con-\ntinuous. The gap decreases monotonically with the bias\npotential and finallycollapses. The collapseofthe FQHE\ngap occurs at a non-zero angular momentum, L≈2, re-\nsulting in the formation of the ground state with a finite\nmomentum. Experimental observation of this transition\nwill provide unique opportunities, for the first time, to\nstudy the phase transition from an incompressible liq-\nuid to a possible charge density wave state and then to\nanother incompressible state.\nWe wish to thank David Abergel for very helpful dis-\ncussions. The work has been supported by the Canada\nResearch Chairs Program and the NSERC Discovery\nGrant.\n[‡] Electronic address: tapash@physics.umanitoba.ca\n[2] K.S. Novoselov, et al., Nature 438, 197 (2005); Y. Zhang,\nJ.W. Tan, H.L. St¨ ormer, and P. Kim, ibid.438, 201\n(2005).\n[3] D.S.L. Abergel, V. Apalkov, J. Berashevich, K. Ziegler,\nand T. Chakraborty, Adv. Phys. (in press) (2010).\n[4] P.R. Wallace, Phys. Rev. 71, 622 (1947).\n[5] J.W. McClure, Phys. Rev. 104, 666 (1956).\n[6] K.S. Novoselov, et al., Nat. Phys. 2, 177 (2006); E. Mc-\nCann and V. Falko, Phys. Rev. Lett. 96, 086805 (2006);\nE. McCann, Phys. Rev. B 74, 161403 (2006); T. Ohta,\nA. Bostwick, T. Seyller, K. Horn, E. Rotenberg, Science\n313, 951 (2006); E.V. Castro, et al., Phys. Rev. Lett. 99,\n216802 (2007).\n[7] J.M. Pereira, Jr., F.M. Peeters, and P. Vasilopoulos,\nPhys. Rev. B 76, 115419 (2007).\n[8] D.S.L. Abergel and T. Chakraborty, Phys. Rev. Lett.\n102, 056807 (2009).\n[9] T. Chakraborty, and P. Pietil¨ ainen, The Quantum HallEffects(Springer, New York, 1995), 2nd edition; T.\nChakraborty, Adv. Phys. 49, 959 (2000).\n[10] D.C. Tsui, H.L. St¨ ormer, and A.C. Gossard, Phys. Rev.\nLett.48, 1559 (1982); R.B. Laughlin, ibid.50, 1395\n(1983).\n[11] V.M. Apalkov, and T. Chakraborty, Phys. Rev. Lett. 97,\n126801 (2006).\n[12] K.I. Bolotin, F. Ghahari, M.D. Shulman, H.L. St¨ ormer,\nand P. Kim, Nature 462, 196 (2009); see also, X. Du, et\nal.,ibid.462, 192 (2009).\n[13] M. Nakamura, E.V. Castro, and B. Dora, Phys. Rev.\nLett.103, 266804 (2009); Y. Zhao, P. Cadden-Zimansky,\nZ. Jiang, and P. Kim, ibid.104, 66801 (2010).\n[14] F.D.M. Haldane, Phys. Rev. Lett. 51, 605 (1983).\n[15] M.O. Goerbig, R. Moessner and B. Doucot,Phys. Rev. B\n74161407(R) (2006).\n[16] G. Fano, F. Ortolani, and E. Colombo, Phys. Rev. B 34,\n2670 (1986)."    },    {        "title": "1003.2094v1.The_entanglement_of_few_particle_systems_when_using_the_local_density_approximation.pdf",        "content": "arXiv:1003.2094v1  [cond-mat.other]  10 Mar 2010Theentanglement offew-particle systemswhen using the loc al-densityapproximation\nJ P Coe∗and I D’Amico†\nDepartment of Physics, University of York, York YO10 5DD, Un ited Kingdom.\nInthischapter wediscuss methods tocalculate theentangle ment ofasystemusingdensity-functional theory.\nWe firstly introduce density-functional theory and the loca l-density approximation (LDA). We then discuss\nthe concept of the ‘interacting LDA system’. This is charact erised by an interacting many-body Hamiltonian\nwhich reproduces, uniquely and exactly, the ground state de nsity obtained from the single-particle Kohn-Sham\nequationsofdensity-functionaltheorywhenthelocal-den sityapproximationisused. Wemotivatewhythisidea\ncan be useful for appraising the local-density approximati on in many-body physics particularly with regards to\nentanglementandrelatedquantuminformationapplication s. Usinganiterativescheme,wefindtheHamiltonian\ncharacterisingtheinteractingLDAsysteminrelationtoth etestsystemsofHooke’satomandhelium-likeatoms.\nThe interacting LDA system ground state wavefunction is the n used to calculate the spatial entanglement and\ntheresultsarecomparedandcontrastedwiththeexactentan glement forthetwotestsystems. ForHooke’s atom\nwe also compare the entanglement to our previous estimates o f an LDA entanglement. These were obtained\nusing a combination of evolutionary algorithm and gradient descent, and using an LDA-based perturbative\napproach. We finally discuss if the position-space informat ion entropy of the density—which can be obtained\ndirectly from the system density and hence easily from densi ty-functional theory methods—can be considered\nas aproxy measure for the spatial entanglement for the tests ystems.\nI. INTRODUCTION\nThe mapping of an interacting many-body system to a collecti on of effective single particle equations is a ubiquitous an d\npowerful approach in theoretical physics. This is perhaps m ost notably embodied in density-functional theory (DFT) wh ere\na non-interacting many-body system, the Kohn-Sham system [ 1], is proven to have exactly the same ground state density as\nthe original interacting system. By solving a set of single- particle Kohn-Sham (KS) equations, the groundstate densit y can be\ncalculatedand fromthis in principleall the ground-statep ropertiesof the interactingmany-bodysystem [2]. The KS eq uations\nwouldgivetheexactdensityduetotheinclusionintheireff ectivesingle-particlepotentialoftheso-called“exchan ge-correlation”\npotential(vxc). In practice vxcis an unknownfunctionalofthe densityso approximationsfo rvxchavebeen designed. The most\npopular and often effective method is the local-density app roximation (LDA). DFT has been so far one of the most powerful\nmethodsto calculate many-bodypropertiesof complexsyste ms, and its efficiencylies in the observationthat modelling many-\nparticlesystemsexactlyisgenerallycomputationallyint ractableas,forexample,oneisdealingwithamany-bodywav efunction\nwhosestoragerequiresexcessiveamountsofmemory. Solvin gasystemofsingle-particleequationsordealingwiththed ensity,\nwhichisafunctionofonlyonevariable,representsindeeda computationallyattractivealternative. Howeverwe donot knowin\ngeneralhowtoexpresssomegroundstateproperties,suchas theentanglement,intermsofthegroundstateparticledens ity,and\nthisimpliesthatcommonapproximationschemesusedwithin DFTdonothaveastraightforwardextensiontothecalculati onsof\ntheseproperties. Herewepresentaschemetocalculatethee ntanglementassociatedwiththeLDAapproximationandinve stigate\nhow well it compares with the exact entanglement. Our test-b ed systems are helium-like atoms and Hooke’s atoms. This\ninformationcouldbeusefulwhenconsideringwhichapproxi mationtoinvoketomodelthesuitabilityofsystemsasacomp onent\nof a quantuminformationprocessing device. As the entangle mentis consideredas one of the main resourcesfor these devi ces,\nthe knowledge of how much entanglement could be created in a c ertain system appears essential for quantum information\npurposes. In addition comparison of the entanglement assoc iated with a DFT approximation to the exact entanglement off ers\nanotherwaytoappraisetheaccuracyofthe approximation.\nIn this chapter we firstly introduce DFT and the LDA. We explai n the concept of the ‘interacting LDA system’ (i-LDA) and\nhowthismaybeusedtocalculatetheentanglementcorrespon dingtotheuseoftheLDA.Wethenintroduceourtest-bedsyst ems,\nHooke’satomandthehelium-likeatoms. Nextweconsiderhow thesesystemscanbesolvedusingnumerically‘exact’metho ds\nand also approximately using the LDA within DFT. The exact an d i-LDA spatial entanglement results are then displayed and\ncompared. We contrast these results with the position space informationentropy which can easily be expressed as a funct ional\nof the density—andhence calculated using traditionalDFT s chemes—andinvestigateits usefulnessas a proxymeasure fo r the\nentanglement. Finally,wecomparethei-LDAentanglementt oourpreviousbestapproximationstoanLDAentanglement,f ound\nusingacombinationofevolutionaryalgorithmandgradient descent,andanLDA-basedperturbativeapproach[3].\n∗Electronic address: jpc503@york.ac.uk\n†Electronic address: ida500@york.ac.uk2\nII. DENSITY-FUNCTIONALTHEORY\nDirectly solving the many-body Schr¨ odinger equation is a f ormidable, arduous task and doing so accurately for systems of\nmorethanafewelectronsremainsoutsidetherealmsofconte mporarycomputing. Thiscanbepartlyilluminatedbyconsid ering\nthewavefunctiononameshof 10003pointsforoneelectron;forthesameaccuracywith Nelectronsonewouldrequire 10003N\npoints, clearly an excessive amount of memory as Nbecomes large. Yet nature, in a sense performs similar calcu lations all\nthe time, but these are too complex to be reproduced by ‘class ical’ computation. Quantum computing offers some hope in th e\nsimulation of quantum systems (see for example Ref. [4] and R ef. [5]). However, unless quantum computers of a substantia l\nnumberof qubitsare realised, simulating large systems may not be feasible. There is though a powerfuland importantmet hod\nthatallowsclassical computingtoefficientlymodelmany-b odyelectronsystemsthroughthedensity: density-functio naltheory.\nThe density depends upon only one co-ordinate (so only 10003mesh points in our example) therefore by using this, rather\nthan the many-body wavefunction, solving many-body system s can be made computationally more tractable. In its simples t\nformulationdensity-functionaltheoryexpressesobserva blesof theground-stateofan interactingelectronsystem a sfunctionals\noftheground-statedensity,\nn(r1) =N/integraldisplay\n|Ψ(r1...rN)|2dr2...drN (1)\nandisexactin principledueto atheorembyHohenbergandKoh n[2] whichwe presentnext.\nA. The Hohenberg-KohnTheorem\nInanelectronsystem withHamiltonian\nˆH=ˆT+ˆVee+/summationdisplay\nivext(ri) (2)\nwith a non-degenerate ground state, the ground state densit y uniquely determines the ground state many-body wavefunct ion\nand therefore the external potential. This is known as the Ho henberg-Kohn theorem [2]. The usual proof shows how the\ndensityuniquelydeterminesthewavefunction[6]. Assumet hereexisttwoHamiltonians H1andH2withdifferentground-state\nwavefunctions, |Ψ1/an}b∇acket∇i}htand|Ψ2/an}b∇acket∇i}htrespectively,that giverise tothesame density. Then\n/an}b∇acketle{tΨ1|H2|Ψ1/an}b∇acket∇i}ht=/an}b∇acketle{tΨ1|H2−H1|Ψ1/an}b∇acket∇i}ht+/an}b∇acketle{tΨ1|H1|Ψ1/an}b∇acket∇i}ht\n=/an}b∇acketle{tΨ1|/summationdisplay\ni(v2,ext(ri)−v1,ext(ri))|Ψ1/an}b∇acket∇i}ht+E1\n=/integraldisplay\n(v2,ext(r)−v1,ext(r))n(r)dr+E1. (3)\nThenbythevariationalprinciple\n/integraldisplay\n(v2,ext(r)−v1,ext(r))n(r)dr+E1>E2, (4)\nwithE2thegroundstate relatedto H2.\nThiscanbe repeatedbut with 1and2exchangedtogive\n/an}b∇acketle{tΨ2|H1|Ψ2/an}b∇acket∇i}ht=/an}b∇acketle{tΨ2|H1−H2|Ψ2/an}b∇acket∇i}ht+/an}b∇acketle{tΨ2|H2|Ψ2/an}b∇acket∇i}ht\n=/an}b∇acketle{tΨ2|/summationdisplay\ni(v1,ext(ri)−v2,ext(ri))|Ψ2/an}b∇acket∇i}ht+E23\n=/integraldisplay\n(v1,ext(r)−v2,ext(r))n(r)dr+E2. (5)\nBy thevariationalprinciple\n/integraldisplay\n(v1,ext(r)−v2,ext(r))n(r)dr+E2>E1. (6)\nAddingtheinequalitiesgives\nE1+E2>E1+E2. (7)\nThisinequalityisafallacysotheinitialassumptionmusth avebeenincorrect: theredonotexisttwogroundstatewavef unctions\nthatgiveriseto thesamedensity. Hencebyreductioadabsur dumthedensityuniquelydeterminesthewavefunction.\nAswe mayexpresstheexternalpotentialas\n−ˆTψ\nψ−ˆVeeψ\nψ+E=/summationdisplay\nivext(ri) (8)\nthenthedensitybydeterminingthewavefunctionalsodeter minestheexternalpotential(uptoanadditiveconstant),a ndtherefore\nall ofthepropertiesofthesystem.\nAlthough the Hohenberg-Kohn theorem proves the existence o f the one-to-one relationship between ground state densiti es\nand wavefunctionsit does not suggest how to construct the de nsity nor find the ground-state properties of the system from the\ndensity. ToachievethistheKohn-Shamequations[1] areuse d.\nB. TheKohn-Shamequations\nThe Kohn-Sham(KS) equationsallow the ground-statedensit y to be foundefficientlyand the system energyto be expressed\nusing the density. The KS equations are derived by consideri ng the system energy as a functional of the density E[n], this is\nminimisedsubjectto aconstraintincorporating N,thenumberofelectrons,\n/integraldisplay\nn(r)dr=N. (9)\nUsingLagrangemultiplierswiththefunctionalderivative\nδF[n(y)]\nδn(x)= lim\nǫ→0F[n(y)+ǫδ(y−x)]−F[n(y)]\nǫ(10)\ngives\nδE[n]\nδn(r)−µ= 0. (11)\nAs the expression of the kinetic energy as a functional of the density for an interacting electron system is unknown, Kohn and\nShamre-wrotetheexactenergyfunctionalintermsofthekin eticenergyofnon-interactingelectrons TNI[n],theHartreeenergy\nU[n], and the potential energy from the electron density. This re quires the introduction of a functional correctional term, the\nexchange-correlationenergy Exc[n]. Theexpressionfor E[n]thenbecomes\nE[n] =TNI[n]+U[n]+/integraldisplay\nvext(r)n(r)dr+Exc[n] (12)4\nwhere\nU[n] =1\n2/integraldisplay /integraldisplayn(r)n(r′)\n|r−r′|drdr′. (13)\nUsingLagrangemultipliersnowgives\nδTNI[n]\nδn(r)+veff(r)−µ= 0. (14)\nHere\nveff(r) =vH(r;[n])+vext(r)+vxc(r;[n]) (15)\nwheretheydefinetheexchange-correlationpotential\nvxc(r;[n]) =δExc[n]\nδn(r)(16)\nandtheHartreepotential\nvH(r;[n]) =/integraldisplayn(r′)\n|r−r′|dr′. (17)\nNowEq.14canbereadasanon-interactingsystem—theKSsyst em—withapotential veff. Sotofindthedensityandhence\ntheenergythe KSequationsmust besolvedself-consistentl y:\n/parenleftbigg\n−1\n2∇2+veff(r)/parenrightbigg\nφi(r) =ǫiφi(r) (18)\nand\nn(r) =N/summationdisplay\ni=1|φi(r)|2. (19)\nAs\nTNI[n] =N/summationdisplay\ni=1/integraldisplay\nφ∗\ni(r)/parenleftbigg\n−1\n2∇2φi(r)/parenrightbigg\ndr (20)\nthentheenergy E[n]maybecalculatedfromthedensityusing Exc.\nThis scheme would be exact but the general functional form of Excis unknown hence it has to be approximated. One of\nthe simplest yet often effective approximations is using th e fact that, for systems with slowly varying density, the exc hange-\ncorrelationdensitycanbelocallyapproximatedbythatofa uniformelectrongas. Thisisknownasthelocaldensityappr oxima-\ntion.\nC. The local-densityapproximation\nThelocal-densityapproximationfortheexchange-correla tionenergydependsonlyuponthedensityasopposedtogener alised\ngradientapproximationswhichincludefirst andhigherderi vatives. Itmaybewrittenusingtheform\nExc=/integraldisplay\ndrexc(n(r)). (21)\nThe exchange-correlationenergy density may be separated i nto its exchangeand correlation parts, exc=ex+ec. Then, for a\nthree-dimensionalhomogeneouselectrongas, exisknownexactly[7]\nex=−3\n4/parenleftbigg3\nπ/parenrightbigg1/3\nn4\n3, (22)5\nwhileechas to be approximated. We use ecas calculated numerically by Perdew-Wang [8] by fitting the p arametersA,α,\nβ1...β4to the results of Monte-Carlo simulations of the homogeneou selectron gas [9]. For zero spin polarisation this has the\nform\nec=−2An(1+αrs)log/parenleftBigg\n1+1\n2A(β1r1/2\ns+β2rs+β3r3/2\ns+β4r2s)/parenrightBigg\n, (23)\nwherersisthe radiusofthespherewhosevolumeisequaltothevolume perelectronin ahomogeneouselectrongas,\nrs=/parenleftbigg3\n4πn(r)/parenrightbigg1\n3\n. (24)\nTheaboveapproximationisexactforthehomogeneouselectr ongassowouldbeexpectedtobemoreaccurateforsystemsclo se\ntothislimit.\nIII. THEINTERACTINGLDA SYSTEM\nWewishtocomparetheentanglementcalculatedusingalocal -densityapproximationtotheexactentanglement,andalth ough\nin theory the density determines all the ground-stateprope rtiesof the system it is unknownhow to use the density to repr esent\ncertain observables, including the entanglement. One meth od to calculate the entanglement within LDA is to construct t hat\nmany-body interacting system whose ground state wavefunct ion reproduces exactly the LDA density. We named this ‘the\ninteractingLDAsystem’(i-LDA)[3]. By theHohenberg-Kohn theorem[2] thissystemisunique.\nGivenatest-bedinteractingmany-bodysystemforwhichgro und-stateproperties,suchastheentanglement,canbecalc ulated\nexactly,bycomparingtheirvalueswiththeonesobtainedus ingthecorrespondingi-LDAwavefunction,wecaninferhowp recise\ntheLDA canbeasan approximationtothosequantities. Notic e thatthe externalpotentialscharacterisingtheinitial m any-body\nsystem and the i-LDA system are different, and that the LDA density is at the same time an approximationto the den sity of the\ntest-bedsystem andtheexactgroundstate densityofthei-L DA system .\nSuch a route to assess system properties within the LDA is com putationally intensive, but, for small test systems, allow s a\nway to investigatethe accuracyof the LDA for quantitiesfor which functionalexpressionsin termsof the density are cur rently\nunknown. Inadditionthisprocedureoffersanovelwaytoapp raisetheLDAbyconsideringthedifferencesbetweentheexa ctand\ni-LDAexternalpotentials,therebyfurnishingresearcher swithspatiallyresolvedinformationaboutanapproximati on’saccuracy.\nA similarproceduretotheonedescribedcanbeappliedtooth erapproximationswithinDFT.\nWe introduced a method to approximate the i-LDA wavefunctio n in Ref. [3], then in Ref. [10] we developed an iterative\nschemetofindtheHamiltonianofthei-LDAsystem‘exactly’( withinnumericalaccuracy). Thenthe‘exact’i-LDAwavefun ction\ncould then be derived. This is the scheme we use in this chapte r to allow the entanglement corresponding to the LDA to be\nascertained.\nIV. THETEST-BEDSYSTEMS\nTheheliumatomcomprisestwoelectronsheldbyanuclearcha rgearisingfromtwoprotonsinthenucleus. ByusingtheBorn -\nOppenheimerapproximationthemotionofthenucleusmaybei gnored. Thesystemgeneralisestohelium-likeatomsbycons id-\neringa nuclearchargeof Zgivingriseto thefollowingnon-relativisticHamiltonian inatomicunits (e=/planckover2pi1=m= 4πǫ0= 1)\nH=−1\n2∇2\n1−1\n2∇2\n2−Z\nr1−Z\nr2+1\n|r1−r2|, (25)\nwherer=|r|. WithZ= 2thisistheHamiltonianfortheheliumatom. Interestinglya lthoughthehydrideatom( Z= 1)exists,\nanditsenergymaybecalculated,Bakeret al [11] showedthat fornuclearchargesbelow Z≈0.911the systemisunbound.\nHooke’s atom is an interacting system of two electrons held b y a harmonic confining potential with Hamiltonian in atomic\nunitsof\nH=−1\n2∇2\n1−1\n2∇2\n2+1\n2ω2r2\n1+1\n2ω2r2\n2+1\n|r1−r2|. (26)\nHereωis the characteristic frequencyof the confiningpotential. The exact spatial entanglementand its approximationswith in\nLDAhavepreviouslybeenevaluatedforHooke’satominRef. [ 3].6\nV. METHODOF SOLUTION\nWesolvethemany-bodySchr¨ odingerequation,using‘exact ’diagonalisation(EXD)withanefficientchoiceofbasis. Fo rtwo\nelectrons in a spherically symmetric potential the ground s tate can only be a function of the distance of each electron fr om the\noriginand the angle betweenthe electronvectors. Hence we m ay find the groundstate using exact diagonalisationwith a ba sis\ninthreedimensions: r1,r2,θ.\nWe use thebasis\nφijl=Ri(r1)Rj(r2)1√\n4π/radicalbigg\n2l+1\n4πPl(cos(θ)), (27)\nwhere we employ the hydrogen-like wavefunction Ri(r) =Qi(r)e−αr. The choice of αis influenced by the strength of the\nnuclear charge Z. Here theQi(r)are polynomials of degree icreated via the Gram-Schmidt procedure such that the Riare\northonormal, i.e.,/integraltext∞\n0Ri(r)Rj(r)r2dr=δij. The Legendre polynomials Plare orthogonal to each other and we explicitly\ninclude the normalisation factor for integration over all a ngles. We intend to calculate the matrix elements of the Hami ltonian\nwithrespectto the φijlandφi′j′l′basisfunctionsEq.27. Theexternalpotentialis spherical lysymmetricthereforethe potential\nintegralsbecomejust 1D integralsand are only non-zerowhen l=l′. Thisis propitiousas theyare the onlyintegralsthat have\ntobere-calculatedeachtimein theiterativescheme.\nThekinetictermintheintegrandis\n∇2\ni=1\nr2\ni∂\n∂rir2\ni∂\n∂ri−L2\ni\nr2\ni(28)\nwhereLiis theangularmomentumoperator. As\nPl(cos(θ)) =4π\n(2l+1)m=+l/summationdisplay\nm=−lYlm(θ1,φ1)Y∗\nlm(θ2,φ2), (29)\nthen\nL2\niPl=l(l+1)Pl. (30)\nHencethekinetictermsarealso 1Dintegralsin r1andr2whenl′=l,otherwisetheintegralsarezero.\nThe Coulomb interaction integral is more complicated. Firs t we expand the Coulomb interaction term using the Legendre\npolynomials\n1\n|r1−r2|=∞/summationdisplay\nk=0(r<)k\n(r>)k+1Pk(cos(θ)), (31)\nwherer<= min{r1,r2}andr>= max{r1,r2}.\nNextweuse a result[12] foran integrationofthreeLegendre polynomials\n/integraldisplay1\n−1Pl′(z)Pk(z)Pl(z)dz= 2/parenleftbigg\nl′k l\n0 0 0/parenrightbigg2\n. (32)\nIfl′> k+lthen the polynomial of degree k+lcan be formed from Legendre polynomials each of degree less t hanl′\nhence the integral is zero, similarly for k > l+l′andl > k+l′. Here the 3jsymbol (see for example Ref. [13]) is zero for\nl′+k+l= 2p+1wherepisaninteger. While for l′+k+l= 2p\n/parenleftbigg\nl′k l\n0 0 0/parenrightbigg\n= (−1)p/radicalbig\n∆(l′kl)p!\n(p−l′)!(p−k)!(p−l)!, (33)\nwhere\n∆(abc) =(a+b−c)!(b+c−a)!(c+a−b)!\n(a+b+c+1)!. (34)\nHencetheintegralfortheCoulombinteractionwhenswitchi ngtosphericalpolarintegration,includingthenormalisa tion,and\nintegratingoveranglesis7\n/an}b∇acketle{tφi′j′l′|1\n|r1−r2||φijl/an}b∇acket∇i}ht=\n√\n2l′+1√\n2l+1l+l′/summationdisplay\nk=0/parenleftbigg\nl′k l\n0 0 0/parenrightbigg2/integraldisplay∞\n0dr1Ri′(r1)Ri(r1)\n/parenleftbig1\nrk−1\n1/integraldisplayr1\n0Rj′(r2)Rj(r2)rk+2\n2dr2+\nrk+2\n1/integraldisplay∞\nr1Rj′(r2)Rj(r2)1\nrk−1\n2dr2/parenrightbig\n. (35)\nAs the 3j symbol is zero for k > l+l′then there are at most l+l′double integrals, but the other constraints then produce\na zero 3j symbol for many of these. Hence efficiencyis increas ed: we have a small sum of double integrals rather than a tripl e\nintegraltocompute.\nWe also considerthe single particleKS equation(Eq.18) whe revext(r) =−Z/rforhelium-likeatomsand vext(r) =ωr2/2\nforHooke’satom. Forapproximating vxcwe usevLDA\nxc=vLDA\nx+vLDA\nc. We employthe fit ofPerdewandWang [8] (seeEq.23)\nforvLDA\ncwhile the expression for vLDA\nxis that of Wigner and Seitz [7] (see Eq. 22). We consider only s pherically symmetric\npotentials so the KS equation reduces to a one dimensional pr oblem. We solve this self consistently to obtain ntarget, the LDA\napproximationtothe groundstateelectrondensity.\nThis‘target’densityisthenusedin theiterativeschemede velopedin Ref.[10] whichallowstofindthe externalpotenti alfor\nthe interacting many-body system of ground state density ntarget, in this case the i-LDA system. This scheme is based on the\nrelation\nvi+1\next(r1) =1\nni(r1)|Ei|[ni(r1)−ntarget(r1)]+vi\next(r1). (36)\nHereEiis the energy of the interacting system with external potent ial/summationtext\njvi\next(rj): this many-body system is solved at\nevery step by using the aforementioned method to give the man y-body wavefunction from which we compute the den-\nsityni. We aid convergence for both test-bed systems by mixing vi+1\nextwith80%ofvi\next, and iterate until the error/integraltextd3r|ni(r)−ntarget(r)|//integraltextd3rntarget(r)hasreachedadesiredlevelonthechosenintegrationrange.\nWenotethattheinversionschemeEq.36couldalsobeusedtod erivetheunknownpotentialrelatedtoagenericdensity ntarget\nobtained,e.g.,fromexperimentalmeasurements.\nVI. SPATIALENTANGLEMENTCALCULATION\nFor the systems we are considering, the ground state wave-fu nction can be factorised in orbital and spin part. The spin pa rt\nis a singlet which is maximally entangled and constant. We th erefore ignore the spin entanglement and investigate the sp atial\nentanglement,i.e.,theparticle-particleentanglements temmingfromthecontinuousspatial degreesoffreedom.\nWe use thetwo-electronwavefunctionto calculatethelinea rentropyofthereduceddensitymatrix\nL= 1−Trρ2\nA (37)\nas a measure of the spatial entanglement. Here ρA=TrB|ψ/an}b∇acket∇i}ht/an}b∇acketle{tψ|is found by tracing out the system density matrix over the\ndegreesoffreedomofone ofthe two subsystems. The linear en tropywas, perhaps,first put forwardas a measureof purityof a\nquantumstate byZurek,HabibandPaz[14]. Thepurityofther educeddensitymatrixcanbeshowntogiveanindicationofth e\nnumberandspreadofthetermsintheSchmidtdecompositiono fthestateandthereforeameasureoftheentanglement. Here ,in\nthecontinuouscase, wecalculatetheentanglementusing\nρA(r1,r2) =/integraldisplay\nΨ∗(r1,r3)Ψ(r2,r3)dr3, (38)\nρ2\nA(r1,r2) =/integraldisplay\nρA(r1,r3)ρA(r3,r2)dr3, (39)\nand\nTrρ2\nA=/integraldisplay\nρ2\nA(r,r)dr. (40)\nWe will investigatethe change in spatial entanglementwith the strength of the confiningpotential, which is characteri sedby\nωorthe nuclearcharge ZforHooke’sandhelium-likeatomsrespectively.8\nVII. HELIUM-LIKEATOMS\nFor helium-like atoms the (numerically) exact and the i-LDA entanglement are depicted in Fig. 1. There we see that the\nentanglement increases as the nuclear charge decreases whi ch may be attributed to the increase in the ratio of the electr on-\nelectron interaction to the energy from the external potent ial. A similar situation was observed in the spatial entangl ement\nof Hooke’s atom [3]. We note that the spatial entanglement is low for the parameters considered, when compared with the\ntheoretical maximum of L= 1. The ‘exact’ and i-LDA entanglement are close and the gap bet ween them increases as the\nweight of the particle-particle interactions increase. Th is fits in with the LDA performingbetter when many-bodyinter actions\nareweak: astheconfiningpotentialincreases,many-bodyin teractionswill becomelessimportantrelativetoit, until thesystem\nmay be fairly well approximatedby a non-interactingHamilt onian thereby giving rise to a product state solution. We not e that\nZ (e) L\n 1.4  1.6  1.8  2  2.2  2.4  2.6  2.8  3 0.02\n 0.015\n 0.01\n 0.005\n 0‘Exact’\ni−LDA 0.025\nFIG. 1: The linear entropy Lof the reduced density matrix versus the nuclear charge Zas a measure of entanglement for helium-like atoms\nand the corresponding i-LDAsystems.\nalthoughhelium-likeatomshavea boundstate until Z≈0.911we wereunableto numericallysolvethe Kohn-Shamequations\nwith the LDA satisfactorily for Z <1.5. We hypothesisethat for the exact solution with Z/lessorsimilar0.911a transition to a resonance\nstate—as studied recently for two electrons in a sphericall y symmetric square well [15] and the spherical helium atom [1 6]—\nwould be expected to occur. This would result in a superposit ion of one electron constrained by the nucleus with the other\nunbound. Suchawavefunctionwouldbeexpectedtohaveaspat ial entanglementof L= 1/2.\nVIII. HOOKE'SATOM\nFor Hooke’s atom (Fig. 2) we see that the accuracy of the entan glement when using the LDA is high, but decreases a little\nasωdecreases and interactions, as quantified by the ratio of the electron-electron interaction to the energy from the exter nal\npotential, becomelarger. The valueof the i-LDA entangleme ntforω= 0.00584,lower than the valuefor ω= 0.00957,seems\nanomalousas does the changefrom an overestimateto an under estimateof the exact entanglement. We note that ω= 0.00584\nhas the poorest match ( 0.053%error) between the i-LDA estimate and the actual LDA density of theωdisplayed and is the\nsmallestωfor which we could achieve a good match with a reasonable basi s size of 83wavefunctions. Therefore due to a\nseeminglyhighsensitivityoftheentanglementtothewavef unctionyetarelativeinsensitivityofthedensitytothewa vefunction,\nwe may have crossed a thresholdso the small error in reproduc ingthe LDA density is now not small enoughto preventa large\nerror when calculating the LDA entanglement. However we als o note that 0.053%error still represents a very good accuracy\nin reproducing the LDA density, and that when employing an ev en larger basis the difference between old and new density\nestimates was just 0.089%. A basis of 83was also sufficient to calculate the exact entanglement to 3significant figures. In\naddition we saw in previous work [3] that the difference betw een exact and LDA energies exhibited a non-trivial relation ship\nwithω, and a perturbedLDA-basedapproachto approximateHooke’s atommany-bodywavefunctionalso producesa decrease\nof the entanglement in a similar range of ω(see Fig. 3). These observationsmay imply that a non-trivia l relationship between\nωand LDA-based estimates of the entanglement could be a real e ffect. We suggest that although the accuracy of the i-LDA\nentanglementforthetwosmallest valuesof ωmaybequestionable,thecorrecti-LDAentanglementforthe sevaluesshouldnot\nbetoodifferentandwe areconfidentoftheaccuracyofthei-L DAentanglementforlargervaluesof ω.9L\nω(Hartrees) 0.03\n 0.025\n 0.02\n 0.015\n 0.01\n 0.005\n 0i−LDAExact\n 0.001  0.01  0.1  1\nFIG. 2: The linear entropy Lof the reduced density matrix versus ωas a measure of entanglement for Hooke’s atom and the corresp onding\ni-LDAsystem.\nA. Comparison withotherLDA-based entanglementestimates\nInpreviouswork[3] wedeviseddifferentwaystoderiveanLD A-basedapproximationtotheentanglement.\nThe first is a combination of gradient descent and evolutiona ry algorithms to derive an approximation for the i-LDA wave-\nfunction from which to calculate the entanglement. The main limitation of this approach is that the trial i-LDA wavefunc tion\nwas based on a specific functional form. This variational-ty pe approach included some parameters to be varied to improve\nperformance.\nThe second approach to derive an LDA-based approximation fo r the interacting many-body wavefunction is based on a\nperturbativescheme whose zero-orderHamiltonianinclude sthe Kohn-Shameffective potential (and hence, through vxc, partly\nthe many-bodyinteraction) with the LDA approximationfor t he exchange-correlationterm. This approachimprovesas hi gher\nordersaretakenintoaccount,butafirst-orderwavefunctio nwouldstillfailtoreproducetheexactentanglementwhenm any-body\ninteractionsbecomedominant.\nWe see in Fig. 3 that, in the intermediate and strong particle -particle interaction regime, the exact i-LDA entanglemen t for\nHooke’s atom reproduces the system exact entanglement much more accurately than any of the approximations mentioned\nabove. Approximationslabelled as EA1 and EA2 were achieved by using a combination of gradient descent and evolutionary\nalgorithms [3]. In EA1 we sought to minimise the difference b etween the density arising from the trial wavefunction and t he\nexact density and then, within the wavefunctionsgivingan a ccurate enoughmatch, we choose the wavefunctioncorrespon ding\nto the lowest ground state energy. The trial wavefunctions i n this case are restricted to wavefunctions which can be sepa rated\nin centre of mass and relative motion components. In EA2 we so ught to minimise the difference between the density arising\nfromthetrial wavefunctionandthe exactdensitytogetherw ith a minimisationof theenergyofthe trialwavefunction. T hetrial\nwavefunctions in this case used a finite number of functions i n terms ofr1,r2, andθfrom a general basis for two electrons\nin a spherically symmetric potential. The gradient descent allows the basis coefficients for the wavefunction that give s a local\nminimumtobefoundwhiletheevolutionaryalgorithmallows moreoftheparameterspacetobeexploredsoinprinciple,gi vena\nlargeenoughbasisandenoughtime,thecombinationshouldfi ndtheglobalminimum. Withourfinitebasisandlimitedrun-t ime\nwe see that, although this method gives the trend of the i-LDA entanglement, the approximate i-LDA entanglement from the\nevolutionaryalgorithmismuchhigherthantheactuali-LDA entanglement.\nAs mentioned above, in [3] we also considered a perturbative approach based on the LDA-KS equations, whose related\ninteractingwavefunctionconvergestowardtheexactwavef unctionashigherordersareconsidered. Tofirstorderwewou ldhope\nfor a good approximationto the exact entanglement that perh aps leans towards the i-LDA entanglement. We see in Fig. 3 tha t\ntheentanglementresultsfromfirst orderbecomesmuchgreat erthanthei-LDAentanglement—butlesssothantheevolutio nary\nalgorithm—for stronger interactions, as quantified by the r atio of expectation of the interaction energy to the potenti al energy.\nHowever, the perturbation method allows an easier and more e fficient way to approximate the entanglement within an LDA-\nrelated scheme, whose first order is accurate for low interac tions; the exact i-LDA entanglement, although more accurat e, is\ninaccessible without repeated complicated calculations: the inversion scheme requires solving the many-body equati ons more\ntimesthanwouldbeneededtoascertaintheexactentangleme nt.10\n(Hartrees)ωL\n 0.001  0.01  0.1  1 0 0.04 0.08 0.12 0.16 0.2\nEA2EA1Perturbed LDA KSi−LDAExact\nFIG. 3: The linear entropy Lof the reduced density matrix as a measure of the entanglemen t obtained using the exact, i-LDA, first order\nperturbation of the LDAKSequations, and evolutionary algo rithm (EA1and EA2)many-body wavefunctions.\nIX. POSITIONSPACEINFORMATIONENTROPY\nWe also investigatethe position-spaceinformationentrop yofthedensity\nSn=−/integraldisplay\nn(r)lnn(r)dr. (41)\nasaproxymeasurefortheentanglement. Thishasbeenusedto studytheentanglementoftheMoshinskyatom[17]andmayals o\nbethoughtofasazeroth-orderapproximationtothevonNeum annentropyofthereduceddensitymatrix,wheretheoff-dia gonal\ntermsofthe reduceddensitymatrixareset to zero[3]. Asthe vonNeumannentropyof thereduceddensitymatrixmaybeused\nas a measure of the entanglement and has been seen to behave si milarly toLfor Hooke’s atom [3], it would be hoped that Sn\nshouldcapturesomeofthebehaviourofthe entanglement.\nSn\nZ (e) 2  2.8 1 2 3 4 5 6 7\n 1.4  1.6  1.8  2.2  2.4  2.6  3‘Exact’\nLDA\nFIG.4: The positionspace information entropy Snfor the ‘exact’ andLDAdensities forhelium-like atoms.\nWe see in Fig. 4 that Sncaptures the monotonically decreasing trend of the exact an d i-LDA entanglement. The Snfor the\nLDA is greaterthan the exact Snbut the differencebetween the two doesnot seem to increasea sZdecreasesas is the case for\ntheexactandi-LDAentanglement. Thebehaviourshownby Snisalso morelinearwithincreasing Zthanthe entanglement.\nFor Hooke’s atom we previously noted that Sndid not model the entanglement behaviour well over a much lar ger range of11\nconfiningpotentials[3]1. WeseeinFig.5that,overasmallerrangeof ω,Snreproducesthegeneraltrendoftheentanglementbut\nSnfromLDAisalwaysgreaterthantheexact Snwhichappearstonotbethecasefortheentanglement(seeFig .2). Inaddition\nwe see that Snis almost a straight line when plotted against ωon a logarithmic scale, this is in contrast to the entangleme nt\nwhichcanbeseentoincreaselessfast as ωdecreasesevenforthesmall rangeof ωconsideredhere.Sn\nω (Hartrees) 0.001  0.01  0.1  1 6 10 14 18 22\nLDAExact\nFIG.5: The position space information entropy Snfor the ‘exact’ andLDA densities for Hooke’s atom.\nX. CONCLUSION\nWehavecalculatedthespatialentanglementoftheelectron sinhelium-likeatomscorrespondingtotheexactsystemand tothe\nuse of the local-density approximationwithin density-fun ctionaltheory using the concept of the interacting LDA syst em. The\nentanglementwasseen todecreasewith increasingnuclearc hargeZ. Themaximumentanglementfoundwasonlyafraction ∼\n2.5%ofthetheoreticalmaximum. Theentanglementofthei-LDAsy stemwasseentoreproducethequalitativeandquantitative\nbehaviour of the exact entanglement with relatively good ac curacy, although the discrepancy between the measures incr eased\nas the nuclear charge diminished. We also compared the exact spatial entanglement of Hooke’s atom to the entanglement of\nthe related i-LDA system and saw that the LDA was a reasonably good approximation to the entanglement, but the accuracy\ndecreasedasthestrengthoftheconfiningpotentialdecreas edandthei-LDAentanglementbecamenon-monotonicatverys mall\nconfiningpotential.\nWe also saw that previously considered approximations to th e exact or i-LDA entanglement were not so reliable when the\nparticle-particle interaction became dominant, but noted that they captured the trend of the entanglement, were succe ssful at\nstrongconfiningpotentialsandwereoftenmuchsimplercomp utationally. Inparticulartheaccuracyoftheperturbativ eapproach\nbasedontheLDA-KSequationscouldbesystematicallyimpro vedbygoingbeyondfirst order.\nWe consideredthe position space informationentropyof the density,and show that it was able to efficientlygive the gene ral\ntrendin theentanglementbutnotitsdetailedbehaviour.\nFrom our test-bed calculations on few-particle systems it w ould appear that the LDA may reproduce the entanglement be-\nhaviour well, even for systems with a highly inhomogeneousd ensity and fairly strong interactions. More studies are nec essary\ntoascertainif thisconclusionholdsformany-particlesys tems.\n1Label on y-axis in [3],Figs. 5 and 12, should read Sn/ln2.12\nXI. ACKNOWLEDGEMENT\nWe acknowledgefundingfromEPSRC grantEP/F016719/1.\n[1] Kohn Wand ShamL J 1965 Phys.Rev. 1401133\n[2] Hohenberg Pand Kohn W1964 Phys. Rev. 136B864\n[3] Coe JP,Sudbery Aand D’AmicoI2008 Phys.Rev. B 77205122\n[4] Zalka C1998 Fortschr. Phys. 46877\n[5] NielsenMAand Chuang I L2000 Quantum Computation and Quantum Information (Cambridge UniversityPress)\n[6] Capelle K2006 BrazilianJournal of Physics 361318\n[7] WignerE Pand SeitzF1933 Phys. Rev. 43804\n[8] Perdew JP andWangY 1992 Phys. Rev.B 4513244\n[9] CeperleyD MandAlder B J1980 Phys. Rev. Lett. 45566–569\n[10] Coe JP,Capelle K andD’AmicoI 2009 Phys.Rev. A 79032504\n[11] Baker J D,FreundD E,HillR Nand MorganIII JD 1990 Phys. Rev.A 411247\n[12] Mavromatis H Aand Alassar RS1999 Applied Mathematics Letters 42101–105\n[13] MessiahA 1999 Quantum Mechanics (Dover Publications, Inc.)\n[14] ZurekWH,Habib SandPaz J P1993 Phys. Rev.Lett. 701187\n[15] Ferr´ onA,Osenda Oand SerraP2009 Phys.Rev. A 79032509\n[16] Osenda O and SerraP2007 Phys. Rev. A 75042331\n[17] AmovilliC andMarch N H2004 Phys. Rev. A 69054302"    },    {        "title": "1004.3026v1.Subgraph_densities_in_signed_graphons_and_the_local_Sidorenko_conjecture.pdf",        "content": "arXiv:1004.3026v1  [math.CO]  18 Apr 2010Subgraph densities in signed graphons\nand the local Sidorenko conjecture\nL´ aszl´ o Lov´ asz∗\nInstitute of Mathematics, E¨ otv¨ os Lor´ and University\nBudapest, Hungary\nJune 14, 2018\nContents\n1 Introduction 1\n1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.2 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2 Density inequalities for signed graphons 4\n2.1 Ordering signed graphons . . . . . . . . . . . . . . . . . . . . . . 4\n2.2 Edge weighting models . . . . . . . . . . . . . . . . . . . . . . . . 5\n2.3 Inequalities between densities . . . . . . . . . . . . . . . . . . . . 7\n2.4 Special graphs and examples . . . . . . . . . . . . . . . . . . . . 8\n2.5 The Main Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n3 Local Sidorenko Conjecture 16\n3.1 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\n3.2 Graphic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n1 Introduction\nLetFbe a bipartite graph with knodes andledges and let Gbe any graph\nwithnnodes andm=p/parenleftbign\n2/parenrightbig\nedges. Sidorenko [5, 6] conjectured that the number\nof copies of FinGis at leastpl/parenleftbign\nk/parenrightbig\n+o(plnk) (where we consider kandlfixed,\nandn→ ∞). In a weaker form, this was also conjectured by Simonovits [8].\n∗Research supported by OTKA Grant No. 67867 and ERC Grant No. 2 27701.\n1One can get a cleaner formulation by counting homomorphisms instea d of\ncopies ofF. Let hom(F,G) denote the number of homomorphisms from Finto\nG. Since we need this notion for the case when FandGare multigraphs, we\ncount here pairs of maps φ:V(F)→V(G) andE(F)→E(G) such that\nincidence is preserved: if i∈V(F) is incident with e∈E(F), thenφ(i) is\nincident with ψ(e). We will also consider the normalized version t(F,G) =\nhom(F,G)/nk. IfFandGare simple, then t(F,G) is the probability that a\nrandom map φ:V(F)→V(G) preserves adjacency. We call this quantity the\ndensity ofFinG.\nIn this language, the conjecture says that t(F,G)≥t(K2,G)|E(F)|(this is an\nexact inequality, no error terms.) We can formulate this as an extre mal result\nin two ways: First, for every graph G, among all bipartite graphs with ledges,\nit is the graph consisting of disjoint edges (the matching) that has t he least\ndensity inG. Second, for every bipartite graph F, among all graphs on nnodes\nand edge density p, the random graph G(n,p) has the smallest density of Fin\nit (asymptotically, with large probability).\nSidorenko proved his conjecture in a number special cases: for tr eesF, and\nalso for bigraphs Fwhere one of the color classes has at most 4 nodes. Since\nthen, the only substantial progress was that Hatami [2] proved t he conjecture\nfor cubes.\nSidorenko gave an analytic formulation of this conjecture, which is p erhaps\neven cleaner, and which we will also use. Let Fbe a bipartite graph with\nbipartition ( A,B). For each edge e∈E(F), leta(e) andb(e) be the endpoints\nofeinAandB, respectively. Assign a real variable xito eachi∈Aand a\nreal variable xjto eachj∈B. LetW: [0,1]2→R+be a bounded measurable\nfunction, and define\nt(F,W) =/integraldisplay\n[0,1]V(F)/productdisplay\ne∈E(F)W(xa(e),yb(e))/productdisplay\ni∈Adxi/productdisplay\nj∈Bdyj.\nEverygraph Gcanbe representedbyafunction WG: LetV(G) ={1,...,n}.\nSplit the interval [0 ,1] intonequal intervals J1,...,J n, and forx∈Ji,y∈Jj\ndefineWG(x,y) =1ij∈E(G). (The function obtained this way is symmetric.)\nThen we have\nt(F,G) =t(F,WG).\nIn this analytic language, the conjecture says that for every bipa rtite graph\nFand function W∈ W+,\nt(F,W)≥t(K2,W)|E(F)|. (1)\nSince both sides are homogeneous in Wof the same degree, we can scale Wand\nassume that\nt(K2,W) =/integraldisplay\n[0,1]2W(x,y)dxdy= 1.\n2Thenwewanttoconcludethat t(F,W)≥1. In otherwords, thefunction W≡1\nminimizest(F,W) among all functions W≥0 with/integraltext\nW= 1.\nThe goal of this paper is to prove that this holds locally, i.e., for funct ions\nWsufficiently close to 1. Most of the time we will work with the function\nU=W−1, which can negative values. Most of our work will concern estimate s\nfor the values t(F′,U) for various (bipartite) graphs F′. This type of question\nseems to have some interest on its own, because it can be considere d as an\nextension of extremal graph theory to signed graphs.\n1.1 Notation\nFor each bigraph, we fix a bipartition and specify a first and second b ipartition\nclass. So the complete bigraphs Ka,bandKb,aare different. If Fis a bigraph,\nthen we denote by F⊤the bigraph obtained by interchanging the classes of F.\nWe have to consider graphs that are partially labeled. More precisely , ak-\nlabeled graph Fhasasubset S⊆V(F) ofkelementslabeled 1 ,...,k(it can have\nany number of unlabeled nodes. For some basic graphs, it is good to in troduce\nnotation for some of their labeled versions. Let Pndenote the unlabeled path\nwithnnodes (so, with n−1 edges). Let P′\nndenote the path Pnwith one of its\nendpoints labeled. Let P′′\nndenote thePnwith both of its endpoints labeled. Let\nCndenote the unlabeled cycle with nnodes, and let C′\nnbe this cycle with one\nof its nodes labeled. Let Ka,bdenote the unlabeled complete bipartite graph;\nletK′\na,bdenote the complete bipartite graph with its a-element bipartition class\nlabeled. Note that K2,2∼=C4, butK′\n2,2andC′\n4are different as partially labeled\ngraphs.\nThe most important use of partial labeling is to define a product: ifFandG\narek-labeled graphs, then FGdenotes the k-labeled graph obtained by taking\ntheir disjoint union and identifying nodes with the same label.\nWe setI= [0,1]. For ak-labeled graph F, (F)unlis the graph obtained by\nunlabeling. We set F1∗F2= (F1F2)unl, andF∗2=F∗F.\nLetWdenote the set of bounded measurable functions U:I2→R;W+is\nthe set of bounded measurable functions U:I2→R+, andW1is the set of\nmeasurable functions U:I2→[−1,1]. Every function U∈ Wdefines a kernel\noperatorL1(f)→L1(f) by\nf/mapsto→/integraldisplay\nIU(.,y)f(y)dy.\nForU,W∈ W, we denote by U◦Wthe function\n(U◦W)(x,y) =/integraldisplay\nIU(x,z)W(z,y)dz\n(this corresponds to the product of UandWas kernel operators). For every\nW∈ W, we denote by W⊤the function obtained by interchanging the variables\ninW.\n31.2 Norms\nWe consider various norms on the space W. We need the standard L2andL∞\nnorms\n/bar⌈blU/bar⌈bl2=/parenleftBig/integraldisplay\nI2U(x,y)2dxdy/parenrightBig2\n,/bar⌈blU/bar⌈bl∞= sup ess |U(x,y)|.\nFor graph theory, the cut norm is very useful:\n/bar⌈blU/bar⌈bl/square= sup\nS,T⊆I/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nS×TU(x,y)dxdy/vextendsingle/vextendsingle/vextendsingle.\nThis norm is only a factor less than 4 away from the operator norm of Uas a\nkernel operator L∞(I)→L1(I).\nThe functional t(F,U) can be used define further useful norms. It is trivial\nthatt(C2,U)1/2=/bar⌈blU/bar⌈bl2. The value t(C2r,U)1/(2r)is ther-thSchatten norm of\nthe kernel operator defined by U. It was proved in [1] that for U∈ W1,\n/bar⌈blU/bar⌈bl4\n/square≤t(C4,U)≤ /bar⌈blU/bar⌈bl/square.\nThe other Schatten norms also define the same topology on W1as the cut norm\n(cf. Corollary 2.16).\nIt is a natural question for which graphs does t(F,W)1/|E(F)|or\nt(F,|W|)1/|E(F)|define a norm on W. Besides even cycles and complete bi-\npartite graphs, a remarkable class was found by Hatami: he proved that\nt(F,|W|)1/|E(F)|is a norm if Fis a cube. He in fact proved that Sidorenko’s\nconjecture is true whenever Fis such a “norming” graph. However, a charac-\nterization of such graphs is open.\n2 Density inequalities for signed graphons\n2.1 Ordering signed graphons\nFor two bipartite multigraphs FandG, we say that F≤Gift(F,U)≤t(G,U)\nfor allU∈ W1. We say that G≥0 ift(G,U)≥0 for allU∈ W1. Note that\nifUis nonnegative, then trivially G⊆Fimplies that t(F,U)≤t(G,U); but\nsince we allow negative values, such an implication does not hold in gener al. For\nexample,F≥0 cannot hold for any bigraph Fwith an odd number of edges,\nsince thent(F,−U) =−t(F,U).\nWe start with some simple facts about this partial order on graphs.\nProposition 2.1 IfFandGare nonisomorphic bigraphs without isolated nodes\nsuch thatF≤G, then|E(F)| ≥ |E(G)|,|E(G)|is even, and G≥0. Further-\nmore,|t(F,U)| ≤t(G,U)for allU∈ W1.\nThe proof of this is based on a technical lemma, which is close to facts that\nare well known, but not in the exact form needed here.\n4Lemma 2.2 LetFandGbe nonisomorphic graphs without isolated nodes.\nThen for every U∈ W1andε >0there exists a function U′∈ W1such\nthat/bar⌈blU−U′/bar⌈bl∞<εandt(F,U′)/\\⌉}atio\\slash=t(G,U′).\nProof. First we show that if FandGare two graphs without isolated nodes\nsuch thatt(F,W) =t(G,W) for every W∈ W1, thenF∼=G. Consider the\nfunctionU=1x,y≤1/2. Thent(F,U) = 2−|V(F)|, sot(F,U) =t(G,U) implies\nthat|V(F)|=|V(G)|. Using the function U≡1/2, we get similarly that\n|E(F)|=|E(G)|. Using this, we get (by scaling W) thatt(F,W) =t(G,W) for\neveryW∈ W.\nFor every multigraph Hwe have\nt(F,H) =t(F,WH) =t(G,WH) =t(G,H),\nand hence it follows that\nhom(F,H) =t(F,H)|V(H)||V(F)|=t(G,H)|V(G)||V(F)|= hom(G,H).\nFrom this it follows by standard arguments that F∼=G(e.g., we can apply\nTheorem 1(iii) of [3] to the 2-partite structures ( V,E,J), whereG= (V,E) is a\nmultigraph and Jis the incidence relation between nodes and edges).\nSinceFandGare non-isomorphic, this argument shows that there exists a\nfunctionW∈ W1such thatt(F,W)/\\⌉}atio\\slash=t(G,W). The values t(F,(1−s)U+sW)\nandt(F,(1−s)U+sW) are polynomials in sthat differ for s= 0. Therefore,\nthere is a value 0 ≤s≤εfor which they differ. Since (1 −s)U+sW∈ W1and\n/bar⌈blU−((1−s)U+sW)/bar⌈bl∞=s/bar⌈blU−W/bar⌈bl∞≤ε, this proves the lemma. /square\nProof of Proposition 2.1. Applying the definition of F≤GwithU= 1/2,\nwe get that 2−|E(F)|≤2−|E(G)|, and hence |E(F)| ≥ |E(G)|. The relation\nF≤Gimplies that t(F,U)2=t(F,U⊗U)≤t(G,U⊗U) =t(G,U)2also\nholds, so |t(F,U)| ≤ |t(G,U)|for allU∈ W1. By Lemma 2.2, Ucan be\nperturbed by arbitrarily little to get a U′∈ W1witht(F,U′)/\\⌉}atio\\slash=t(G,U′), then\nt(F,U′)< t(G,U′) and|t(F,U′)| ≤ |t(G,U′)|imply that t(G,U′)>0. Since\nU′is arbitrarily close to U, this implies that t(G,U)≥0, and soG≥0. Since\nthis holds for Ureplaced by −U, it follows that Gmust have an even number\nof edges. /square\n2.2 Edge weighting models\nWe need the following generalization of Cauchy–Schwarz:\nLemma 2.3 Letf1,...,f n:Ik→Rbe bounded measurable functions, and\nsuppose that for each variable there are at most two function s which depend on\nthat variable. Then /integraldisplay\nIkf1...fn≤ /bar⌈blf1/bar⌈bl2.../bar⌈blfn/bar⌈bl2.\n5This will follow from an inequality concerning a statistical physics type\nmodel. Let G= (V,E) be a multigraph (without loops), and let for each i∈V\nletfi∈L2(IE) such that fidepends only on the variables xjwhere edge jis\nincident with node i. Letf= (fi:i∈V), and define\ntr(G,f) =/integraldisplay\nIE/productdisplay\ni∈Vfi(x)dx\n(where the variables correspondingto the edges not incident with iare dummies\ninfi).\nLemma 2.4 For every multigraph gand assignment of functions f,\ntr(G,f)≤/productdisplay\ni∈V/bar⌈blfi/bar⌈bl2.\nProof. Byinductiononthechromaticnumberof G. LetV1,...,V rbethecolor\nclasses of an optimal coloring of G. LetS1=V1∪···∪V⌊r/2⌋andS2=V\\S1.\nLetE0be the set of edges between S1andS2, and letEibe the set of edges\ninduced by Si. Letxibe the vector formed by the variables in Ei. Then\ntr(G,f) =/integraldisplay\nIE0/parenleftBigg/integraldisplay\nIE1/productdisplay\ni∈S1fi(x)dx1/parenrightBigg/parenleftBigg/integraldisplay\nIE2/productdisplay\ni∈S2fi(x)dx2/parenrightBigg\ndx0.\nThe outer integral can be estimated using Cauchy-Schwarz:\ntr(G,f)2≤/integraldisplay\nIE0/parenleftBigg/integraldisplay\nIE1/productdisplay\ni∈S1fi(x)dx1/parenrightBigg2\ndx0\n×/integraldisplay\nIE0/parenleftBigg/integraldisplay\nIE2/productdisplay\ni∈S2fi(x)dx2/parenrightBigg2\ndx0. (2)\nLetG1be defined as the graph obtained taking a disjoint copy ( S′\n1,E′\n1) of the\ngraph (S1,E1), and connect each node i∈S1to the corresponding node i′∈S′\n1\nby as many edges as those joining itoS2isG. Note that these newly added\nedges correspond to the edges of E0in a natural way. We assign to each node\nthe same function as before, and also the same function (with differ ently named\nvariables for the edges in E′\n1) toi′. Then the first factor in (2) can be written\nas /integraldisplay\nIE0/integraldisplay\nIE1/integraldisplay\nIE′\n1/productdisplay\ni∈S1∪S′\n1fi(x)dx1dx0= tr(G1,f).\nWe defineG2analogously, and get that the second factor in (2) is just tr( G2,f).\nSo we have\ntr(G,f)2≤tr(G1,f)tr(G2,f) (3)\nNext we remark that G1andG2have chromatic number at most ⌈r/2⌉, and\nso ifr>2, then we can apply induction and use that\ntr(Gj,f)≤/productdisplay\ni∈V(Gj)/bar⌈blfi/bar⌈bl2=/productdisplay\ni∈Sj/bar⌈blfi/bar⌈bl2\n2.\n6Ifr= 2, thenGjhas edges connecting pairs i,i′only, and so\ntr(Gj,f) =/productdisplay\ni∈Sj/bar⌈blfi/bar⌈bl2\n2.\nIn both cases, the inequality in the lemma follows by (3). /square\n2.3 Inequalities between densities\nLetF1andF2be twok-labeled graphs. Then the Cauchy–Schwarz inequality\nimplies that for all U∈ W,\nt(F1∗F2,U)2≤t(F∗2\n1,U)t(F∗2\n2,U). (4)\nWith the notation introduced above, this can be written as\n(F1∗F2)2≤F∗2\n1F∗2\n2. (5)\nThis also implies that for each k-labeled graph F,\nF∗2≥0. (6)\nLetFsubdenote the subdivision of graph Fwith one new node on each edge.\nLemma 2.5 IfF≤G, thenFsub≤Gsub.\nProof. For everyU∈ W,\nt(Fsub,U) =t(F,U◦U⊤)≤t(G,U◦U⊤) =t(Gsub,U).\n/square\nLemma 2.6 LetFbe a bigraph, let S⊆V(F), and letH1,...,H mbe the\nconnected components of F\\S. Assume that each node in Shas neighbors in\nat most two of the Hi. LetFidenote the graph consisting of Hi, its neighbors\ninS, and the edges between HiandS. Let us label the nodes of Sin everyFi.\nThen\nF2≤m/productdisplay\ni=1F∗2\ni.\nProof. LetF0denote the subgraph induced by S, and consider the nodes of\nF0labeled 1,...,k; we may assume that these nodes are labeled the same way\nin everyFi. Then using that |tx1...xk(F0,U)| ≤1, we get\n|t(F,U)|=/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nIkm/productdisplay\ni=0tx1...xk(Fi,U)dx1...dxk/vextendsingle/vextendsingle/vextendsingle\n≤/integraldisplay\nIkm/productdisplay\ni=1|tx1...xk(Fi,U)|dx1...dxk.\nLemma 2.3 implies the assertion. /square\nWe formulate some special cases.\n7Corollary 2.7 IfFcontains two nonadjacent nodes of degree at least 2, then\nF≤C4.\nMore generally,\nCorollary 2.8 Letv1,...,v kbe independent nodes in Fwith degrees d1,...,d k\nsuch that no node of Fis adjacent to more than 2of them. Then\nF2≤k/productdisplay\ni=1K2,di≤Ck\n4,\nAhanging path system in a graphFis a set{P1,...,P m}of openly disjoint\npaths such that the internal nodes of each Pihave degree 2, and at most two of\nthem start at any node. The valueof a hanging path system is the total number\nof their internal nodes.\nCorollary 2.9 LetFbe a bigraph that contains a hanging path system with\nlengthsr1,...,r m. Then\nF2≤m/productdisplay\ni=1C2ri.\nCombining with Corollary 2.15, we get the following bound, which we will\nuse:\nCorollary 2.10 LetFbe a simple graph that contains a hanging path system\nof lengths between 2andrand value 2r+a−2,a≥0. ThenF≤C2rCa/2\n4.\nCorollary 2.11 LetFbe a graph and S⊆V(F). LetF0be obtained by deleting\nthe edges within S, and labeling the nodes in S. Then\nF≤(F∗2\n0)1/2.\n2.4 Special graphs and examples\nLemma 2.12 LetU∈ W1. Then the sequence (t(C2k,U) :k= 1,2,...)is\nnonnegative, logconvex, and monotone decreasing.\nWith the notation introduced above, we have C2≥C4≥C6≥ ··· ≥0.\nProof. We have\nt(Ca+b,U) =/integraldisplay\nI2txy(P′′\na,U)txy(P′′\nb,U).\nTakinga=b=k, nonnegativity follows. Applying Cauchy–Schwarz,\nt(Ca+b,U)≤t(C2a,U)1/2t(C2b,U)1/2.\nThis implies logconvexity. Since the sequence remains bounded by 1, it follows\nthat the sequence is monotone decreasing. /square\n8Lemma 2.13 Letr1,r2,...,r kbe positive integers, and r=r1+···+rk. Then\nC2\nr≤C2r1...C2rk.\nCorollary 2.14\nC2k+2≤C2kC1/2\n4.\nCorollary 2.15 If1≤r1,...,r n≤rand/summationtext\ni(ri−1) =k(r−1), then\nk/productdisplay\ni=1C2ri≤Ck\n2r.\nCorollary 2.16 For allk≥2,\nCk−1\n4≤C2k≤Ck/2\n4.\nWe can get similar bounds for paths, of which we only state two, which will\nbe needed. Recall that Pndenotes the path with nnodes andn−1 edges.\nLemma 2.17 For alla,b≥1, we have\n(a)Pa+b+1≤P1/2\n2a+1P1/2\n2b+1;\n(b)P2a+b+1≤P2a+1C1/4\n4b.\nProof. SincePa+b+1=P′\na+1∗Pb+1, the first inequality follows by (5). To get\nthe second, we use the first to get\nP2a+b+1≤P1/2\n2a+1P1/2\n2a+2b+1.\nCutP2a+2b+1into pieces Pa+1,P2b+1andPa+1, and apply Lemma 2.6; we get\nP2a+2b+1≤P2a+1C1/2\n4b,\nand hence\nP2a+b+1≤P1/2\n2a+1/parenleftbig\nP2a+1C1/2\n4b/parenrightbig1/2=P2a+1C1/4\n4b.\n/square\nLemma 2.18 LetU∈ W 1. Then for every h≥1, the sequence\n(t(Kh,2k,U) :k= 1,2,...)is nonnegative, logconcave and monotone decreas-\ning.\nProof. The proof is similar, based on the equation\nt(Kh,a+b,U) =/integraldisplay\nIhtx1...xh(K′\nh,a,U)tx1...xh(K′\nh,b,U)dx1...dxh.\n/square\nFor complete bipartite graphs, however, we don’t have a bound simila r to\nCorollary 2.16, at least as long as we restrict ourselves to simple grap hs (see\nExample 1). But we do have the following inequality.\n9Lemma 2.19 For alln≥3, we have\nKn,n≤K2,nC1/2\n2.\nProof. LetHbe the 2-labeled graph obtained from Kn,nby deleting an edge\nand labeling its endpoints. Then Kn,n= (K′′\n2H)unl, and hence\nK2\nn,n≤(K′′\n2)∗2H∗2=C2H∗2.\nNow taking two unlabeled nodes from one color class from one copy of Hand\ntwo unlabeled nodes from the other color class from the other copy , we get a\nset of 4 independent nodes of degree nsuch that no three have a neighbor in\ncommon. Hence by Corollary 2.8,\nH∗2≤K2\n2,n,\nwhich proves the Lemma. /square\nExample 1 LetU: [0,1]2→[−1,1] be defined by\nU(x,y) =/braceleftBigg\n−1,ifx,y≥1/2,\notherwise.\nThen it is easy to calculate that for all n,m≥1,\nt(Kn,m,U) =1\n4.\nLemma 2.20 For allU∈ Wandx∈I,\n0≤tx(C′\n2r,U)≤t(C4r−4,U)1/2.\nProof. The first inequality follows from the formula\ntx(C′\n2r,U) =/integraldisplay\nItux(P′′\nr+1,U)2du.\nFor the second, write\ntx(C′\n2r,U) =/integraldisplay\nI2U(x,u)tuv(P′′\n2r−1,U)U(v,x)dudv,\nand apply Cauchy–Schwarz:\ntx(C′\n4,U)2≤/integraldisplay\nI2U(x,u)2U(v,x)2dudv/integraldisplay\nI2tuv(P′′\n2r−1,U)2dudv\n=tx(C′\n2,U)2t(C4r−4,U)≤t(C4r−4,U).\n/square\n10Lemma 2.21 For allU∈ W,k≥4andx,y∈I,\n|txy(P′′\nk,U)| ≤t(C4k−12,U)1/4.\nProof. We can write\ntxy(P′′\nk,U) =/integraldisplay\nU(x,u)tuy(P′′\nk−1,U)du.\nHence by Cauchy–Schwarz,\ntxy(P′′\nk,U)2≤/integraldisplay\nU(x,u)2du/integraldisplay\ntuy(P′′\nk−1,U)2du\n≤/integraldisplay\ntuy(P′′\nk−1,U)2du=ty(C′\n2k−2,U)du.\nApplying Lemma 2.20 the proof follows. /square\n2.5 The Main Bounds\nOur main Lemma is the following.\nLemma 2.22 LetFbe a bipartite graph with all degrees at least 2, with girth\n2r, which is not a single cycle or a complete bipartite graph. Th enF≤C2rC1/4\n4.\nBefore proving this lemma, we need some preparation. Let Tbe a rooted\ntree. By its min-depth we mean the minimum distance of any leaf from t he\nroot. (As usual, the depthofTis the maximum distance of any leaf from the\nroot.) For a rooted tree T, we denote by T2the graph obtained by taking two\ncopies ofTand identifying leaves corresponding to each other.\nLemma 2.23 LetTbe a tree with min-depth hand depthg. ThenT∗2contains\na hanging path system with value at least g+max(0,h−3), in which the paths\nare not longer than max(g,2).\nProof. The proof is by induction on |V(T)|. We may assume that the root\nhas degree 1, else we can delete all branches but the deepest from the root. Let\nadenote the length of the path PinTfrom the root rto the first branching\npoint or leaf v.\nIfPendsataleaf, then thewholetreeisapathoflength a=g=h. Ifa= 1,\nwe get a hanging path in T∗2of length 2, and so of value 1 = 1+max(0 ,−1).\nIfa≥2, then we can even cut this into two, and get two hanging paths in T∗2\nof lengtha, which has value 2 a−2≥a+max(0,a−3).\nIfPends at a branching point, then we consider two subtrees F1,F2rooted\natv(there may be more), where F1has depthg−a. Clearly,F1has mind-depth\nat leasth−aandF2has min-depth and depth at least h−a. By induction,\nF∗2\n1andF∗2\n2contain hanging path systems of value g−a+max(0,(h−a)−3)\n11andh−a+ max(0,(h−a)−3), respectively. The two systems together have\nvalue at least g+h−2a, and they form a valid system since v(and its mirror\nimage) are contained in at most one path of each system. If a= 1, we are done,\nsince clearly h≥2 and sog+h−2≥g+max(0,h−3).\nAssume that a≥2. LetF3be obtained from F2by deleting its root. By\ninduction,F∗2\n1contains hanging path systems of value g−a+max(0,h−a−3),\nandF∗2\n3contains a hanging path system of value h−a+max(0,h−a−4). We\ncan addPand its mirror image, to get a hanging path system of value\n(g−a)+(h−a−1)+max(0,h−a−3)+max(0,h−a−4)+2(a−1)\n≥(g−a)+(h−a−1)+2(a−1) =g+h−3 =g+max(0,h−3),\nsinceh≥a+1≥3. We know that every path constructed lies in the tree or its\nmirror image, except for the paths in the case g= 1, which are of length 2. /square\nProof of Lemma 2.22. We distinguish several cases.\nCase 1.r= 2. By hypothesis, Fis not a complete bipartite graph, and\nhencewecanchoosenonadjacentnodes uandvfromdifferentbipartitionclasses.\nLetNdenote the set of neighbors of u,|N|=d, and letF0denote the graph\nF−uwith the neighbors of ulabeled. Then F∼=F0∗K′\nd,1, and hence by (7),\nF2≤F∗2\n0(K′\nd,1)∗2=F∗2\n0Kd,2≤F∗2\n0C4.\nNow letv1andv2be the two copies of vinF2\n0, andw, any third node in the\nsame bipartition class. These three nodes have no neighbor in commo n, so by\nCorollary 2.8, we get that F∗2\n0≤C3/2\n4, and soF≤C5/4\n4.\nCase 2.Fis disconnected. If one of the components is not a single cycle,\nwe can replace Fby this component. If Fis the disjoint union of single cycles,\nthenF≤C2\n2r≤C2rC4.\nSo we may assume that Fis connected. Then it must have at least one node\nof degree larger than 2.\nCase 3.Fhas at most one node of degree larger than 2 in each color class.\nLetu1andu2be two nodes, one in each color class, such that all the other\nnodes have degree 2. Then Fmust consist of one or more odd paths connecting\nu1andu2, and even cycles attached at u1and/oru2.\nIf there are even cycles attached at u1and also at u2, thenFhas a hanging\npath system consisting of 4 paths of length r, and soF≤C2\n2r≤C2rC1/2\n4by\nLemma 2.6. If (say) u1but notu2has a cycle attached, then there are at least\ntwo paths connecting u1andu2, and one of them has length at least r. This\nimplies that F≤C3/2\n2r≤C2rC1/2\n4.\nSowemay assumethat Fconsistsofopenly disjoint paths connecting u1and\nu2. SinceFis not a single cycle, there are at least three paths. Let a1≤a2≤a3\n12be their lengths. Clearly a1+a2≥2r. Ifa2≥r+1, then we have two hanging\npaths of length r+ 1, which implies that F≤C2r+2≤C2rC1/2\n4. So we may\nassume that a1=a2=a3=r. Ifr≥4, then we can select two of the paths\nand path of length 2 disjoint from them, which gives F≤C2rC1/2\n4.\nSo we get to the special case when Fconsists of 3 or more paths of length\n3 connecting u1andu2. In this case, we use Lemma 2.21:\nt(F,U) =/integraldisplay\nI2txy(P′′\n4,U)3dxdy≤t(C4,U)1/4/integraldisplay\nI2txy(P′′\n4,U)2dxdy\n=t(C6,U)t(C4,U)1/4.\nCase 4. Suppose that there are two nodes u1,u2in the same bipartition\nclass ofFof degree at least 3.\nLetS1be the set of nodes in Fwithd(x,u1)≤min(r−2,d(x,u2)−2), and\nletS′\n1be the set of neighbors of S1inF. We define S2andS′\n2analogously. Let\nFibe the subgraph induced by Si∪S′\ni.\nClaim 1Fiis a tree with endnode set S′\ni. Everyx∈S′\nisatisfiesd(x,u1) =\nmin(r−1,d(x,u2)).\nFrom the fact that Fhas girth 2rit follows that Fiis a tree. The nodes\ninSiare not endnodes of Fi, since their degree in Fis at least 2 and all their\nneighbors are nodes of Fi. It is also trivial that the nodes in S′\niare endnodes.\nLetx∈S′\ni, thenx /∈Siand hence d(x,u1)≥min(r−1,d(x,u2)−1). But\nd(x,u1) andd(x,u2) have the same parity, and hence it follows that d(x,u1)≥\nmin(r−1,d(x,u2)). On the other hand, xhas a neighbor y∈Si, and hence\nd(x,u1)≤d(y,u1) + 1≤r−1, andd(x,u1)≤d(y,u1) + 1≤d(y,u2)−1≤\nd(x,u2). This implies that d(x,u1)≤min(r−1,d(x,u2)), which proves the\nClaim.\nClaim 2 There is no edge between S1andS2.\nIndeed, suppose that x1x2is such an edge, xi∈Si. Thend(x1,u1)<\nd(x2,u2), which by parity means that d(x1,u1)≤d(x2,u2)−2. But then\nd(x2,u1)≤d(x1,u1)+1≤d(x2,u2)−1≤d(x2,u1), showing that x2/∈S2.\nConsider the nodes of FiinS′\nias labeled. Lemma 2.6 implies that\nF2≤F2\n1F2\n2. (7)\nHence to complete the proof, it suffices to show that\nF2\ni≤C2rC1/4\n4. (8)\nThis will follow by Corollary 2.10, if we construct in Fia hanging path system\nof paths of length at most rwith value 2 r−1.\n13Claim 3 Lety/\\⌉}atio\\slash=xbe two leaves of F1. Thend(r,x)+d(r,y)+d(x,y)≥2r.\nIfd(r,x) =r−1 ord(r,y) =r−1 then this is trivial, so suppose that\nd(r,x),d(r,y)≤r−2. Then by Claim 1, we must have d(x,u2) =d(x,u1) and\nd(y,u2) =d(y,u1). Going from xtou2toyand back to xinF, we get a closed\nwalk of length d(r,x)+d(r,y)+d(x,y), which contains a cycle of length no more\nthan that, which implies the inequality in the Claim.\nClaim 4 Each ofF1andF2contains a hanging path system of paths with length\nat mostr, with value 2r−2. At least one of them contains such a system with\nvalue2r−1.\nTo prove this, we need to distinguish two cases.\nCase 4a. All branches of F1are single paths. Let a1≤ ··· ≤adbe their\nlengths. Claim 3 implies that a1+a2≥r, soa2≥r/2. The graph F2\n1consists of\npathsQ1,...,Q dof length 2a1,...,2adconnecting u1and its mirror image u′\n1.\nSelect subpaths of length rfromQ2andQ3, this gives a hanging path system\nof value 2r−2. Ifa1≥2, then we can add to this a path of length 2 from Q1\nnot containing its endpoints, and we get a path system of value 2 r−1. So we\nmay assume that a1= 1. Thena2≥r−1>r/2, and so 2a2,2ar>r. Thus we\ncan select the paths of length rfromQ2andQ3so that one of them misses u1\nand the other one misses u′\n1. The we can add Q1to the system, and conclude\nas before.\nCase 4b. At least one of the branches of F1, sayA, is not a single path.\nLetabe the length of the path Qfrom the root u1to the first branch point v.\nLetT1,T2be two subtrees of Arooted atv, of depthd1andd2. LetBandC\nbe two further branches, of depth bandc, respectively, where b≥c. By Claim\n3, we have\nd1+d2+a≥randb+c≥r.\nWe start with a simple computation showing that we can get a hanging p ath\nsystem inF2\n1of value 2r−2. Ifa= 1, then we choose a hanging path system\nfromT2\n1of valued1, fromT2\n2of valued2, fromB2of valueband fromC2of\nvaluec. This is a total of\nd1+d2+b+c≥2r−1.\nIfa≥2, then we choose a hanging path system from T2\n1of valued1, from\n(T2−v)2of valued2−1, fromB2of valueband from (C−u1)2of valuec−1.\nLeaving out vfromT2andu1fromCallows us to add Qand its mirror image\nof value 2(a−1). This is a total of\nd1+d2−1+2(a−1)+b+c−1≥2r+a−4≥2r−2. (9)\n14If equality holds in all estimates, then d1+d2+a=r,b+c=r, anda= 2. It\nalso follows that b≤3, or else we get a larger system in B. Note that the depth\nofAis at leasta+1 = 3, and c≤r/2≤b≤3.\nIfBis a single path, then we can select a hanging path of length rfrom\nB2, of valuer−1> b−1, and we have gained 1 relative to the previous\nconstruction. So we may assume that Bis not a single path. Then applying the\nsame argument as above with AandBinterchanged, we get that b= 3, and\nthe depth of Ais also 3. Hence d1=d2= 1 andr=d1+d2+a= 4. It follows\nthatc=r−b= 1, soCconsists of a single edge.\nIfu1has degree larger than 3, then applying the argument to A,Band a\nfourth branch D, we get that Dmust have depth 1, but this contradicts Claim\n3. Hence the degree of u1is 3.\nIfAhas at least 3 leaves, then these must be connected to u2by disjoint\npaths of length 3. Since u2must be connected to the endpoint of Cas well by\nClaim 1, we get that u2has degree at least 4, and so F2≥C2rC1/2\n4.\nSoAand similarly Bhave two leaves, and F1is a 10-node tree consisting of\na path with 5 nodes and 2 endnodes hanging from its endnodes and 1 f rom its\nmiddle node. F2must be the same, or else we are done. There is only one way\nto glue two copies of this tree together at their endnodes to get a g raph of girth\n8, and this yields the subdivision of K3,3(by one node on each edge). To settle\nthis single graph, we use that\nK3,3≤C1/2\n2K3,2≤C1/2\n2C4\nby Lemmas 2.19 and 2.18, and so by Lemma 2.5, we have\nF=Ksub\n3,3≤(Csub\n2)1/2Csub\n4=C1/2\n4C8.\nThus we know that F2\n1F2\n2≥C2r, and for at least one of them F2\ni≥C2rC1/2\n4,\nwhich implies that F2≥F2\n1F2\n2≥C2rC1/4\n4. /square\nLemma 2.6 implies that if Fis a graph with two nonadjacent nodes u,vof\ndegree 1, then F≤P3. We need a stronger bound:\nLemma 2.24 LetFbe a graph with two nonadjacent nodes u,vof degree 1,\nwhich is not a star and has at least 3edges. Then F≤P3C1/4\n4.\nProof. Case 1. First,supposethat Fhastwononadjacentnodes u,vofdegree\n1 whose neighbors u′andv′are different. If there is a node w/\\⌉}atio\\slash=u,v,u′,v′of\ndegreed≥2, then we can apply Lemma 2.6 to the stars of u,vandw, to get\nF≤P3K1/2\n2,d≤P3C1/2\n4.\nIfthere is anode w/\\⌉}atio\\slash=u,v,u′,v′ofdegree1, then asimilarapplication ofLemma\n2.6 gives that\nF≤P3/2\n3≤P3C1/4\n4.\n15Finally, ifV(F) ={u,v,u′,v′}, thenF=P4, and the bound follows from\nLemma 2.17(b).\nCase 2. Suppose that all nodes of Fof degree 1 have a common neighbor w.\nLetF0denote the subgraph obtained by deleting the nodes of degree 1. S ince\nFis not a star, F0must have at least two edges. If F0has a node not adjacent\ntow, then we conclude similarly as above. So suppose that wis adjacent to all\nthe other nodes of F0. LetF1denote the subgraph formed by the edges incident\nwithw(a star), with the nodes in F0−wlabeled. /square\nLemma 2.25 LetFbe a bigraph with exactly one node of degree 1and with\ngirth2r. Then\nF≤1\n2(C2r+P3)C1/8\n4.\nProof. Letvbe the unique node of degree 1. We can write F∼=F0∗P′\n2, where\nF0is a 1-labeled graph in which all nodes except possibly the labeled node v\nhave degrees at least 2. By (7), we get that F2≤(P′\n2)∗2F∗2\n0∼=P3F∗2\n0. Here\nF∗2\n0is a graph with girth 2 rand all degrees at least 2. Hence Lemma 2.22, we\ngetF2≤P3C2rC1/4\n4. Thus\n|t(F,U)| ≤/radicalbig\nt(P3,U)t(C2r,U)t(C4,U)1/8\n≤1\n2(t(C2r,U)+t(P3,U))t(C4,U)1/8.\n/square\n3 Local Sidorenko Conjecture\nThe Sidorenko Conjecture asserts that t(F,W) is minimized by the function\nW≡1 among all functions W≥0 with/integraltext\nW= 1. The following theorem\nasserts that this is true at least locally.\nTheorem 3.1 LetF= (V,E)be a simple bigraph. Let W∈ Wwith/integraltext\nW= 1,\n0≤W≤2and/bar⌈blW−1/bar⌈bl/square≤2−8m. Thent(F,W)≥1.\nProof. We may assume that Fis connected, since otherwise, the argument\ncan be applied to each component. Let U=W−1, then we have the expansion\nt(F,W) =/summationdisplay\nF′t(F′,U), (10)\nwhereF′ranges over all spanning subgraphs of F. Since isolated nodes can\nbe ignored, we may instead sum over all subgraphs with no isolated no des\n(including the term F′=K0, the empty graph). One term is t(K0,U) = 1,\nand every term containing a component isomorphic to K2is 0 sincet(K2,U) =/integraltext\nU= 0.\n16Based on (6), we can identify two special kinds of nonnegative term s in (10),\ncorresponding to copies of P3and to cycles in F. We show that the remaining\nterms do not cancel these, by grouping them appropriately.\n(a) For each node i∈V, let/summationtext\n∇(i)denote summation over all subgraphs F′\nwith at least two edges that consist of edges incident with i. Letdidenote the\ndegree ofiinF, assume that di≥2, and sett(x) =tx(K′\n2,U). Then using that\nt(x)≥ −1 and Bernoulli’s Inequality,\n/summationdisplay\n∇(i)t(F′,U) =/integraldisplay\nIdi/summationdisplay\nk=2/parenleftbiggdi\nk/parenrightbigg\nt(x)kdx=/integraldisplay\nI(1+t(x))di−1−dit(x)dx\n≥/integraldisplay\nI(1+t(x))(1+(di−1)t(x))−1−dit(x)dx\n=/integraldisplay\nI(di−1)t(x)2dx= (di−1)t(P3,U).\nHence the terms in (10) that correspond to stars sum to at least\n/summationdisplay\nstarst(F′,U)≥(di−1)t(P3,U) = (2m−n)t(P3,U).\n(b) Next, consider those terms F′with at least two endnodes that are not\nstars. For such a term we have\n|t(F′,U)| ≤t(P3,U)t(C4,U)1/4≤2−2mt(P3,U)\n(if there are two nonadjacent endpoints, then the left hand side is 0; else, this\nfollows from Lemma 2.24). The sum of these terms is, in absolute value , at most\n2m2−2mt(P3,U)<t(P3,U).\n(c) The next special sum we consider consists of complete bipartite graphs\nthat are not stars. Fixing a subset Awith|A| ≥2 in the first bipartition class\nofFwithh≥2 common neighbors, and fixing the variables in A, the sum over\nsuch complete bigraphs with Aas one of the bipartition classes is\nh/summationdisplay\nj=2/parenleftbiggh\nj/parenrightbigg/parenleftBigg/integraldisplay\nI/productdisplay\ni∈AU(xi,y)dy/parenrightBiggj\n≥(h−1)/parenleftBigg/integraldisplay\nI/productdisplay\ni∈AU(xi,y)dy/parenrightBigg2\nby the same computation as above. This gives that this sum is nonneg ative.\n(d) IfF′has all degrees at least 2 and girth 2 r, and it is not a single cycle\nor complete bipartite, then F′≤C2rC1/4\n4by Lemma 2.22, and so\n|t(F′,U)| ≤t(C2r,U)t(C4,U)1/4≤2−2mt(C2r,U).\nSo if we fix rand sum over all such subgraphs, we get, in absolute value, at\nmost\n2m2−2mt(C2r,U)<1\n2t(C2r,U).\n17(e) Finally, if F′has exactly one node of degree 1 and girth 2 r, then by\nLemma 2.25\n|t(F′,U)| ≤1\n2(t(P3,U)+t(C2r,U))t(C4,U)1/8\n≤2−m−1(t(P3,U)+t(C2r,U)).\nIf we sum over all such subgraphs F′, then we get less than t(P3,U) +\n1\n2/summationtext\nr≥2t(C2r,U).\nThe sum in (a) is sufficient to compensate for the sum in (b) and the fir st\nterm in (e), while the sum over cycles compensates for the sum in (d) and the\nsecond sum in (e). This proves that the total sum in (10) is nonnega tive./square\n3.1 Variations\nWe could do the computations above more carefully, and use the Neu mann-\nSchatten norm t(C4,U)1/4instead of the cut norm in the statement. The best\none can achieve this way is to replace the bound of 2−8mby about 2−m:\nTheorem 3.2 LetF= (V,E)be a simple bigraph. Let W∈ Wwith/integraltext\nW= 1\nand0≤W≤2andt(C4,W)≤2−4m. Thent(F,W)≥1.\nOne can combine the conditions and assume a bound on /bar⌈blW−1/bar⌈bl∞. It\nfollows from the Theorem that /bar⌈blW−1/bar⌈bl∞≤2−8msuffices. Going through the\nsame arguments (in fact, in a somewhat simpler form) we get:\nTheorem 3.3 LetF= (V,E)be a simple bigraph. Let W∈ Wwith/integraltext\nW= 1\nand/bar⌈blW−1/bar⌈bl∞≤1/(4m). Thent(F,W)≥1.\nThe condition that /bar⌈blW−1/bar⌈bl∞≤1/(4m) implies trivially that 0 ≤W≤2. It\nwouldbe interestingtoget ridofthecondition that W≤2under anappropriate\nbound on /bar⌈blW−1/bar⌈bl/square. We can only offer the following result.\nTheorem 3.4 LetF= (V,E)be a simple bigraph with medges, let 0< ε <\n2−1−8m, and letW∈ Wsuch that/integraltext\nW= 1,/integraltext\nS×TW≤2λ(S)λ(T)whenever\nλ(S),λ(T)≥2−4/ε2, and/bar⌈blW−1/bar⌈bl/square≤2−1−8m. Thent(F,W)≥1−ε.\nProof. For every function W∈ Wand partition P={V1,...,V k}ofIinto\na finite number of measurable sets with positive measure, let WPdenote the\nfunction obtained by averaging Wover the partition classes; more precisely, we\ndefine\nWP(x,y) =1\nλ(Vi)λ(Vj)/integraldisplay\nVi×VjW(u,v)dudv\nforx∈Viandy∈Vj.\n18The Weak Regularity Lemma of Frieze and Kannan in the form used in [1]\nimplies that there is a partition PintoK≤24l2/ε2equal measurable sets such\nthat the function WPsatisfies\n/bar⌈blWP−W/bar⌈bl/square≤ε\nm,\nand hence by the Counting Lemma 4.1 in [4],\n|t(F,WP)−t(F,W)| ≤ε.\nClearly/integraltext\nWP= 1,WP≥0, and for all x∈Viandy∈Vj,\nWP(x,y) =1\nλ(Vi)λ(Vj)/integraldisplay\nVi×VjW(u,v)dudv≤2.\nFurthermore,\n/bar⌈blWP−1/bar⌈bl/square≤ /bar⌈blWP−W/bar⌈bl/square+/bar⌈blW−1/bar⌈bl/square≤2−8m,\nThus Theorem 3.1 implies that t(F,WP)≥1, and hence t(F,W)≥t(F,WP)−\nε≥1−ε. /square\n3.2 Graphic form\nWe end with a graph-theoretic consequence of Theorem 3.1.\nCorollary 3.5 LetFbe a bipartite graph with nnodes andmedges, and let G\nbe a graph with Nnodes andM=p/parenleftbigN\n2/parenrightbig\nedges. Letε>0. Assume that\n/vextendsingle/vextendsingleeG(S,T)−p|S|·|T|/vextendsingle/vextendsingle≤(2−8mp−ε)N2\nfor allS,T⊆V(G), and\neG(S,T)≤2p|S|·|T|\nfor allS,T⊆V(G)with|S|,|T| ≥2−4m2/ε2N. Then\nt(F,G)≥pl−ε.\nProof. This follows by applying Theorem 3.4 to the function WG/p./square\nReferences\n[1] C. Borgs, J.T. Chayes, L. Lov´ asz, V.T. S´ os, and K. Veszterg ombi: Con-\nvergent Graph Sequences I: Subgraph frequencies, metric prop erties, and\ntesting,Advances in Math. (2008), 10.1016/j.aim.2008.07.008.\n19[2] H. Hatami: Graph norms and Sidorenko’s conjecture, Israel J. of Math.\n(to appear),\nhttp://arxiv.org/abs/0806.0047\n[3] L. Lov´ asz: On the cancellation law among finite relational struct ures,Pe-\nriodica Math. Hung. 1(1971), 145-156.\n[4] L. Lov´ asz, B. Szegedy: Limits of dense graph sequences, J. Comb. Theory\nB96(2006), 933–957.\n[5] A.F. Sidorenko: Inequalities for functionals generated by bipart ite graphs\n(Russian) Diskret. Mat. 3(1991),50–65;translationin Discrete Math. Appl.\n2(1992), 489–504.\n[6] A.F. Sidorenko: A correlation inequality for bipartite graphs, Graphs and\nCombin. 9(1993), 201–204.\n[7] A.F. Sidorenko: Randomness friendly graphs, Random Struc. Alg. 8, 229–\n241.\n[8] M. Simonovits: Extremal graph problems, degenerate extrema l problems,\nand supersaturated graphs, in: Progress in Graph Theory , NY Academy\nPress (1984), 419–437.\n20"    },    {        "title": "1004.4542v1.Spin_density_matrices_for_nuclear_density_functionals_with_parity_violations.pdf",        "content": "arXiv:1004.4542v1  [nucl-th]  26 Apr 2010Spin density matrices for nuclear density\nfunctionals with parity violations\nB. R. Barrett\nPhysics Department, University of Arizona,\nTucson, AZ 85721, USA\nB. G. Giraud\nInstitut de Physique Th´ eorique, DSM, CE Saclay,\n91191 Gif-sur-Yvette, France\nOctober 18, 2018\nAbstract\nThespindensitymatrix(SDM)usedinatomicandmolecular ph ysicsis\nrevisited for nuclear physics, in the context of the radial d ensity functional\ntheory. The vector part of the SDM defines a “hedgehog” situat ion, which\nexists only if nuclear states contain some amount of parity v iolation.\nPACS:21.10.-k, 21.10.Dr, 21.10.Hw, 21.60.-n\n1 Introduction\nThe subject of density functionals (DFs) in nuclear physics [1, 2] is p resently\nreceiving intense attention. One of its difficulties is the handling of inte ractions\nthat depend on spins. While there is a priori no theorem preventing a the-\nory with simple densities from accommodating the influence of spin dep endent\nforces, it is likely that a generalization of density profiles to “spin den sity” ones\nshould, in practice, make the construction of a DF easier. The purp ose of this\npaper is see whether one can adapt to nuclear physics the same con cept [3, 4]\nas that used for many years in atomic and molecular physics.\nGiven the creation and annihilation operators, a†\n/vector rσanda/vector rσ,respectively, of\na nucleon with spin σ=±1\n2at position /vector r,and given a density operator Din\nmany-body space, the SDM, ¯ ρ(/vector r),is defined by its matrix elements,\nρσσ′(/vector r) = Tra†\n/vector rσa/vector rσ′D. (1)\nIn the following, we shall take advantage of the recent proof [5], b ased upon\nthe rotational invariance of the nuclear Hamiltonian, that the nucle ar DF is a\nscalar, namely a radial density functional (RDF); accordingly, it is u nderstood\n1in the following, unless explicitly stated otherwise, that the density o perator,\nD,in many-body space, is a scalar under rotations. Since practical ca lculations\nfor a DF can eventually result in Kohn-Sham (KS) potentials [6], the ap proach\ndescribed by the present paper, with its explicit treatment of spin, might give\nindications for the spin-orbit term in KS equations.\nThe basic formalism for the SDM is explained in Sec. 2. A mandatory\ngeneralisation of the formalism is explained in Sec. 3. An illustrative exa mple\nis provided in Sec. 4. We conclude in Sec. 5.\n2 Basic formalism\nWe first relate the local creation and annihilation operators to thos e of anℓs\nshell model,\na†\n/vector rσ=/summationdisplay\nnℓmϕnℓ(r)Y∗\nℓm(ˆr)a†\nnℓmσ, a/vector rσ′=/summationdisplay\nn′ℓ′m′ϕn′ℓ′(r)Yℓ′m′(ˆr)an′ℓ′m′σ′.(2)\nHere the wave functions, ϕnℓ(r)Yℓm(ˆr),represent the orbitals created by the\noperators a†\nnℓm,with real radial form factors ϕnℓ(r).The summation,/summationtext\nnℓm,\nruns over a complete basis of orbitals, assumed to be discrete for t he sake of\nsimplicity. A generalization with continuum orbitals brings no difficulty ex cept\nfor slightly less simple notations. Isospin labels are understood.\nWe then rearrange the products, a†\n/vector rσa/vector rσ′,into their scalar and vector parts\nin spin space with the usual Clebsch-Gordan coefficients,\nA/vector rSMS=/summationdisplay\nσσ′(−)1\n2−σ′∝an}bracketle{t1\n2σ1\n2−σ′|SMS∝an}bracketri}hta†\n/vector rσa/vector rσ′, (3)\nThis gives, after inserting Eqs. (2),\nA/vector rSMS=/summationdisplay\nℓmℓ′m′Y∗\nℓm(ˆr)Yℓ′m′(ˆr)Bℓmℓ′m′SMS(r), (4)\nwith\nBℓmℓ′m′SMS(r) =\n/summationdisplay\nnn′σσ′ϕnℓ(r)ϕn′ℓ′(r)(−)1\n2−σ′∝an}bracketle{t1\n2σ1\n2−σ′|SMS∝an}bracketri}hta†\nnℓmσan′ℓ′m′σ′.(5)\nNext we recouple the orbital momenta carried by the operators Bℓmℓ′m′SMS,\nCℓℓ′LMSM S(r) =/summationdisplay\nm1m2(−)ℓ′−m2∝an}bracketle{tℓm1ℓ′−m2|LM∝an}bracketri}htBℓm1ℓ′m2SMS(r),(6)\nso that\nA/vector rSMS=/summationdisplay\nℓmℓ′m′Y∗\nℓm(ˆr)Yℓ′m′(ˆr)(−)ℓ′−m′/summationdisplay\nLM∝an}bracketle{tℓmℓ′−m′|LM∝an}bracketri}htCℓℓ′LMSM S(r).\n(7)\n2Upon taking advantage of the relations i) between spherical harmo nics,\nYℓm(ˆr)Y∗\nℓ′m′(ˆr) = (−)m′/summationdisplay\nλµ/radicalbigg\n(2ℓ+1)(2ℓ′+1)(2λ+1)\n4π×\n/parenleftbigg\nℓ ℓ′λ\n0 0 0/parenrightbigg /parenleftbigg\nℓ ℓ′λ\nm−m′µ/parenrightbigg\nY∗\nλµ(ˆr), (8)\nand ii) between Wigner 3 j-coefficients and Clebsch-Gordan ones,\n∝an}bracketle{tlml′−m′|LM∝an}bracketri}ht= (−)ℓ−ℓ′+M√\n2L+1/parenleftbigg\nℓ ℓ′L\nm−m′−M/parenrightbigg\n,(9)\nthe orthogonality between Wigner 3 j-coefficients,\n/summationdisplay\nmm′(2L+1)/parenleftbigg\nℓ ℓ′L\nm−m′−M/parenrightbigg /parenleftbigg\nℓ ℓ′λ\nm−m′µ/parenrightbigg\n=δLλδ−M µ,(10)\nsimplifies Eq. (7) into,\nA/vector rSMS=/summationdisplay\nℓℓ′LM(−)L−MYL−M(ˆr)×\n(−)ℓ′/radicalbigg\n(2ℓ+1)(2ℓ′+1)\n4π/parenleftbigg\nℓ ℓ′L\n0 0 0/parenrightbigg\nCℓℓ′LMSM S(r).(11)\nIn Eq. (11), we used the facts that all numbers, ℓ,ℓ′,L,M,are integers and that\nthe 3j-coefficient,/parenleftbigg\nℓ ℓ′L\n0 0 0/parenrightbigg\n,vanishes unless ℓ+ℓ′+Lis even.\nFinally, a recoupling of total orbital momentum and total spin yields,\nDLSJµ(r) =/summationdisplay\nMMS∝an}bracketle{tLMSM S|Jµ∝an}bracketri}ht ×\n/bracketleftBigg/summationdisplay\nℓℓ′(−)ℓ′/radicalbigg\n(2ℓ+1)(2ℓ′+1)\n4π/parenleftbigg\nℓ ℓ′L\n0 0 0/parenrightbigg\nCℓℓ′LMSM S(r)/bracketrightBigg\n,(12)\nso that,\n/summationdisplay\nℓℓ′(−)ℓ′/radicalbigg\n(2ℓ+1)(2ℓ′+1)\n4π/parenleftbigg\nℓ ℓ′L\n0 0 0/parenrightbigg\nCℓℓ′LMSM S(r) =\n/summationdisplay\nJµ∝an}bracketle{tLMSM S|Jµ∝an}bracketri}htDLSJµ(r). (13)\nIn terms of DLSJµ(r),the scalar or vector operators for the SDM now read\nA/vector rSMS=/summationdisplay\nLM(−)LY∗\nLM(ˆr)/summationdisplay\nJµ∝an}bracketle{tLM SM S|Jµ∝an}bracketri}htDLSJµ(r).(14)\n3With scalar density matrices Din many-body space, there will be vanishing\ntraces, Tr DLSJµ(r)D,unlessJ=µ= 0.In this case, the corresponding\nClebsch-Gordan coefficient becomes,\n∝an}bracketle{tLM SM S|00∝an}bracketri}ht=δLSδM−MS(−)S+MS\n√\n2S+1, (15)\nso that the SDM scalar or vector elements reduce to,\nTrA/vector rSMSD=(−)MSY∗\nS−MS(ˆr)√\n2S+1TrDSS00(r)D=YSMS(ˆr)√\n2S+1TrDSS00(r)D.\n(16)\n3 Generalization\nTwo very different spin profiles emerge from the study made in Sec. 2 . For the\nfirst of them, namely, for S= 0,the result is simple, since, necessarily in this\ncase,ℓandℓ′are equal,\nD0000(r) =1√\n8π/summationdisplay\nnn′ℓmσϕnℓ(r)ϕn′ℓ(r)a†\nnℓmσan′ℓmσ. (17)\nFor the second profile, i.e.,forS= 1,spherical symmetry is ensured by the\nfact that all three spherical harmonics are multiplied by the same, r adial form\nfactor, which we denote ρhh(r) in the following; we have a ‘hedgehog” situation.\nHere we mean hedgehog-like in the sense that the vector spin field ha s only a\nradial dependency. It must be noticed, however, that only those pairs of particle\norbital momenta {ℓ,ℓ′},where|ℓ−ℓ′| ≤1,can couple to L= 1.Ifℓ=ℓ′,the\n3j-coefficient,/parenleftbigg\nℓ ℓ′L\n0 0 0/parenrightbigg\n,vanishes identically, since ℓ+ℓ′+1 becomes odd.\nConversely, if ℓ−ℓ′=±1,the correspondingproducts of operators, a†\nnℓman′ℓ′m′,\nhave an odd parity. Since parity violations in nuclear states are most often\ntoo tiny to be observable, the density operators Dof interest always have an\neven parity. Therefore, if the traces, Tr Cℓℓ±11−MS1MS(r)D,do not vanish\ncompletely, then they will detect parity violations in D.A basic RDF, that uses\nD0000only, has no easy signature for parity violations. It is the occurren ce of\na tiny, but non-vanishing profile from D1100that allows a more elaborate RDF\ntheory to explicitly accommodate parity violations.\nFor the sake of completeness, we show in Eq. (18) this “hedgehog” operator,\nDhh(r)≡D1100/√\n3,the trace of which with Dis the coefficient of Y1MS(ˆr) in\nEq. (16). It reads, upon taking advantage of Eqs. (5), (6), (12 ) and (16),\nDhh(r) =1\n3√\n4π/summationdisplay\nnn′ℓmℓ′m′σσ′MSϕnℓ(r)ϕn′ℓ′(r)(−)1+MS−m′+1\n2−σ′×\n/radicalbig\n(2ℓ+1)(2ℓ′+1)/parenleftbigg\nℓ ℓ′1\n0 0 0/parenrightbigg\n∝an}bracketle{tℓmℓ′−m′|1−MS∝an}bracketri}ht ×\n∝an}bracketle{t1\n2σ1\n2−σ′|1MS∝an}bracketri}hta†\nnℓmσan′ℓ′m′σ′. (18)\n4A natural way to enlarge the theory to cases where the S= 1 form factor\nis not tiny consists in embedding the nucleus in an external field, H1,that\nsimultaneously breaks the rotational symmetry and the parity. To avoid loosing\nthe advantage of an RDF, i.e.,the reduction of three-dimensional calculations\nto one-dimensional ones, the symmetry breaking can be chosen as a minimal\none, in the following way. Let H1be a negative parity operator, bounded\nfrom below, that transforms as a vector under rotations. There is no need to\nassume that H1is only made of local fields, H1=/summationtext\nih1(/vector ri,σi),where/vector riandσi\ndenote the position and spin of the ith nucleon; any complicated H1is allowed\nfor the argument to come. What counts is that the extended Hamilt onian,\nH′=H+H1,which is bounded from below, now contains, besides the basic\nscalar and positive parity H,a vector and negative parity component H1.Then\nwe use the “constrained search” definition [7] of a DF,\nF[¯ρ] = Inf D→¯ρTrH′D, (19)\nwhere now Dis generalized into an arbitrary density operator, without symme-\ntry properties. Here the symbol, D →¯ρ,means that the minimization of the\nenergy is performed over subsets in the Dspace that show a given spin density\nmatrix ¯ρ.Then the same argument, as that used in [5], to restrict Dto be a\nrotation scalar, can be extended to restrict Dto be a mixture D01of a scalar\nand a vector. Next one can take advantage of Eq. (14) and derive F[¯ρ] from\nthosefewandradialprofile operators, DLSJµ(r), where the conditions, S= 0,1\nandJ= 0,1 give limits to Lvia the usual triangular rules.\nTo conclude this Sec., we note that a spin density DF is usually not very\nuseful for an isolated nucleus, but becomes legitimate for a non-iso lated one.\n4 Toy model for an illustrative example\nConsider a fictitious16O nucleus made of a full 0 sshell and an almost full 0 p\nshell and driven by a harmonic oscillator Hamiltonian,\nH0=/summationdisplay\nnlmστ/parenleftbigg\n2n+ℓ+3\n2/parenrightbigg\na†\nnℓmστanℓmστ=/summationdisplay\nnljµτ/parenleftbigg\n2n+ℓ+3\n2/parenrightbigg\nb†\nnℓjµτbnℓjµτ.\nHere, temporarily, the isospin label, τ=±1\n2,is explicit. The relation between\nℓsandjjcreation operators (and, similarly, for annihilation ones) in this toy\nmodel reads,\nb†\nnℓjµτ=/summationdisplay\nmσ∝an}bracketle{tℓm1\n2σ|jµ∝an}bracketri}hta†\nnℓmστ, a†\nnℓmστ=/summationdisplay\njµ∝an}bracketle{tℓm1\n2σ|jµ∝an}bracketri}htb†\nnℓjµτ.(20)\nA Slater determinant, |φ∝an}bracketri}ht,will describe this nucleus for our model. Assume that\na perturbation of the harmonic oscillator slightly mixes the 0 p1\n2orbitals with\nthe 1s1\n2orbitals. The mixtures read,\nβ†\nj=1\n2,µ,τ= cosε b†\n0p1\n2,µ,τ+sinε b†\n1s1\n2,µ,τ. (21)\n5We keep intact a core, made of the 0 sand 0p3\n2orbitals. The 16-body operator,\nD=|φ∝an}bracketri}ht∝an}bracketle{tφ|,is still a scalar under rotations, but it has now a negative parity\ncomponent at first order in ε.Such a state, and similar density matrices, would\njustify the use of a “spin RDF”, with two profiles.\nLet|0∝an}bracketri}htdenote the fully closed 0 sand 0pshells. At first order in ε,the wave\nfunction under consideration is, |φ∝an}bracketri}ht=|0∝an}bracketri}ht+ε/summationtext\nτ|τ∝an}bracketri}ht,with\n|τ∝an}bracketri}ht=/parenleftBig\nb†\n1s1\n2,1\n2,τb0p1\n2,1\n2,τ+b†\n1s1\n2,−1\n2,τb0p1\n2,−1\n2,τ/parenrightBig\n|0∝an}bracketri}ht. (22)\nProtons and neutrons will give equal matrix elements; hence, within an inessen-\ntialfactorof2, isospinlabelsandsummationscanagainbe omitted. N oticealso,\nincidentally, that the particle-hole states |τ∝an}bracketri}ht,shown in Eq. (22), do not repre-\nsent center-of-mass spurious shifts; the latter induce dipoles, n ot monopoles, in\nthe one-particle-one-hole space.\nIn thejjrepresentation, we obtain for the S= 0 case\nD0000(r) =1√\n8π/summationdisplay\nnn′ℓjϕnℓ(r)ϕn′ℓ(r)/summationdisplay\nµb†\nnℓjµbn′ℓjµ, (23)\nand the scalar profile, Tr D0000|φ∝an}bracketri}ht∝an}bracketle{tφ|,has a vanishing contribution from the\nfirst-order matrix elements, ∝an}bracketle{t0|D0000|τ∝an}bracketri}ht=∝an}bracketle{tτ|D0000|0∝an}bracketri}ht= 0,because of the re-\nstrictionto equal valuesof ℓ. The zeroth-orderprofilefrom the 0 s- and0p-shells,\nrespectively, is obviously\nD0000(r)∝ϕ2\n00(r)+3ϕ2\n01(r) =/parenleftBig\n2π−1\n4e−1\n2r2/parenrightBig2\n+3/parenleftBig\n23\n23−1\n2π−1\n4re−1\n2r2/parenrightBig2\n,\n(24)\nwith an inessential coefficient,/radicalbig\n2/π,omitted for simplicity.\nAgain for the jjrepresentation, we find for the S= 1 (hedgehog) case,\nDhh(r) =1\n3√\n4π/summationdisplay\nnn′ℓℓ′jj′µµ′mm′σσ′MSϕnℓ(r)ϕn′ℓ′(r)(−)1+MS−m′+1\n2−σ′×\n/radicalbig\n(2ℓ+1)(2ℓ′+1)/parenleftbigg\nℓ ℓ′1\n0 0 0/parenrightbigg\n∝an}bracketle{tℓmℓ′−m′|1−MS∝an}bracketri}ht ×\n∝an}bracketle{t1\n2σ1\n2−σ′|1MS∝an}bracketri}ht∝an}bracketle{tℓm1\n2σ|jµ∝an}bracketri}htb†\nnℓjµ∝an}bracketle{tℓ′m′1\n2σ′|j′µ′∝an}bracketri}htbn′ℓ′j′µ′,(25)\nwhich reduces into,\nDhh(r) =1√\n4π/summationdisplay\nnn′ℓℓ′j(−)j−1\n2/radicalbig\n(2ℓ+1)(2ℓ′+1)ϕnℓ(r)ϕn′ℓ′(r)\n/parenleftbigg\nℓ ℓ′1\n0 0 0/parenrightbigg /braceleftbigg\nl l′1\n1\n21\n2j/bracerightbigg/summationdisplay\nµb†\nnℓjµbn′ℓ′jµ, (26)\nwhere{ }is a Wigner 6 jsymbol. The equalities, j=j′andµ=µ′,reflect the\nfact that the LScoupling used in the previous section, Sec. 3, boils down to\n60.5 1.0 1.5 2.0 2.5 3.0r0.51.01.52.02.5D0000\nFigure 1: Scalar profile of the toy model\n0.5 1.0 1.5 2.0 2.5 3.0r\n/Minus0.20.20.40.6Dhh\nFigure 2: Vector profile of the toy model\ntotal spin J= 0,as demanded by the scalar nature of the many-body density\noperator D.Accordingly, in a jjscheme, both the particle and the hole total\nspin labels must be equal.\nThe zeroth-order matrix element in εthat results from Eqs. (22) and (26),\n∝an}bracketle{t0|Dhh|0∝an}bracketri}ht,trivially vanishes. Upon a simple inspection of the first-order matrix\nelements, ∝an}bracketle{t0|b†\nnℓjµbn′ℓ′jµb†\n1s1\n2νb0p1\n2ν|0∝an}bracketri}ht,it is seen that the only non-vanishing\ncontributions come from the cases, {nℓjµ}= 0p1\n2νand{n′ℓ′jµ}= 1s1\n2ν, be-\ncause of the restrictions on the values of ℓandℓ′. Hereνdenotes the magnetic\nlabel of both the particle and the hole in Eq. (22).The two values of νin|τ∝an}bracketri}htgive\nthe same contribution, similarly to the two isospin components. With a global\nfactor, 4/radicalbig\n3/π/parenleftbigg\n1 0 1\n0 0 0/parenrightbigg /braceleftbigg\n1 0 1\n1\n21\n21\n2/bracerightbigg\n,omitted, the S= 1 form factor for\nthe toy model reads,\nDhh(r)∝ϕ01(r)ϕ10(r) =π−1\n2/parenleftBig\n23\n23−1\n2re−r2\n2/parenrightBig/bracketleftbigg\n61\n2/parenleftbigg\n1−2\n3r2/parenrightbigg\ne−r2\n2/bracketrightbigg\n.(27)\nFigures 1 and 2 show these scalar and vector profiles, respectively , in ar-\n7bitrary units to avoid unessential global coefficients and because w e prefer to\ncompare shapes. There is no need to stress how different their sha pes are.\n5 Discussion\nWe set out to investigate the possible role of the spin density matrix in the con-\nstruction of the density functional for nuclei. Such spin densities h ave played an\nimportant rolein atomic and molecularphysics. However,the severe constraints\nof rotational invariance and parity for nuclei led to the result that the vector\npart of the spin density essentially vanishes in a nuclear DF that prop erly takes\ninto account such symmetries, namely, in an RDF. Thus, there is no w ay, in\nthis approach, to explicitly describe spin properties in a nuclear RDF. On the\nother hand, the vector part becomes a signature of parity violatio n allowed in\nthe RDF theory. We were able to legitimize the use of a spin density RDF , at\nthe cost of introducing an external perturbation that has negat ive parity and\ntransforms as a vector. Future studies are needed to understa nd the role of the\nspin-density-matrix formalism, when symmetries are broken by ext ernal forces.\nAcknowledgments : B.R.B.and B.G.G. thank B.K. Jennings and T. Papen-\nbrock for stimulating and helpful discussions. B.R.B. and B.G.G. also th ank\nTRIUMF, Vancouver, B. C., Canada, for its hospitality, where part of this work\nwas done. The Natural Science and Engineering Research Council o f Canada is\nthanked for financial support. TRIUMF receives federal funding via a contribu-\ntion agreement through the National Research Council of Canada . B.R.B. also\nthanks Institut de Physique Th´ eorique, CEA Saclay, France, for its hospital-\nity, where part of this work was carried out, and acknowledges par tial support\nby NSF grants PHY-0555396 and PHY-0854912 and by Institut de P hysique\nTh´ eorique, CEA Saclay.\nReferences\n[1] UNEDF SciDAC Collaboration, www.unedf.org/\n[2] J. E. Drut, R. J. Furnstahl, and L. Platter, Prog. Part. Nucl. P hys.64, 120\n(2010) and references cited therein; arXiv:nucl-th:0906.1463 (20 09),\n[3] O. Gunnarson and B. J. Lundqvist, Phys. Rev. B 13, 4274 (1976).\n[4] A. G¨ orling, Phys. Rev. A 47, 2783 (1993).\n[5] B.G. Giraud, Phys. Rev. C 78, 014307 (2008).\n[6] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). .\n[7] M. Levy, PNAS 766062 (1979); E.H. Lieb, Int. J. Quant. Chem. 24, 243\n(1983).\n8"    },    {        "title": "1004.5054v1.The_scaling_of_the_density_of_states_in_systems_with_resonance_states.pdf",        "content": "arXiv:1004.5054v1  [cond-mat.other]  28 Apr 2010The scaling of the density of states in systems with\nresonance states\nFederico M Pont, Omar Osenda, and Pablo Serra\nFacultad de Matem´ atica, Astronom´ ıa y F´ ısica, Universidad Nacion al de C´ ordoba,\nand IFEG-CONICET, Ciudad Universitaria, X5016LAE C´ ordoba, Ar gentina\nE-mail:serra@famaf.unc.edu.ar\nE-mail:osenda@famaf.unc.edu.ar\nE-mail:serra@famaf.unc.edu.ar\nAbstract. Resonance states of a two-electron quantum dot are studied usin g a\nvariational expansion with both real basis-set functions and comp lex scaling methods.\nWe present numerical evidence about the critical behavior of the d ensity of states in\nthe region where there are resonances. The critical behavior is sig naled by a strong\ndependence of some features of the density of states with the ba sis-set size used to\ncalculateit. Theresonanceenergyandlifetimeareobtainedusingth escalingproperties\nof the density of states\nPACS numbers: 31.15.ac,03.67.Mn,73.22.-fThe scaling of the density of states 2\n1. Introduction\nResonance states are slowly decaying scattering states charact erized by a large but\nfinite lifetime [1, 2]. There is a host of established methods that allow th e calculation\nof the energy and lifetime associated to the resonance [1, 2, 3, 4, 5 , 6, 7] . In many\ncases of interest where complex scaling (analytic dilatation) techniq ues can be applied,\nresonances energies and lifetimes show up as real and imaginary par t, respectively,\nof isolated complex eigenvalues of the rotated Hamiltonian [1]. Though conceptually\nsimple, complex scaling is difficult to implement in molecular systems, for t his kind\nof systems the analytical continuation is obtained using absorbing p otential methods\n[8], which can be roughly grouped in two categories: the absorbing po tentials based\non negative imaginary potentials (NIPs), and the potentials derived from the complex\nscaling based theories (see [8] and References therein).\nDespitethewealthofmethodsavailabletodealwiththecalculationof theresonance\nenergy the subject constantly receives attention and new metho ds are proposed. The\nnew proposals arise, mainly, because almost all the methods have so me drawbacks or\nbecause new concepts are applied to the problem.\nRecently Pont et al[9] and Ferr´ on et al[10] have used the Fidelity [11] and the von\nNeumann entropy [12], respectively, to obtain the real part of the resonance energy of\na two electron quantum dot. Both concepts, borrowed from the q uantum information\ntheory, provided accurate methods for the calculation of resona nce-state properties. In\nparticular the method proposed by Pont et al[9] have some points in common with\nthe work of Kar and Ho [7], both methods rest on the availability of a mo noparametric\nfamily of basis sets, where each basis set corresponds to a particu lar value of a free\nparameter.\nKarandHo[7]calculateavariationalapproximationfortheproblemw hichprovides\nan approximate spectrum. With the spectrum they construct the density of states\n(DOS) of the problem and, after this step, it is possible to find the co mplex eigenenergy\nof the resonance fitting the DOS. Their method is quite general but the fit that is\nnecessary to obtain the resonance energy is by no means unambigu ously defined. In\ncontradistinction the methods proposed by Pont et al[9] do not require any fitting, but\ntoproducetheresonanceenergyforagivenproblem(withitscorr esponding parameters)\nit is necessary to find an adequate value of the basis-set free para meter by bisection.\nIn this work we will present a method to find the energy of a resonan ce state using\nthe DOS but without resort to any fitting or bisection. We will study t he resonance\nstate that arise in a two electron quantum dot when its ground stat e looses stability and\nenters into the continuum. In Section 2 we briefly present the mode l and the variational\nmethodused tofindanapproximatespectrum. InSection3theDOS isobtainedandthe\nenergyandlifetimeoftheresonancestatearefoundedusingfinite sizescalingtechniques.\nFinally, in Section 4 we discuss our results and draw some conclusions.The scaling of the density of states 3\n2. The model and basic results\nThere are many models of quantum dots, with different symmetries a nd interactions\n(see [4, 10] and References therein). In this work we consider a mo del with spherical\nsymmetry, with two electrons interacting via the Coulomb repulsion. The main results\nshould not be affected by the particular potential choice as it is alrea dy known that the\nnear threshold behavior and other critical quantities (such as the critical exponents of\nthe energy and other observables) are mostly determined by the r ange of the involved\npotentials [13]. Therefore to model the dot potential we use a shor t-range potential\nsuitable to apply the complex scaling method. After this consideratio ns we propose the\nfollowing Hamiltonian Hfor the system\nH=−/planckover2pi12\n2m∇2\nr1−/planckover2pi12\n2m∇2\nr2+V(r1)+V(r2)+e2\n|r2−r1|, (1)\nwhereV(r) =−(V0/r2\n0) exp(−r/r0),rithe position operator of electron i= 1,2;r0\nandV0determine the range and depth of the dot potential. After re-sca ling withr0, in\natomic units, the Hamiltonian of Eq. (1) can be written as\nH=−1\n2∇2\nr1−1\n2∇2\nr2−V0e−r1−V0e−r2+λ\n|r2−r1|, (2)\nwhereλ=r0.\nWe choose the exponential binding potential to take advantage of its analytical\nproperties. In particular this potential is well behaved and the ene rgy of the resonance\nstates can be calculated using complex scaling methods. So, besides its simplicity, the\nexponential potential allows us to obtain independently the energy of the resonance\nstate and a check for our results. The threshold energy, ε, of Hamiltonian Eq. (2), that\nis, the one-body ground state energy can be calculated exactly [14 ] and is given by\nJ2√\n2ε/parenleftBig/radicalbig\n2V0/parenrightBig\n= 0, (3)\nwhereJν(x) is the Bessel function.\nThe discrete spectrum and the resonance states of the model giv en by Eq. (2) can\nbe obtained approximately using L2variational functions [4], [6]. So, if |ψj(1,2)∝angb∇acket∇ightare\nthe exact eigenfunctions of the Hamiltonian, we look for variational approximations\n|ψj(1,2)∝angb∇acket∇ight ≃ |Ψj(1,2)��angb∇acket∇ight=M/summationdisplay\ni=1c(j)\ni|Φi∝angb∇acket∇ight, c(j)\ni= (c(j))i;j= 1,···,M.(4)\nwhere the |Φi∝angb∇acket∇ightmust be chosen adequately and Mis the basis set size.\nThefunctions |Φi∝angb∇acket∇ightarechosenaseigenfunctions ofthetotalangularmomentumwith\nzeroeigenvalue. Theradialpartofthebasisfunctionsareconstr uctedbysymmetrization\nof the one-body Slater functions\nφ(α)\nn(r) =/bracketleftbiggα2n+3\n(2n+2)!/bracketrightbigg1/2\nrne−αr/2. (5)The scaling of the density of states 4\nWiththeseconstraintsthefunctions |Φi∝angb∇acket∇ighthaveonlythreerelevantquantumnumbers\nn1,n2andl, wheren1andn2are related to the radial part of the function, and lto the\nangular one [9]. The basis set is characterized by a non-linear parame terα, then we use\nthe notation |Φi∝angb∇acket∇ight → |n1,n2;l;α∝angb∇acket∇ight. So, we get that/vextendsingle/vextendsingle/vextendsingleΨ(α)\nj(1,2)/angbracketrightBig\n=/summationdisplay\nn1n2lc(i),(α)\nn1n2l|n1,n2;l;α∝angb∇acket∇ight, (6)\nwheren1≥n2≥l≥0, then the basis set size is given by\nM=N/summationdisplay\nn1=0n1/summationdisplay\nn2=0n2/summationdisplay\nl=01 =1\n6(N+1)(N+2)(N+3), (7)\nso we refer to the basis set size using both NandM. The matrix elements of the kinetic\nenergy, the Coulombic repulsion between the electrons and other m athematical details\ninvolving the functions |n1,n2;l;α∝angb∇acket∇ightare given in references [9],[15], [16].\n11.522.533.5\nλ-1.5-1-0.50\nE\n11.522.533.5\nλ-1.5-1-0.50\n(a) (b)\nFigure 1. (color on-line) (a) the figure shows the behavior of the variational\neigenvalues E(α)\nj(λ) (black lines) for N= 14 and non-linear parameter α= 2. The red\ndashed line corresponds to the threshold energy ε≃ −1.091. Note that the avoided\ncrossings between the variational eigenvalues are fairly visible. (b) The figure shows\nthe same variational eigenvalues that (a) (black lines) and the ener gy calculated using\nthe complex scaling method (green line) for a parameter θ=π/10.\nResonance states have isolated complex eigenvalues, Eres=Er−iΓ/2,Γ>0,\nwhose eigenfunctions are not square-integrable. These states a re considered as quasi-\nbound states of energy Erand inverse life time Γ. For the Hamiltonian Eq. (2), the\nresonance energies belong to the interval ( ε,0) [2].\nThe resonance states can be analyzed using the spectrum obtaine d with a basis\nofL2functions (see [10] and References therein). The levels above the threshold have\nseveral avoided crossings that “surround” the real part of the energy of the resonance\nstate. The presence of a resonance can be made evident looking at the eigenvalues\nobtained numerically. Figure 1 shows a typical spectrum obtained fr om the variational\nmethod. This figure shows the behavior of the variational eigenvalu esE(α)\njas functionsThe scaling of the density of states 5\nof the parameter λ, the results were obtained using N= 14 andα= 2.0. The value of α\nwas chosen in order to obtain the best approximation for the energ y of the ground state\nin the region of λwhere it exists. The figure shows clearly that for λ<λth≃1.54 there\nis only one bound state. Above the threshold there is not a clear cut criteria to choose\nthe value of the non-linear parameter, and the variational approx imation provides a\nfinite number of solutions with energy below zero. However, it is poss ible to calculate\nEr(λ) calculating E(α)\njfor many different values of the variational parameter (see Kar\nand Ho [7]).\nFigure 2 shows the numerical results for the second eigenvalue E(α)\n2(λ), for different\nvalues of the variational parameter α. As can be seen from Figure 2, there are three\nregions, in each one of them the curve for a given value of αis basically a straight line\nand the slope is different in each region. A feature that appears rat her clearly is that,\nfor fixedλ, the density of levels per unit energy is not uniform, despite that th e curves\nE(αi)\nj(λ) are drawn for forty equally spaced αi’s between α= 2.0 andα= 6.0. This\nfact has been observed previously [17] and the DOS can be written in terms of two\ncontributions, a localized one and an extended one. The localized DOS is attributed to\nthe presence of the resonance state, conversely the extended DOS is attributed to the\ncontinuum of states between ( ε,0).\n1 1.5 2 2.5 3\nλ-1-0.8-0.6-0.4\nE1α=6.0\nα=1.5\nEres\nFigure 2. (color on-line) The second variational state energy vsλ, for different values\nof the variational parameter α. From bottom to top αincreases its value from α= 2\n(dashed blue line) to α= 6 (dashed orange line). The real part of the resonance\neigenvalue obtained using complex scaling ( θ=π/10) is also shown (green line).The scaling of the density of states 6\n3. The density of states\nThe localized DOS ρ(E) can be expressed as [7, 17]\nρ(E) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂E(α)\n∂α/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n. (8)\nSince we are dealing with a variational approximation, we calculate\nρ(E(αi)\nj(λ)) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleE(αi+1)\nj(λ)−E(αi−1)\nj(λ)\nαi+1−αi−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n. (9)\nIn complex scaling methods the Hamiltonian is dilated by a complex facto rη= ˜αe−iθ.\nAs was pointed out long time ago by Moiseyev and coworkers, the par ameter ˜αis\nequivalent to our parameter α[18]. Besides, the DOS attains its maximum at optimal\nvalues ofαandErthat could be obtained with a self-adjoint Hamiltonian without\nusing complex scaling methods [19]. So, locating the position of the re sonance using\nthe maximum of the DOS is equivalent to the stabilization criterium used in complex\ndilationmethods that requires theapproximate fulfillment of the com plex virial theorem\n[20].\nFigure3shows the typical behavior of ρj(E)≡ρ(E(αi)\nj(λ))forseveral eigenenergies,\nwithλ= 2.25. The real and imaginary parts of the resonance’s energy, Er(λ) and\nEi(λ) =−Γ/2respectively, canbeobtainedfrom ρ(E), seeforexample[7]andreferences\ntherein.\nThe values of Er(λ) and Γ(λ) are obtained performing a nonlinear fitting of ρ(E),\nwith a Lorentzian function,\nρ(E) =ρ0+A\nπΓ/2\n[(E−Er)2+(Γ/2)2]. (10)\nOne of the drawbacks of this method results evident: for each λthere are several\nρj(E) (in fact one for each variational level, see Figure 3), and since eac hρj(E) provides\na valueforEj\nr(λ)andΓj(λ)onehasto choosewhich oneisthebest. KarandHo [7] solve\nthis problem fitting all the ρj(E) and keeping as the best values for Er(λ) and Γ(λ) the\nfitting parameters with the smaller χ2value. At least for their data the best fitting (the\nsmallerχ2) usually corresponds to the larger n. This fact has a clear interpretation,\nif the numerical method approximates Er(λ) withE(α)\nn(λ) a largenmeans that the\nnumerical method is able to provide a large number of approximate lev els, and so the\ncontinuum of states between ( ε,0) is “better” approximated.\nIn most numerical studies it is assumed that using the largest possib le basis set size\nleads to the best results. Indeed, most usually this assumption is co rrect but taking in\nconsideration only the results obtained for the largest basis set siz e does not provide\nany insight about the accuracy of the result.\nThe Finite Size Scaling method (FSS) provides a systematic way to ana lyze the\ndata generated using different basis set sizes and has been extens ively used in atomic\nphysics [21], statistical mechanics [22], etc. The FSS method require s a scaling functionThe scaling of the density of states 7\n-0.8 -0.75 -0.7\nE20406080100120140\nρ(E)n=2\nn=3\nn=4\nn=5\nFigure 3. (color on-line) The DOS ρ(E) forλ= 2.25 and basis set size N= 14. The\nresults were obtained using Eq. (8) and correspond, from top to b ottom, to the second\n(black line), third (dashed red line), fourth (green line) and fifth (d ashed blue line)\nlevels.\nthat reduces the data corresponding to different basis set sizes t o a single “universal”\nset. Moreover, as we will show, the FSS allows to obtain the real and imaginary parts\nof the resonance energy using any ρj(E).\nWe choseρ3(E) to apply the FSS method, but using any other ρjshould lead to the\nsame conclusions. As we have said previously the functional form of the scaling function\nis unique, independently of the actual value of λ. Figure 4(a) shows ρ3(E) for different\nvalues ofλand several basis set sizes. For a fixed value of N, sayN= 16 the Figure\n4(a) shows seven peaks, each one corresponding to a given value o fλ. The values of\nλwere chosen to sample different regimes of the resonance, from sm all values -sharp\nresonances- to larger ones -broad resonances. The broadening of the peaks corresponds\nroughly to the broadening of the resonance or, equivalently, to lar ger values of Γ.\nFigure 4 (b) shows the scaled data of panel (a), for each λthe curves corresponding\nto different values of Ncollapse into a single one which is independent of N.\nThe form of the scaling function can be inferred from the behavior o f the maximum\nvalue of the function ρ3(E) for different basis set sizes. Figure 5 shows the behavior\nof ln(ρ3max)vsln(N), forNlarge enough the maximum values behave as a power law.\nWith this numerical evidence we propose\nρ(E)∝mapsto−→N−β(λ)ρ(E) (11)\nas the scaling function for the DOS. As can be seen in Figure 4 (b), th e collapse of\nthe data is excellent. The exponent βdepends on λand its value must be found\nfor each case. So, for a given λthe real part of the complex resonance energy\ncorresponds to the value where the DOS attains its maximum, Er(λ) =Eρmax, whereThe scaling of the density of states 8\n-0.75 -0.5 -0.25\nE1101001000ρ(Ε)N=16\nN=14\nN=12\nN=10\n-0.75 -0.5 -0.25\nE1101001000Ν−β(λ)ρ(E)N=16\nN=14\nN=12\nN=10\nβ=0.75\nβ=0.812(a) (b)\nFigure 4. (color on-line)(a) The DOS ρ3(E) for (from left to right) λ=\n2.25,2.50,2.75,3.00,3.25 and 3 .50 and different basis set sizes. The results were\nobtained using Eq. (8). (b) Data collapse obtained using the scaling f unction\nN−β(λ)ρ(E)\nρmax= max Eρ(E) =ρ(Eρmax). At least from our data it is clear that the localization\nof the maximum of the DOS is a very slowly function of the basis set size , see Figure 5\n(b).\n2.2 2.4 2.62.8 3\nln(N)44.14.24.34.44.5ln(ρmax)y=2.298 + 0.75 x\n0.05 0.06 0.07 0.08 0.09 0.1\n1/N-0.7477-0.7476-0.7475-0.7474-0.7473-0.7472-0.7471-0.747\nΕr\n(a) (b)\nFigure 5. (color on-line) (a) log−logplot of the maximum of the density of states\nagainst the basis-set size N. (b)Ervs. 1/N. For both panels the data were obtained\nwithλ= 2.25.\nOn the following we will argue about the form of the scaling function ne cessary to\nobtain the imaginary part of the resonance energy, Γ /2. As we have said previously, the\nreal part of the resonance energy is located between ( ε,0), and as log as Γ →0 when\nλ→λ+\nc, the scaling function proposed for the imaginary part of the reson ance energy\nshould be zero at λcor, equivalently, Er(λc) =ε, andEi(λc) =−Γ(λc)/2 = 0. Besides,The scaling of the density of states 9\nin some examples the life time of a resonance is proportional to the DO S, Γ∝1\nρ, so we\ntested our numerical results using\nE(N)\ni(λ) =−κ(E(N)\nr(λ)−Ec)Nδ1\nρ(N)\nmax, (12)\nwhere we have made explicit the superscript Nto emphasize the dependency with the\nbasis set size, kappa is a positive constant, and δis a exponent to be determined. As\na first guess we used δ= 0.8, since this figure is very close to the exponents founded\nstudying the scaling of the maximum of the DOS. Then, we observed t hat\nE(N)\ni(λ′′)\nE(N)\ni(λ′)≃E(CS)\ni(λ′′)\nE(CS)\ni(λ′), (13)\ni.e.the ratio between the imaginary parts of two resonance energies, corresponding to\nλ′′andλ′chosen arbitrarily, calculated using the Complex Scaling (CS) method and\nEq. (12) is fairly the same . Regrettably this finding does not provide the value of\nthe constant κ. Just to compare with the result of Complex Scaling methods [9], we\ncalculatedκimposing that E(N)\ni(λ) =E(CS)\ni(λ) for one particular λ. Figure 6 shows\nthe values obtained for E(N)\niobtained using the recipe described above and Eq. (12),\nthe agreement between values corresponding to different basis se t sizes is excellent, and\nthe same can be said with respect to the values obtained from Comple x Scaling, see\nFigure 6 (a). Despite that the value of κwas obtained using a particular value of Nand\nλour numerical findings strongly support the functional form prop osed in Eq. (12), in\nFigure 6 (b) we present a comparison between the E(N)\ni(λ)’s and the Complex Scaling\nresults, the agreement is better than 1% for all the values of λandNanalyzed.\n-0.9-0.75 -0.6 -0.45 -0.3-0.15\nEr-0.08-0.06-0.04-0.020\nEi\nN=14  ; Complex Scaling        .\nN=16\nN=14\nN=12\nN=10\n-0.9-0.75 -0.6 -0.45 -0.3-0.15\nEr-0.002-0.00100.0010.002Ei(N) - Ei(CS)\nData collapse Eq. (12)\n(a) (b)\nFigure 6. (color on-line) (a) Resonance pole string curve obtained from comp lex\nscaling and basis-set size scaling. A mean value of δ= 0.8 was chosen for the scaling\nexponent. (b) The energy discrepancies between the two method s. The relative error\nis less than 1% for the calculated values.The scaling of the density of states 10\n4. Summary and conclusions\nSo far we have presented numerical evidence of the critical behav ior of the density of\nstatesnearthepeakassociatedtothepresenceofaresonance state. Thecriticalbehavior\nshould be understood in the sense of FSS method, i.ethere is a strong dependency of\nthe density of states with the basis set size used to obtain it. The cr itical behavior\nalso manifest itself through the scaling properties of some quantitie s, in particular the\nmaximum value attained by the density of states for fixed λandN.\nDespite that we have been able to propose a scaling function that pr ovides the\nimaginary part of the resonance energy, the proposal is made only in phenomenological\nterms and, more importantly, relays in other methods to be quantit ative. Our efforts\nwill be directed to put the scaling law Eq.(12) over a more formal basis , which hopefully\nwill allow us to obtain the constants κandδusing only scaling arguments.\nThe critical behavior of the density of states studied in this work, t ogether with\nthe behavior observed in the fidelity [9], points that the analogy betw een the “line” of\nresonance states and a “critical line” (in the sense of critical phen omena) could be more\nexploited and it is worth of further exploration.\nAcknowledgments\nWe would like to acknowledge SECYT-UNC, CONICET and FONCyT for pa rtial\nfinancial support of this project.\nReferences\n[1] N Moiseyev, Phys. Rep. 302, 211 (1998).\n[2] W P Reinhardt and S Han, Int. J. Quantum Chem. 57, 327 (1996).\n[3] Y Sajeev and N Moiseyev, Phys. Rev. B 78, 075316 (2008).\n[4] M Bylicki, W Jask´ olski, A Stach´ ow and J Diaz, Phys. Rev. B 72, 075434 (2005).\n[5] J Dubau and I A Ivanov, J. Phys. B: At. Mol. Opt. Phys. 313335 (1998).\n[6] A T Kruppa and K Arai, Phys. Rev. A 59, 3556 (1999).\n[7] S Kar and Y K Ho, J. Phys. B: At. Mol. Opt. Phys. 37, 3177 (2004).\n[8] Y Sajeev, V Vysotskiy, L S Cederbaum, and N Moiseyev, J. Chem. Phys.131, 211102 (2009)\n[9] F M Pont, O Osenda, P Serra and J H Toloza, arXiv:1003.0468 [Phys. Rev. A (accepted for\npublication 2010)].\n[10] A Ferr´ on, O Osenda and P Serra, Phys. Rev. A 79, 032509 (2009).\n[11] P Zanardi and N Paunkovi´ c, Phys. Rev. E 74, 031123 (2006).\n[12] M A Nielsen and I L Chuang, Quantum Computation and Quantum Information , Cambridge\nUniversity Press; first edition (September 2000).\n[13] F M Pont and P Serra, J. Phys A: Math. Theor. 41, 275303 (2008).\n[14] A Galindo and P Pascual. Quantum Mechanics I ( Springer, 1991).\n[15] O Osenda, P Serra and S Kais, Int. J. Quantum Inf. 6, 303 (2008).\n[16] P Serra and S Kais, Chem. Phys. Lett. 372, 205 (2003).\n[17] V A Mandelshtam, T R Ravuri and H S Taylor, Phys. Rev. Lett. 70, 1932 (1993).\n[18] N Moiseyev, C Corcoran, Phys. Rev. A 20, 814 (1979).\n[19] N Moiseyev, S Friedland, Phys. Rev. A 22, 618 (1980).The scaling of the density of states 11\n[20] N Moiseyev, S Friedland, P R Certain, J. Chem. Phys. 78, 4739 (1981)\n[21] S Kais and P Serra, Adv. Chem. Phys. 125, 1 (2003), and references therein.\n[22] MEFisher,In Critical Phenomena , Proceedingsofthe51stEnricoFermiSummerSchool,Varenna,\nItaly, M. S. Green, ed. (Academic Press, New York 1971)."    },    {        "title": "1004.5260v2.Ground_state_at_high_density.pdf",        "content": "arXiv:1004.5260v2  [cond-mat.other]  20 Jul 2011Published in Commun. Math. Phys. 305, 657-710 (2011)\nGround state at high density\nAndr´ as S¨ ut˝ o\nResearch Institute for Solid State Physics and Optics\nHungarian Academy of Sciences\nP. O. B. 49, H-1525 Budapest, Hungary\nE-mail: suto@szfki.hu\nAbstract\nWeak limits as the density tends to infinity of classical grou nd states of integrable pair potentials\nare shown to minimize the mean-field energy functional. By st udying the latter we derive global\nproperties of high-density ground state configurations in b ounded domains and in infinite space. Our\nmain result is a theorem stating that for interactions havin g a strictly positive Fourier transform the\ndistribution of particles tends to be uniform as the density increases, while high-density ground states\nshow some pattern if the Fourier transform is partially nega tive. The latter confirms the conclusion\nof earlier studies by Vlasov (1945), Kirzhnits and Nepomnya shchii (1971), and Likos et al.(2007).\nOther results include the proof that there is no Bravais latt ice among high-density ground states of\ninteractions whose Fourier transform has a negative part an d the potential diverges or has a cusp at\nzero. We also show that in the ground state configurations of t he penetrable sphere model particles\nare superimposed on the sites of a close-packed lattice.\nPACS: 61.50.Ah, 02.30.Nw, 61.50.Lt\nContents\n1. Introduction\n2. Best superstability constant\n3. Infinite-density limit of the ground state energy and the f ree energy per pair\n4. Infinite-density ground state and the energy functional i n finite volume\n4.1 Bounded interactions\n4.2 Unbounded interactions\n4.3 Fourier representation of the energy functional\n5. Stability conditions for pair potentials\n6. Stationary points of the energy functional\n7. Ground state configurations in infinite space\n8. Infinite-density ground state in infinite space\n9. Infinite-density ground states for v≥0\n10. Uniformity vs nonuniformity of high-density ground sta tes\n11. Interactions without Bravais lattice ground states at h igh density\n11.1 Compensation by higher harmonics\n11.2 Potentials with a cusp at zero\n12. The penetrable sphere model1 Introduction\nThe presentpaper is a continuation ofmy earlierworkon groundsta tes of bounded Fourier-transformable\npair potentials [1, 2]. Bounded or integrable interactions appear in qu antum physics for example in\nBogoliubov’stheoryof the Bose gas, which is based on models expres sedin terms of the Fourier transform\nof the pair potential. Such interactions play also a central role in clas sical soft matter physics [3]. Those\nstudied in [1, 2] had a nonnegativeFouriertransformof compact su pport. In 2007Likosand his coworkers\npublished a mean-field study of the case when the Fourier transfor m of the pair potential has a negative\npart [4]. If the system discussed in [1, 2] showed already very peculia r properties at high densities,\naccording to these authors a partly negative Fourier transform in duced an even more curious behavior,\nthe particles having the tendency to form clusters on the sites of a lattice whose lattice constant would be\ndetermined by the (negative) minimum of the Fourier transform, an d only the population of the clusters\nwould increase with the density. The question then arises whether it is possible to prove rigorously this\nkind of behavior. When, in the fall of 2008, I presented preliminary r esults on this problem and on its\nquantum mechanical counterpart [5] at a Montreal meeting in math ematical physics, Valentin Zagrebnov\nkindly informed me that, without knowing it, I worked on the theory o f coherent crystals, a subject\npromoted by Russian physicists a long time ago.\nIn effect, maybe the first attempt at a theory of crystallization wa s due to Vlasov. In his paper [6]\nVlasov studied the solutions of the today called Vlasov equation\n−∂f\n∂t=v·∂f\n∂r+1\nmF·∂f\n∂v(1.1)\nfor the one-particle distribution f(r,v,t). There is no collision term, but the equation contains a self-\nconsistent force field\nF(r,t) =−∂\n∂r/integraldisplay\nu(r−r′)/integraldisplay\nf(r′,v,t)dvdr′(1.2)\ninduced by a translation invariant pair potential u. Considered on the torus or in the entire space,\nthis equation has spatially homogeneous static solutions f0(v). Vlasov investigated his equation after\nlinearizing it about f0. Besides many others, he asked the question whether there may e xist spatially\nperiodic static solutions. If yes, they could be associated with crys tals. In his analysis, however, he\narrived at the false conclusion that for the existence of such a solu tion/integraltext\nu(r)dr<0 must hold. Today we\nknow that there can be no equilibrium (crystalline or other) phase un der the effect of such an interaction:\nthe system collapses, the grand-canonicalpartition function dive rges in finite volumes [7]. The possibility\nof a static periodic solution for a (super)stable uis, nonetheless, there, and to see how such a solution\nemerges, we can follow Vlasov’s reasoning almost up to the end. Given the velocity profile f0, one\nsubstitutes f(r,v,t) =f0(v)+φ(r,v,t) into Eq. (1.1) and keeps only the terms linear in φ:\n∂φ\n∂t+v·∂φ\n∂r−1\nm∂f0\n∂v·∂(u∗Φ)\n∂r= 0, (1.3)\nwhere\nΦ(r,t) =/integraldisplay\nφ(r,v,t)dv. (1.4)\nWith the ansatz\nφ(r,v,t) =eiωt−ik·rgk(v), (1.5)\none obtains\ngk(v) =/hatwideu(k)k·∂f0\n∂v\nm(k·v−ω)/integraldisplay\ngk(v′)dv′, (1.6)\nwhere/hatwideuis the Fourier transform of u. Integration over vyields\n1 =/hatwideu(k)\nm/integraldisplayk·∂f0\n∂v\nk·v−ωdv (1.7)\n2which implicitly determines the dispersion relation ω(k). With the choice f0(v) =h(v2) this further\nsimplifies, and for a static solution ( ω= 0),kmust satisfy the equation\n1 =2/hatwideu(k)\nm/integraldisplay\nh′(v2)dv. (1.8)\nIn three dimensions, for a bounded and sufficiently fast decaying hthis becomes\n1 =−2π/hatwideu(k)\nm/integraldisplay∞\n0h(x)√xdx. (1.9)\nBecauseh≥0, we see that a solution for kis possible only if /hatwideuhas a negative part! Making the\nsubstitutions f0(v) =h(v2) andω= 0 also in gk, we obtain an approximate static solution of Eq. (1.1)\nin the form\nf(r,v) =h(v2)+h′(v2)/summationdisplay\nckcosk·r, (1.10)\nwith the sum running over the vectors kwhich solve Eq. (1.9), and the real ck(incorporating the\nconstant (2 /m)/hatwideu(k)/integraltext\ngk) chosen so that f≥0. To obtain (1.10) we have supposed that u(x) =u(−x),\nwhich implies /hatwideu(k) =/hatwideu(−k) real and (1.9) holding simultaneously for ±k. Ifuis integrable, then /hatwideu(k)\nis continuous and decays at infinity; thus, for any “natural” uthe sum in (1.10) is finite. Moreover,\n/hatwideu(0)>0 for a superstable interaction, therefore /hatwideu(k)≤0 can be only if |k| ≥k0>0. Let now f0be the\nMaxwell distribution, i.e.,\nh(x) =ρ/parenleftbiggβm\n2π/parenrightbigg3/2\ne−βmx/2, (1.11)\nwhereρis the density and βis the inverse temperature. Then h′(x) =−(βm/2)h(x), and (1.8) becomes\n−βρ/hatwideu(k) = 1. (1.12)\nThis equation has a solution if βρ≥(βρ)c, where\n(βρ)c=−1\nmin{/hatwideu(k)}. (1.13)\nIf the minimum of /hatwideuis nondegenerate, at the critical point the resulting distribution is s patially periodic.\nAsβρincreases, the solutions for kshift away from the minimizer of /hatwideu, and the solution of the linearized\nequation is, in general, almost periodic. With the Maxwell distribution ( 1.10) becomes\nf(r,v) =f0(v)/bracketleftbigg\n1−βm\n2/summationdisplay\nckcosk·r/bracketrightbigg\n. (1.14)\nIn order to keep fnonnegative, as βincreases, the coefficients ckmust scale as β−1.\nThe right condition for the formation of “coherent crystals” appe ared only much later, in papers by\nKirzhnits and Nepomnyashchii [8, 9]. The term referred to (hypoth etical) crystals of mobile particles as\nin Vlasov’s theory, capable of ballistic motion which must be coherent if it preserves long-range order.\nThe context was somewhat different, these authors were interes ted in the crystallization of a quantum\nliquid. The starting point was a Hamiltonian the ground state of which w as determined in the Hartree\napproximation. This analysis showed that the lowest-energy solutio n of the Hartree equation can be\nperiodic only if the Fourier transform of the pair potential is partially negative, and the periodicity\nis then given by the wave vector at which the Fourier transform is min imal. These authors, just as\nVlasov, were fully aware of the extraordinary properties of coher ent crystals, even more pronounced in\nthe quantum than in the classical case, and described them almost in the same terms as Likos et al.[4].\nThe mean-field or density-functional approach of Likos et al., the partial differential equation method\nof Vlasov, and the Hartree approximation of Kirzhnits and Nepomny ashchii are effective one-particle\ntheories, and all arrive at the same conclusion. One can have little do ubt in their truth. The problem is\n3more difficult than the case treated in [1, 2], and a new idea is necessar y to obtain some progress in the\nrigorous theory. The idea presented and exploited in this paper is th at of an infinite-density ground state\n(IDGS). Workingin a finite domain (on a torus here), the N-particleground states arearrangementsof N\npoints that minimize the interaction energy. To each N-point subset one can assign a discrete measure,\nand an IDGS is the weak limit of a sequence of discrete measures asso ciated with N-particle ground\nstates, when Ngoes to infinity. The ground state energy and energy per particle d iverge in this limit\nbut, if the interaction is integrable, the energy per pair is converge nt. The limit is proportional to the\nbest superstability constant, the largest number that can multiply N2/Vin a lower bound on the energy\nofNparticles in a cube of volume V. The two notions, that of an IDGS and of the best superstability\nconstant, can be related via an energy functional written for nor malized measures on the torus. It is\nshown that any IDGS is a minimizer of the energy functional, and the v alue of the minimum is the best\nsuperstability constant. The task is then to find these minimizers, b ecause they can provide information\non ground state configurations at high but finite densities. Most re sults on IDGS’s will be obtained\nby writing the energy functional in Fourier representation; an exc eption is the penetrable sphere model\npresented at the end of the paper (for another example see Ref. [10]). Our main result is a confirmation\nof the conclusion of the earlier works [4, 6, 8, 9]: if the Fourier trans form of the pair potential has a\nnegative part, the distribution of particles in high-density ground s tates is nonuniform. In some cases,\ne.g. when the potential diverges at the origin or has a cusp there, t his nonuniform distribution cannot\nbe approached through Bravais lattices. Except for the penetra ble sphere model, from this analysis we\ncannot predict the ground state configurationsat high densities w ith the precision obtained in Refs. [1, 2].\nThe lack of precise information is true also for interactions with a str ictly positive Fourier transform,\nbut we can at least assert that, in contrast with the former, the a symptotic distribution of particles\nas the density increases tends to be uniform. Since there is a contin ued interest in the Gaussian core\nmodel [11]-[16], even this weak result may be of some value.\nIt is to be emphasized that the limit of infinite particledensity is not at all unusual in classicalphysics.\nWhenever a continuum theory is applied to describe a system of class ical pointlike particles, tacitly\nthis limit is used. Theories of classical fluids, the rigorous van der Waa ls theory of liquid-vapor phase\ntransition [17, 18, 19], the continuum theories of droplet formation [20, 21] and liquid-gas interfaces [22]\nare based on different free energy functionals defined on continuo us mass densities. These theories can\nbe derived through scaling limits in which the particle density tends to in finity while the mass of the\nparticles and the interactions among them tend to zero in order to k eep the mass and energy densities\nfinite. In this work we do not have to scale the mass because it enter s only the kinetic energy which\nvanishes in classical ground states (gravitating systems are out o f the scope of this study), but we do\nscale the interaction: considering the energy per pair of particles c orresponds to scaling the pair potential\nby dividing it with NasNtends to infinity in a fixed volume. The crucial difference compared wit h\nthe continuum theories cited above is the extension of the energy f unctional to discrete distributions.\nStarting with them and by controlling their convergence in the limit of in finite density we can obtain\ninformation on ground state configurations at finite densities.\nThe paper is organized as follows. Section 2 fixes the conditions on th e interaction and the basic\nnotations, and contains the definition of the best superstability co nstant together with some preliminary\nresults on it. Throughout the paper we deal with both bounded and unbounded interactions. The results\nwill be more complete for bounded interactions, and the reason of t his appears already in Sections 2\nand 3. For bounded interactions the self-energy is finite, and the e nergy per pair with and without the\nself-energy converges to the best superstability constant as Ngoes to infinity from above and from below,\nrespectively. This permits to prove that the convergence is unifor m in the volume – a fact that we do\nnot know for unbounded interactions. In Section 3 we still show tha t the infinite-density limit of the free\nenergy per pair is independent of the temperature and is the same a s that of the ground state energy.\nIn Section 4 we define the infinite-density ground states as weak limit s of Dirac combs associated with\nN-particle ground states, introduce the energy functional and pr ove two theorems, one for bounded and\nanother for unbounded interactions. They show that IDGS’s minimiz e the energy functional, and the\nminimum is the best superstability constant. From the Fourier repre sentation of the energy functional,\nintroduced in Section 4.3, we derive some stability conditions in Section 5. Section 6 contains a complete\n4description of the stationary points of the energy functional. The ir knowledge is important because there\ncan be minimizers among them. The truly challenging problem is to find th e ground state configurations\n(GSC) in infinite space. The definition, based on local stability, is reca lled in Section 7. Here we prove\ntwo lemmas, the first relating periodic configurations and minimizers o f the energy density, the second\nestablishing the relation between periodic GSC’s in infinite space and GC S’s on tori. The analysis of\nthe problem in infinite space is continued in Section 8. We introduce the notion of an IDGS in infinite\nspace and formulate as a conjecture a commutative diagram relatin g four objects: GSC and IDGS in\nfinite domains and in infinite space. The main result of this section is Pro position 8.2 which gives a\nnew expression of the best superstability constant. Sections 9 – 1 2 present applications. In Section 9 we\ngive a rather complete description of IDGS’s in the case when the Fou rier transform of the interaction is\nnonnegative. Section 10 contains the main theorem proving the asy mptotic uniformity or non-uniformity\nof GSC’s in infinite space for the case of a strictly positive or partially n egative Fourier transform,\nrespectively. In Section 11 we discuss two special classes of intera ctions with a partly negative Fourier\ntransform. In the first one for any nonzero kthe sum of the Fourier transform over integer multiples of\nkis nonnegative. In the second case the Fourier transform is ultimat ely positive and slowly decaying, so\nthat the pair potential either diverges at zero or has a cusp there . In both cases we show that no Bravais\nlattice can be an IDGS, and high-density GSC’s are different from Bra vais lattices. On the contrary,\nbounded pair potentials that are flat or nesting at the origin or have a dominantly negative Fourier\ntransform at large wave vectors may prefer the accumulation of p articles on the sites of a lattice which is\nindependent of the density. In Section 12 we demonstrate this pro perty on the so-calledpenetrable sphere\nmodel in which the interaction is a repulsive square core potential. Th e paper ends with an Appendix.\n2 Best superstability constant\nConsider the problem of the ground state of a system of classical id entical particles confined in a fixed\nbounded domain Λ of volume Vand interacting via a translation-invariant pair interaction u(x−y) =\nu(y−x). Assume uto be integrable, bounded outside the origin, strongly tempered, s uperstable and\nlower semicontinuous (see below). In the simplest case Λ is a d-dimensional cube of side length Ltaken\nwith periodic boundary conditions, and uis also made periodic. The usual way to achieve this is to\nreplaceu(x) by\nuΛ(x) =/summationdisplay\nn∈Zdu(x+Ln). (2.1)\nThis series is absolutely convergent for any L>0 and any x(outside the set LZdwhenu(0) =∞) ifu\nis strongly tempered. The general definition of strong temperedn ess is\n|u(x)|<C|x|−d−η(2.2)\nfor|x|> r0, with some r0,C,η >0. Ifuhas no hard core, r0= 0 can be taken. Condition (2.2) also\nguearantees that uΛconverges to upointwise as Ltends to infinity. A ground state of Nparticles in Λ\nis anyN-point configuration ( r)N= (r1,...,rN) minimizing the N-particle interaction energy\nUΛ(r)N=/summationdisplay\ni<juΛ(ri−rj). (2.3)\nThe minimum energy will be denoted by E0(N). Because of the shift-invariance of u, any translate of a\nground state is a ground state. To make sure that (2.3) can be minim ized for all N, we will ask uto be\nlower semicontinuous [23].\nA way to understand the behavior of soft matter at high density is t o study the limit of infinite\ndensity. Since the particles have no hard core, this limit can be given a meaningful definition. The limit\nρ=N/V→ ∞will be realized by letting Ndiverge in a fixed volume. To characterize a ground state, a\nsuitable scaling of UΛis necessary. If uis bounded then\nǫ(r)N=V\nN(N−1)UΛ(r)N (2.4)\n5remains bounded in this limit whatever the sequence of N-point configurations is. Moreover,\nǫN= min\n(r)N⊂Λǫ(r)N=V\nN(N−1)E0(N) (2.5)\nis bounded as Ngoesto infinity even if uis unbounded but is integrableand bounded from below. Indeed,\ndivide Λ into Ncubes of volume V/Nand choose ( r)Nto be the centers of the cubes. Then the second\nsum in\nUΛ(r)N=N\n2V/summationdisplay\ni/summationdisplay\nj/ne}ationslash=iuΛ(ri−rj)V\nN(2.6)\nis a Riemann-sum, therefore\nǫ(r)N=1\n2/integraldisplay\nu(x)dx+o(1) (N→ ∞) (2.7)\nis an upper bound to ǫN. We conclude that the right quantity to look at is ǫ(r)Nthat, with a slight\nabuse, we call the pair energy.\nThe search for the ground state in the limit of Ngoing to infinity is closely related to finding the\nbest superstability constant foru. The notion of superstability was introduced by Ruelle [24, 7]. Stability\nmeans the existence of a constant Bsuch that for any Nthe ground state energy is bounded below by\n−BN; superstability means that the ground state energy density (ene rgy per volume) increases with the\nparticle density at least quadratically. If uis integrable and superstable, to leading order in the density\nthe increase is quadratic.\nDefinition 2.1 The best superstability constant for a superstable interac tionuis the supremum of the\npositive numbers Csuch that for any large enough cube Λof volumeV\nUΛ(r)N≥CN2/V (2.8)\nholds for every Nabove a possibly Λ-dependent value and every (r)N⊂Λ.\nLemma 2.1 The best superstability constant for uis\nC[u] = liminf\nV→∞CΛ[u] (2.9)\nwhere the limit is taken over cubes of increasing volume Vand\nCΛ[u] = liminf\nN→∞ǫN. (2.10)\nProof.Take anyC <C[u]. From the definition of liminf it follows that CΛ[u]>Cfor Λ large enough,\nsay, Λ⊃ΛC. It also follows that there exists some NΛsuch that\nN−1\nNǫN>Cif Λ⊃ΛCandN >N Λ (2.11)\nwhich is (2.8). On the other hand, if C > C[u] then there exist arbitrarily large domains Λ such that\nC >C Λ[u], and an infinite sequence Njof positive integers (which may depend on Λ) such that\nlim\nj→∞ǫNj=CΛ[u]. (2.12)\nThus,\nǫNj<C (2.13)\nforjlarge enough, which is the negation of (2.8). /square\n6Remarks. (i) In the definition of superstability [7] there is an additional term −BNon the right-hand\nside of (2.8). We can add such a term with any B≥0 without violating the inequality, it even allows for\nthe extension of the inequality to every N, cf. Eq. (2.31) below. Obviously, no choice of Bpermits to\nincrease the coefficient of N2and attain a superstability constant larger than C[u] defined by (2.9). (ii)\nLater on, Λ will be a general parallelepiped. The periodized interactio n can be defined, and the analogue\nof Lemma 2.1 can be proven in the following form. Let Λ 0be any nondegenerate parallelepiped, and for\ns >0 considersΛ0={sx:x∈Λ0}. Then the best superstability constant is liminf s→∞CsΛ0[u]. The\nlimit actually exists,\nC[u] = lim\nΛ→∞CΛ[u] (2.14)\nif Λ tends to infinity in the Fisher sense [7], and is the same as that one o btains with the use of uinstead\nof the periodized uΛ. Based on the strong temperedness of the interaction, a proof in analogy with the\nproof of existence of the thermodynamic limit of the free energy [7] could be done, but in this paper\n(2.14) is considered as a hypothesis.\nAn integrable interaction uhas a bounded continuous Fourier transform decaying at infinity [23 ]. We\nwill denote it by v. Thus,\nv(k) =/integraldisplay\nu(r)e−ik·rdr. (2.15)\nIf/summationtext\nk∈Λ∗|v(k)|<∞then/summationtext\nk∈Λ∗v(k)eik·ris a continuous function and\nuΛ(r) =1\nV/summationdisplay\nk∈Λ∗v(k)eik·r(2.16)\nalmost everywhere; if uis continuous, equality holds everywhere. Here Λ∗= (2π/L)Zdif Λ is a cube of\nside length L. Superstability implies that v(0)>0 and alsou(0)>0 anduΛ(0)>0.\nProposition 2.1 CΛ[u]≤v(0)/2and thusC[u]≤v(0)/2.\nProof.This follows from Eq. (2.7). An alternative proof is obtained by compu ting the average of the\npotential energy,\n1\nVN/integraldisplay\nΛNUΛ(r)Nd(r)N=v(0)N(N−1)\n2V. (2.17)\nThe minimum is smaller than the average,\nN(N−1)\nVǫN=E0(N)≤v(0)N(N−1)\n2V(2.18)\nand hence\nCΛ[u]≤v(0)/2.✷ (2.19)\nThe following simple observation helps to see the possibility of superpo sition of particles in some\nground state configurations. Given ( r)M, for any integer n≥1 we define the nM-point configuration\n(r)n\nMbyntimes repeating ( r)M,\nrmM+j=rj, m= 0,...,n−1, j= 1,...,M. (2.20)\nThen, for any bounded complex-valued function fonRd,\n1\n(nM)2nM/summationdisplay\ni,j=1f(ri−rj) =1\nM2M/summationdisplay\ni,j=1f(ri−rj). (2.21)\nWhenuis bounded, we apply this identity to f=VuΛ/2. Introducing\nω(r)N=V\n2N2N/summationdisplay\ni,j=1uΛ(ri−rj) =N−1\nNǫ(r)N+VuΛ(0)\n2N(2.22)\n7and\nωN= min\n(r)N⊂Λω(r)N=N−1\nNǫN+VuΛ(0)\n2N, (2.23)\nand recalling that m|nfor integers m,ndenotes that mis a divisor of n, we obtain the following.\nLemma 2.2\nω(r)n\nM=ω(r)M (2.24)\nand, hence,\nωN1≥ωN2≥ωN3≥ ···ifN1|N2|N3|···. (2.25)\nProof.Equation (2.24) repeats (2.21). Let ( r)Njbe aNj-particle ground state. Then\nωNj+1≤ω(r)Nj+1/Nj\nNj=ω(r)Nj=ωNj.✷ (2.26)\nProposition 2.2 Ifuis bounded,\nCΛ[u] = inf\nNωN. (2.27)\nProof.\nωN−ǫN=1\nN/parenleftbiggVuΛ(0)\n2−ǫN/parenrightbigg\n(2.28)\ngoes to zero as Nincreases, so the statement is\nliminf\nN→∞ωN= inf\nNωN. (2.29)\nNowinf NωN≤liminf N→∞ωN; supposing astrictinequality, becauseof(2.25)there wouldbe ase quence\nNjtending to infinity such that\nωNj≤ωN1<liminf\nN→∞ωN (2.30)\ncontradicting the definition of liminf. ✷\nA direct consequence of Equation (2.27) is that\nUΛ(r)N≥ −uΛ(0)\n2N+CΛ[u]N2\nV(2.31)\nforallN. Another consequence is as follows.\nCorollary 2.1 Ifω(r)M=CΛ[u], then(r)n\nMis anM-particle ground state for all n≥1.\nThe simplest realization of this situation is provided by the penetrable sphere model and is presented in\nSection 12. For a less trivial example see Ref. [10].\n3 Infinite-density limit of the ground-state energy and the f ree\nenergy per pair\nProposition 3.1 The sequence ǫNis convergent.\nProof.We give two different proofs.\n(i) The first proof works for ubounded. In this case ωN−ǫNtends to zero with increasing N, therefore\nthe convergence of ǫNis equivalent to\nlim\nN→∞ωN= inf\nNωN=CΛ[u]. (3.1)\n8NowCΛ[u] is the smallest accumulation point of the sequence ωN, and suppose there is another one, ω0.\nLetδ=1\n3(ω0−CΛ[u]) and define two subsequences, KmandLnvia the inequalities ωKm≤CΛ[u]+δand\nωLn≥ω0−δ; thus,ωLn−ωKm≥δfor allm,n. Both subsequences are infinite and, because of Eq. (2.25),\nℓK1∈ {Km}for all integers ℓ≥1. Therefore, for any nthere is anmsuch that 0 <Ln−Km<K1and,\nthus,\nE0(Ln)≤E0(Km)+1\n2K1(K1−1+Km)∝⌊ar⌈⌊luΛ∝⌊ar⌈⌊l∞. (3.2)\nThe right member of this inequality is an upper bound to the energy of aLn-particle configuration that\none obtains from a Km-particle ground state by adding Ln−Kmparticles. Dividing by L2\nn/Vwe find\nωLn−ωKm=O(L−1\nn) (3.3)\ncontradicting ωLn−ωKm≥δ.\n(ii) The second proof works even if uis unbounded. It is based on the monotonic increase of the ground\nstate energy per pair,\nE0(N+1)\nN(N+1)≥E0(N)\nN(N−1); (3.4)\nsee also Kiessling [25]. If uis integrable then ǫN=VE0(N)/N(N−1) is bounded above by v(0)/2, cf.\nEq. (2.18), so its limit as Ngoes to infinity exists,\nlim\nN→∞ǫN= supǫN=CΛ[u]. (3.5)\nAs to Eq. (3.4), let N≥2 andRbe any sequence of N+1 points of Λ (repetition allowed). Then\nUΛ(R) =/summationdisplay\n(x,y)⊂RuΛ(x−y) =1\nN−1/summationdisplay\nS⊂R,|S|=N/summationdisplay\n(x,y)⊂SuΛ(x−y) =1\nN−1/summationdisplay\nS⊂R,|S|=NUΛ(S),(3.6)\nbecause for any two-point subsequence ( x,y)⊂Rwe can choose N−1 differentN-point subsequences\nS⊂Rcontaining ( x,y). Since the sum over ShasN+1 terms,\nmin\nR⊂Λ,|R|=N+1UΛ(R)≥N+1\nN−1min\nS⊂Λ,|S|=NUΛ(S) (3.7)\nwhich is just (3.4). ✷\nWe have nowhere used the periodicity of the interaction, therefor e Proposition 3.1 applies also to\nuinstead ofuΛ. It holds true also for unstable interactions, when CΛ[u]<0.Ifuis bounded then\nPropositions 2.2 and 3.1 yield\nsupǫN=CΛ[u] = infωN. (3.8)\nCorollary 3.1 Ifuis bounded then the convergence of ǫ⌊ρV⌋andω⌊ρV⌋toCΛ[u]asρtends to infinity\nis uniform in V.\nProof.\n0≤max/braceleftbig\nω⌊ρV⌋−CΛ[u],CΛ[u]−ǫ⌊ρV⌋/bracerightbig\n≤ω⌊ρV⌋−ǫ⌊ρV⌋≤uΛ(0)\n2ρ≤u(0)\nρ(3.9)\nifρandVare large enough. Here we used Eq. (2.28) and superstability for th e third and strong\ntemperedness for the fourth inequality. ✷\nThe uniform convergenceof ǫ⌊ρV⌋toCΛ[u] probablyholds true alsoif udivergesat the origin. If Λ is a\ncubeofside LandRisanyN-pointconfigurationthenthereisan x∈Λsuchthatdist( x,R)≥L/(2N1/d).\nUsing this fact, for a radial uone can easily prove that\nǫN≤ǫN+1≤ǫN+cρ−1u/parenleftbigg1\n2ρ1/d/parenrightbigg\n(3.10)\n9if Λ is large enough. Here ρ=N/Vandc= 1 ifu(x) is a monotonic function of |x|close to zero. The\ntermρ−1u(ρ−1/d/2) is the analogue of ρ−1u(0). Because uis integrable, it tends to zero as ρgoes to\ninfinity. However, the right member of (3.10) may not be an upper bo und toCΛ[u] and, therefore, it\ncannot play the role of ωNin the bounded case.\nIt is natural to ask how the free energy per pair behaves as the de nsity tends to infinity. Let\nZΛ,N=1\nVN/integraldisplay\nΛNe−βUΛ(r)Nd(r)N≡e−βFN(β). (3.11)\nFor the free energy per pair, fN(β), we have\nǫN≤fN(β)≡VFN(β)\nN(N−1)≤v(0)\n2(3.12)\nwhere the upper bound results from Jensen’s inequality. We are inte rested in the limit of fN(β) asβor\nNgoes to infinity. The usual definition of the partition function is VN/N! times the expression (3.11).\nHowever, computing −(V/βN(N−1))lnZΛ,Nwith the modified definition or with (3.11) yields the same\nresult for both limits. At positive temperatures, as Ngoes to infinity, fN(β)−ǫNremains positive only\nif the entropy is negative and of order N2. This means that the level set\n{(r)N∈ΛN:UΛ(r)N≤E0(N)+N2/β}\nshould be of Lebesgue measure ∼e−cN2with some c >0. Under some natural conditions on the\ninteraction we shall find the opposite result, an infinite-density limit o f the free energy per pair that\nagrees with the limit of ǫN, i.e.,CΛ[u].\nTheorem 3.1 Letusatisfy one of the following conditions.\n(i)uis bounded and\n|u(x)−u(y)| ≤(|u(x)|+|u(y)|)/parenleftbigg|x−y|\nr0/parenrightbiggα\n(3.13)\nwith somer0>0and0<α≤1.\n(ii) There exist constants u0>0,r0>0,c>0,0<ζ <d and0<α≤1such that\nu(x)≥u0/parenleftbiggr0\n|x|/parenrightbiggd−ζ\nif|x| ≤r0 (3.14)\nand\n|u(x)−u(y)| ≤c(|u(x)|+|u(y)|)/parenleftbigg|x−y|\nmin{|x|,|y|,r0}/parenrightbiggα\n. (3.15)\nThen\nlim\nβ→∞fN(β) =ǫN,lim\nN→∞fN(β) = lim\nN→∞ǫN=CΛ[u]allβ >0. (3.16)\nProof.(i) Using (3.13),\n|uΛ(x)−uΛ(y)| ≤2∝⌊ar⌈⌊luΛ∝⌊ar⌈⌊l/parenleftbigg|x−y|\nr0/parenrightbiggα\n(3.17)\nwhere\n∝⌊ar⌈⌊luΛ∝⌊ar⌈⌊l= sup\nx/summationdisplay\nn∈Zd|u(x+Ln)|. (3.18)\nLet(r)Nbeagroundstateconfiguration[hence, UΛ(r)N=E0(N)]and(r+δr)N= (r1+δr1,...,rN+δrN)\nan arbitrary perturbation of it. Then\nUΛ(r+δr)N−E0(N)≤ ∝⌊ar⌈⌊luΛ∝⌊ar⌈⌊l/parenleftbigg\n2max\ni|δri|\nr0/parenrightbiggα\nN2≤β−1(3.19)\n10if for alli\n|δri| ≤r0\n2/parenleftbigg1\nβ∝⌊ar⌈⌊luΛ∝⌊ar⌈⌊lN2/parenrightbigg1\nα\n≡r0\n2εβ,N. (3.20)\nUsing Eqs. (3.11), (3.12), (3.19) and (3.20),\n1≥e−βN(N−1)[fN(β)−ǫN]/V≥cd\ne/bracketleftbigg(r0εβ,N/2)d\nV/bracketrightbiggN\n(3.21)\nwherecdis a dimension-dependent constant. Taking the logarithm, dividing by βN(N−1)/Vand letting\neitherβorNgo to infinity we find Eq. (3.16).\n(ii) Let ( r)Nbe a ground state configuration and let rij=ri−rj. Because of Eq. (2.18) and the lower\nboundedness of u,u(rij+Ln)≤u1N2holds with some u1>0 for any n∈Zd. By decreasing, if\nnecessary,r0, (3.14) is valid also with u1replacingu0and implies\n|rij+Ln| ≥r0N−2\nd−ζ. (3.22)\nLetxn=rij+δrij+Lnandyn=rij+Lnwhereδrij=δri−δrj, and choose\n|δri| ≤r0\n4/parenleftBigg\n1\nu2βN4+2\nd−ζ/parenrightBigg1\nα\n(3.23)\nfor everyi, whereu2= 3cu1. Ifu2βN4≥1, this inequality implies\n|δrij| ≤r0\n2N−2\nd−ζ. (3.24)\nThen\nmin{|xn|,|yn|,r0} ≥r0\n2N−2\nd−ζ (3.25)\nand by (3.15),\n|u(xn)−u(yn)| ≤c(|u(xn)|+|u(yn)|)/parenleftbigg2|δrij|\nr0N−2\nd−ζ/parenrightbiggα\n. (3.26)\nWhen passing to uΛwe have to estimate/summationtext\nn|u(xn)|and/summationtext\nn|u(yn)|. It is easily seen that only a single\nterm can be large, the sum of the rest is of order 1 as Ngoes to infinity. The possible large term is\nbounded by u1N2. Summation over i,jbrings in another factor N2, and forNorβlarge enough we end\nup with a generous upper bound\nUΛ(r+δr)N−E0(N)≤u2/parenleftbigg\n4max\ni|δri|\nr0/parenrightbiggα\nN4+2\nd−ζ≤β−1. (3.27)\nThe rest of the proof is as in the bounded case. ✷\nRemarks. (i) For the validity of (3.16) the continuity of uwas essential. We shall see that for\nthe penetrable sphere model at least the first of Eqs. (3.16) fails. Condition (3.13) is a sort of strong\nH¨ older-continuity (because u(x) decays as |x|grows) which guearantees the ordinary H¨ older-continuity\nofuΛ. In the case of a free boundary condition ordinary H¨ older-contin uity would suffice. (ii) There is\na gap between interactions that are bounded or diverge algebraica lly at the origin. Interactions with\na logarithmic divergence, occurring in some cases between colloidal p articles [26], allow particles to be\nmuch closer to each other and, thus, the ground state configura tions to be strongly inhomogeneous. One\ncannot exclude that instead of (3.20) one should impose |δrij|=O(e−cN). Then, one could still prove\nthe first of Eqs. (3.16) but not the second one, because of an ent ropy∝ −N2at positive temperatures.\n114 Infinite-density ground state and the energy functional in fi-\nnite volume\n4.1 Bounded interactions\nIn this section we show that it is possible to obtain CΛ[u] and a good approximation of high-density\nground states via the minimization of an energy functional. For boun ded interactions we can write ω(r)N\nin the form\nω(r)N=1\n2V/integraldisplay\nΛ2uΛ(x−y)dµ(r)N(x)dµ(r)N(y) (4.1)\nwhere\nµ(r)N=V\nNN/summationdisplay\nj=1δrj (4.2)\nis ameasureon Λ oftotalweight V(henceforth, a normalizedmeasure); δxis the Dirac delta at x. We call\nµ(r)Nthe measure associated with ( r)N. The discrete measures of the form (4.2) – sums of Dirac deltas\nwith equal weights, sometimes called Dirac combs – are dense among t he normalized Borel measures\nrelative to the weak topology, cf. Lemma A.1. Therefore, when Ngoes to infinity and RNis aN-particle\nconfiguration, the sequence µRNof associated measures can converge weakly to any normalized mea sure.\nWe shall use this property to introduce the notion of an infinite-den sity ground state. Let MΛdenote\nthe set of normalized Borel measures on Λ. A sequence µn∈ MΛconverges vaguely to a µ∈ MΛif for\nevery real continuous function fof compact support\nlim\nn→∞µn(f) =µ(f), (4.3)\nwhereµ(f) :=/integraltext\nfdµ. The convergence is in the weak sense if (4.3) holds for every bound ed continuous\nf. Because Λ is compact, every continuous function is bounded and o f compact support, so the two types\nof convergences are the same. Henceforth µn⇀µwill denote that µis the vague limit of the sequence\nµn. For the reader’s convenience, in the Appendix we show that any infi nite sequence µn∈ MΛhas a\nweakly convergent subsequence (i.e., MΛis compact in the weak topology), and any µ∈ MΛis the weak\nlimit of a sequence of discrete measures of the form (4.2).\nDefinition 4.1 µ∈ MΛis an infinite-density ground state (IDGS) of uΛif there is a sequence µRmof\nmeasures associated with configurations Rmsuch that |Rm|goes to infinity with mand\nµRm⇀µ, lim\nm→∞ǫ(Rm) =CΛ[u]. (4.4)\nNote thatRmmay not be a sequence of ground state configurations.\nProposition 4.1 The set of IDGS’s is nonempty.\nProof.Given any sequence {RN}∞\n1ofN-particle ground states, ǫ(RN) =ǫNtends toCΛ[u], cf. Eq. (3.5).\nMoreover, the sequence µRNof associated measures has at least one weak limit point which is, ther efore,\nan IDGS. ✷\nMotivated by Eq. (4.1), we define an energy functional on MΛby the equation\nI[µ] =1\n2V/integraldisplay\nΛ2uΛ(x−y)dµ(x)dµ(y) =1\n2Vµ∗/tildewideµ(uΛ). (4.5)\nHere we use the notations of Ref. [27]: for a complex function f,/tildewidef(x) :=f(−x),/tildewideµ(f) :=µ(/tildewidef). The\nmeasureV−1µ∗/tildewideµis the autocorrelationof µ. It is defined on Λ −Λ (= Λ in the case of periodic boundary\nconditions when Λ is a torus) by\nµ∗/tildewideµ(f) =/integraldisplay\nΛ−Λf(z)d(µ∗/tildewideµ)(z) :=/integraldisplay\nΛ2f(x−y)dµ(x)dµ(y), (4.6)\n12wherefis any continuous periodic function of period cell Λ. Thus,\nµ∗/tildewideµ=/integraldisplay\nΛ2δx−ydµ(x)dµ(y) =/integraldisplay\nΛ2δx+ydµ(x)d/tildewideµ(y). (4.7)\nFor general results about the autocorrelation and its Fourier tra nsform see [27, 28]. Equation (4.1) can\nbe rewritten as ω(r)N=I[µ(r)N].I[µ] is meaningful also for unbounded interactions, but it can be finite\nonly ifµis a continuous measure. For example, for the Lebesgue measure λ,I[λ] =v(0)/2. InI[µ] one\ncan recognize the interaction energy common to all mean-field theo ries, either quantum as the Hartree\napproximation [8], or classical as the mean-field theory of fluids [4, 12 ]. The difference in our use of it\nis that, while in mean-field theories µis tacitly supposed to be absolutely continuous, here no a priory\nassumption is made on it. Let\nI0= inf\nµ∈MΛI[µ]. (4.8)\nFor bounded interactions\nI0≤inf\nN,(r)NI[µ(r)N] = inf\nNωN=CΛ[u]. (4.9)\nBefore proving the opposite inequality in Theorem 4.1, let us note tha t the correspondence between a\nfinite configuration ( r)Nand the associated measure (4.2) is not one-to-one. The configur ation (r)n\nM\ndefined by Eq. (2.20) is different from ( r)Mwhosen-times repetition gives rise to it. This is in contrast\nwith\nµ(r)n\nM=µ(r)M, (4.10)\nimplied by Eq. (2.24). Thus, any discrete measure is associated with a n infinity of finite configurations.\nThis observation leads to the counterpart of Corollary 2.1.\nProposition 4.2 Suppose that µ(r)Mis a minimizer of the energy functional. Then for any positiv e\nintegern,(r)n\nMis anM-particle ground state of uΛ.\nProof.Because of the identities (2.24) and (4.10),\nω(r)n\nM=I[µ(r)M] =I0≤I[µ(x)nM] =ω(x)nM (4.11)\nand, hence,\nUΛ(r)n\nM≤UΛ(x)nM (4.12)\nfor anynM-particle configuration ( x)nM.✷\nIn this way, if a minimizer of I[µ] happens to be a discrete measure concentrated e.g. on the vert ices of\na lattice inside Λ, then the particles pile up on the lattice sites already in finite-density ground states.\nFor a Λ-periodic bounded Borel function fand aµ∈ MΛ, let\nIf[µ] =1\n2Vµ∗/tildewideµ(f). (4.13)\nI[µ]≡IuΛ[µ] in this notation.\nLemma 4.1 Ifuis bounded then I[µ]is continuous in the weak topology, i.e., I[µn]→I[µ]ifµn⇀µ.\nProof.Let firstube continuous. If µn⇀µthenµn∗/tildewiderµn⇀µ∗/tildewideµ, therefore\nµn∗/tildewiderµn(f)→µ∗/tildewideµ(f) (4.14)\niffis continuous on Λ. Applying this to f=uΛthe result follows. Suppose now that uis bounded. For\nanyε >0 we can choose a real continuous fon Λ and an integer m(depending also on f) such that\n∝⌊ar⌈⌊luΛ−f∝⌊ar⌈⌊l∞<εand|If[µ]−If[µn]|<εforn≥m. Then|IuΛ[µ]−IuΛ[µn]|<3εforn≥mand, hence,\nlim\nn→∞|IuΛ[µ]−IuΛ[µn]| ≤3ε. (4.15)\nThis being true for every ε>0, the above limit actually vanishes. ✷\n13Theorem 4.1 Ifuis bounded, then\nI0=CΛ[u], (4.16)\nthe infimum I0is attained, and I[µ] =I0if and only if µis an IDGS.\nRemark. Because of Eq. (2.27), Equation (4.16) states that the infimum of I[µ] over allµ∈ MΛis the\nsame as the infimum over discrete measures associated with finite-d ensity ground states. This does not\nmean that the IDGS cannot be a continuous measure, see the fort hcoming sections.\nProof.Chooseµm∈ MΛsuch thatI[µm] tends toI0asmgoes to infinity ( µmmay be the same for each\nm). Because the set of discrete measures (4.2) is dense in MΛandIis continuous, µmcan be chosen to\nbe of the form (4.2); hence, I[µm]≥CΛ[u]. Ifµis any weak limit point of µmthen by the continuity of\nI,\nCΛ[u]≥I0= lim\nm→∞I[µm] =I[µ]≥CΛ[u]. (4.17)\nThis proves (4.16) and that µis an IDGS. Starting with an IDGS µand choosing µm=µRmwhereRm\nis the defining sequence (4.4), Eq. (4.17) shows that µis a minimizer of I.✷\nRemark. We have found that weak limits of N-particle ground states are minimizers of I, but have not\nproved that every minimizer=IDGS can be obtained as such a limit.\n4.2 Unbounded interactions\nWe restrict the discussion to pair potentials such that u(x)→ ∞asx→0 anduis bounded otherwise.\nDefinition 4.1 and Proposition 4.1 are valid in this case. Because the not ion of an IDGS is independent of\nthe energy functional, the divergence of I[µ] on point measures does not allow to directly conclude that\nIDGS’s are continuous measures. Below we present a proof which is in dependent of I.\nProposition 4.3 Letube integrable and u(x)→ ∞asx→0. Then any IDGS µis purely continuous.\nProof.Take any x0∈suppµ. LetRmbe the defining sequence of µ, cf. Definition 4.1. Because\nuΛ(0) =∞, all points of Rmare distinct. Choose any ε >0 and letCbe an open ball of diameter ε\ncentered at x0. By approximating the characteristic function of Cwith continuous functions one can see\nthat\nlim\nm→∞µRm(C) =µ(C).\nIt follows that for mlarge enough µRm(C)≥1\n2µ(C) and, because µRm({x}) =V/|Rm|forx∈Rm,\n|Rm∩C| ≥µ(C)|Rm|\n2V. (4.18)\nAs a consequence,\nUΛ(Rm∩C)≥µ(C)|Rm|\n4V/parenleftbiggµ(C)|Rm|\n2V−1/parenrightbigg\ninf\n|x|<εuΛ(x) (4.19)\nand\nUΛ(Rm)≥ |Rm|2/braceleftbigg/bracketleftbiggµ(C)2\n8V2−µ(C)\n4V|Rm|/bracketrightbigg\ninf\n|x|<εuΛ(x)−supu−\nΛ\n2/bracerightbigg\n(4.20)\nwhereu−\nΛ(x) =−min{uΛ(x),0}. Becauseuis bounded below and strongly tempered, sup u−\nΛis finite.\nDividing by |Rm|(|Rm|−1)/Vand letting mtend to infinity,\nCΛ[u]≥µ(C)2\n8Vinf\n|x|<εuΛ(x)−Vsupu−\nΛ\n2(4.21)\nor\nµ(C)2≤4V(2CΛ[u]+Vsupu−\nΛ)\ninf|x|<εuΛ(x). (4.22)\n14The upper bound vanishes as εgoes to zero, proving that µ({x0}) = 0.✷\nThe extension of Theorem 4.1 to unbounded interactions is not immed iate because IDGS’s are weak\nlimits of discrete measures which make I[µ] diverge. We therefore introduce auxiliary functionals which\ncan give finite values on point measures. For a positive Klet\n/tildewideIK[µ] =1\n2V/integraldisplay\n{(x,y):uΛ(x−y)≤K}uΛ(x−y)dµ(x)dµ(y) (4.23)\nand\n/tildewideI[µ] = lim\nK→∞/tildewideIK[µ] = sup\nK/tildewideIK[µ]. (4.24)\nIfuis bounded,/tildewideI[µ]≡I[µ]. Ifuis unbounded, /tildewideI[µ]≤I[µ]. In contrast to I,/tildewideIcan be finite on point\nmeasures if u(x) tends to infinity when x→0 butuis bounded outside the origin, in which case we also\nhave\n/tildewideI[µ] =1\n2Vlim\nε→0/integraldisplay\ndΛ(x,y)≥εuΛ(x−y)dµ(x)dµ(y) (4.25)\nwheredΛ(x,y) = min n∈Zd|x−y+Ln|. The usefulness of /tildewideIis based on the easily verifiable fact that\n/tildewideI[µ(r)N] =ǫ(r)N, and on/tildewideI[µ] =I[µ] (maybe infinite) for continuous measures.\nLemma 4.2 Ifµis continuous then I[µ] =/tildewideI[µ]≤ ∞.\nProof.Let\ngn=1\n2V(µ×µ)({(x,y)∈Λ×Λ :uΛ(x−y)>n}), (4.26)\na monotone decreasing sequence. By the definition of the Lebesgu e integral,\nI[µ] = lim\nK→∞/bracketleftBig\n/tildewideIK[µ]+KgK/bracketrightBig\n=/tildewideI[µ]+ lim\nK→∞KgK, (4.27)\nimplying that the limit exists ( ≤ ∞). Ifµhas a point part then\n2Vlim\nK→∞gK= (µ×µ)/parenleftBigg∞/intersectiondisplay\nn=1{uΛ>n}/parenrightBigg\n=/summationdisplay\nx∈Λµ({x})2>0, (4.28)\nthereforeI[µ] =∞, but/tildewideI[µ] is finite if e.g. µis concentrated on a finite number of points. Suppose that\nµis continuous, then lim gK= 0. If lim KgK= 0, we have the claimed equality. If lim KgK≥c >0,\nintroduce\n/tildewideI+\nK[µ] =/tildewideIK[µ]−/tildewideI0[µ] =/tildewideIK[µ]+Iu−\nΛ[µ]. (4.29)\nBecauseuΛis bounded below, the second term is finite. Now\n/tildewideI+\nK[µ]≥K−1/summationdisplay\nn=1n(gn−gn+1) =K/summationdisplay\nn=1(gn−gK)≥K′/summationdisplay\nn=1(gn−gK) (4.30)\nfor anyK′≤K, therefore\nlim\nK→∞/tildewideI+\nK[µ]≥K′/summationdisplay\nn=1gn (4.31)\nfor anyK′. It follows that\nlim\nK→∞/tildewideI+\nK[µ]≥∞/summationdisplay\nn=1gn=∞ (4.32)\nbecausegn>c/2nfornlarge enough. Thus, in this case I[µ] =/tildewideI[µ] =∞.✷\n15Lemma 4.3 Ifµis an IDGS then /tildewideI[µ]≤CΛ[u].\nProof.LetRmbe the defining sequence of µ. By Lemma 4.1 and Theorem 4.1,\n/tildewideIK[µ] = lim\nm→∞/tildewideIK[µRm]≤lim\nm→∞/tildewideI[µRm] = lim\nm→∞ǫ(Rm) =CΛ[u]. (4.33)\nTaking the supremum over Kgives the result. ✷\nLemma 4.4 CΛ[u]≤I0.\nProof.Define\nuK\nΛ(x) = min{uΛ(x),K}. (4.34)\nFor anyNthere exists some KNsuch thatRNis aN-particle ground state of UΛif and only if it is also\naN-particle ground state of UK\nΛfor anyK≥KN, and\nEK\n0(N) =UK\nΛ(RN) =UΛ(RN) =E0(N). (4.35)\nFor example, KN=E0(N)+(N2/2)supu−\nΛis a possible choice. Indeed, uK\nΛ≤uΛ, thereforeEK\n0(N)≤\nE0(N). It follows that if K≥KNandRNis aN-particle ground state of either UΛorUK\nΛ, then\nuK\nΛ(x−y) =uΛ(x−y)≤K\nfor every pair {x,y} ⊂RN, otherwise\nUΛ(RN)≥UK\nΛ(RN)>E0(N).\nThis means, however, that RNis a common ground state of UΛandUK\nΛ, and (4.35) holds true. With\nthe aboveKNandRN, in an obvious notation, cf. (4.13),\nI0\nuKN\nΛ=CΛ[uKN]≥ǫKN\nN=ǫN, (4.36)\nthe three relations holding due to Eqs. (4.16), (3.5) and (4.35), res pectively. Letting Ntend to infinity,\nor taking the supremum over N,\nI0≡I0\nuΛ≥lim\nN→∞I0\nuKN\nΛ≥CΛ[u], (4.37)\nwhere the first inequality follows from Eq. (4.34). ✷\nCombining Proposition 4.3 and Lemmas 4.2, 4.3 and 4.4, for any IDGS µwe find\n/tildewideI[µ] =CΛ[u] =I0=I[µ] (4.38)\nand, therefore, the following result.\nTheorem 4.2 Ifuis integrable, bounded outside the origin, and u(x)→ ∞asx→0, thenI0=CΛ[u],\nthe infimum is attained, and any IDGS is continuous and minimi zesI.\nRemark to Sections 4.1 and 4.2. In the proof of Theorems 4.1 and 4.2 we have not used that uΛwas a\nperiodized interaction. Therefore, the theorems are valid in the ca se when Λ is any compact Lebesgue-\nmeasurable set and the interaction is the original, non-periodic u.\n164.3 Fourier representation of the energy functional\nMost of the results on IDGS’s will be found by writing I[µ] in Fourier representation. This is obtained\nby substituting the expansion (2.16) into Eq. (4.5) and integrating t erm by term:\nI[µ] =1\n2\nv(0)+/summationdisplay\n0/ne}ationslash=k∈Λ∗v(k)|/hatwideµ(k)|2\n=1\n2V/hatwideµ∗/tildewideµ(v). (4.39)\nHere\n/hatwideµ(k) =1\nV/integraldisplay\nΛe−ik·xdµ(x), (4.40)\nso that|/hatwideµ(k)| ≤/hatwideµ(0) = 1, and\n1\nV/hatwideµ∗/tildewideµ=/summationdisplay\nk∈Λ∗|/hatwideµ(k)|2δk. (4.41)\nWe do not need the equality (4.39) to hold for all µ∈ MΛ. We are interested in IDGS’s, and I[µ] =\nCΛ[u]≤v(0)/2 for any of them. It suffices, therefore, that the Fourier expan sion (4.39) converges to\nI[µ], whenever I[µ]≤v(0)/2. In the case v≥0,I[µ]≤v(0)/2 holds (with equality) if and only if µis\nan IDGS; later on, we will discuss this case in detail. Let\nv+(k) = max{0,v(k)}, v−(k) =−min{0,v(k)}. (4.42)\nIfv−∝\\⌉}atio\\slash= 0,thentheIDGS’sareamongthose µmakingtherightmemberofEq.(4.39)absolutelyconvergent\nand satisfying/summationdisplay\n0/ne}ationslash=k∈Λ∗v−(k)|/hatwideµ(k)|2−/summationdisplay\n0/ne}ationslash=k∈Λ∗v+(k)|/hatwideµ(k)|2≥0. (4.43)\nFor a stable interaction the left member of this inequality cannot exc eedv(0), cf. Proposition 5.1. The\nverity of (4.39) in this case can be seen as follows. Given a t>0, consider the Gaussian\nGt(x) = (4πt)−d/2e−|x|2/4t(4.44)\nand its periodization, called the periodic heat kernel,\nGt\nΛ(x) =/summationdisplay\nn∈ZdGt(x+Ln) =1\nV/summationdisplay\nk∈Λ∗e−t|k|2eik·x, (4.45)\nwhere the right member is obtained by the Poisson summation formula (or the Fourier expansion of the\nperiodic function on the left). If µ∈ MΛ, thenGt\nΛ∗µ∈ MΛas well. [f∗µ(x) :=/integraltext\nf(x−y)dµ(y).]\nMoreover,Gt\nΛ∗µis absolutely continuous, and its Radon-Nikodym derivative\nφ(x) =/summationdisplay\nk∈Λ∗e−t|k|2/hatwideµ(k)eik·x(4.46)\nis a real entire function of each component of x. These properties guarantee that the energy functional\nevaluated on Gt\nΛ∗µsatisfies (4.39),\nI[Gt\nΛ∗µ] =1\n2/summationdisplay\nk∈Λ∗v(k)e−2t|k|2|/hatwideµ(k)|2. (4.47)\nBecause the left member is finite and the right member is absolutely co nvergent at t= 0, the regularity\nof Abel summability implies that Eq. (4.47) survives the t→0 limit and yields (4.39).\n17The form (4.39) of I[µ] reveals an important property of the minimizers, hidden in the defin ing\nequation (4.5). For stable interactions 0 ≤I0≤v(0)/2 but, because Λ∗is a set of density ∼Ld, for most\nµ∈ MΛone findsI[µ] of the order of some power of L. For example,\nI[Vδx] =1\n2/summationdisplay\nk∈Λ∗v(k) =VuΛ(0)/2. (4.48)\nThus, one may expect that when Λ increases, the minimizers are amo ngµ’s such that for some volume-\nindependent constant cand for any K >0\n/vextendsingle/vextendsingle/vextendsinglesupp/hatwideµ/intersectiondisplay\n{k∈Λ∗:|k| ≤K}/vextendsingle/vextendsingle/vextendsingle≤cKd. (4.49)\nIn Section 6 we shall meet some special sets of this kind, namely, add itive subgroups of Λ∗. These\nwill be shown to support the Fourier transform of the stationary p oints ofI. Whenv≥0 oru(x)≥\nconst×e−α|x|2, at least the uniform local boundedness as Λ → ∞of both the autocorrelation and its\nFourier transform (4.41) is easily seen. We return to this question in Section 8.\n5 Stability conditions for pair potentials\nFrom Theorems 4.1 and 4.2 we can deduce stability conditions on u. Ifuis nonnegative then it is\nobviously stable; if u≥0 and is integrable then its Fourier transform vis a positive definite function (a\nfunction of positive type) [29], therefore v(0)−|v(k)| ≥0 for all k∈Rd. Due to Theorems 4.1 and 4.2,\nsimilar conditions can be obtained without assuming u≥0.\nProposition 5.1 Letube an integrable pair potential with u(−x) =u(x). ThenI0=I0(Λ)≥0for any\ncubeΛis sufficient and necessary for stability, and I0(Λ)>0for allΛis sufficient and necessary for\nsuperstability. Some particular necessary conditions for the stability of uare as follows.\n(i)\n2v(0)+v(k)≥0,allk∈Rd. (5.1)\n(ii) Ifuis radial and v(k)≤0for|k| ≥k0then\nv(0)+ndmin\n|k|≥k0v(k)≥0 (5.2)\nwherendis the number of nearest neighbors in a d-dimensional close-packed lattice ( nd= 6,12,24,40,...\nford= 2,3,4,5,...).\n(iii) Ifuis bounded, radial and u(r)≤0for|r| ≥r0then\nu(0)+ndmin\n|r|≥r0u(r)≥0. (5.3)\nRemark. Ruelle’s example of a catastrophic potential ([7], Section 3.2.3) violate s the inequality (5.3).\nProof.Because for uboundedI0= infN,(r)Nω(r)N, the condition I0≥0 is part of Proposition 3.2.2 of\nRuelle [7]. The role of the sign of I0foruunbounded is also clear from I0=CΛ[u]. Nowv(−k) =v(k)\nreal, and assertions (i)-(ii) are nontrivial if vtakes on negative values.\n(i) Suppose that v(k)<0 for some k∝\\⌉}atio\\slash=0and consider the measure\ndµk(x) = (1+cos k·x)dx. (5.4)\nThe form (4.39) of I[µ] shows (cf. also Proposition 9.1) that I[µk] =v(0)/2 +v(k)/4 which must be\nnonnegative if uis stable.\n(ii) Let min |k|≥k0v(k) =v(km). Choose Λ and a Bravais lattice B(lattice, for mathematicians) such\nthat the reciprocal lattice B∗(dual lattice, defined by k·r∈2πZfork∈B∗andr∈B) is close-\npacked,B∗⊂Λ∗and the nearest neighbor distance of B∗is|km|. Then, for ( r)N=B∩Λ, by a simple\ncomputation\nI[µB]≤1\n2[v(0)−nd|v(km)|], (5.5)\n18and the left member is nonnegative if uis stable. Here I[µB] =I[µB∩Λ] which is independent of Λ\nprovided that Λ is a period cell for B, see also Eq. (9.11).\n(iii) Condition (5.3) is the dual of (5.2), obtained from the Poisson sum mation formula\n/summationdisplay\nr∈Bu(r) =ρ(B)/summationdisplay\nk∈B∗v(k) = 2ρ(B)I[µB]. (5.6)\nHereρ(B) denotes the density of B. Let min |r|≥r0u(r) =u(rm). Choose Bto be close-packed with\nnearest-neighbor distance |rm|. Then, in case of stability,\n0≤I[µB] =1\n2ρ(B)/summationdisplay\nr∈Bu(r)≤1\n2ρ(B)[u(0)−nd|u(rm)|] (5.7)\nwhich was to be shown. ✷\n6 Stationary points of the energy functional\nThe natural candidates for IDGS’s are the stationary points of I[µ]. To motivate the definition of\nstationary points, we first consider absolutely continuous normaliz ed measures,\ndµ(x) =f2(x)dx, (6.1)\nwherefis a real function. We have to minimize I[µ] with respect to funder the condition/integraltext\nΛf2(x)dx=\nV. Via functional derivation of I[µ] we obtain\nf(x)/bracketleftbigg/integraldisplay\nΛuΛ(x−y)f2(y)dy−c/bracketrightbigg\n= 0. (6.2)\nHerecis a Lagrange multiplier. Another way to write this equation is\n/integraldisplay\nΛuΛ(x−y)f2(y)dy=ciff(x)∝\\⌉}atio\\slash= 0. (6.3)\nAny absolutely continuous µwith a density f2solving Eq. (6.2) is a stationary point of the energy\nfunctional. Because with such an f\nI[µ] =1\n2V/integraldisplay\nΛ2uΛ(x−y)f2(y)f2(x)dydx=c\n2, (6.4)\namong the solutions of (6.2) one has to choose the one giving the sma llest constant c. A solution of (6.2)\nisf2(x)≡1 which defines the Lebesgue measure and yields c=v(0) and, hence, an already familiar\nupper bound I0≤v(0)/2, cf. Eqs. (2.19), (4.9). We shall see that the Lebesgue measure is the only\nabsolutely continuous measure among the stationary points of the functionalI. By generalizing (6.3) we\narrive at the following definition.\nDefinition 6.1 Aµ∈ MΛis a stationary point of If,µ∈ MΛst(f), if\n(f∗µ)(x) =/integraldisplay\nΛf(x−y)dµ(y) = constant forx∈suppµ. (6.5)\nµ∈ MΛis a stationary point (of I),µ∈ MΛst, if it is a stationary point of Iffor every continuous\nfunctionf.\n19Remarks. (i) One could ask f∗µto be constant only µ-almost everywhere, but the above choice is\nsufficient and more convenient for our purposes. (ii) The name is jus tified by the following property. Let\nµ∈ MΛst(f) and letν∈ MΛ,ν≪µ. Then\n/integraldisplay\nΛ(f∗µ)dµ=/integraldisplay\nΛ(f∗µ)dν, (6.6)\nand therefore\nIf[(1−ε)µ+εν]−If[µ] =ε2(If[ν]−If[µ]) (6.7)\nfor any 0 ≤ε≤1. Soµmay not minimize If, not even among the measures absolutely continuous with\nrespect to it, but if we perturb µwith aν≪µin the order of ε,Ifchanges only in the order of ε2.\nMΛst⊂ MΛst(f), and we shall give two examples showing that the first can be a prop er subset of\nthe second. MΛstis nonempty, the Lebesgue measure is a stationary point. Intuitive ly, it appears to be\nclear thatµis a stationary point if it is uniform on its support, and supp µ−x≡ {y−x|y∈suppµ}is\nthe same for all x∈suppµ. The theorem below provides the proof.\nFort∈Λ and a function gon Λ define Ttgby\nTtg(x) =g(x���t), (6.8)\nand forµ∈ MΛ,Ttµ∈ MΛby\nTtµ(g) =µ(T−tg). (6.9)\nThen for Borel sets D⊂Λ,\nTtµ(D) =Ttµ(1D) =µ(D−t). (6.10)\nLemma 6.1 Any translate of a stationary point of Ifis a stationary point of If.\nProof.Supposeµ∈ MΛst(f). LetA= suppµ, and (f∗µ)(x)≡cforx∈A. Then (f∗Ttµ)(x)≡cfor\nx∈A+tand, because A+t= suppTtµ, we conclude that Ttµ∈ MΛst(f).✷\nThe set Λ = LTdis an additive group, Λ −Λ = Λ. A closed subset Gof Λ is a closed additive\nsubgroup if G−G⊂G. A Haar measure µonGis a nonzero measure of support Gwhich is invariant\nunder shifts with elements of G; that is,Ttµ=µfor every t∈G,µis uniformly distributed on G. The\nnormalized Haar measure is unique.\nTheorem 6.1 Aµ∈ MΛis a stationary point if and only if it is a translate of the nor malized Haar\nmeasure on a closed additive subgroup of Λ.\nExamples. Some simple closed additive subgroups with their normalized Haar meas ures are\n{0}withVδ0, (6.11)\nΛ with λd (6.12)\nand /parenleftBig\nB(n)∩Λ(n)/parenrightBig\n×Λ(d−n)(6.13)\nwith\nµB(n)∩Λ×λd−n≡λn(Λ(n))\n|B(n)∩Λ(n)|/summationdisplay\nr∈B(n)∩Λ(n)δ(n)\nr×λd−n (6.14)\nwheren= 1,...,d,λnandδ(n)\nrare thendimensional Lebesgue and Dirac measures, respectively, Λ(n)is\nthe projection of Λ to the first ncoordinates and B(n)is an-dimensional Bravais lattice commensurate\nwith Λ(n). Any Bravais lattice can appear as a stationary point if we replace Λ = LTdby a suitable\nparallelepiped.\nIn the following lemma we give a full description of closed additive subgr oups of Λ and the Haar\nmeasures on them. Related results can be found in Ref. [27]. Recall t hat Λ∗= (2π/L)ZdandV=Ld.\n20Lemma 6.2 LetGbe a closed additive subgroup of Λwith normalized Haar measure µ(i.e.,µ(G) =V),\nand let\nG∗={k∈Λ∗|k·x∈2πZ,allx∈G}. (6.15)\nThenG∗is an additive subgroup of Λ∗, and/hatwideµis a Haar measure (the counting measure) on G∗,\n/hatwideµ(k) =/summationdisplay\nK∈G∗δk,K. (6.16)\nConversely, if G∗is an additive subgroup of Λ∗with the counting measure /hatwideµon it, then\nG={x∈Λ|k·x∈2πZ,allk∈G∗} (6.17)\nis a closed additive subgroup of Λ, and the measure µdefined by/hatwideµas its Fourier transform is a Haar\nmeasure on Gwithµ(G) =V.\nProof.The (closed) subgroup property is obvious in both cases. Let now Gbe a closed additive subgroup\nof Λ, either given initially or constructed as in (6.17), G∗the dual of G, andµthe normalized Haar\nmeasure on G. We show that /hatwideµis just (6.16). For any k∈Λ∗andy∈G,\n/hatwideµ(k) =1\nV/integraldisplay\nGe−ik·xdµ(x) =1\nV/integraldisplay\nGe−ik·xdµ(x+y)\n=1\nV/integraldisplay\nGe−ik·(x−y)dµ(x) =eik·y/hatwideµ(k). (6.18)\nThen either/hatwideµ(k) = 0 or k·y∈2πZfor ally∈G, that is, k∈G∗. In the second case /hatwideµ(k) = 1, proving\nthe claim. ✷\nThus, we have obtained a simple characterization of closed additive s ubgroups of the torus Λ as the\ndual sets, in the sense of Eq. (6.17), of intersections of linear sub spaces of Rdwith the lattice Λ∗. Indeed,\neach additive subgroup of Λ∗is such an intersection. We now see that Λ is the only closed additive\nsubgroup of positive Lebesgue measure, therefore the Lebesgu e measure is the only absolutely continuous\nstationary point of I.\nProof of Theorem 6.1. 1. Letµbe the normalized Haar measure on an additive subgroup G, extended\nto Λ withµ(Λ\\G) = 0. For arbitrary Borel sets D1⊂GandD2⊂Λ\\G, and any x∈G,\nµ(D1−x) =µ(D1), µ(D2−x) =µ(D2) = 0.\nHence, for any Borel set D⊂Λ and any x∈G,µ(D−x) =µ(D) or, equivalently,\n(1D∗µ)(x) =µ(D) for any x∈G. (6.19)\nBecause any continuous function is the limit of a sequence of finite line ar combinations of characteristic\nfunctions, (6.5) holds for every continuous f; hence,µ∈ MΛst.\n2. Suppose that µ∈ MΛstand letA= suppµ. Although the characteristic functions are discontinu-\nous, Eq. (6.5) extends to them through limits. For example, the cha racteristic functions of open sets are\nlimits of monotone increasing sequences of continuous functions. F or an arbitrary Borel set D⊂Λ,\nµ(D+x)≡λDfor anyx∈A (6.20)\nis then obtained by using the upper regularity of µ.\n(i) Suppose that 0∈A. Thenµ(D+x) =µ(D) for allD⊂Λ Borel sets and all x∈A; in particular,\nforD=A−x,\nµ(A−x) =µ(A) =µ(A∩A−x) =V (6.21)\nfor allx∈A. SinceA−xis closed together with A, necessarily A−x=Afor allx∈Aand, hence,\nA−A=A. Thus,A= suppµis an additive subgroup of Λ and µis a shift-invariant measure on it.\n(ii) If0/∈A, choose any t∈ −Aand replace AbyA+tandµbyTtµ. Because Ttµ∈ MΛst, the\nargument (i) yields that Ttµis the normalized Haar measure on the additive group A+t, andµis a\ntranslate of this measure. ✷\n21Corollary 6.1 Letµ,ν∈ MΛbe two measures with coinciding supports. If both are statio nary points\nthenµ=ν.\nProof.suppµ= suppνis an additive group and µandνare normalized Haar measures on it. But the\nnormalized Haar measure is unique, whence µ=ν.✷\nRemarks. (i) Here are two examples showing that MΛst/subsetornotdbleqlMΛst(uΛ).\n1. A two-dimensional example is the Dirac comb µHconcentrated on the honeycomb lattice. It is not a\nstationary point because, if x1andx2are nearest neighbors, supp µH−x2=−suppµH+x1. However,\nuΛ(−x) =uΛ(x) impliesµH∈ MΛst(uΛ).\n2. Ifuis radial and Λ = [ −L/2,L/2]d,uΛhas a cubic symmetry. Then for any 0 <a≤L/4\nµ=V\n2dd/summationdisplay\nj=1/parenleftbig\nδ−aej+δaej/parenrightbig\n(6.22)\nis stationary for uΛ, but not for functions not having a cubic symmetry.\n(ii) If the periodic boundary condition is replaced by a free one, the s et of stationary points changes\ncompletely. The reason is that now uitself, and not the periodized uΛ, appears in the energy functional;\nin fact, Λ may not be a parallelepiped. The additive groups are replace d by others. For a radial uthe\nstationary points of Iuare uniform distributions on 1, 2,..., d-1 dimensional spheres, on the v ertices of\nregular polyhedra, in 3 dimensions also on the 60 vertices of the socc er ball (fullerene), and others. The\nsupports are now sets that are invariant under transformations forming finite point groups or groups\nof rotation, and the stationary points correspond to Haar measu res on those groups. As Λ increases,\nthe dependence of the ground states on the boundary condition s hould continuously disappear in the\nbulk. For example, on parallelepipeds there should be a continuous tr ansition between IDGS’s for free\nand periodic boundary conditions. Such a transition cannot take pla ce on stationary points for both\ntypes of boundary conditions. Intuition suggests that stationar y points have better chance with periodic\nboundary conditions.\n(iii) Ifuis integrable but u(x)→ ∞asx→0thenI[µ] =∞for anyµhaving a point part. However,\ndepending on the rate of divergence of u(x),I[µ] can be finite in some continuous stationary points, and\nthese are to be considered as possible IDGS’s. If u(x)∼ |x|−d+ζnear the origin with 0 < ζ < d then\nuis integrable in d−ndimensions for n < ζ, so the energy functional is finite on the corresponding\nmeasures (6.14). Some interactions not allowing stationary points o fIto be IDGS’s will be presented in\nTheorem 11.2. If 0 <ζ <1,I[µ] can be finite only if the Hausdorff dimension of supp µis greater than\nd−ζ >d−1. According to our characterization given in Lemma 6.2, the only sta tionary point providing\na finite energy for such an interaction is the Lebesgue measure. In Section 9 we will show, however, that\nthe Lebesgue measure does not minimize Iif the Fourier transform of uis partly negative. Thus, for\nu∈L1(Rd) diverging at zero as |x|−d+ζwith 0<ζ <1 and having a partly negative Fourier transform,\nno stationary point is an IDGS. Theorem 11.2 will cover this case.\n7 Ground state configurations in infinite space\nFrom Refs. [1, 2] let us recall the definition of a ground state config uration in Rd. To begin with, a\nconfiguration is an equivalence class of point sequences, two seque nces being equivalent if they differ\nonly in a permutation of the entries. We must consider sequences ins tead of sets because particles can\nbe placed in the same point if the interaction is bounded. For an infinite configuration Xand a finite\nconfiguration Rlet\nU(R|X) =U(R)+I(R,X) (7.1)\nwith\nU(R) =/summationdisplay\n(r,r′)⊂Ru(r−r′), I(R,X) =/summationdisplay\nr∈R,x∈Xu(r−x). (7.2)\n22Definition 7.1 Letube a strongly tempered interaction. Given a real number µ, an infinite configuration\nXis a (grand canonical) ground state configuration for chemic al potential µ(aµGSC) if for any finite\nconfiguration Xf⊂X,U(Xf)is finite, the sum I(Xf,X\\Xf)is absolutely convergent, and for any finite\nconfiguration R⊂Rd,\nU(R|X\\Xf)−µ|R| ≥U(Xf|X\\Xf)−µ|Xf|. (7.3)\nXis a (canonical) ground state configuration (GSC) if (7.3) ho lds true for every Rsuch that |R|=|Xf|.\nThus, a ground state configuration is a locally stable configuration, in the sense that no local modification\ncan decrease its energy. Equation (7.3) must hold also when RorXfis the empty set. Hence, if Xis a\nµGSC then\nU(Xf|X\\Xf)−µ|Xf| ≤0\nfor any finite Xf⊂X. Ifuis not strongly tempered, absolute convergence of I(Xf,X\\Xf) must be\nreplaced by some weaker condition, e.g. Abel summability, see [2]. The above definition and the two\nlemmas below apply also to interactions that are not integrable at the origin or have a hard core.\nThe existence of GSC’s and µGSC’s is by no means obvious. For certain interactions they can be\nconstructed explicitly [1, 2, 10, 30, 31], for some others only their e xistence is proven [32, 33]. In this\npaper we assume that for the class of interactions considered, hig h-density GSC’s do exist; at least, all\nstatements about GSC’s in infinite space are subject to this proviso .\nDefinition 7.2 Forx∈Rd, letCxdenote the unit ball centered at x. An infinite configuration Xis\nlocally uniformly finite (l.u.f.), if there exists an intege rmsuch that for any x∈Rd,\n|X∩Cx| ≤m. (7.4)\nEquivalently, for any compact K∈Rdthere is an integer mKsuch that\n|X∩(K+x)| ≤mK,anyx∈Rd. (7.5)\nIn [2] we proved the absence of metastability in the case of strongly tempered interactions, by showing\nthat a necessary condition for a l.u.f. configuration to be a GSC is to m inimize the energy density among\nl.u.f. configurations of the same density. Here we prove that this co ndition is also sufficient among\nperiodic arrangements (while it is obviously insufficient among l.u.f. confi gurations).\nThe general form of a periodic configuration is\nX=J/unionmultidisplay\nj=1(B+yj), (7.6)\nwhereB=Za1+···+Zad,ai∈Rdarelinearlyindependent, yjarenotnecessarilydifferent d-dimensional\nvectors, and in this paper we use/unionmultitextto indicate that in the union coinciding points occur with repetition.\nThe choice of Bcan be made unique by asking Jto be minimal. Bthus obtained is the maximal group\nof period vectors of X, and is denoted by B(X).\nTheparticledensity andthe energyperparticleofaninfinite configu rationXaredefined, respectively,\nas\nρ(X) = lim\nΩ→Rd|X∩Ω|\nV(Ω), ep(X) = lim\nΩ→RdU(X∩Ω)\n|X∩Ω|, (7.7)\nprovided that the limits exist. Here Ω is a bounded Lebesgue-measur able domain and V(Ω) is its volume.\nThe energy density (energy per volume) of Xise(X) =ρ(X)ep(X). If the particle density is kept fixed,\nwe can work with any of epore. IfXis a periodic configuration, then ρ(X) =|X∩Λ|/V(Λ) where Λ is\nany period-parallelepiped. If, moreover, uis strongly tempered, cf. Eq. (2.2), and Ω tends to Rdin the\nVan Hove sense [7], then\nep(X) =U(X0)\n|X0|+1\n2|X0|∞/summationdisplay\nm=1I(X0,Xm) =U(X0)\n|X0|+I(X0,X\\X0)\n2|X0|. (7.8)\n23HereXm(m= 0,1,...) are non-overlapping translates of any finite X0⊂Xsuch thatX=∪∞\nm=0Xm.\nThe simplest choice is X0= (y1,...,yJ), cf. (7.6). However, X0can be the content of an arbitrarily\nlarge period cell, and then comparison of (7.8) and (7.7) shows that\nI(X0,X\\X0) =o(|X0|). (7.9)\nSinceuis superstable, for some b≥0 andρ(X)>b/C[u] we have\n|X0| ≤U(X0)\n−b+C[u]ρ(X)(7.10)\nand, consequently,\nlim\n|X0|→∞I(X0,X\\X0)\nU(X0)= 0. (7.11)\nThe first part of the following lemma was proven by Sinai for a lattice g as [34] and by Radin in one\ndimensional continuous space [35].\nLemma 7.1 Let the interaction be strongly tempered, and let Xbe a periodic configuration. (i) If X\nminimizes the energy density among periodic configurations of the same density and of period vectors\nbelonging to B(X), thenXis a GSC. (ii) If Xis a GSC then it minimizes the energy density among\nl.u.f. configurations of the same density (provided that the respective densities exist).\nProof.(i) Suppose that Xis not a GSC; then there exist Xf⊂XandR⊂Rdfinite sequences such that\nR∩X=∅,|R|=|Xf|, and\n∆ =U(R|X\\Xf)−U(Xf|X\\Xf)<0. (7.12)\nWe construct a periodic Ywhose period vectors form a subgroup of those of X, such that ρ(Y) =ρ(X)\nandep(Y)<ep(X). For an odd integer n, let\nX0=X∩Λn,Λn=/braceleftBiggd/summationdisplay\ni=1xiai:−n−1\n2≤xi<n−1\n2/bracerightBigg\n. (7.13)\nThis is the union of the content of ndunit cells of X. DefineY0=X0\\Xf∪R. Choosenso large that\nXf∪R⊂Λn. LetYbe the periodic extension of Y0,\nY=/uniondisplay\nt∈nB(Y0+t) =/unionmultidisplay\ny∈Y0(nB+y), (7.14)\nto be compared with\nX=/uniondisplay\nt∈nB(X0+t) =/unionmultidisplay\nx∈X0(nB+x). (7.15)\nIt is straightforward to verify that\n∆ =U(Y0|X\\X0)−U(X0|X\\X0). (7.16)\nClearly,ρ(X) =ρ(Y) and from (7.8) we find\n|X0|[ep(Y)−ep(X)] = ∆+1\n2[I(Y0,Y\\Y0)−I(Y0,X\\X0)]+1\n2[I(X0,X\\X0)−I(Y0,X\\X0)].(7.17)\nNowX0\\Xf=Y0\\Rimplies\nI(X0,X\\X0)−I(Y0,X\\X0) =I(Xf,X\\X0)−I(R,X\\X0), (7.18)\n24and, usingI(A,B+t) =I(B,A−t),\nI(Y0,Y\\Y0)−I(Y0,X\\X0) =I(R,Y\\Y0)−I(Xf,Y\\Y0). (7.19)\nThe four infinite sums in the right member of the last two equations ar e absolutely convergent and tend\nto zero asntends to infinity, because the distances of RandXftoX\\X0and toY\\Y0diverge with\nn. Thus, ifnis large enough, ep(Y)<ep(X).\n(ii) Earlier it was shown (Proposition, Ref. [2]) that if XandYare two l.u.f. configurations of equal\ndensity and ep(X)>ep(Y), thenXis not a GSC. Since periodic configurations are l.u.f., the second part\nof the claim follows. ✷\nFor a general parallelepiped Λ spanned by the vectors li(i= 1,...,d), let\nuΛ(r) =/summationdisplay\nt∈BΛu(r+t) (7.20)\nwhere\nBΛ=/braceleftBiggd/summationdisplay\ni=1nili:n1,...,n d∈Z/bracerightBigg\n. (7.21)\nLemma 7.2 Letube strongly tempered and let Xbe a periodic configuration. (i) If Xis a GSC of u,\nthenX∩Λis a ground state of uΛon any period parallelepiped ΛofX. (ii) IfX∩Λis a ground state\nofuΛon any large enough period parallelepiped ΛofX, thenXis a GSC of u.\nProof.(i) IfXis a GSC, then it minimizes epamong periodic configurations of density ρ(X). In\nparticular, if R⊂Λ,|R|=|X∩Λ|andYis the periodic extension of Rfrom Λ, then ep(X)≤ep(Y).\nComparison with Eq. (7.8) written for a periodic configuration Zof period cell Λ and X0=Z∩Λ shows\nthat\neΛ\np(Z∩Λ) :=UΛ(Z∩Λ)\n|Z∩Λ|=ep(Z)−1\n2/summationdisplay\n0/ne}ationslash=r∈BΛu(r). (7.22)\nApplying (7.22) with Z=XandY, one finds\neΛ\np(R)−eΛ\np(X∩Λ) =ep(Y)−ep(X)≥0. (7.23)\n(ii) We prove that if Yis a periodic configuration, B(Y)⊂B(X) andρ(Y) =ρ(X), thenep(Y)≥ep(X).\nLet Λ be a period cell of Y(and, hence, of X) chosen so large that X∩Λ is a ground state of uΛ. Then\ninequality (7.23) holds true with R=Y∩Λ.✷\n8 Infinite-density ground state in infinite space\nLetMdenote the family of (positive) Borel measures on Rd.\nDefinition 8.1 LetXnbe a sequence of infinite configurations with existing densit yρnand energy per\nparticleep(Xn)which tend to infinity with n. Suppose that\nlim\nn→∞ep(Xn)/ρn=C[u] (8.1)\nand there is some nonzero µ∈ Msuch that\nµXn=ρ−1\nn/summationdisplay\nx∈Xnδx⇀µ. (8.2)\nThenµis called an infinite-density ground state in Rd.\n25In contrast with IDGS’s in finite volumes, the existence of IDGS’s in infi nite space is not a priori\nguearanteed. The ultimate goal would be to prove the following comm utative diagram:\nConjecture 8.1\nGSC inΛΛ→∞− −−− →GSC inRd\nρ→∞/arrowbt/arrowbtρ→∞\nIDGS in Λ− −−− →\nΛ→∞IDGS in Rd(8.3)\nIfuis superstable and strongly tempered then high-density GSC ’s and IDGS’s in infinite space exist. Any\nIDGS in Rdcan be obtained as the vague limit of both a sequence of period ic extensions of IDGS’s in\nincreasing volumes and a sequence of Dirac combs (8.2) assoc iated with GSC’s of increasing density. For\nany high-density GSC X, the associated measure µXis the vague limit of a sequence µYn, whereYnis\nthe periodic extension of a ground state of uΛnin some parallelepiped Λn, andρ(Yn) =ρ(X).\nIf the conjecture holds true then information on GSC’s in Rdcan be obtained by studying IDGS’s in\nfinite volume. In the case when v≥0 and of compact support the results of Refs. [1, 2] and those of\nSection 9 below prove the conjecture but nothing new can be obtain ed about GSC’s. If v≥0 but its\nsupport is noncompact, the existence of IDGS’s in infinite space still follows from CΛ[u]≡C[u] =v(0)/2;\nthe Lebesgue measure is one of the IDGS’s (the only one if v >0) and Section 9 gives a fairly complete\ndescription of them, cf. Corollary 9.1. Concerning the general cas e, below we present some clarification\nand partial result. Let Λ be any parallelepiped centered at zero,\nΛ =/braceleftBiggd/summationdisplay\ni=1xili:−1/2≤xi<1/2,i= 1,...,d/bracerightBigg\n(8.4)\nwhereliare linearly independent vectors. If nis a positive integer, then nΛ ={nx:x∈Λ}is the union\nofndtranslates of Λ:\nnΛ =nd/uniondisplay\nj=1(Λ+tj) (8.5)\nwhere\ntj=1\n2d/summationdisplay\ni=1mjili, mj1,mj2,...,m jd∈ {−n+1,−n+3,...,n−3,n−1}. (8.6)\nDefinition 8.2 Letnbe a positive integer. The periodic extension EnfromMΛtoMnΛis a map\nassigning\nEnµ=nd/summationdisplay\nj=11Λ+tjTtjµ (8.7)\ntoµ∈ MΛ; that, is,\nEnµ(D) =µ(D−tj), j= 1,...,nd(8.8)\nfor Borel sets D⊂Λ+tj. The periodic extension EfromMΛtoMis a map assigning\nEµ=/summationdisplay\nt∈BΛ1Λ+tTtµ (8.9)\ntoµ∈ MΛ.\nLemma 8.1 Eµis the vague limit of E2n+1µasntends to infinity.\n26Proof.Letf∈Cc(Rd). Comparing (8.9) with (8.7) one obtains\nEµ(f) = lim\nn→∞E2n+1µ(f).✷ (8.10)\nRecall that Λ∗={/summationtextnil′\ni:ni∈Z,i= 1,...,d}, wherel′\ni·lj= 2πδij. Clearly,\nΛ∗⊂(nΛ)∗=n−1Λ∗. (8.11)\nLemma 8.2 Ifnis a positive integer and k∈Λ∗then\n/hatwidestEnµ(k) =/braceleftbigg/hatwideµ(k)ifnis odd\ns(k)/hatwideµ(k)ifnis even(8.12)\nwheres(k)∈ {−1,1}. Ifk∈(nΛ)∗\\Λ∗then/hatwidestEnµ(k) = 0. Furthermore,\n/hatwiderEµ(k) = 1Λ∗(k)/hatwideµ(k),k∈Rd. (8.13)\nProof.Using the definition of Enµand (4.40),\n/hatwidestEnµ(k) =1\nndV(Λ)/integraldisplay\nnΛe−ik·xEnµ(dx) =1\nndV(Λ)nd/summationdisplay\nj=1/integraldisplay\nΛe−ik·(x+tj)µ(dx) =1\nndnd/summationdisplay\nj=1e−ik·tj/hatwideµ(k).(8.14)\nThe average of the complex units is 1, ±1, or 0, depending on the case. Taking the limit n→ ∞gives\nthe result for /hatwiderEµ(k).✷\nBelow we use the notations IΛ[µ] andI0\nΛfor the energy functional (4.5) and its infimum (4.8), and\nǫΛ(r)N,ǫΛ,N,ωΛ(r)NandωΛ,Nfor (2.4), (2.5), (2.22) and (2.23), respectively, to indicate the de pendence\non Λ. Expanding the sum that defines uΛ,\nIΛ[µ] =1\n2/integraldisplay\nRdu(r)dEγΛ(r) =1\n2EγΛ(u) (8.15)\nwhereγΛ=µ∗/tildewideµ/Vforµ∈ MΛ.\nProposition 8.1 For any positive integer n,CnmΛ[u](m= 1,2,...)is a monotone decreasing sequence.\nProof.Let Λm=nmΛ. From (8.15) or from Lemma 8.2 and the form (4.39) of the energy f unctional, it\nfollows that for any µ∈ MΛm\nIΛm+1[Enµ] =IΛm[µ]. (8.16)\nBy Theorems 4.1 and 4.2 then\nCΛm+1[u] =I0\nΛm+1≤I0\nΛm=CΛm[u].✷ (8.17)\nFor a stable interaction CΛ[u]≥0 for all Λ, therefore the limit of CnmΛ[u] asmtends to infinity exists\nand is bounded below by the best superstability constant C[u]. As noted earlier, the limit must, in fact,\nbeC[u]. Now we prove a different form of C[u].\nProposition 8.2 Ifuis bounded then\nC[u] = lim\nρ→∞1\nρinf{ep(X) :Xperiodic,ρ(X) =ρ}. (8.18)\n27Remark. Here we have not supposed that there are periodic GSCs for arbitr arily high densities. If this is\nthe case, the infimum in (8.18) is attained, and C[u] = lim n→∞ep(Xn)/ρ(Xn) whereXnis any sequence\nof periodic GSCs with ρ(Xn) tending to infinity.\nProof.(i)Firstwefindasequence Xnofperiodicconfigurationssuchthat ρ(Xn)divergesand ep(Xn)/ρ(Xn)\ntends toC[u]. Let Λ nbe any sequence of parallelepipeds such that CΛn[u] converges to C[u]. For each\nnletRnm(m= 1,2,...) be a sequence of ground state configurations in Λ n,|Rnm|going to infinity with\nm. According to Equation (3.5), lim m→∞ǫΛn(Rnm) =CΛn[u]. Therefore, one can choose a subsequence\nRnmn≡Rnsuch that lim ǫΛn(Rn) = limωΛn(Rn) =C[u]. LetXndenote the periodic extension of Rn.\nThus,Rn=Xn∩Λnand|Rn|/V(Λn) =ρ(Xn)→ ∞. From Eq. (7.22), for any periodic configuration X\nof period parallelepiped Λ\nep(X)\nρ(X)=|X∩Λ|−1\n|X∩Λ|ǫΛ(X∩Λ)+1\nρ(X)/summationdisplay\n0/ne}ationslash=r∈BΛu(r) =ωΛ(X∩Λ)−u(0)\n2ρ(X).(8.19)\nApplying this formula to Xnand Λ n, we obtain lim n→∞ep(Xn)/ρ(Xn) =C[u].\n(ii) Suppose there is a sequence Ynof periodic configurations with respective period cells Λ nsuch that\nρ(Yn) tends to infinity and\nlim\nn→∞ep(Yn)\nρ(Yn)=C0<C[u].\nWe can always choose the sequence Λ nso that it tends to infinity in Fisher’s sense. We may also suppose\nthatYn∩Λnis a ground state of uΛn, otherwise we can replace Ynby the periodic extension of an\n|Yn∩Λn|-particle ground state of uΛn, cf. Eq. (7.23). For any ε >0 there is a Nsuch that for any\nn≥N,/vextendsingle/vextendsingle/vextendsingle/vextendsingleep(Yn)\nρ(Yn)−C0/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ε\n3,/vextendsingle/vextendsingle/vextendsingle/vextendsingleǫΛn(Yn∩Λ)−ep(Yn)\nρ(Yn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ε\n3, CΛn[u]−ǫΛn,|Yn∩Λ|≤ε\n3.\nThe second and third inequalities use Eq. (8.19) and Corollary 3.1, res pectively. By assumption,\nǫΛn(Yn∩Λ) =ǫΛn,|Yn∩Λ|,\ntherefore\n|CΛn[u]−C0| ≤εifn≥N.\nThus,CΛn[u] converges to C0<C[u], but this contradicts the hypothesis (2.14). ✷\nRemarks. (i) Without the hypothesis (2.14), from (2.9) we still find that equat ion (8.18) holds true if\nlimρ→∞is replaced by liminf ρ→∞. (ii) The first equality of Equation (8.19) applies also to unbounded\ninteractions. Equation (8.18) is presumably valid in this case as well, bu t in the absence of a proof of\nCorollary 3.1 for unbounded interactions we cannot prove it.\nFor the sequence Xnused in the proof of Proposition 8.2, Eq. (8.1) holds true. However, the proof of\nEq. (8.2) is missing. We may suppose that Λ n= (2ln+1)Q0, whereQ0is the unit cube centered at 0and\nthe integers lntend to infinity with n.Xncan be selected so that |Xn∩Q0|= max t∈Rd|(Xn+t)∩Q0|,\nthenµXn(Q0)≥V(Q0) = 1 because µXn(Λn) =V(Λn). The task would be to prove that µXn(Q0) is a\nbounded sequence. If this holds, one could find a vaguely converge nt subsequence of µXntending to a\nnonzero measure which is by definition an IDGS in Rd. The same IDGS in Rdcould also be obtained as\nthe vague limit of a sequence of IDGS’s µn∈ MΛnchosen such that µn(Q0)≥V(Q0).\nLetGbe a locally compact Abelian group (in this paper Gis a torus or G=Rd). A measure µonG\nis called shift-bounded if for any compact A⊂Gthere is a constant αAsuch that for all a∈G,\nµ(A+a)≤αA. (8.20)\nSuppose again that Λ nis a sequence of increasing parallelepipeds such that CΛn[u] converges to C[u].\nFor anynletµnbe an IDGS in Λ n. Hence, with the notation γn=V(Λn)−1µn∗/tildewiderµn,\n1\n2lim\nn→∞γn(uΛn) =1\n2lim\nn→∞Eγn(u) =1\n2lim\nn→∞/hatwidestEγn(v) =1\n2lim\nn→∞/hatwiderγn(v) =C[u]. (8.21)\n28Eγnis periodic and, thus, shift-bounded. /hatwidestEγnis a positive measure which is the Fourier transform\nof a measure, therefore it is also shift-bounded, see [27] (Propos ition 4.9) or [28] (Proposition 3.3). A\nnecessary condition for (8.2) to hold is that these sequences are u niformly shift-bounded. In a special\ncase the proof is given in the following lemma.\nLemma 8.3 Ifu(x)≥ce−α|x|2for somec,α >0, then both sequences Eγnand/hatwidestEγnare uniformly\nshift-bounded.\nProof.LetAbe a unit cube. If g(k) =Ke−|k|2/4αwithK=/parenleftBig/integraltext1\n0e−q2/4αdq/parenrightBig−d\nthen 1 A∗g≥1A.˘f\ndenoting the inverse Fourier transform of f,\n/summationdisplay\nk∈Λ∗n∩A+a|/hatwiderµn(k)|2=/hatwidestEγn(A+a)≤/hatwidestEγn(1A+a∗g) =Eγn(˘1A+a˘g)≤Eγn(|˘1A+a|˘g)\n≤K\n(2π)d(π/α)d/2/integraldisplay\ne−α|x|2dEγn(x)≤β/integraldisplay\nu(x)dEγn(x)≤βC[u] (8.22)\nfornsufficiently large ( β=K\nc(2π)d(π/α)d/2). The passageto any compact Ais obvious, so /hatwidestEγnis uniformly\nshift-bounded. From (8.22) or with a similar independent argument o ne can show that for any Schwartz\nfunctionf,/hatwidestEγn(f)(n= 1,2,...) is a bounded sequence. We use this fact to prove that Eγnis uniformly\nshift-bounded. Let again Abe a unit cube and g(x) as before; then there is some constant cgsuch that\nEγn(A+a)≤Eγn(1A+a∗g) =/hatwidestEγn(/hatwide1A+a/hatwideg)≤/hatwidestEγn(|/hatwider1A|/hatwideg)≤/hatwidestEγn(/hatwideg)≤cgalln.✷(8.23)\nAn example to this lemma is the Gaussian pair potential for which, as me ntioned already, the unique\nIDGS is the Lebesgue measure. Other examples are pair potentials w ith a Gaussian lower bound and a\npartly negative Fourier transform – the latter condition holds, for example, if ∆ u(0) = 0, cf. Eq. (11.18).\nUnfortunately, uniform shift-boundedness of the sequences Eγnand/hatwidestEγndoes not allow one to conclude\nthat the sequence µnis uniformly shift-bounded. Unbounded but o(/radicalbig\nV(Λn)) weight rearrangements of\nµndo not affect uniform shift-boundedness of Eγn.\n9 Infinite-density ground states for v≥0\nThe most complete information about IDGS’s can be obtained when th e Fourier transform vof the\ninteraction is nonnegative. The minimum of I[µ] is attained on any µnot contributing to the sum in\nEq. (4.39), and its value is I0=v(0)/2, thus,CΛ[u] =v(0)/2 =C[u] for every Λ.\nProposition 9.1 The Lebesgue measure λis an IDGS in Λif and only if v≥0, and then it yields\nI[λ] =C[u] =v(0)/2. (9.1)\nIfv>0, the Lebesgue measure is the unique IDGS.\nProof.Fork∈Λ∗,\n/hatwideλ(k) =δk,0, (9.2)\nandλis the only element of MΛwith Fourier transform δk,0. Soλis an IDGS, and is the only one if v\nis strictly positive. It remains to prove that the Lebesgue measure is not an IDGS if vis partly negative.\nIn this case consider\ndµq(x) = (1+cos q·x)dx (9.3)\nand choose q∈Λ∗such thatv(q) = min k∈Λ∗v(k)<0. Then\n/hatwiderµq(k) =δk,0+1\n2(δk,q+δk,−q) (9.4)\n29and, recalling that v(−k) =v(k),\nI0≤I[µq] =v(0)\n2+v(q)\n4<v(0)\n2=I[λ], (9.5)\nsoλis not an IDGS. ✷\nLet us make a small detour here to explain the choice of µq. The energy functional I[µ] is the diagonal\npart of a quadratic form on MΛ. The operator underlying it has nice properties.\nProposition 9.2 Letube an integrable even pair potential, u(−x) =u(x). OnL2(Λ)define an integral\noperatorAby setting\n(Aφ)(x) =/integraldisplay\nΛuΛ(x−y)φ(y)dy. (9.6)\nThenAis a bounded self-adjoint operator with eigenvalues {v(k) :k∈Λ∗}and eigenvectors ψk(x) =\nexp(ik·x).\nProof.Ifk∈Λ∗then\nv(k) =/integraldisplay\nu(x)e−ik·xdx=/integraldisplay\nΛuΛ(x)e−ik·xdx. (9.7)\nSubstituting φ=ψkinto Eq. (9.6) gives Aψk=v(k)ψk.✷\nThus,µqis an absolutely continuous measure composed of the lowest-lying re al eigenvector of Athat we\nmust mix with the constant eigenvector in order to make the sum non negative.\nIfv≥0 and takes on the zero value, besides λmany other IDGS’s appear. We consider two specific\ncases.\nProposition 9.3 Suppose that v≥0andv(k) = 0for|k| ≥K0, whereK0<∞. LetΛbe a paral-\nlelepiped,Bany Bravais lattice such that Λis a period cell for B, and letqB∗denote the nearest-neighbor\ndistance of B∗, the reciprocal lattice of B. IfqB∗≥K0then\nµB∩Λ=V\n|B∩Λ|/summationdisplay\nx∈B∩Λδx (9.8)\nis an IDGS.\nRemarks. (i) For the finite point configurations associated with µB∩Λ, the proposition partly repeats the\nresults of [1]. It provides also an example to Proposition 4.2. (ii) µB∩Λextends periodically to Rdinto\nµB=ρ(B)−1/summationdisplay\nx∈Bδx, (9.9)\nand the energy functional shown below actually depends only on µB.\nProof.ForµB∩Λgiven by (9.8) and for k∈Λ∗,\n/hatwideµB∩Λ(k) = 1B∗(k) =/summationdisplay\nK∈B∗δk,K. (9.10)\nThis implies\nI[µB∩Λ] =1\n2\nv(0)+/summationdisplay\n0/ne}ationslash=K∈B∗v(K)\n≡I[µB] (9.11)\nTheshortestnonzerovectorin B∗hasalength qB∗, atandabovewhich v(k) vanishes,therefore I[µB∩Λ] =\nv(0)/2 =I0.✷\n30The second special case is an interaction with a “soft mode”, v(k) = 0 for |k|=k0andvis positive\notherwise. One may think of the example\nv(k) =c1e−c2|k|(|k|−k0)2(9.12)\nwith some positive constants c1,c2,k0, but the precise form of vplays no role. Choose Λ so that Λ∗\ncontains a vector qof lengthk0; then there are at least two vectors, ±qof lengthk0. The following\nproposition is an immediate consequence of the discussion above. Th e measureµqappearing in it is an\nexample for an IDGS that is not a stationary point of I.\nProposition 9.4 Ifv(k)≥0andv(k) = 0for|k|=k0, the measure µqof Eq. (9.3) with |q|=k0is an\nIDGS.\nA particularity of the case v≥0 is that convex combinations of IDGS’s are also IDGS’s.\nLemma 9.1 Ifv≥0thenI[µ]is a convex functional on MΛ, that is, if µ,ν∈ MΛand0<α,β < 1,\nα+β= 1then\nI[αµ+βν]≤αI[µ]+βI[ν]. (9.13)\nProof.This immediately follows from the form (4.39) of I[µ] and the convexity of |z|2on the complex\nplane:\n|αz1+βz2|2≤(α|z1|+β|z2|)2≤α|z1|2+β|z2|2.✷ (9.14)\nProposition 9.5 Ifv≥0, the set of IDGS’s is convex.\nProof.Letµ,ν∈ MΛbe two IDGS’s and 0 <α,β < 1,α+β= 1.\nI0≤I[αµ+βν]≤αI[µ]+βI[ν] =v(0)/2 =I0, (9.15)\nthereforeαµ+βνis an IDGS. ✷\nCorollary 9.1 Ifv≥0then the periodic extension of any IDGS from any parallelepi pedΛis an IDGS\ninRd. IDGS’s in Rdform a convex set. If v>0, the Lebesgue measure is the unique IDGS in Rd.\n10 Uniformity vs nonuniformity of high-density ground stat es\nProposition 9.1 has an immediate implication on the distribution of partic les in high-density ground\nstates in Λ. If µis an IDGS and Rmis a sequence of Nm-particle ground states in Λ such that µRm\nweakly converges to µ, then\nlim\nm→∞|Rm∩D|\nρ(Rm)=µ(D) (10.1)\nfor any open D⊂Λ, whereρ(Rm) =Nm/V. Equation (10.1) is a direct consequence of the definition of\nan IDGS. If v >0 thenλis the unique IDGS. Thus, for any sequence RNofN-particle ground states\nµRNconverges weakly to the Lebesgue measure λ, resulting\nlim\nN→∞|RN∩D|\nρ(RN)=λ(D). (10.2)\nOn the other hand, if vis partly negative, the asymptotic distribution of particles in any gro und state is\nnecessarily inhomogeneous. We shall prove an analogous result for ground state configurations in infinite\nspace.\nIn Refs. [1, 2] we gave a presumably complete description of defect -free high-density GSC’s of inter-\nactions with a nonnegative Fourier transform of compact support . If the Fourier transform is strictly\npositive, a prominent example of which is the Gaussian pair potential, a rigorous identification of GSC’s\n31is still missing and, if the decorrelation conjecture of Torquato and Stillinger [13] is valid, the task is near\nto impossible in high dimensions. One is then reduced to some probabilist ic description, and such an ap-\nproach may be helpful already for d<8. According to another conjecture of Torquato and Stillinger [14],\nfor the Gaussian potential at d≤8 the close-packed Bravais lattice and its reciprocal lattice should b e\nthe unique GSC at low and high densities, respectively. However, this conjecture was disproved by Cohn\nand Kumar [15], who found uniform close-packed periodic structure s of lower energy at low densities\nin 5 and 7 dimensions. The result below shows the fundamental differe nce between interactions with a\nstrictly positive or a partly negative Fourier transform. It may also be helpful in the numerical search\nfor low-energy arrangements at high densities.\nTheorem 10.1 For arbitrary d≥1, letube a bounded, integrable, strongly tempered pair potential .\nSuppose that there exists a sequence of periodic GSC’s Xnof respective densities ρntending to infinity,\nand let\nµXn=ρ−1\nn/summationdisplay\nx∈Xnδx. (10.3)\nWriteXnin the form (7.6),\nXn=Jn/unionmultidisplay\nj=1(Bn+yj), (10.4)\nwhereBn=B(Xn) =/summationtextd\ni=1Zani.\n(i) Let the Fourier transform vof the potential be strictly positive. Suppose that the sequ ence of lattice\nconstants |ani|(i= 1,...,d;n= 1,2,...)is bounded. Then µXnconverges to the Lebesgue measure in\ndistribution sense, that is, for any Schwartz function f:Rd→C,\nlim\nn→∞/integraldisplay\nf(x)dµXn(x) =/integraldisplay\nf(x)dx. (10.5)\n(ii) Ifvis partly negative, then (10.5) does not hold for any subsequ ence ofµXn.\nRemarks. The restriction to bounded interactions is for technical reasons. The result implies that if\nvis partly negative, any (vaguely or in distribution sense) convergen t subsequence of µXntends to\nan IDGS in infinite space which is different from the Lebesgue measure . The IDGS’s are expected to\nbe periodic or almost periodic in this case. Accordingly, the distributio n of particles in high-density\nGSC’s shows a pattern corresponding to the infinite-density limit. On the other hand, if v>0 then the\ndistribution of particles is asymptotically uniform. We cannot prove t hat the set of lattice constants is\nalways bounded, but the opposite, the divergence of at least one o f thedlattice constants, leaves only\ntwo possibilities. The first is that Jn, the number of particles in the primitive cell (called complexity),\ntends to infinity. The second is that Jnis bounded, implying that Xnfalls into the union of lower than\nd-dimensional substructures of diverging separation and density. In the case of the Gaussian potential, in\n5 and 7 dimensions, a new numerical result [16] suggests a uniaxially an isotropic high-density GSC with\na stronger compression along the distinguished axis than perpendic ular to it. According to part (i) of\nthe theorem, the other lattice constants cannot remain bounded if the anisotropy is to survive the limit\nof infinite density.\nProof.(i) Let Λ nbe a period parallelepiped of Bnand, hence, of Xn. Then, by Lemma 7.2, Xn∩Λnis\na ground state of uΛn. From Eqs. (2.28), (3.8) and (9.1),\n0<ω|Xn∩Λn|−v(0)\n2<uΛn(0)\n2ρn. (10.6)\nOn the other hand,\nω|Xn∩Λn|=I[µXn] =v(0)\n2+1\n2/summationdisplay\n0/ne}ationslash=k∈B∗nv(k)|/hatwideµXn(k)|2, (10.7)\n32where\n/hatwideµXn(k) =J−1\nnJn/summationdisplay\nj=1eik·yjifk∈B∗\nn. (10.8)\nThus,/summationdisplay\n0/ne}ationslash=k∈B∗nv(k)|/hatwideµXn(k)|2<uΛn(0)\nρn<2u(0)\nρn(10.9)\nfornlarge enough; moreover, for any bounded set D\n/summationdisplay\n0/ne}ationslash=k∈B∗\nn∩D|/hatwideµXn(k)|2<2u(0)\nρninfk∈Dv(k). (10.10)\nTake anyf∈ S(Rd) (function of rapid decrease). Using ρn=Jnρ(Bn), the Poisson summation formula\nyields/integraldisplay\nfdµXn=ρ−1\nn/summationdisplay\nx∈Xnf(x) =/integraldisplay\nf(x)dx+/summationdisplay\n0/ne}ationslash=k∈B∗n/hatwidef(k)/hatwideµXn(k). (10.11)\nBecause the set of period lengths of B∗\nn(n= 1,2,...) is separated from zero, the sum over k∝\\⌉}atio\\slash=0tends\nto zero asngoes to infinity. Indeed, for m∈Zd, let\nQm={k∈Rd:mi≤ki≤mi+1(i= 1,...,d)}\\{0}. (10.12)\nThen∪mQm=Rd\\{0}and|B∗\nn∩Qm| ≤lwith somelindependent of nandm. Sincevis continuous\nand strictly positive, for any M >0 there exists an ε>0 such that v(k)≥εifk∈ ∪|m|≤MQm. Write\n/summationdisplay\n0/ne}ationslash=k∈B∗\nn=/summationdisplay\nk∈B∗n∩(∪|m|≤MQm)+/summationdisplay\nk∈B∗n∩(∪|m|>MQm). (10.13)\nUsing (10.10), by Cauchy’s inequality\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nk∈B∗n∩(∪|m|≤MQm)/hatwidef(k)/hatwideµXn(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≤2u(0)\nρnε/summationdisplay\nk∈B∗n∩(∪|m|≤MQm)|/hatwidef(k)|2≤2u(0)lMd∝⌊ar⌈⌊l/hatwidef∝⌊ar⌈⌊l2\n∞\nρnε(10.14)\nwhich tends to zero as n→ ∞. On the other hand,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nk∈B∗n∩(∪|m|>MQm)/hatwidef(k)/hatwideµXn(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤l/summationdisplay\n|m|>Mmax\nk∈Qm|/hatwidef(k)|. (10.15)\nThe sum on the right-hand side is convergentand tends to zero as Mtends to infinity, proving Eq. (10.5).\n(ii) FixQ= [−L/2,L/2]dand define Nn=⌊ρnLd⌋. LetQn= [−Ln/2,Ln/2]dwhereLd\nn=Nn/ρn,\nthen 0≤Ld−Ld\nn< ρ−1\nn. Letq∈Q∗= (2π/L)Zdbe the minimizer of v(k) inQ∗, and let qn=\n(L/Ln)q∈Q∗\nn= (2π/Ln)Zd. Because Ln→L,qntends to qasnincreases. Let µRn∈ MQnbe\naNn-point approximation of µqn, cf. Eq. (9.3), constructed according to Lemma A.1. Let Znbe the\nperiodic extension of Rn. Thenρ(Zn) =ρn, and by part (ii) of Lemma 7.1,\nlim\nn→∞ep(Xn)\nρn≤lim\nn→∞ep(Zn)\nρn=v(0)\n2−|v(q)|\n4. (10.16)\nApplying Eq. (8.19) to Xn,\nep(Xn)\nρn+u(0)\n2ρn=ωΛn(Xn∩Λn) =1\n2V(Λn)/integraldisplay\nΛn/integraldisplay\nRdu(x−y)dµXn(y)dµXn(x), (10.17)\n33where Λ nis any period parallelepiped of Xn. Suppose that µXntends toλon Schwartz functions. If\nu∈ S(Rd), this would yield v(0)/2 instead of (10.16). If uis not a Schwartz function, replace it by u∗Gt,\ncompute the limit of ep(Xn)/ρnand letttend to zero. However,\n/integraldisplay\nRd(u∗Gt)(x−y)dy=v(0) (10.18)\nfor anyt≥0; thus, again, we obtain a contradiction. ✷\n11 Interactions without Bravais lattice ground states at hi gh\ndensity\nFor certain potentials having a partially negative Fourier transform we can complete the information\nabout GSC’s given in part (ii) of Theorem 10.1. Namely, we will show that for these potentials no\nperiodic GSC Xcan be a singly or multiply occupied Bravais lattice if ρ(X) is large enough, that is,\nJ >1 must hold and all yjcannot coincide [cf. (7.6)]. This result is valid in any dimension. Our meth od\nis to exclude Bravais lattices first as IDGS’s and then as GSC’s on the t orus and in infinite space at high\nbut finite densities. We note that one-dimensional examples of inter actions having a nondegenerate GSC\nwith two points in the unit cell [30, 36] and others with no periodic GSC [3 7] existed already thirty years\nago, see also Radin’s review [38].\n11.1 Compensation by higher harmonics\nIn the following example the negative contribution to the energy com ing from a wave vector kis com-\npensated by the positive contribution coming from integer multiples o fk.\nTheorem 11.1 Consider a bounded integrable pair potential having a partl y negative Fourier transform\nvwith the property that for any k∝\\⌉}atio\\slash= 0\n∞/summationdisplay\nn=1v(nk)≥0. (11.1)\nThen no Bravais lattice can be an IDGS: if Bis a Bravais lattice and the parallelepiped Λis a period\ncell forB, thenI[µB∩Λ]> I0. Furthermore, at finite, high enough densities there is no Br avais lattice\namong the ground state configurations on tori and in infinite s pace, and no lattice tower of the form\nBJ=J/unionmultidisplay\nj=1B (11.2)\nwhereBis a Bravais lattice and J≥1can be a GSC if ρ=Jρ(B)is high enough. How large ρmust be\ndepends on the details of the interaction. As an example, sup pose that\n(i) there exist 0<k1<k2≤2k1such thatv(k)<0fork1<|k|<k2andv(k)≥0otherwise,\n(ii)v(2k)≥ |v(k)|fork1<|k|<k2, and\n(iii)v(k) = 0for|k| ≥3qwith someqsuch thatk1<q<k 2.\nThen there is no Bravais lattice GSC if\nρ≥(3/2)d−1(q/π)d.\nProof.Consider Equation (9.11). If K∈B∗thennK∈B∗as well. Summing over B∗along lattice\nhalf-lines, the contribution of each partial sum is nonnegative, whic h shows that I[µB∩Λ]≥v(0)/2. On\nthe other hand, choosing any qsuch thatv(q)<0,I0≤I[µq] =v(0)/2−|v(q)|/4, cf. Eq. (9.5), thus,\nµB∩Λis not an IDGS. The absence of Bravais lattice ground states at high but finite densities is true\nboth in finite and infinite volume. To see this, suppose that Λ is a period cell ofB. Then\nI[µBJ∩Λ] =ω(BJ∩Λ) =ω(B∩Λ)≥v(0)\n2. (11.3)\n34On the other hand, if ρis high enough then according to Lemma A.1 and due to the continuity o fI[µ]\none can choose a configuration R=/parenleftbig\nr1,...,r|BJ∩Λ|/parenrightbig\n⊂Λ such that\n|I[µR]−I[µq]|<|v(q)|/8\nand therefore\nI[µR] =ω(R)<v(0)\n2−|v(q)|\n8≤ω(BJ∩Λ)−|v(q)|/8. (11.4)\nThis shows that BJ∩Λ cannot be a ground state on Λ. In infinite space we compare the en ergies per\nparticleep(BJ) andep(Rper) whereRperis the periodic extension of R. NowBJandRperhave the same\ndensity, and Eqs. (8.19) and (11.4) imply ep(Rper)<ep(BJ). From Lemma 7.1 we conclude that BJis\nnot a GSC.\nConsider the specific example. The idea is to find a periodic discrete po int approximation Xof the\ndensity 1+cos q·xsuch thatω(X∩Λ)<v(0)/2. The computation is done by using the expression\nω(r)N=1\n2/summationdisplay\nk∈Λ∗v(k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nNN/summationdisplay\nn=1eik·rn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (11.5)\nseeEqs.(2.22)and (2.16). Select a qwhoselength qsatisfiescondition (iii). Choose Lsothatq= (2π/L)l\nwherelis an integer. Let ej,j= 1,...,dbe the Cartesianunit vectorschosen in such a waythat q=qe1,\nand\nΛ =\n\nd/summationdisplay\nj=1xjej: 0≤xj<L\n\n. (11.6)\nFor a positive integer l⊥, witha=L/l⊥define the lattice\nBq=\n\n2π\nqn1e1+ad/summationdisplay\nj=2njej: (n1,...,n d)∈Zd\n\n(11.7)\nand, for any integer M≥2, the periodic configuration\nX=M/uniondisplay\nm=1(Bq+ym) (11.8)\nwhere\nym=m−1\nMπ\nqe1, m= 1,...,M. (11.9)\nWith this choice of Landa, the reciprocal lattice\nB∗\nq=\n\nn1q+2π\nad/summationdisplay\nj=2njej: (n1,...,n d)∈Zd\n\n(11.10)\nis a subset of Λ∗= (2π/L)Zd. We show that for ρ≥(3/2)d−1(q/π)dthe energy per particle of Xis lower\nthan that of any BJof the same density. Now\n1\n|X∩Λ|/summationdisplay\nx∈X∩Λeik·x=1\n|Bq∩Λ|/summationdisplay\nr∈Bq∩Λeik·r1\nMM/summationdisplay\nm=1eik·ym\n=/summationdisplay\nK∈B∗qδk,K1\nMM/summationdisplay\nm=1eik·ym(11.11)\n35ifk∈Λ∗, therefore the sum (11.5) reduces to B∗\nq. The contribution of k=0toωisv(0)/2. Fork=±q\nwe have/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nMM/summationdisplay\nm=1eik·ym/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=2\nM2/parenleftbig\n1−cosπ\nM/parenrightbig/braceleftbigg=1\n2, M = 2\n→/parenleftbig2\nπ/parenrightbig2, M→ ∞(11.12)\nand the function decreases monotonically with Mto its limit at M=∞. Fork=±2q, exp(ik·ym) runs\nover theMth roots of unity and, therefore,/summationtextM\nm=1eik·ym= 0.Under the condition that all other vectors\nofB∗\nqhave a length not smaller than 3 q,\nω(X∩Λ) =v(0)\n2−2|v(q)|\nM2/parenleftbig\n1−cosπ\nM/parenrightbig. (11.13)\nThe density of Xis\nρ=Mq\n2πad−1. (11.14)\nIn one dimension B∗\nq=qZ, so the condition of the validity of (11.13) is satisfied, and the smalles t density\nto which (11.13) applies is q/π, obtained with M= 2. The density can be increased only by increasing\nM, therefore the pair energy (11.13) increases with the density, bu t for allρ≥q/π\nω(X∩Λ)≤v(0)\n2−/parenleftbigg2\nπ/parenrightbigg2\n|v(q)|, (11.15)\nwhere the upper bound is obtained by substituting (11.12) with its limit atM=∞. Ford≥2 we can\nreach the smallest value for both ω(X∩Λ) andρif we choose the smallest M, that is,M= 2, and the\nlargestacompatible with the condition of validity of (11.13). For athe condition implies 2 π/a≥3q, so\nits largest allowed value is a= 2π/3q. To summarize, with (8.19) we find that the energy per particle of\nany infinite-volume ground state configuration Gof densityρ≥(3/2)d−1(q/π)dsatisfies the inequality\nep(G)≤ρv(0)\n2−u(0)\n2−ρ|v(q)|\n2≤ep(BJ)−ρ|v(q)|\n2(d≥2) (11.16)\nand\nep(G)≤ρv(0)\n2−u(0)\n2−4ρ|v(q)|\nπ2≤ep(BJ)−4ρ|v(q)|\nπ2(d= 1) (11.17)\nifBis any Bravais lattice and J≥1 such that Jρ(B) =ρ.✷\n11.2 Potentials with a cusp at zero\nIf the integral/integraltext\nv(k)|k|2dkis absolutely convergent, then uis twice continuously differentiable at 0, and\n−∆u(0) = (2π)−d/integraldisplay\nv(k)|k|2dk. (11.18)\nIf\nlim\nK→∞/integraldisplay\n|k|<Kv(k)|k|2dk=∞, (11.19)\nthenuhas a cusp at zero. We consider as a special case of a cusp if u(r)→ ∞asr→0. The condition\n(11.20) below implies that uhas a cusp at zero.\nTheorem 11.2 Letv−be nonzero and of compact support. Suppose that there exist K,c>0such that\nv(k)≥c\n|k|d+2for|k| ≥K. (11.20)\n36LetΛbe any parallelepiped. Then, no Bravais lattice commensura te withΛcan be an IDGS in Λ. In\nparticular:\n(i) Ifd>1and\nc\n|k|d+2≤v(k)≤c′\n|k|n+2+εat|k| ≥K (11.21)\nfor some positive integer n < d, positivec′and0< ε <1, then IDGS’s may be singular continuous\nmeasures of the form\nµ0=λd−n×µB(n)∩Λ(n), (11.22)\ncf. Eq. (6.14), in which case high-density ground state confi gurations in Λare strongly anisotropic, and\nmay or may not be Bravais lattices.\n(ii) Ifd≥1and\nv(k)≥c\n|k|3for|k| ≥K, (11.23)\nthen no IDGS is a stationary point of the energy functional. F urthermore, high-density ground state\nconfigurations are different from Bravais lattices and their towers (11.2) in Λand, ifuis bounded, also\nin infinite space.\nRemark. At the end of Section 6 we discussed the case when u(x)∼ |x|−d+ζwith 0< ζ < d near the\norigin. By a Tauberian argument v(k)∼ |k|−ζat infinity. If also v−∝\\⌉}atio\\slash= 0, thend≥4 and 3< ζ < d is\ncovered by (11.21), and 0 <ζ <min{d,3}is covered by (11.23). If 0 <ζ≤1 then/summationtext∞\nn=1v(nk) =∞for\nall nonzero k, which is a special case also of Theorem 11.1.\nOne-dimensional example to (ii). Let\nv(k) =1\nk2e−a/k2+(1−k2)e−k2. (11.24)\nFora≥3 this function takes on negative values on two symmetric bounded in tervals and decays as 1 /k2\nat infinity. Thus, uis bounded but has a cusp at zero, and\n/summationdisplay\nk∈Λ∗|/hatwideµ(k)|2<∞\nmust hold for an IDGS. Hence, any IDGS is absolutely continuous in Λ = [−L/2,L/2] and is different\nfrom the Lebesgue measure, because v−∝\\⌉}atio\\slash= 0. High-density GSC’s form a non-arithmetic sequence.\nProof.Let us return to the tempered measure Gt\nΛ∗µand Equation (4.47). For any t>0,\nd\ndtI[Gt\nΛ∗µ] =−/summationdisplay\nk∈Λ∗v(k)|k|2e−2t|k|2|/hatwideµ(k)|2\n=/summationdisplay\nk∈Λ∗v−(k)|k|2e−2t|k|2|/hatwideµ(k)|2−/summationdisplay\nk∈Λ∗v+(k)|k|2e−2t|k|2|/hatwideµ(k)|2,(11.25)\nbecause both sums are absolutely convergent. Now lim t→0dI[Gt\nΛ∗µ]/dtexists, it can be −∞for a\ngeneralµ, but it must be finite nonnegative if µis an IDGS. Thus, for any IDGS µinMΛ,\n/summationdisplay\nk∈Λ∗v+(k)|k|2|/hatwideµ(k)|2≤/summationdisplay\nk∈Λ∗v−(k)|k|2|/hatwideµ(k)|2≤/summationdisplay\nk∈Λ∗v−(k)|k|2<∞. (11.26)\nLetBbe a Bravais lattice commensurate with Λ and let µB∩Λbe the associated measure, cf. Eq. (9.8).\nThen /summationdisplay\nk∈Λ∗v+(k)|k|2|/hatwiderµB∩Λ|(k)|2=/summationdisplay\nk∈B∗v+(k)|k|2=∞ (11.27)\nbecause of (11.20), so µB∩Λis not an IDGS.\n37Combining (11.26) with (11.20), we find that\n/summationdisplay\nk∈Λ∗|/hatwideµ(k)|2\n|k|d<∞ (11.28)\nmust hold for any IDGS µ. Thus, either /hatwideµis nonvanishing on a lower than d-dimensional subset of Λ∗or\nit decays to zero.\n(i) In this case/summationdisplay\nk∈Λ∗:k1=...=kd−n=0v+(k)|k|2<∞, (11.29)\ntherefore a measure of the form (11.22) [which is a stationary point ofI, c.f. Eq. (6.14)] cannot be\nexcluded from being an IDGS. Indeed,\n/summationdisplay\nk∈Λ∗v+(k)|k|2|/hatwiderµ0(k)|2=/summationdisplay\nk′∈(B(n))∗v+(0d−n,k′)|k′|2<∞. (11.30)\nNowµ0can be obtained as the weak limit of a sequence of measures associat ed with Bravais lattices:\nthe component µB(n)is unchanged in this limit, while λd−nis approximated by a sequence of discrete\nmeasures associated with Bravais lattices that fill in Λ(d−n)more and more densely. The approximating\nmeasures can, thus, be associated with more and more anisotropic Bravais lattices.\n(ii) Combining (11.26) with (11.23) one finds\n/summationdisplay\nk∈Λ∗|/hatwideµ(k)|2\n|k|<∞ (11.31)\nfor any IDGS µ. Apart from the Lebesgue measure, no stationary point of the en ergy functional satisfies\nEq. (11.31). Moreover, in contrast with λandµ0, aµ∝\\⌉}atio\\slash=λsatisfying (11.31) cannot be obtained as the\nweak limit of measures associated with Bravais lattices. Indeed, for any Bravais lattice B,/hatwiderµB∩Λ(k) = 1\non a sublattice of Λ∗; therefore, if µ∝\\⌉}atio\\slash=λwas a weak limit of measures associated with Bravais lattices,\nthen/hatwideµ(k) = 1 would be on lattice, and (11.31) would fail. It follows that there ex ists someρΛsuch that\nifBis a Bravais lattice, Λ is a period cell for Bandρ(BJ) =J|B∩Λ|/V≥ρΛ, thenBJ∩Λ is not a\nground state of uΛ. In one and two dimensions a pair potential can be bounded and satis fy condition\n(11.23). Then, due to the uniform convergence proven in Corollary 3.1,ρΛcan be chosen independent of\nΛ, and we can conclude that in infinite space BJis not a GSC of u.✷\n12 The penetrable sphere model\nSo far we have only given one example, Proposition 9.3, illustrating Cor ollary 2.1 or Proposition 4.2.\nHowever, in that case GSC’s of the form of lattice towers — particles superimposed on lattice sites —\nare submerged in a continuum of other GSC’s. The truly interesting s ituation is when a lattice tower\nground state occurs in a non-degenerate manner. Intuition to fin d such interactions may be based on\nbothr- andk-space considerations. The k-space representation (4.39) of I[µ] suggests that IDGS’s are\nto be looked for among Dirac combs concentrated on Bravais lattice s ifv(k)≤0 for|k|large enough.\nChoosing the density of the lattice Blarge enough, apart from /hatwiderµB(0) = 1,/hatwiderµB(k) will be nonzero only\nwherev(k)≤0, with its modulus equal to 1 on the lattice B∗. Thus, one may expect that I[µ] is\nminimized by some high-density µB, see also Eq. (5.5). If vhas a unique minimum at km, the simplest\nguess is that B∗is close-packed with a nearest-neighbor distance (close to) km. Thinking in r-space,\nthe best candidates are potentials that are flat or even nesting at zero, that is, having ∆ u(0)≥0. The\ninteractions u(x)∼exp(−α|x|β) withβ >2, studied by Likos et al.[4], belong to this class. Intuitively,\nsuch an interaction may prefer the formation of lattice towers. Th e large negative part of vin such a\ncase, seen on (11.18), supports this intuition. Below we discuss the penetrable sphere model which is the\n38simplest although somewhat pathological example. A one-dimensiona l family of pair potentials giving\nrise to superimposed particles in ground state configurations is pre sented in a separate publication [10].\nThe particles in the penetrable sphere model interact via the pair po tential\nu(x) =/braceleftbigg\nu0>0 for|x|<d0\n0 for |x| ≥d0.(12.1)\nWe defineuto be lower semicontinuous. This potential is applied in soft matter ph ysics to model a\nsystem of interpenetrating micelles in a solvent [39]-[42]. The interact ion has a partly negative Fourier\ntransform. In 3 dimensions it reads ( k=|k|)\nv(k) =4πu0d3\n0\n(kd0)2/parenleftbiggsinkd0\nkd0−coskd0/parenrightbigg\n. (12.2)\nNotethatvisnot absolutelyintegrable. Thefirstand deepest minimum of v(k) isatk=km= 6.12/d0. If\nthis wasto determine the periodicity ofthe ground state then the g round state would be the dual ofan fcc\nlattice of lattice constant km, that is, a bcc lattice of nearest-neighbor distance√\n6π/km= (7.70/6.12)d0,\ncf. Eq. (15) of Ref. [2]. Instead, we will see that the ground state is an fcc lattice of nearest-neighbor\ndistanced0.\nWe show that with a proper choice of Λ, the ground state configura tions can be given for every\nparticle number N. Inddimensions choose a1,...,adsuch that |aj|=d0and the vectors ajgenerate a\nd-dimensional close-packed lattice,\nB=/braceleftBiggd/summationdisplay\n1njaj:n1,...,n d∈Z/bracerightBigg\n. (12.3)\nChoose Λ to be a period cell for Bcontaining |B∩Λ|=Mpoints ofBand having a side length ≥2d0\nin every direction. The volume of Λ is\nV=M|det[a1...ad]|, (12.4)\nso the volume of the unit cell of Bis\nρ−1\nB=V/M=|det[a1...ad]|=/braceleftbigg√\n3d2\n0/2 if d=2\nd3\n0/√\n2 if d=3.(12.5)\nThe property of Bis that for the given density ρBit has the largest nearest-neighbor distance or for\nthe given nearest-neighbor distance d0the largest density among Bravais lattices. We make also the\nhypothesis (trivial in two and proven in three dimensions) that Brealizes the densest packing of hard\nspheres of diameter d0. Note that with the choice of Λ we exclude the occurrence of other close-packed\n(e.g. hcp) structures which can be ground states in suitable domain s or in infinite space.\nIfN <M, the ground state in Λ is continuously degenerate, any configurat ion with all inter-particle\ndistances ≥d0is a ground state. Below we describe the ground state configuratio ns (GSC) for N≥M,\nmodulo cyclic translations in Λ.\nProposition 12.1 The IDGS is B∩Λ,\nCΛ[u] =ω(B∩Λ). (12.6)\nForNfinite, letn≥1be an integer and let nM≤N≤(n+1)M. Then there are/parenleftbigM\nN−nM/parenrightbig\nGSC’s in Λ.\nIn each of them (n+1)M−Npoints ofB∩Λare occupied by nparticles and N−nMpoints ofB∩Λ\nare occupied by n+1particles. Moreover,\nǫN=u0nV\nN−1/bracketleftbigg\n1−(n+1)M\n2N/bracketrightbigg\n, (12.7)\nso that\nCΛ[u] = lim\nN→∞ǫN=u0\n2ρB=u0\n2×/braceleftbigg√\n3d2\n0/2if d=2\nd3\n0/√\n2if d=3.(12.8)\n39Proof.The proof goes by induction over N. Because each side of Λ has a length ≥2d0, only a single\nterm ofuΛcan be nonvanishing, thus, uΛ(r) =u0or 0. IfN=M,B∩Λ has zero energy, therefore it is a\nGSC. Any perturbation of B∩Λ creates at least one pair of particles of distance <d0, therefore of energy\nu0. Thus,Bis the unique GSC in Λ. Suppose we know the result up to N, wherenM≤N <(n+1)M,\nand want to prove it for N+1. Adding a particle to a ground state of Nparticles costs the less if it is\nplaced on a point of B∩Λ occupied by nparticles. The increase in energy is nu0; at any other place it\nis at least twice as much. No relaxation of the configuration thus obt ained can decrease the energy, so it\nis a ground state for N+1 particles. The ground state energy for Nparticles is\nE0(N) = (N−nM)n(n+1)\n2u0+[(n+1)M−N]n(n−1)\n2u0=nu0[N−n+1\n2M].(12.9)\nDividing by N(N−1)/VyieldsǫN.✷\nAn unpleasant feature of the penetrable sphere model is the disco ntinuity of its energy at zero tem-\nperature when N≥M. The values of UΛ(r)Ncan only be integer multiples of u0taken from the set\n{E0(N),...,N(N−1)u0/2}. Therefore,\ne−β[FN(β)−E0(N)]≡1\nVN/integraldisplay\nΛNe−β[UΛ(r)N−E0(N)]d(r)N\n=N(N−1)/2/summationdisplay\nm=m0e−βmu0λdN(Am)\nVN. (12.10)\nHereλdNdenotes the dN-dimensional Lebesgue measure,\nAm=/braceleftbig\n(r)N∈ΛN:UΛ(r)N=E0(N)+mu0/bracerightbig\n, (12.11)\nand\nm0= min{m:λdN(Am)>0}. (12.12)\nNowA0is the set of translates of B∩Λ increased by the finite number of possible assignments of N\nparticles to Msites, so that λd(A0)>0, but its higher than d-dimensional Lebesgue measures vanish.\nOn the other hand,\nλdN(AN(N−1)/2) =Vλd(Bd0/2)N−1>0, (12.13)\nwhereBd0/2={x∈Rd:|x|<d0/2}. Therefore 0 <m0≤N(N−1)/2. From (12.10) it follows that\ne−βm0u0λdN(Am0)\nVN≤e−β[FN(β)−E0(N)]≤e−βm0u0. (12.14)\nTaking the logarithm and dividing by −β,\nm0u0≤FN(β)−E0(N)≤m0u0+1\nβln/bracketleftbig\nVN/λdN(Am0)/bracketrightbig\n. (12.15)\nLettingβtend to infinity in (12.15) and in the mean energy\nEN(β) =−∂lnZΛ,N\n∂β=E0(N)+/summationtextN(N−1)/2\nm=m0mu0e−βmu0λdN(Am)\n/summationtextN(N−1)/2\nm=m0e−βmu0λdN(Am), (12.16)\nwe find the following.\nProposition 12.2 For the penetrable sphere model in a finite volume,\nlim\nβ→∞FN(β) = lim\nβ→∞EN(β) =E0(N)+m0u0, (12.17)\nwherem0is given by Eq. (12.12).\nFrom the point of view of thermodynamics, it is, therefore, more ad equate to consider E0(N)+m0u0as\nthe ground state energy. The limit of fN(β) asNgoes to infinity depends on the limit of m0/N2and on\nthe largeNbehavior of λdN(Am0), and remains to be answered.\n40Note added\nAfter the submission of this paper I learned that a part of the tools I developed for studying ground\nstate configurations at high densities existed already in abstract p otential theory. The first result dates\nfrom 1923 and is due to Fekete [43]. In an article about the distributio n of the roots of polynomials of\ninteger coefficients, Fekete considered the quantity dn(S), the maximum of the geometric mean of the\nn(n−1)/2 distances between points in an n-point set, chosen from an infinite compact set S. He proved\nthatdn(S) was monotone decreasing as nincreased, and called the limit the transfinite diameter of S. If\nwe take the pair potential u(x−y) =−ln|x−y|, then the ground state energy per pair of nparticles\ninSis−lndn(S). Thus, Fekete’s result is the proof of the monotone increase of t he ground state energy\nper pair [equation (3.4)] and of Proposition 3.1 in this special case. Wh at I called the best superstability\nconstant is minus the logarithm of the transfinite diameter. The firs t proof of the monotone increase\nof the ground state energy per pair for a general symmetric kern elu(x,y) was given by Choquet [44],\nwho also proved a theorem corresponding to the present Theorem 4.2. The best superstability constant\nappears in [44] under the name of Fekete constant. The analogous theorem for a kernel which is bounded\non the diagonal (Theorem 4.1 here) can be found in a recent paper b y Farkas and Nagy [45]. Early results\non abstract potential theory are summarized in Fuglede’s paper [46 ].\nAcknowledgements\nI thank Szil´ ard R´ ev´ eszfor drawing my attention to related resu lts in abstract potential theory. This work\nwas supported by OTKA Grants K67980 and K77629.\nAppendix\nIn this appendix we collect a few results on the convergence of meas ures on compact sets. Let Λ ⊂Rd\nbe compact and let MΛbe the set of normalized Borel measures on Λ.\nLemma A.1 Anyµ∈ MΛis the weak limit of a sequence {µRN}∞\nN=1of discrete measures of the form\n(4.2).\nProof.We may suppose that Λ is a cube, otherwise, we take a cube that cov ers Λ and extend µwith zero\nvalue to a measure on the cube (cf. Lemma A.4.) We may also suppose t hat Λ is a unit cube. Given\nµ∈ MΛ(thus,µ(Λ) = 1), divide Λ into a disjoint union of ndsemi-open cubes Qn\njof side length 1 /n,\nwheren=⌊log2(N+1)⌋. InQn\njchooseNjpoints{rn\nji}Nj\ni=1whereNj=⌊Nµ(Qn\nj)⌋orNj=⌈Nµ(Qn\nj)⌉in\nsuch a way that/summationtext\njNj=N. Thus,\n|µ(Qn\nj)−Nj/N| ≤1/N. (A.1)\nDefine\nµRN=1\nNnd/summationdisplay\nj=1Nj/summationdisplay\ni=1δrn\nji(A.2)\nwhereδris the Dirac measure at r. For any continuous function f, let\n∝a\\}⌊ra⌋k⌉tl⌉{tf∝a\\}⌊ra⌋k⌉tri}htn,j=1\nµ(Qn\nj)/integraldisplay\nQn\njfdµ. (A.3)\nBecause Λ is compact, fis uniformly continuous, so there is some δnwhich goes to zero as nincreases,\nsuch that for each j /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nNjNj/summationdisplay\ni=1f(rn\nji)−∝a\\}⌊ra⌋k⌉tl⌉{tf∝a\\}⌊ra⌋k⌉tri}htn,j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δn. (A.4)\n41It follows that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nfdµRN−/integraldisplay\nfdµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δn+nd/summationdisplay\nj=1/vextendsingle/vextendsingle/vextendsingle/vextendsingleNj\nN−µ(Qn\nj)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|∝a\\}⌊ra⌋k⌉tl⌉{tf∝a\\}⌊ra⌋k⌉tri}htn,j|. (A.5)\nThe second term is O(nd/N), so both terms tend to zero as Ngoes to infinity. Thus, µRN⇀µ.✷\nNext, we show that given any infinite sequence of normalized measur es on a compact set, one can\nselect a subsequence converging weakly to a normalized measure. W e start with the compact set being a\nunit cube Λ equipped with periodic boundary conditions ( d-dimensional torus).\nThe Fourier transform (4.40) of a µ∈ MΛis a function of positive type [29] on Λ∗= 2πZd, meaning\nthat for any integer l, anyk1,...,klin Λ∗and any complex numbers z1,...,z l\nl/summationdisplay\ni,j=1/hatwideµ(ki−kj)zizj≥0. (A.6)\nIt follows among others that\n|/hatwideµ(k)| ≤/hatwideµ(0) = 1. (A.7)\nLemma A.2 From any sequence µn∈ MΛone can select a subsequence µnisuch that/hatwideµniconverges\npointwise to the Fourier transform /hatwideµof aµ∈ MΛ. We say that µniconverges to µin Fourier transform.\nProof.Because Λ∗is a countable set, we can use the diagonal process [23] to choose a subsequence/hatwideµni\nwhich converges pointwise to some function ψon Λ∗,\nlim\ni→∞/hatwideµni(k) =ψ(k),everyk∈Λ∗. (A.8)\nFixing any integer l,k1,...,klin Λ∗and complex numbers z1,...,z l, writing down Eq. (A.6) for /hatwideµni\nand taking the limit i→ ∞, we find\nl/summationdisplay\ni,j=1ψ(ki−kj)zizj≥0. (A.9)\nThus,ψis a function of positive type, ψ(0) = 1, and by Bochner’s theorem [29], ψ=/hatwideµ, the Fourier\ntransform of a probability measure µon Λ.✷\nLemma A.3 µnconverges to µin Fourier transform if and only if it converges weakly to µ.\nProof.Ifµn⇀µthen\nlimµn(f) =µ(f) forf(x) = sink·x,cosk·x, (A.10)\ntherefore lim/hatwideµn(k) =/hatwideµ(k) for allk∈Λ∗.\nSuppose now that µ,µn∈ MΛ,/hatwideµ= lim/hatwideµn. Consider the trigonometric polynomials,\np(x) =/summationdisplay\nk∈Λ∗akeik·x(A.11)\nwhereak∝\\⌉}atio\\slash= 0 for a finite number of k. Any continuous function fon Λ is the uniform limit of a sequence\npmof the form (A.11). Given any ε>0, we fix an mso that∝⌊ar⌈⌊lf−pm∝⌊ar⌈⌊l∞≤ε, and choose Nεsuch that\nforn>N ε\n|µ(pm)−µn(pm)| ≤/summationdisplay\nk∈Λ∗|am\nk||/hatwideµ(k)−/hatwideµn(k)| ≤ε. (A.12)\nBecause |µ(f−pm)| ≤εand|µn(pm−f)| ≤ε, we obtain\n|µ(f)−µn(f)| ≤3εifn>N ε, (A.13)\nproving that µnconverges weakly to µ.✷\nThe extension of the above result to the general case, when the c ompact domain may not be a cube\nand the boundary condition may not be periodic, is immediate.\n42Lemma A.4 LetΛ0⊂Rdbe a compact Borel set and let MΛ0denote the set of normalized Borel\nmeasures on Λ0. Given any infinite sequence µn∈ MΛ0, one can select a subsequence converging weakly\nto someµ∈ MΛ0.\nProof.Take a cube Λ ⊃Λ0. Letµ′\nn∈ MΛwith suppµ′\nn⊂Λ0be the extension of µnto Λ. Constructing\na normalized limit measure µ′∈ MΛaccording to Lemma A.2, it is clear that supp µ′⊂Λ0, therefore\nµ=µ′|Λ0∈ MΛ0. 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Chechkin‡\nSchool of Chemistry, Tel Aviv University, Ramat Aviv, Tel Av iv 69978, Israel and\nInstitute for Theoretical Physics NSC KIPT, Akademicheska ya st. 1, Kharkov 61108, Ukraine\n(Dated: October 30, 2018)\nWe discuss the existence of stationary states for subharmon ic potentials V(x)∝ |x|c,c <2, under\naction of symmetric α-stable noises. We show analytically that the necessary con dition for the\nexistence of the steady state is c >2−α. These states are characterized by heavy-tailed probabili ty\ndensity functions which decay as P(x)∝x−(c+α−1)for|x| → ∞ , i.e. stationary states posses a\nheavier tail than the corresponding α-stable law. Monte Carlo simulations confirm the existence o f\nsuch stationary states and the form of the tails of correspon ding probability densities.\nPACS numbers: 05.40.Fb, 05.10.Gg, 02.50.-r, 02.50.Ey,\nI. INTRODUCTION\nThe dynamics of many physical systems can be reduced\nto a standard model of a “particle” (relevant coordinate\nx) under action of a deterministic force f(x) and the\nnoise force including the action of all neglected degrees\nof freedom. The mathematical tool for such a description\nis given by a Langevin equation, which in the overdamped\nlimit typically takes the form\n˙x(t) =f(x) +ζ(t). (1)\nIn Eq. (1) f(x) stands for the deterministic force, while\nζ(t) is the stochastic force — noise — describing inter-\nactions of a test particle with its complex surrounding.\nCommonly, it is assumed that the stochastic force is of\nthe white type, for example representing the particle’s\ncollisions with molecules of the bath. Following General-\nized Central Limit Theorem, a large number of indepen-\ndent collisions will lead to white noise of the stable type,\ni.e. either to Gaussian noise or to more general L´ evy\nnoise. The presence of fluctuations distributed according\nto L´ evy laws have been observed in various situations\nin physics, chemistry or biology [1, 2], paleoclimatology\n[3] or economics [4]. The situations pertinent to heavy-\ntailed distributions appear in context of different models\n[5–8], and are analyzed in an increasing number of studies\n[9–23].\nThe action of white L´ evy noise leads to increments\nof the stochastic process which are distributed accord-\ning toα-stable L´ evy type distributions. Symmetric L´ evy\n∗bartek@th.if.uj.edu.pl\n†igor.sokolov@physik.hu-berlin.de\n‡achechkin@kipt.kharkov.uadistributions pα(x;σ) are characterized by their Fourier-\ntransforms (characteristic functions of the distributions)\nφ(k) =/integraltext∞\n−∞eikxpα(x;σ)dxbeing [24–26]\nφ(k) = exp [−σα|k|α]. (2)\nThe parameter α(whereα∈(0,2]) is the stability index\nof the distribution describing, for α <2, the asymptotic\ndecay of its tails\npα(x)∝ |x|−(1+α). (3)\nFinally,σis the scale parameter which controls the over-\nall distribution width. The Gaussian distribution corre-\nsponds to a special case of a L´ evy stable law with α= 2\nwhen√\n2σis interpreted as the standard deviation of the\ndistribution. In more general cases, for α < 2, the vari-\nance ofα-stable densities diverges. For α <1, the mean\nvalue ofα-stable densities also does not exist.\nThe present work addresses properties of L´ evy flights\nin external potentials. Theoretical descriptions of such\nsystems is based on the Langevin equation and/or\nFokker-Planck equation, which is of the fractional order\n[27]. The research performed here extends earlier studies\n[9, 19–21] where analysis of L´ evy flights in harmonic and\nsuperharmonic potentials has been presented. The dis-\ncussion conducted there did not account for the problem\nof existence of stationary states in potential less steep\nthan parabolic, which is the main topic of the present\nwork. This issue is addressed here by the use of analyti-\ncal arguments and Monte Carlo simulations.\nThe model under discussion is presented in Section II.\nSection III discusses obtained results. The paper is closed\nwith concluding remarks (Section IV).2\n 0 1 2 3\n-2 -1  0  1  2V(x)\nx|x|1.5\n|x|    \nFIG. 1. Exemplary subharmonic potentials: V(x) =|x|and\nV(x) =|x|1.5used for examination of the problem of existence\nof stationary states.\nII. MODEL\nAn overdamped Brownian-L´ evy particle moves in an\nexternal potential V(x), therefore Eq. (1) takes the form\n˙x(t) =−V′(x) +ζ(t), (4)\nwhereV(x) represents a subharmonic single well poten-\ntialV(x) =|x|cwith 0< c < 2, see Fig. 1, and ζ(t) de-\nnotes a L´ evy stable white noise process [9, 10, 15, 28, 29].\nThe position of a random walker can be calculated by\nmeans of the stochastic integration of Eq. (4) [25, 30]\nx(t) =x(0)−/integraldisplayt\n0V′(x(s))ds+/integraldisplayt\n0ζ(s)ds\n=x(0)−/integraldisplayt\n0V′(x(s))ds+Lα(t). (5)\nThe integral/integraltextt\n0ζ(s)ds≡Lα(t) defines an α-stable L´ evy\nprocessLα(t) [9, 10, 15, 28, 31] which is driven by a\nL´ evy stable noise ζ(t). Increments ∆ Lα(∆t) =Lα(t+\n∆t)−Lα(t) of theα-stable L´ evy process are distributed\naccording to the α-stable density with the stability index\nα. If the time step is set to ∆ t, the appropriate α-stable\ndistribution is pα(��Lα;σ(∆t)1/α) [25, 26, 32, 33].\nThe equation (4) is associated with the following frac-\ntional Fokker-Planck equation (FFPE) [34–38]\n∂P(x,t)\n∂t=/bracketleftbigg∂\n∂xV′(x,t) +σα∂α\n∂|x|α/bracketrightbigg\nP(x,t) (6)\nwhere the fractional (Riesz-Weyl) derivative is defined\nby the Fourier transform [9, 19, 20, 39] F/bracketleftBig\n∂α\n∂|x|αf(x)/bracketrightBig\n=\n−|k|αF[f(x)].The (space) fractional derivative in\nEq. (6) describes long jumps which are distributed ac-\ncording to the α-stable density, see Eq. (2) and Refs. [1,\n35, 37, 40]. The Langevin equation (4) provides a\nstochastic representation of the fractional Fokker-Planck100101102103104\n101102103qy-q1-y\nt10-410-310-210-1100\n102103104105106Fc (x)=1-F(x)\nx\n100101102103104105106\n101102103104105qy-q1-y\nt10-410-310-210-1100\n101102103104105106Fc (x)=1-F(x)\nx\n10-1100101\n101102103qy-q1-y\nt10-410-310-210-1100\n10-1100101102103Fc (x)=1-F(x)\nx\nFIG. 2. Left column presents interquantile distance ( q0.9−\nq0.1,q0.8−q0.2,q0.7−q0.3andq0.6−q0.4, from top to bottom)\nas a function of time t. Solid lines present t1/αscaling of\nthe interquantile distance, which is observed in the force f ree\ncase. Right column demonstrates complementary cumulative\ndistributions at the end of simulation. Solid lines present\nx−αdecay. Various rows correspond to different potentials:\nV(x) = 0 (top panel), V(x) =|x|(middle panel) and V(x) =\nx2/2 (bottom panel). The stability index αand the scale\nparameter σare set to α= 0.9 andσ= 1 respectively.\nequation (6). In other words, it describes an evolution\nof a single realization of the stochastic process {x(t)}.\nFrom ensemble of trajectories it is possible to analyze\nfurther properties of the system, in particular a proba-\nbility density P(x,t) of finding a random walker at time\ntin the neighborhood of x, see Eq. (6). More details on\nnumerical scheme of integration of stochastic differential\nequations with respect to α-stable noises can be found in\n[14, 15, 25, 26, 32].\nIn the following sections properties of stationary prob-\nability distributions for single-well systems perturbed by\nsymmetric L´ evy noises are discussed. In the limiting\ncases of parabolic and quartic potentials the performed\nsimulations corroborate earlier theoretical findings [9, 19–\n21, 41].\nIII. RESULTS\nOnly in the limited number of special cases it is possi-\nble to find stationary probability densities P(x) analyt-\nically by solving Eq. (6). Therefore, the construction of\nstationary densities rely on numerical methods. In such\na case, there are two frameworks possible: one option is3\n10-1100101\n101102103104105qy-q1-y\nt100101102103\n101102103104105qy-q1-y\nt\n10-1100101102\n101102103104105qy-q1-y\nt100101102103104\n101102103104105qy-q1-y\nt\nFIG. 3. Interquantile distance ( q0.9−q0.1,q0.8−q0.2,q0.7−q0.3\nandq0.6−q0.4, from top to bottom) as a function of time tfor\nV(x) =|x|1.5with different values of the stability index α=\n{1.5,1.1}(left column, from top to bottom) and α={0.9,0.8}\n(right column, from top to bottom). The scale parameter σ\nis set to σ= 1.\nto discretize Eq. (6), see [19, 21, 42–45], alternatively one\nmight use a Monte-Carlo method based on the simula-\ntion of the Langevin equation (4), see [25, 26, 30, 32, 46].\nHere, we rely on Monte Carlo simulations. Such a choice\nis based on statistical properties of searched densities. As\nwe proceed to show, for subharmonic potentials, if sta-\ntionary states exist, the tail of the corresponding prob-\nability density function P(x) is “fatter” than the tail of\nthe corresponding α-stable density. In such a case con-\nfining the numerical solution of equation (6) to a finite\ninterval may introduce uncontrollable errors. Therefore,\nin our simulations, we rely solely on stochastic represen-\ntation of the fractional Fokker-Planck equation (6) which\nis provided by Eq. (4).\nIn theα= 2 case (Gaussian noise) the stationary solu-\ntion for Eq. (6), which in this case reduces to a “normal”\nFokker-Planck equation, has the Boltzmann-Gibbs form\nP(x)∝exp/bracketleftbigg\n−V(x)\nσ2/bracketrightbigg\n(7)\nand exists for any power-law potential V(x) such that\nlim|x|→∞V(x) = +∞. The situation drastically changes\nfor L´ evy noises with α < 2. Stationary states always\nexist in potentials steeper than the parabolic one [19–\n21, 41]. However, as it will be shown this is no more the\ncase for subharmonic potentials.\nA. Parabolic and quartic potentials\nThere are two special cases of power-law potentials for\nwhich the stationary solution can be given in a closed\nform: the case of a parabolic potential (for any 0 < α/lessorequalslant\n2) and the case of a quartic potential for α= 1 and10-410-310-210-1100\n10-210-1100101102103Fc (x)=1-F(x)\nx10-310-210-1100\n10-210-1100101102103104105106Fc (x)=1-F(x)\nx\n10-410-310-210-1100\n10-210-1100101102103104105Fc (x)=1-F(x)\nx10-210-1100\n10-210-1100101102103104105106Fc (x)=1-F(x)\nx\nFIG. 4. Complementary cumulative distributions at the end\nof simulation for V(x) =|x|1.5with different values of the\nstability index α={1.5,1.1}(left column, from top to bot-\ntom) and α={0.9,0.8}(right column, from top to bottom).\nThe scale parameter σis set to σ= 1. Solid lines present\nx−(α−0.5)decay predicted by Eq. (21).\nα= 2. These cases will serve as a benchmark for our\nfurther considerations.\nIn order to find stationary solution of the fractional\nFokker-Planck equation (6) we rewrite it in the Fourier\nspace\n∂ˆP(k,t)\n∂t=ˆU(k)ˆP(k,t)−σα|k|αˆP(k,t). (8)\nIn Eq. (8) ˆU(k) denotes the operator related to the\nFourier representation of the potential V(x) [21].\nFor the parabolic potential V(x) =x2/2 the expression\nforˆU(k) reads ˆU(k) =−k∂\n∂k. For symmetric α-stable\nnoises, the stationary probability density fulfills [19, 20]\n∂ˆP(k)\n∂k=−σαsignk|k|α−1ˆP(k). (9)\nThe solution of Eq. (9) is\nˆP(k) = exp/bracketleftbigg\n−σα|k|α\nα/bracketrightbigg\n, (10)\ni.e., the stationary solution is a symmetric L´ evy distribu-\ntion, see Eq. (2), characterized by a different value of the\nscale parameter than the noise in Eq. (4). The scale pa-\nrameter of the stationary density is σ/α1/α. Forα < 2,\nthe variance of the stationary solution diverges and the\nparabolic potential is not sufficient to produce bounded\nstates, i.e. states characterized by a finite variance [9, 19–\n21, 40].\nFor the quartic potential V(x) =x4/4 the expression\nforˆU(k) reads ˆU(k) =k∂3\n∂k3. The stationary density\nfulfills\n∂3ˆP(k)\n∂k3=σαsignk|k|α−1ˆP(k). (11)4\n100101102103104105106\n10-510-410-310-210-1| q...|\n1/t10-410-310-210-1100\n10-210010210410610810101012Fc (x)=1-F(x)\nx\n10-310-210-1100101102103\n10-510-410-310-210-1| q...|\n1/t10-310-210-1100\n10-2100102104106108Fc (x)=1-F(x)\nx\n10-510-410-310-210-1100\n10-510-410-310-210-1| q...|\n1/t10-310-210-1100\n10-2100102104106108Fc (x)=1-F(x)\nx\nFIG. 5. Quantile lines qyand|q1−y|(y={0.9,0.8,0.7,0.6},\nfrom top to bottom) as a function of 1 /t(left column) and\ncomplementary cumulative distributions at the end of simu-\nlation (right column) for V(x) =|x|1.5with the value of the\nstability index α= 0.7 and the scale parameter σ:σ= 1 (top\npanel),σ= 0.1 (middle panel) and σ= 0.01 (bottom panel).\nIn the right column, solid lines present x−(α−0.5)decay pre-\ndicted by Eq. (21).\nForα= 1, the solution of Eq. (11) reads [19, 20]\nˆPα=1(k) =2√\n3exp/bracketleftbigg\n−σ1/3|k|\n2/bracketrightbigg\ncos/bracketleftBigg√\n3σ1/3|k|\n2−π\n6/bracketrightBigg\n.\n(12)\nThe formula for the corresponding stationary density\nP(x) [19–21, 41] in the real space is\nPα=1(x) =1\nπw[1−(x/w)2+ (x/w)4], (13)\nwherew=σ1/3. The stationary solution (13) of Eq. (11),\nas well as the steady state (10) with α <2, is no longer\nof the Boltzmann-Gibbs type. This is typical for system\ndriven by L´ evy white noises with the stability index α <2\n[47]. The stationary state, see Eq. (13), has the asymp-\ntotic power-law dependence, i.e. P(x)∝ |x|−4. Further-\nmore, the stationary solution (13) is bimodal, with modal\nvalues located at x=±σ1/3/√\n2 [19–21].\nB. Subharmonic potentials\n1. Analytical results.\nSubharmonic potentials ( V(x) =|x|cwith 0< c < 2,\nsee Fig. 1) interpolate between the force free case ( c= 0)10-1100101\n101102103104105qy-q1-y\nt10-1100101102\n101102103104105qy-q1-y\nt\n10-1100101102\n101102103104105qy-q1-y\nt10-1100101102103\n101102103104105qy-q1-y\nt\nFIG. 6. Interquantile distance ( q0.9−q0.1,q0.8−q0.2,q0.7−q0.3\nandq0.6−q0.4, from top to bottom) as a function of time t\nforV(x) =|x|with different values of the stability index α=\n{1.7,1.6}(left column, from top to bottom) and α={1.5,1.4}\n(right column, from top to bottom). The scale parameter σ\nis set to σ= 1.\nand the harmonic potential ( c= 2). As it was indicated\nin the previous subsection for the harmonic potential, the\nstationary state is an α-stable L´ evy type distribution, see\nEqs. (2) and (10). In the force free case the stationary\nsolutions do not exist. The time dependent solutions\nare just L´ evy stable distributions with the growing scale\nparameter σ, i.e.σ(t) =σt1/α. This observation suggests\nthat for intermediate values of the exponent c(0< c < 2)\nthere should be a transition between the situation when\nthe stationary state exists and the situation when the\nstationary state is absent.\nIt is possible to obtain a necessary condition for the\nexistence of the stationary state using qualitative argu-\nments based on the FFPE (6). If a stationary state for\nEq. (6) exists, it satisfies\n/bracketleftbiggd\ndxV′(x) +σαdα\nd|x|α/bracketrightbigg\nP(x) = 0. (14)\nAssuming that tails of the distribution are given by a\npower-law, i.e. P(x)∝ |x|−νas|x| → ∞ , the first term\nin Eq. (14) behaves as\nd\ndx/bracketleftbig\nc|x|c−1P(x)/bracketrightbig\n≃xc−2−ν. (15)\nThe fractional derivative, see Eq. (14), is a nonlocal op-\nerator. The asymptotics of the fractional derivative of\na probability density has a universal behavior indepen-\ndent of the particular form of this density. Indeed, its\nbehavior for x→ ∞ is governed by the behavior of its\nFourier-transform −|k|αˆP(k) fork→0. Due to the nor-\nmalization condition ˆP(k= 0) = 1. The inverse Fourier\ntransform results in\ndα\nd|x|αP(x)≃x−1−α. (16)5\nIn order to satisfy Eq. (14), both terms (15) and (16)\nhave to represent the same x-dependence. Consequently,\nwe getc−2−ν=−1−αand\nν=c+α−1. (17)\nTherefore a stationary state is asymptotically character-\nized by a power-law\nP(x)∝ |x|−(c+α−1), (18)\ni.e. by the same expression as for the superharmonic\npotentials [21]. The probability density function P(x)\nhas to be integrable, which corresponds to ν >1. Thus,\nthe necessary condition for the existence of the steady\nstate is\nc >2−α (19)\nor\nα >2−c. (20)\nFor the parabolic potential ( c= 2), the exponent νre-\nproduces the decay of the L´ evy distribution ( ν= 1 +α),\nsee Eq. (10). For the quartic potential ( c= 4) with α= 1\nwe getν= 4, as follows from Eq. (13). The complemen-\ntary cumulative density function ( Fc(x) = 1−F(x) =\n1−/integraltextx\n−∞P(x′)dx′) is thus asymptotically characterized\nby a power-law decay\nFc(x)∝x−(c+α−2). (21)\nHaving the condition (19), it is also possible to deter-\nmine the dependence of the width of the stationary dis-\ntribution w(σ) on the noise strength σ. Assuming that\nP(x) =w−1f(x/w) can be expressed through a univer-\nsal function f(ξ) of a new, dimensionless, length vari-\nableξ=x/w (this assumption is reasonable due to the\nscale-free nature of the power-law potential), we change\nthe length variable to ξ=x/w. ForV(x) =|x|c(i.e.\nV′(x)∝c|x|c−1,x∝ne}ationslash= 0), we get\nwc−1w−2d\ndξ/bracketleftbig\nc|ξ|c−1f(ξ)/bracketrightbig\n+σαw−1−αdα\nd|ξ|αf(ξ) = 0.\n(22)\nThe universal rescaled equation for f(ξ)\nd\ndξ/bracketleftbig\nc|ξ|c−1f(ξ)/bracketrightbig\n+dα\nd|ξ|αf(ξ) = 0 (23)\ncan only be obtained if the width of distribution w(σ) is\nthe solution of the following equation\nwc−3=σαw−1−α. (24)\nCondition (24) assures that prefactors of both terms in\nEq. (22) scale similarly when σis changed. Thus, from\nEq. (24) one obtains\nw(σ)∝σα/(c+α−2). (25)In the absence of the noise ( σ→0) the solution P(x)\ncorresponds to the particle’s position in the minimum of\nthe potential, P(x) =δ(x), i.e.w(σ) = 0. For c+α <2\nthe exponent in Eq. (25) is negative, and the distribu-\ntion’s width w(σ) diverges for σ→0 which corresponds\nto a non-physical situation. Consequently, no station-\nary state is possible. On the contrary, for c+α > 2\nthe exponent in Eq. (25) is positive and the distribution\nwidth is the increasing function of the scale parameter σ.\nTherefore, the necessary condition of the existence of the\nsteady state, see Eq. (19), is confirmed by the condition\n(25).\nNote that the cases of parabolic and quartic potentials\nconfirm Eq. (25). For the parabolic potential ( c= 2)\nwe getw(σ)∝σ, see Eq. (10). In the case of quartic\npotential ( c= 4) with α= 1 we get w(σ)∝σ1/3, see\nEq. (13). Moreover, it is also possible to discuss how\ndoes the relaxation time to the equilibrium depends on\nthe scale parameter σ. To do this, we turn to Eq. (6) and\nrescale both the length ( ξ=x/w) and the time variables\n(τ=t/τ0(σ)), so that the rescaled solution to Eq. (6) is\na universal function. The rescaling of tcorresponds to\nτ0(σ)∝[w(σ)]3−c=σα(3−c)/(c+α−2). (26)\nIt indicates that the dependence on the scale parameter\nσfor the stability index αclose to 2 −cgets extremely\nsharp. In other words, when αapproaches 2 −c(from\nabove) the relaxation time grows and diverges at α=\n2−c.\nThe condition (19) for the existence of the steady state\ncan be corroborated by the following qualitative argu-\nment based on the decomposition of the noise into a\n“background” noise with finite variance, and “bursts” of\narbitrary amplitude [48, 49].\nLet us consider the discretized Langevin scheme with\nthe time step of integration ∆ tand the characteristic am-\nplitude of the L´ evy jumps ǫ=σ(∆t)1/α. Additionally, let\nus introduce a cutoff level Λ, so that only the jumps with\nthe amplitude a >Λ are considered as bursts. The over-\nall situation is then considered as pertinent to a pertur-\nbation of the equilibrium distribution of particles in the\npotential V(x) under the action of the background noise\n(i.e. the Boltzmann distribution in the potential V(x) at\nthe temperature kT≃Λ) by these distinct bursts. The\nwidth of this distribution Wdepends on the cutoff level\nΛ and on the type of the potential. Moreover, we assume\nthat only very large bursts, with a≫Λ, can be respon-\nsible for the absence of a stationary distribution. There-\nfore, the finite width of the background distribution can\nbe neglected when considering the action of such bursts.\nThe distribution of the burst amplitudes is then given by\nthe asymptotic behavior of tails of the symmetric L´ evy\ndistribution\np(a)≃ǫα\nx1+α=σα∆t\nx1+α. (27)\nA burst of amplitude abrings the particle to a position\nx≈a, and the typical time necessary to return to the6\ninterval of the characteristic width Wis given by the\nsolution of the ordinary equation of motion\n˙x(t) =−V′(x). (28)\nForc∝ne}ationslash= 2, the return time T(a) into the equilibrium\ndomain, i.e. |x|< W , from the initial position x=ais\ngiven by\nT(a) =1\nc(2−c)/bracketleftbig\na2−c−W2−c/bracketrightbig\n. (29)\nFor subharmonic potentials, 0 < c < 2, the return time\nT(a) is dominated by the initial position a. The lead-\ning contribution to T(a) is provided by the first term of\nEq. (29), i.e. a2−c/[c(2−c)]. Therefore, instead of the\nreturn time to the equilibrium domain it is possible to\nconsider the return time to the origin, which is finite and\nslightly larger than the return time to the characteristic\ndomain.\nIt is reasonable to assume, that the steady state does\nnot exist, if the time T(a) to return from the point a\nis larger than the typical time between two bursts of the\namplitude aor larger, and might exist in the opposite sit-\nuation. The probability to have a burst of the amplitude\naor larger is\nP(a) = 2/integraldisplay∞\nap(a′)da′≃/parenleftBigǫ\na/parenrightBigα\n, (30)\nso that the typical time between two such events is\nT≃∆t/parenleftBiga\nǫ/parenrightBigα\n. (31)\nComparing TwithT(a) we conclude that the inequality\nT(a)< T applies and the stationary states are possible\nfor\nc >2−α. (32)\nFor superharmonic potentials, c >2, the situation is\ninverse to the situation for subharmonic potentials. The\nlarger the initial displacement ais, the faster is the return\nto theW-domain. For the increasing initial displacement\na, the typical return time essentially tends to a constant\nvalueW2−c/[c(c−2)], while the time between two subse-\nquent bursts grows with their amplitude a. Comparing T\nwithT(a), we conclude that stationary states for super-\nharmonic potentials exist for every value of the stability\nindexα. The harmonic potential, c= 2, is the limiting\ncase for which the return time shows not a power-law but\na logarithmic dependence. In such a case, the very same\narguments proofing existence of stationary states hold.\n2. Numerical results.\nNumerical results were obtained by Monte Carlo sim-\nulations of Eq. (4) with the time step of integration∆t= 10−2and averaged over N= 106realizations\n[14, 15, 25, 26, 32]. Initially, at time t= 0 a test parti-\ncle was located at the origin, i.e. x(0) = 0. Due to the\nsymmetry of the noise and of the potential, stationary\nstates are symmetric with respect to the origin; the me-\ndian and modal values of stationary densities are located\nat the origin.\nOur test of stationarity is based on the quantiles of dis-\ntributions. Quantiles of stationary distributions remain\nconstant. Consequently, plotting quantile values as func-\ntions of time (quantile lines) gives us the key to analyze\nstationarity. More precisely, if the quantile lines are par-\nallel to the abscissa, it means that the stationary state\nhas been reached.\nIn order to verify performance of the test based on\nquantile lines we have constructed quantile lines for L´ evy\nflights in the parabolic potential. In such a case the sta-\ntionary state exists and is given by the α-stable density,\nsee Eq. (10). Therefore, quantile lines should be paral-\nlel to the abscissa. Analogously, interquantile distance\nshould not change over time. This behavior is indeed\nobserved, e.g. in the bottom panel of Fig. 2.\nFurthermore, Fig. 2 compares interquantile distance\n(left column) and complementary cumulative distribu-\ntionsFc(x,t) = 1−F(x,t) = 1−/integraltextx\n−∞P(x′,t)dx′(right\ncolumn) for the stability index α= 0.9 and various po-\ntentials: V(x) = 0 (top panel), V(x) =|x|(middle panel)\nandV(x) =x2/2 (bottom panel). From these cases only\ntheV(x) =x2/2 case corresponds to the situation when\nstationary state exists. Top panel of Fig. 2 corresponds\nto the force free case in which no stationary density ex-\nists. In such a case, the interquantile distance grows like\nt1/α. The middle panel of Fig. 2 corresponds to the in-\ntermediate case, i.e. V(x) =|x|. Numerical simulations\nindicate that for V(x) =|x|andα= 0.9 no stationary\nstate exists as well, what is coherent with analytical ar-\nguments given above. Note that in all three cases the\ndecay of the probability densities follows the one of the\nnoise, albeit on different reasons. For free motion, this\ndecay follows the one of the noise due to the stability of\nthe process. For the case V(x) =|x|, where according to\nanalytical result the stationary state is absent, numerical\nsimulations suggest that the tail of the distribution func-\ntion is still dominated by the properties of the noise, and\nfollowsFc(x)∝x−αpattern. Finally, for the parabolic\npotential it is essentially given by Fc(x)∝x−(c+α−2), i.e.\nagain by Fc(x)∝x−αforc= 2.\nConsecutive figures present results for subharmonic po-\ntentials. Figs. 3–5 present results for V(x) =|x|1.5.\nFigs. 3–4 present: interquantile distance as a function\nof timet(Fig. 3) and complementary cumulative dis-\ntributions ( Fc(x) = 1−F(x)) at the end of simulation\n(Fig. 4). Various panels present result for various val-\nues of the stability index α:α={1.5,1.1}(left column,\nfrom top to bottom) and α={0.9,0.8}(right column,\nfrom top to bottom). Finally, Fig. 5 presents quantile\nlines (left column) and complementary cumulative dis-\ntributions (right column) for α= 0.7 with various scale7\n10-410-310-210-1100\n10-210-1100101102103104Fc (x)=1-F(x)\nx10-310-210-1100\n10-210-1100101102103104Fc (x)=1-F(x)\nx\n10-310-210-1100\n10-210-1100101102103104Fc (x)=1-F(x)\nx10-210-1100\n10-210-1100101102103104Fc (x)=1-F(x)\nx\nFIG. 7. Complementary cumulative distributions at the end\nof simulation for V(x) =|x|with different values of the sta-\nbility index α={1.7,1.6}(left column, from top to bottom)\nandα={1.5,1.4}(right column, from top to bottom). The\nscale parameter σis set to σ= 1. Solid lines present x−(α−1)\ndecay predicted by Eq. (21).\nparameters σ:σ= 1 (top panel), σ= 0.1 (middle panel)\nandσ= 0.01 (bottom panel).\nFigures 6–8 present results for V(x) =|x|. Figs. 6–\n7 present: interquantile distance as a function of time\nt(Fig. 6) and complementary cumulative distributions\n(Fc(x) = 1−F(x)) at the end of simulation (Fig. 7). Vari-\nous panels present result for various values of the stability\nindexα:α={1.7,1.6}(left column, from top to bottom)\nandα={1.5,1.4}(right column, from top to bottom).\nFinally, Fig. 8 presents quantile lines (left column) and\ncomplementary cumulative distributions (right column)\nforα= 1.3 with various scale parameters σ:σ= 1 (top\npanel) and σ= 0.1 (bottom panel).\nThe stability index αin Figs. 3–8 is chosen in such a\nway that Eq. (19) holds true. Even in this case the nu-\nmerical verification whether stationary states exist or do\nnot exist, due to possible slow convergence to a station-\nary state, is not trivial, see discussion of Figs. 5 and 8\nbelow.\nThe probability densities for the system described by\nEqs. (4) and (6) are symmetric with respect to the origin.\nTherefore the quantile lines for qyand|q1−y|overlap and\nare not distinguishable in left panels of Figs. 5 and 8.\nForV(x) =|x|1.5, Fig. 3 indicates that for sufficiently\nlarge values of the stability index αstationary densities\nexist. Although all the values of stability indices shown\ncorrespond to the domain where stationary states are pre-\ndicted to exist, the convergence to these states becomes\nslower when αapproaches 2 −c= 0.5 forV(x) =|x|1.5.\nFig. 6 shows the very similar behavior for V(x) =|x|.\nHere again the relaxation time to the equilibrium gets\nlarger when αapproaches 2 −c= 1.\nFigures 4 and 7 present complementary cumulative dis-\ntributions in the situation when Eq. (19) is valid. There-\nfore, according to Eq. (21), complementary cumulative10-1100101102103\n10-510-410-310-210-1| q...|\n1/t10-410-310-210-1100\n10-210-1100101102103104105106Fc (x)=1-F(x)\nx\n10-310-210-1\n10-510-410-310-210-1| q...|\n1/t10-310-210-1100\n10-210-1100101102103104Fc (x)=1-F(x)\nx\nFIG. 8. Quantile lines qyandq1−y(y={0.9,0.8,0.7,0.6},\nfrom top to bottom) as a function of 1 /t(left column) and\ncomplementary cumulative distributions at the end of simu-\nlation (right column) for V(x) =|x|with the value of the\nstability index α= 1.3 and the scale parameter σ:σ= 1 (top\npanel) and σ= 0.1 (bottom panel). In the right column, solid\nlines present x−(α−1)decay predicted by Eq. (21).\ndistributions should asymptotically decay as x−(c+α−2).\nIndeed, in Figs. 4 and 7 the decay predicted by Eq. (21)\nis observed.\nFigures 5 ( α= 0.7 andV(x) =|x|1.5) and 8 ( α= 1.3\nandV(x) =|x|) present quantile lines and complemen-\ntary cumulative distributions for different values of the\nscale parameter σ. The quantile lines are plotted as func-\ntions of 1 /tin order to elucidate convergence. Thus, in\nFig. 5 the convergence of q0.9is not achieved over the time\nof simulation for σ= 1, but is approached for σ= 0.01.\nCorrespondingly, the tail of the complementary cumula-\ntive density differs from theoretical predictions for σ= 1\nand approaches its theoretical asymptotics for σ= 0.01.\nThe same is true for the data in Fig. 8, where the con-\nvergence is achieved already for σ= 0.1. This indicates\nthat the problem of the slow convergence to the station-\nary state has to be taken seriously.\nIV. SUMMARY AND CONCLUSIONS\nWe considered the problem of existence of stationary\nstates in power-law subharmonic potentials V(x) =|x|c,\nc <2, under the action of L´ evy-stable noise character-\nized by the stability index α. The scaling analysis of the\nfractional Fokker-Planck equation leads us to the conclu-\nsion that such states exist if α >2−c. This conclusion\nis corroborated by an alternative argument based on the\ndecomposition of the L´ evy noise. For subharmonic po-\ntentials, the probability density functions of stationary\nstates are characterized by the asymptotic power-law de-\ncayP(x)∝ |x|−ν=|x|−(c+α−1)for|x| → ∞ , which is\nslower than decay of the tails of the corresponding L´ evy\ndistributions. Monte Carlo simulations of the Langevin8\nequation confirm these analytical findings, at least for α\nnot too close to 2 −c. The convergence to the stationary\nstate for αapproaching 2 −cis very slow, which poses\nconsiderable numerical problems, since both long times\nand very small values of scaling parameter σare required.\nHowever, for σsmall enough the tendency to converge is\nstill observable.\nACKNOWLEDGMENTS\nThe authors acknowledge the financial support by\nDFG within SFB555. AVC acknowledges financial sup-port from European Commission via MC IIF, grant\n219966 LeFrac. 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Gen.\n39, L237 (2006)."    },    {        "title": "1006.3232v1.Signature_of_pseudogap_formation_in_the_density_of_states_of_underdoped_cuprates.pdf",        "content": "arXiv:1006.3232v1  [cond-mat.supr-con]  16 Jun 2010Signature of pseudogap formation in the density of states of underdoped cuprates\nA. J. H. Borne1,2,3, J. P. Carbotte4,5, and E. J. Nicol1,2∗\n1Department of Physics, University of Guelph, Guelph, Ontar io, Canada N1G 2W1\n2Guelph-Waterloo Physics Institute, University of Guelph, Guelph, Ontario, Canada N1G 2W1\n3PHELMA Grenoble INP, Minatec, 3 Parvis Louis N´ eel, BP 275, 3 8016, Grenoble, Cedex 1, France\n4Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1 and\n5The Canadian Institute for Advanced Research, Toronto, Ont ario, Canada M5G 1Z8\n(Dated: August 7, 2018)\nThe resonating valence bond spin liquid model for the underd oped cuprates has as an essen-\ntial element, the emergence of a pseudogap. This new energy s cale introduces asymmetry in the\nquasiparticle density of states because it is associated wi th the antiferromagnetic Brillouin zone.\nBy contrast, superconductivity develops on the Fermi surfa ce and this largely restores the particle-\nhole symmetry for energies below the superconducting energ y gap scale. In the highly underdoped\nregime, these two scales can be separately identified in the d ensity of states and also partial density\nof states for each fixed angle in the Brillouin zone. From the t otal density of states, we find that the\npseudogap energy scale manifests itself differently as a fun ction of doping for positive and negative\nbias. Furthermore, we find evidence from recent scanning tun neling spectroscopy data for asymme-\ntry in the positive and negative bias of the extracted ∆( θ) which is in qualitative agreement with\nthis model. Likewise, the slope of the linear low energy dens ity of states is nearly constant in the\nunderdoped regime while it increases significantly with ove rdoping in agreement with the data.\nPACS numbers: 74.72.-h,74.20.Mn,74.55.+v\nI. INTRODUCTION\nBCS theory modified to account for d-wave symme-\ntry of the superconducting order parameter has provided\na solid basis for a first understanding of the properties\nof the cuprates around optimum and in the overdoped\nregime. However, the underdoped region of the phase\ndiagram provides challenges to such a simple approach.\nThese systemspresent manyfeatures that havebeen con-\nsidered anomalous and are not part of simple BCS the-\nory. While the subject remains controversial and very\ndifferentideashavebeen put forwardtounderstandthese\nanomalies, a recentlydeveloped model by Yang, Rice and\nZhang (YRZ)1which is based on a spin liquid resonating\nvalence bond approach, has had considerable success in\nthis direction. It isverydifferentfromthe preformedpair\nmodel2whereasuperconductinggapformsatahightem-\nperature, the pseudogap temperature T∗, and the super-\nconductivity appears only at a lower temperature Tcas a\nresult of the onset of phase coherence. The YRZ model\nhas a central element, a new energy scale, the pseudogap\n∆pg, which is responsible for changes in the electronic\nstructure of the normal state above Tc. The pseudogap\nis distinct from the superconducting gap ∆ scand the su-\nperconductingstateisconceivedasformingfromthisnew\nnormal pseudogap phase which is quite different from an\nordinary Fermi liquid state. In the YRZ model, the large\nFermi surface of Fermi liquid theory (FLT) reconstructs\ninto hole and electron pockets as a result of the growth\nin pseudogap at doping levels x, in the underdoped re-\ngion of the cuprate phase diagram, with x < x cwhere\nxcis the doping associated with a quantum critical point\n(QCP).\nThe model is related but different from other compet-ing order proposals such as D-density wave formation3,4\nand has the desirable property that, in its final form,\nit remains simple and has been successfully applied to\nthe calculation of many superconducting properties5–12.\nBesides a self-energy which accounts for the formation\nof a pseudogap on the antiferromagnetic Brillouin zone\n(AFBZ) boundary, the model has Gutzwiller factors\nmodifying the underlying band structure parameters.\nThese narrow the bands as correlation effects become\nmore important. Also, a Gutzwiller factor accounts for\nthe loss of coherence which greatly reduces the weight of\nthe remaining quasiparticle peak as the incoherent back-\nground increases. These elements account for the ap-\nproach to the Mott insulating state which is best under-\nstoodnearhalf-fillinginalocalizedpictureofelectrondy-\nnamics. The hopping from one site to another is blocked\nby a largeHubbard Uwhich describes the energycost for\ndouble occupancy. The pseudogap and AFBZ then plays\na role similar to a band gap at the Brillouin zone in or-\ndinary band theory but with essential differences. For\nexample, in YRZ, the bands are not filled rigidly with\ndecreasing doping but instead undergo profound changes\nas the pseudogap increases.\nAmong the properties already calculated and com-\npared with experiment are Raman5,6, specific heat7, op-\ntical properties8, aspects of angular resolved photoemis-\nsionspectroscopy(ARPES)andscanningtunnelingspec-\ntroscopy (STS)9including the checkerboard pattern10,\nAndreev tunneling11and also the penetration depth12.\nEach of these properties show behaviors which cannot\nbe understood in d-wave BCS nor in it extensions to in-\nclude inelastic scattering13–20, anisotropy21–25, or strong\ncoupling effects rooted in Eliashbergtheory24–26. Among\nthe previously considered anomalous properties that are\nnow understood are the two distinct gap scales seen in2\nRaman spectra. The B 2gpeak decreases with decreas-\ning doping while the B 1gscale increases instead5,6. The\nnormalized jump in the specific heat drops rather precip-\nitously as xdecreases towards the bottom of the super-\nconductingdome7. New structuresareseenin the optical\nself-energy and the two energy scales found in the partial\noptical sum as a function of energy areunderstood8. The\nrapid drop in the superfluid density at zero temperature\nwith decreasing xwhile the slope of the low tempera-\nture linear in Tlaw is relatively only weakly changed is\nalso shown12. Encouraged by these successes, here we\nconsider the quasiparticle density of states (DOS).\nIn Sec. II, we present the formalism of YRZ1for the\nelectronspectraldensityin the underdoped regime. Itin-\ncludes pseudogap formation below a QCP at a doping of\nx=xc= 0.2. We describe how this new energy scale\nmodifies the electronic structure from the usual large\nFermi surface of FLT to Luttinger pockets. There are\ntwo energy branches in the theory E±\nk, withE−\nkgiving\na hole pocket and E+\nkan electron pocket, the latter only\nin a restricted doping range just below xc. For very un-\nderdoped samples, only the hole pocket remains. In Sec.\nIII, we present our results for the quasiparticle DOS with\nand without superconductivity and also break up the re-\nsults into their partial contributions from each of the two\nenergy branches separately. We discuss how the pseudo-\ngap alone introduces an asymmetry between positive and\nnegative biases in the DOS N(ω) and how superconduc-\ntivity overrides this effect and so restores particle-hole\nsymmetry at small energies of order the superconducting\ngap energy ∆0\nsc. At higher energy, asymmetry remains.\nWe also show that modifications in the DOS introduced\nby superconductivity which are confined to a range of a\nfew times ∆0\nscin the case of a Fermi liquid extend instead\nover the range of the pseudogap energy scale in YRZ. In\nSec. IV, we consider decomposing the total quasipar-\nticle DOS into partial contributions from fixed angle θ\nmeasured from ( π,π) in the upper right quadrant of the\nBrillouin zone. These partial distributions can display\nseveral structures. Nevertheless, upon examination we\nare able to define for each direction θan energy gap asso-\nciated with a specific peak in the partial DOS. This peak\nwhich is taken mainly, but not always, to correspond to\nthe smallest energy closest to ω= 0, is sometimes the\nsuperconducting gap peak but can also be a pseudogap\npeak. The energies extracted in this way show consid-\nerable anisotropy between positive and negative biases\nwhich is a fundamental characteristic of the model used\nhere. We find evidence in experiment for this anisotropy\nand provide a comparison with STS data. In Sec. V, we\ndiscuss the limit of low bias where the DOS is linear in ω.\nWe recover the FL result with an extra Gutzwiller factor\nreflecting the strong correlations in the system. Section\nVI contains a summary and conclusions.II. FORMALISM\nThe spectral function for the coherent part of the elec-\ntronic propagatorin the model of Yang, Rice and Zhang1\ntakes the form\nA(k,ω) =/summationdisplay\nα=±gt(x)Wα\nk[(uα\nk)2δ(ω−Eα\nS)+(vα\nk)2δ(ω+Eα\nS)],\n(1)\nwhereEα\nS=/radicalbig\n(Eα\nk)2+∆2sc(k) are the quasiparticle en-\nergies in the superconducting state with ( uα\nk)2and (vα\nk)2\nthe corresponding Bogoliubov amplitudes. In Eq. (1),\ntheWα\nk(α=±) are the weighting factors of the YRZ\ntheory which involve the input pseudogap ∆ pg(k) and do\nnot change in the superconducting state. They depend\non the electronic band structure energies ξkas well as\nthe Umklapp surface energy ξ0\nk. Specifically,\nW±\nk=1\n2/parenleftBigg\n1±˜ξk\nEk/parenrightBigg\n. (2)\nwith˜ξk= (ξk+ξ0\nk)/2 andEk=/radicalBig\n˜ξk2+∆2pg(k). In\nthe nonsuperconducting state, the energies have two\nbranches E±\nk= (ξk−ξ0\nk)/2±Ekand in the supercon-\nducting state, the Bogoliubov weights are given in terms\nof these by\n(uα\nk)2=1\n2/parenleftbigg\n1+Eα\nk\nEα\nS/parenrightbigg\n, (3)\n(vα\nk)2=1\n2/parenleftbigg\n1−Eα\nk\nEα\nS/parenrightbigg\n. (4)\nIn the YRZ paper, the band energies, taken to\ninclude up to third nearest neighbor hopping, are\nξk=−2t(x)(coskxa+coskya)−4t′(x)coskxacoskya−\n2t′′(x)(cos2kxa+cos2kya)−µp, whereµpis the chem-\nical potential adjusted to obtain the correct number\nof electrons, through a procedure based on Luttinger’s\ntheorem. The Umklapp surface energy is where ξ0\nk=\n−2t(x)(coskxa+coskya) equals zero. Here ais the two-\ndimensional CuO 2plane lattice parameter. The form of\nthe hopping coefficients t(x),t′(x), andt′′(x) are fixed\nin the YRZ model1and will not be changed in this work\nexcept to note that t0enters as a proportionality factor\nin these hoppings and consequently all our results scale\nbyt0and so this parameter can be varied at will. The\nGutzwiller factor gt(x) which appears as a simple mul-\ntiplicative factor in Eq. (1) provides a measure of the\nremaining quasiparticle strength in the coherent part of\nthe Green’s function. Along with a second Gutzwiller\nfactorgs(x), it also enters the band structure, through\nt(x),t′(x) andt′′(x), which provides narrower bands as\nthe doping xis reduced towards the Mott insulating\nstate at half filling. Specifically, gt(x) = 2x/(1+x) and\ngs(x) = 4/(1+x)2.3\nFor the input superconducting gap, YRZ take\n∆sc(k) =∆0\nsc(x)\n2(coskxa−coskya) (5)\nand likewise the same form for ∆ pg(k) with the gap am-\nplitude ∆0\npg(x) replacing the superconducting gap am-\nplitude in Eq. (5). Both have the simplest d-wave form\ncharacterizedby the lowestharmonichaving the required\nsymmetry. The amplitudes in Eq. (5) and equivalently\nfor the pseudogap are\n∆0\nsc(x) = 0.14t0[1−82.6(x−0.2)2] (6)\n∆0\npg(x) = 3t0(0.2−x), (7)\nwhere both are taken to be proportional to t0and the\nQCP at which the pseudogap gap becomes nonzero is\nxc= 0.2, where the superconducting gap is also taken to\nhave its optimum value. This last condition can easily\nbe relaxed to have the maximum gap at 0.16 instead (as\nin experiment).\nIn Fig. 1, we show how the Fermi surface contours\nevolve with doping. At the QCP ( x= 0.2) there is no\npseudogapin the YRZmodel andthe Fermisurfaceisthe\nusual large open surface of Fermi liquid theory (FLT). As\nxis lowered into the underdoped regime, the Fermi sur-\nface reconstructs and Luttinger contours of zero energy\nemerge. They correspond to either E−\nk= 0 orE+\nk= 0.\nThe solutions for the first case give the hole pockets cen-\ntered on the nodal direction and these exist for θ=π/4\ntoθhas indicated in the figure. For the case of x= 0.19\nbut not for the other values of doping shown, there is an\nadditional electronpocket ( E+\nk= 0) locatedin the region\nbetween the AFBZ boundary (dashed red line) and the\nBrillouin zone which extends from θ=θetoθ= 0, where\nθis an angle measured from the origin ( π,π) in the right\nhand upper quadrant of the Brillouin zone as shown. In\nboth casesforholeandelectroncontours, solutionstothe\nequation E±\nk= 0 when they exist (for a given angle θ) al-\nways come in pairs. For some angles however, there is no\nsolution at all. When two solutions exist, the backside of\nthe Luttinger pocket closest to the AFBZ boundary (red\ndashed line) has only a small weight W±\nkas compared\nwith the front part which is orientated towards the cen-\nter of the Brillouin zone for the hole pocket and oppo-\nsitely for the electron pocket. Both of these have weight\nof order one and this is the piece of the Fermi surface\n(FS) which corresponds to the Fermi liquid (FL) when\nthe pseudogap goes to zero. The backsides instead go\ninto the AFBZ boundary and have weight exactly equal\nto zero in this same limit. As seen in the figure, when x\nmoves towards the Mott insulating state, the Luttinger\nhole pocket becomes increasingly short with θhmoving\ntowardthenodaldirectionandthisistheagencywhereby\nthe metallicity of the material is increasingly reduced.\nThe number of states with significant weighting and zero\nexcitation energy is reduced. The approach to half fill-\ning has a progressively stronger detrimental effect on theakxaky\nx=0.10\nx=0.16\nx=0.190 0.1 0.2 0.3\nx00.20.40.6\n∆0\nSC\n∆0\nPG(π,π)\n(π,0)(0,π)\n(0,0)Gap sizes\nθhθe\nFIG. 1: (Color online) The reconstructed Fermi surface con-\ntours for three values of doping: x= 0.10 (brown), 0.16 (yel-\nlow), and 0.19 (black), shown in the upper right quadrant\nof the square Brillouin zone. The black contours have both\nhole and electron Luttinger pockets (located around the nod al\nand antinodal directions, respectively). The angles measu red\nfrom (π,π) which define the end of these pockets are θhand\nθe, respectively. The inset shows the superconducting dome\nand pseudogap line as a function of doping defined via ∆0\nsc\nand ∆0\npgin the YRZ model.\ndynamicsofthe chargecarriersdue tothe increasedmag-\nnitude of the pseudogap which has opened on the AFBZ\nboundary.\nAn important point to note is that the branch E−\nkcor-\nresponds to negative energy except for momenta forming\nthe hole pocket, while E+\nkcorresponds to positive ener-\ngies except for momenta inside the electron pocket.\nIII. RESULTS FOR THE DENSITY OF\nQUASIPARTICLE STATES\nIn Fig. 2, we show the quasiparticle DOS correspond-\ning to each band separately in frames (b) and (c), N−(ω)\nandN+(ω), respectively, withthetotal N(ω) =N−(ω)+\nN+(ω) given in frame (a). Here, N±(ω) is the sum over\nthe Brillouin zone of gt(x)W±\nkδ(ω−E±\nk). The double-\ndashed dotted black curve for N−(ω) gives results for\nx= 0.2 which has no pseudogap and corresponds to\nthe usual Fermi liquid band structure. If to this we add\nthe contribution from N+(ω) in frame (c), we obtain the\nusual FL result shown as the double-dashed dotted black\ncurve for the total DOS in (a). It extends over an energy\nrange of order several t0and has a van Hove singularity4\n00.20.40.6N(ω)\nx=0.10\nx=0.14\nx=0.16\nx=0.18\nx=0.2000.20.4N-(ω)\n-0.5 -0.25 0 0.25 0.5\nω/t000.20.4N+(ω)(a) Total\n(b) E- only\n(c) E+ only\npocketelectronhole\npocketPG only\nFIG. 2: (Color online) Density of quasiparticle states N(ω)\nin the pseudogap state as a function of ωin units of t0. (a)\nshows the total DOS, and (b) and (c) give the partial results\nfrom the E−\nkandE+\nkbranches separately. The shaded yellow\nregion in (b) and (c) identifies the contribution of hole and\nelectron pockets, respectively.\nat an energy slightly above −0.25t0. The energy scale\non which the DOS can vary significantly, however, is set\nby the band width except for the rapid variation around\nthe van Hove singularity, but we will not emphasize this\naspect. Returning to the frames (b) and (c) of Fig. 2,\nthe areas shaded in yellow for emphasis correspond, re-\nspectively, to the contributions for hole and electron Lut-\ntinger pockets. Note in particular that the shaded region\nforN+(ω) exists only for dopings near optimum which\nis the only regime for which electron pockets appear. As\nthe doping is decreased towards the Mott transition, a\ngap forms in this band above ω= 0 and there is a sharp\nrise in DOS at some finite frequency above which states\nare seen to pile up before N+(ω) returns to a value closer\nto its no pseudogap value. The energy associated with\nthe sharp rise in N+(ω) can be identified as an effective\npseudogap value, it is different from the input pseudo-\ngap although it is of the same order in magnitude. In\nfact, the net effect of the input pseudogap can be a de-\npression which can even fall at negative energies as can\nbe seen clearly in the composite curve shown in the top00.20.40.6N(ω)\nx=0.10\nx=0.14\nx=0.16\nx=0.18\nx=0.2000.20.4N-(ω)\n-0.5 -0.25 0 0.25 0.5\nω/t000.20.4N+(ω)(a) Total\n(b) E- only\n(c) E+ onlySC + PG\nFIG. 3: (Color online) Density of quasiparticle states N(ω)\nwhen both the superconducting and pseudogap are included,\nas a function of ωin units of t0. Similar to Fig. 2, (a) shows\nthe total DOS and (b) and (c) give the partial results for E−\nk,S\nandE+\nk,Sbranches, respectively.\nframe (a) for x= 0.18. The DOS coming from the neg-\native branch E−\nkalso shows a clear effect of pseudogap\nformationwhencomparedwiththe double-dasheddotted\nblack curve. We see a shift of the main van Hove singu-\nlarity (present in the FL case) to lower energies, followed\nby a depression at higher energies beyond a second rela-\ntively smaller van Hovestructure. The position in energy\nofthis second structure canbe takenas a secondmeasure\nof an effective pseudogap.\nIn the upper curve for the total DOS, the two scales\nidentified as due to the pseudogap combine to give a dip\nin the FL DOS which initially, for small pseudogap, is\nconfined to the region of negative energies and is not\nat the Fermi surface as is often assumed in phenomeno-\nlogical models of the pseudogap. This dip grows with\ndecreasing xboth in range over which it extends and in\ndepth. For small xit ranges over a large energy region\nandbecomesadominant featurein the DOSwhich isalso\ngreatly reduced around the Fermi energy ω= 0. It is im-\nportant to emphasize that the pseudogap can introduce\nsignificant anisotropy between positive and negative val-5\n-0.5 0 0.5\nω/t000.10.20.30.40.50.6N(ω)SC only\nSC + PGx=0.16\n-0.11\n0.23-0.12\n0.120.08-0.08\n0.11-0.23 -0.22-0.30\n0.28\n0.300.22\nFIG. 4: (Color online) Comparison of the total DOS N(ω)\nversusωin units of t0for doping x= 0.16. Superconductivity\nis included in both curves. The dashed blue curve includes\nthe pseudogap.\nuesofωbeyondthe relativelymild particle-holeasymme-\ntry ofthe startingFL band structure. This is anessential\nelement of the YRZ model which would not arise if the\npseudogap opened on the Fermi surface rather than on\nthe AFBZ boundary. In this regard Fig. 3 is particularly\nrelevant. Itshowsourresultswhen, inadditiontoapseu-\ndogap, we include a superconducting gap. Because this\nsecond gap opens on the Fermi surface it produces a new\ntotal DOS which is much more particle-hole symmetric\naboutω= 0 than was the underlying pseudogap-only\nDOS, for energies of order of the superconducting gap.\nBefore emphasizing this important point further, we note\nthat the opening of the superconducting gap has pushed\nsome spectral weight in N+(ω) to lowernegative energies\ndue to the Bogoliubov coherence factors characteristic of\nCooper pairing. The peak at the gap at negative energies\nis quite significant in magnitude.\nIn Fig. 4, we compare results of calculations for x=\n0.16 when only a superconducting gap is present (solid\nredcurve) andwhen in addition there isalsoa pseudogap\n(dashed blue curve). For the solid curve, the supercon-\nducting coherence peaks fall symmetrically at ω=±0.11\neven though the underlying FL DOS has a van Hove sin-\ngularity which appears only at negative bias. There is\nas well a second peak at ω=±0.23 which is the nor-\nmal state van Hove singularity slightly shifted by the su-\nperconductivity and now appearing at both positive and\nnegative biases although its positive bias image is greatly\nsuppressed in amplitude. More surprisingly for the pseu-\ndogap case, the symmetry between positive and negative\nωremains in that there are peaks at ω=±0.08 and\n±0.12 as well as at ω=±0.22 and even at ω=±0.30.\nBut of course the magnitude of each peak in a given pair\ncan be very different. The asymmetry prominent in the\nnonsuperconductingstatehasbeen greatlysuppressedby\nthe onset of the superconducting gap. Finally note that0.1 0.12 0.14 0.16 0.18 0.2x00.10.20.30.4∆/t0∆0\nSC(x)\n∆0\nPG(x)\n∆1(SC+PG)\n∆2+(SC+PG)\n∆2-(SC+PG)\n∆1(SC)\n∆1(SC)*cos(2 θh)\nFIG. 5: (Color online) Energy scales extracted from total\nquasiparticle densityof states as afunction of doping x. Black\nsolid and dashed curves are for comparison and are the ampli-\ntudes of the input gap ∆0\nsc(x) and ∆0\npg(x), respectively. The\nsolid red squares are taken from the superconducting coher-\nence peak position when the pseudogap is set to zero and the\nopen red squares are cos(2 θh) times the value of the solid red\nsquares. With the pseudogap on, the positive and negative\nbiases give different energy scales associated with resolva ble\npeaks in N(ω). The green triangles are the superconducting\ngap peaks at larger dopings x/greaterorsimilar0.16 and the solid blue dots\nare secondary superconducting gap peaks associated with th e\nend of the Luttinger hole pocket. For x/lessorsimilar0.16, the filled\nand open triangles are the pseudogap peaks for positive and\nnegative bias, respectively.\nat low bias, the dashed blue curve and solid red curve\nagree very well and the introduction of the pseudogap\nhas had no effect on this region of energy. We will return\nto this issue later in Sec. V.\nHavingjust emphasized the restorationofparticle-hole\nsymmetry through superconductivity, we explore next\nthe asymmetry which nevertheless remains in the quasi-\nparticle DOS and which can be traced to the pseudo-\ngap. In Fig. 5, the solid black curve gives the input\nsuperconducting gap amplitude ∆0\nsc(x) as a function of\ndoping from x= 0.2 (the QCP) to x= 0.1, close to\nthe end of the superconducting dome for the parameters\nused by YRZ. The solid red squares were obtained in the\nFL calculations based on the underlying large Fermi sur-\nfaces assuming ∆0\npg(x) = 0 at all dopings. These points\nare taken from the energy of the superconducting coher-\nence peaks in these calculations and fall slightly below\nthe input values for ∆0\nsc(x). This is expected since the\npeak in the DOS is representative of the extremal value\nof the superconducting gap on the Fermi surface rather\nthan in the Brillouin zone and these are slightly different.\nWhen both the superconductivity and the pseudogap are\npresent, the peak structure in N(ω) becomes more com-\nplex. The lowest energy peaks in the top frame of Fig. 3\nare associated with the superconducting gap. Starting\nwith the most underdoped case first, solid blue curve for\nx= 0.1, we see suppressed coherence peaks as compared6\nto their magnitude in the optimum case x= 0.2 (black\ndouble dashed-dotted curve). Also, the energy at which\nthey occur is considerablyless than the value ofthe input\ngap amplitude. We plot these in Fig. 5 as the solid blue\ncircles and see that they trace a dome, but in all cases\nfall considerably below the input gap amplitude curve\n(solid black line). These points correspond to the gap at\nthe edge of the Luttinger hole pocket and are therefore\nconsiderably smaller in energy than ∆0\nsc(x). In fact, we\nshow as open red squares, the product of the solid red\nsquares times cos(2 θh) (see Fig. 1) and these agree very\nwell with the solid blue circles. In all cases, the solid\nand open green triangles for positive and negative bias,\nrespectively, are the energies associated with the second\nsignificant or resolvable peak closest to ω= 0. These\npoints fall very close to the superconducting coherence\npeak energies of the corresponding FL for x/greaterorsimilar0.16 (the\ntriangles are underneath the solid squares for x≥0.17 in\nFig. 5) and are the same for positive and negative bias\nω,i.e.we have particle-hole symmetry in this case. We\nalso conclude that for this range of doping, these peaks\nare the primary superconducting coherence peaks even\nwhen the pseudogap is present and further they remain\nlargely unmodified from the FL ∆0\npg(x) = 0 case. In\nthis sense for these cases, the superconducting gap has\nlargely overridden the effect of the pseudogap. This all\nchanges for x/lessorsimilar0.16. In those cases, the energy of the\nsecond peak is quite different for positive and negative\nbias and these pseudogap peaks exhibit a great deal of\nanisotropy. This second energy scale is also seen in Ra-\nman scattering27where the nodal B2ggeometry probes\nthe superconducting gap scale and the B1gantinodal ge-\nometry, the pseudogap. As we have seen here, when op-\ntimum doping is approached, the two scales are found to\nmerge into a single superconducting gap scale. What is\ndifferent, however, is that the DOS peak structure can\nbe exploited to get information on the asymmetric effect\nof the pseudogap between positive and negative energies.\nOur results in Fig. 5 show that this can be considerable\nand that the effect sets in quite abruptly around x= 0.16\nwhen only the hole Luttinger pocket remains.\nFig. 6 shows results for the DOS at two dopings x=\n0.14 (left frames) and x= 0.18 (right frames). The top\nframes compare results for normal (red dashed curve)\nand superconducting state (black solid curve). The yel-\nlowsolidshadingandbluehatchedregionshelponetosee\nthat our calculations conserve the number of states be-\ntweenthenormalandsuperconductingcaseastheymust.\nWhatisclearfromthefigureandwhatwewishtoempha-\nsize here is that when there is no pseudogap, the states\nlost below the superconducting gap are largely recovered\njust above the coherence peak except for a small shift in\nthe van Hove (VH) singularity which introduces a slight\ncomplication that would not be present in models with\na constant DOS. When the pseudogap is included there\nis further readjustment of spectral weight introduced by\nthe transition to the superconducting state. The energy\nrangeoverwhich there remainssignificantchangesis now00.20.40.6\nSuperconducting state\nNormal statex=0.14 x=0.18\n-0.2500.2500.20.4\n-0.2500.25ω/t0N(ω)PG off PG off\nPG on PG on(a) (b)\n(c) (d)SCSCVHVH\nSCSC\nSCSC\nSCSCVHVH\nFIG. 6: (Color online) The quasiparticle DOS N(ω) with the\npseudogap off [(a) and (b)] and on [(c) and (d)] for two doping\nvaluesx= 0.14 (left frames) and 0.18 (right frames). Com-\nparison is made between the normal state (red dashed curve)\nwhich has no superconductivity and the case with supercon-\nductivity(blacksolidcurve). Theyellowsolidshadingand the\nblue hatched shading help to see the conservation of states.\nset by the pseudogap scale. Here we are not emphasizing\nthe slight complication brought about by the existence of\na van Hove singularity in the underlying band structure\nchosen in the YRZ model.\nIV. DECOMPOSITION OF DOS IN ANGLES IN\nTHE BRILLOUIN ZONE\nWe turn next to the decomposition of the total DOS\ninto partial contributions coming from different angular\nslices in the Brillouin zone. Such a decomposition has\nbeen considered by Pushp et al.28in relation to their\nexperimental scanning tunneling spectroscopy (STS) re-\nsults. Theyeffectivelydecomposethetotal N(ω)bywrit-\ning\nN(ω)≡/integraldisplay2π\n0dθ\n2πN(ω,θ), (8)\nwhere for N(ω,θ), they take the explicit analytic func-\ntional form\nN(ω,θ) =Re/braceleftbiggω−iΓ/radicalbig\n(ω−iΓ)2−∆(θ)2/bracerightbigg\nW(θ),(9)\nwhere Γ is a smearing parameter to be varied to get a\nbest fit of the data along with the value of the gap ∆( θ)\nand weighting factor W(θ). In this way, a gap scale ∆( θ)\ncan be obtained as afunction of θ. Inspired by the above,7\n-0.2500.25N(ω,θ)(a) x=0.12\n-0.2500.25(b) x=0.18\nθ=π/4\nθ=10π/100\nθ=5π/100\nθ=0SC + PG\nω/t0\nFIG. 7: (Color online) The partial DOS N(ω,θ) as a function\nofωin units of t0for two values of doping (a) x= 0.12 and\n(b)x= 0.18. From top to bottom, the curves are shown from\nθ=π/4 toθ= 0 in steps of π/100.\nwe decompose the full density of states\nN(ω) =/summationdisplay\nkA(k,ω), (10)\nwhereA(k,ω) is given in Eq. (1), into an integration\nover the magnitude of momentum kfor fixed angle θ\nin the Brillouin zone. This gives N(ω,θ) directly and\nresults are presented in Fig. 7. Frame (a) is for doping\nx= 0.12 and (b) for x= 0.18. Twenty-six values of\nθare shown between 45◦and 0◦as measured from the\n(π,π) point of the Brillouin zone (see Fig. 1). The nodal\ndirection (top curves) show small coherence peaks due to\nsuperconductivity which are centered about ω= 0 and\nare particle-hole symmetric. At higher positive energies,\nwe see that there are two more pseudogap peaks while at\nnegative bias, a van Hove singularity is seen.\nIn Fig. 8, we show results for E+\nkandE−\nkas a function\nof absolute value of momentum |k|=kmeasured from\n(π,π) for several values of angle θmeasured similarly as\nthe angles shown in Fig. 1 and used for Fig. 7. Doping0 1 2 3 4\n|k|-2-1012Ek+, Ek-\n0.1 0.15 0.2x0.20.30.4EPG, Diracfrom fig. 7\n[4t''(x)-µp]/2\nx=0.14θ=π/4\nθ=0θ=15π/100\nθ=10π/100Ek+\nEk-\n(π,π)Ek=0.24\n|k|=2.45 (nodal)\nFIG. 8: (Color online) The variation of the energies E+\nkand\nE−\nkwith magnitude of momentum k=|k|, measured from\n(π,π), for various fixed values of the angle θdefined in Fig. 1\nas indicated in the figure. The inset gives the energy of the\npseudogap Dirac point as a function of doping x. Results for\nour complete numerical calculations from curves as shown in\nFig 7 are compared with the approximate formula Edirac=\n(4t′′−µp)/2.\nwas set at x= 0.14. The pseudogap Dirac point falls at\nθ=π/4 andk= 2.45 in units of 1 /a(solid black curves)\nand this corresponds to E+\nk=E−\nk= 0.24 in units of the\nunrenormalized nearest neighbor hopping parameter t0.\nAs the angle θis increasedawayfromthe nodal direction,\nthe energies E+\nkandE−\nkno longer meet but split and\nthere is a clear gap between them. Both move down in\nenergy but not by the same amount. Also, the maximum\nand minimum corresponding to a given pair of curves\ndo not fall at exactly the same value of k. In the above\nsense, thetwovanHovesingularitiesatissuewhichdefine\nthe pseudogap peaks in N(θ,ω) of Fig. 7 are not entirely\nsymmetric. The energies of the Dirac point can easily\nbe traced as a function of doping. It corresponds to the\npointE+\nk=E−\nkwhich occurs for ξk+ξ0\nk= 0,i.e.Ek=/radicalBig\n˜ξ2\nk+∆2pg(k) = 0 for θ=π/4, and the details of the\ndispersion curves determine the critical value of k≡kc.\nForx= 0.14, thiskc= 2.45 in Fig. 8 which is somewhat\nlarger than the value π/√\n2 which corresponds to kx=\nky=π/2. For this latter point, the dispersion curves\nof YRZ are particularly simple and only third neighbor\nhopping survives. Taking Ek= 0 but using kx=ky=\nπ/2 to evaluate the remaining expression, provides an\napproximate estimate for the energy of the Dirac point\nEdiracas [4t′′(x)−µp]/2. This estimate is shown in the\ninset of Fig. 8 to be quite adequate and shows how Edirac\nmoves towards zero energy as xdecreases, as is also seen\nin Fig. 7 where the pseudogap Dirac point is seen as the\nclosing of the pseudogap coherence peaks at θ=π/4.8\nReturning now to this Fig. 7, we note that as θis de-\ncreased towards the antinodal direction, it remains pos-\nsible to identify unambiguously a lowest energy peak and\nthe magnitude of the energy at which these peaks occur\nis recordedon Fig. 9 as a gap in units of t0for each angle.\nFor optimum or even near optimum doping, the resulting\ncurveis closeto asimple cos(2 θ) curveand representsthe\nsuperconducting gap. However, it should be noted that\nthe curves for x= 0.18, 0.17 and 0.16 all show an ad-\nditional structure between 10◦and 15◦and the overall\ncurve does deviate visibly from a simple cos(2 θ). This is\nnot surprising since near x/lessorsimilar0.2, the Fermi surface, as\nwe haveshownin Fig. 1, has reconstructedfrom the large\nopen surface of FLT to Luttinger surfaces (holes about\nthe nodalregionandelectronsaboutthe antinodalpoint)\nand the partial DOS N(ω,θ) has some knowledge of this\nfact. Consequently, the extracted energy scale is not a\npure superconducting gap scale. Nevertheless, the dis-\ntortions from a pure cos(2 θ) are not large and the curves\nshowalmostperfectparticle-holesymmetry(comparethe\ntop and bottom frame of Fig. 9 for x≥0.16).\nThe situation is quite different and more interesting\nfor low dopings approaching closer to the Mott transi-\ntion. In that case, only a Luttinger hole pocket remains\nas shown in Fig. 1. When superconductivity is not con-\nsidered, there are zero energy excitations along the Lut-\ntinger contours which are real Fermi surfaces and these\nare gapped by superconductivity. But for angles smaller\nthanθh, no true Fermi surface exists and consequently,\nthe peaks seen in N(ω,θ) [Fig. 7(a)] no longer have their\norigin in ∆ sc(k) but are related rather to the pseudo-\ngap ∆ pg(k). This can be easily traced in Fig. 7(a) for\nx= 0.12. In the nodal direction, the peaks nearest ω= 0\nare the superconducting coherence peaks and these be-\ncome more prominent as the gap opens up. But eventu-\nally, particularly on the positive bias side, they start to\nloseintensity while atthe sametime, the pseudogappeak\nat higher energy remains quite intense and it is this peak\nthat must eventually be taken if we are to characterize\nN(ω,θ) for smaller values of θwith a single energy scale\nas we are doing here. It is clear that we need to jump\nfrom one energy scale to the other. On the negative bias\nside, however, the progression with decreasing θremains\nsmooth. We always take the large intensity peak closest\ntotheorigin ω= 0. ThesefactsarereflectedinourFig.9.\nIn the top frame for x≤0.14, we see two very distinct\ngap scales: a superconducting one for θnear 45◦and a\npseudogap one for θgoing towards zero. The curves have\na kink at an angle corresponding to θ=θh, the end of\nthe Luttinger pocket, but otherwise they show a rather\nsmooth behavior. On the other hand, for positive biases\n(lower frame of Fig. 9) there is a clear jump in the curves\nas we transfer from superconducting to pseudogap scale\nand this is followed by a progressive drop to lower values\nasθdecreases towards zero in sharp contrast to the top\nframe for negative ω.\nTo make clearer that our results for the frequency de-\npendence of N(ω,θ) cannot be fit by the simple for-00.10.20.3x=0.10\nx=0.12\nx=0.14\nx=0.16\nx=0.17\nx=0.18\nx=0.20\n0 10 20 30 40\nθ (deg)00.10.20.3PG + SC\nPG + SC positive biasnegative bias∆/t0(a)\n(b)\nFIG. 9: (Color online) The energy gap read off the curves\nforN(ω,θ) of the type shown in Fig. 7 as a function of θ\nin the Brillouin zone measured from ( π,π). The top frame is\nfor negative bias (occupied states) andthe bottom for posit ive\nbias (unoccupiedstates). Notetheisotropy betweenthese t wo\nsets of datafor angles around 45◦orπ/4(nodal direction) and\nthe large anisotropy at small angles (antinodal).\nmula of Eq. (9), we have used this formula along with\nthe gaps presented in Fig. 9 to recalculate a total DOS\nN(ω) according to Eq. (8) with weights W(θ) set equal\nto one for simplicity. Numerical results are presented in\nFig. 10(b). For ease of comparison, we have reproduced\nin Fig. 10(a) the full DOS already presented in Fig. 3 on\nwhich our angular decomposition is based. For small val-\nues ofω, we see a great deal of agreement between these\ntwo sets of figures. Near optimum doping only the su-\nperconducting gap scaleis prominent while for the highly\nunderdopedregimeboth superconductingandpseudogap\nscales are clearly seen. At larger energies important dif-\nferences arise and these have their origin in the failure\nof Eq. (9) with a single energy scale to capture all the\ndetails present in the partial densities of states of Fig. 7.\nIn particular, the van Hove singularity seen on the nega-\ntive bias side in the top frame is completely missed in the\nlower frame. While this figure speaks to the limitation in\nthe method used to extract energy gaps from STS data28\nit also shows that important qualitative and even semi-9\n00.20.40.6N(ω)x=0.10\nx=0.14\nx=0.16\nx=0.20\n-0.5 0 0.5\nω/t00123Neff(ω)PG+SC\nPG+SC(a)\n(b)\nFIG. 10: (Color online) Comparison of (a) the original DOS\nN(ω) from Fig. 3 with (b) the quasiparticle DOS Neff(ω) ver-\nsusωinunitsof t0as definedbyEqs.(8)-(9). Both (a)and(b)\nrefer to the state with superconductivity and the pseudogap\npresent.\nquantitative information can be extracted in this way.\nSome details are certainly missed but other important\nfeatures are quite prominently seen such as the pseudo-\ngap scale and its asymmetry. One could improve the\nagreement between the top and bottom frame in Fig. 10\nby including weights W(θ) or multiplying Neff(ω) by an\nenvelope function, which partially accounts for the pres-\nence of the van Hove singularity present in our model\nband structure, but this is not our aim here.\nIn Fig. 11, we provide a direct comparisonbetween our\nresults for the asymmetry between positive and negative\nbiasofthederivedangulardependentgapandthedataof\nPushp et al.28for the same quantity. Plotted is the ratio\nof ∆−bias/∆+biasfrom our Fig. 9 for x= 0.12 (squares)\nand 0.14 (circles). Pushp et al.28show several curves\nin their figure S3 for a range of data taken from several\nspots on their UD58 sample. As the negative bias data is\nfairly uniform, we took points from alongthe trend ofthe\ndata. For the positive bias, there is a range of values for\neach angle and so we took the highest and lowest values\nto form separately the ratio with the negative bias data.\nThis is shown as the blue triangles and green diamonds,0 10 20 30 40\nθ (deg)0.511.52∆- bias/∆+ bias\nx=0.12\nx=0.14\nPushp Bi2212, lower values\nPushp Bi2212, upper values-100 0 100\nBias (mV)00.511.5Normalized conductancePushp Bi2212\nNSC+PG\nFIG. 11: (Color online) The ratio of ∆ −bias/∆+biasversus\nangle, takenfrom Fig. 9for x= 0.12and0.14. Thetheoretical\nwork is compared with the data of Pushp et al.28which is\nbounded by the diamonds and triangles (see text for details) .\nInset: a comparison between the theory and experiment for\nthe normalized conductance (thick black curve) of Pushp et\nal. and a DOS curve from our calculations (thin red curve).\nrespectively. As we wish to concentrate on the asym-\nmetry, only the data for angles less than 30◦are shown.\nDue to the expected symmetry for angles in the nodal re-\ngion, the data ratio towards the nodal region have been\nnormalized to one to facilitate comparison with theory\neven though the ratio in the data was slightly greater\nthan one. Our procedure is not entirely rigorous and in\nthe hands of the experimentalists there might be a more\naccurate analysis of the result. Nevertheless, while the\nindividual curves for ∆ +and ∆ −are quite different in\nshape as compared with our theoretical results, we find\nthat the ratio shows the same qualitative trend as the-\nory. There is a significant dip around 15-20◦followed\nby a rise as the antinodal region is approached. This\ncomparison provides evidence that the pseudogap forms\nasymmetrically about the Fermi surface as in our model.\nIn the inset to Fig. 11, we show results for the DOS in\nthe resonating valence bond spin liquid compared with\nUD35 data from Pushp et al.28(their figure S4). We\nusedx= 0.11 to match the Tcreduction from optimum\nand have added broadening and a linear incoherent back-\nground to provide a better fit to the data. This linear\nbackground does not alter the structures due to the two\nenergy gaps in the model. The agreement is good in\nthe low energy region shown and it again clearly reveals\nasymmetry between positive and negative biases.\nV. ZERO FREQUENCY LIMIT\nWe finally turn to the slope of the DOS at ω→0\nshownin Fig. 12(a). In YRZ theory, the superconducting\nDirac point [shown schematically by Fig. 12(b)], on the10\n0 0.05 0.1\nω/t000.050.10.15N(ω)\n0.15 0.2 0.25 0.3x02468slopes=gt/(πv∆vF)\ncorrected s\nexperimentsx=0.14\nSC + PG(a)\nFIG. 12: (Color online) (a) The slope of N(ω) normalized to\nthat for optimal doping versus xcompared with the datafrom\nPushpet al.28. Inset: N(ω) forx= 0.14 with both the su-\nperconducting gap and pseudogap present showing the linear\nbehavior at low ωin agreement with Eq. (11) which is shown\nas the red dashed line. (b) A schematic of the superconduct-\ning Dirac cone as it grows in energy out of the nodal region on\nthe heavily weighted side of the Luttinger hole pocket. The\nheavy solid black curve is the Luttinger contour for x= 0.14.\nheavily weighted side of the Luttinger hole pocket in the\nnodal direction, is not affected by pseudogap formation.\nFor small energies ω, it is clear from the cone shown in\nFig. 12, that the only excited states available are those\nnear the bottom tip of the Dirac cone. We can easily\nshow that, as for an ordinary FL29, the slope is given by\ns=gt\nπv∆vF(11)\nwithN(ω) =s|ω|. Here, vFis the Fermi veloc-\nity at the Dirac point and v∆=|∇∆sc(k)|kF=\n(∆0\nsc/√\n2)|sin(kFx)|, is the superconducting gap velocity\nat this same point. In the inset of Fig. 12(a), we show\nresults of complete numerical calculations of the DOS for\nthe case of x= 0.14 with both pseudogap and supercon-\nducting gap included (solid black curve) and compare\nwith the results of Eq. (11) (dashed red line). We seegood correspondence.\nFormula (11) is the same as would hold in a FL with\ntwo very important differences: gtand a possible varia-\ntion ofv∆in this model. In Eq. (11), the Gutzwiller co-\nherence factor gt(x) carries the information on how much\nweight remains in the coherent part of the Green’s func-\ntion when correlation effects are included, with the rest\nshifted to incoherent processes. This factor is a crucial\npart of our present approach but does not enter FLT.\nAn important consequence of this fact is that it reduces\nthe rise in the slope in the highly underdoped regime of\nthe cuprate phase diagram as compared with a pure FL\napproach. It provides a factor of 2 x/(1 +x) while the\ngap velocity is proportional to Tc. Pushp et al.28found\nin STS data that the slope was nearly constant over a\nsignificant doping range below optimum. Matching their\nreduction from optimum of the Tcof their samples with\noursuperconductingdomein orderto determinethe rela-\ntion to the doping xused in our model, we plot the slope\nof their data normalized to the value at optimum versus\nx. As there was a range of values in their data for a\nparticular doping, we have done our best to indicate this\nas a point with a bracketing bar. Comparison with this\ndata (solid black circles in Fig. 12) shows that, around\nx≃0.10 which is the doping in our calculations that\ncorresponds to a reduction of Tcby a factor of 3 below\nits value at optimum, the predicted slope represented by\nthe dashed red curve starts to increase while experiment\ndoes not. This could mean that in reality the Gutzwiller\nfactor is a more strongly decaying function of xthan the\nonewehaveused. Alternatively,broadeningwillhavethe\ntendency to decrease the slope. However, so far we are\nneglecting another important effect associated with the\nYRZmodel. Inthismodelthegapto Tcratio2∆0\nsc/kBTc,\nwhich for simplicity we have fixed at a value of 6 in all\nour calculations, is known to vary importantly with x.\nIn very recent work, Schachinger and Carbotte30have\nsolved a generalized BCS gap equation with the pseudo-\ngap and Fermi surface reconstruction fully accounted for\nand have found that this ratio changes from its canoni-\ncal value of 4 .3 atx= 0.2 (optimum) to ∼6.5 or even\nhigher towards the end of the dome as the Mott insulator\nand antiferromagnetism is approached. Accounting for\nthis reduces the variation in slope between optimum and\nhighly underdoped by ∼50%, bringing our calculations\nmuch closer to experiment as indicated by the red dotted\ncurve in Fig. 12(a). The overall agreement between the-\nory and the data is very good. It is important to stress\nthatgt(x) provides an important factor in Eq. (11) which\nbrings theory much closer to experiment than what one\nwould find in a FL approach and this also is true as well\nfor the variation of gap to critical temperature ratio.\nVI. SUMMARY AND CONCLUSIONS\nWe have computed the total quasiparticle DOS in the\nresonating valence bond spin liquid model1of the under-11\ndopedcuprates. Whensuperconductivityisnotincluded,\nthe formation of the pseudogap, which provides a mech-\nanism for Fermi surface reconstruction, modifies the un-\nderlying Fermi liquid DOS in an asymmetric way with\nrespect to the Fermi energy ( ω= 0). For small values of\nthe pseudogap just below the critical doping associated\nwith a QCP, the resulting depression in N(ω) is confined\nto negative energies. As doping is reduced towards the\nMott insulating state, the depression deepens, covers a\nlarger range in energy and spans positive as well as nega-\ntive biases but its effect remains asymmetric. However, if\nsuperconductivity is also included, particle-hole symme-\ntry is restoredat lowerenergies oforder ∆0\nsc(x), although\nbeyond this range the asymmetry associated with the\npseudogap remains. This effect is traced to the fact that\nthe pseudogap is associated with the antiferromagnetic\nBrillouin zone rather than the Fermi surface where the\nsuperconducting gap opens.\nOne can trace peaks in the total quasiparticle DOS\nwhich are, at optimum doping and just below, coher-\nence peaks due to superconductivity but these evolve for\nx/lessorsimilar0.16 into pseudogap peaks which are distinctly dif-\nferent for positive and negative biases. This asymmetry\nis an intrinsic part of the YRZ model and can be used\nto test its validity. We are also able to trace a second\nset of peaks associated with the superconducting gap\nwhich however originate from the end of the Luttinger\nhole pocket at an angle θhin the Brillouin zone. In the\nheavily underdoped region of the phase diagram, these\npeaks are the only ones that can be identified as due\npurely to the superconducting gap. These signatures are\nrelatively weak, however, as found in, for example, the\nexperiments of Boyer et al.31, in comparison with those\nseen in an ordinary d-wave BCS superconductor. As a\nfunction of doping, the energy corresponding to these co-\nherence structures follow the dome associated with the\ncritical temperature although the scale of the effective\ngap involved is reduced.\nWhen the total DOS is decomposed into partial contri-\nbutionsfromeachangle θintheBrillouinzoneseparately,\nN(ω,θ), it is found that at some angles N(ω,θ) can show\ncomplex behavior as a function of energy ωand even dis-\nplayseveralpeaks. Suchdistributionscannotbemodeled\nby the simple function of Eq. (9). Nonetheless, we find\nthat we could identify, in a fairly unambiguous fashion,\na single peak in these distributions which was both close\nto the origin ω= 0 and had significant amplitude. This\nprovides a single energy scale for each direction θ. At\nangles near the nodal direction, this scale is associated\nwith the superconducting gap but as θis decreased it\nprogresses into a peak associated with the pseudogap en-ergy. In some cases the transition from one scale to the\nother is smooth while in others it can be abrupt and also\nsomewhat ambiguous. What is found is that the super-\nconducting gap peak at low energies rapidly loses inten-\nsity while the pseudogap peak at higher energies is quite\nintense and so dominates over the lower energy structure\nand thus must be chosen as characteristic of a single gap\nscale for this angle θin the Brillouin zone. This gap\ncan however differ considerably in size depending on the\nsign of the bias involved. For energy below the super-\nconducting gap scale, there is little asymmetry but this\nchangesradicallyastheantinodaldirectionisapproached\nin the highly underdoped regime where the pseudogap is\nprobed. Ourfindingshaveimplicationsforthe analysisof\nSTSdatawhen onewishestoextractdirectionalinforma-\ntion from such experiments.28Comparison with the data\nof Pushp et al. for both N(ω) and ∆ −bias(θ)/∆+bias(θ)\nshows indeed an asymmetry in the DOS which is in qual-\nitative agreement with the YRZ model.\nAnother resultis that the zeroenergyslope ofthe DOS\nis importantly reduced from its FL value in YRZ theory\nbecause of the appearance of a Gutzwiller factor gt(x)\nwhich is a rapidly decreasing function as xdecreases and\nrepresentsthe reduction, due to correlationeffects, of the\ncoherentpartofthechargecarrierGreen’sfunction. This\nfactor can partially compensate for the rapid increase in\nslope that would occur due to the appearance of the gap\nvelocityv∆in the denominator of the expression for the\nslope in ordinary BCS theory. However, this factor is\nmodulated by a rapid increase in the value of gap to crit-\nical temperature found to be the direct consequence of\nan increase in the pseudogap in the theoretical work of\nSchachinger and Carbotte.30Without these two effects\nthere would be a serious conflict between theory and the\nexperimental results of Pushp et al.28who find a slope\nwhich remains fairly constant in the underdoped regime.\nWhile a comparison with the existing data and YRZ pre-\ndictionsgivesqualitativeagreement,thereareindications\nthat the Gutzwiller factor gt(x) and/or the gap to Tcra-\ntio may in fact vary more rapidly with decreasing xthan\nindicated by theory.\nAcknowledgments\nWe thank James LeBlanc for his assistanceand helpful\ndiscussions. 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Hudson, Nature Phys.\n3, 802 (2007)."    },    {        "title": "1006.4503v1.Density_functional_theory_calculation_of_ground_state_energy__dipole_polarizability_and_hyperpolarizability_of_a_confined_helium_atom.pdf",        "content": "arXiv:1006.4503v1  [physics.atom-ph]  23 Jun 2010Density functional theory calculation of ground state ener gy,\ndipole polarizability and hyperpolarizability of a confine d helium\natom.\nSubhajit Waugha, Avijit Chowdhuryb, Arup Banerjeea,\n(a) Laser Physics Application Division, Raja Ramanna Centre for Adv anced Technology\nIndore 452013, India\n(b) BARC Training School at RRCAT, Raja Ramanna Centre for Adva nced Technology\nIndore 452013, India\nAbstract\nWe calculate ground-state energies and densities of a heliu m atom confined in an impenetrable\nspherical box within density functional theory. These calc ulations are performed by variationally\nsolving Kohn-Sham equation with the ground-state orbital e xpanded in terms of Slater-type or-\nbitals. Using the ground-state densities we then calculate static linear polarizability and nonlinear\nhyperpolarizability and study their variation with the rad ius of confinement. We find that polar-\nizability decreases monotonically with decreasing confine ment radius and the hyperpolarizability\nnot only decreases but also undergoes a change in sign in the s trong confinement regime.\nPACS numbers: 31.15Bs, 31.15Ew, 36.40Vz,\n1I. INTRODUCTION\nThe properties of atoms and molecules undergo drastic change whe n they are spatially\nconfined in either penetrable or impenetrable cavity as compared to their free counterparts.\nIn recent years the topic of confined atoms has been attracting lo t of attention and it has\nbecome a field of active research [1]. The main reason for this intere st in spatially confined\natoms and molecules is their applicability to several problems of physic s and chemistry. For\nexample, the model of an atom confined in an impenetrable sphere ha s been employed to\nsimulate the effect of high pressure on the physical properties of a toms, ions, and molecules\n[2–4]. The study of confined atoms provides insight into various prop erties of quantum\nnanostructures like quantum dots or artificial atoms [5, 6]. For mo re detailed discussion on\nthese applications we refer the reader to review articles [7–9].\nRecently numerous studies on helium atom confined in an impenetrable spherical cavity\n[4, 10–20] and in a penetrable spherical cavity [21, 22] as well have been reported in the\nliterature. Heliumatombeingthesimplestmany-electronsystem, th econfinedversionofthis\natomprovidesalucidwaytostudytheeffectofconfinement onthee lectroncorrelationwhich\narises due to coulomb interaction between the two electrons and pa uli exclusion principle.\nBesides helium atom, few studies on some more confined many-electr on atoms up to neon\natom have also been reported in the literature [11, 23, 24]. We note h ere that most of the\nstudiesonconfinedheliumatommainlyconsidered theground-state electronicproperties. In\ncontrastnotmanystudieshavebeencarriedoutontheelectricre sponsepropertieslikedipole\npolarizability and hyperpolarizablity of this system. To the best of ou r knowledge results for\nthe linear dipole polarizability of confined helium atom were reported on ly in Refs. [4, 23].\nHowever, no study devoted to the calculation of hyperpolarizability of confined helium atom\nexists in the literature. The main aim of this paper is to carry out calcu lation of not only\nlinear dipole polarizability but also nonlinear hyperpolarizability of confin ed helium atom\nand study the evolution of these quantities with the size of the cavit y (or strength of the\nconfinement). We wish to point out here that the calculation of both linear polarizabilty\n(α) and third-order hyperpolarizability ( γ) of confined hydrogen atom were carried out in\nRef. [25]. It was shown that both αandγstrongly depend upon the radius of confinement\nand moreover, γchanges sign and becomes negative under strong confinement. Th us it will\nbe interesting to find out when γof confined helium atom undergo reversal of sign.\n2In this work we carry out calculation of electric response propertie s by employing density\nfunctional theory (DFT) based variation-perturbation approac h [26]. To carry out these\ncalculations we need to have ground-state densities of confined he lium atom. This task\nhas been accomplished by employing a variational approach involving m inimization of the\nground-state energy fuctional within the realm of DFT. A brief des cription of the theoretical\nmethods employed for calculations of both ground-state densities and electrical response\nproperties are presented in Section 2. The Section 3 is devoted to t he discussion of results\nand paper is concluded in Section 4\nII. METHOD OF CALCULATIONS\nWe begin this section with a brief description of the method for obtat ining ground-state\ndensity of confined helium atom within the realm of DFT. The Kohn-Sha m (KS) equation\nof DFT are obtained by minimizing the energy functional (in atomic unit s) [27]\nEKS[ρ] =Ts[ρ]+J[ρ]+Exc[ρ]+/integraldisplay\nvext(r)ρ(r)dr. (1)\nHereTs[ρ] denote the kinetic energy functional of non-interacting particle s and in terms of\nsingle-particle orbitals ψi(r) it is represented as\nTs[ρ] =N/summationdisplay\ni=1/integraldisplay\nψ∗\ni(r)(−1\n2∇2)ψi(r)dr. (2)\nThese orbitals yield density of interacting system via\nρ(r) =N/summationdisplay\ni=1/summationdisplay\ns|ψi(r,s)|2. (3)\nIn Eq. (1) J[ ρ] represent the classical part of the electron-electron repulsion , Exc[ρ] is the\nexchange-correlation functional and the last term corresponds to the contribution due to the\nexternal potential vext(r). For confined atom vext(r) has two parts namely: (1) the nuclear\npotential −Z/r(where Z is the nuclear charge) and (2) the confining potential vconfdue to\nan impentrable spherical box of radius rcof the form\nvconf(r) =\n\n0r<rc\n∞r≥rc(4)\n3The minimization of E KS[ρ] with respect to single-particle orbital ψi(r) satisfying the con-\ndition given by Eq. (3) leads to so-called Kohn-Sham equation which is t he workhorse of\nDFT. In this paper we carry out this minimization explicitly by using appr opriate varia-\ntional forms for the single-particle orbitals. For this purpose we ex pand the single-particle\norbital of a confined helium atom as\nψ1s(r) =/summationdisplay\niciχi(r)fc(r) (5)\nwhere{ci}are the variational parameters which are determined by minimization of the\nground-state energy (Eq. (1)), f c(r) is the cut-off function which takes care of the confine-\nment boundary condition of density vanishing at r=rc, and the basis function χi(r) is given\nby the product of a Slater-type orbital (STO) for the radial part and a spherical harmonic\nfunction Y lm(θ,φ) for the angular part as\nχ(r) =Rnl(r)Ylm(θ,φ). (6)\nThe radial function R nl(r) is given by\nRnl(r) =rn−1e−ζr(7)\nwithnandζrepresenting orbital parameters which we choose from Ref. [11]. F or cut-off\nfunction f c(r) we choose both linear\nfc(r) =/parenleftbigg\n1−r\nrc/parenrightbigg\n(8)\nand the quadratic\nfc(r) =/parenleftbigg\n1−r2\nr2c/parenrightbigg\n(9)\nforms and investigate their performance in yielding ground-state e nergies of confined helium\natom.\nHaving described the variational approach for obtaining ground-s tate density of confined\nhelium atom we next briefly outline the method adopted in this paper to calculate static\nlinear (α) and nonlinear ( γ) polarizabilities. The response properties mentioned above are\ncalculatedbyemployingvariation-perturbation(VP)approachwith inDFT.Indensitybased\nVP, energy to order (2n + 1) is determined by the perturbation exp ansion of the density\ncorrect to order nonly. Further, the even-order energy correction E(2n+2)is minimum for\n4the exact induced density ρ(n+1), if expansion up to order nis known exactly. For details of\nthe VP approach within DFT, we refer the reader to reference [26]. For our purpose here\nit is sufficient to note that αandγare calculated from the second-order ∆ E(2)and the\nfourth-order ∆ E(4)change in energies respectively, by employing relations\nα=−2∆E(2)\nγ=−24∆E(4)(10)\nThese energy changes are in turn obtained variationally by minimising\nE(2)=/integraldisplay\nv(1)(r1)ρ(1)(r1)dr1+1\n2/integraldisplayδ(2)F[ρ0]\nδρ(r1)δρ(r2)ρ(1)(r1)ρ(1)(r2)dr1dr2,(11)\nwith respect to ρ(1), and\nE(4)=1\n2/integraldisplayδ(2)F[ρ0]\nδρ(r1)δρ(r2)ρ(2)(r1)ρ(2)(r2)dr1dr2\n+1\n2/integraldisplayδ(3)F[ρ0]\nδρ(r1)δρ(r2)δρ(r3)ρ(1)(r1)ρ(1)(r2)ρ(2)(r3)dr1dr2dr3\n+1\n24/integraldisplayδ(4)F[ρ0]\nδρ(r1)δρ(r2)δρ(r3)δρ(r4)ρ(1)(r1)ρ(1)(r2)ρ(1)(r3)ρ(1)(r4)\n×dr1dr2dr3dr4. (12)\nwith respect to ρ(2). Here v(1)(r) is the applied (external) perturbation. F[ ρ] is a universal\nfunctional of the density and it is given by the sum of the kinetic, Har tree and the exchange-\ncorrelation energies of the electrons. All the functional derivativ e in the equations above\n(Eqs.(11) and (12)) are evaluated at the ground-state density ρ0. For an atom placed in a\nstatic electric field Ealong z-axis the variational ansatz for ρ(1)andρ(2)are\nρ(1)(r) = ∆ 1(r)cosθρ0(r),\nρ(2)(r) = [∆ 2(r)+∆3(r)cos2θ]ρ0(r)+λρ(0)(r) (13)\nwhere\n∆i(r) =air+bir2+cir3+···,i= 1,2,3··· (14)\nwith a i, bi···being the variationals parameters. λis fixed for each set of parameters\nby the second-order normalization condition/integraltext\nρ(2)(r)dr= 0. Notice that the first-order\nnormalization condition/integraltext\nρ(1)(r)dr= 0 is automatically satisfied by ρ(1)(r) in Eq.(13).\nWe have used five parameters for ∆ 1and eight parameters each for ∆ 2and ∆ 3. Adding\n5more parameters does not affect the results significantly indicating their convergence. For\nevaluating the functional derivatives of the the exchange and cor relation energies, we use\nthe Dirac functional [28] for the exchange contribution and Gunna rsson-Lundquist (GL)\nparametrization [29] for the correlation energy within local-density approximation (LDA).\nIn the next section we discuss results obtained by us using above-m entioned methods.\nIII. RESULTS AND DISCUSSIONS\nWe begin this section with the discussions of the results for ground- state energy of a\nconfined helium atom obtained by us to assess their accuracies. In t his connection we note\nthat DFT based results for confined helium atom have already been r eported in Ref. [17]\nwhich were obtained by numerically integrating the KS equation with Dir ichilet boundary\ncondition [30]. In order to establish the accuracy of our variational results we compare them\nwith those of Ref. [17]. First of all we note that we perform calcualt ion with both linear\nand quadartic cut-off functions as given by Eqs. (8) and (9) respe ctively. We find that the\nground-state energies for several values of confinement radius rcobtained with quadratic\ncut-off function are close but slightly lower than the corresponding results obtained with\nlinear cut-off functions. Therefore, in the following we report resu lts only with quadratic\ncut-off functions.\nIn table I we present the results for the energies for the ground- state1S(1s2) of a confined\nhelium atom as a function of confinement radius rc. In this table we present the results for\nthe case of exact exchange (EXX) energy, which are obtained by s ubstituting EX=−EH/2\n(exact for two-electron systems) and E C= 0. This case corresponds to Hatree-Fock (HF)\napproximation and we compare our EXX results with those of Ref. [1 1]. We also present\nthe results obtained with exchange-only (XO) with E C= 0 and exchange-correlation (XC)\nenergies within LDA in second and third columns respectively. These r esults are compared\nwith the corresponding numbers of Ref. [17] which were obtained b y numerically solving\nthe Konn-Sham equation with Dirchilet boundary condition. In order to assess the accuracy\nof DFT based results we also display results obtained via correlated w avefunction based\ncalculation with 7-parameter Hylleraas expansion [18] in the last colum n of Table I. First\nwe note that the results for the case of EXX energy obtained by us match very well up to\n4-th decimal place with the results of Ref. [11] for all values of rc. This establishes the\n6accuracy of the variational method employed by us. The XO-LDA re sults obtained by us\nare close but slightly higer than the corresponding EXX values as long asrc≤1.0a.u..\nOn the other hand, for rc>1.5a.u.we find that trend is just reverse. Moreover, XO-\nLDA results are also quite close to the data available for the range rc= 2−6a.u.in Ref.\n[17]. With the inclusion of correlation energy term within LDA the groun d-state energies of\nconfined helium atom reduce slightly as compared to the correspond ing XO numbers. Our\nXC results match well with those of Ref. [17] and the difference in the two results are mainly\ndue to the use of two different XC functionals for the calculations. W e note here that in\nRef. [17] Pewrdew-Wang form of the correlation functional [31] alo ng with Dirac form for\nthe exchange energy functional has been employed whereas we em ploy GL parametrization\nfor the correlation part [29]. With the inclusion of correlation, the gr ound-state energy of\na confined helium atom decreases relative to the corresponding XO- LDA valuses as long as\nenergies remainpositive. Forconfinement radii withnegative groun d-stateenergies inclusion\nof correlation leads to lowering of the ground-state energy. Similar trend is also observed\nwith Perwdew-Wang XC functional.\nThe comparison of DFT based results with the corresponding Hyllera as wavefunction\nbased numbers clearly shows the EXX results are the closest to the latter. From these\nresults we conclude that the contribution of correlation energy bo th in strong ( rc≤1a.u.)\nand weak (rc≥1a.u.) confinemnets is not very significant to the total energy of a confi ned\nhelium atom. Thus for confined helium atom it is possible to get sufficient ly accurate results\nfor the ground-state energy provided exchange part of the ene rgy is accurately taken into\naccount.\nHaving established the accuracy of the DFTbased results for theg round-stateenergy ofa\nconfined helium atom we now present the results for static polarizab ilityαand second-order\nhyperpolarizabilty γof confined helium atom by employing the ground-state densities whic h\nare obtained via above-mentioned variational ground-state ener gy calculations. In Table\nII we compile the results for both αandγcorresponding to several values of rcobtained\nwith EXX, XO-LDA and XC-LDA ground-state densities. It can be se en from Table II\nthat for all the three energy functionals αdecreases monotonically with decrease in rcand\nappproaches zero for very small value of rc. This trend is in agreement with the results\nof Ref. [23]. We also note that with increase in rcthe values of αcorrectly tend to the\nrespective free atom cases which are αEXX= 1.33a.u.,αXO= 1.77a.u., andαXC= 1.63\n7a.u.[32]. It is also interesting to note from Table II that for rc<1.5a.u.the values of\npolarizability αobtained with three differnt densities are almost identical. Therefor e, like\nground-state energy of a confined helium atom its polarizability too d oes not have a strong\ndependence on the correlation energy specially in the strong confin ment regime. Next we\nfocus our attention on the results for the hyperpolarizability γof a confined helium atom,\nwhich are to our knowledge not reported earlier. The results for γalso correctly converge\nto their respective free atom values with the increase in rc. These values are γEXX= 36.2\na.u.,γXO= 114.2a.u., andγXC= 88.15a.u.[32]. Like polarizability, the values of γalso\nshow a monotonic decreasing trend, however, the decrease in the values ofγare more rapid\nas comapred to the polarizablity. With the increase in the compressio n, that is decrease\ninrcthe values of hyperpolarizability γnot only approach zero but also undergo a change\nin sign. The change in sign occurs at different value of rcfor three different ground-state\ndensities employed in this paper. The change in the sign of hyperpolar izabilty of a strongly\nconfined hydrogen atom has already been discussed in Ref. [25]. We note here that the\nchange in sign of γis akin to the term hypopolarizability which was coined by Coulson et\nal. [33]. The change in sign of γwas later discussed by Langhoff et al. [34] in connection\nwith study of hyperpolarizability of high Z ions isoelectronic with Na and Mg series. They\nconcluded that the compact electronic charge density resulting fr om the increasing value of\nZ within isoelectonic series is responsible for reversal of sign of γ. Similarly in a confined\natom the charge density becomes highly compact with decreasing ra dius of confinement\nthereby yielding a negative hyperpolarizability. Finally, we note that u nlike polarizabilty\nthe results for γobtained with different densities vary for all values of rc. This indicates\nthat hyperpolarizability γdepends crucially on the nature of ground-state density employed\nfor the calculation.\nIV. CONCLUSION\nIn this paper we have calculated the ground-state energies and th e densities of a helium\natom confined in an impentrable spherical box for various values of r adius of confinement\nwithin DFT. We use three different exchange-correlation energies n amely, exact exchange\nequivalent to HF level, exchange-only at LDA level, and correlation inc luded with exchange\nat LDA level to obtain the ground-state properties. Using these g round-state densities we\n8perform calculations of static linear and nonlinear electric response propertise of confined\nheliumatomandstudytheirvariationwiththestrengthoftheconfin ment. Theground-state\nenergies and densities are obtained by variationally solving the KS equ ation of DFT with\nthe variational form for the ground-state orbital expanded in te rms of STOs. The results\nobtained by us are quite accurate and compare well with the already published data. We\nfind that energies of a confined helium atom are not affected by the in clusion of correlation\nenergy especially in the strong confiment regimes. With the increase in the value of radius\nof confinement the differences between the results for energy ob tained by employing three\ndifferent densities increase indicating the importance of correlation energy for determining\nthe energy of a free helium atom. The linear polarizability αof a confined helium atom\ndecreases with the decrease in the value of confinement radius app roacing the value of zero\nin the strong confinement regime. Results of our calculations clearly demonstrate that for\nrc<1.5a.u.the values of αdo not depend much on the correlation energy. On the other\nhand, value of hyperpolarizability γshows a decreasing trend with decreasing rcwhich\napproaches the value of zero rapidly and then changes sign on furt her decreasing rc. This\nchange in sign of γis attributed to charge density becoming highly compact in the case o f\nstrong confinement.\nAcknowledgments\nWe dedicate this paper to Prof. K. D. Sen who introduced us to the fi eld of confined\natoms. S. W. and A. B. wish to thank Dr. S. C. Mehendale for his cons tant support\n9[1] S. A. Cruz, Ed. The Theory of Confined Quantum Systems Adva nces in Quantum Chemistry\n57 (2009) and refeerences therein.\n[2] A. Michels, J. de Boer, and A. Bijl, Physica (Amsterdam) 4 (1937) 981.\n[3] A. Sommefeld and H. Welker, Ann. Phys., Lpz. 32 (1938) 56.\n[4] C. A. Ten Seldam and S. R. de Groot, Physica 18 (1952) 904.\n[5] T. Sako and G. H. F. Diercksen, J. Phys. B At. Mol. Opt. Phys . 36 (2003) 1433 .\n[6] T. Sako and G. H. F. Diercksen, J. Phys. B At. Mol. Opt. Phys . 36 (2003) 1681 .\n[7] V. K. Dolmatov, A. S. Baltenkov, J. -P. Connerade, and S. M anson, Radiation Physics and\nChemistry 70(2004) 417 and references theirin.\n[8] J. -P. Connerade and P. Kengkan, Proc. Idea-Finding Symp . Frankfurt Institute for Advanced\nStudies, (2003) p. 35.\n[9] W. Jaskolski, Phys. Rep. 271 (1996) 1 .\n[10] B. M. Gimarc, J. Chem. Phys. 47 (1967) 5110.\n[11] E. V. Ludena, J. Chem. Phys. 69 (1978) 1170.\n[12] E. V. Ludena and M. Greogri, J. Chem. Phys. 71 (1979) 2235 .\n[13] C. Joslin and S. Goldman, J. Phys. B: At. Mol. Opt. Phys. 2 5 (1992) 1965.\n[14] N. Aquino, A. F-Riveros, J. F. Rivas-Silva, Phys. Lett. A 307 (2003) 326.\n[15] S. H. Patil and Y. P. Varshni, Can. J. Phys./Rev. Can. Phy s. 82 (2004) 647.\n[16] A. Banerjee, C. Kamal, and A. Chowdhury, Phys. Lett. A 35 0 (2006) 121 .\n[17] N. Aquino, J. Garza, A. Flores-Riveros, J. F. Rivas-Sil va, and K. D. Sen, J. Chem. Phys. 124\n(2006) 054311.\n[18] A. Flores-Riveros, N. Aquino, H. E. Montgomery Jr., Phy s. Lett. A 374 (2010) 1246 .\n[19] C. Laughlin and S. I. Chu, J. Phys. A: Math. Theor. 42 (200 9) 265004.\n[20] H. E. Montgomery Jr, N. Aquino, and A. Flores-Riveros, P hys. Lett. A 374 (2010) 2044.\n[21] J. L. Martin and S. A. Cruz, J. Phys. B: At. Mol. Opt. Phys. 224 (1991) 2899.\n[22] J. L. Martin and S. A. Cruz, J. Phys. B: At. Mol. Opt. Phys. 25 (1992) 4365.\n[23] M. van Faassen, J. Chem Phys 131 (2009) 104108.\n[24] C. Diaz-Garcia and S. A. Cruz, Int. J. Quant. Chem. 108 (2 008) 1572.\n[25] A. Banerjee, K. D. Sen, J. Garza, and R. Vargas, J. Chem. P hys. 116 (2002) 4054.\n10[26] M. K. Harbola and A. Banerjee, Phys. Lett. A 222 (1996) 31 5; A. Banerjee and M.K. Harbola,\nPramana J. Phys. 49 (1997) 455; A. Banerjee and M.K. Harbola, Eur. Phys. J. D 1 (1998)\n265.\n[27] R.G. Parr and W. Yang, Density Functional Theory of Atom s and Molecules, Oxford Univer-\nsity Press, New York, 1989.\n[28] P. A. M. Dirac, Proc. Camb. Phil. Soc. 26 (1930) 376.\n[29] O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13 (1976) 4274.\n[30] J. Garza, R. Vargas, and A. Vela, Phys. Rev. E (1998) 3949 .\n[31] J. P. Perdew and Y. Wang, Phys. Rev. B 45 (1992) 13244.\n[32] A. Banerjee and M. K. Harbola, Phys. Rev. A 60 (1999) 3599 .\n[33] C. A. Coulson, A. Maccoll, and L. E. Sutton, Trans . Farad ay soc. 61 (1952) 106.\n[34] P. W. Langhoff, J. D. Lyons, and R. P. Hurst, Phys. Rev. 148 ( 1966) 18.\n11TABLE I: Ground-state energies of confined helium atom as a fu nction of confinment radius rc\n(in atomic units) for the case of exact exchange (EXX), excha nge-only within LDA (XO-LDA)\nand exchange-correlation within LDA (XC-LDA). The numbers in the parenthesis are taken from\nalready published data. The results in the last column are ta ken from [18]\nrcEXX XO-LDA XC-LDA Correlated-Hylleraas\n(Ref.[11]) (Ref. [17]) (Ref. [17]) (Ref. [18])\n0.522.79096 23.32202 23.099 22.7419\n(22.79095) - -\n0.613.36683 13.81792 13.605 13.3187\n(13.36682) - -\n0.77.97302 8.36716 8.164 7.9258\n(7.97302) - -\n0.84.65736 5.00895 4.8133 4.6110\n(4.65737) - -\n0.92.50942 2.82805 2.369 2.4638\n(2.50944) - -\n1.01.06120 1.35362 1.17040 1.063\n(1.01624) - -\n1.5-1.86422 -1.64897 -1.81185 -1.9066\n(-1.86422) - -\n2.0-2.56257 -2.39363 -2.53480 -2.6035\n(-2.56253) (-2.38363) (-2.50589)\n3.0-2.83078 -2.68201 -2.82256 -2.8715\n(-2.83083) (-2.68210) (-2.79608)\n4.0-2.85854 -2.71807 -2.85552 -2.8994\n(-2.85852) (-2.71813) (-2.82970)\n5.0-2.86138 -2.72288 -2.85856 -2.9026\n(-2.86134) (-2.72290) (-2.83387)\n6.0-2.86162 -2.72346 -2.86000 -2.9032\n(-2.86151) (-2.72354) (-2.83439)\n12TABLE II: Static polarizability αand hyperpolarizability γof a confined helium atom for various\nvalues of confinemnt radius rc. All numbers are in atomic unit\nEXX XO-LDA XC-LDA\nrc α γ α γ α γ\n1.0 0.043 -1.05×10−40.043 -1.16×10−40.043 -8.78×10−5\n1.1 0.059 -1.72×10−40.060 -2.04×10−40.060 -1.36×10−4\n1.2 0.079 -2.15×10−40.080 -2.97×10−40.080 -1.46×10−4\n1.3 0.102 -1.24×10−40.103 -3.12×10−40.103 -2.08×10−6\n1.4 0.129 3.54×10−30.131 -7.24×10−50.131 5.22×10−4\n1.5 0.160 1.56×10−30.162 7.74×10−40.162 1.87×10−3\n2.0 0.360 0.0603 0.374 0.051 0.372 0.0634\n2.5 0.606 0.487 0.647 0.469 0.641 0.530\n3.0 0.839 1.900 0.934 2.158 0.916 2.324\n4.0 1.161 10.358 1.392 14.862 1.336 14.674\n5.0 1.286 23.381 1.641 45.029 1.544 40.552\n6.0 1.322 35.766 1.763 93.056 1.628 75.265\n13"    },    {        "title": "1007.2998v1.Searching_for_topological_density_wave_insulators_in_multi_orbital_square_lattice_systems.pdf",        "content": "arXiv:1007.2998v1  [cond-mat.str-el]  18 Jul 2010Searching fortopologicaldensity waveinsulators inmulti -orbital square latticesystems\nBohm-Jung Yang1and Hae-Young Kee1,2,∗\n1Department of Physics, University of Toronto, Toronto, Ont ario M5S 1A7, Canada\n2Canadian Institute for Advanced Research, Quantum Materia ls program, Toronto, Ontario M5G 1Z8, Canada\n(Dated: February 28, 2022)\nWestudytopologicalpropertiesofdensitywavestateswith brokentranslationalsymmetryintwo-dimensional\nmulti-orbitalsystemswithaparticularfocusont 2gorbitalsinsquarelattice. Duetodistinctsymmetryproper ties\nof d-orbitals, a nodal charge or spin density wave state with Dirac points protected by lattice symmetries can\nbe achieved. When an additional order parameter with opposi te reflection symmetry is introduced to a nodal\ndensitywavestate,thesystemcanbefullygappedleadingto abandinsulator. Amongthose,topologicaldensity\nwave (TDW) insulators can be realized, when an effective sta ggered on-site potential generates a gap to a pair\nof Dirac points connected by the inversion symmetry which ha ve the same topological winding numbers. We\nalso present a mean-field phase diagram for various density w ave states, and discuss experimental implications\nof our results.\nPACS numbers:\nI. INTRODUCTION\nIdentifyingtopologicalinsulatorshasbeen one of the most\nfascinating research fields in contemporary condensed mat-\nter physics.1–3Topological insulators have a bulk gap like\nband insulators, but are distinguished by topologically pr o-\ntected conductingedge states preservingtime-reversalin vari-\nance. Inparticular,two-dimensionaltopologicalinsulat orsare\nknown as quantum spin Hall insulators with finite counter-\npropagatingspin currents on the edge, analogousto quantum\nHall states. Haldane4proposed that the fictitious magnetic\nfluxes in the honeycomb lattice lead to the quantum anoma-\nlous Hall insulator (or Chern insulator). Generalizing Hal -\ndane’smodelincludingtimereversalinvariantspin-orbit cou-\npling,itwastheoreticallyshownthatsuchaquantumspinHa ll\ninsulator can exist in graphene.5,6A two-dimensional semi-\nconductorsystemwithauniformstraingradientwasalsopro -\nposed to be a candidate.7Later, the predicted edge states in\nHgCeTe quantumwell systems8wereexperimentallyverified\nwhich confirmed the existence of two-dimensional topologi-\ncal insulators.9\nThe topological insulators in these systems normally exist\nduetostrongspin-orbitcoupling.5,10Whenthespin-orbitcou-\npling preserves spin rotational symmetry about an axis, the\ncounter-propagating edge modes which carry opposite spin\nquantum numbers result in quantum spin Hall insulators. It\nwas shown that these modes are protected by time reversal\nsymmetryeven in the absence of spin rotational invariance.10\nItwasfurtherpointedoutthataneffectivespin-orbitcoup ling\ntermcanbegeneratedbyspontaneousspinrotationalsymme-\ntrybreakinginanextendedHubbardmodelonthehoneycomb\nlattice.11In these studies, the structure of the honeycomblat-\ntice plays an important role, as the tight binding model on\nthis lattice possesses two Dirac points at the Brillouin zon e\ncorners. Thereforein low energydescription, variousgapp ed\ninsulating phases proximate to the Dirac semi-metal can be\nunderstood in terms of mass perturbations to gapless Dirac\nparticles. For instance, the fictitious magnetic fluxes intr o-\nduced by Haldane generate a mass term that has the opposite\nsignsatthetwoDiracpointsleadingtoaninsulatorwithfini tequantizedHallconductivity. TheDiracHamiltonianapproa ch\nfurther provides a framework to understand the time-revers al\ninvariantZ2topologicalinsulators.10\nWhile systems on the honeycomb lattice such as graphene\nnaturally support two-dimensional massless Dirac particl es\nin the bare band structures, this is not the case in a simple\nsquare lattice system which is an effective model for abun-\ndant layered perovskite materials in nature. In this respec t,\nit is interesting to note that the recently proposed nodal de n-\nsity wave state12exhibits gapless Dirac particles via broken\ntranslational symmetry. This proposal was made in the con-\ntext of iron pnictide systems, where d-orbitalsof t 2gbandsin\nan effectivelytwo-dimensionalsquare lattice give rise to sev-\neral Fermi pockets with interesting topological propertie s. In\nthis system, the spin density wave instability with the finit e\nordering wave vector Q= (π,0)(or(0,π)) leads to band\ntouchings between the kstates with the momentum differ-\nence ofQ. In general, the degeneracies at the band touching\npointsdisappearbecause of the finite overlapmatrixbetwee n\nthe degenerate states induced by the density wave order pa-\nrameter. However, in multi-orbital systems, because of the\ndistinct symmetry properties of orbitals, the degeneracie s at\nsomebandtouchingpointsareprotectedleadingtonodalden -\nsitywavestates,whichisgenerallyvalidforanydensitywa ve\norders.\nIn this work, we ask if topological insulators can be\nemerged by gapping nodal points turning the system from\nnodaldensitywave statesto topologicaldensitywave (TDW)\ninsulators. To find such a TDW insulator, we first inves-\ntigate the properties of the nodal density wave states. We\nfindthatonegeneralandimportantcharacteristicoftheDir ac\nparticles in nodal density wave states is that a pair of Dirac\nHamiltonians connected by the inversion symmetry have the\nsame topological winding numbers. Thus an effective stag-\ngeredon-site potentialgeneratinga mass term, which has th e\nsame signs at the inversion symmetric nodal points, induces\nTDW insulators. This can be contrasted with the topologi-\ncal properties of the Dirac particles in the honeycomb latti ce\nwherethe DiracHamiltoniansat thetwoinversionsymmetric\nnodal points have the opposite winding directions.13,14Thus2\nthe massterminducedby,forexample,a staggeredsublattic e\nchemicalpotential,whichhasthesamesignsat thetwoDirac\npointswouldgenerateatopologicallytrivialbandinsulat oras\nshownin graphenesystem.5,15\nThe rest of the paper is organized as follows. In Sec. II,\nwe first considera simpletwo-bandmodelHamiltoniancom-\nposedofdxzanddyzorbitalsonthesquarelattice. Afterclas-\nsifyingallpossiblechargeandspindensitywaveorderpara m-\neterswiththeorderingwavevector Q= (π,0)basedontheir\ntransformationpropertiesunderlattice symmetries, we es tab-\nlish general relations between the locations of Dirac nodes\nand orderparametersymmetriesin Sec. III. The fact that dxz\nanddyzorbitalshavetheoppositeeigenvaluesunderreflection\nsymmetriesalonghighsymmetrydirectionsinthemomentum\nspace,playsthekeyrolefortheemergenceofDiracpoints. I n\naddition to the Dirac points coming from the Brillouin zone\nfolding, additional contributions from quadratic band deg en-\neracy splitting are also discussed. In Sec. IV, topological\nproperties of gapped density wave phases with two order pa-\nrameters with the opposite reflection symmetries are studie d.\nFully-gapped insulating phases can be obtained by introduc -\ning two density wave order parameterswhich have the oppo-\nsite eigenvalues under reflection symmetries. Among them,\na certain combination turns the system to a TDW insulator.\nIn Sec. V, the mean field phase diagram including the TDW\nphase is presented, which is obtained by solving an extended\nHubbard model Hamiltonian with orbital degeneracy. Topo-\nlogical density wave states in three orbital systems are dis -\ncussed in Sec. VI. Straightforward extension to three-orbi tal\nsystemsshowsthegeneralapplicabilityofthe ideawepursu e\nin this work to obtain topological insulators in multi-orbi tal\nsystems. Finally,weconcludeinSec. VII.\nII. TWOBANDHAMILTONIANAND SYMMETRIESOF\nORDER-PARAMETERS\nA. Tight-bindingHamiltonian\nWe considera tight bindingHamiltonianon the square lat-\ntice with two orbital ( dxz,dyz) degrees of freedom at each\nsite. A generic Hamiltonian which contains all possible hop -\npingprocessesallowedbylatticesymmetriesisgivenby\nH0=/summationdisplay\nk,σψ†\nk,σH(k)ψk,σ,\n(1)\nwhere\nH(k) = (ε+(k)−µ)1+ε−(k)τ3+εxy(k)τ1.\n(2)\nHere a two-component field ψ†\nk,σ=[d†\nxz,σ(k),d†\nyz,σ(k)]de-\nscribes the creation of particles with dxzanddyzorbital fla-\nvors with spin σ, and the Pauli matrix τconnects these twoorbitalstates. Intheabove,\nε+(k) =−(t1+t2)(coskx+cosky)−4t3coskxcosky,\nε−(k) =−(t1−t2)(coskx−cosky),\nεxy(k) =−4t4sinkxsinky.\n(3)\nDiagonalization of H(k)gives rise to the following two\nbanddispersions,\nE±(k) =ε+(k)−µ±/radicalBig\nε2\n−(k)+ε2xy(k).\n(4)\nIn addition to time-reversal symmetry T, the Hamiltonian\nH0has theC4point group symmetry, which consists of the\nfour-fold rotation Cπ\n2, the inversion I, and the two reflec-\ntionsPxandPymappingxto−xandyto−y, respectively.\nEach symmetry operation transforms a two-component field\nψσ(kx,ky)inthefollowingway,\nCπ\n2:ψσ(kx,ky)→iτ2ψσ(−ky,kx),\nPx:ψσ(kx,ky)→ −τ3ψσ(−kx,ky),\nPy:ψσ(kx,ky)→τ3ψσ(kx,−ky),\nI:ψσ(kx,ky)→ −τ0ψσ(−kx,−ky).\n(5)\nIf we choosethe hoppingparametersin sucha way as t1=-\n1.0,t2=1.3,t3=t4= -0.85, the H0works as an effective\ntwo-band Hamiltonian describing the Fe-pnictide systems.16\nGiven the hopping parameters above, the Fermi surface con-\nsists of two hole pockets and two electron pockets when the\nsystem is near half-filling.12,16A pair of electron and hole\npockets are connected by a nesting wave vector Q= (π,0)\n(or(0,π)), which drives various density wave instabilities.17\nHere we choose Q= (π,0)18and perform a detailed study\nabout the band structures of density wave ground states con-\nsideringall possibledensitywave orderparameters.\nB. Symmetryof order parameters\nWe considervariouson-site densitywave orderparameters\nandinvestigatetheirsymmetryproperties. Sincewe havetw o\norbitals per site, there are 4 different on-site charge dens ity\nwave(CDW)stateswiththeorderingwavevector Q= (π,0),\nwhichare givenby\nˆDi=1\nN/summationdisplay\nk2/summationdisplay\na,b=1d†\na,σ(k)[τi]abdb,σ(k+Q),(6)\nwherea,bare indices describing the dxz(a=1) ordyz(a=2)\norbital states. Ncounts the number of unit cells in the sys-\ntem. Similarly,wealsodefinespindensitywave(SDW)states\nchoosingthe spinorderingdirectionalongthe z-axis,3\n(a) D  - CDW (b) M  - SDW 3 2\nFIG. 1: (Color online) Description of representative densi ty wave\nordering patterns with the ordering wave vector Q= (π,0). (a)\nD3charge density wave ( D3-CDW) ordering. dxzanddyzor-\nbitals align alternatively along the xdirection while local charge\nand spin densities are uniform. (b) M2spin density wave ( M2-\nSDW) ordering. Here orbital and spin orderings occur at the s ame\ntime. A solid circle represents the d+orbital defined as d+=\n(dxz+idyz)/√\n2whileadottedcircleindicatesthe d−orbitalgiven\nbyd−= (dxz−idyz)/√\n2. The arrows inside circles describe spin\nordering.\nˆMi=1\nN/summationdisplay\nk2/summationdisplay\na,b=1d†\na,σ1(k)[τi]ab[sz]σ1σ2db,σ2(k+Q).(7)\nThese8orderparametersrepresentdistinctphaseswithdif -\nferent broken symmetries. For example, the D3CDW order\nparametercorrespondsto/summationtext\ni(−1)ix(ni,xz−ni,yz)whereni,a\nisthedensityofelectronswith theorbital aat thesitei. Thus\nit is characterized by the relative density difference betw een\ntwoorbitals(orbitalordering),whichalternatesalongth exdi-\nrection,whilekeepingthetotaldensity (ni,xz+ni,yz)uniform\non every site as shown in Fig. 1 (a). This breakstranslationa l\nsymmetry doubling the unit cell along xdirection. On the\nother hand, the M2SDW order parameterdescribed in Fig. 1\n(b)correspondstoastaggeredspin-orbitcoupling. Thisis be-\ncauseM2can be written as/summationtext\ni(−1)ixSi,zLi,zwhereLi,zis\nproportional to (di,xz+idi,yz)†(di,xz+idi,yz)−(di,xz−\nidi,yz)†(di,xz−idi,yz) = 2id†\ni,xzdi,yz+h.c.. However, un-\nliketheuniformspin-orbitalcoupling/summationtext\niSi,zLi,z,theM2isa\nstaggeredspin-orbitcouplingwith alternatingsigns alon gthe\nxdirection. It breaks spin-rotational and translational sy m-\nmetriesbutpreservestimereversalsymmetry. Inaddition, D0\nandM0describe conventional charge and spin density wave\nstates, respectively. It was found that M0describes the lead-\ningdensitywave instabilityinFe-pnictides.12\nTheabove8orderparameterscanbedistinguishedbytheir\ntransformationpropertiesunderlatticesymmetries. Thes ym-\nmetries of density wave order parameters are summarized in\nTableI. Notethateverydensitywavestatehasevenparityun -\nder the inversion symmetry. Moreover, all the diagonal den-\nsity wave states Di(orMi) withi= 0 or3 areevenunderthe\ntwo reflection symmetries while the other off-diagonal den-\nsity wave states with i= 1 or 2 are odd under the reflections.D0D1D2D3M0M1M2M3\nPx+--++--+\nPy+--++--+\nI++++++++\nT++-+--+-\nTABLEI: Symmetry of density wave order parameters. Here ‘+’ (‘-\n’) indicates ‘even’ (‘odd’) symmetry of the order parameter s under\nthe corresponding symmetryoperation.\nThese symmetry propertiesof density wave order parameters\nstrongly constrain the location of Dirac nodes generated by\nthe Brillouin zone folding and the winding numbers around\nDirac nodes in the momentum space, which are discussed in\ndetailinthe followingsection.\nIII. NODAL DENSITYWAVEPHASES\nOne intriguing property of the Q= (π,0)density wave\ngroundstates isthat a largenumberofDirac nodesemergein\nthe band structure.12The numbers and locations of the nodal\npoints dependon band dispersionsand the symmetries of the\norderparameters.\nThere are two different sources generating nodal points in\ngeneral. One way is via introducinga density wave orderpa-\nrametercarryingafinitemomentum. ThisinducesaBrillouin\nzone folding which generates several band touching points.\nIn most cases, the degeneracy at the band touching point is\nlifted because the density wave order parameter induces a fi-\nnite overlap between the pair of states touching at a point.\nHenceforth a band gap opens up. However, when the band\ntouching occurs at a high symmetry point in the Brillouin\nzone, the overlap matrix vanishes due to the lattice symme-\ntries,generatingsymmetryprotectednodalpoints.\nThesecondgroupofnodalpointscomefromthesplittingof\nquadraticband touchingpoints, which exist in the bare band -\nstructure. Becauseoftheunderlyingfour-foldrotational sym-\nmetry, the original hopping Hamiltonian in Eq.(1) supports\nquadraticbandcrossingpoints.12,19–22Theintroductionofthe\ndensity wave order parameter carrying a finite momentum\nsplits a quadratic band touching point into two Dirac points\nalong high symmetry directions in the momentum space. In\nthe following, we discuss in detail the relation between the\norder parameter symmetry and the locations of Dirac points\nderived from these two different sources in separate subsec -\ntions.\nA. Dirac nodesgenerated byBrillouinzone folding\nWe first focus on the generation of Dirac nodes along the\nky-axis. InFig.2(a)(Fig.2(b)),weplottheenergydispersio n\nof the two bands given in Eq. (4) along the kx= 0(kx=π)\ndirection. Since the εxyterm in Eq. (3), which describes the\nhybridization between dxzanddyzorbitals, vanishes along\nthekx= 0axis, the upper and lower bands in Fig. 2(a) are4\n(a) (b) \n(c) k y -π E(k) \nπ k yπ -π E(k) \n-π k y π E(k) For k  = 0 x For k  = πx \nFIG. 2: (Color online) (a) Band dispersion along kx= 0for the\nhoppingHamiltonianinEq.(1). Hereweusearedsolid(blued otted)\nline to indicate a band which is odd (even) under the Pxreflection\nsymmetry. (b) Band dispersion along the kx=πdirection. (c) Band\nstructure along kx=0 after the unit cell doubling due to the Q=(π,0)\ndensity wave ordering. Four bands in (a) and (b) meet and disp erse\ntogether along kx=0 after the Brillouin zone folding. Note that the\nzone folding generates 8 band touching points. Black solid ( dotted)\ncircles indicate the band touching points between two bands having\nthe same (opposite) Pxeigenvalues.\njustdyzanddxzbands, respectively. For kx= 0,H(k)is\ninvariantunderthe Pxreflectionsymmetrywhichtransformsa\nmomentumkxto−kx. Thereforeeachbandisaneigenstateof\nPxwitheigenvaluesof ±1. Thisisconsistentwiththefactthat\ndxz(dyz)orbitalisodd(even)under Px. Similar analysiscan\nalso be applied to the two bandsdispersing along the kx=π\ndirection. Since H(k)hasPxsymmetry along kx=π, the\ntwobandsalsohavedefinite Pxeigenvalues. InFig.2,the Px\neven (odd) bands are represented by blue dotted (red solid)\nlines.\nOnceweintroduceadensitywaveorderparameterwiththe\nordering wave vector Q= (π,0), the unitcell doubles along\nthex-direction, which leads to the Brillouin zone folding in\nthemomentumspace. ThuswithinthereducedBrillouinzone,\nwe have four bands dispersing along the ky-axis. Note that\nthe zone folding generates8 band touchingpoints, which are\nindicatedbycirclesinFig.2(c). Herethebandtouchingpoi nt\nbetween two bands with the same (opposite) Pxeigenvalues\nisencircledbya solid(dotted)circle.\nThe degeneracy between two states, |ψ1(k)/an}b∇acket∇i}htand|ψ2(k+\nQ)/an}b∇acket∇i}ht, touching at the momentum kafter the Brillouin zone\nfolding,is liftedwhenthe matrixelementofthedensitywav e\norder parameter ˆDibetween these two states is finite, that is,\n/an}b∇acketle{tψ1(k)|ˆDi|ψ2(k+Q)/an}b∇acket∇i}ht /ne}ationslash= 0. Thereforeiftheorderparameter\nˆDi(orˆMi) isPxeven, the degeneracyis lifted when the two\ndegeneratebandshavethesame Pxeigenvalues. However,the\nnodal pointremainsgaplessif the two degeneratebandshave\ntheopposite Pxeigenvalues.-(a) \n(b) -π k y π E(k) \n-π k y π E(k) \nFIG.3: (Coloronline)Thebandstructuresofthe Q= (π,0)density\nwave ground states along the ky-axis. (a) For Pxeven density wave\nstates. (b)For Pxodd density wave states.\nOn the other hand, if the density wave order parame-\nterˆDi(orˆMi) isPxodd, the full Hamiltonian is not in-\nvariant under Pxanymore. However, even in this case\n/an}b∇acketle{tψ1,s1(k)|ˆDi|ψ2,s2(k+Q)/an}b∇acket∇i}ht= 0in the weak coupling limit,\nifs1=s2wheresrefers toPxeigenvalues. Namely, the\nmatrix element of Di, which is odd under Px, vanisheswhen\nthetwo degenerateeigenstateshavethe same Pxeigenvalues.\nTo understand this point clearly, let us define the eigenvect or\n|φ(0)\nn,s(k)/an}b∇acket∇i}htof the hopping Hamiltonian H0with the even ( s=\n+)orodd(s=-)Pxeigenvalues. Here nisabandindex. Now\nwe turn on a small density wave order parameter ˆDiwhich\nis odd under Px. SincePxeigenvalue is not a good quantum\nnumber,|φ(0)\nn,s(k)/an}b∇acket∇i}htcan be contaminatedby the states with the\noppositePxeigenvalue |φ(0)\nm,¯s(k+Q)/an}b∇acket∇i}ht, leadingto\n|φ(0)\nn,s(k)/an}b∇acket∇i}htˆDi−→|ψn,s(k)/an}b∇acket∇i}ht=|φs(k)/an}b∇acket∇i}ht+|φ¯s(k+Q)/an}b∇acket∇i}ht,\nwhere|φs(k)/an}b∇acket∇i}ht=/summationtext\nncn|φ(0)\nn,s(k)/an}b∇acket∇i}htis a linear combination of\nthe states with the Pxeigenvalue of s, while|φ¯s(k+Q)/an}b∇acket∇i}ht=/summationtext\nnc′\nn|φ(0)\nn,¯s(k+Q)/an}b∇acket∇i}htis a linear combination of the states with\nthe opposite Pxeigenvalue of ¯s. Notice that |φs(k)/an}b∇acket∇i}htand\n|φ¯s(k+Q)/an}b∇acket∇i}hthave a momentum difference given by the order-\ningwavevector Qcarriedbythedensitywaveorderparameter\nˆDi. Because of the fact that the two componentsof the wave\nfunction with the opposite Pxeigenvalues have the momen-\ntum difference given by Q, it is straight forward to show that\n/an}b∇acketle{tψ1,s1(k)|ˆDi|ψ2,s2(k+Q)/an}b∇acket∇i}ht= 0ifs1=s2.\nTherefore the nodal point remains gapless if the order pa-\nrameterˆDiisPxeven(Pxodd)whilethetwogeneratedbands\nhave the opposite (same) Pxeigenvalues. It means that 4\nnodal points among the 8 band touching points remain gap-5\n(a) (b) \n(c) k x -π E(k) \nπ k x 2π  0 E(k) \n-π k x π E(k) For k  = 0y For k  = 0y \nFIG.4: (Color online) (a) Band dispersion along ky= 0centered at\nk=(0,0). (b) Band dispersion along ky= 0centered at k=(π,0). (c)\nBandstructurealong ky=0aftertheunitcelldoubling. Fourbandsin\n(a)and(b) meetanddisperse together along ky=0aftertheBrillouin\nzone folding. Here we use a red solid (blue dotted) line to ind icate\na band which is odd (even) under the Pysymmetry. Black solid\n(dotted)circlesindicatethebandtouchingpointsbetween twobands\nhaving the same (opposite) Pyeigenvalues.\nless independent of the condition that the order parameter i s\nevenoroddunder Pxreflectionsymmetry.\nIn Fig. 3 we plot the band structure of the density wave\ngroundstates along the ky-axis. Fig. 3 (a) correspondsto the\ndensity wave orders D0,D3,M0,M3, which are Pxeven,\nwhile Fig. 3 (b) describes the band structure for the other or -\nder parameters, D1,D2,M1,M2, which are odd under Px\nsymmetry. Notice that nodal points show opposite behavior\nfor these two different classes of order parameters. Namely ,\nwhena nodalpointsremainsgaplessforoneorderparameter,\nit is gapped out for the other order parameter which has the\noppositePxeigenvalue.\nWecanextendthesameanalysistounderstandnodalpoints\nlying along the kx-axis. In this case we have Pyreflection\nsymmetrymapping yto−y. However,comparedtotheprevi-\nous analysis for nodal points on the ky-axis, there is one im-\nportant difference in this case. Before the unitcell doubli ng,\nwe have two bands dispersing along the kx-axis. The Bril-\nlouin zone folding induces overlaps of these two bands with\nthemselves. InFig. 4, weplotthe dispersionofthetwo bands\nalong thekx-axis centered at k= (0,0)(Fig. 4(a)) and at\nk= (π,0)(Fig. 4(b)). The 4 bands after the zone folding\ndisplayed in Fig. 4(c) can be obtained by superposing the 4\nbandsinFig.4(a)and(b). InFig.4(c),weplotthebandstruc -\nture fromkx=−πtokx=πfor convenience although the\nfirst Brillouin zone is from kx=−π/2tokx=π/2. Note\nthat in Fig. 4(c) the location of solid and dotted circles are\ninterchanged compared to those in Fig. 2(c). Because of this\ndifference, the location of Dirac nodes along the kxandky\naxesalso showthe oppositebehaviors.B. Dirac nodesfrom quadraticbandcrossing\nThe band structure of the two-band hopping Hamiltonian\nH0in Eq. (1) supports two quadratic band crossing points at\nk= (0,0)andk= (π,π).12,23Splitting of these quadratic\nbandcrossingpointsgeneratesadditionalDiracpoints,wh ich\ncontribute additional Chern numbers for various insulatin g\nphases.\nExpanding the Hamiltonian H(k)in Eq. (2) near k=\n(0,0), we obtain the following low energy effective Hamil-\ntonian,\nHeff=/integraldisplay\nd2kψ†(k)Hquad(k)ψ(k), (8)\ninwhich\nHquad(k) =α(k2\nx+k2\ny)ˆτ0+βkxkyˆτ1+γ(k2\nx−k2\ny)ˆτ3,(9)\nwhereα= (t1+t2+4t3)/2,β=−4t4,andγ= (t1−t2)/2.\nNontrivialtopologicalpropertyofthequadraticbandcros sing\npointisreflectedinthewindingnumber Nw,whichisdefined\nas,20\nNw≡1\nπi/contintegraldisplay\nCdk·/an}b∇acketle{tψ†(k)|∇k|ψ(k)/an}b∇acket∇i}ht,(10)\nwhere|ψ(k)/an}b∇acket∇i}htis a Bloch wave function correspondingto one\nof the bands involved in the band touching and Cis a closed\nloop in the momentum space encircling the band crossing\npoint. Aquadraticbandcrossingpointcontributes NW=±2,\nwhichistwicelargerthanthewindingnumberaroundaDirac\npoint.19–21,24\nAdding a generic perturbation given by V=/summationtext3\ni=1miˆτi,\nthe degeneracy at the quadratic band crossing point can be\nlifted.m2term breaks time-reversal symmetry and the de-\ngeneracy is lifted by opening a gap. On the other hand, m1\nandm3terms that break 4-fold rotational symmetry, split the\nquadraticbandtouchingpointintotwo Diracpoints.19–21\nNowwe considertheeffectofthe Q= (π,0)densitywave\norderings on the degeneracy lifting at quadratic band cross -\ning points. Since the density wave orderparameterscarry th e\nmomentum Q= (π,0), they cannot couple to the degenerate\nstates atk= (0,0)(ork= (π,π)) at first order. The low-\nest order contribution to degeneracy lifting at quadratic b and\ntouching points starts from the second order processes. We\nfirst considerchargedensitywaveorderparametersgivenby ,\nHCDW=/summationdisplay\nk,σψ†\nk,σˆDψk+Q,σ, (11)\nwhereˆD=/summationtext3\ni=0Diˆτi. Treating the above H CDWas a per-\nturbation,thestandardsecondorderperturbationtheoryg ives\nrise to the following effectiveHamiltonian near the quadra tic\nbandtouchingpointat k= (0,0),\nHeff\nquad=Hquad(k)+Hmass,\n=Hquad(k)+3/summationdisplay\ni=0m(i)\nΓˆτi, (12)6\n(b) (c) \nkk\nxy\nkk\nxy\nkk\nxy(d) (a) \nFIG. 5: (Color online) Distribution of Dirac points for the D3(or\nM3) density wave state. (a) Four bands withinthe reduced Brill ouin\nzone. The energy eigenvalue Ei(k)for a band isatisfiesE1(k)≥\nE2(k)≥E3(k)≥E4(k). (b) Dirac points between the band 1 and\n2, (c) between band 2 and 3, (d) between band 3 and 4. Blue (Red)\ndots indicates Dirac points coming from the Brillouin zone f olding\n(splittingquadratic band touching points).\ninwhich\nm(0)\nΓ=λ{(D2\n1+D2\n2)(t1+t2+4t3)\n+(D0+D3)2(t2+2t3)+(D0−D3)2(t1+2t3)},\nm(1)\nΓ=2λD1{D3(−t1+t2)+D0(t1+t2+4t3)},\nm(2)\nΓ=2λD2{D3(−t1+t2)+D0(t1+t2+4t3)},\nm(3)\nΓ=λ{(D2\n1+D2\n2)(t1−t2)\n+(D0+D3)2(t2+2t3)−(D0−D3)2(t1+2t3)},\n(13)\nwhereλ=−1/{8(t1+2t3)(t2+2t3)}. Notethat aslongas\nonly one of the order parameters has finite magnitude while\nall the other order parameters are zero, m(1)\nΓ=m(2)\nΓ= 0.\nIn other words, if Dn/ne}ationslash= 0for a givennwhile all the other\nDi/negationslash=n= 0, the quadratic band crossing point always splits\nintotwoDirac pointsalongthemainaxes.\nCombining the contributions both from the Brillouin zone\nfolding and from the splitting of quadratic band crossing\npoints, we show the distribution of Dirac points for a D3(or\nM3)densitywavegroundstateinFig.5. Therearefourbands\nwithin the reduced Brillouin zone as shown in Fig. 5(a). We\nassign a band index isuch that the energy eigenvalue Ei(k)\nof the band isatisfiesE1(k)≥E2(k)≥E3(k)≥E4(k).\nThe location of Dirac points between the upper two bands\n(band1and2) are indicatedin Fig. 5(b). Similarly, the Dirac\npoints between the middle (bottom) two bands are described\nin Fig. 5(c) (Fig. 5(d)). Notice that there are many Dirac\ntouching points between the bands. Blue dots indicate thenodal points coming from the Brillouin zone folding induced\nby theQ= (π,0)density wave states. On the other hand,\nred dots result from the splitting of quadratic band touchin g\npoints. Two quadratic band touching points generate four\nDirac points lying on the y-axis. With the understanding of\ntheoriginandlocationsofDiracpoints,belowwediscussho w\ntoachieveTDWinsulators.\nIV. TOPOLOGICALPROPERTIESOF THEGAPPED\nDENSITYWAVE PHASES\nA single density wave order parameter induces a metallic\nphase with many Dirac points. The locations of Dirac points\nare determined by the transformation properties of the orde r\nparametersunderthereflectionsymmetries PxandPy. There-\nfore to get an insulating phase, two coexisting density wave\nstates, in which one is even and the other is odd under the\nPxandPysymmetries, are required. In addition, according\nto the order parameter symmetries summarized in Table I, if\ntime-reversalinvarianceisimposed,thereareonlyfourdi ffer-\nentwaysofchoosingapairofdensitywaveorderparameters,\nwhich give rise to a gapped phase. The four pairs of time\nreversal invariant density wave order parameterswith the o p-\nposite transformation properties under the reflections Pxand\nPy, are given by ( D3,D1), (D3,M2), (D0,D1), and (D0,\nM2).\nSince thezcomponent of the spin, Szis conserved, the\nChern number N(n)\nC,σis well defined for each band in a fully\ngappedphase.25–27HereN(n)\nC,↑(N(n)\nC,↓) is the Chern numberof\nthenth spin-up (spin-down) band. For every pair of the den-\nsity wave order parameters generating a fully gapped phase,\nthe four bands within the reduced Brillouin zone are well-\nseparatedfromeach otherwith a finite gapbetweenanypairs\nofthe bands. Eachbandis distinguishedbytheindex nrang-\ning from 1 to 4 as the energy decreases. The Chern number\nN(n)\nC,σofthenthbandwith thespin σisdefinedas,\nN(n)\nC,σ=1\n2π/integraldisplay\nRBZd2kF(n)\nσ(k), (14)\nwhere the momentum space Berry curvature F(n)\nσ(k)\nfor thenth band with the spin σis defined as\nF(n)\nσ(k)≡∂kxA(n)\ny,σ(k)−∂kyA(n)\nx,σ(k)in which\nthe Berry potential A(n)\nµ,σ(k)is given by A(n)\nµ,σ(k) =\n−i/an}b∇acketle{tΦ(n)\nσ(k)|∂kµ|Φ(n)\nσ(k)/an}b∇acket∇i}ht.28,29Here the Bloch wave function\n|Φ(n)\nσ(k)/an}b∇acket∇i}htisdefinedwithinthereducedBrilluinzone(RBZ).\nExplicit computationof the Chern numbers using Eq. (14)\nshows that every band of the insulating density wave phase\nwithfiniteD3andM2,hasanonzeroChernnumberasshown\nin Table II. On the other hand, every band has zero Chern\nnumber for the other three gapped phases defined with a pair\nof nonzero order parameters given by ( D3,D1), (D0,D1),\nand(D0,M2).7\nNC,↑NC,↓\nband 1+ 1- 1\nband 2- 3+3\nband 3+ 3- 3\nband 4- 1+1\nTABLE II: The Chern number of each band for the gapped density\nwave ground state withfinite D3andM2.\n-(a) (b) \n-π k y π E(k) \n-π k y π E(k) D  only 3 D  + M 2 3\nFIG. 6: (Color online) (a) Band structure of the D3nodal density\nwavestatealongthe kyaxis. (b)Bandstructureofthegappeddensity\nwave phase withfinite D3andM2.\nA. Topological propertiesof thetopological densitywave\ngroundstate withfinite D3andM2\nNontrivial topological properties of the fully gapped den-\nsity wave phase with nonzero D3andM2can be understood\nin the following way. We first consider the charge density\nwavestatewiththefinite D3orderparameter. The D3density\nwave phase supportsmany Dirac pointswhose distribution is\ndescribedinFig.5. Nowweturnonasmall M2whichinduces\ngap opening at each nodal point, leading to the fully gapped\ninsulatingphase,whichisdescribedinFig.6. Thedegenera cy\nliftingateachnodalpointcanbeunderstoodasa resultofth e\nmass perturbation, induced by the finite M2, to the gapless\nDirac particles. The nonzero Chern number of each band is\nobtainedby addingup the Chern numbercontributionsof the\nmassiveDiracparticlesderivedfromthecorrespondingban d.\nWe first focus on the two Dirac nodes lying along the ky\naxis between the band 3 and 4 shown in Fig. 5(d). The ef-\nfective Dirac Hamiltonian can be obtained by linearizing th e\nHamiltonian near the two nodal points sitting at the momen-\ntumk=k+andk−. Tosimplifythecomputationalprocedures,\nthe hopping parameters tiare slightly shifted from the ini-\ntial values given in Sec. IIA to t1=−t2=t3=t4=-1.0.\nThis small parameter change does not affect the topological\nproperties of the gapped phase but shifts the nodal points to\nk±= (0,±π\n2)making analytical analysis simpler. The ef-\nfective Hamiltonian describing the low energy fermions nea r\nthesetwo nodalpointsisgivenby\nHDirac\neff=/summationdisplay\nµ=±/integraldisplay\nd2qΨ†\nµ,σ(q)HDirac\nµ,σ(q)Ψµ,σ(q),(15)where\nHDirac\nµ,σ(q) =−µ{(2+8/radicalbig\n4+D2\n3)qyˆτ3+4|D3|/radicalbig\n4+D2\n3qxˆτ1}.\n(16)\nHere the momentum qof the Hamiltonian HDirac\nµ,σ(q)is mea-\nsured with respect to the degeneracy point at k=kµ(µ=\n+,−). Thetwo-componentfermionfield Ψµ,σ(q)isgivenby\nΨµ,σ(q) =/parenleftbigg\nα1dyz,σ(kµ+q)+α2dyz,σ(kµ+Q+q)\nβ1dxz,σ(kµ+q)+β2dxz,σ(kµ+Q+q)/parenrightbigg\n(17)\nwhere the constant coefficients αiandβisatisfyα2\n1+α2\n2=\nβ2\n1+β2\n2= 1. Explicitly, αiandβi(i=1,2)aregivenby\nα1= (2+/radicalBig\n4+D2\n3)//radicalbigg\n8+2D2\n3+4/radicalBig\n4+D2\n3,\nα2=D3//radicalbigg\n8+2D2\n3+4/radicalBig\n4+D2\n3,\nβ1= (2−/radicalBig\n4+D2\n3)//radicalbigg\n8+2D2\n3−4/radicalBig\n4+D2\n3,\nβ2=D3//radicalbigg\n8+2D2\n3−4/radicalBig\n4+D2\n3.\nNoticethatthefirst(second)componentof Ψµ,σ(q)isderived\nentirelyfromthe dyz(dxz) orbital.\nNowwe includethe M2spindensitywaveorderparameter\nwhichgeneratesamassterminthe lowenergylimit givenby\nHmass\nM2=/summationdisplay\nµ=±/integraldisplay\nd2qΨ†\nµ,σ(q){−2|D3|M2σ\nD3/radicalbig\n4+D2\n3ˆτ2}Ψµ,σ(q).\n(18)\nFor the given spin σ, the mass term has the same magni-\ntude and sign at the two Dirac points. At each Dirac point,\nthis mass term opens a gap and contributes to the Chern\nnumberNC,σ= +M2σ\n2|M2|for the upper band (band 3) and\nNC,σ=−M2σ\n2|M2|for the lower band (band 4).24,30–32Adding\nthe Chern number contributions from the two Dirac points,\nthe total Chern numberof the band 4 with the spin σis given\nbyN(4)\nC,σ=−sgn(M2)σ, which is consistent with the result\nobtained from the integration of the Berry curvature over th e\nreduced Brillouin zone using Eq. (14). (See Table II.) In the\ncase of the band 3, the Chern number is determined after in-\ncludingtheadditionalcontributionsfromthenodalpoints be-\ntweentheband2andband3.\nSimilarly, the trivial topological property of the gapped\nphasewithfinite D3andD1canalsobeunderstoodbyapply-\ning the same analysis. For the D3nodal density wave state,\nthe smallD1charge density wave order parameter generates\nmassperturbationsto Dirac particles, whichcan be describ ed\nbythefollowingHamiltonian,\nHmass\nD1=/summationdisplay\nµ=±/integraldisplay\nd2qΨ†\nµ,σ(q){2|D3|D1\nD3/radicalbig\n4+D2\n3ˆτ1}Ψµ,σ(q).\n(19)8\nBandfilling SpinChernNumber ( CS)Z2invariant ( ν)\n3\n4- 2 1\n1\n2+ 4 0\n1\n4- 2 1\nTABLE III: Spin Chern numbers and topological Z2invariants for\nthe gapped density wave ground state withfinite D3andM2.\nNotice that this term just induces the shifting of the nodal\npointsawayfromthe ky-axis. OnceaDiracpointmovesaway\nfromthereflectionsymmetryaxis,thedegeneracyattheband\ntouching point is no longer protected by the symmetry and a\ngapopens,becausethedensitywaveorderparameterssuppor t\nfinite matrix elements between the two bands touching at the\nnodal point. Since the D1charge density wave order param-\neter does not generate a mass term to the Dirac Hamiltonian,\nithasnocontributiontotheChernnumberleadingtothezero\nChernnumbersofall bands.\nWeapplysimilaranalysistoeveryDiracpointderivedfrom\nthe Brillouin zone folding for all pairs of density wave orde r\nparameters generating fully gapped phases. In all cases, it is\nconfirmed that the Chern number of each band obtained by\nsumming up the Chern number contributions from the Dirac\npoints is identical to the result obtained by the integratio n of\ntheBerrycurvaturein themomentumspace.\nIn addition, the coexisting D3andM2density wave order\nparametersalsoliftthedegeneraciesofthetwoquadraticb and\ntouching points at k= (0,0)and(0,π)leading to a fully\ngappedbandstructure. In contrast to the case of Dirac point s,\ntheChernnumberobtainedbyliftingaquadraticbanddegen-\neracy is two times larger than the contribution from a single\nDirac point. However, since the two quadratic band crossing\npointslead to the Chern numbercontributionswith the oppo-\nsite signs, the net effect of the two quadratic band touching\npointsvanishes.\nSincethefourbandsarewellseparatedfromeachotherfor\nthetopologicaldensitywavephasewith D3/ne}ationslash= 0andM2/ne}ationslash= 0,\nifthemagnitudeoftheorderparameter M2islargeenough,an\ninsulating ground state is obtained whenever the Fermi leve l\nlies in the gap between two neighboring bands. Therefore\ntherearethreedifferentinsulatingphases,in principle, when-\nevertheFermilevelliesbetweentheband nandn+1(n=1,2,\n3)correspondingtothebandfillingfactor Nfilling= (4−n)/4.\nThetopologicalpropertyoftheinsulatingphasecanbeexpl ic-\nitly characterized by computing topological invariants. W e\nfirst consider the spin Chern number CSwhich is defined in\nthefollowingway,\nCS=/summationdisplay\nn∈occ{N(n)\nC,↑−N(n)\nC,↓}, (20)\nwhere the summation includes all the occupied bands. When\ntheSzis conserved, the spin Chern number CSis quan-\ntized and characterizesthe two dimensionaltopologicalin su-\nlators.25InTable.IIIweshowthespinChernnumbersforthe\ninsulatingphases. ItisinterestingthatthespinChernnum bers\nare nonzero for all cases. Therefore as long as the zcompo-\nnent of the spin is conserved, we can obtain the topologicalinsulator with finite spin hall conductivity for every quart er\nfilling. However,in the presence of spin non-conservingper -\nturbations and disorders, the spin Chern number is not well-\ndefined and conserved only modulo 4.27,33In other words,\nwhentheSzisnotconserved,thehalf-filledsystemisequiva-\nlenttothephasewithzerospinChernnumber,whichisnoth-\ning but a topologically trivial band insulator. However, it is\nimportant to notice that even in this case the system remains\nas a topologicalinsulatorwith nonzero Z2topologicalinvari-\nantfor1/4and3/4fillings.\nWe also compute the Z2topological invariant νshown in\nTable III, which can be used to distinguish topological in-\nsulators (ν= 1) from trivial band insulators ( ν= 0) for\ngenerictime-reversalinvariantsystems. Since the system has\ntheinversionsymmetry,the Z2invariantcanbeobtainedfrom\nthe parity eigenvalues ξm(Γl)of the occupied bands with\nthe band indices mat the time-reversal invariant momenta\nΓl.34Using the reciprocallattice vectors Gi(i=1,2), the four\ntime reversal invariant momenta can be written as Γl=n1n2\n=(n1G1+n2G2)/2withn1,2= 0,1. Explicitlythe Z2topo-\nlogicalinvariant νisgivenby\n(−1)ν=/productdisplay\nni=0,1/productdisplay\nmξm(Γn1n2), (21)\nwheretheparityeigenvaluesoftheoccupiedbandsatthefou r\ntime reversal invariant momenta are multiplied. Because of\nthe time reversal symmetry, each band is doubly degenerate\nat the time reversal invariant momentum and every Kramers\ndoublet share the same inversion parity. The Z2topologi-\ncal invariant counts the parity of one state for each Kramers\npair.34\nAs shown in Table. III, the Z2topological insulators exist\nwhen the band filling is one-quarter or three-quarter. How-\never, the 3/4 filled case requires unreasonably large M2to\nachieve an insulating phase. This is because, as shown in\nFig. 6, the overall structures of the band 1 and 2 are in par-\nallel. Toopenafullgapbetweentheuppertwobands(band1\nand2),themagnitudeofthe M2densitywaveorderparameter\nshould be as large as their bandwidth. Therefore the quarter -\nfilledsystemisthemostfavorablefortherealizationofthe Z2\ntopologicalinsulator.\nTo support further the nontrivial topological properties o f\nthe topological density wave phase with D3/ne}ationslash= 0andM2/ne}ationslash=\n0, we compute the edge state spectrum by considering the\nHamiltonian on a strip geometry, which is infinite in the x-\ndirection but finite in the y-directionwith open boundariesat\ny=1 andy=Ny. HereNyindicates the numberof lattice sites\nin they-direction. The energy spectrum of the system with\nNy= 40is describedin Fig. 7, which showsthe existence of\nrobust gapless edge states traversing between the lower two\nbands (band 3 and 4) and the middle two bands (band 2 and\n3). The upper two bands (band 1 and 2) are not well sep-\narated forD3= 0.85andM2= 0.75, which are used to\nobtainthe energyspectrum. Each edge state representedby a\nreddottedlineisdoublydegenerate,onewithspin-upandth e\nother with spin-down. For the 1/4-filling with the chemical\npotentiallying in the gap between the band 3 and 4, there are\ntwo gaplessedgesstates oneachboundarypropagatingin the9\nk x E \nπ −π 0 0 \n-8 -4 4 \nFIG. 7: (Color online) Energy spectrum for the topological d ensity\nwave ground state with D3= 0.85andM2= 0.75in a strip ge-\nometry. Here we use open boundary conditions with Ny=40 sites\nalong the y-direction. The red dotted lines stand for edge states, all\nof which are doubly degenerate. For 1/2-filling, the number o f edge\nstates istwice largerthan thatfor 1/4-filling.\nopposite directions with the opposite spin quantum numbers .\nOn the other hand, for the 1/2-filling, there are four gapless\nedge states on each boundaryconsistent with the fact that th e\nspin Chern number CS= 4whose magnitude is twice larger\nthan that for the 1/4-filling with CS=−2. Therefore for\nthe collinearspin orderingwith M2/ne}ationslash= 0whichconserves Sz,\nthere are robust gapless edge states for both the quarter-fil led\nandhalf-filledsystems.\nB. Comparison tothehoneycomb lattice\nItisinterestingthatasimpleon-sitedensitywaveorderpa -\nrameter can generate insulating phases with nontrivial top o-\nlogicalproperties. Thisresultcanbecontrastedwiththet opo-\nlogicalinsulatoronthehoneycomblatticewherecomplexse c-\nondneighborhoppingprocessesarerequiredtoobtainatopo -\nlogical insulator while the simple on-site staggered chemi cal\npotentialgivesrisetoatrivialbandinsulator.4,5,10Itisthedis-\ntinct topological properties of the Dirac particles in the D3\nnodaldensitywavephaseinthe d-orbitalsystem,whichmake\nitpossibletorealizethetopologicalinsulatorbyintrodu cinga\nsimpleon-siteorderparameter M2.\nIn this subsection, we discuss the topological property of\nthe Dirac particles in the D3nodal density wave state in de-\ntail and compareit with the topologicalpropertyof the Dira c\nparticlesonthehoneycomblattice. Intheforthcomingdisc us-\nsion we neglect the spin degreesof freedomand focuson the\ncondition under which the insulating phase possesses a finit e\nChern number, which is nothing but a Chern insulator. Once\nwe find the condition to obtain a Chern insulator, the time\nreversal invariant topological insulator can be realized b y su-\nperposing two Chern insulators with spin-up and spin-down(b) \nkk\nxy(a) \nkk\nxy\nFIG. 8: (Color online) The winding directions of the /vectord(q)vector\naround the Dirac point. Black arrows describe the two compon ent\n/vectord= (dx,dy)vector in Eq.(23) along the circular path around the\nDirac point at the center. (a) For the two Dirac points betwee n the\nband3andband 4inthenodal densitywave statewiththefinite D3.\nThe/vectordvector has the same winding direction around the two Dirac\npoints. (b) For the two Dirac points on the graphene system. T he\n/vectordvector has the opposite winding directions around the two Di rac\npoints.\nparticles,respectively.\nThe topological property of the Dirac particles in the D3\nnodal density wave state can be understood in the following\nway. The low energy Hamiltonian HDirac\n±in Eq.(16) for the\nDirac particles near the momentum k=k±, where the band\ntouchingpointsbetweenthe band3 and4 locate,canbe writ-\ntenas\nHDirac\n±(q) =h±,x(q)ˆτ1+h±,y(q)ˆτ3.(22)\nSince/radicalBig\nh2\n±,x(q)+h2\n±,y(q)is nonzero away from the de-\ngeneracy point, a two component vector /vectord±(q)with the unit\nlengthcanbedefinedas\n/vectord±(q) = (d±,x(q),d±,y(q))≡(h±,x(q),h±,y(q))/radicalBig\nh2\n±,x(q)+h2\n±,y(q).(23)\nAlongthecircle CRsatisfyingq2\nx+q2\ny=R2/ne}ationslash= 0withthede-\ngeneracypointat thecenter,the2Dunitvector /vectord±(q)defines\na mapfromthe circle CRto the unitcircle S1. Since the fun-\ndamentalgroup π1(S1) =Z,the2Dunitvector /vectord±(q)hasan\ninteger-valuedtopological invariant, which is nothing bu t the\nwinding number Nwdefined in Eq.(10). In terms of the 2D\nunit vector/vectord±(q), the winding number Nwcan be rewritten\nas,\nNw=1\n2π/contintegraldisplay\nCRdθˆz·(/vectord×d/vectord\ndθ) (24)\nwhere the loop integral is defined along the circle where the\nmomentum q=Reiθ.3510\nIn Fig. 8(a) we describe the directions of the two-\ncomponent/vectordvector along the circular path around the Dirac\npoints for the D3nodal density wave state. Notice that the /vectord\nvectorhasthe samewindingdirectionwiththe windingnum-\nber ofNw=1 around the two Dirac points. The two Dirac\npoints share the same winding number because of the con-\nstraintimposedbylatticesymmetries. Sincethefirst(seco nd)\ncomponent of the two-component Dirac field Ψµ(q)is given\nby thedyz(dxz) orbital state, Ψµ(q)transforms to - Ψ¯µ(−q)\nundertheinversionsymmetryduetotheoddparityof dxzand\ndyzorbitals. Here ¯µhasthe oppositesign of µ. Thisimposes\nthefollowingconstraintonthepairofDiracHamiltoniansr e-\nlatedbytheinversionsymmetry,\nHDirac\n+(q) =HDirac\n−(−q). (25)\nThis constraint guaranteesthe same winding numbersfor the\ntwo Dirac Hamiltonians HDirac\n+andHDirac\n−. It is important to\nnoticethateverypairofDiracHamiltoniansrelatedbythei n-\nversionsymmetrysatisfies the same constraint fornodal den -\nsity wavephases.\nThe fact that a pair of Dirac Hamiltonians related by the\ninversion symmetry have the same winding numbers is the\ndistinct topological property of the Dirac particles in the\nnodal density wave states, distinguishable from the topolo g-\nicalpropertiesoftheDiracparticlesinthehoneycomblatt ice.\nIn this system, the two Dirac points at the corners of the first\nBrillouinzonehavethe oppositewindingdirections,which is\ndescribed in Fig. 8(b). Since the inversion symmetry inter-\nchangesthe two sublattices of the honeycomblattice, each o f\nwhich comprises one component of the Dirac fermion field\nΨµ(q), the Dirac fermion field Ψµ(q)transforms, for exam-\nple, toˆτxΨ¯µ(−q)under the inversion symmetry. This im-\nposes the following constraint to the two Dirac Hamiltonian s\nrelatedbythe inversionsymmetry,\nHDirac\n+(q) =ˆτxHDirac\n−(−q)ˆτx. (26)\nThe additional Pauli matrix reverses the winding direction of\none of the /vectordvectors, leading to the two Dirac Hamiltoni-\nans with the opposite winding numbers. Therefore these two\nDirac points can be pair-annihilated when they are brought\ntogetherbyperturbations.13,14\nThe relative winding numbers of the pair of the Dirac\nHamiltoniansrelatedbytheinversionsymmetry,stronglyc on-\nstrain the topological properties of the insulating phases ob-\ntained by mass perturbations to the Dirac particles. The in-\ntroductionofa constantmass term Hmass\n±=mˆτ2to the Dirac\nHamiltonian in Eq. (22) gives rises to the third component\ndz(q)of the corresponding /vectordvector.19,25Explicitly, for the\nmassiveDiracHamiltoniangivenby\nHDirac(q) =hx(q)ˆτ1+hy(q)ˆτ3+mˆτ2,(27)\nthe3Dunitvector /vectord3Disdefinedas\n/vectord3D(q) =(dx(q),dy(q),dz(q))\n≡(hx(q),hy(q),m)/radicalBig\nh2x(q)+h2y(q)+m2. (28)If we introduce, for instance, a positive mass term to the\ntwo Dirac Hamiltonians corresponding to the D3nodal den-\nsitywavephasedescribedinFig.8(a),bothofthe /vectord3Dvectors,\nwhich have positive zcomponents, move along the northern\nhemisphere as the momentum qsweeps over the two dimen-\nsional momentum space. The net solid angles subtended by\nthese two/vectord3Dvectors over the entire momentum space are\nthe same, each of which covers 2π. At this point, it is useful\nto take into accountthe followingrelation betweenthe Cher n\nnumberofthevalencebandandthe3Dunitvector /vectord3Dforthe\ntwobandHamiltonianin Eq.(27),19\nNC=/integraldisplayd2k\n4π/vectord3D·(∂kx/vectord3D×∂ky/vectord3D).(29)\nThe above identity implies that the Chern number counts the\nnumber of times the 3D unit vector /vectord3Dwinding around the\nunit sphere over the Brillouin zone torus. Therefore when a\nconstantmasstermisaddedtothetwoDiracpointsconnected\nby the inversion symmetry, the Chern number NC=±1if\nthe two Dirac points have the same winding numbers, which\nisrealizedin thenodaldensitywavegroundstate.\nIn contrast, the net solid angles covered by the two /vectord3D\nvectors in the honeycomb lattice have the same magnitudes\nbut with the opposite signs. Therefore the total solid angle\ncovered by these two /vectord3Dvectors vanishes. The vanishing\nChern number contributions from the two Dirac points leads\nto the topologically trivial insulating phase when the simp le\nmass term in Eq. (27) is introduced. This contrasting behav-\nior of the pair of 3D unit vectors /vectord3Din these two systems\nresults in the distinct topological properties of the insul ating\nphases, the topological insulator in the d-orbital system and\nthe topologically trivial band insulator in the honeycombl at-\ntice when constant mass terms with the same signs are added\ntothepairofDiracpointsconnectedbytheinversionsymme-\ntry. However,ingrapheneifweintroducemasstermswiththe\noppositesignsat the two Dirac points, a topologicalinsula tor\nwitha finite Chernnumbercanbe obtained,whichisrealized\nby consideringthe complexsecondnearest neighborhopping\nprocessesonthe honeycomblattice.\nV. MEANFIELDTHEORY\nNow we address the question whether the TDW insulator\nwithfiniteD3andM2canbeachievedinrealsystems. Inpar-\nticular, by taking into account the interactions between el ec-\ntrons,weinvestigatetheconditionstorealizetheTDWinsu la-\ntor via spontaneous symmetry breaking. Previous mean field\nstudies on the two-orbital Hubbard model with the hopping\nHamiltonian H0in Eq. (1) show that the leading instability\nof the system is the uniform spin density wave phase ( M0-\nSDW)whichisdescribedbynonzero M0. However,thereex-\nistanotherdensitywaveorderparametersincludingimagin ary\nchargeandspindensitywavestates,whicharecompetingwit h\ntheuniformspindensitywavestate( M0-SDW)withsmallen-\nergy differences.17,36,37Therefore if we include longer range\ninteractionswhicharenotincludedinthemulti-orbitalon -site11\nM  - SDW TDW \n(D  + M  ) \nD  - CDW M  - SDW \n0\n2 32\n3\nU26 7 8V2,eff \n-1 3\n2\n1\n0\nFIG. 9: (Color online) Mean field phase diagram. Here we set\nU= 5,JH= 0and plot the ground state phase diagram varying\nthe inter-orbital on-site repulsion U2and the effective next nearest\nneighbor repulsion V2,eff=V2B−2V2A. There is a finite range\nbetweenD3charge density wave phase ( D3-CDW) and M2spin\ndensity wave phase ( M2-SDW) where the topological density wave\nphase (TDW)becomes theground state. Solid(dotted)linesi ndicate\nthe first(second) order phase transitions.\nHubbard Hamiltonian, another competing ground states, for\nexample, the topological density wave state (TDW), can be-\ncomethegroundstate replacingthe M0-SDWstate.\nIncluding on-site and inter-site electron-electron inter ac-\ntions,thefullHamiltonianisgivenby\nHfull=H0+Honsite+Hintersite (30)\ninwhich\nHonsite=U/summationdisplay\ni/summationdisplay\na=1,2ni,a,↑ni,a,↓+U2/summationdisplay\nini,1ni,2\n+JH/summationdisplay\ni/summationdisplay\nσ1,σ2d†\ni,1,σ1d†\ni,2,σ2di,1,σ2di,2,σ1\n+JH/summationdisplay\ni(d†\ni,1,↑d†\ni,1,↓di,2,↓di,2,↑+H.c.),(31)\nand\nHintersite=V1A/summationdisplay\n/angbracketleftij/angbracketright/summationdisplay\na=1,2ni,anj,a+V1B/summationdisplay\n/angbracketleftij/angbracketrightni,1nj,2\n+V2A/summationdisplay\n/angbracketleft/angbracketleftij/angbracketright/angbracketright/summationdisplay\na=1,2ni,anj,a+V2B/summationdisplay\n/angbracketleft/angbracketleftij/angbracketright/angbracketrightni,1nj,2,\n(32)\nwhereH0indicates the hopping Hamiltonian in Eq.(1). For\nthe on-site interactions described by Honsite, the intra-orbital\nrepulsionU, the inter-orbital repulsion U2, and the Hund’s\ncouplingJHare considered. In the Hintersitedescribing\nthe inter-site Coulomb interactions, V1A(V1B) indicates the\nnearest-neighborCoulombrepulsionbetweenelectronsint hesame (different) kinds of orbitals. Finally, V2A(V2B) indi-\ncates the next nearest-neighbor Coulomb repulsion between\nelectronsin thesame(different)kindsoforbitals.\nToinvestigatetheexistenceofthetopologicaldensitywav e\nstate (TDW) with D3/ne}ationslash= 0andM2/ne}ationslash= 0and its competition\nwiththeuniformspindensitywavephase( M0-SDW),weap-\nply a mean field approximationto the Hamiltonian Hfull. The\nresultingmean-fieldHamiltonianisgivenby\nHMF=H0+Nǫ0\n+/summationdisplay\nk∈RBZ,σ[A11,σ{d†\n1,σ(k)d1,σ(k+Q)+d†\n1,σ(k+Q)d1,σ(k)}\n+A22,σ{d†\n2,σ(k)d2,σ(k+Q)+d†\n2,σ(k+Q)d2,σ(k)}\n+A12,σ{d†\n1,σ(k)d2,σ(k+Q)+d†\n1,σ(k+Q)d2,σ(k)}\n+A21,σ{d†\n2,σ(k)d1,σ(k+Q)+d†\n2,σ(k+Q)d1,σ(k)}],\n(33)\ninwhich\nǫ0=U\n8(M2\n0−D2\n3)+U2\n8(M2\n2+2D2\n3)\n+JH\n8(M2\n0−D2\n3−M2\n2)+(V2A−V2B\n2)D2\n3,(34)\nand\nA11,σ=U\n4(D3−σM0)−U2\n2D3\n−(2V2A−V2B)D3+JH\n4(−D3+σM0),\nA22,σ=−U\n4(D3+σM0)+U2\n2D3\n+(2V2A−V2B)D3+JH\n4(D3+σM0),\nA12,σ=A∗\n21,σ=U2\n4iσM2−JH\n4iσM2. (35)\nNote that the nearest neighbor Coulomb repulsions V1Aand\nV1Bdo not contribute to the mean field Hamiltonian be-\ncause the order parameters have the ordering wave vector\nQ= (π,0).\nTheorderparameters M0,M2,D3aredeterminedbysolv-\ningthefollowingself-consistentequations,\nM0=1\nN/summationdisplay\nr=(rx,ry)/summationdisplay\nσ=±(−1)rxσ/an}b∇acketle{td†\nr,1,σdr,1,σ+d†\nr,2,σdr,2,σ/an}b∇acket∇i}ht,\nM2=1\nN/summationdisplay\nr=(rx,ry)/summationdisplay\nσ=±(−1)rxσ/an}b∇acketle{tid†\nr,1,σdr,2,σ−id†\nr,2,σdr,1,σ/an}b∇acket∇i}ht,\nD3=1\nN/summationdisplay\nr=(rx,ry)/summationdisplay\nσ=±(−1)rx/an}b∇acketle{td†\nr,1,σdr,1,σ−d†\nr,2,σdr,2,σ/an}b∇acket∇i}ht.\n(36)\nThe chemical potential µis also determined self-consistently\ntosatisfy thehalf-fillingcondition.\nThe resulting mean field phase diagram is shown in Fig. 9.\nHere we choose U= 5,JH= 0and compute the ground12\nstate phase diagram as a function of the inter-orbital on-si te\nrepulsionU2and effective next nearest neighbor repulsion\nV2,eff≡V2B−2V2A. In the absence of the inter-site inter-\nactionsV2,eff= 0, the uniformspin density wave phase ( M0-\nSDW) dominates the phase diagram consistent with the pre-\nvious studies. However, the inter-orbital on-site repulsi onU2\nsuppresses the uniform spin density wave states ( M0-SDW)\nwhich is diagonal in the orbital space, but promotes the M2-\nSDW,whichisoff-diagonalintheorbitalspace,tothegroun d\nstate. On the other hand, the D3charge density wave phase\n(D3-CDW) is strongly affected by the next nearest neighbor\nCoulomb repulsions V2AandV2B. In particular, the V2A,\nthe next nearest neighbor repulsion between the electrons i n\nthe same kinds of orbitals, strongly favors the D3-CDW be-\ncause the staggered orbital ordering described by D3-CDW\ncanavoidtheenergycostcomingfrom V2A. Noticethatthere\nis a finite range in the parameter space where both M2-SDW\nandD3-CDW are nonzero realizing the topological density\nwavephase(TDW).\nTheabovemeanfieldphasediagramisobtainedforthehalf\nfilled case where the topological property of the TDW phase\nisnotrobustagainstperturbationsbreaking Szsymmetry. On\nthe other hand, the TDW insulator at 1/4 filling maintains its\ntopological properties as long as time reversal symmetry is\npreserved. It was shown, in the study of the single orbital\nextendedHubbardmodel,thatthenextnearestneighborinte r-\naction (V2Afor our model) stabilizes a stripe pattern charge\nordering with the momentum Q= (π,0).38This occurs at\n1/4 filling when the on-site Hubbard interaction Uis much\nstronger than the hopping amplitude tsatisfyingU >> t, so\nthat double occupancy is almost frozen. In our model, there\nare two orbitals of dxzanddyz. Similar to the single orbital\ncase, we find that the next nearest neighbor interaction stab i-\nlizesD3order. Therefore, when the on-site intra- and inter-\norbital interactionssatisfy U,U2≫t, theD3-CDW ordering\nshouldbefavoredat 1/4filling.\nTo get a TDW insulator, finite M2is required in addi-\ntion toD3. As shown in Sec. II B, the M2order order\nparameter is equivalent to a staggered spin-orbit coupling ,\n∼/summationtext\ni(−1)ixSi,zLi,z. One can show that when spin-orbitin-\nteraction is present, M2term can be induced as long as D3\nsets in, since D3leads to unequal density between dxzand\ndyzorbitals. Therefore,weexpectthattheTDWinsulatorcan\nbe obtained by tuning V2BwhenU,U2≫tat 1/4 filling in\nthepresenceofthespin-orbitcoupling.\nVI. TOPOLOGICALINSULATORSIN THREE-BAND\nSYSTEMS\nIn the preceding sections, we have focused on a two-band\ntight-bindingHamiltonian, which consists of dxzanddyzor-\nbitals. However,the main idea for realizing topologicalin su-\nlatorsusing two densitywave orderparameterswith opposit e\nsymmetries under reflections is valid in general and applica -\nble to more realistic multi-orbital systems. Here we extend\nouranalysistoathree-bandmodelcomposedof dxz,dyz,and\ndxyorbitals. In particular, we apply our idea to a more real-t1t2t3t4t5t6t7t8∆xy\n0.02 0.06 0.03 -0.01 0.2 0.3 -0.2 0.12 0.4\nTABLE IV: Parameters for the three-band tightbinding Hamil to-\nnian.43\nistic model Hamiltonian relevant to the iron pnictide syste m,\nwhich is a representative itinerant multi-band system mani -\nfesting a density wave ground state with the ordering wave\nvectorQ= (π,0).36,39–42Here we adopt the three-orbital\nmodelproposedbyDaghoferetal.,39whichcapturesthemain\nphysicalpropertiesoftheFe-pnictidesystems. Inthemome n-\ntumspace,theeffectivethree-bandtight-bindingHamilto nian\nisgivenby\nH3band=/summationdisplay\nk,σ/summationdisplay\nµ,νd†\nµ,σ(k)Tµν(k)dν,σ(k),(37)\nwhere\nT11= 2t2coskx+2t1cosky+4t3coskxcosky−µc,\nT22= 2t1coskx+2t2cosky+4t3coskxcosky−µc,\nT33= 2t5(coskx+cosky)+4t6coskxcosky−µc+∆xy,\nT12=T21= 4t4sinkxsinky,\nT13= (T31)∗= 2it7sinkx+4it8sinkxcosky,\nT23= (T32)∗= 2it7sinky+4it8sinkycoskx.(38)\nHere we use the unfolded Brillouin zone satisfying −π <\nkx,ky≤πasbefore,whichcorrespondstooneironatomper\nunit cell. In real iron pnictidematerials, the unit cell con tains\ntwo Fe atoms due to the bucklingof the As atoms. Therefore\ntheunit translationsalongthe x(Tx) andy(Ty) directionsby\nthenearestneighborFe-Fedistance,arenotthesymmetries of\nthe system. However,as pointedout in Ref. 40, the system is\ninvariant under the translations combined with the reflecti on\nPzwith respect to the xyplane, i.e.PzTxandPzTy. Then\ntheeigenstatescanbelabeledbyapseudo-crystalmomentum\ncorrespondingto the eigenvalues of the combined operation s\nPzTxandPzTywith one iron atom per unit cell. We use this\npseudo-crystal momentum to label states for the momentum\nspacerepresentationoftheHamiltonianinEq.(37).\nInEq.(37)and(38), µ=1,2,3indicate dxz,dyz,anddxyor-\nbitals,respectively. ∆xyrepresentstheatomicpotentialof dxy\norbitalrelativeto dxzanddyzorbitals. Thechemicalpotential\nis given byµc. The hopping parameters are displayed in Ta-\nble IV, whichare determinedin Ref. 39. Forthe 2/3 filling,39\nthere are two hole pockets near the Γpoint and two electron\npockets at the XandYpoints, which are consistent with the\nLDAcalculationsandARPES measurementforLaOFeAs.\nDensity wave order parameters with the momentum\nQ=(π,0)canbedescribedbythefollowingHamiltonian,13\nD11D22D33DR\n12DI\n12DR\n13DI\n13DR\n23DI\n23\nPxPz+++----++\nPyPz+++--++--\nI+++++----\nT++++-+-+-\nTABLE V: Symmetry of the charge density order parameters wit h\nthemomentum ( π,0). Here‘+’(‘-’)indicates‘even’(‘odd’)parityof\norder parameters under the corresponding symmetry operati on. The\ncomplex off-diagonal components Dij(i/negationslash=j) are decomposed as\nDij=DR\nij+iDI\nij.\nM11M22M33MR\n12MI\n12MR\n13MI\n13MR\n23MI\n23\nPxPz+++----++\nPyPz+++--++--\nI+++++----\nT----+-+-+\nTABLEVI: Symmetry of the spin density order parameters with the\nmomentum ( π,0). Here ‘+’ (‘-’) indicates ‘even’ (‘odd’) parity of\norder parameters under the corresponding symmetry operati on. The\ncomplex off-diagonal components Mij(i/negationslash=j) are decomposed as\nMij=MR\nij+iMI\nij.\nˆHCDW=/summationdisplay\ni,σ3/summationdisplay\na,b=1(−1)ixDabd†\ni,a,σdi,b,σ,\nˆHSDW=/summationdisplay\ni,σ1σ23/summationdisplay\na,b=1(−1)ixMabd†\ni,a,σ1sz\nσ1σ2di,b,σ2.(39)\nTheorderparameterrepresentedby3 ×3Hermitianmatrix ˆD\n(ˆM) hasnineindependentcomponents Dij(Mij). Thetrans-\nformation properties of these density wave order parameter s\nunder the reflections PxPzandPyPz, inversionI, and time-\nreversalTare summarized in Table V and VI. Notice that\nthe reflections PxandPyare not the symmetries of the sys-\ntem. The Hamiltonian is invariant only under the combined\ntransformations PxPzandPyPz.\nTo obtain a nodal spin density wave ground state, we con-\nsider the simplest uniform charge density wave order param-\neter,ˆDuniform≡diag[d0,d0,d0] with the finite diagonal com-\nponents ofD11=D22=D33=d0. This generates many\nDirac points along an axis with a reflection symmetry in the\nmomentumspace, whenevera band touchingoccursbetween\ntwo bands with opposite reflection parities. Now let us in-\ntroduceanotherdensitywaveorderparametertogetagapped\nphase. Toopenafullgapbetweenneighboringbandsweneed\nadensitywaveorderparameterwhichisoddunderthereflec-\ntion symmetries PxPzandPyPz. Imposing the time reversal\nsymmetry, the imaginary part of the spin density wave order\nparameterMI\n12≡Im(M12), is the unique choice to obtain a\ntopologicalinsulator.\nIn Fig. 10 (a) we plot the band structure of the uniform\nchargedensitywavestatewithnonzero Duniform/ne}ationslash= 0alongthe\nkyaxis. SinceDuniformpreservesthe PxPzreflectionsymme-(a) \n(b) -π k y π E(k) \n-π k y π E(k) + \n+ - \n- \n- - \nFIG. 10: (Color online) The band structures of the density wa ve\nstates with the momentum k=(π,0) along kyaxis. (a) For the uni-\nform charge density wave state with Duniform= 0.5. The locations\nof Dirac nodes are indicated by red dotted circles. The parit ies of\nthe bands under the PxPzsymmetry are indicated by + (even) and -\n(odd). (b) A density wave phase with Duniform/negationslash= 0andMI\n12/negationslash= 0at\nthe same time. Here we take Duniform=0.5 and MI\n12=0.1. Each band\niswell-separated from the other bands.\nMI\n12= 0.1MI\n12= 0.2MI\n12= 0.3\nCS,10 -2 -2\nCS,20 +2 +2\nCS,30 0 -4\nCS,4-2 -2 +2\nCS,5+2 +2 +2\nCS,60 0 0\nTABLE VII: The spin Chern numbers of the bands for a gapped\ndensity wave phase with Duniform/negationslash= 0andMI\n12/negationslash= 0. Here we set\nDuniform= 0.5and change the magnitude of MI\n12.\ntry,eachbandhasadefinitereflectionparityunder PxPz. The\nreflectionparitiesofthebandsarealsoindicatedinFig.10 (a).\nNoticethatnodalpointsexistbetweenapairofthebandswit h\noppositereflection parities, which are indicatedby red dot ted\ncircles in Fig. 10 (a). However, once we introduce a nonzero\nMI\n12, these nodal points disappear and a fully gapped phase\nwithwell-separatedbandsemerges. Thebandstructureofth e\nresultinggappedphaseisdescribedinFig. 10(b).\nToinvestigatethetopologicalpropertyofthegappedphase ,\nwe compute the spin Chern numbers of the bands. Since\nthezcomponent of the spin Szis still conserved, the spin\nChern number is a well-defined quantity. In Table VII, we\nshow the distribution of the spin Chern numbers for several14\nvalues ofMI\n12supporting fully gapped phases. Here CS,n\nindicates the spin Chern number of the nth band defined as\nCS,n=NC,n,↑−NC,n,↓whereNC,n,↑(=-NC,n,↓) denotes\nthe Chern number of the nth spin-up band. We label that the\nband1 has the highest energyand the band index nincreases\nas the energyeigenvaluedecreases. In the case of the gapped\nphase that is obtained by adding a small MI\n12on the nodal\nchargedensity wave state with Duniform/ne}ationslash= 0, only the 4th and\n5th band support nonzero spin Chern numbers shown in the\n2nd columnof Table VII. Interestingly,however,as the mag-\nnitude ofMI\n12increases, band gap closing and reopeningoc-\ncur successively. For instance, for the uniform density wav e\nstate withDuniform= 0.5, the first gap closing happens be-\ntween the band 1 and 2 for MI\n12≈0.15. AsMI\n12increases\nfurther, another fully gapped phase is obtained with the spi n\nChernnumbersdisplayedinthe3rdcolumnofTableVII. Itis\ninteresting to notice that after the gap closing and reopeni ng\nprocess,the numberofthebandssupportingfinitespinChern\nnumbershasincreased. Similargapclosinghappensagainfo r\nMI\n12≈0.22leading to the redistribution of the spin Chern\nnumbersshown in the last column of Table VII. Note that all\ncaseswith1/3fillinggiveTDWinsulators,whileTDWphase\nwith5/6fillingoccursonlyforthe MI\n12= 0.2and0.3.\nVII. SUMMARYANDDISCUSSION\nIn this paper, we investigate theoretically if topological in-\nsulatorscanbeachievedfroma nodaldensitywavestate with\nbrokentranslationalsymmetry. Whileanonzerodensitywav e\norder parameter in general opens a gap between the degener-\nate states connected by the ordering wave vector, nodal den-\nsity wave phases occur in multi-orbital systems via transla -\ntional symmetry breaking due to the distinct symmetry prop-\nerties of orbitals. Such a nodal density wave state supports a\nlargenumberofDirac nodesbetweenneighboringbands. We\nhaveexplicitlyprovedthatapairofinversionsymmetricDi rac\npoints share the same topological winding numbers in nodal\ndensity wave states contraryto the Dirac pointsin the honey -\ncomb lattice. If we introduce an additional order parameter\nwhosetransformationpropertyunderreflectionsymmetries is\noppositetothatoftheunderlyingorderparameter,thesyst em\ncan be a gapped insulator at certain filling factors. Among\nthose insulators, time-reversal invariant TDW insulators with\nhelicaledgestatesareidentified.\nThe existence of a nodal density wave ground state is\nexperimentally verified in a recent ARPES measurementon BaFe 2As244and quantum oscillation experiments on\nBaFe2As2and SrFe 2As2.45–47It is interesting to notice that,\naccordingto these experimentalstudies, the velocityof Di rac\nfermions is estimated to be 14 - 20 times slower than that\nin graphene.47This implies that the Dirac fermions in nodal\ndensitywavestatesaremoresusceptibletointeractioneff ects.\nHowever, according to our mean field calculation, it seems\nto be difficult to realize quantum spin Hall insulators in Fe\npnictidessystem,asitfavorsaconventionalspindensityw ave\nstate (M0).\nOur results in general imply that transition metal materi-\nals with two-dimensional square lattice structure possess ing\npartially filled t2gorbitals are good candidates for TDW in-\nsulators. In particular, in the case of the effective two-or bital\n(three-orbital) model, the 1/4 filled (1/3 filled) system is t he\nmost promising for the realization of TDW insulators. How-\never, to make a predictionon real materials with layered per -\novskite structure, it is important to generalize our study t o\nthreedimensionalsystemstakingintoaccountinterlayerc ou-\nplings. Stacking of two dimensional TDW insulators simply\nleads to a weak topologicalinsulator.48Therefore identifying\nthree dimensionalTDW phaseswith a nontrivialstrong topo-\nlogicalinvariantinthelayeredperovskitestructureisan inter-\nestingbutchallengingfuturework.\nFinally, it is worthwhile to comment the consequences of\nrelaxing the constraint of time-reversal invariance. When\nan imaginary charge density wave state ( D2) breaking time-\nreversal symmetry occurs in the presence of a nodal density\nwavestate,agappedtopologicalphasewithtopologicallyp ro-\ntected edge modes can be developed. 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Kee, and Y. B. Kim, Europhysics Letters, 88,\n17004 (2009).\n50Z. -J. Yao, J. -X. Li, Q. Han, and Z. D. Wang, arXiv:1003.1660\n(unpublished)."    },    {        "title": "1008.1087v1.The_integrated_density_of_states_of_the_random_graph_Laplacian.pdf",        "content": "The integrated density of states of the random graph Laplacian\nT. Aspelmeier1, 2and A. Zippelius1, 3\n1Max Planck Institut f ur Dynamik und Selbstorganisation, Bunsenstr. 10, 37073 G ottingen, Germany\n2Scivis GmbH, Bertha-von-Suttner-Str. 5, 37085 G ottingen, Germany\n3Institut f ur Theoretische Physik, Georg-August Universit at G ottingen,\nFriedrich-Hund-Platz 1, 37077 G ottingen, Germany\nWe analyse the density of states of the random graph Laplacian in the percolating regime. A\nsymmetry argument and knowledge of the density of states in the nonpercolating regime allows us\nto isolate the density of states of the percolating cluster (DSPC) alone, thereby eliminating trivially\nlocalised states due to \fnite subgraphs. We derive a nonlinear integral equation for the integrated\nDSPC and solve it with a population dynamics algorithm. We discuss the possible existence of a\nmobility edge and give strong evidence for the existence of discrete eigenvalues in the whole range\nof the spectrum.\nThe eigenvalue spectrum of sparse random matri-\nces is a fascinating and largely unsolved problem with\nwidespread applications ranging from transport in disor-\ndered systems, graph theory and optimization problems\nto nuclear physics and QCD [1, 2]. In this paper we con-\nsider a prototype of such a random matrix, the Laplace\noperator on a mean-\feld random graph with Nnodes, i.e.\na graph where a link between two arbitrary sites is either\npresent with probability p= 2c=N or not present with\nprobability 1\u0000p. The constant 2 cis the mean connec-\ntivity of the graph. The Laplace operator on this graph\nis a matrix \u0000 ijwhere fori6=j\u0000ij=\u00001 if the nodes i\nandjof the graph are connected and \u0000 ij= 0 otherwise,\nwhile on the diagonal \u0000 ii=\u0000P\nj6=i\u0000ij. The entries on\nthe diagonal are thus correlated to the random entries\noutside the diagonal.\nEven though the computation of the density of states\nfor a mean-\feld random graph has been reduced to an in-\ntegral equation [3{5], a complete solution is still missing.\nIn the limit of in\fnite coordination, c!1 , Wigner`s\nsemi-circle law is recovered. For any \fnite c, Lifshitz tails\nwere shown to exist in the integrated density of states [6].\nBeyond these asymptotic results a number of approxima-\ntions have been used to compute the spectrum approx-\nimately, such as e\u000bective medium theory, single defect\napproximation, moment expansions and numerical diag-\nonalisation [7{9].\nIn this paper we show \frst that the density of states\nof the percolating cluster (DSPC) can be isolated using\na symmetry argument and our knowledge of the density\nof states below the percolation threshold. Second, we\nderive an integral equation for the integrated density of\nstates which can be solved reliably with a population\nalgorithm. The numerical solution reveals jumps in the\nintegrated DSPC in the whole range of the spectrum,\ncalling in question the existence of a mobility edge.\nModel and symmetry : The model shows a percola-\ntion transition at ccrit= 1=2. Below this concentration\nthere is no macroscopic cluster and almost all \fnite clus-\nters are trees. The average number of tree clusters Tnwithnnodes is given in the macroscopic limit by [10]\nlim\nN!1Tn(2c)\nN=\u001cn(2c) =nn\u00002(2ce\u00002c)n\n2cn!: (1)\nIn particular the total number of clusters per particle is\n\u001ctot(2c) = 1\u0000c. Forc <1=2, the spectrum consists of\na very complicated, but countable set of \u000e-peaks which\ncan be calculated iteratively [11]. Above the percola-\ntion threshold c>1=2 a percolating cluster coexists with\nmany \fnite clusters, which are also trees. The fraction\nof sites in the macroscopic cluster, Q(c), is the solution\nof 1\u0000Q(c) = exp(\u00002cQ(c)). Alternatively we rewrite\nQ(c) = 1\u0000x(c)\n2cand obtain x(c) as the solution of\nx(c)e\u0000x(c)= 2ce\u00002c: (2)\nThis equation has two solutions, a trivial one with x(c) =\n2cand a nontrivial one with x(c) = 2c\u0003such thatc\u0003>1\n2\nifc <1\n2and vica versa. This nontrivial solution allows\none to establish a symmetry for the number of trees above\nand below the percolation threshold. Using Eq. (2), we\ncan rewrite Eq. (1) according to\n\u001cn(x(c)) =2c\nx(c)\u001cn(2c) or\u001cn(2c\u0003) =c\nc\u0003\u001cn(2c):(3)\nHence the number of trees above the percolation thresh-\nold is simply related to the number of trees below the\npercolation threshold.\nDensity of states This relation allows us to com-\npute the density of states of the percolating cluster alone ,\nwhich is the quantity of primary interest. It is known\nthat the density of states, D(\n), of the in\fnite cluster\ncontains at least some \u000epeaks, so that a population\ndynamics algorithm [9] cannot be applied. In this pa-\nper we instead compute the integrated density of states,\n\u0001(\n) =R\n0d\n0D(\n0), which according to measure the-\nory [12], may be decomposed into a singular part and\nan absolutely continuous part. The absolutely contiuous\npart may itself consist of two contributions, eigenvalues\nstemming from localised eigenvectors which happen to liearXiv:1008.1087v1  [cond-mat.dis-nn]  5 Aug 20102\ndense and each have vanishing weight in the thermody-\nnamic limit, and the eigenvalues stemming from nonlo-\ncalised eigenvectors. We may thus write\n\u0001(\n) = \u0001 loc,disc (\n) + \u0001 loc,cont (\n) + \u0001 nonloc (\n):(4)\nOften in random matrix problems there is a mobility\nedge, i.e. a value \n 0such that all eigenvectors corre-\nsponding to eigenvalues \n <\n0are localised and all\neigenvectors corresponding to \n >\n0are nonlocalised\n(or vice versa). We will show here that such a sharp edge\ndoes not seem to exist in our problem as we \fnd dis-\ncrete eigenvalues even in the region where nonlocalised\neigenvectors lie. This shows that a naive approach using\nthe inverse participation ratio in order to \fnd the on-\nset of nonlocalised eigenvectors will not work since the\nlocalised eigenvectors do not disappear where the nonlo-\ncalised ones start, so the inverse participation ration will\nnot drop down to 0 as it would at a true mobility edge.\nThe density of eigenvalues, f\nigN\ni=1, of the Laplacian\nmatrix \u0000 is de\fned by\nD(\n;c) = lim\nN!11\nNNX\ni=1\u000e(\n\u0000\ni) = lim\nN!11\nNTr\u000e(\n\u0000\u0000)\n(5)\nHere\u0001denotes the average over all realizations of connec-\ntivity for a given c. To compute the density of eigenvalues\nwe introduce the resolvent\nG(\n;c) = lim\nN!11\nNTr1\n\u0000\u0000\n(6)for complex argument \n = \r+i\u000f;\u000f > 0. In the limit\n\u000f!0, we recover the spectrum from the imaginary part\nof the resolvent according to\nD(\n;c) =1\n\u0019lim\n\u000f#0=G(\n +i\u000f;c): (7)\nIn the percolating regime c\u00151\n2the macroscopic cluster\ncoexists with many \fnite ones. In order to study localised\nstates of the macroscopic cluster it is essential to isolate\nthe density of states of the macroscopic cluster only. We\nuse Eqs. (2) and (3) to decompose the resolvent into two\ncontributions, one from the percolating cluster and one\nfrom the \fnite clusters:\nG(\n;c) =Gperc(\n;c) +c\u0003\ncG(\n;c\u0003) (8)\nThe above relation allows us to compute the density of\nstates of the percolating cluster alone from the full den-\nsity of states above, G(\n;c), and the density of states\nbelow the percolation threshold, G(\n;c\u0003) forc\u0003<1\n2,\nwhich is known [11].\nThe average over all realizations of connectivity is per-\nformed with the replica trick, resulting in a nonlinear\nintegral equation for gc;\n(\u001a) (cf. Eqs. (16) and (17) in\nRef. [3])\ngc;\n(\u001a) = 2cexp\u001a\n\u0000i\u001a2\n2\u001b\n+ 2ice\u00002cZ1\n0dx\u001aI1(i\u001ax) exp\u001a\n\u0000i\n2(\u001a2+x2) +i\n2x2+gc;\n(x)\u001b\n(9)\nwithgc;\n(0) = 2c. Here I\u0017(z) are the modi\fed Bessel\nfunctions of the \frst kind. The solution of Eq. (9) yields\nthe resolvent [3]\nG(\n;c) =\u00001 +i\n2cZ1\n0d\u001a\u001agc;\n(\u001a) (10)\nNonpercolating regime : The analytical solution of\nEq. (9) for c<1=2 was given in [11]. Brie\ry, it is\ngc;\n(\u001a) = 2cX\nkakexp(\u0000i\n2zk\u001a2) = 2cZ1\n\u00001dac;\n(\u0015)e\u0000i\n2\u0015\u001a2\n(11)\nwith coe\u000ecients akandzkto be de\fned below. The sum\ncan also be expressed as a Riemann-Stieltjes integral with\nweight function ac;\n(\u0015) =P\nkak\u0012(\u0015\u0000zk) (\u0012(\u0001\u0001\u0001) is the\nHeaviside function). This formulation will be useful later.The coe\u000ecients akandzkcan be grouped in in\fnitely\nmany \\classes.\" The coe\u000ecients of class n+ 1 are given\nrecursively by the relations\na(n+1)\n(lk)=e\u00002cY\nk(2ca(n)\nk)lk\nlk!(12)\nz(n+1)\n(lk)=\n\u0000P\nklkz(n)\nk\n\n\u00001\u0000P\nklkz(n)\nk(13)\n(the upper index denotes the class). Class 0 contains\nonly one element, namely a(0)\n0=e\u00002candz(0)\n0=\n\n\u00001.\nThe index on the left hand side is a \fnite sequence ( lk)\nof nonnegative integers. In order to proceed to the next\nstage of the recursion, ( lk) must be mapped to a natural\nnumbermsince for the calculation of, say, z(n+1)\n(lk)it is\nnecessary to access the coe\u000ecients z(n)\nmof the previous3\niteration. Such a mapping is possible because the set of\nsequencesf(lk)gis countably in\fnite. It was shown in\n[11] that the correct way to do the mapping is to choose\nm=P\nklkMkand to letM(formally) tend to in\fnity at\nan appropriate point. Note that class n+ 1 also contains\nall coe\u000ecients from class n. See [11] for details.\nThese classes give us an in\fnite hierarchy of coe\u000e-\ncients, each recursion step adding in\fnitely many coef-\n\fcients to the previous ones. Note that the coe\u000ecients\nin classnconstructed in this way are notan approxima-\ntion but constitute (part of) the exact solution of Eq. (9).\nOnly the total weight of coe\u000ecientsP\nkakfalls short of\n1 when stopping the recursion at a \fnite n. This solu-\ntion of Eq. (9) leads to a density of states consisting of \u000e\npeaks which are located at those \n where zk= 0.\nPercolating regime : In complete analogy to the re-solvent, we can decompose gc;\n(\u001a) =gperc\nc;\n(\u001a) +gc\u0003;\n(\u001a)\ninto a part gperc\nc;\n(\u001a) pertaining to the in\fnite cluster and\na part which is equal to the (known) solution gc\u0003;\n(\u001a)\nof the integral equation at c\u0003. This decomposition has\nthe advantage that we can directly obtain the density of\nstates of the in\fnite cluster by deriving an equation for\ngperc\nc;\n(\u001a), which in analogy to Eq. (11) can be represented\nas\ngperc\nc;\n(\u001a) = (2c\u00002c\u0003)Z1\n\u00001dbc;\n(\u0015)e\u0000i\n2\u0015\u001a2: (14)\nThe function bc;\n(\u0015) is normalized such thatR1\n\u00001dbc;\n(\u0015) = 1 and the \\initial condition\" gc;\n(0) = 2c\nis being taken care of by the prefactor 2 c\u00002c\u0003. This\nansatz is plugged into the integral Eq. (9) to yield\nbc;\n(\u0015) = 2c\u00031X\nn=0an1X\nM=1(2c\u00002c\u0003)M\u00001\nM!Z1\n\u00001dbc;\n(\u00151)\u0001\u0001\u0001Z1\n\u00001dbc;\n(\u0015M)\u0002\u0012 \n\u0015\u0000\"\n1\u00001\n1\n1\u0000zn+PM\ni=1\u0015i#!\n:(15)\nThis equation can be solved numerically by running a\npopulation dynamics algorithm for the coe\u000ecients zkat\nc\u0003below the critical point in parallel to a population\ndynamics for a \u0015-population at cabove the critical point,for which the zkand their weights akare needed as input.\nGivenac\u0003;\n(\u0015) andbc;\n(\u0015) the integrated DSPC can\nbe computed according to (see [13]):\n2\u0001perc(\n;c) =\u0012\n1 +Z1\n\u00001dbc;\n(\u0015)\u0012\nsgn(\u0015\n\u0015\u00001) +csgn(\u0015\u00001)\u0013\n\u0000(c\u0000c\u0003)Z1\n\u00001dbc;\n(\u0015)dbc;\n(\u00150)sgn(\u0015\n\u0015\u00001\u0000\u00150)\n+c\u0003Z\ndac\u0003;\n(\u0015) sgn(1\n\u0015\u00001)\u0000c\u0003Z\n(dac\u0003;\n(\u0015)dbc;\n(\u00150) +dbc;\n(\u0015)dac\u0003;\n(\u00150))sgn(\u0015\n\u0015\u00001\u0000\u00150)\u0013\n:(16)\nDiscussion : We have developed a systematic ap-\nproach to compute the integrated DSPC. We stress that\nthe (nonintegrated) DSPC could not be reliably obtained\nfrom a population dynamics algorithm. Naively it is\ngiven in terms of bc;\n(\u0015) by\nDperc(\n;c) =1\n\u0019=Z1\n0d\u001a\u001aZ1\n\u00001dbc;\n(\u0015)e\u0000i\n2\u0015\u001a2\n=Z1\n\u00001dbc;\n(\u0015)\u000e(\u0015) (17)\nThe above density of states would be very simple if\nbc;\n(\u0015) was di\u000berentiable with respect to \u0015, as it would\nthen be equal to ( bc;\n)0(0). It is however known that\nthe density of states of the percolating cluster contains\n\u000epeaks also for c>1=2. Hence the position and weight\nof the\u000epeaks can not reliably be obtained from a popu-lation dynamics algorithm { a problem already encoun-\ntered in the nonpercolating regime. It is thus essential to\nobtain the integrated density of states directly from pop-\nulation dynamics without going to the density of states\n\frst and integrating, and this is what Eqs. (15) and (16)\nprovide.\nThe \fnal equation for the integrated DSPC, Eq. (16),\nreveals much about the eigenvalues of the percolating\ncluster. We can, for example, track down the origin\nof the\u000epeaks in the density of states of the perco-\nlating cluster. Suppose for the moment that bc;\n(\u0015)\nis continuous as a function of \n. Then the integrals\noverdbc;\n(\u0015) certainly do not generate any jumps in\n\u0001perc(\n;c) due to the continuity in \n. However, we\nknow that ac\u0003;\n(\u0015) =P\nnan\u0012(\u0015\u0000zn) is discontinuous\nas a function of \n since the zndepend on \n, and in-4\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8\n 0 0.5 1 1.5 2 2.5 3ΔpercΩc=1/2c=2\n 0.124 0.1245 0.125 0.1255 0.126 0.1265 0.127 0.1275 0.128\n 0.998 0.999 1 1.001 1.002ΔpercΩc=2\nFigure 1: The integrated density of states at c=1\n2and c= 2.\nThe bottom \fgure shows an enlargement of the curve for c= 2\naround \n = 1.\nspection of Eq. (16) shows that this leads to a jump in\n\u0001perc(\n) if the location znof a jump moves from \u00001\nto +1when \n is increased in\fnitesimally (it does not\nlead to a jump if znmoves across 0 or 1, as could be\nsuspected at \frst sight, since the various contributions\ncancel in these cases). Eq. (13) shows that zncan and\ndoes indeed pass 1as \n is varied. While the peaks of\nthe density of states of the \fnite clusters are located at\nthose \n for which zn= 0, the peaks for the percolating\ncluster are located where zn=1. Since the zeros and\nthe poles of znnecessarily alternate when \n is varied, it\nfollows that the peaks in the percolating cluster lie dense\nif the peaks in the \fnite clusters lie dense. In this sense\nthere is no mobility edge in the percolating cluster since\nisolated and thus localized eigenvalues exist throughout\nthe whole range of 0 \u0014\n<1.\nUnfortunately, this argument only strictly holds ifbc;\n(\u0015) is indeed continuous in \n. If it is not, cancel-\nlations might occur which could reduce (or even remove)\nthe peaks from the percolating cluster. It is shown in [13]\nthat indeed bc;\n(\u0015) is not continuous but the argument\npresented there also shows that a complete removal of\npeaks would seem an extremely fortuitous cancellation.\nIn order to check these results we have performed popu-\nlation dynamics simulations of the integrated DSPC ac-\ncording to Eq. (15). Fig. 1 shows the integrated DSPC\ndirectly at the critical point c=1\n2and deep inside the\npercolating regime at c= 2. Some jumps are clearly ob-\nserved forc=1\n2, and they are located at the predicted\npositions. The most prominent ones occur at \n = 1\nwhere the coe\u000ecient z=\n\n\u00001=1, or at \n =3\u0006p\n5\n2\nwherez=\n\u0000\n\n\u00001\n\n\u00001\u0000\n\n\u00001=1. Forc= 2 the integrated den-\nsity of states in Fig. 1 looks smooth. This is, however,\nnot the case as the close-up in the bottom part of Fig. 1\nreveals. The jump at \n = 1 which is very pronounced at\nc=1\n2is also present at c= 2.\nOur work is based on ideas by Kurt Broderix who died\non May 12, 2000. We thank Peter M uller and Reimer\nK uhn for valuable discussions.\n[1] M. Mehta, Random Matrices, Academic Press, Boston\n(1991)\n[2] T. Guhr, A. M uller-Groeling, H.A. Weidenm uller, Phys.\nRep. 299, 190 (1998)\n[3] A. J. Bray and G. J. Rodgers, Phys. Rev. B 38, 11461\n(1988)\n[4] Y. V. Fyodorov and A. D. Mirlin, J. Phys. A: Math. Gen.\n24, 2219 (1991)\n[5] O. Khorunzhy, M. Shcherbina, and V. Vengerovsky, J.\nMath. Phys. 45, 1648 (1994)\n[6] O. Khorunzhy, W. Kirsch and P. M uller, Ann. Appl.\nProbab. 16, 295 (2006)\n[7] G. Biroli and R. Monasson, J. Phys. A 32, L255 (1999)\n[8] M. Bauer and O.Golinelli, J. Stat. Phys. 103, 301 (2001)\n[9] R. K uhn, J. Phys. A: Math. Gen. 41, 295002 (2008)\n[10] P. Erd} os and A. R\u0013 enyi, Magyar Tud. Akad. Mat. Kut.\nInt. K} ozl. 5, 17 (1960); reprinted in: The Art of Counting\nedited by J. Spencer (MIT Press, Cambridge, MA) (1973)\n[11] K. Broderix, T. Aspelmeier, A.K. Hartmann and A. Zip-\npelius, Phys.Rev. E 64, 021404 (2001)\n[12] M. Reed and B. Simon, Methods of Modern Mathemati-\ncal Physics, Vol. 1: Functional Analysis, Academic Press,\nNew York (1972)\n[13] See EPAPS Document No. [number will be inserted by\npublisher] for technical details. For more information on\nEPAPS, see http://www.aip.org/pubservs/epaps.html."    },    {        "title": "1008.1816v1.Chirality__charge_and_spin_density_wave_instabilities_of_a_two_dimensional_electron_gas_in_the_presence_of_Rashba_spin_orbit_coupling.pdf",        "content": "arXiv:1008.1816v1  [cond-mat.mes-hall]  11 Aug 2010Chirality, charge and spin-density wave instabilities of a two-dimensional electron gas\nin the presence of Rashba spin-orbit coupling\nGeorge E. Simion and Gabriele F. Giuliani\nPhysics Department, Purdue University\n(Dated: June 9, 2018)\nWe show that a result equivalent to Overhauser’s famous Hart ree-Fock instability theorem can\nbe established for the case of a two-dimensional electron ga s in the presence of Rashba spin-obit\ncoupling. In this case it is the spatially homogeneous param agnetic chiral ground state that is\nshown to be differentially unstable with respect to a certain class of distortions of the spin-density-\nwave and charge-density-wave type. The result holds for all densities. Basic properties of these\ninhomogeneous states are analyzed.\nI. INTRODUCTION\nRecent interest in the properties of the quasi-two dimensional elec tron and hole devices in the presence of structural\n(Rashba-Bychkov)1,2or intrinsic (Dresselhaus)3spin orbit has brought to the fore the problem of the interacting\nchiral electron liquid. It is therefore important to revisit several o f the fundamental notions of many-body theory\nfor this intriguing system. The purpose of present paper is to begin a theoretical exploration of the relevance and\nspecial properties of a class of spatially non homogeneous spontan eously broken symmetry states of the electron\nliquid in the presence of Rashba spin-orbit coupling in two dimensions. S pecifically we will focus our attention on spin\ndensity and charge density wave type states henceforth referr ed to for simplicity as SDW and CDW. SDW and CDW\nstates, originally conceived by A. W. Overhauser,4,5are generally stabilized by the electron-electron interaction and\nare characterized by spatial oscillations of the spin density, the ch arge density, or both. In the absence of spin-orbit\ncoupling, one can begin to describe SDW and CDW states by simply cons idering the electron number density for both\nspin projections:\nn↑=n\n2+Acos/parenleftbigg\n/vectorQ·/vectorR+φ\n2/parenrightbigg\n, (1)\nn↓=n\n2+Acos/parenleftbigg\n/vectorQ·/vectorR−φ\n2/parenrightbigg\n. (2)\nIn these expressions the wave vector /vectorQspans the Fermi surface, i.e. does satisfy the condition |/vectorQ| ≃2kF. A CDW\ncorrespondsto φ= 0, while a SDW state obtains for φ=π. Mixed state are also possible: one such state is beautifully\nrealized in chromium.6–8As we will show, in the presence of linear Rashba spin orbit the corres ponding distorted\nstates are characterized by a more complex spatial dependence o f the number density, the spin density and, where\nappropriate, a chiral density. As a first step towards establishing the fundamental properties of SDW- and CDW-like\nstates in the presence of spin-orbit interaction we present here a generalization of the famous Overhauser’s Hartree-\nFock (HF) instability theorem. The latter represents an important exact result in many-body theory for it establishes\nthat, within HF, the homogeneous paramagnetic plane wave state d oes not represents a minimum of the energy for an\notherwise uniform electron gas for it can be always variationally bett ered by a suitably constructed distorted chiral\nSDW or CDW.9\nThe paper is structured as follows: In Section II we discuss the rele vant aspects of the theory of a two dimensional\nnon-interacting electron gas in the presence of Rashba spin-obit c oupling. Section III briefly discusses useful notions\nof the electron-electron interaction within Hartree-Fock approx imation. Section IV is dedicated to the actual proof of\nthe theorem and contains the main results. Finally the last Section co ntains the conclusions while a number of useful\nmathematical relations are derived in the two Appendices.\nII. TWO DIMENSIONAL ELECTRON GAS IN THE PRESENCE OF RASHBA SP IN-ORBIT\nIn the presence of linear Rashba spin-orbit coupling, the one-part icle hamiltonian can be written as follows:\nˆH0=/vector p2\n2m+α/vector p·(/vector σ׈z), (3)2\nwhere ˆzis the unit direction along the z-axis, the motion taking place in the x,yplane.\nThe non interacting problem can be readily diagonalized to obtain the e nergy spectrum and the eigenfunctions:\nEk,µ=¯h2k2\n2m−αµk , (4)\nand\nψ/vectork,+=1√\n2Lei/vectork·/vector r/parenleftbigg1\n−ieiφ/vectork/parenrightbigg\n, (5)\nψ/vectork,−=1√\n2Lei/vectork·/vector r/parenleftbigg\n−ie−iφ/vectork\n1/parenrightbigg\n, (6)\nwhereµ=±labels a state’s chirality andφ/vectorkis the angle spanned by the x-axis and the two dimensional wave vector\n/vectork. A schematic of the lower energy sector of the spectrum is plotted in Fig. 1.\nFIG. 1: Non interacting energy spectrum in the presence of li near Rashba spin-orbit coupling.\nBy making use of the states of Eqs. (5) and (6) as a basis for a seco nd quantization representation, the familiar\nfully interacting electron gas hamiltonian reads:\nˆH=/summationdisplay\n/vectork,µE/vectork,µˆb/vectork,µˆb†\n/vectork,µ+1\n2L2/summationdisplay\n/vector q,/vectork1,/vectork2/summationdisplay\nµ1,µ2,µ3,µ4vqΦ/vectork1,/vectork2,/vector q\nµ1µ2µ3µ4ˆb†\n/vectork1+/vector q,µ1ˆb†\n/vectork2−/vector q,µ2ˆb/vectork2,µ3ˆb/vectork1,µ4, (7)\nwhere the phase factor Φ/vectork1,/vectork2,/vector q\nµ1µ2µ3µ4is defined as follows:\nΦ/vectork1,/vectork2,/vector q\nµ1µ2µ3µ4=iµ1+µ2−µ3−µ4\n2ei/parenleftbig1−µ1\n2φ/vectork1+/vector q+1−µ2\n2φ/vectork2−/vector q−1−µ3\n2φ/vectork2−1−µ4\n2φ/vectork2/parenrightbig\n×\n1\n4/bracketleftbigg\n1+µ1µ4ei/parenleftbig\nφ/vectork1−φ/vectork1+/vector q/parenrightbig/bracketrightbigg/bracketleftbigg\n1+µ2µ3ei/parenleftbig\nφ/vectork2−φ/vectork2−/vector q/parenrightbig/bracketrightbigg\n. (8)\nUncorrelated many body wavefunctions for the system at hand ca n be represented by Slater determinants con-\nstructed by occupying any combinations of the chiral states Eqs. (5) and (6).3\nIII. HARTREE-FOCK THEORY OF A TWO-DIMENSIONAL ELECTRON LIQ UID IN THE\nPRESENCE OF RASHBA SPIN-ORBIT COUPLING\nAn accurate description of a realistic electronic system requires th at electron-electron interaction be taken into\naccount. A first step towards developing such a many-body theor y is to investigate the results of a mean-field\napproach. The main idea behind the mean field procedure is to find an e ffective Hamiltonian which is quadratic in\nthe electron creation and annihilation operators and can therefor e be easily diagonalized. Within the HF theory,\nthe ground state is approximated by a single Slater determinant mad e out of single particle wavefunctions, which\nin turn, are determined by imposing the requirement that the expec tation value of the Hamiltonian over the Slater\ndeterminant be a minimum10. Using these wavefunctions as our basis set, a standard Wick deco upling procedure10\nallows us to determine the effective HF potential. It can be easily prov ed that the non-interacting chiral states are\nindeed among the solutions of the corresponding HF equations. In t his case, the HF potential is diagonal in wave\nvectors and chiral indices:\nVHF\n/vectorkµ,/vectork′µ′=−δ/vectork,/vectork′δµ,µ′\n2L2/summationdisplay\n/vector κνv/vectork−/vector κ/bracketleftbig\n1+µνcos(φ/vectork−φ/vector κ)/bracketrightbig\n. (9)\nThe corresponding HF eigenvalues are given by:\nǫ/vectorkµ=¯h2k2\n2m−αµk−1\n2L2/summationdisplay\n/vector κνv/vectork−/vector κ/bracketleftbig\n1+µνcos(φ/vectork−φ/vector κ)/bracketrightbig\n. (10)\nAn evaluation of the Fermi energy of the two sub-bands leads to an interesting problem. Since one band will, in\ngeneral, acquire more exchange energy than the other, this may r esult (in a first iteration) in two different Fermi\nlevels. In order to equalize them (for elementary stability reasons) , electrons from one subband will have to be moved\nto the other. This is the phenomenon of repopulation .\nThe spatially homogeneous chiral states are just one of the possib le Hartree-Fock solutions. A detailed analysis\nof the possible solutions corresponding to symmetric occupations in momentum space can be done by systematically\nminimizing the total energy as a function of spin orientation and gene ralized chirality of the system11. More general\nsolutions correspond to non-symmetric occupations of the single p article chiral states. The problem has been studied\nand the corresponding very interesting phase diagram has been ex plored11,12. As we will presently discuss there also\nexists an interesting class of spatially non-homogenous solutions to the problem.\nIV. PROOF OF THE INSTABILITY THEOREM\nWe will proceed by showing that it is always possible to lower the energy of the homogeneous paramagnetic chiral\nground state by introducing a suitable real space distortion which is periodic with wave vector /vectorQ= 2kFˆx. The general\napproach follows that of Fedders and Martin13and is based on an Ansatz which represents a generalization of that\ngiven by Giuliani and Vignale for the case of the three dimensional elec tron gas.10\nLet us consider first the putative HF ground state of our many-bo dy system |ΦS/angbracketright. A complete, and, as we shall\nsee convenient, description of this determinantal state can be ac hieved in terms of the corresponding single-particle\ndensity matrix elements here given by:\nραβ=/angbracketleftbig\nΦS/vextendsingle/vextendsingleˆa†\nαˆaβ/vextendsingle/vextendsingleΦS/angbracketrightbig\n, (11)\nwhere here αandβlabel the one-particle states which are used to build the Slater dete rminant.\nNow, within the space of Slater determinants, any slightly modificatio n of the state |ΦS/angbracketrightcan be described in terms\nof a corresponding infinitesimal change of the single-particle densit y matrix elements. Let us indicate such a change\nbyδραβ. At this point the next task consists in trying to evaluate the chang e of the total HF energy in terms of these\nquantities.\nSince|ΦS/angbracketrightis a solution of the HF equations,14the first order variation in the energy must vanish so that the\nproblem at hand is reduced to determining the sign of the energy cha nge to second order in the δραβ’s. The relevant\nexpression is therefore given by:10\n∆(2)EHF=1\n2/summationdisplay\n(α,β)δραβǫβ−ǫα\nnα−nβδρβα+1\n2/summationdisplay\n(α,δ)/summationdisplay\n(β,γ)(vαβγδ−vαβδγ)δραδδρδγ, (12)4\nwhere the notation ( α,β) means that only states situated on opposite sides of the Fermi se a are considered in the\nsummation. Inthis formula,weindicatetheHartree-Fockeigenvalu esasǫαandthecorrespondingoccupationnumbers\nasnα.\nThe next step in our procedure consists in constructing a Slater de terminant for which the HF energy is lower than\n/angbracketleftΦS|H|ΦS/angbracketright. In order to do, we follow Overhauser’s idea and choose the new one -particle states to be suitable linear\ncombinations of chiral plane waves states situated near opposite p oints on the Fermi surface, the distortion being\nlimited to a very narrow strip. The width of this strip will play the role of a variational parameter. We then carefully\ndevise an expression for the wave vector dependent amplitude of t he coupling between the plane waves and construct\nthe corresponding Slater determinant. In the last step, we calcula te the change of the Hartree-Fock energy due to\nthis perturbation to leading order in the distortion amplitude from Eq . (12).\nFIG. 2: Case of unit chirality when the states of the lower chi ral subband are occupied up to the lower threshold of the uppe r\nsubband. The arrows indicate the states that are coupled by t he distortion along the kxaxis.\nA. Instability for the case of chirality equal one\nBecause it presents a formally simpler problem, the first case to be t reated is that in which only the lower subband\nis occupied while the upper one is just about to be filled.15This situation is depicted in Fig. 2.\nLet us build the new trial wavefunctions as mixtures of the wavefun ctions corresponding to wave vector /vectorkwith\nthose corresponding to wave vector /vectork±/vectorQ, i.e.\nΨ/vectork≃ψ/vectork,++A/vectork+/vectorQ,/vectorkψ/vectork+/vectorQ,++A/vectork−/vectorQ,/vectorkψ/vectork−/vectorQ,+. (13)\nHere, as anticipated, |/vectorQ|= 2k+\nFas shown in Fig. 2.\nIn evaluating the HF energy change, only the states situated on op posite sides of the Fermi sea are relevant. We\nwill consider/vextendsingle/vextendsingle/vextendsingle/vectork±/vectorQ/vextendsingle/vextendsingle/vextendsingle> k+\nFandk < k F. Here, the occupation numbers are n/vectork=θ(kF−k), while the amplitude\nsatisfies the condition: A/vectork±/vectorQ,/vectork=A/vectork,/vectork±/vectorQ. The only non-zero matrix elements of δˆρhave the form:\nδρ/vectork+/vectorQ\n2,/vectork−/vectorQ\n2=δρ/vectork−/vectorQ\n2,/vectork+/vectorQ\n2=/parenleftig\nn/vectork+/vectorQ\n2+n/vectork−/vectorQ\n2/parenrightig\nA/vectork. (14)\nThese wavefunctions indeed describe SDW/CDW-like states. A simple calculation shows that retaining only the\nlinear order in the amplitude of the distortion, the spin and the charg e densities exhibit spatial oscillations with wave\nvector 2k+\nF. Specifically:\nδSx(/vector r)\n¯h=/summationdisplay\n/vectorkA/vectork/bracketleftig/parenleftig\ncosφ/vectork+/vectorQ\n2−cosφ/vectork−/vectorQ\n2/parenrightig\nsin/vectorQ·/vector r+/parenleftig\nsinφ/vectork+/vectorQ\n2+sinφ/vectork−/vectorQ\n2/parenrightig\ncos/vectorQ·/vector r/bracketrightig\n,\nδSy(/vector r)\n¯h=/summationdisplay\nkA/vectork/bracketleftig/parenleftig\nsinφ/vectork+/vectorQ\n2−sinφ/vectork−/vectorQ\n2/parenrightig\nsin/vectorQ·/vector r−/parenleftig\ncosφ/vectork+/vectorQ\n2+cosφ/vectork−/vectorQ\n2/parenrightig\ncos/vectorQ·/vector r/bracketrightig\n,5\nδSz(/vector r)\n¯h=/summationdisplay\nk2A/vectork/bracketleftig\n1−cos/parenleftig\nφ/vectork+/vectorQ\n2−φ/vectork−/vectorQ\n2/parenrightig/bracketrightig\ncos/vectorQ·/vector r ,\nδn(/vector r)=/summationdisplay\nk4A/vectork/bracketleftig\n1+cos/parenleftig\nφ/vectork+/vectorQ\n2−φ/vectork−/vectorQ\n2/parenrightig/bracketrightig\ncos/vectorQ·/vector r . (15)\nThe change in the HF energy is obtained by substituting the express ion of the non-zero density matrix elements\nfrom Eq. (14) into Eq. (12). The resulting expression can be expre ssed as:\n∆(2)EHF= ∆(2)\n0EHF+∆(2)\nHEHF+∆(2)\nXEHF, (16)\nwhere we have defined\n∆(2)\n0EHF=/summationdisplay\n/vectork/parenleftig\nn/vectork+/vectorQ\n2+n/vectork−/vectorQ\n2/parenrightigǫ/vectork−/vectorQ\n2,+−ǫ/vectork+/vectorQ\n2,+\nn/vectork+/vectorQ\n2−n/vectork−/vectorQ\n2A2\nk, (17)\n∆(2)\nHEHF=1\n2/summationdisplay\n/vectork,/vector p/parenleftig\nv/vectork+/vectorQ\n2,+,/vector p−/vectorQ\n2,+,/vector p+/vectorQ\n2,+,/vectork−/vectorQ\n2,++v/vectork−/vectorQ\n2,+,/vector p+/vectorQ\n2,+,/vector p−/vectorQ\n2,+,/vectork+/vectorQ\n2,+/parenrightig\n×/parenleftig\nn/vectork+/vectorQ\n2+n/vectork−/vectorQ\n2/parenrightig/parenleftig\nn/vector p+/vectorQ\n2+n/vector p−/vectorQ\n2/parenrightig\nA/vectorkA/vector p, (18)\n∆(2)EHF\nX=−1\n2/summationdisplay\n/vectork,/vector p/parenleftig\nv/vectork+/vectorQ\n2,+,/vector p−/vectorQ\n2,+,/vectork−/vectorQ\n2,+,/vector p+/vectorQ\n2,++v/vectork−/vectorQ\n2,+,/vector p+/vectorQ\n2,+,/vectork+/vectorQ\n2,+,/vector p−/vectorQ\n2,+/parenrightig\n×/parenleftig\nn/vectork+/vectorQ\n2+n/vectork−/vectorQ\n2/parenrightig/parenleftig\nn/vector p+/vectorQ\n2+n/vector p−/vectorQ\n2/parenrightig\nA/vectorkA/vector p. (19)\nThe Hartree and exchange terms in Eqs. (18)-(19) contain combin ations of the matrix elements of the electron-\nelectron interaction. By employing Eq. (8), after simple algebraic ma nipulations, we obtain:\nv/vectork+/vectorQ\n2,+,/vector p−/vectorQ\n2,+,/vector p+/vectorQ\n2,+,/vectork−/vectorQ\n2,++v/vectork−/vectorQ\n2,+,/vector p+/vectorQ\n2,+,/vector p−/vectorQ\n2,+,/vectork+/vectorQ\n2,+=\nvQ\n2L2/bracketleftig\n1+cos/parenleftig\nφ/vectork+/vectorQ\n2−φ/vectork−/vectorQ\n2/parenrightig\n+cos/parenleftig\nφ/vector p+/vectorQ\n2−φ/vector p−/vectorQ\n2/parenrightig\n+ cos/parenleftig\nφ/vectork+/vectorQ\n2−φ/vectork−/vectorQ\n2+φ/vector p+/vectorQ\n2−φ/vector p−/vectorQ\n2/parenrightig/bracketrightig\n,(20)\nand\nv/vectork+/vectorQ\n2,+,/vector p−/vectorQ\n2,+,/vectork−/vectorQ\n2,+/vector p+/vectorQ\n2,++v/vectork−/vectorQ\n2,+,/vector p+/vectorQ\n2,+,/vectork+/vectorQ\n2,+,/vector p−/vectorQ\n2,+=\nv|/vectork−/vector p|\n2L2/bracketleftig\n1+cos/parenleftig\nφ/vector p+/vectorQ\n2−φ/vectork+/vectorQ\n2/parenrightig\n+cos/parenleftig\nφ/vectork−/vectorQ\n2−φ/vector p−/vectorQ\n2/parenrightig\n+ cos/parenleftig\nφ/vector p+/vectorQ\n2−φ/vectork+/vectorQ\n2+φ/vectork−/vectorQ\n2−φ/vector p−/vectorQ\n2/parenrightig/bracketrightig\n.(21)\nIn order to explicitly evaluate the change in the Hartree-Fock ener gy, we need to assume a specific expression for\nthe distortion amplitudes. As a first condition, we will perturb only a n arrowregion near the Fermi surface. Following\nthe same pattern of the proof of reference10, we propose for the present problem the following educated variat ional\nguess:\nA/vectork=\n\n(bk+\nF)3\n2\nln2\nb/vextendsingle/vextendsingle/vextendsinglen/vectork+/vectorQ\n2−n/vectork−/vectorQ\n2/vextendsingle/vextendsingle/vextendsingle\nk3\n2/radicalbig\n|sinφ/vectork|\n|cosφ/vectork|, bk+\nF<k<ςbk+\nF\n0, otherwise,(22)\nwhereb≪1 is our small parameter and the second arbitrary ς >1. This expression is intentionally chosen to display\nsingularities for k= 0 andφk= 0. These singularities are crucial to the present proof.\nWe now notice that in the expressions of the interaction matrix eleme nts, there appear factors of the type:\ncos/parenleftig\nφ/vector p+/vectorQ\n2−φ/vectork+/vectorQ\n2/parenrightig\n=/parenleftbig\npcosφ/vector p+k+\nF/parenrightbig/parenleftbig\nkcosφ/vectork+k+\nF/parenrightbig\n+pksinφ/vector psinφ/vectork/radicalig\np2+(k+\nF)2+2k+\nFpcosφ/vector p/radicalig\nk2+(k+\nF)2+2k+\nFkcosφ/vectork. (23)6\nSince we are only interested in the leading order expansion with respe ct tob, these cosines can be simply taken to be\nequal to unity, since in the region where the amplitude of the distort ion is non-vanishing, both p/k+\nFandk/k+\nFare of\norderb. Accordingly we will assume\ncos/parenleftig\nφ/vector p+/vectorQ\n2−φ/vectork+/vectorQ\n2/parenrightig\n≃1+O/parenleftbig\nb2/parenrightbig\n. (24)\nAt this point, we recall that /vectork+/vectorQ\n2and/vectork−/vectorQ\n2must lie on opposite sides of the Fermi sea (i.e |/vectork−/vectorQ\n2|< k+\nFand\n|/vectork+/vectorQ\n2|>k+\nF), which implies that/vextendsingle/vextendsinglecosφ/vectork/vextendsingle/vextendsingle>b\n2.\nWe can now proceed to the evaluation of the three components of ∆(2)EHFfrom Eqs. (18)-(19) to leading order in\nb.\nFor ∆(2)\n0EHFthe first step is to calculate ǫ/vectork−/vectorQ\n2,+−ǫ/vectork+/vectorQ\n2,+. Using (10), with energies expressed in Ry, x=k\nkFand\nx′=k′\nkF, we have:\nǫ/vectork−/vectorQ\n2−ǫ/vectork+/vectorQ\n2=\n4u2\nr2s−4˜αu\nrs−1\nπrs2π/integraldisplay\n01/integraldisplay\n0x′(1+cosϕ′)dx′dφ′\n/radicalbig\nu2+x′2−2ux′cosϕ′\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nu=√\n1+x2−2xcosφ/vectork\n−\n4u2\nr2s−4˜αu\nrs−1\nπrs2π/integraldisplay\n01/integraldisplay\n0x′(1+cosϕ′)dx′dφ′\n/radicalbig\nu2+x′2−2ux′cosϕ′\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nu=√\n1+x2+2xcosφ/vectork.\n(25)\nThe quadratures appearing in this expression can then be manipulat ed by making use of the results of Appendix A.\nThe result is:\nǫ/vectork−/vectorQ\n2−ǫ/vectork+/vectorQ\n2=−16xcosφ\nr2s/parenleftbigg\n1−˜αrs\n2−rs\n2πln|ξxcosφ|/parenrightbigg\n, (26)\nwhere the logarithmic term accounts for the divergence of the der ivative of the HF single particle energy near the\nFermi level. Here, ξis a constant approximately equal to 0 .51.\nBy substituting (26) and (22) in (17) we obtain:\n∆(2)\n0EHF=16b3L2k2\nF/parenleftbig\nln2\nb/parenrightbig2r2sπ2ςb/integraldisplay\nbdx\nxarccosb\n2/integraldisplay\n0dϕsinϕ\ncosϕ/parenleftbigg\n1−˜αrs\n2−rs\n2πln|ξxcosϕ|/parenrightbigg\n, (27)\nan expression that, to leading order in b, reduces to:\n∆(2)\n0EHF≃48Nb3lnς\nrsπ2. (28)\nwhereNis the number of particles.\nThe Hartree term ∆(2)\nHEHF(containing vQ) can in turn be evaluated as follows. By making use of the assumed\namplitude in (22), we write:\n∆(2)\nHEHF=16Nb3\nπ2/parenleftbig\nln2\nb/parenrightbig2rsbς/integraldisplay\nbdx√xbς/integraldisplay\nbdy√yarccosb\n2/integraldisplay\n0dϕx√sinϕx\ncosφxarccosb\n2/integraldisplay\n0dϕy/radicalbigsinϕy\ncosϕy. (29)\nAt this point, using the result (B5), the leading order in bof this quantity is given by:\n∆(2)\nHEHF≃64Nb4(√ς−1)2\nπ2rs. (30)7\nThe last term of (16), the exchange energy contribution, is clearly negative and therefore will certainly lower the\nenergy. Its evaluation is formidable, for it involves several complica ted and seemingly difficult quadratures. Rather\nthan attempting to actually calculate it, we will establish a lower limit for its magnitude.\nWe will restrict ourselves to the region in which both angles φ/vectorkandφ/vector pare in the first quadrant. Since this excludes\nsome contributions of the same sign, the exchange energy will be un derestimated. We therefore have:\n/vextendsingle/vextendsingle/vextendsingle∆(2)\nXEHF/vextendsingle/vextendsingle/vextendsingle≥2Nb3\nπ2/parenleftbig\nln2\nb/parenrightbig2rsbς/integraldisplay\nbdxbς/integraldisplay\nbdyarccosb\n2/integraldisplay\n0dϕxarccosb\n2/integraldisplay\n0dϕy1/radicalbig\nx2+y2−2xycos(ϕx−ϕy)1√x/radicalbig\n|sinϕx|\n|cosϕx|1√y/radicalbig\n|sinϕy|\n|cosϕy|.\n(31)\nIt is simple to see that this expression will turn out to be proportiona l to (lnb)3. This is due to the presence of three\nsingularities in the denominator of the integrand. Of these one stem s from the divergence of the Coulomb potential,\nwhile the other two come from the upper limit of the angular integratio ns.\nAnother simplification is provided by the use of the inequality:\n1/radicalbig\nx2+y2−2xycos(ϕx−ϕy)≥1/radicalbig\nx2+y2−2xysinϕxsinϕy. (32)\nUsing the same changes of variable as in Appendix B, i.e. tx= tan(φx/2) andty= tan(φy/2), we can rewrite this\nintegral as follows:\n/vextendsingle/vextendsingle/vextendsingle∆(2)\nXEHF/vextendsingle/vextendsingle/vextendsingle≥16Nb3\nπ2/parenleftbig\nln2\nb/parenrightbig2rsbς/integraldisplay\nbdx√xbς/integraldisplay\nbdy√y/radicalbig\n2−b\n2+b/integraldisplay\n0dtx/radicalbig\n2−b\n2+b/integraldisplay\n0dty1/radicalig\nx2+y2−8xytxty\n(1+t2x)(1+t2y)1\n1−t2x/radicaligg\ntx\n1+t2x1\n1−t2y/radicaligg\nty\n1+t2y.\n(33)\nIt is clear now that the main contribution to the integral comes from the region around the upper limit of integration\nfor bothtxandty. In order to retain the leading order term, a good approximation will be to replace both t’s with\n1−b/2 in the denominator of the first square root. In this way, we can se parate the angular integrations in (31) to\nobtain:\n/vextendsingle/vextendsingle/vextendsingle∆(2)\nXEHF/vextendsingle/vextendsingle/vextendsingle≥2Nb3\nπ2/parenleftbig\nln2\nb/parenrightbig2rsbς/integraldisplay\nbdx√xbς/integraldisplay\nbdy√y1/radicalig\nx2+y2−2xy/parenleftbig\n1−b2\n8/parenrightbig21−b\n2/integraldisplay\n0dtx1−b\n2/integraldisplay\n0dty1\n1−t2x/radicaligg\ntx\n1+t2x1\n1−t2y/radicaligg\nty\n1+t2y.\n(34)\nThe last two integrals are evaluated using (B5) and other changes o f variables ( x=ub,y=vb) :\n/vextendsingle/vextendsingle/vextendsingle∆(2)\nXEHF/vextendsingle/vextendsingle/vextendsingle≥8Nb3\nπ2rs√ς/integraldisplay\n1du√ς/integraldisplay\n1dv1/radicalig\n(u2−v2)2+u2v2b2\n2. (35)\nThe last quadrature is calculated in (B8), leading us to a very simple ine quality for the exchange contribution:\n/vextendsingle/vextendsingle/vextendsingle∆(2)\nXEHF/vextendsingle/vextendsingle/vextendsingle≥8Nb3\nπ2rslnςln1\nb. (36)\nThis term contains a logarithmic factor ln1\nb, which allows the negative change in the exchange contribution to co ntrol\nall the remaining terms. This concludes the proof for this case.\nThe same chain of arguments does apply to the case in which the gene ralized chirality is greater than one. The\ncoupling that produces this kind of instability is schematically shown in F ig. 3. All the formulas we derived in the\nprevious case do still apply. The only difference lies in the lower integra tion limits of Eq. (25), but no relevant\ncontribution ensues from this. The matrix elements related to the H artree and exchange contributions are the same,\nand, as a consequence, the leading order approximation is identical.8\nFIG. 3: HF instability: symmetry breaking coupling for the c ase in which only the lower subband is occupied.\nB. Instability for the case of chirality less than one\nThe argument of the previous Section can be applied when the chiralit y is less than one, i.e. when both chiral\nsubbands are occupied. We can try to break the symmetry by coup ling states with the same chirality as well as states\nwith different chiralities. When same chirality states are coupled, the re is nothing new, as one simply just adds a\nchirality index to the various quantities. In this case, the wave vect ors characterizing the oscillations are given by:\nQµ= 2kµ\nFand the trial wavefuntions can be written as:\nΨ/vectorkµ≃ψ/vectorkµ+A/vectork+/vectorQµ,/vectorkµψ/vectork+/vectorQµµ+A/vectork−/vectorQµ,/vectorkµψ/vectork−/vectorQµµ. (37)\nThis type of coupling is depicted in Fig. 4.\nThe corresponding distortion of the components of the spin densit y and the number density can be again calculated\nup to the first order in the amplitudes:\nδSx(/vector r) = ¯h\n/summationdisplay\nk,µA/vectorkµ/parenleftbigg\ncosφ/vectork+/vectorQµ\n2−cosφ/vectork−/vectorQµ\n2/parenrightbigg\nsin/vectorQµ·/vector r+/summationdisplay\nk,µA/vectorkµ/parenleftbigg\nsinφ/vectork+/vectorQµ\n2+sinφ/vectork−/vectorQµ\n2/parenrightbigg\ncos/vectorQµ·/vector r\n,(38)\nδSy(/vector r) = ¯h\n/summationdisplay\nk,µA/vectorkµ/parenleftbigg\nsinφ/vectork+/vectorQµ\n2−sinφ/vectork−/vectorQµ\n2/parenrightbigg\nsin/vectorQµ·/vector r−/summationdisplay\nk,µA/vectorkµ/parenleftbigg\ncosφ/vectork+/vectorQµ\n2+cosφ/vectork−/vectorQµ\n2/parenrightbigg\ncos/vectorQµ·/vector r\n,(39)\nδSz(/vector r) = 2¯h\n/summationdisplay\nk,µA/vectorkµ/parenleftbigg\n1−cos/parenleftbigg\nφ/vectork+/vectorQµ\n2−φ/vectork−/vectorQµ\n2/parenrightbigg/parenrightbigg\ncos/vectorQµ·/vector r\n, (40)\nδn(/vector r) = 4/summationdisplay\nk,µA/vectorkµ/parenleftbigg\n1−cos/parenleftbigg\nφ/vectork+/vectorQµ\n2−φ/vectork−/vectorQµ\n2/parenrightbigg/parenrightbigg\ncos/vectorQµ·/vector r . (41)\nWe proceed in this case by choosing an amplitude not unlike the one ass umed above:\nA/vectorkµ=\n\n(bkµ\nF)3\n2\nln2\nb/vextendsingle/vextendsingle/vextendsinglen/vectork+/vectorQ\n2µ−n/vectork−/vectorQ\n2µ/vextendsingle/vextendsingle/vextendsingle\nk3\n2/radicalbig\n|sinφ/vectork|\n|cosφ/vectork|, bkµ\nF<k<ςbkµ\nF\n0, otherwise.(42)\nThe proof of the corresponding instability theorem proceeds then in exactly the same manner.\nAs anticipated, there is also not much difference when we try to coup le states with opposite chirality (see Fig. 5).\nAlthough some of the expressions involved in the derivation do chang e, the main features of the argument remain\nunchanged. The coupling vector in this case is given by Q=k+\nF+k−\nF. Here, we try to find a lower energy state\nby coupling wavefunctions with wave vector /vectorkwith those with wave vector /vectork±/vectorQand opposite chirality. The trial\nwavefunctions then read:\nΨ/vectork+≃ψ/vectork++A/vectork+/vectorQ,/vectorkψ/vectork+/vectorQ,−+A/vectork−/vectorQ,/vectorkµψ/vectork−/vectorQ,−. (43)9\nForA/vectork±/vectorQ,/vectork=A/vectork,/vectork±/vectorQthe only non zero variations of the matrix elements of the single-par ticle density matrix\noperator acquire the following form:\nδρ/vectork+/vectorQ\n2,µ,/vectork−/vectorQ\n2,−µ=δρ/vectork−/vectorQ\n2,µ,/vectork+/vectorQ\n2,−µ=/parenleftig\nn/vectork+/vectorQ\n2,µ+n/vectork−/vectorQ\n2,−µ/parenrightig\nA/vectork. (44)\nIn this case, the new state is characterized by a similar spin and dens ity modulation:\nδSx(/vector r) = ¯h/summationdisplay\nkA/vectork/bracketleftig\ncos/parenleftig\n/vectorQ·/vector r−φ/vectork+/vectorQ\n2−φ/vectork−/vectorQ\n2/parenrightig\n+cos/vectorQ·/vector r/bracketrightig\n, (45)\nδSy(/vector r) = ¯h/summationdisplay\nkA/vectork/bracketleftig\nsin/parenleftig\n−/vectorQ·/vector r+φ/vectork+/vectorQ\n2+φ/vectork−/vectorQ\n2/parenrightig\n+sin/vectorQ·/vector r/bracketrightig\n, (46)\nδSz(/vector r) = ¯h/summationdisplay\nkA/vectork/bracketleftig\nsin/parenleftig\n/vectorQ·/vector r−φ/vectork−/vectorQ\n2/parenrightig\n+sin/parenleftig\n/vectorQ·/vector r−φ/vectork+/vectorQ\n2/parenrightig/bracketrightig\n, (47)\nδn(/vector r) = 2/summationdisplay\nkA/vectork/bracketleftig\nsin/parenleftig\n/vectorQ·/vector r−φ/vectork−/vectorQ\n2/parenrightig\n−sin/parenleftig\n/vectorQ·/vector r−φ/vectork+/vectorQ\n2/parenrightig/bracketrightig\n. (48)\nThe corresponding terms in the Hartree-Fock energy change are :\n∆(2)\n0EHF=/summationdisplay\n/vectorkµ/parenleftig\nn/vectork+/vectorQ\n2,µ+n/vectork−/vectorQ\n2,−µ/parenrightigǫ/vectork−/vectorQ\n2µ−ǫ/vectork+/vectorQ\n2−µ\nn/vectork+/vectorQ\n2−µ−n/vectork−/vectorQ\n2µA2\nk, (49)\n∆(2)\nHEHF=1\n2/summationdisplay\n/vectork/vector pµν/parenleftig\nv/vectork+/vectorQ\n2,µ,/vector p−/vectorQ\n2ν,/vector p+/vectorQ\n2,−ν,/vectork−/vectorQ\n2,−µ+v/vectork−/vectorQ\n2,µ,/vector p+/vectorQ\n2,ν,/vector p−/vectorQ\n2,−ν,/vectork+/vectorQ\n2,−µ/parenrightig\n/parenleftig\nn/vectork+/vectorQ\n2,µ+n/vectork−/vectorQ\n2,−µ/parenrightig/parenleftig\nn/vector p+/vectorQ\n2ν+n/vector p−/vectorQ\n2−ν/parenrightig\nA/vectorkµA/vector pν, (50)\n∆(2)\nXEHF=−1\n2/summationdisplay\n/vectork/vector pµν/parenleftig\nv/vectork+/vectorQ\n2µ,/vector p−/vectorQ\n2,−ν,/vectork−/vectorQ\n2,−µ,/vector p+/vectorQ\n2,ν+v/vectork−/vectorQ\n2,−µ,/vector p+/vectorQ\n2,ν,/vectork+/vectorQ\n2µ,/vector p−/vectorQ\n2,−ν/parenrightig\n/parenleftig\nn/vectork+/vectorQ\n2,µ+n/vectork−/vectorQ\n2,−µ/parenrightig/parenleftig\nn/vector p+/vectorQ\n2,ν+n/vector p−/vectorQ\n2,−ν/parenrightig\nA/vectorkµA/vector pν. (51)\nThe relevant interaction matrix elements which appear in the expres sion of the Hartree term become:\n2vQ\nL2/parenleftig\nsinφ/vector p+/vectorQ\n2−sinφ/vector p−/vectorQ\n2/parenrightig/parenleftig\nsinφ/vectork−/vectorQ\n2−φ/vectork+/vectorQ\n2/parenrightig\n, (52)\nwhile those determining the exchange contribution reduce to:\nv|/vectork−/vector p|\nL2/parenleftig\ncos/parenleftig\nφ/vector p+/vectorQ\n2−φ/vectork+/vectorQ\n2/parenrightig\n+cos/parenleftig\nφ/vector p−/vectorQ\n2−φ/vectork−/vectorQ\n2/parenrightig\n+\n+cos/parenleftig\nφ/vector p+/vectorQ\n2−φ/vectork−/vectorQ\n2/parenrightig\n+cos/parenleftig\nφ/vector p−/vectorQ\n2−φ/vectork+/vectorQ\n2/parenrightig\n−\n−cos/parenleftig\nφ/vectork+/vectorQ\n2−φ/vectork−/vectorQ\n2/parenrightig\n−cos/parenleftig\nφ/vector p+/vectorQ\n2−φ/vector p−/vectorQ\n2/parenrightig\n+\n+cos/parenleftig\nφ/vector p+/vectorQ\n2−φ/vectork+/vectorQ\n2+φ/vectork−/vectorQ\n2−φ/vector p−/vectorQ\n2/parenrightig\n+1/parenrightig\n. (53)\nAs in the previous cases, we assume the following coupling amplitude:\nAµ\n/vectork=\n\n(bkµ\nF)3\n2\nln2\nb/vextendsingle/vextendsingle/vextendsinglen/vectork+/vectorQ\n2,µ−n/vectork−/vectorQ\n2,−µ/vextendsingle/vextendsingle/vextendsingle\nk3\n2/radicalbig\n|sinφ/vectork|\n|cosφ/vectork|, bk+\nF<k<ςbk+\nF,|cosφ/vectork|<b\n2\n0, otherwise.(54)\nThefirsttermintheHartree-Fockenergy(17)hasapositivecont ributionwhichisproportionalto b3. Thisoriginates\nfrom the same logarithmic factor associated with the divergence of the derivative of the single-particle energy at the10\nFIG. 4: Symmetry breaking coupling of states with the same ch irality.\nFIG. 5: States with opposite chiralities coupled to obtain a HF instability\nFermi level. The Hartree term introduces higher order terms in bdue to the presence of the sine factors in its matrix\nelements. Finally, the leading order contribution to the exchange ma trix elements is 4 v|/vectork−/vector p|. By evaluating integrals\nsimilar to those in Eq. (31) we again obtain a negative energy change o f orderb3ln(1/b).\nV. CONCLUSIONS\nWe have been able to formally construct a number of distorted chira l states which, irrespective of the electron\ndensity, have, within mean field, a lower energy than the spatially hom ogeneous paramagnetic chiral HF ground\nstate, thereby affording a rigorous proof of a generalization of Ov erhauser’s Hartree-Fock instability theorem to a two\ndimensional electron liquid in the presence of linear Rashba-Dresselh aus spin-orbit coupling. It is important to notice\nthat, as mentioned in Section III, to establish the instability we have not allowed for momentum space repopulation:\ninclusion of this phenomenon would have further lowered the energy of the trial states while greatly increasing the\ndifficulty of the analysis.\nIt is worth mentioning that the states that have been analyzed in th is paper differ in a number of ways form the\noriginal spin/charge density waves proposed by Overhauser. Our states are chiral density states, that display both\nspin and charge modulations. The presence of charge modulations c annot be ignored (as commonly done in the case\nwithout spin-orbit) and the Hartree term has to be evaluated explic itly. The exchange gain is shown to be larger than\nthe kinetic energy plus the ensuing Hartree terms. In this respect the difference with the plain electron liquid is quite\nmarked for it is precisely the electrostatic term that is believed to be fatal to the charge density waves in that case.\nWe prove that this is not the case for the chiral waves. The main rea son is that the exchange energy behaves quite\ndifferently as compared to that in the absence of spin-orbit interac tion. In particular it does not necessarily favor a\nhomogeneous chiral instability as shown in Ref.11. Our calculations show that in a vast class of inhomogeneous states ,\nthe exchange energy can induce a chiral instability for all densities o f the electron liquid. Of particular importance is\nthat the chiral states coupled by the type of mean fields explored h ere do always have opposite spins. The fact that11\nsuch couplings are possible for all values of the spin-orbit coupling is a nother consequence of the non analyticity of\nthe properties of the electron liquid on this parameter.\nOur results only represent the first step in understanding non hom ogeneous states in this interesting many-body\nsystem. HowtheinclusionofcorrelationswillmodifytheHFscenariois ofcourseamostimportantquestion. Advances\nin this respect can in principle be pursued by followingthe programout lined in Reference 16. Such a study will require\na vastly more complicated analysis. An alternative route is to make us e of perturbative techniques to establish the\nrole of correlation effects in the high density limit. Part of this progra m has been carried out in Reference 17 for\nhomogeneous states.\nAcknowledgments\nThe authors would like to acknowledge useful discussions with A. W. O verhauser. GS work was partially supported\nby a grant from the Purdue Research Foundation.\nAppendix A: Elliptic integral expansion\nFor the purpose of our calculations the following expression must be evaluated in the limit of small u:\n2π/integraldisplay\n01/integraldisplay\n0x′(1+cosϕ)dx′dϕ/radicalbig\nz2+x′2−2zx′cosϕ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=1+u\n−2π/integraldisplay\n01/integraldisplay\n0x′(1+cosϕ)dx′dϕ/radicalbig\nz2+x′2−2zx′cosϕ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=1−u. (A1)\nAn obvious problem is the presence of singularities in the integrand fo rz= 1.\nLet us define:\nA(z) =2π/integraldisplay\n01/integraldisplay\n0x′(1+cosϕ)dx′dϕ/radicalbig\nz2+x′2−2zx′cosϕ. (A2)\nOf course,Ais not differentiable in z= 1. Still we have to evaluate∂A(z)\n∂zand expand it in an asymptotic series\naroundz= 1.\n∂\n∂zA(z) =∂\n∂z\nz2π/integraldisplay\n01/z/integraldisplay\n0y(1+cosϕ)dydϕ/radicalbig\n1+y2−2ycosϕ\n\n=2π/integraldisplay\n01/z/integraldisplay\n0y(1+cosϕ)dydϕ/radicalbig\n1+y2−2ycosϕ\n+z∂\n∂z\n2π/integraldisplay\n01/z/integraldisplay\n0y(1+cosϕ)dydϕ/radicalbig\n1+y2−2ycosϕ\n. (A3)\nThe derivative in the second term can be evaluated in the following way :\n∂\n∂z\n2π/integraldisplay\n01/z/integraldisplay\n0y(1+cosϕ)dydϕ/radicalbig\n1+y2−2ycosϕ\n12\n=−1\nz22π/integraldisplay\n0y(1+cosϕ)dϕ/radicalbig\n1+y2−2ycosϕ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\ny=1\nz=\n=−1\nz22π/integraldisplay\n0(1+cosϕ)dϕ/radicalbig\n1+z2−2zcosϕ. (A4)\nWe can integrate this expression over its angular part so that the r esult is expressed as:\n2π/integraldisplay\n0(1+cosϕ)dϕ/radicalbig\n1+z2−2zcosϕ=−2|z−1|\nzE/parenleftigg\n−4z\n(z−1)2/parenrightigg\n+2\nz(z+1)2\n|z−1|K/parenleftigg\n−4z\n(z−1)2/parenrightigg\n,\n(A5)\nwhereEandKare the elliptic integrals of first and second kind defined as in18,19.\nThe asymptotic expansion of the elliptic integrals around z= 1 leads to:\n∂A(z)\n∂z=ρ−12ln2+4+4ln |z−1|, (A6)\nwhereρ=2π/integraltext\n01/integraltext\n0y(1+cosϕ)dydϕ√\n1+y2−2ycosϕ≃5.6639.\nIntegrating over zwe finally obtain:\nA(1+u)−A(1−u)≃2(ρ−12ln2+4ln |u|)u=\n≃8u(ln|ξu|), (A7)\nwithξ≃0.51.\nAppendix B: Evaluation of useful quadratures\nWe begin by calculating here the leading order term in the expansion of the expression\nI=arccosb\n2/integraldisplay\n0√sinϕ\ncosϕdϕfor b≪1, (B1)\n.\nExpanding the upper limit and setting t= tanϕ\n2, the integral becomes:\nI≃2√\n21−b\n2/integraldisplay\n0dt\n1−t2/radicalbigg\nt\n1+t2\n≃√\n21−b\n2/integraldisplay\n0dt/radicalbigg\nt\n1+t2/parenleftbigg1\n1−t+1\n1+t/parenrightbigg\n, (B2)\nThe divergence in the limit b→0 stems only from the first term and we therefore proceed to try is olating the\nsingularity:13\nI≃√\n21−b\n2/integraltext\n0dt/radicalig\nt\n1+t21\n1−t+√\n21/integraltext\n0dt1\n1+t/radicalig\nt\n1+t2+O(b)≃\n≃ −√\n21−b\n2/integraltext\n0dt/radicalig\nt\n1+t2(ln(1−t))′+0.5256+O(b).(B3)\nAn integration by parts is used in the remaining integral to further is olate the singular contribution:\nI≃ −/radicalbigg\n2t\n1+t2(ln(1−t))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−b\n2\n0\n+√\n21−b\n2/integraldisplay\n0dtd\ndt/parenleftigg/radicalbigg\nt\n1+t2/parenrightigg\nln(1−t)+0.5256+O(b).\n(B4)\nAt this point the non singular second term is calculated numerically so t hat the final result is:\nI≃lnb+0.9475. (B5)\nThe following integral is used in the proof of the HF instability theorem :\nJ=√ς/integraldisplay\n1√ς/integraldisplay\n1dudv/radicalig\n(u2−v2)2+u2v2b2\n2, (B6)\nforb≪1.\nBecause the singular behavior originates from the region where u≃v, in order to find the leading order term, we\napproximate u2−v2with 2u(u−v). Since the function is symmetric with respect to the interchange o fuandvwe\ncan use the relation√ς/integraltext\n1du√ς/integraltext\n1dv f(u,v) = 2√ς/integraltext\n1duu/integraltext\n1dv f(u,v).\nThen:\nJ≃√ς/integraldisplay\n1du\nuu/integraldisplay\n1dv/radicalig\n(u−v)2+u2b2\n8≃ −√ς/integraldisplay\n1du\nuln/parenleftbiggbu\n2√\n2/parenrightbigg\n+√ς/integraldisplay\n1du\nuln/parenleftigg\nu−1+/radicalbigg\n(u−1)2+u2b2\n8/parenrightigg\n. (B7)\nThe integrand of the first term has a logarithmic singularity in the limit o f smallbwhile the second one is instead\nanalytic in this limit. The main contribution to the integral is therefore :\nJ≃ −√ς/integraldisplay\n1du\nuln/parenleftbiggbu\n2√\n2/parenrightbigg\n≃ −lnςlnb . (B8)\n1E. I. Rashba, Sov.Phys. Solid State 1, 368 (1959).\n2E. I. Rashba, Sov.Phys. Solid State 2, 1224 (1960).\n3G. Dresselhaus, Phys. Rev. 100, 580 (1955).14\n4A. W. Overhauser, Phys. Rev. Lett 4, 462 (1960).\n5A. W. Overhauser, Phys. Rev. 128, 1437 (1962).\n6A. W. Overhauser and A. Arrott, Phys. Rev. Lett. 4, 226 (1960).\n7L. M. Corliss, J. M. Hastings, and R. J. Weiss, Phys. Rev. Lett .3, 211 (1959).\n8C. G. Shull and M. K. Wilkinson, Rev. Mod. Phys. 25, 100 (1953).\n9Stability of a CDW state can only be achieved if the associate d Hartree term is cancelled by a corresponding deformation\nof the neutralizing background. The simplest such case bein g represented by the case of the so called deformable jellium . A\nrather interesting case is discussed in20.\n10G. F. Giuliani and G. Vignale, Quantum Theory of The Electron Liquid (Cambridge University Press, 2005).\n11S. Chesi and G. F. Giuliani, Phys. Rev. B 75, 155305 (2007).\n12S. Chesi and G. F. Giuliani, Phys. Rev. B 75, 153306 (2007).\n13P. Fedders and P. C. Martin, Phys. Rev. 143, 245 (1966).\n14That the states of Eqs. (5) and (6) are solutions of the Hartre e-Fock problem was shown in Ref.11.\n15Notice that in this case, the generalized chirality, as defin ed for instance in Ref.11, also equals one.\n16A. W. Overhauser, Phys. Rev. 167, 691 (1968).\n17S. Chesi and G. F. Giuliani, to be published.\n18Http://functions.wolfram.com/08.04.02.0001.01 .\n19Http://functions.wolfram.com/08.05.02.0001.01 .\n20G. F. Giuliani and A. W. Overhauser, Phys. Rev. B 20, 1328 (1979)."    },    {        "title": "1008.4817v1.Lifshitz_tails_estimate_for_the_density_of_states_of_the_Anderson_model.pdf",        "content": "arXiv:1008.4817v1  [math-ph]  27 Aug 2010LIFSHITZ TAILS ESTIMATE FOR THE DENSITY OF STATES\nOF THE ANDERSON MODEL\nJEAN-MICHEL COMBES, FRANC ¸OIS GERMINET, AND ABEL KLEIN\nAbstract. We prove an upper bound for the (differentiated) density of st ates\nof the Anderson model at the bottom of the spectrum. The densi ty of states is\nshown to exhibit the same Lifshitz tails upper bound as the in tegrated density\nof states.\n1.Introduction and main result\nWe consider the Anderson model, the simplest random Schr¨ odinger operator,\ngiven by the random Hamiltonian\nHω:=−∆+Vωonℓ2(Zd), (1.1)\n∆ is the d-dimensional discrete Laplacian operator and Vωis the random potential\ngiven by Vω(j) =ωjforj∈Zd, whereω={ωj}j∈Zdis a family of independent\nidentically distributed random variables whose common probability dist ribution µ\nis non-degenerate, with support bounded from below, and has a bo unded density\nρ. (The requirement infsupp µ >−∞is equivalent to requiring Hωto be bounded\nfrom below with probability one.) Note that\nVω=/summationdisplay\nj∈ZdωjΠj, (1.2)\nwhere Π jis the orthogonal projection onto δj, the delta function at site j.\nThe integrated density of states (IDS) of this celebrated model is known to\nexhibit Lifshitz tails behavior as the bottom of its spectrum (e.g., [CaL , K, PF]), a\nproperty that can be interpreted as a first signature of localizatio n.\nIn the physics literature there is much interest on the density of th e IDS, the\ndensity of states (DOS). There is an implicit assumption that the IDS N(E) is\nabsolutely continuous, and hence has a density n(E) given by its almost everywhere\nderivative. Very few mathematical results are available for the DOS of random\nSchr¨ odinger operators. For nice enough models, like the Anderso n model as above,\nthe existence of the DOS is a consequence of the celebrated Wegne r estimate (see\n(2.4) below), which also shows the DOS to be bounded. We note that w hile the\nabsolute continuity of the IDS has been known for a long time for the Anderson\nmodel [W, FS], this is a more recent result for models in the continuum [C oH,\nCoHK]. In dimension d= 1 this DOS is known to be regular [ST, CK, VSW]. For\narbitrary dimension d, the DOS of the Anderson model described above is known\n2000Mathematics Subject Classification. Primary 82B44; Secondary 47B80, 60H25.\nJM.C. and F.G. were supported by ANR BLAN 0261. A.K was suppor ted in part by NSF\nGrant DMS-1001509.\n12 J.-M. COMBES, F. GERMINET, AND A. KLEIN\nto be regular at high disorder, a result obtained using supersymmet ric methods by\n[BoCKP]. Regularity of the DOS is an open question for models in the con tinuum.\nIn this article, we show that the DOS exhibits the same Lifshitz tails up per\nbound as the IDS. To our best knowledge, this is the first time that s uch a bound\nis proved for the DOS of a random Schr¨ odinger operator.\nThere is another reason to study the DOS n(E). As proved by Minami [M],\nin the localization region the properly rescaled eigenvalues of a discre te Anderson\nHamiltonian are distributed as a Poisson point process with intensity n(E). Thus\nour results (see (2.13)) estimate the intensity of these Poisson po int processes.\nAn Anderson Hamiltonian Hωis aZd-ergodic family of random self-adjoint\noperators. It follows from standard results (cf. [KM]) that ther e exists fixed subsets\nΣ, Σpp, Σacand Σ scofRso that the spectrum σ(Hω) ofHω, as well as its pure\npoint, absolutely continuous, and singular continuous components , are equal to\nthese fixed sets with probability one. This non-random spectrum is g iven by\nΣ =σ(−∆)+supp µ= [0,4d]+suppµ. (1.3)\nNote infΣ = infsupp µ >−∞.\nIf we set ω′\nj=ωj−infsuppµ, the new common probability distribution µ′\nsatisfies infsupp µ′= 0, and we have Hω=Hω′+infsupp µ. Thus, without loss of\ngenerality we assume from now on that\ninfΣ = infsupp µ= 0. (1.4)\nLifshitz tails and localization are known to hold at the bottom of the sp ectrum\n[FMSS, DK, K]. If the support of µis also bounded from above, then so is Σ, and\nLifshitz tails and localization also hold at the top, i.e., upper edge, of th e spectrum.\nIn this case the results of this paper also hold at the top of the spec trum.\nThe integrated density of states (IDS), N(E), can be written as (e.g., [K])\nN(E) =E/braceleftbig\ntr/parenleftbig\nΠ0χ]−∞,E](Hω)Π0/parenrightbig/bracerightbig\n=E/braceleftbig\ntr/parenleftbig\nΠ0χ[0,E](Hω)Π0/parenrightbig/bracerightbig\n.(1.5)\nThe Anderson model is known to satisfy the following Lifshitz tails est imate,\nwhich asserts that the IDS has an exponential fall off as one appro aches the edges\nof Σ. At the bottom of the spectrum, i.e., at energy E= 0, the IDS satisfies (e.g.,\n[K])\nlim\nE↓0log|logN(E)|\nlogE≤ −d\n2. (1.6)\nEquality is actually known to hold in (1.6). Since N(E) is an increasing function, it\nhas a derivative n(E) :=N′(E) almost everywhere, which is the density of states.\nWe prove\nTheorem 1.1. LetHωbe an Anderson Hamiltonian. Then there exists a Borel\nsetN ⊂[0,1]of zero Lebesgue measure, such that\nlim\nE↓0;E/∈Nlog|logn(E)|\nlogE≤ −d\n2. (1.7)\nRemark 1.2. The same Lifshitz tails estimate holds for models in the cont inuum;\nsee[CoGK3] .\nThe proof of Theorem 1.1 takes advantage of a new double averagin g procedure\nintroducedin[CoGK2, Theorem2.2]to extractbetter controlont heconstantinthe\nWegner estimate. Theorem 1.1 will be an immediate consequence of Th eorem 2.1.LIFSHITZ TAILS FOR THE DOS 3\n2.Finite volume operators, the integrated density of states, and\nthe Wegner estimate\nFinite volume operators will be defined for finite boxes\nΛ = Λ L(j) :=j+/bracketleftbig\n−L\n2,L\n2/bracketleftbigd, (2.1)\nwherej∈ZdandL∈2N,L >1. Given such Λ, we will consider the ran-\ndom Schr¨ odinger operator H(Λ)\nωonℓ2(Λ) given by the restriction of the Anderson\nHamiltonian Hωto Λ with periodic boundary condition. To do so, we identify Λ\nwith the torus Zd/LZdin the usual way, and define finite volume operators\nH(Λ)\nω:=−∆(Λ)+V(Λ)\nωonℓ2(Λ), (2.2)\nwhere∆(Λ)istheLaplacianonΛwithperiodicboundarycondition, andtherandom\npotential V(Λ)\nωis the restrictionof Vω(Λ)to Λ, where, given ω={ωi}i∈Zd, we define\nω(Λ)=/braceleftBig\nω(Λ)\ni/bracerightBig\ni∈Zdby\nω(Λ)\ni=ωiifi∈Λ,\nω(Λ)\ni=ω(Λ)\nkifk−i∈LZd.(2.3)\nThe finite volume random operator H(Λ)\nωis covariantwith respect to translations\nin the torus. If B⊂Ris a Borelset, we write P(Λ)\nω(B) :=χB/parenleftBig\nH(Λ)\nω/parenrightBig\nandPω(B) :=\nχB(Hω) for the spectral projections.\nThe finite volume operator H(Λ)\nωis a finite dimensional operator, and hence its\n(ω-dependent) spectrum consists of a finite number of isolated eigen values with\nfinite multiplicity. These finite volume operators satisfy a Wegner est imate [W, FS]\n(see also [CoGK1]), which provides informations on the number of eige nvalues in a\ngiven energy interval: given E0>0, there exists a constant KW(E0), independent\nof Λ, such that for all intervals I⊂[0,E0] we have\nE/braceleftBig\ntrP(Λ)\nω(I)/bracerightBig\n≤KW(E0)/bardblρ/bardbl∞|I||Λ|. (2.4)\nThe Wegner estimate is an an immediate consequence of the following s pectral\naveraging property (e.g., [CoHK, CoGK1]) : for any k∈Λ,\nEωk/braceleftBig\n/a\\}bracketle{tδk,P(Λ)\nω(I)δk/a\\}bracketri}ht/bracerightBig\n=Eωk/braceleftBig\ntrΠkP(Λ)\nω(I)Πk/bracerightBig\n≤ /bardblρ/bardbl∞|I|,(2.5)\nwhere (δk)k∈Zdstands for the canonical basis (so Π k=|δk/a\\}bracketri}ht/a\\}bracketle{tδk|). It follows that for\nthe Anderson model (2.4) holds with\nKW(E0)≤1 for all E0>0. (2.6)\nWe shall prove that at the bottom of the spectrum, in the Lifshitz t ails region, the\nconstant KW(E0) falls off in the same way as the IDS.\nWe set\nN(Λ)\nω(E) :=|Λ|−1trχ]−∞,E]/parenleftBig\nH(Λ)\nω/parenrightBig\n, (2.7)\nand recall that (e.g., [CaL, K, PF]) for P-a.e.ωwe have, using the fact that N(E)\nis a continuous function by (2.4),\nN(E) = lim\nL→∞N(ΛL(0))\nω(E) for all E∈R, (2.8)4 J.-M. COMBES, F. GERMINET, AND A. KLEIN\nwhereN(E) is the integrated density of states (IDS) given in (1.5)). Setting\nN(Λ)(E) :=E/braceleftBig\nN(Λ)\nω(E)/bracerightBig\n=E/braceleftBig\ntr/parenleftBig\nΠjχ]−∞,E](H(Λ)\nω)Πj/parenrightBig/bracerightBig\nforj∈Λ,(2.9)\nthe last equality holding in view of the periodic boundary condition, it fo llows that\nN(E) = lim\nL→∞N(ΛL(0))(E) =E/braceleftBig\ntr/parenleftBig\nΠ0χ]−∞,E](H(ΛL(0))\nω)Π0/parenrightBig/bracerightBig\n(2.10)\nfor allE∈R.\nCombining (2.4) and (2.10), we conclude that, if N(E) is differentiable at E,\nwhich is true for a.e. E, we have\nn(E) :=N′(E)≤KW(E)/bardblρ/bardbl∞. (2.11)\nMoreover, to obtain (2.11), it suffices to have the Wegner estimate (2.4) for boxes\nΛLn(0) with Ln→ ∞. Thus Theorem 1.1 follows immediately from the following\nresult.\nTheorem 2.1. LetHωbe an Anderson Hamiltonian. Then there is an energy\nE0>0, and for a.e. E∈]0,E0]and allε∈]0,d\n2[there exists a scale L(E,ε)<∞,\nsuch that given L∈4NwithL≥L(E,ε), the Wegner estimate (2.4)holds in all\nboxesΛ = Λ Lfor all intervals I⊂[0,E]with a constant KW(E)satisfying\nKW(E)≤Cd,εe−E−d\n2+ε, (2.12)\nfor some constant Cd,ε<∞. As a consequence, we have\nn(E)≤Cd,ε/bardblρ/bardbl∞e−E−d\n2+εfor a.e. E∈]0,E0]. (2.13)\n3.The proof of Theorem 2.1\nWe borrow from [CoGK3] the following observation.\nLemma 3.1 ([CoGK3]) .LetH=H0+W, whereH,H0are semi-bounded self-\nadjoint operators, say H,H0≥ −Θfor some Θ>0, such that (H+Θ+1)−pis\na trace class operator for some p >0, andWis a bounded self-adjoint operator.\nGivenE0∈R, letf,gbe bounded Borel measurable nonnegative functions such tha t\nf=χ(−∞,E0]fandχ(−∞,E0]≤g≤1. Thenf(H)Wandf(H)Wg(H0)are trace\nclass operators, and\ntrf(H)W≤trf(H)Wg(H0). (3.1)\nThe proof is elementary. It consists in proving that tr f(H)W(1−g(H0))≤0,\nusingW=H−H0.\nProof of Theorem 2.1. All the operators in this proofs will be finite volume opera-\ntors on a box Λ = Λ L(0) as defined in (2.2)-(2.3). We will require L∈4N.\nWe setω⊥\n0=ω\\{ω0}andHω⊥\n0=Hω−ω0Π0. GivenE >0, we fix a C∞real-\nvalued non-increasing function fEonR, such that fE(t) = 1 for t≤E,fE(t) = 0\nfort≥2E, and/vextendsingle/vextendsinglef(j)(t)/vextendsingle/vextendsingle≤CE−jfor allt∈Randj= 1,2,...,2d+3, where Cis\na constant independent of E. We let ˜P0=˜P(Λ)\nω⊥\n0,E=fE(H(Λ)\nω⊥\n0).LIFSHITZ TAILS FOR THE DOS 5\nGiven an interval I⊂[0,E], it follows from Lemma 3.1 that\ntrP(Λ)\nω(I)Π0≤trP(Λ)\nω(I)Π0˜P0=/summationdisplay\nk∈ΛtrP(Λ)\nω(I)Π0˜P0Πk\n≤/summationdisplay\nk∈Λ/bardblP(Λ)\nω(I)Π0/bardbl2/bardblP(Λ)\nω(I)Πk/bardbl2/bardblΠ0˜P0Πk/bardbl (3.2)\n≤1\n2/summationdisplay\nk∈Λ/braceleftBig\ntr(Π0P(Λ)\nω(I)Π0)+tr(Π kP(Λ)\nω(I)Πk)/bracerightBig\n/bardblΠ0˜P0Πk/bardbl.\nNext, for any given k(including k= 0) there exists k0∈Zd, so that, defining the\nsublattice Γ k=k0+(4Z)d⊂Zd, we have 0 ,k/\\e}atio\\slash∈Γ. Since we required L∈4N, we\nhave|Γ∩Λ|= 4−d|Λ|. We define ωΓ={ωγ}γ∈Γ. We have, using the fast decay\nof the kernel of smooth functions of Schr¨ odinger operators (e .g. [GK]), using the\nnotation /a\\}bracketle{tk/a\\}bracketri}ht=/radicalBig\n1+|k|2, and letting rstand for either 0 or k,\n/bardblΠ0˜P0Πk/bardbl ≤ /bardblΠ0˜P0Πk/bardbl1\n2/bardblΠr˜P0/bardbl1\n2=/bardblΠ0˜P0Πk/bardbl1\n2/bardblΠr˜P0Πr/bardbl1\n4\n≤Cd\nEd+3\n2/a\\}bracketle{tk/a\\}bracketri}htd+1/parenleftBig\ntrΠr˜P0Πr/parenrightBig1\n4(3.3)\n≤Cd\nEd+3\n2/a\\}bracketle{tk/a\\}bracketri}htd+1/parenleftbigg\ne2tEtrΠre−tH(Λ)\nω⊥\n0Πr/parenrightbigg1\n4\n≤Cd\nEd+3\n2/a\\}bracketle{tk/a\\}bracketri}htd+1/parenleftbigg\ne2tEtrΠre−tH(Λ)\nωΓkΠr/parenrightbigg1\n4\n,\nfor allt >0, where we used r/\\e}atio\\slash∈Γkand a positivity preserving argument like\n[BGKS, Lemma 2.2] to get the last inequality. Hence, again letting rstand for\neither 0 or k, sor/\\e}atio\\slash∈Γk, using the spectral averaging (2.5) with the bound (2.6)\nand (3.3) leads to\nE/parenleftBig\ntr/braceleftBig\nΠrP(Λ)\nω(I)Πr/bracerightBig\n/bardblΠ0˜P0Πk/bardbl/parenrightBig\n≤Cd\nEd+3\n2/a\\}bracketle{tk/a\\}bracketri}htd+1Eω⊥r/braceleftBigg/parenleftbigg\ne2tEtrΠre−tH(Λ)\nωΓkΠr/parenrightbigg1\n4\nEωr/braceleftBig\ntrΠrP(Λ)\nω(I)Πr/bracerightBig/bracerightBigg\n≤Cd\nEd+3\n2/a\\}bracketle{tk/a\\}bracketri}htd+1/bardblρ/bardbl∞|I|et\n2E/parenleftBigg\nEωΓk/braceleftbigg\ntrΠre−tH(Λ)\nωΓkΠr/bracerightbigg1\n4/parenrightBigg\n≤Cd\nEd+3\n2/a\\}bracketle{tk/a\\}bracketri}htd+1/bardblρ/bardbl∞|I|et\n2E/parenleftbigg\nEωΓk/braceleftbigg\ntrΠre−tH(Λ)\nωΓkΠr/bracerightbigg/parenrightbigg1\n4\n. (3.4)\nThanks to (3.2) and (3.4), it now suffices to bound the quantity\nEωΓk/braceleftbigg\ntrΠre−tH(Λ)\nωΓkΠr/bracerightbigg\nwithr= 0,k. (3.5)\nTo alleviate notations, we write from now on Γ = Γ k. For all t >0 we have\ntrΠre−tHωΓΠr≤trΠrχ]−∞,4E](H(Λ)\nωΓ)Πr+e−4tEtrΠr\n= trΠ rχ]−∞,4E](H(Λ)\nωΓ)Πr+e−4tE. (3.6)6 J.-M. COMBES, F. GERMINET, AND A. KLEIN\nBy the Γ-ergodicity of H(��)\nωΓ, and taking advantage of the periodic boundary con-\ndition, we have\nE/braceleftBig\ntrΠrχ]−∞,4E](H(Λ)\nωΓ)Πr/bracerightBig\n=1\n|Γ∩Λ|/summationdisplay\nγ∈(r+Γ)∩ΛEωΓ/braceleftBig\ntrΠγχ]−∞,4E](H(Λ)\nωΓ)Πγ/bracerightBig\n≤4d\n|Λ|EωΓ/braceleftBig\ntrχ]−∞,4E](H(Λ)\nωΓ)/bracerightBig\n. (3.7)\nWenowusetheLifshitztailsestimatefor HωΓtobound EωΓ/braceleftBig\ntrχ]−∞,4E](H(Λ)\nωΓ)/bracerightBig\n.\nNote that Γ is a strict sublattice of Zd, so we lack the so called covering condition.\nThe Lifshitz tails estimate (1.6) is nevertheless valid for HωΓ(e.g. [K]), and it\nimplies that for all ε∈]0,d\n2[ there is an energy Eε>0 such that (on Zd)\nNΓ(E) =EωΓ/braceleftbig\ntrχ]−∞,E](HωΓ)/bracerightbig\n≤e−E−d\n2+εfor allE≤Eε.(3.8)\nGiven a box Λ L= ΛL(0), we set, similarly to (2.7) and (2.9),\nN(ΛL)\nωΓ(E) =|ΛL|−1trχ]−∞,E](H(ΛL)\nωΓ), N(ΛL)\nΓ(E) =EωΓ/braceleftBig\nN(ΛL)\nωΓ(E)/bracerightBig\n.(3.9)\nSinceHωΓis Γ-ergodic, for P-a.e.ωΓwe have [CaL, PF]\nNΓ(E) = lim\nL→∞N(ΛL)\nωΓ(E) for a.e. E∈R, (3.10)\nso we conclude\nNΓ(E) = lim\nL→∞N(ΛL)\nΓ(E) for a.e. E∈R. (3.11)\n(Compare with (2.8) and (2.10), where convergence holds for all E. The difference\nis the lack of a Wegner estimate for HωΓ.) It follows that for a.e. E∈]0,Eε] there\nexistsL(E,ε)<∞such that for all L≥L(E,ε) we have\nN(ΛL)\nΓ(E)≤2NΓ(E)≤2e−E−d\n2+ε. (3.12)\nThus, for a.e. E∈]0,1\n4Eε] andL≥L(E,ε), combining (3.6), (3.7), and (3.12),\nwe get\nE/braceleftBig\ntrΠr˜P(ΛL)\n0Πr/bracerightBig\n≤2·4de−(4E)−d\n2+ε+e−4tE. (3.13)\nChoosing t=tEby\ne−4tE= 2·4de−(4E)−d\n2−1+ε,i.e.,t= (4E)−d\n2−1+ε−(1+2d)(log2)(4 E)−1,(3.14)\nand letting E′\nε= max/braceleftbig\nE≤1\n4Eε;tE>0/bracerightbig\n, we conclude that for a.e. E∈]0,E′\nε] and\nL≥L(E,ε) we have\nE/braceleftBig\ntrΠr˜P(ΛL)\n0Πr/bracerightBig\n≤4d+1e−(4E)−d\n2+ε= 2e−4tEE. (3.15)\nCombining (3.2), (3.4), and (3.15), we conclude that, for a.e. E∈]0,E′\nε] andL≥\nL(E,ε) we have, for all intervals I⊂[0,E],\nE/braceleftBig\ntrPω(ΛL)(I)Π0/bracerightBig\n≤/summationdisplay\nk∈ΛCd\nEd+3\n2/a\\}bracketle{tk/a\\}bracketri}htd+1/bardblρ/bardbl∞|I|e1\n2tEE/parenleftbig\n2e−4tEE/parenrightbig1\n4(3.16)\n≤C′\ndE−d−3\n2e−1\n2tEE/bardblρ/bardbl∞|I|=C′′\ndE−d−3\n2e−1\n8(4E)−d\n2+ε/bardblρ/bardbl∞|I|.\nTheorem 2.1 follows. /squareLIFSHITZ TAILS FOR THE DOS 7\nReferences\n[BGKS] Bouclet, J.M., Germinet, F., Klein, A., Schenker, J. : Linear response theory for mag-\nnetic Schr¨ odinger operators in disordered media. J. Funct . Anal.226, 301-372 (2005)\n[BoCKP] Bovier, A., Campanino, M., Klein, A., Perez, J.F.: S moothness of the density of states\nin the Anderson model at high disorder, Commun. Math. Phys. 114, 439-461 (1988)\n[CK] Campanino, M., Klein, A.: A supersymmetric transfer ma trix and differentiability of\nthe density of states in the one-dimensional Anderson model . Comm. Math. Phys. 104,\n227-241 (1986)\n[CaL] Carmona, R, Lacroix, J.: Spectral theory of random Schr¨ odinger operators . Boston:\nBirkha¨ user, 1990\n[CoGK1] Combes, J.M., Germinet, F., Klein, A.: Generalized eigenvalue-counting estimates for\nthe Anderson model, J. Stat. Phys. 135, 201-216 (2009)\n[CoGK2] Combes, J.M., Germinet, F., Klein, A.: Poisson stat istics for eigenvalues of continuum\nrandom Schr¨ odinger operators, Analysis and PDE 3, 49-80 (2010)\n[CoGK3] Combes, J.M., Germinet, F., Klein, A.: Local Wegner estimates forrandom Schr¨ odinger\noperators, Minami estimate and Lifshitz behavour of the den sity of states. In prepara-\ntion.\n[CoH] Combes, J.M., Hislop, P.D.: Localization for some con tinuous, random Hamiltonians\nin d-dimension. J. Funct. Anal. 124, 149-180 (1994)\n[CoHK] Combes, J.M., Hislop, P.D., Klopp, F.: Optimal Wegne r estimate and its application\nto the global continuity of the integrated density of states for random Schr¨ odinger\noperators. Duke Math. J. 140, 469-498 (2007)\n[DK] von Dreifus, H., Klein, A.: A new proof of localization i n the Anderson tight binding\nmodel. Commun. Math. Phys. 124, 285-299 (1989)\n[FMSS] Fr¨ ohlich, J., Martinelli, F., Scoppola, E., Spence r, T.: Constructive proof of localization\nin the Anderson tight binding model. Commun. Math. Phys. 101, 21-46 (1985)\n[FS] Fr¨ ohlich, J., Spencer, T.: Absence of diffusion with An derson tight binding model for\nlarge disorder or low energy. Commun. Math. Phys. 88, 151-184 (1983)\n[GK] Germinet, F, Klein, A.: Operator kernel estimates for f unctions of generalized\nSchr¨ odinger operators. Proc. Amer. Math. Soc. 131, 911-920 (2003)\n[K] Kirsch, W.: An invitation to random Schr¨ odinger operat or, with an Appendix by Frdric\nKlopp, Panoramas et synthses 25 (2008), 1-119\n[KM] Kirsch, W., Martinelli, F.: On the ergodic properties o f the spectrum of general random\noperators. J. Reine Angew. Math. 334, 141-156 (1982)\n[M] Minami, N.: Local fluctuation of the spectrum of a multidi mensional Anderson tight\nbinding model. Comm. Math. Phys. 177, 709-725 (1996)\n[PF] Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators . Heidelberg:\nSpringer-Verlag, 1992\n[ST] Simon, B., Taylor, M.: Harmonic analysis on SL(2 ,R) and smoothness of the density of\nstates in the one-dimensional Anderson model. Comm. Math. P hys.101, 1-19 (1985)\n[St] Stollmann, P.: Caught by disorder. Bound States in Rand om Media. Birka¨ user, 2001\n[VSW] Shubin, C. , Vakilian, R. , Wolff, T.: Some harmonic anal ysis questions suggested by\nAnderson-Bernoulli models. Geom. Funct. Anal. 8, 932–964 (1998)\n[W] Wegner, F.: Bounds on the density of states in disordered systems, Z. Phys. B 449-15\n(1981)\n(Combes) CPT, CNRS, Luminy Case 907, Cedex 9, F-13288 Marseille, Franc e\nE-mail address :combes@cpt.univ-mrs.fr\n(Germinet) Universit ´e de Cergy-Pontoise, CNRS UMR 8088, IUF, D ´epartement de\nMath´ematiques, F-95000 Cergy-Pontoise, France\nE-mail address :germinet@math.u-cergy.fr\n(Klein) University of California, Irvine, Department of Mathematic s, Irvine, CA\n92697-3875, USA\nE-mail address :aklein@uci.edu"    },    {        "title": "1009.1115v2.Information_geometry_of_density_matrices_and_state_estimation.pdf",        "content": "arXiv:1009.1115v2  [quant-ph]  2 Jun 2011Information geometry of density matrices and state estimat ion\nDorje C. Brody\nMathematical Sciences, Brunel University, Uxbridge UB8 3P H, UK\nGiven a pure state vector |x∝an}b∇acket∇i}htand a density matrix ˆ ρ, the function p(x|ˆρ) =∝an}b∇acketle{tx|ˆρ|x∝an}b∇acket∇i}ht\ndefines a probability density on the space of pure states para meterised by density\nmatrices. The associated Fisher-Rao information measure i s used to define a uni-\ntary invariant Riemannian metric on the space of density mat rices. An alternative\nderivation of the metric, based on square-root density matr ices and trace norms,\nis provided. This is applied to the problem of quantum-state estimation. In the\nsimplest case of unitary parameter estimation, new higher- order corrections to the\nuncertainty relations, applicable to general mixed states , are derived.\nPACS numbers: 03.65.Ta, 03.67.-a\nThe structure of the space Dof density matrices is of considerable interest in quantum\ninformation theory and related disciplines such as quantum tomogra phy and quantum sta-\ntistical inference. While substantial progress has been made in unr avelling the structure\nofD[1–5], the subject remains difficult, partly because of the complicate d structure of D,\nwhich is a manifold with a boundary given by degenerate density matric es. The purpose of\nthe present paper is to reveal the existence of a remarkably simple dual metric structure over\nthe space of pure and mixed states, and to apply this to obtain new h igher-order corrections\nto Heisenberg uncertainty relations for general mixed states.\nThe idea of the dual metric structure can be sketched as follows. I f|x∝an}b∇acket∇i}htdenotes a nor-\nmalisedpurestatevector and ˆ ρageneric density matrix, thentheexpectation ∝an}b∇acketle{tx|ˆρ|x∝an}b∇acket∇i}htdefines\na probability density function either on the space of pure states pa rameterised by density\nmatrices, or on the space of density matrices parameterised by pu re states. Hence, methods\nof information geometry [6–8] can be employed to define a pair of Riem annian metrics, one\nfor the pure state space and one for the space of density matrice s.\nAnalternativeformulationistousethetracenormandmatrixalgebr aassociatedwiththe\nHermitiansquare-rootmap ˆ ρ→ˆξ=√ˆρ. Itisshownthatif ˆξisparameterisedbyapurestate\nso thatˆξ= ˆρ, then the associated information metric is the Fubini-Study metric; whereas if\nˆξis parameterised by a generic density matrix, then we recover the in formation metric on D.\nApart fromthese two extremes, however, there are many other cases of considerable interest.\nTypically, in a problem of quantum state estimation, the density matr ix ˆρ(θ) depends on one\nor several unknown parameters {θi}that are to be estimated. The Fisher-Rao information\nmetric 4tr[ ∂iˆξ(θ)∂jˆξ(θ)], where ∂i=∂/∂θi, then determines the covariance lower bound of\nthe parameter estimates. In the simplest case in which ˆ ρ(θ) represents a one-parameter\nunitary curve in the space of density matrices (e.g., θrepresents time), the quantum Fisher-\nRao metric coincides with the “skew information” defined by Wigner an d Yanase [9]. By\nthis device we are able to extend the results of [10–15] on quantum statistical inference. In\nparticular, we derive: (a) a variance lower bound for general mixed states, which is sharper\nthan the previously known ones that are expressed in terms of the skew information; (b)\nthe quantum-mechanical analogue of the conditional variance rep resentation in terms of\nthe skew information measures; and (c) higher-order correction s to the uncertainty lower2\nbound for mixed states, whereby the concept of higher-order sk ew information measures are\nintroduced.\nConsider a finite, n-dimensional quantum system. The space of pure states Γis the space\nof rays through the origin of C(n), which is just the complex projective space equipped with\nthe unitary invariant Fubini-Study metric. We let Ddenote the space of density matrices on\nC(n). Every density matrixdefines apositive scalarfunction ∝an}b∇acketle{tx|ˆρ|x∝an}b∇acket∇i}htonΓ, andthis, properly\nnormalised with respect to the Fubini-Study volume element, yields a p robability density on\nΓ. Since, by the real polarisation identity, a density matrix is uniquely d etermined by the\ntotality of its expectation values, these probability distributions ar e parameterised by Din\na one-to-one fashion. Hence we can use the Hellinger distances bet ween these distributions\nand obtain an information metric onDto investigate properties of D, which are directly\nrelated with the physical significance of density matrices. The dual picture emerges from\nthe converse construction: Given any |x∝an}b∇acket∇i}ht ∈Γ, one obtains a nonzero nonnegative function\n∝an}b∇acketle{tx|ˆρ|x∝an}b∇acket∇i}htonD, which defines, when properly normalised, a set of probability distrib utions on\nD. This likewise yields an information metric on Γ. By considerations of symmetry and\ninvariance, using the unitary structure of Cn, one can perceive that this a priorinew metric\nonΓcan only be the Fubini-Study metric.\nIf one assumes the Schr¨ odinger equations of motion |x(t)∝an}b∇acket∇i}ht= exp(−iˆHt)|x(0)∝an}b∇acket∇i}htonΓand\nthe Heisenberg equations of motion ˆ ρ(t) = exp(−iˆHt)ˆρ(0)exp(i ˆHt) onD, then the following\nconclusions can be drawn. The evolution on Γis isometric with respect to the Fubini-\nStudy metric (known as Wigner’s theorem), and, moreover, the ev olution on Dis isometric\nwith respect to the information metric. Thus, we have a complete re ciprocity between the\nSchr¨ odinger dynamics on Γ, equipped with the Fubini-Study metric, and the Heisenberg\ndynamics on D, endowed with the information metric.\nThe components of these informationmetrics, with respect to a giv en choice ofcoordinate\nsystem, can be calculated by use of the standard expression for t he Fisher-Rao metric. Let\nuswrite{xµ}µ=1,...,2n−2forthecoordinateson Γwithvolume element d vx, and{ρa}a=1,...,n2−1\nfor the coordinates on Dwith volume element d Vρ. Likewise, we write p(x|ˆρ)∝ ∝an}b∇acketle{tx|ˆρ|x∝an}b∇acket∇i}htfor\nthe normalised density on ΓandP(ˆρ|x)∝ ∝an}b∇acketle{tx|ˆρ|x∝an}b∇acket∇i}htfor the normalised density on D. Then\nthe information metric GabonDis given by\nGab(ˆρ) =/integraldisplay\nΓ1\np(x|ˆρ)∂p(x|ˆρ)\n∂ρa∂p(x|ˆρ)\n∂ρbdvx. (1)\nSimilarly, the information (Fubini-Study) metric gµνonΓis given by\ngµν(x) =/integraldisplay\nD1\nP(ˆρ|x)∂P(ˆρ|x)\n∂xµ∂P(ˆρ|x)\n∂xνdVρ. (2)\nWe therefore have two statistical manifolds, Γwith the metric gµν, andDwith the metric\nGab; each of these is represented by probability distributions on the ot her.\nThe duality of information metrics leads to integral representation s for observable ex-\npectations. Let us consider trace-free observables ˆA. Defining A(x) =∝an}b∇acketle{tx|ˆA|x∝an}b∇acket∇i}htand\nA(ˆρ) = tr(ˆρˆA), the dual formulae read\nA(ˆρ) =/integraldisplay\nΓA(x)p(x|ˆρ)dvxandA(x) =/integraldisplay\nDA(ˆρ)P(ˆρ|x)dVρ. (3)\nThe first formula is a special case of Gibbons’ explicit formula for the trace of a product of\ntwo observables [16]. The second is proved using Gibbons’ formula alo ng with invariance\narguments.3\nWe remark that the space of density matrices Dis a manifold with a piecewise smooth\nboundary, the latter consisting of the degenerate density matric es, i.e. those with at least\none zero eigenvalue. The structure of Dis thus analogous to that of the unit n−1 simplex,\nwhich is the space of probability distributions on a set of npoints. Information geometry\nis indeed applicable, mutatis mutandis , to manifolds with boundary. Thus, defining the\ninformation metric on Din the above described manner, one obtains a volume element with\nthe necessary invariance properties to prove the dual integral f ormulae. In particular, the\nHeisenberg dynamics on Ddefines a Killing field with respect to this metric, which leaves\nthe set of degenerate matrices (i.e. the boundary) invariant, tha t is, the Killing field is\ntangent to the boundary.\nAn alternative approach is to label density matrices by their Hermitia n square roots.\nThe significance of the square-root map in the context of classical probability is outlined in\nRao’s seminal paper [6]; also elaborated in [8]. The key formulation here in the quantum\ncontext is to consider not arbitrary square root of the density ma trix, but the Hermitian\nsquare root, which provides the natural extension of the classica l theory. Thus, we write\nˆρ=ˆξˆξ, where ˆξis an arbitrary Hermitian matrix such that tr( ˆξ2) = 1. If ˆ ρis a pure\nstate so that ˆ ρ=ˆξ=|x∝an}b��acket∇i}ht∝an}b∇acketle{tx|, then the quantum Fisher-Rao information metric is given by\ngµν= 4tr(∂µˆξ∂νˆξ), where ∂µ=∂/∂xµ, which is just the Fubini-Study metric (2). On the\notherhand, if ˆ ρisagenericmixed statethequantum Fisher-Raometricis Gab= 4tr(∂aˆξ∂bˆξ),\nwhere∂a=∂/∂ρa, and this is just the information metric (1).\nAs an illustration, consider the 2 ×2 case. A generic Hermitian matrix ˆξcan be expressed\nin the form\nˆξ=/parenleftbigg\nt+z x−iy\nx+iy t−z/parenrightbigg\n. (4)\nFor a pure state we write ( x1,x2) = (θ,φ), where t=1\n2,x=1\n2sinθcosφ,y=1\n2sinθsinφ,\nandz=1\n2cosθ. Calculating gµν= 4tr(∂µˆξ∂νˆξ), we obtain the usual spherical metric for the\nspace of pure states. For a mixed state we write ( ρ1,ρ2,ρ3) = (u,θ,φ), wheret= (1\n2−u2)1/2,\nx=usinθcosφ,y=usinθsinφ, andz=ucosθ. Calculating Gab= 4tr(∂aˆξ∂bˆξ) we obtain\nthe usual hyperbolic metric for a ball of radius1\n2(interior of the Bloch sphere).\nThe richness of the geometry of the space of density matrices, ev en in the 2 ×2 example,\nis elucidated further by the following analysis. We square (4) to obta in\nˆρ=/parenleftbigg\nt2+x2+y2+z2+2tz 2t(x−iy)\n2t(x+iy)t2+x2+y2+z2−2tz/parenrightbigg\n.\n(5)\nBy construction ˆ ρis positive; thus we need only consider the trace condition\ntr(ˆρ) = 1 = ⇒t2+x2+y2+z2=1\n2. (6)\nTherefore, under the Hermitian square-root map the space of 2 ×2 density matrices is\nmapped to a sphere S3⊂R4. Making use of the trace condition (6) we can write\nˆρ=/parenleftbigg1\n2+2tz2tx−i(2ty)\n2tx+i(2ty)1\n2−2tz/parenrightbigg\n, (7)4\nwheretis determined by ( x,y,z) via (6). Clearly, under the reflection ( t,x,y,z)→\n(−t,−x,−y,−z) we have ˆ ρ→ˆρ. This, however, is not the only degeneracy. While a density\nmatrix has a unique positive semidefinite square root ˆξand a unique negative semidefi-\nnite square root −ˆξ, there also exist square roots which are neither positive nor negat ive\nsemidefinite.\nTo work out the roots, suppose that a density matrix\nˆ̺=/parenleftbigg1\n2+a b−ic\nb+ic1\n2−a/parenrightbigg\n(8)\nis given. How many points on S3give rise to the same ˆ ̺of (8)? By comparing (7) and (8)\nwe obtain:\n2tx=b,2ty=c,2tz=a. (9)\nTofindtheintersectionof(6)and(9)weeliminate( x,y,z)using(9)toobtain4 t4−2t2+R2=\n0, where R2≡a2+b2+c2(clearly, 0 ≤R2≤1\n4). This quartic equation admits in general\nfour solutions:\nt=±/radicalBigg\n1±√\n1−4R2\n4. (10)\nWe see, therefore, that the space of an equivalence class on S3is already quite nontrivial.\nTherearethreedistinct cases: (i) R= 0(so that ˆ ̺is‘fullymixed’, i.e. proportionaltothe\nidentity); (ii) R=1\n2(so that ˆ̺is ‘pure’); and (iii) 0 < R <1\n2(so that ˆ̺is ‘generic’). In case\n(i), we have either t=±1/√\n2 ort= 0 (double root). For t=±1/√\n2 we have an antipodal\npair onS3, whose coordinatesare ( t,x,y,z) = (1√2,0,0,0)and(−1√2,0,0,0), that give rise to\nthesamedensity matrix; for t= 0wehave an S2degreesoffreedomgivenby x2+y2+z2=1\n2;\nall the points on this S2determine the density matrix proportional to the identity. In case\n(ii), we have t=±1\n2, and hence ( t,x,y,z) =±(1\n2,b,c,a). The totality of pure states on\nS3is then given by a pair of two-spheres S2associated with t=1\n2,x2+y2+z2=1\n4and\nt=−1\n2,x2+y2+z2=1\n4. In case (iii), the generic case, we have four distinct values for t\ngiven by (10). That is, two antipodal pairs on S3give rise to the same density matrix (8).\nTherefore, the space of mixed-state density matrix is given by the quotient of S3by the\nidentification of two antipodal pairs. These configurations are sch ematically illustrated in\nfigure 1.\nThe utility of information geometry is not merely confined to the stud y of the structure of\nquantum state spaces; information geometry plays a crucial role in the theory of statistical\nestimation. In the quantum context, typically one has a system cha racterised by a density\nmatrix ˆρ(θ) that depends on a family of unknown parameters {θi}. To determine the state\nof the system one constructs a family of observables {ˆΘi}, corresponding to the estimators of\n{θi}; by measuring these observables one estimates the unknown para meters. The degrees\nof error in these estimates are characterised by the covariance m atrix tr[( ˆΘi−θi)(ˆΘj−\nθj)ˆρ(θ)], where for simplicity we have assumed that the estimators {ˆΘi}are unbiased so\nthat tr(ˆΘiˆρ(θ)) =θi. The lower bound of the covariance matrix is determined by the inver se\nof the quantum Fisher-Rao metric gij= 4tr(∂iˆξ(θ)∂jˆξ(θ)). In many applications the trace\ncan be computed explicitly to determine the lower bounds for the est imates.5\nfully \nmixed t\npure \nstates \nS2S3\nFIG. 1: (colour online) Manifold of 2×2density matrices . Thesphere S3ofradius1 /√2constitutes\nthe covering space for the space of 2 ×2 density matrices. If we regard the t-axis as defining the\n‘poles’, then the density matrix ˆ ̺=1\n21corresponds to the ‘north’ and the ‘south’ poles, as well as\nthe equator. Moving down from the north pole, at t=±1\n2we have the latitudinal ‘circles’ S2given\nbyx2+y2+z2=1\n4, corresponding to the pure-state boundary. Crossing this b oundary by further\nreducing the value of twe see a ‘mirror image’, where starting from pure state one ap proaches the\nfully-mixed state ˆ ̺=1\n21ast→0. The northern hemisphere of the S3is thus partitioned into two\nparts; the northern cap t≥1\n2and the latitudinal band 0 ≤t <1\n2. Each point is identified with an\nantipodal point in the southern hemisphere, but in addition the points above and below t=1\n2are\nalso identified in pairs; and finally, the equator is identifie d as a single point, also to be identified\nwith the two poles.\nAs a concrete example, let us work out the variance lower bound ass ociated with the\nestimation of unitary (e.g., time) parameter. The problem is as follows . We have a system\nprepared in an initial state ˆξ0=ˆξ(0), which evolves under the Hamiltonian ˆHso that\nˆξt= exp(−iˆHt)ˆξ0exp(iˆHt). (11)\nThe task is to estimate how much time has elapsed since its initial prepa ration. Let ˆTbe an\nunbiased estimator for the time parameter tso that tr( ˆTˆξtˆξt) =t(strictly speaking, all we\nneedisalocallyunbiasedestimatorsatisfying ∂ttr(ˆTˆξtˆξt) = 1)[17]. Ifwewrite ˜T=ˆT−t1for\nthe mean-adjusted estimator, then the variance of ˆTis ∆T2= tr(˜T2ˆξtˆξt). By differentiating\nthe trace condition tr( ˆξtˆξt) = 1 intand writing ˆξ′\nt=∂tˆξt, we obtain tr( ˆξ′\ntˆξt) = 0. Hence by6\ndifferentiating tr( ˜Tξtξt) = 0 intand using ˜T′=−1we find\ntr[(˜Tˆξt+ˆξt˜T)ˆξ′\nt] = 1. (12)\nRecall that the matrix Schwarz inequality applied to a pair of Hermitian operators ˆXand\nˆYstates that [tr( ˆXˆY)]2≤tr(ˆX2)tr(ˆY2). Squaring (12) and substituting ˆX=˜Tˆξt+ˆξt˜T\nandˆY=ˆξ′\ntin the Schwarz inequality, we deduce a version of the quantum Cram´ er-Rao\ninequality\n∆T2+δT2≥1\n2tr(ˆξ′\ntˆξ′\nt), (13)\nwhereδT2= tr(ˆTˆξtˆTˆξt)−t2, and where the denominator on the right side is half the\nFisher-Rao metric. By use of the unitary evolution ˆξ′\nt=−i(ˆHˆξt−ˆξtˆH) we find that\ntr(ˆξ′\ntˆξ′\nt) = 2/bracketleftBig\ntr(ˆH2ˆξ2\nt)−tr(ˆHˆξtˆHˆξt)/bracketrightBig\n. (14)\nThe right side of (14) is twice the skew information introduced by Wign er and Yanase [9].\nIt is interesting that Fisher introduced the left side of (14), in the classical context, as\nthe amount of extractable information concerning the value of t[18], whereas Wigner and\nYanase introduced the right side of (14), in the quantum context, as an a priorimeasure of\ninformation contained in the state concerning “not easily measured quantities” (such as ˆT)\n[9]. In special cases for which ˆξtis a pure state satisfying ˆξ2\nt=ˆξt, the skew information is\nmaximised and (14) coincides with twice the energy variance 2∆ H2, whileδT2= 0, and we\nrecover the uncertainty relation for the unitary parameter estim ation as in [13]. For mixed\nstates, however, we obtain\n∆T2+δT2≥1\n4(∆H2−δH2). (15)\nWe note that for an arbitrary observable ˆHthe Wigner-Yanase skew information Iρis\ndefined by the expression\nIρ(H) = tr(ˆH2ˆρ)−tr(ˆH/radicalbig\nˆρˆH/radicalbig\nˆρ), (16)\nwhereas the quantity δHintroduced above, which might appropriately be called the skew\ninformation of the second kind, is given by\nδH2= tr(ˆH/radicalbig\nˆρˆH/radicalbig\nˆρ)−/bracketleftBig\ntr(ˆHˆρ)/bracketrightBig2\n. (17)\nIt follows that the total variance takes the form ∆ H2=Iρ(H)+δH2, which can be inter-\npreted as the quantum analogue of the conditional variance formu la (that the total variance\nis the sum of the expectation of the conditional variance and the va riance of the conditional\nexpectation). In particular, it is easily shown that ∆ T2≥δT2≥0. By taking the upper\nbound for δT2, one obtains ∆ T2≥1/4tr(ˆξ′\ntˆξ′\nt), which is the uncertainty relation based on\nthe skew information obtained by Luo [14] (see also [19]). This bound is not tight because\nthe Luo relation gives ∆ T2∆H2≥1/8 for pure states, whereas (13) gives ∆ T2∆H2≥1/4.7\nIf we combine (15) with its dual inequality ∆ H2+δH2≥1/4(∆T2−δT2), we obtain a less\ntight but more symmetric expression\n∆T2∆H2≥1\n4+δT2δH2, (18)\nwhich implicitly appears in [20].\nBy exploiting the utility of information geometry, we can go beyond th e lowest-order\nterm in uncertainty relations. To this end, we remark that the Fishe r-Rao metric that\ndetermines the uncertainty lower bound is just the squared ‘velocit y’ of the evolution of\nquantum states. In the case of a pure state, the fact that squa red velocity of the state\nevolution is given by the energy uncertainty is known as the Anandan -Aharonov relation\n[21]. The observation of Luo can be paraphrased by saying that this velocity, in the case of\na generic mixed state, is given by the skew information. In addition to the squared velocity,\nwe consider the squared ‘acceleration’ ˆ αt=ˆξ′′\nt−tr(ˆξ′′\ntˆξt)ˆξt, which determines the curvature\nγ2= 16tr(ˆαtˆαt)/tr(ˆξ′\ntˆξ′\nt) of the curve in the space of square-root density matrices. The\nsquared acceleration determines the next-order correction to t he uncertainty relation. More\ngenerally, we can obtain arbitrarily many higher-order corrections by considering higher-\norder derivative terms, thus generalising the quantum Bhattacha rrya bounds derived in\n[13, 22, 23] for pure states to general density matrices.\nTo illustrate how higher-order corrections to uncertainty relation s can be obtained sys-\ntematically, let us examine the second-order and the third-order c orrections here. For this\npurpose, it will be convenient to relax the unit trace condition trˆ ρ= 1 but merely require\ntrˆρ∝ne}ationslash= 0 andtrˆ ρ <∞, andimpose unit trace attheend ofthe calculation. Wethen define t he\nsymmetric function t(ˆξ) = tr(ˆξˆTˆξ)/tr(ˆξˆξ) ofˆξand consider the gradient ∇tof this function\nin the space of square-root density matrices, evaluated at tr( ˆξˆξ) = 1. A short calculation\nthen shows that the squared magnitude (in the Hilbert-Schmidt nor m) of the gradient is\n|∇t|2= 2(∆T2+δT2). On the other hand, the squared magnitude of the gradient is clea rly\nlarger than (or equal to) the squared magnitude of its component in the direction of ˆξ′\nt, and\nthus ∆T2+δT2≥1/2tr(ˆξ′\ntˆξ′\nt), where we have made use of the relation i[ ˆH,ˆT] = 1. This\nprovides an alternative derivation of the quantum Cram´ er-Rao ine quality.\nThenextordercorrectiontotheuncertaintylowerboundformixe dstatesisobtainedfrom\nthe component of ∇tin the direction of ˆ αt, given by tr(( ˆTˆξt+ˆξtˆT)ˆαt)/tr(ˆαtˆαt). Hence if the\ncurvature γof the curve ˆξtis large, the correction is small. The numerator, however, in gen-\neraldepends on ˆT. Toseekaboundthatisindependent of ˆTwethusconsider thethird-order\nterm, i.e. the component of the gradient in the direction of ˆβt=ˆξ′′′\nt−(tr(ˆξ′′′\ntˆξ′\nt)/tr(ˆξ′\ntˆξ′\nt))ˆξ′\nt.\nThis is given by tr(( ˆTˆξt+ˆξtˆT)ˆβt)/tr(ˆβtˆβt). Although the calculation of this term is straight-\nforward, the resulting expression is lengthy and will be omitted here . It suffices to note that\neach of the odd-order corrections involves commutators betwee nˆTandˆHkfor some k, and\nthus is independent of ˆT; whereas all even-order corrections are in general dependent o n\nˆT. Furthermore, terms appearing in the corrections have specific s tatistical interpretations\nin terms of the quantum extensions of central moments of the Ham iltonian. In the pure\nstate limit where ˆξ2\nt=ˆξt, these expressions reduce to the “classical” central moments, in a\nmanner analogous to the reduction of the Wigner-Yanase skew info rmation to the second\ncentral moment ∆ H2. We are thus able to obtain various higher-order generalisations of\nthe skew information, which appear to be entirely new quantities in th e study of statistical\nproperties of mixed states in quantum mechanics. These quantities might appropriately be\ncalledquantum skew moments .8\nI thank E. J. Brody, E. M. Graefe, L. P. Hughston, M. F. Parry, a nd the participants of\nthe third international conference on Information Geometry and Its Applications, Leipzig\n2010, for stimulating discussions.\n[1] H¨ ubner, M. “Explicit computation of the Bures distance for density matrices” Phys. Lett.\nA163, 239 (1992).\n[2] Petz, D. and Sud´ ar, C. “Geometry of quantum states” J. Math. Phys. 37, 2662 (1996).\n[3] Uhlmann, A. “Spheres and hemispheres as quantum state sp aces”J. Geom. Phys. 18, 76\n(1996).\n[4] Bengtsson, I. and Zyczkowski, K. Geometry of Quantum States: An Introduction to Quantum\nEntanglement (Cambridge: Cambridge University Press 2007).\n[5] Hansen, F. “Metric adjusted skew information” Proc. Nat. Acad. Sci. 105, 9909 (2008).\n[6] Rao, C. R. “Information and the accuracy attainable in th e estimation of statistical parame-\nters”Bull. Calcutta Math. Soc. 37, 81 (1945).\n[7] Amari, S. and Nagaoka, H. Methods of Information Geometry (Oxford: Oxford University\nPress 2000).\n[8] Brody, D. C. and Hook, D. W. “Information geometry in vapo ur-liquid equilibrium” J. Phys.\nA42, 023001 (2009).\n[9] Wigner, E. P. and Yanase, M. M. “Information contents of d istributions” Proc. Nat. Acad.\nSci.49, 910 (1963).\n[10] Braunstein, S. L. and Caves, C. M. “Statistical distanc e and the geometry of quantum states”\nPhys. Rev. Lett. 72, 3439 (1994).\n[11] Fujiwara, A. and Nagaoka, H. “Quantum Fisher metric and estimation for pure state models”\nPhys. Lett. A201, 119 (1995).\n[12] Caianiello, E. R. and Guz, W. “Quantum Fisher metric and uncertainty relations” Phys. Lett.\nA126, 223 (1988).\n[13] Brody, D. C. and Hughston, L. H. “Geometry of quantum sta tistical inference” Phys. Rev.\nLett.77, 2851 (1996).\n[14] Luo, S. “Wigner-Yanase skew information and uncertain ty relations” Phys. Rev. Lett. 91,\n180403 (2003).\n[15] Luo, S. “Logarithm versus square root: Comparing quant um Fisher information” Commun.\nTheo. Phys. 47, 597 (2007).\n[16] Gibbons, G. W. “Typical states and density matrices” J. Geom. Phys. 8, 147 (1992).\n[17] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland Pub-\nlishing Company, Amsterdam, 1982).\n[18] Fisher, R. A. “Theory of statistical estimation” Proc. Camb. Phil. Soc. 22, 700 (1925).\n[19] Gibilisco, P., Hiai, F. and Petz, D. Covariance, quantu m Fisher information, and the uncer-\ntainty relations IEEE Trans. Inf. Theory 55, 439 (2009).\n[20] Luo, S. “Heisenberg uncertainty relation for mixed sta tes”Phys. Rev. A72, 042110 (2005).\n[21] Anandan, J. and Aharonov, Y. “Geometry of quantum evolu tion”Phys. Rev. Lett. 65, 1697\n(1990).\n[22] Brody, D. C. and Hughston, L. H. “Generalised Heisenber g relations for quantum statistical\nestimation” Phys. Lett. A236, 257 (1997).\n[23] Brody, D. C. and Hughston, L. H. “Statistical geometry i n quantum mechanics” Proc. R. Soc.9\nLondonA454, 2445 (1998)."    },    {        "title": "1010.0584v1.Wigner_function_evolution_in_self_Kerr_Medium_derived_by_Entangled_state_representation.pdf",        "content": "arXiv:1010.0584v1  [quant-ph]  25 Sep 2010Wigner function evolution in self-Kerr Medium derived by En tangled state\nrepresentation\nLi-yun Hu, Zheng-lu Duan, Xue-xiang Xu, and Zi-sheng Wang\nCollege of Physics & Communication Electronics,\nJiangxi Normal University, Nanchang 330022, China\n(Dated:)\nBy introducing the thermo entangled state representation, we convert the calculation of Wigner\nfunction (WF) of density operator to an overlap between ”two pure” states in a two-mode enlarged\nFock space. Furthermore, we derive a new WF evolution formul a of any initial state in self-Kerr\nMedium with photon loss and find that the photon number distri bution for any initial state is\nindependent of the coupling factor with Kerr Medium, where t he number state is not affected by\nthe Kerr nonlinearity and evolves into a density operator of binomial distribution.\nKeywords: Wigner function, Kerr Medium, entangled state representat ion\nPACS numbers:\nI. INTRODUCTION\nNonclassicality of optical fields has been a topic of\ngreat interest in quantum optics and quantum informa-\ntionprocessing[1], whichisusuallyassociatedwithquan-\ntum interference and entanglement. The phase space\nWigner function (WF) [2, 3] of quantum states of light\nis a powerful tool for investigating such nonclassical ef-\nfects. The WF was first introduced by Wigner in 1932 to\ncalculate quantum corrections to a classical distribution\nfunction of a quantum-mechanical system. The partial\nnegativity of the WF is indeed a good indication of the\nhighly nonclassical character of the state [4] and moni-\ntors a decoherence process of a quantum state, e. g. the\nexcited coherent state in both photon-loss and thermal\nchannels [5, 6], the single-photon subtracted squeezed\nvacuum state in both amplitude decay and phase damp-\ning channels [7], and so on [8–12].\nNonlinear interaction of light in a medium provides a\nveryuseful frameworkto study variousnonclassicalprop-\nerties of quantum states of radiation. The Kerr medium\nis one of the simplest nonlinearity, which shall allow us\nto investigatethe full time-dependent WF dynamics with\nor without a quantum noise. Recently, a Fokker-Planck\nequation for the WF evolution in a noisy Kerr medium\n(χ(3)nonlinearity) is presented [13]. Then the authors\nnumerically solved this equation assuming coherent state\nas an initial condition and discussed its dissipation ef-\nfects. However, for any initial condition, as far as we\nare concerned, there is no report about the WF evolu-\ntion. On the other hand, by using the thermo entan-\ngled state representation (TESR) we solved various mas-\nter equations to obtain density operators with an infinite\noperator-sum representation [14] and then revealed that\nthe WF of density operator can be expressed as an over-\nlap between two pure states (see Eq. (20) below) [15].\nThis brings much convenience to calculate time evolu-\ntion of WFs when quantum decoherence happens. Thus\nthe TESR is beneficial to quantum decoherence theory.\nIn this paper, we shall appeal the TESR |η/an}bracketri}htto treat\nthe WF evolution at any initial condition in self-KerrMedium with photon loss and present a new formula to\ncalculate time evolution of the WF for quantum deco-\nherence. In addition, based on the derived WF evolution\nformula, we shall deduce the photon number distribu-\ntion for any initial state in presence of Kerr interaction,\nwhere the photon number distribution is independent of\nthe coupling factor χthat is relativeto the Kerrmedium.\nAs examples, the WF formula is applied to the cases\nof initial coherent state, and number state, respectively.\nConclusions are involved in the last section.\nII. BRIEF REVIEW OF THERMO ENTANGLED\nSTATE REPRESENTATION\nWe begin with briefly reviewing the thermo entangled\nstate representation (TESR). On the basis of Umezawa-\nTakahash thermo field dynamics (TFD) [16–18] we con-\nstructed the TESR in the doubled Fock space [19–21],\n|η/an}bracketri}ht=D(η)|η= 0/an}bracketri}ht\n= exp/bracketleftbigg\n−1\n2|η|2+ηa†−η∗˜a†+a†˜a†/bracketrightbigg/vextendsingle/vextendsingle0,˜0/angbracketrightbig\n,(1)\nwhereD(η) =eηa†−η∗ais the displacement operator, ˜ a†\nis a fictitious mode accompanying the real photon cre-\nation operator a†,/vextendsingle/vextendsingle0,˜0/angbracketrightbig\n=|0/an}bracketri}ht/vextendsingle/vextendsingle˜0/angbracketrightbig\n,and/vextendsingle/vextendsingle˜0/angbracketrightbig\nis annihilated\nby ˜awith the relations/bracketleftbig\n˜a,˜a†/bracketrightbig\n= 1 and/bracketleftbig\na,˜a†/bracketrightbig\n= 0. The\nstructure of |η/an}bracketri}htis similar to that of the EPR eigenstate\nshown in Ref. [19]. Operating aand ˜aon|η/an}bracketri}htin Eq.(1)\nwe can obtain the eigen-equations of |η/an}bracketri}ht,\n(a−˜a†)|η/an}bracketri}ht=η|η/an}bracketri}ht,(a†−˜a)|η/an}bracketri}ht=η∗|η/an}bracketri}ht,\n/an}bracketle{tη|(a†−˜a) =η∗/an}bracketle{tη|,/an}bracketle{tη|(a−˜a†) =η/an}bracketle{tη|.(2)\nNote that/bracketleftbig\n(a−˜a†),(a†−˜a)/bracketrightbig\n= 0,thus|η/an}bracketri}htis the\ncommon eigenvector of ( a−˜a†) and (˜a−a†).Us-\ning the normally ordered form of vacuum projector/vextendsingle/vextendsingle0,˜0/angbracketrightbig/angbracketleftbig\n0,˜0/vextendsingle/vextendsingle=: exp/parenleftbig\n−a†a−˜a†˜a/parenrightbig\n: and the technique of\nintegration within an ordered product (IWOP) of opera-\ntors [22–24], we can easily prove that |η/an}bracketri}htis complete and2\northonormal,\n/integraldisplayd2η\nπ|η/an}bracketri}ht/an}bracketle{tη|= 1,/an}bracketle{tη′|η/an}bracketri}ht=πδ(η′−η)δ(η′∗−η∗).\n(3)\nIt is easily seen that |η= 0/an}bracketri}hthas the properties\n|η= 0/an}bracketri}ht=ea†˜a†/vextendsingle/vextendsingle0,˜0/angbracketrightbig\n=∞/summationdisplay\nn=0|n,˜n/an}bracketri}ht,(4)\n(wheren= ˜n, and ˜ndenotes the number in the fictitious\nHilbert space) and\na|η= 0/an}bracketri}ht= ˜a†|η= 0/an}bracketri}ht,\na†|η= 0/an}bracketri}ht= ˜a|η= 0/an}bracketri}ht, (5)\n/parenleftbig\na†a/parenrightbign|η= 0/an}bracketri}ht=/parenleftbig\n˜a†˜a/parenrightbign|η= 0/an}bracketri}ht.\nNote that density operators ρ(a†,a) are defined in the\nreal space which are commutative with operators (˜ a†,˜a)\nin the tilde space.\nIn a similar way, we can introduce the state vector |ξ/an}bracketri}ht\nconjugated to |η/an}bracketri}ht, defined as\n|ξ/an}bracketri}ht=D(ξ)e−a†˜a†/vextendsingle/vextendsingle0,˜0/angbracketrightbig\n= exp/parenleftbigg\n−1\n2|ξ|2+ξa†+ξ∗˜a†−a†˜a†/parenrightbigg/vextendsingle/vextendsingle0,˜0/angbracketrightbig\n= (−1)a†a|η=−ξ/an}bracketri}ht, (6)\nwhich also possesses orthonormal and complete proper-\nties\n/integraldisplayd2ξ\nπ|ξ/an}bracketri}ht/an}bracketle{tξ|= 1,/an}bracketle{tξ′|ξ/an}bracketri}ht=πδ(ξ′−ξ)δ(ξ′∗−ξ∗).(7)\nIII. MASTER EQUATION FOR A SELF-KERR\nINTERACTION\nIn the Markov approximation and interaction picture\nthe master equation for a dissipative cavity with Kerr\nmedium has the form [25, 26]\ndρ\ndt=−iχ/bracketleftBig/parenleftbig\na†a/parenrightbig2,ρ/bracketrightBig\n+γ/parenleftbig\n2aρa†−a†aρ−ρa†a/parenrightbig\n,(8)\nwhereγis decaying parameter of the dissipative cavity,\nχis coupling factor depending on the Kerr medium. Mil-\nburn and Holmes [27] solved this equation by changing\nit to a partial differential equation for the Q-function\nand for an initial coherent state. Here we will solve the\nmaster equation by virtue of the entangled state repre-\nsentation and present the infinite sum representation of\ndensity operator.\nOperatingthebothsidesofEq.(8)onthestate |η= 0/an}bracketri}ht,\nletting|ρ/an}bracketri}ht=ρ|η= 0/an}bracketri}ht(Here one should understand the\nsingle-mode density operator ρin the left of Eq.(8) as the\ndirect product ρ⊗˜Iwhenρacts onto the two-mode state|η= 0/an}bracketri}ht=ea†˜a†/vextendsingle/vextendsingle0,˜0/angbracketrightbig\n, where˜Iis the identity operator in\nthe auxiliary mode.), and using Eq.(5) we can convert\nthe master equation in Eq. (8) into the following form,\nd\ndt|ρ/an}bracketri}ht=/braceleftBig\n−iχ/bracketleftBig/parenleftbig\na†a/parenrightbig2−/parenleftbig\n˜a†˜a/parenrightbig2/bracketrightBig\n+γ/parenleftbig\n2a˜a−a†a−˜a†˜a/parenrightbig/bracerightbig\n|ρ/an}bracketri}ht,(9)\ni.e., an evolution equation of state vector |ρ/an}bracketri}ht. Its solution\nis then of the form\n|ρ/an}bracketri}ht=e−iχt/bracketleftBig\n(a†a)2−(˜a†˜a)2/bracketrightBig\n+γt(2a˜a−a†a−˜a†˜a)|ρ0/an}bracketri}ht,(10)\nwhere|ρ0/an}bracketri}ht=ρ0|η= 0/an}bracketri}ht, ρ0is an initial density operator.\nThe advantage of using thermo field notation over more\ntraditionalalgebraicmanipulationwith superoperatorsis\nthat in many situations (and, particularly, ones of our in-\nterest) it enables to simplify, make more illustrative and\nless cumbersome finding the solution (10) and estima-\ntion of time-dependent matrix elements. In particular,\nit allows to represent in a simple form a factorization of\nthe superoperator exp {···}into multipliers with easily\nestimated actions on the number states [28].\nBy introducing the following operators,\nK0=a†a−˜a†˜a, Kz=a†a+˜a†˜a+1\n2, K−=a˜a,(11)\nwhich satisfy [ K0,Kz] = [K0,K−] = 0,we can rewrite\nEq.(10) as\n|ρ/an}bracketri}ht=e{−iχt[K0(2Kz−1)]+γt(2K−−2Kz+1)}|ρ0/an}bracketri}ht\n= exp[iχtK0+γt]\n×exp/braceleftbigg\n−2t(γ+iχK0)/bracketleftbigg\nKz+−γ\nγ+iχK0K−/bracketrightbigg/bracerightbigg\n|ρ0/an}bracketri}ht.\n(12)\nWith the aid of the operator identity [30]\neλ(A+σB)=eλAexp/bracketleftbig\nσB/parenleftbig\n1−e−λτ/parenrightbig\n/τ/bracketrightbig\n= exp/bracketleftbig\nσB/parenleftbig\neλτ−1/parenrightbig\n/τ/bracketrightbig\neλA,(13)\nwhich is valid for [ A,B] =τB,and noticing [ Kz,K−] =\n−K−,we can reform Eq.(12) as\n|ρ/an}bracketri}ht= exp[iχtK0+γt]exp[Γ zKz]exp[Γ −K−]|ρ0/an}bracketri}ht,\n(14)\nwhere\nΓz=−2t(γ+iχK0),Γ−=γ(1−e−2t(γ+iχK0))\nγ+iχK0.(15)\nFrom Eq.(14) we can obtain the infinite operator-sum\nform ofρ(t), (see Appendix A)\nρ(t) =∞/summationdisplay\nm,n,l=0Mm,n,lρ0M†\nm,n,l, (16)3\nwhere the two operators Mm,n,landM†\nm,n,lare respec-\ntively defined as\nMm,n,l≡/radicalBigg\nΛlm,n\nl!e−iχtm2−γtm|m/an}bracketri}ht/an}bracketle{tm|al,\nM†\nm,n,l≡\n\n/radicalBigg\nΛln,m\nl!e−iχtn2−γtn|n/an}bracketri}ht/an}bracketle{tn|al\n\n†\n.(17)\nAlthough Mm,n,lis not hermite conjugate to M†\nm,n,l, the\nnormalization still holds, (/summationtext∞\nm,n,l=0M†\nm,n,lMm,n,l= 1,\nsee Appendix B [29]) i.e., they are trace-preserving in a\ngeneral sense, so Mm,n,landM†\nm,n,lmay be named the\ngeneralized Kraus operators.\nIV. EVOLUTION OF WIGNER FUNCTION\nFOR SELF-KERR CHANNEL\nInthissection, weconsiderWignerfunction’stimeevo-\nlution in the self-Kerrmedium channel. For this purpose,\nweshallderiveanewexpressionofWignerfunctioninthe\nTESR. According to the definition of Wigner function of\ndensity operator ρ,\nW(α,α∗) = Tr[∆( α,α∗)ρ], (18)\nwhere ∆( α,α∗) is the single-mode Wigner operator [2,\n30], whose explicit normally ordered form is [31]\n∆(α,α∗) =1\nπ:e−2(a†−α∗)(a−α): =1\nπD(2α)(−1)a†a.\n(19)\nBy using /an}bracketle{t˜n|˜m/an}bracketri}ht=δn,m(n= ˜n,m= ˜m) and noticing (6)\nas well as |ρ/an}bracketri}ht=ρ|η= 0/an}bracketri}ht,we can reform Eq.(18) as\nW(α,α∗) =∞/summationdisplay\nm,n/an}bracketle{tn,˜n|∆(α,α∗)ρ|m,˜m/an}bracketri}ht\n=1\nπ/an}bracketle{tη= 0|D(2α)(−1)a†a|ρ/an}bracketri}ht\n=1\nπ/an}bracketle{tη=−2α|(−1)a†a|ρ/an}bracketri}ht\n=1\nπ/an}bracketle{tξ= 2α|ρ/an}bracketri}ht, (20)\nwhere Eq. (20) is the Wigner function formula in thermo\nentangled state representation, with which the Wigner\nfunction of density operator is simplified as an overlap\nbetween two “pure states” in enlarged Fock space, rather\nthan using ensemble average in the system-mode space.\nThis will brings much convenience to calculate the time\nevolution of Wigner functions when quantum decoher-\nence happens.\nProjecting (14) on the entangled state representation\n1\nπ/an}bracketle{tξ=2α|,and inserting the completeness relation (7), we\nfind\nW(α,α∗,t) = 4/integraldisplayd2β\nπG(α,β,t)W(β,β∗,0),(21)whereW(α,α∗,t) andW(β,β∗,0) are the Wigner func-\ntions at the evolving time tand initial time, respectively,\nand\nG(α,β,t) =/an}bracketle{tξ=2α|exp[iχtK0+γt]\n×exp[ΓzKz]exp[Γ −K−]/vextendsingle/vextendsingleξ′\n=2β/angbracketrightbig\n.(22)\nIt is convenient to the matrix element in (22) according\ntothe two-modeFockspace. Thusthe /an}bracketle{tξ=2α|isexpanded\nas\n/an}bracketle{tξ|=/angbracketleftbig\n0,˜0/vextendsingle/vextendsingle∞/summationdisplay\nm,n=0am˜an\nm!n!Hm,n(ξ∗,ξ)e−|ξ|2/2.(23)\nBy using the two-mode Fock state |m,˜n/an}bracketri}ht=\na†m˜a†n/√\nm!n!/vextendsingle/vextendsingle0,˜0/angbracketrightbig\n, we get\n/an}bracketle{tξ|m,˜n/an}bracketri}ht=Hm,n(ξ∗,ξ)e−|ξ|2/2/√\nm!n!,(24)\nwhereHm,n(ξ∗,ξ) is the two-variable Hermite poly-\nnomials [32, 33]. Inserting the complete relation/summationtext∞\nm,n=0|m,˜n/an}bracketri}ht/an}bracketle{tm,˜n|= 1,after a long but straight calcu-\nlation, then the Wigner function’s evolution is given by\n(see Appendix C)\nW(α,α∗,t) =∞/summationdisplay\nm,n=0Cm,n(α,α∗,t)Em,n,(25)\nwhere Λ m,n≡γ(1−e−2t(γ+iχ(m−n)))\nγ+iχ(m−n),\nCm,n(α,α∗,t)\n≡e−iχt(m2−n2)−γt(m+n)e−2|α|2\nm!n!(Λm,n+1)(m+n+2)/2Hm,n(2α∗,2α),(26)\nand\nEm,n= 4/integraldisplayd2β\nπW(β,β∗,0)e2(Λm,n���1)\nΛm,n+1|β|2\n×Hm,n/parenleftBigg\n2β/radicalbig\nΛm,n+1,2β∗\n/radicalbig\nΛm,n+1/parenrightBigg\n.(27)\nIt is obvious that, when χ= 0,the case of photon loss,\nΛm,n→(1−e−2γt) =Tand Eq.(25) just does reduce to\n(see Appendix E)\nW(α,α∗,t) =2\nT/integraldisplayd2β\nπe−2\nT|α−βe−γt|2\nW(β,β∗,0),\n(28)\nwhich is just the evolving formula of Wigner function for\namplitude-damping channel. While for γ= 0,without\nphoton-loss, Eq.(25) reduces to\nW(α,α∗,t)\n=∞/summationdisplay\nm,n=0exp/bracketleftbig\n−iχt/parenleftbig\nm2−n2/parenrightbig/bracketrightbig\nm!n!e2|α|2Hm,n(2α∗,2α)\n×4/integraldisplayd2β\nπe−2|β|2Hm,n(2β,2β∗)W(β,β∗,0).(29)4\nV. PHOTON NUMBER DISTRIBUTION IN\nPRESENCE OF KERR INTERACTION\nNow we consider photon number (PN) distribution in\npresence of Kerr medium. According to the TFD, we can\nreform the PN p(n) =tr[ρ|n/an}bracketri}ht/an}bracketle{tn|] as\np(n) =/an}bracketle{tn|ρ|n/an}bracketri}ht=∞/summationdisplay\nm=0/an}bracketle{tn,˜n|ρ|m,˜m/an}bracketri}ht\n=/an}bracketle{tn,˜n|ρ|η= 0/an}bracketri}ht=/an}bracketle{tn,˜n|ρ/an}bracketri}ht,(30)\nwhich is converted to the matrix element /an}bracketle{tn,˜n|ρ/an}bracketri}htin the\ncontext of thermo dynamics. Then using the complete-\nness of/an}bracketle{tξ|and Eq.(30) as well as Eq.(20), we have\np(n) =/integraldisplayd2ξ\nπ/an}bracketle{tn,˜n|ξ/an}bracketri}ht/an}bracketle{tξ|ρ/an}bracketri}ht\n=/integraldisplay\nd2ξ/an}bracketle{tn,˜n|ξ/an}bracketri}htW(α=ξ/2,α∗=ξ∗/2)\n= 4π/integraldisplay\nd2αW|n/an}bracketri}ht/an}bracketle{tn|(α,α∗)W(α,α∗),(31)\nwhereW|n/an}bracketri}ht/an}bracketle{tn|(α,α∗) =(−1)n\nπe−2|α|2Ln(4|α|2) is the\nWigner function of number state |n/an}bracketri}ht/an}bracketle{tn|as shown in\n[34, 35]. Thus one can calculate the PN by combining\nEqs.(25) and (31).\nNext we evaluate the PN for the above decoherence\nmodel in Eq.(8). Substituting Eq.(25) into Eq.(31), we\nhave\np(s) = 4π∞/summationdisplay\nm,n=0/integraldisplay\nd2αW|s/an}bracketri}ht/an}bracketle{ts|(α,α∗)Cm,n(α,α∗,t)Em,n\n=∞/summationdisplay\nm,n=04πe−iχt(m2−n2)−γt(m+n)\nm!n!(Λm,n+1)(m+n+2)/2Em,n·Fm,n,\n(32)\nwhere\nFm,n≡/integraldisplay\nd2αe−2|α|2W|s/an}bracketri}ht/an}bracketle{ts|(α,α∗)Hm,n(2α∗,2α)\n=s!\n4δm,sδn,s. (33)\nThen substituting Eqs.(33) and (27) into (32) yields\np(s) =4(−1)se2γt\n(2e2γt−1)s+1/integraldisplay\nd2βexp/braceleftBigg\n−2|β|2\n2e2γt−1/bracerightBigg\n×Ls/parenleftBigg\n4e2γt|β|2\n2e2γt−1/parenrightBigg\nW(β,β∗,0),(34)\nwhich corresponds to the photon number of density op-\nerator in the amplitude-damping quantum channel [15].\nFrom Eq.(34), it is easily to see that, for any initial state,\nthe photon number distribution p(s) is independent of\nthe coupling factor χthat is relativeto the Kerrmedium,\nas respected in [36].VI. EVOLUTION OF QUANTUM STATES\nThe phase space Wigner distribution function descrip-\ntion of quantum states of light is a powerful tool to inves-\ntigate nonclassical effects, such as quantum interference\nand entanglement. In this section, as the applications of\nWF evolution formula, we take two special initial states\nas examples.\n(1) When the initial state is the coherent state |z/an}bracketri}ht,\nwhose WF is given W(β,β∗) =1\nπe−2|β−z|2,thus substi-\ntuting into Eq.(27) yields (see Appendix G)\nEm,n=1\nπ(Λm,n+1)m+n+2\n2e(Λm,n−1)|z|2zmz∗n,(35)\nso\nW(α,α∗,t)\n=e−2|α|2\nπ∞/summationdisplay\nm,n=0zmz∗n\nm!n!e−iχt(m2−n2)−γt(m+n)\n×e(Λm,n−1)|z|2Hm,n(2α∗,2α), (36)\nwhich is a new expression of the evolution of WF for\nany initial state. In particular, when γ= 0,without the\ndissipation, Eq.(36) reduces to\nW(α,α∗,t)\n=e−|z|2\nπe2|α|2∞/summationdisplay\nm,n=0zmz∗n\nm!n!e−iχt(m2−n2)Hm,n(2α∗,2α),(37)\nwhich is identical to Eq.(7) in Ref.[13], where Eq.(7) is\nused to make the numerical calculation since it is much\nmorerapidthan the otherexpressionEq.(6). Inaddition,\nfrom Eq.(36) one can see that the WF can be obtained\nvery quickly when the dissipation cannot be negligible.\nFurther when χt= 2π, Eq.(37) just returns to the WF\nof the initial coherent state.\nFig.1 presents the plots of the WF for different pa-\nrameters χtandα= 2.From Fig.1, one can see that\nthe WF turns into an ellipse and squeezing appears in\nan appropriate direction. Then the ellipse changes into\na banana shape and a tailor of the interference fringes\nappears where the distribution takes the negative values.\n(2) Another example is number state, where the WF\nof number state |s/an}bracketri}htis given by\nWs(β,β∗,0) =1/s!\nπe−2|β|2Hs,s(2β,2β∗)\n=(−1)s\nπe−2|β|2Ls(4|β|2),(38)\nsubstitutingitinto(27)andusingthegeneratingfunction\nof two-variable Hermite polynomials [see below Eq.(F3)],\nwe have\nEm,n=s!\nπΛs−m\nm,n(Λm,n+1)m+1\n(s−m)!δm,n,(39)5\nFIG. 1: (Color online) The WF for different parameters χt\nandα= 2,(a)χt= 0,(b)χt= 0.04,(c)χt= 0.06,(d)χt=\n0.08,(e)χt= 0.1,(f)χt= 0.2.\nthus the evolution of WF for |s/an}bracketri}htis\nWs(α,α∗,t)\n=s/summationdisplay\nm=0/parenleftbiggs\nm/parenrightbigg\ne−2mγt/parenleftbig\n1−e−2γt/parenrightbigs−mWm(α,α∗,0),(40)\nwhich is the WF ofnumber state in the photon-losschan-\nnel and indicates that the number state is not affected\nby the Kerr nonlinearity. In particular, when γ= 0,or\nt= 0,Eq.(40) just reduces to the WF of number state.\nFrom Eq.(40), on the other hand, it is found that the\nnumber state evolves into a density operator of binomial\ndistribution (a mixed state) if e−2γtis a binomial param-\neter.\nVII. CONCLUSIONS\nIn summary, by converting Wigner function for quan-\ntum state into an overlap between two ”pure states” in\na two-mode enlarged Fock space, we investigate the WF\nevolution of any initial condition in self-Kerr Medium\nwith photon loss and present a new formula for calculat-\ning time evolution of the WF for quantum decoherence.\nBased on the derived WF evolution formula, in addition,\nwe discuss the photon number distribution for any initial\nstate in presence of Kerr interaction. It is found that thephoton number distribution is independent of the cou-\npling factor χin correlation with the Kerr medium, as\nexpected by people. As applications, furthermore, the\ntwo cases of initial coherent state and number state are\nconsidered. It is shown that the coherent state can be\nsqueezed due to the presence of Kerr medium, while the\nnumber state is not affected by the Kerrnonlinearity and\nevolves into a density operator of binomial distribution\n(a mixed state) with e−2γtbeing a binomial parameter.\nACKNOWLEDGEMENTS Work supported by a\ngrant from the Key Programs Foundation of Ministry of\nEducationofChina(No. 210115)andtheResearchFoun-\ndation of the Education Department of Jiangxi Province\nof China (No. GJJ10097).\nAppendix A: Derivation of Eq.(16)\nIn order to obtain the infinite operator-sum form of\nρ(t) from Eq.(14), using the completeness relation of\nFock state in the enlarged space/summationtext∞\nm,n=0|m,˜n/an}bracketri}ht/an}bracketle{tm,˜n|=\n1 and noticing a†l|n/an}bracketri}ht=/radicalBig\n(l+n)!\nn!|n+l/an}bracketri}ht, we have\n|ρ/an}bracketri}ht=eiχtK0+γteΓzKzeΓ−K−|ρ0/an}bracketri}ht\n=eiχtK0+γteΓzKz∞/summationdisplay\nm,n=0|m,˜n/an}bracketri}ht/an}bracketle{tm,˜n|eΓ−K−|ρ0/an}bracketri}ht\n=∞/summationdisplay\nm,n=0e−iχt(m2−n2)−γt(m+n)\n×|m,˜n/an}bracketri}ht/an}bracketle{tm,˜n|eΛm,na˜a|ρ0/an}bracketri}ht, (A1)\nwhere\nΛm,n=γ(1−e−2t(γ+iχ(m−n)))\nγ+iχ(m−n).(A2)\nFurthermore, using the relations\n/an}bracketle{tn|η= 0/an}bracketri}ht=|˜n/an}bracketri}ht,|m,˜n/an}bracketri}ht=|m/an}bracketri}ht/an}bracketle{tn|η= 0/an}bracketri}ht,(A3)\nwe find\n/an}bracketle{tm,˜n|alρ0a†l|η= 0/an}bracketri}ht=/an}bracketle{tm|/an}bracketle{t˜n|alρ0a†l|η= 0/an}bracketri}ht\n=/an}bracketle{tm|alρ0a†l(/an}bracketle{t˜n|η= 0/an}bracketri}ht)\n=/an}bracketle{tm|alρ0a†l|n/an}bracketri}ht. (A4)\nThus Eq.(A1) becomes\n|ρ/an}bracketri}ht=∞/summationdisplay\nm,n,l=0Λl\nm,n\nl!e−iχt(m2−n2)−γt(m+n)\n×|m,˜n/an}bracketri}ht/an}bracketle{tm|alρ0a†l|n/an}bracketri}ht\n=∞/summationdisplay\nm,n,l=0/radicalbig\n(n+l)!(m+l)!√\nn!m!l!Λ−lm,n\n×e−iχt(m2−n2)−γt(m+n)|m,˜n/an}bracketri}htρ0,m+l,n+l,(A5)6\nwhereρ0,m+l,n+l≡ /an}bracketle{tm+l|ρ0|n+l/an}bracketri}ht.Using Eq.(A3)\nagain, we see\n|ρ/an}bracketri}ht=∞/summationdisplay\nm,n,l=0/radicalbig\n(n+l)!(m+l)!√\nn!m!l!Λ−lm,n\n×e−iχt(m2−n2)−γt(m+n)ρ0,m+l,n+l|m/an}bracketri}ht/an}bracketle{tn|η= 0/an}bracketri}ht.\n(A6)After depriving |η= 0/an}bracketri}htfrom the both sides of Eq.(A6),\nthe solution of master equation (8) appears as an infinite\noperator-sum form\nρ(t) =∞/summationdisplay\nm,n,l=0/radicalbigg\n(n+l)!(m+l)!\nn!m!Λl\nm,n\nl!e−iχt(m2−n2)−γt(m+n)|m/an}bracketri}ht/an}bracketle{tm+l|ρ0|n+l/an}bracketri}ht/an}bracketle{tn|\n=∞/summationdisplay\nm,n,l=0Λl\nm,n\nl!e−iχt(m2−n2)−γt(m+n)|m/an}bracketri}ht/an}bracketle{tm|alρ0a†l|n/an}bracketri}ht/an}bracketle{tn|. (A7)\nNote that the factor ( m−n) appears in the denominator\nof Λm,n(see Eq.(A2)), (this is originated from the non-\nlinear term of/parenleftbig\na†a/parenrightbig2) so that it is impossible to move all\nn−dependent terms to the right of alρ0a†l. Fortunately,\nwe can formally express Eq.(A7) as Eq.(16).\nAppendixB: Proof of normalization for the gen-\neralized Kraus operators\nUsing the operator identity\neλa†a=: exp/bracketleftbig/parenleftbig\neλ−1/parenrightbig\na†a/bracketrightbig\n: and the IWOP tech-\nnique, we can prove that\n∞/summationdisplay\nm,n,l=0M†\nm,n,lMm,n,l\n=∞/summationdisplay\nn,l=0(n+l)!\nn!(1−e−2tγ)l\nl!e−2nγt|n+l/an}bracketri}ht/an}bracketle{tn+l|\n=∞/summationdisplay\nn,l=0(1−e−2tγ)l\nl!a†le−2γta†a|n/an}bracketri}ht/an}bracketle{tn|al\n=∞/summationdisplay\nl=0(1−e−2tγ)l\nl!: exp/bracketleftbig/parenleftbig\ne−2γt−1/parenrightbig\na†a/bracketrightbig/parenleftbig\na†a/parenrightbigl: = 1,\n(B1)from which one can see that the normalization still holds,\ni.e., they are trace-preserving in a general sense, so\nMm,n,landM†\nm,n,lmay be named the generalized Kraus\noperators.\nAppendix C: Derivation of Eq.(25)\nUsing Eq.(24) and (A1) as well as (A2), Eq.(22) can\nbe rewritten as\nG(α,β,t) =∞/summationdisplay\nm,n=0e−iχt(m2−n2)−γt(m+n)/an}bracketle{tξ=2α|m,˜n/an}bracketri}ht/an}bracketle{tm,˜n|eΛm,na˜a/vextendsingle/vextendsingleξ′\n=2β/angbracketrightbig\n=∞/summationdisplay\nm,n=0e−2|α|2\n√\nm!n!e−iχt(m2−n2)−γt(m+n)Hm,n(2α∗,2α)∞/summationdisplay\nl=0Λl\nm,n\nl!/an}bracketle{tm,˜n|al˜al/vextendsingle/vextendsingleξ′\n=2β/angbracketrightbig\n=e−2|β|2−2|α|2∞/summationdisplay\nm,n=0e−iχt(m2−n2)−γt(m+n)\nm!n!Hm,n(2α∗,2α)∞/summationdisplay\nl=0Λl\nm,n\nl!Hm+l,n+l(2β,2β∗).(C1)7\nFurther using a new sum formula (see appendix D)\n∞/summationdisplay\nl=0zl\nl!Hm+l,n+l(x,y)\n=ezxy\nz+1\n(z+1)(m+n+2)/2Hm,n/parenleftbiggx√z+1,y√z+1/parenrightbigg\n,(C2)\nthus Eq.(C1) can be recast into the following form\nG(α,β,t) =∞/summationdisplay\nm,n=0Cm,n(α,α∗,t)e2Λm,n−1\nΛm,n+1|β|2\n×Hm,n/parenleftBigg\n2β/radicalbigΛm,n+1,2β∗\n/radicalbigΛm,n+1/parenrightBigg\n,(C3)whereCm,n(α,α∗,t) is defined in (26). Substituting\nEq.(C3) into (21) yields (25) and (27).\nAppendix D: Derivation of Eq.(C2)\nUsing the integratal expression of two-mode Hermite\npolynomials,\nHm,n(ξ,η) = (−1)neξη/integraldisplayd2z\nπznz∗me−|z|2+ξz−ηz∗,\n(D1)\nwe have\n∞/summationdisplay\nl=0αl\nl!Hm+l,n+l(x,y) =∞/summationdisplay\nl=0αl\nl!(−1)n+lexy/integraldisplayd2z\nπzn+lz∗m+lexp/bracketleftBig\n−|z|2+xz−yz∗/bracketrightBig\n=exy(−1)n/integraldisplayd2z\nπ∞/summationdisplay\nl=0/parenleftBig\n−α|z|2/parenrightBigl\nl!znz∗mexp/bracketleftBig\n−|z|2+xz−yz∗/bracketrightBig\n=exy(−1)n/integraldisplayd2z\nπznz∗mexp/bracketleftBig\n−(α+1)|z|2+xz−yz∗/bracketrightBig\n=eαxy\nα+1\n(α+1)(m+n+2)/2(−1)nexy/(α+1)/integraldisplayd2z\nπznz∗mexp/bracketleftbigg\n−|z|2+xz−yz∗\n√α+1/bracketrightbigg\n=eαxy\nα+1\n(α+1)(m+n+2)/2Hm,n/parenleftbiggx√α+1,y√α+1/parenrightbigg\n, (D2)\nthus we have completed the proof of (C4).\nAppendix E: Derivation of Eq.(28)\nWhenχ= 0,Λm,n→(1−e−2γt) =T,and\nCm,n(α,α∗,t)\n≡exp[−γt(m+n)]\nm!n!(T+1)(m+n+2)/2Hm,n(2α∗,2α)e−2|α|2,(E1)\nthen we have\n∞/summationdisplay\nm,n=0Cm,n(α,α∗,t)Hm,n(2β,2β∗)\n=e−2|α|2\nT+1∞/summationdisplay\nm,n=0/parenleftBig\ne−γt\n√T+1/parenrightBigm+n\nm!n!\n×Hm,n(2α∗,2α)Hm,n(2β,2β∗).(E2)Using the following formula\n∞/summationdisplay\nm,n=0smtn\nm!n!Hm,n(x,y)Hm,n(α,β)\n=1\n1−stexp/bracketleftbiggsxα+tyβ−(xy+αβ)st\n1−st/bracketrightbigg\n,(E3)\nEq.(E2) can be rewritten as\n(E2) =e−2|α|2\n2Te4e−γt√T+1(α∗β+αβ∗)−(α∗α+ββ∗)4e−2γt\nT+1\n1−e−2γt\nT+1 .(E4)\nThus Eq.(25) becomes8\nW(α,α∗,t) = 4/integraldisplayd2β\nπe2T−1\nT+1|β|2∞/summationdisplay\nm,n=0Cm,n(α,α∗,t)Hm,n/parenleftbigg2β√\nT+1,2β∗\n√\nT+1/parenrightbigg\nW(β,β∗,0)\n=2\nTe−2|α|2/integraldisplayd2β\nπexp\n2T−1\nT+1|β|2+2e−γt(α∗β+αβ∗)−2e−2γt/parenleftBig\n|α|2+|β|2\nT+1/parenrightBig\nT\nW(β,β∗,0)\n=R.H.S.of Eq.(28). (E5)\nAppendix F: Derivation of Eq.(33)\nUsing the relation between Hermite polynomial and\nLagurre polynomial,\nLm(xy) =(−1)m\nm!Hm,m(x,y), (F1)\nwe can recast the left of Eq.( 33) into the following form\nFm,n=1\ns!/integraldisplayd2α\nπe−4|α|2(−1)ss!Ls/parenleftBig\n4|α|2/parenrightBig\nHm,n(2α∗,2α)\n=1\ns!/integraldisplayd2α\nπe−4|α|2Hs,s(2α∗,2α)Hm,n(2α∗,2α)\n=1/4\ns!/integraldisplayd2α\nπe−|α|2Hs,s(α∗,α)Hm,n(α∗,α).\n(F2)\nFurther using the generating function of Hm,n(ǫ,ε),\nHm,n(ǫ,ε) =∂m+n\n∂tm∂t′nexp[−tt′+ǫt+εt′]|t=t′=0,(F3)\nand the integration formula,\n/integraldisplayd2z\nπeζ|z|2+ξz+ηz∗=−1\nζe−ξη\nζ,Re(ζ)<0,(F4)\nwe have\n/integraldisplayd2α\nπe−|α|2Hm′,n′(α∗,α)Hm,n(α∗,α)\n=∂m+n\n∂tm∂t′n∂m+n\n∂τm∂τ′nexp[−tt′−ττ′]\n×/integraldisplayd2α\nπexp/bracketleftBig\n−|α|2+(t+τ)α∗+(t′+τ′)α/bracketrightBig/vextendsingle/vextendsingle/vextendsingle\nt=t′=τ=τ′=0\n=∂m′+n′\n∂tm′∂t′n′∂m+n\n∂τm∂τ′nexp[tτ′+τt′]t=t′=τ=τ′=0\n=m!n!δm′,nδn′,m, (F5)thus Eq.(F2) becomes the right hand side of Eq.( 33).\nAppendix G: Derivation of Eq.(35)\nFor this purpose, from Eq.(27) we have\nEm,n= 4/integraldisplayd2β\nπ2e−2y|β|2Hm,n(2xβ,2xβ∗)e−2|β−z|2,\n(G1)\nwhere we have set\ny=1−Λm,n\n1+Λm,n,x=1/radicalbig\nΛm,n+1.(G2)\nUsing Eqs.(F3) and (F4), Eq.(G1) can be recast into the\nfollowing form,9\nEm,n= 4e−2|z|2∂m+n\n∂tm∂t′ne−tt′/integraldisplayd2β\nπ2exp/bracketleftBig\n−2(y+1)|β|2+2β(xt+z∗)+2β∗(xt′+z)/bracketrightBig/vextendsingle/vextendsingle/vextendsingle\nt=t′=0\n=e−2|z|2∂m+n\n∂tm∂t′ne−tt′/integraldisplayd2β\nπ2exp/bracketleftbigg\n−y+1\n2|β|2+β(xt+z∗)+β∗(xt′+z)/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=t′=0\n=1\nπ2\ny+1e−2y\ny+1|z|2∂m+n\n∂tm∂t′nexp/bracketleftbigg\n−/parenleftbigg\n1−2x2\ny+1/parenrightbigg\ntt′+2xz\ny+1t+2xz∗\ny+1t′/bracketrightbigg\nt=t′=0. 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For any bipartite systems, a universal entanglement witnes s of rank-4 for pure\nstatesis obtainedandaclass offiniterankentanglementwit nesses isconstructed. Inaddition,\namethodofdetectingentanglementofastateonlybyentries ofitsdensitymatrixwithrespect\nto some product basis is obtained.\n1.Introduction\nLetHandKbe separable complex Hilbert spaces. Recall that a quantum s tate is an\noperatorρ∈ B(H⊗K) which is positive and has trace 1. Denote by S(H) the set of all states\nonH. IfHandKare finite dimensional, ρ∈ S(H⊗K) is said to be separable if ρcan be\nwritten as\nρ=k/summationdisplay\ni=1piρi⊗σi,\nwhereρiandσiare states on HandKrespectively, and piare positive numbers with/summationtextk\ni=1pi=\n1. Otherwise, ρis said to be inseparable or entangled (ref. [1, 16]). For the case that at least\none ofHandKis of infinite dimension, by Werner [21], a state ρacting onH⊗Kis called\nseparable if it can be approximated in the trace norm by the st ates of the form\nσ=n/summationdisplay\ni=1piρi⊗σi,\nwhereρiandσiare states on HandKrespectively, and piare positive numbers with/summationtextn\ni=1pi=\n1. Otherwise, ρis called an entangled state.\nEntanglement is a basic physical resource to realize variou s quantum information and quan-\ntum communication tasks such as quantum cryptography, tele portation, dense coding and key\ndistribution [16]. It is very important but also difficult to d etermine whether or not a state\nin a composite system is separable or entangled. It is obviou s that every separable state has\na positive partial transpose (the PPT criterion). For 2 ×2 and 2×3 systems, that is, for the\nPACS.03.67.Mn, 03.65.Ud, 03.65.Db.\nKey words and phrases. Quantum states, separability, entanglement witnesses, po sitive linear maps.\nThis work is partially supported by National Natural Scienc e Foundation of China (No. 10771157), Research\nGrant to Returned Scholars of Shanxi (2007-38) and the Found ation of Shanxi University.\n12 XIAOFEI QI AND JINCHUAN HOU\ncase dimH= dimK= 2 or dim H= 2,dimK= 3, a state is separable if and only if it is\na PPT state, that is, has positive partial transpose (see [7, 17]), but the PPT criterion has\nno efficiency for PPT entangled states appearing in the higher dimensional systems. In [3],\nthe realignment criterion for separability in finite-dimen sional systems was found, which says\nthat ifρ∈ S(H⊗K) is separable, then the trace norm of its realignment matrix ρRis not\ngreater than 1. The realignment criterion was generalized t o infinite dimensional system by\nGuo and Hou in [6]. A most general approach to characterize qu antum entanglement is based\non the notion of entanglement witnesses (see [7]). A self-ad joint operator Wacting onH⊗K\nis said to be an entanglement witness (briefly, EW), if Wis not positive and Tr( Wρ)≥0\nholds for all separable states ρ. It was shown in [7] that, a state ρis entangled if and only if\nit is detected by some entanglement witness W, that is, Tr( Wρ)<0. However, constructing\nentanglement witnesses is a hard task. There was a considera ble effort in constructing and\nanalyzing the structure of entanglement witnesses for finit e and infinite dimensional systems\n[2, 4, 14, 15, 20] (see also [10] for a review). Recently, Hou a nd Qi in [14] showed that every\nentangled state can be recognized by an entanglement witnes sWof the form W=cI+T\nwithIthe identity operator, ca nonnegative number and Ta finite rank self-adjoint operator\nand provided a way how to construct them.\nAnother important criterion for separability of states is t he positive map criterion [7, The-\norem 2], which claims that a state ρ∈ S(H⊗K) with dimH⊗K <∞is separable if and only\nif (Φ⊗I)ρ≥0 holds for all positive linear maps Φ : B(H)→ B(K). Hou [13] generalized the\npositive map criterion to the infinite dimensional systems a nd obtained the following result.\nFinite rank elementary operator criterion. ([13, Theorem 4.5]) LetH,Kbe complex\nHilbert spaces and ρbe a state acting on H⊗K. Then the following statements are equivalent.\n(1)ρis separable;\n(2) (Φ⊗I)ρ≥0holds for every finite-rank positive elementary operator Φ :B(H)→ B(K).\nRecall that a linear map Φ : B(H)→ B(K) is an elementary operator if there are operators\nA1,A2,···,Ar∈ B(H,K ) andB1,B2,···,Br∈ B(K,H ) such that Φ( X) =/summationtextr\ni=1AiXBifor\nallX∈ B(H). It is known that an elementary operator Φ is finite rank posi tive if and only\nif there exist finite rank operators C1,...,Ck,D1,···,Dl∈ B(H,K ) such that ( D1,···,Dl)\nis a contractive local combination of ( C1,···,Ck) and Φ(X) =/summationtextk\ni=1CiXC†\ni−/summationtextl\nj=1DjXD†\nj\nfor allX∈ B(H) (ref. [13] and the references therein).\nTherefore, by the finite rank elementary operator criterion , a stateρonH⊗Kis entangled\nif and only if there exists a finite rank positive elementary o perator Φ : B(H)→ B(K) such\nthat (Φ⊗I)ρis not positive. Here Φ must be not completely positive (brie fly, NCP). Thus itDETECTING ENTANGLEMENT BY ENTRIES 3\nis also important and interesting to find as many as possible fi nite rank positive elementary\noperators that are NCP, and then, to apply them to detect the e ntanglement of states. In [18],\nsome new finite rank positive elementary operators were cons tructed and then applied to get\nsome new entangled states that can not be detected by the PPT c riterion and the realignment\ncriterion.\nDue to the Choi-Jamio/suppress lkowski isomorphism, any EW on finite d imensional system H⊗K\ncorresponds to a linear positive map Φ : B(H)→ B(H). In fact, for system H⊗Kof any\ndimension, if Φ : B(H)→ B(H) is a normal positive completely bounded linear map, and if\nρ0is an entangled state on H⊗K, thenW= (Φ⊗I)ρ0is an entanglement witness whenever\nWis not positive (see lemma 2.1). Recall that a linear map ∆ : B(H)→ B (K) is said to\nbe completely bounded if ∆ ⊗Iis bounded; is said to be normal if it is weakly continuous\non bounded sets, or equivalently, if it is ultra-weakly cont inuous (i.e., if {Aα}is a bounded\nnet and there is A∈ B(H) such that /an}bracketle{tx|Aα|y/an}bracketri}htconverges to /an}bracketle{tx|A|y/an}bracketri}htfor any|x/an}bracketri}ht,|y/an}bracketri}ht ∈H, then\n/an}bracketle{tφ|∆(Aα)|ψ/an}bracketri}htconverges to /an}bracketle{tφ|∆(A)|ψ/an}bracketri}htfor any|φ/an}bracketri}ht,|ψ/an}bracketri}ht ∈K. ref. [5, pp.59]).\nThe finite rank elementary operator criterion, together wit h lemma 2.1, gives a way of\nconstructing finite rank entanglement witnesses from finite rank positive elementary operators\nfor both finite and infinite dimensional bipartite systems. I n the present paper, we construct\na rank-4 entanglement witness Wthat has some what “universal” property for pure states\nin any bipartite systems H⊗K. We show that, for such a rank-4 entanglement witness W,\na pure state ρis entangled if and only if there exist unitary operators UonHandVon\nKsuch that Tr(( U⊗V)W(U†⊗V†)ρ)<0. In addition, if ρis a mixed state such that\nTr((U⊗V)W(U†⊗V†)ρ)<0, thenρis 1-distillable (see theorem 2.2). We also construct a\nclass of entanglement witnesses from the finite rank positiv e elementary operators obtained\nin [18] (see theorem 3.1).\nSo far, by our knowledge, there is no methods of recognizing t he entanglement of a state by\nmerely the entries of its density matrix. Another interesti ng result of this paper gives a way\nof detecting the entanglement of a state in a bipartite syste m by only a part of entries of its\ndensity matrix (see theorems 3.2, 3.3). This method is simpl e, computable and practicable\nbecause it provide a way to recognize the entanglement of a st ate by some suitably chosen\nentries of its matrix representation with respect to some gi ven product basis. As an illustra-\ntion, some new examples of entangled states that can be recog nized by this way are proposed,\nwhich also provides some new entangled states that can not be detected by the PPT criterion\nand the realignment criterion (see examples 3.4, 3.5).4 XIAOFEI QI AND JINCHUAN HOU\nRecall that a bipartite state ρis calledn-distillable, if and only if maximally entangled\nbipartite pure states, e.g. |ψ/an}bracketri}ht=1\n2(|11′/an}bracketri}ht+|22′/an}bracketri}ht), can be created from nidentical copies of\nthe stateρby means of local operations and classical communication; i s called distillable if\nit isn-distillable for some n. It has been shown that all entangled pure states are distill able.\nHowever it is a challenge to give an operational criterion of distillability for general mixed\nstates [8]. In [9], it was shown that a density matrix ρis distillable if and only if there are\nsome projectors P,Qthat map high dimensional spaces to two-dimensional ones su ch that\nthe state (P⊗Q)ρ⊗n(P⊗Q) is entangled for some ncopies.\n2.Universal entanglement witnesses for pure states\nIn this section we will give a simple necessary and sufficient c ondition for separability of\npure states in bipartite composite systems of any dimension .\nBefore stating the main result in this section, we give a basi c lemma.\nLemma 2.1. LetH,Kbe complex Hilbert spaces of any dimension and let Φ :B(H)→\nB(H)be a positive normal completely bounded linear map. Then, for any entangled state ρ0\nonH⊗K,W= (Φ⊗I)ρ0is an entanglement witness whenever WonH⊗Kis not positive.\nProof. Because Φ is completely bounded, W= (Φ⊗I)ρ0is a bounded self-adjoint operator\nonH⊗K. Note that B(H) =T(H)∗, whereT(H) denotes the Banach space of all trace class\noperators on Hendowed with the trace norm. Then the normality of Φ implies t hat there\nexists a bounded linear map ∆ : T(H)→ T (H) such that Φ = ∆∗. We claim that ∆ is also\npositive. In fact, for any unit vector |φ/an}bracketri}ht ∈Hand any positive operator A∈ B(H), we have\nTr(A∆(|φ/an}bracketri}ht/an}bracketle{tφ|)) = Tr(Φ(A)(|φ/an}bracketri}ht/an}bracketle{tφ|)) =/an}bracketle{tφ|Φ(A)|φ/an}bracketri}ht ≥0.\nThis implies that ∆( |φ/an}bracketri}ht/an}bracketle{tφ|) is positive for any |φ/an}bracketri}ht. So, ∆ is a positive linear map.\nNow, for any separable state ρ∈ S(H⊗K), we have\nTr(Wρ) = Tr((Φ ⊗I)ρ0·ρ) = Tr(ρ0·(∆⊗I)ρ)≥0\nsince (∆ ⊗I)ρ≥0. So, ifWis not positive, then it is an entanglement witness. /square\nSince every elementary operator is normal and completely bo unded, by Lemma 2.1, if Φ\nis a positive elementary operator and if ρ0is an entangled state, then W= (Φ⊗I)ρ0is an\nentanglement witness whenever Wis not positive. Also note that, if Wis an entanglement\nwitness, then for any positive number b,bWis an entanglement witness, too.\nLetWbe an entanglement witness on H⊗K. We say that Wis universal (for all states) if,\nfor any entangled state ρonH⊗K, there exist unitary operators UonHandVonKsuch that\nTr((U⊗V)W(U†⊗V†)ρ)<0;Wis universal for pure states if, for any entangled pure state ρonDETECTING ENTANGLEMENT BY ENTRIES 5\nH⊗K, there exist unitary operators UonHandVonKsuch that Tr(( U⊗V)W(U†⊗V†)ρ)<\n0.\nThe following is the main result of this section, which gives a universal entanglement witness\nof rank-4 for pure states. Particularly, we conclude that th e separability of pure states can be\ndetermined by a special class of rank-4 witnesses, and every 1-distillable state can be detected\nby one of such rank-4 entanglement witnesses. However, we do not know whether or not there\nexists a universal entanglement witness for all states.\nLetU(H) (resp.U(K)) be the group of all unitary operators on H(resp. onK).\nTheorem 2.2. LetHandKbe Hilbert spaces and let {|i/an}bracketri}ht}dimH≤∞\ni=1and{|j′/an}bracketri}ht}dimK≤∞\nj=1be\nany orthonormal bases of HandK, respectively. Let\nW=|1/an}bracketri}ht|2′/an}bracketri}ht/an}bracketle{t1|/an}bracketle{t2′|−|1/an}bracketri}ht|1′/an}bracketri}ht/an}bracketle{t2|/an}bracketle{t2′|−|2/an}bracketri}ht|2′/an}bracketri}ht/an}bracketle{t1|/an}bracketle{t1′|+|2/an}bracketri}ht|1′/an}bracketri}ht/an}bracketle{t2|/an}bracketle{t1′|. (2.1)\nThenWis an entanglement witness of rank-4. Moreover, the followi ng statements are true.\n(1)Ifρis a pure state, then ρis separable if and only if\nTr((U⊗V)W(U†⊗V†)ρ)≥0 (2 .2)\nhold for all U∈ U(H)andV∈ U(K). SoWis a universal entanglement witness for pure\nstates.\n(2)Letρbe a state. If there exist U∈ U(H)andV∈ U(K)such that Tr((U⊗V)W(U†⊗\nV†)ρ)<0, thenρis entangled and 1-distillable.\nProof. We first prove that Wis an entanglement witness. It is obvious that Wis not\npositive. Define a map Φ : B(H)→ B(H) by\nΦ(A) =E11AE†\n11+E22AE†\n22+E12AE†\n12\n+E21AE†\n21−(E11+E22)A(E11+E22)†(2.3)\nfor everyA∈ B(H), whereEij=|i/an}bracketri}ht/an}bracketle{tj| ∈ B (H). It is obvious that Φ is a positive map because\nthe map\na11a12\na21a22\n/ma√sto→\na22−a12\n−a21a11\n\nonM2(C) is positive. Note that W= 2(Φ⊗I)ρ+, whereρ+=|ψ+/an}bracketri}ht/an}bracketle{tψ+|with|ψ+/an}bracketri}ht=\n1√\n2(|11′/an}bracketri}ht+|22′/an}bracketri}ht). Thus, by Lemma 2.1, Wis an entanglement witness.\nIfρis separable, then Tr(( U⊗V)W(U†⊗V†)ρ)≥0 as (U†⊗V†)ρ(U⊗V) are separable.\nConversely, assume that ρ=|ψ/an}bracketri}ht/an}bracketle{tψ|is inseparable. Consider its Schmidt decomposition\n|ψ/an}bracketri}ht=/summationtextNψ\nk=1δk|k,k′/an}bracketri}ht, whereδ1≥δ2≥ ···>0 with/summationtextNψ\nk=1δ2\nk= 1,{|k/an}bracketri}ht}Nψ\nk=1and{|k′/an}bracketri}ht}Nψ\nk=1\nare orthonormal in HandK, respectively. As |ψ/an}bracketri}htis inseparable, we must have its Schmidt\nnumberNψ≥2. Thusρ=/summationtextNψ\nk,l=1δkδl|k,k′/an}bracketri}ht/an}bracketle{tl,l′|. Up to unitary equivalence, we may assume6 XIAOFEI QI AND JINCHUAN HOU\nthat{|k/an}bracketri}ht}2\nk=1={|i/an}bracketri}ht}2\ni=1and{|k′/an}bracketri}ht}2\nk′=1={|j′/an}bracketri}ht}2\nj=1. Then Tr( Wρ) = Tr(−δ1δ2|11′/an}bracketri}ht/an}bracketle{t11′| −\nδ1δ2|22′/an}bracketri}ht/an}bracketle{t22′|) =−2δ1δ2<0. Hence the statement (1) is true.\nFor the statement (2), assume that there exist U∈ U(H) andV∈ U(K) such that Tr(( U⊗\nV)W(U†⊗V†)ρ)<0. Thenρis entangled. Moreover, ρhas a matrix representation\nρ=/summationdisplay\ni,j,k,lαijkl|Ui/an}bracketri}ht|Vj′/an}bracketri}ht/an}bracketle{tUk|/an}bracketle{tVl′|.\nThus, one gets\n0>Tr((U⊗V)W(U†⊗V†)ρ) = Tr(W(U†⊗V†)ρ(U⊗V))\n= Tr(/summationtext\ni,j,k,lαijkl(|1/an}bracketri}ht|2′/an}bracketri}ht/an}bracketle{t1|/an}bracketle{t2′|−|1/an}bracketri}ht|1′/an}bracketri}ht/an}bracketle{t2|/an}bracketle{t2′|−|2/an}bracketri}ht|2′/an}bracketri}ht/an}bracketle{t1|/an}bracketle{t1′|+|2/an}bracketri}ht|1′/an}bracketri}ht/an}bracketle{t2|/an}bracketle{t1′|)\n·(U†⊗V†)|Ui/an}bracketri}ht|Vj′/an}bracketri}ht/an}bracketle{tUk|/an}bracketle{tVl′|(U⊗V))\n= Tr(/summationtext\ni,j,k,lαijkl(|1/an}bracketri}ht|2′/an}bracketri}ht/an}bracketle{t1|/an}bracketle{t2′|−|1/an}bracketri}ht|1′/an}bracketri}ht/an}bracketle{t2|/an}bracketle{t2′|−|2/an}bracketri}ht|2′/an}bracketri}ht/an}bracketle{t1|/an}bracketle{t1′|+|2/an}bracketri}ht|1′/an}bracketri}ht/an}bracketle{t2|/an}bracketle{t1′|)\n·|i/an}bracketri}ht|j′/an}bracketri}ht/an}bracketle{tk|/an}bracketle{tl′|)\n=−α2211−α1122.\nNow letPandQbe the projectors from HandKonto the two dimensional subspaces spanned\nby{|1/an}bracketri}ht,|2/an}bracketri}ht}and{|1′/an}bracketri}ht,|2′/an}bracketri}ht}, respectively. Then\nTr(P⊗Q)(U⊗V)W(U†⊗V†)(P⊗Q)ρ(P⊗Q)) =−α2211−α1122<0,\nwhich implies that ( P⊗Q)ρ(P⊗Q) is entangled. It follows from [9] that ρis 1-distillable.\nThe proof is complete. /square\n3.Detecting entanglement of states by their entries\nIn this section, we give a method of detecting entanglement o f a state in any bipartite\nsystem only by some entries of its matrix representation.\nLetHandKbe complex Hilbert spaces of any dimension with {|i/an}bracketri}ht}dimH\ni=1 and{|j′/an}bracketri}ht}dimK\nj=1\nbe orthonormal bases of them respectively. Denote Eij=Ei,j=|i/an}bracketri}ht/an}bracketle{tj|, which is an operator\nfromHintoH. Letn≤min{dimH,dimK}be a positive integer. By [18, Remark 5.2], for\nany permutation κof (1,2,···,n), the linear map Φ κ:B(H)→ B(H) defined by\nΦκ(A) = (n−1)n/summationdisplay\ni=1EiiAE†\nii+n/summationdisplay\ni=1Ei,κ(i)AE†\ni,κ(i)−(n/summationdisplay\ni=1Eii)A(n/summationdisplay\ni=1Eii)†(3.1)\nfor everyA∈ B(H) is a positive elementary operator that is not completely po sitive ifκ/ne}ationslash= id.\nThen, for any unitary operators UandVonH, the map ΦU,V\nκdefined by\nΦU,V\nκ(A) = (n−1)/summationtextn\ni=1(VEiiU)A(VEiiU)†+/summationtextn\ni=1(VEi,κ(i)U)A(VEi,κ(i)U)†\n−(/summationtextn\ni=1VEiiU)A(/summationtextn\ni=1VEiiU)†(3.2)DETECTING ENTANGLEMENT BY ENTRIES 7\nfor everyA∈ B(H) is positive, too. Let ρ+=|ψ+/an}bracketri}ht/an}bracketle{tψ+|, where\n|ψ+/an}bracketri}ht=1√n(|1/an}bracketri}ht|1′/an}bracketri}ht+|2/an}bracketri}ht|2′/an}bracketri}ht+···+|n/an}bracketri}ht|n′/an}bracketri}ht).\nThen, by Lemma 2.1, we get a class of entanglement witnesses o f the form\nWU,V\nκ=n(ΦU,V\nκ⊗I)ρ+= (ΦU,V\nκ(Eij)). (3.3)\nNote thatWU,V\nκis of finite rank because ρ+is.\nParticularly, for permutations π,σof (1,2,···,n), ifUandVare the unitary operators\ndefined byU†|i/an}bracketri}ht=|π(i)/an}bracketri}ht,V|i/an}bracketri}ht=|σ(i)/an}bracketri}htfori= 1,2,···nandU†|i/an}bracketri}ht=|i/an}bracketri}ht,V|i/an}bracketri}ht=|i/an}bracketri}htfori>n ,\nthen we have\nΦπ,σ\nκ(A) = ΦU,V\nκ(A) = (n−1)/summationtextn\ni=1Eσ(i),π(i)AE†\nσ(i),π(i)\n+/summationtextn\ni=1Eσ(i),π(κ(i))AE†\nσ(i),π(κ(i))−(/summationtextn\ni=1Eσ(i),π(i))A(/summationtextn\ni=1Eσ(i),π(i))†(3.4)\nfor everyA. And correspondingly, we get entanglement witnesses of the concrete form\nWπ,σ\nκ= (Φπ,σ\nκ(Eij)), (3.5)\nwhere\nΦπ,σ\nκ(Eij) =−Eσ(π−1(i)),σ(π−1(j)) (3.6)\nif 1≤i/ne}ationslash=j≤n,\nΦπ,σ\nκ(Eii) = (n−2)Eσ(π−1(i)),σ(π−1(i))+Eσ(κ−1π−1(i)),σ(κ−1π−1(i)) (3.7)\nif 1≤i≤n, and\nΦπ,σ\nκ(Eij) = 0 (3 .8)\nifi>n orj >n .\nThus we have proved the following result.\nTheorem 3.1. LetHandKbe complex Hilbert spaces of any dimension with {|i/an}bracketri}ht}dimH≤∞\ni=1\nand{|j′/an}bracketri}ht}dimK≤∞\nj=1be orthonormal bases of them respectively. For any positive integer 2≤n≤\nmin{dimH,dimK}and any permutations κ,π,σof(1,2,···,n)withκ/ne}ationslash= id, the finite rank\noperatorWπ,σ\nκdefined by\nWπ,σ\nκ= (n−2)/summationtextn\ni=1|σπ−1(i),i′/an}bracketri}ht/an}bracketle{tσπ−1(i),i′|\n+/summationtextn\ni=1|σκ−1π−1(i),i′/an}bracketri}ht/an}bracketle{tσκ−1π−1(i),i′|\n−/summationtext\n1≤i/negationslash=j≤n|σπ−1(i),i′/an}bracketri}ht/an}bracketle{tσπ−1(j),j′|\nis an entanglement witness.8 XIAOFEI QI AND JINCHUAN HOU\nAssume that dim H= dimK=n. By applying the witnesses Wπ,σ\nκin Theorem 3.1, we\nget a method of detecting the entanglement of states by the en tries of their density matrix.\nWrite the product basis of H⊗Kin the order\n{|e1/an}bracketri}ht=|1/an}bracketri}ht|1′/an}bracketri}ht,|e2/an}bracketri}ht=|2/an}bracketri}ht|1′/an}bracketri}ht,···,|en/an}bracketri}ht=|n/an}bracketri}ht|1′/an}bracketri}ht,|en+1/an}bracketri}ht=|1/an}bracketri}ht|2′/an}bracketri}ht,\n···,|en2−1/an}bracketri}ht=|(n−1)/an}bracketri}ht|n′/an}bracketri}ht,|en2/an}bracketri}ht=|n/an}bracketri}ht|n′/an}bracketri}ht}.(3.9)\nThen every state ρ∈ S(H⊗K) has a matrix representation ρ= (αkl)n2×n2.\nTheorem 3.2. Letρ∈ B(H⊗K)with dimH= dimK=n <∞be a state with the\nmatrix representation ρ= (αkl)n2×n2with respect to the product basis in Eq.(3.9). If there\nexist distinguished positive integers (i−1)n<ki,hi≤in,i= 1,2,···,nsuch that\nn/summationdisplay\ni=1ki=n/summationdisplay\ni=1hi=1\n2n(n2+ 1), (3.10)\nand\n(n−2)n/summationdisplay\ni=1αkiki+n/summationdisplay\ni=1αhihi−/summationdisplay\n1≤i/negationslash=j≤nαkikj<0, (3.11)\nthenρis entangled.\nProof. Eq.(3.10) implies that, there exist permutations π1andσ1such that ( k1,k2−\nn,···,kn−(n−1)n) =π1(1,2,···,n) and (h1,h2−n,···,hn−(n−1)n) =σ1(1,2,···,n).\nIt is clear that π1(i)/ne}ationslash=σ1(i) aski/ne}ationslash=hifor everyi= 1,2,···,n.\nFor any permutations κ,πandσ, by Theorem 3.1, we have\nTr(Wπ,σ\nκρ) = (n−2)/summationtextn\ni=1ασ(π−1(i))+(i−1)n,σ(π−1(i))+(i−1)n\n+/summationtextn\ni=1ασ(κ−1π−1(i))+(i−1)n,σ(κ−1π−1(i))+(i−1)n\n−/summationtextn\ni/negationslash=jασ(π−1(i))+(i−1)n,σ(π−1(j))+(j−1)n.(3.12)\nComparing Eq.(3.11) with Eq.(3.12), we have to find permutat ionsκ,πandσso that\nπ1(i) =σ(π−1(i)) andσ1(i) =σ(κ−1π−1(i)) (3 .13)\nfor eachi, that is,π1=σπ−1andσ1=σκ−1π−1. Takeπ= id. Then we get σ=π1and\nσ1=σκ−1=π1κ−1. Thus,κ=σ−1\n1π1,π= id andσ=π1satisfy Eq.(3.13). With such κ,π\nandσ, by Eqs.(3.11) and (3.12), we have\nTr(Wπ,σ\nκρ) = (n−2)n/summationdisplay\ni=1αkiki+n/summationdisplay\ni=1αhihi−/summationdisplay\n1≤i/negationslash=j≤nαkikj<0.\nHence,ρis entangled with Wπ,σ\nκan entanglement witness for it. /square\nThe general version of Theorem 3.2 is the following result, w hich is applicable for bipartite\nsystems of any dimension.DETECTING ENTANGLEMENT BY ENTRIES 9\nTheorem 3.3. LetHandKbe complex Hilbert spaces with {|i/an}bracketri}ht}dimH≤∞\ni=1and{|j′/an}bracketri}ht}dimK≤∞\nj=1\nbe orthonormal bases of them respectively. Assume that ρis a state on H⊗Kandn≤\nmin{dimH,dimK}is a positive integer. If there exist permutations πandσof(1,2,···,n)\nwithπ(i)/ne}ationslash=σ(i)for anyi= 1,2,···,nsuch that\n(n−2)n/summationdisplay\ni=1/an}bracketle{tπ(i),i′|ρ|π(i),i′/an}bracketri}ht+n/summationdisplay\ni=1/an}bracketle{tσ(i),i′|ρ|σ(i),i′/an}bracketri}ht−/summationdisplay\n1≤i/negationslash=j≤n/an}bracketle{tπ(i),i′|ρ|π(j),j′/an}bracketri}ht<0,(3.14)\nthenρis entangled.\nThe idea of the proof of Theorem 3.3 is the same as that of Theor em 3.2 and we omit it\nhere.\nTheorems 3.2 and 3.3 tell us, some times we can detect the enta nglement of a state by\nsuitably chosen n2+nentries of its matrix representation with respect to some pr oduct basis,\nwheren≤min{dimH,dimK}.\nTo illustrate how to use Theorem 3.2 and Theorem 3.3 to detect entanglement of a state,\nwe give some examples.\nExample 3.4. Letq1,q2,q3be nonnegative numbers with q1+q2+q3= 1 and let a,b,c∈C\nwith|a|2≤q2q3,|b|2≤q2q3,|c|2≤q2q3. Letρbe a state of 3 ×3 system with matrix\nrepresentation\nρ=1\n3\nq10 0 0 q10 0 0 q1\n0q3a0 0 0 0 0 0\n0 ¯a q20 0 0 0 0 0\n0 0 0 q20b0 0 0\nq10 0 0 q10 0 0 q1\n0 0 0 ¯b0q30 0 0\n0 0 0 0 0 0 q3c0\n0 0 0 0 0 0 ¯ c q20\nq10 0 0 q10 0 0 q1\n. (3.15)\nNote that, ρin Eq.(3.15) is a new kind of states, and ρdegenerates to the state as that in\n[18, Example 3.3] when a=b=c= 0.\nWe claim that, if q2<q1orq3<q1, thenρis entangled.\nIn fact, choosing ( k1,k2,k3) = (1,5,9), (h1,h2,h3) = (3,4,8) or (2,6,7), we have\n3/summationdisplay\ni=1αkiki+3/summationdisplay\ni=1αhihi−/summationdisplay\n1≤i/negationslash=j≤3αkikj=1\n3(3q1+ 3q2−6q1) =q2−q1\nor\n3/summationdisplay\ni=1αkiki+3/summationdisplay\ni=1αhihi−/summationdisplay\n1≤i/negationslash=j≤3αkikj=1\n3(3q1+ 3q3−6q1) =q3−q1.10 XIAOFEI QI AND JINCHUAN HOU\nBy Theorem 3.2, we see that ρis entangled if q2<q1orq3<q1.\nIt is clear that the partial transpose of ρin Eq.(3.15) with respect to the first subsystem is\nρT1=1\n3\nq10 0 0 0 0 0 0 0\n0q3¯a q10 0 0 0 0\n0a q20 0 0 q10 0\n0q10q20¯b0 0 0\n0 0 0 0 q10 0 0 0\n0 0 0 b0q30q10\n0 0q10 0 0 q3¯c0\n0 0 0 0 0 q1c q20\n0 0 0 0 0 0 0 0 q1\n.\nParticularly, if we take q1=1\n5,q2=1\n10,q3=7\n10anda=b=c=1\n20, then, by what proved\nabove, we see that ρis PPT entangled because its partial transpose has eigenval ues\n{1\n60(8±√\n61),1\n4,1\n4,1\n60,1\n60,1\n15,1\n15,1\n15}\nthat are all positive.\nExample 3.5. Letρbe a state in 4 ×4 systems with the matrix\nρ=1\n4\nq10 0 0 0 q10 0 0 0 q10 0 0 0 q1\n0q4a0 0 0 0 0 0 0 0 0 0 0 0 0\n0 ¯a q30 0 0 0 0 0 0 0 0 0 0 0 0\n0 0 0 q2q20 0 0 0 q20 0 0 0 q20\n0 0 0 q2q20 0 0 0 q20 0 0 0 q20\nq10 0 0 0 q10 0 0 0 q10 0 0 0 q1\n0 0 0 0 0 0 q4b0 0 0 0 0 0 0 0\n0 0 0 0 0 0 ¯b q30 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 0 q30 0c0 0 0 0\n0 0 0 q2q20 0 0 0 q20 0 0 0 q20\nq10 0 0 0 q10 0 0 0 q10 0 0 0 q1\n0 0 0 0 0 0 0 0 ¯ c0 0q40 0 0 0\n0 0 0 0 0 0 0 0 0 0 0 0 q4d0 0\n0 0 0 0 0 0 0 0 0 0 0 0 ¯d q30 0\n0 0 0 q2q20 0 0 0 q20 0 0 0 q20\nq10 0 0 0 q10 0 0 0 q10 0 0 0 q1\n, (3.16)DETECTING ENTANGLEMENT BY ENTRIES 11\nwhereqi≥0 with/summationtext4\ni=1qi= 1,|a|2,|b|2,|c|2and|d|2are all≤q3q4.ρdefined by Eq.(3.16)\nis also a new example, and when a=b=c=d= 0 we get states in [18, Example 4.4].\nWe claim that, if qi<q1for somei∈ {2,3,4}; or ifqi<q2for somei∈ {1,3,4}, thenρis\nentangled.\nIn fact, we can take\n(k1,k2,k3,k4) = (1,6,11,16) and ( h1,h2,h3,h4) = (2,7,12,13),\nor\n(k1,k2,k3,k4) = (1,6,11,16) and ( h1,h2,h3,h4) = (3,8,9,14),\nor\n(k1,k2,k3,k4) = (1,6,11,16) and ( h1,h2,h3,h4) = (4,5,10,15),\nor\n(k1,k2,k3,k4) = (4,5,10,15) and ( h1,h2,h3,h4) = (1,6,11,16),\nor\n(k1,k2,k3,k4) = (4,5,10,15) and ( h1,h2,h3,h4) = (2,7,12,13),\nor\n(k1,k2,k3,k4) = (4,5,10,15) and ( h1,h2,h3,h4) = (3,8,9,14).\nThen, it follows from the first three choices that\n24/summationdisplay\ni=1αkiki+4/summationdisplay\ni=1αhihi−/summationdisplay\n1≤i/negationslash=j≤3αkikj=qi−q1\nwithi= 2,3,4. Hence, by Theorem 3.2 we see that ρis entangled if there exists some\ni∈ {2,3,4}such thatqi<q1. Similarly, by the last three choices one sees that ρis entangled\nif there exists some i∈ {1,3,4}such thatqi<q2.\nThe kind of states in Eq.(3.16) allow us give some new example s of entangled states that\ncan not be recognized by PPT criterion and the realignment cr iterion. It is obvious that the12 XIAOFEI QI AND JINCHUAN HOU\npartial transpose of ρin Eq.(3.16) with respect to the first subsystem is\nρT1=1\n4\nq10 0 0 0 0 0 q20 0 0 0 0 0 0 0\n0q4¯a0q10 0 0 0 0 0 q20 0 0 0\n0a q30 0 0 0 0 q10 0 0 0 0 0 q2\n0 0 0 q20 0 0 0 0 0 0 0 q10 0 0\n0q10 0q20 0 0 0 0 0 0 0 0 0 0\n0 0 0 0 0 q10 0q20 0 0 0 0 0 0\n0 0 0 0 0 0 q4¯b0q10 0q20 0 0\nq20 0 0 0 0 b q30 0 0 0 0 q10 0\n0 0q10 0q20 0q30 0 ¯c0 0 0 0\n0 0 0 0 0 0 q10 0q20 0 0 0 0 0\n0 0 0 0 0 0 0 0 0 0 q10 0q20 0\n0q20 0 0 0 0 0 c0 0q40 0q10\n0 0 0 q10 0q20 0 0 0 0 q4¯d0 0\n0 0 0 0 0 0 0 q10 0q20d q30 0\n0 0 0 0 0 0 0 0 0 0 0 q10 0q20\n0 0q20 0 0 0 0 0 0 0 0 0 0 0 q1\n\nand that the realignment of ρis\nρR=1\n4\nq10 0 0 0 q4¯a0 0a q30 0 0 0 q2\n0q10 0 0 0 0 0 0 0 0 0 q20 0 0\n0 0q10 0 0 0 0 0 0 0 0 0 q20 0\n0 0 0 q10 0 0 0 0 0 0 0 0 0 q20\n0 0 0 q2q10 0 0 0 0 0 0 0 0 0 0\nq20 0 0 0 q10 0 0 0 q4¯b0 0b q3\n0q20 0 0 0 q10 0 0 0 0 0 0 0 0\n0 0q20 0 0 0 q10 0 0 0 0 0 0 0\n0 0 0 0 0 0 0 q2q10 0 0 0 0 0 0\n0 0 0 0 q20 0 0 0 q10 0 0 0 0 0\nq30 0 ¯c0q20 0 0 0 q10c0 0q4\n0 0 0 0 0 0 q20 0 0 0 q10 0 0 0\n0 0 0 0 0 0 0 0 0 0 0 q2q10 0 0\n0 0 0 0 0 0 0 0 q20 0 0 0 q10 0\n0 0 0 0 0 0 0 0 0 q20 0 0 0 q10\nq4¯d0 0d q30 0 0 0 q20 0 0 0 q1\n.DETECTING ENTANGLEMENT BY ENTRIES 13\nIf we takeq1=1\n20,q2=1\n10,q3=q4=17\n40anda=b=c=d=1\n40,ρis PPT entangled because\nq1<q2and its partial transpose ρT1has eigenvalues\n{0.0054,0.0054,0.0069,0.0069,0.0223,0.0223,0.0235,0.0235,\n0.0821,0.0821,0.1027,0.1027,0.1212,0.1212,0.1359,0.1359}\nthat are all positive. Moreover, the trace norm of the realig nmentρRofρis/bardblρR/bardbl1.= 0.8303<\n1. Hence, we get another example of entangled states that is P PT and cannot be detected by\nthe realignment criterion.\nIt is not difficult to give some examples of applying Theorem 3. 3 to infinite dimensional\nsystems based on examples 3.4 and 3.5.\n4.Conclusions\nLetHandKbe Hilbert spaces and let {|i/an}bracketri}ht}dimH≤∞\ni=1and{|j′/an}bracketri}ht}dimK≤∞\nj=1be any orthonormal\nbases ofHandK, respectively. By the finite rank elementary operator crite rion [13], a state\nρonH⊗Kis entangled if and only if there exists a finite rank positive elementary operator\nΦ :B(H)→ B(K) that is not completely positive such that (Φ ⊗I)ρis not positive. By this\ncriterion and the finite rank positive elementary operators constructed in [18], we construct a\ncollection of finite rank entanglement witnesses.\nBy using these witnesses we obtain a rank-4 entanglement wit nessW=|1/an}bracketri}ht|2′/an}bracketri}ht/an}bracketle{t1|/an}bracketle{t2′| −\n|1/an}bracketri}ht|1′/an}bracketri}ht/an}bracketle{t2|/an}bracketle{t2′|−|2/an}bracketri}ht|2′/an}bracketri}ht/an}bracketle{t1|/an}bracketle{t1′|+|2/an}bracketri}ht|1′/an}bracketri}ht/an}bracketle{t2|/an}bracketle{t1′|which is universal for pure states, that is, for a pure\nstateρ,ρis separable if and only if Tr(( U⊗V)W(U†⊗V†)ρ)≥0 holds for all unitary operators\nUonHandVonK. In addition, for a mixed state ρ, if there exist unitary operators U0onH\nandV0onKsuch that Tr(( U0⊗V0)W(U†\n0⊗V†\n0)ρ)<0, thenρis entangled and 1-distillable.\nAnother interesting result, maybe for the first time, gives a way of detecting the entangle-\nment of a state in H⊗Kby only a part entries of its density matrix. This method is si mple,\ncomputable and practicable. Assume that ρis a state on H⊗Kandn≤min{dimH,dimK}\nis a positive integer. If there exist permutations πandσof (1,2,···,n) withπ(i)/ne}ationslash=σ(i) for\nanyi= 1,2,···,nsuch that\n(n−2)n/summationdisplay\ni=1/an}bracketle{tπ(i),i′|ρ|π(i),i′/an}bracketri}ht+n/summationdisplay\ni=1/an}bracketle{tσ(i),i′|ρ|σ(i),i′/an}bracketri}ht−/summationdisplay\n1≤i/negationslash=j≤n/an}bracketle{tπ(i),i′|ρ|π(j),j′/an}bracketri}ht<0,\nthenρis entangled. Thus we provide a way of detecting the entangle ment of a state by finite\nsuitably chosen entries of its matrix representation with r espect to some product basis. As\nan illustration how to use this method, some new examples of e ntangled states that can be\nrecognized by this way are proposed, which also provides som e new entangled states that can\nnot be detected by the PPT criterion and the realignment crit erion.14 XIAOFEI QI AND JINCHUAN HOU\nReferences\n[1] I. Bengtsson, K. Zyczkowski, Cambridge University Pres s, Cambridge, 2006.\n[2] D. Bruß, J. Math. Phys. 43 (2002) 4237.\n[3] K. Chen, L. Wu, Quant. Inf. Comput 3 (2003) 193.\n[4] D. Chru´ sci´nski and A. Kossakowski, Open Systems and Inf. Dynamics 14 (20 07) 275; D. Chru´ sci´nski and\nA. Kossakowski, J. Phys. A: Math. Theor. 41 (2008) 145301.\n[5] J. Dixmier, Von Neumann Algebras, North-Holland Publis hing Com., Amsterdan, New York, Oxford,\n1981.\n[6] Y. Guo, J. Hou, arXiv:1009.0116v1.\n[7] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 2 23 (1996) 1.\n[8] R. Horodecki, M. Horodecki, P. Horodecki, Phys. Lett. A 2 22 (1996) 21.\n[9] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Let t. 80 (1998) 5239.\n[10] R. Horodecki, P. Horodecki, M. Horodecki, Rev. Mod. Phy s. 81 (2009) 865.\n[11] J. Hou, Sci. in China (ser.A), 36(9) (1993), 1025-1035.\n[12] J. Hou, J. Operator Theory, 39 (1998), 43-58.\n[13] J. Hou, J. Phys. A: Math. Theor. 43 (2010) 385201; arXiv[ quant-ph]: 1007.0560v1.\n[14] J. Hou, X. Qi, Phys. Rev. A 81 (2010) 062351.\n[15] M. A. Jafarizadeh, N. Behzadi, Y. Akbari, Eur. Phys. J. D 55 (2009) 197.\n[16] M. A. Nielsen, I. L. Chuang, Cambridge University Press , Cambridge, 2000.\n[17] A. Peres, Phys. Lett. A 202 (1996) 16.\n[18] X. Qi, J. Hou, Positive finite rank elementary operators and characterizing entanglement of states,\narXiv:1008.3682v2\n[19] S. Simon, S. P. Rajagopalan, R. Simon, Pramana-Journal of Physics, 73(3) (2009) 471-483.\n[20] G. T´ oth, O. G¨ uhne, Phys. Rev. Lett. 94 (2005) 060501.\n[21] R. F. Werner, Phys. Rev. A 40 (1989) 4277.\n(Xiaofei Qi) Department of Mathematics, Shanxi University , Taiyuan 030 006, P. R. of China;\nE-mail address :qixf1980@126.com\nDepartment of Mathematics, Taiyuan University of Technolo gy, Taiyuan 030024, P. R. of\nChina\nE-mail address :jinchuanhou@yahoo.com.cn"    },    {        "title": "1010.3060v2.Computational_Difficulty_of_Computing_the_Density_of_States.pdf",        "content": "arXiv:1010.3060v2  [quant-ph]  20 Jul 2011Computational Difficulty of Computing the Density of States\nBrielin Brown,1,2Steven T. Flammia,2and Norbert Schuch3\n1University of Virginia, Departments of Physics and Compute r Science, Charlottesville, Virginia 22904, USA\n2Perimeter Institute for Theoretical Physics, Waterloo, On tario N2L 2Y5, Canada\n3California Institute of Technology, Institute for Quantum Information, MC 305-16, Pasadena, California 91125, USA\nWe study the computational difficulty of computing the ground state degeneracy and the density\nof states for local Hamiltonians. We show that the difficulty o f both problems is exactly captured by\na class which we call #BQP, which is the counting version of the quantum complexity cla ssQMA.\nWe show that #BQPis not harder than its classical counting counterpart #P, which in turn implies\nthat computing the ground state degeneracy or the density of states for classical Hamiltonians is\njust as hard as it is for quantum Hamiltonians.\nUnderstanding the physical properties of correlated\nquantum many-body systems is a problem of central im-\nportance in condensed matter physics. The density of\nstates, defined as the number of energy eigenstates per\nenergy interval, plays a particularly crucial role in this\nendeavor. It is a key ingredient when deriving many\nthermodynamic properties from microscopic models, in-\ncludingspecificheatcapacity,thermalconductivity,band\nstructure, and (near the Fermi energy) most electronic\nproperties of metals. Computing the density of states\ncanbe a dauntingtaskhowever,asit in principleinvolves\ndiagonalizing a Hamiltonian acting on an exponentially\nlargespace,thoughothermoreefficientapproacheswhich\nmight take advantage of the structure of a given problem\nare not a priori ruled out.\nIn this Letter, we precisely quantify the difficulty of\ncomputing the density of states by using the powerful\ntools of quantum complexity theory. Quantum complex-\nity aims at generalizing the well-established field of clas-\nsical complexity theoryto assessthe difficulty of tasksre-\nlated to quantum mechanical problems, concerning both\nthe classical difficulty of simulating quantum systems as\nwell as the fundamental limits to the power of quan-\ntum computers. In particular, quantum complexity the-\nory has managed to explain the difficulty of computing\nground state properties of quantum spin systems in vari-\noussettings,suchastwo-dimensional(2D)lattices[1]and\neven one-dimensional(1D) chains [2], aswell as fermionic\nsystems [3].\nWe will determine the computational difficulty of two\nproblems: First, computing the density of states of a lo-\ncal Hamiltonian, and second, counting the ground state\ndegeneracy of a local gapped Hamiltonian; in both cases,\nthe result holds even if the Hamiltonian is restricted to\nact on a 2D lattice of qubits, or on a 1D chain. To this\nend, wewill introduce the quantum counting class #BQP\n(sharpBQP),whichconstitutes the naturalcountingver-\nsionoftheclass QMA(QuantumMerlinArthur)whichit-\nself captures the difficulty of computing the ground state\nenergy of a local Hamiltonian [4, 5]. Vaguely speaking,\n#BQPcounts the number of possible “quantum solu-\ntions” to a quantum problem that can be verified using aquantum computer. We show that both problems, com-\nputingthedensityofstatesandcountingthegroundstate\ndegeneracy, are complete problems for the class #BQP,\ni.e., they are among the hardest problems in this class.\nHaving quantified the difficulty of computing the den-\nsity of states and counting the number of ground states,\nwe proceed to relate #BQPto known classical counting\ncomplexity classes, and show that #BQPequals#P(un-\nder weaklyparsimoniousreductions). Here, the complex-\nity class#Pcounts the number of satisfying assignments\nto any efficiently computable boolean function. This can\nbeaveryhardproblemwhichisbelievedtotakeexponen-\ntial time; in particular, it is at least as hard as deciding\nwhether the function has at least one satisfying input,\ni.e., the complexity class NP. Examples for #P-complete\nproblems(i.e., thehardestproblemsinthat class)include\ncounting the number of colorings of a graph, or com-\nputing the permanent of a matrix with binary entries.\nPhrased in terms of Hamiltonians, what we show is that\ncomputing the density of states and counting the ground\nstate degeneracy of a classical spin system is just as hard\nas solving the same problem for a quantum Hamiltonian.\nQuantum complexity classes.— Let us start by intro-\nducing the relevant complexity classes. The central role\nin the following is taken by the verifierV, which veri-\nfies “quantum solutions” (also called proofs) to a given\nproblem. More formally, a verifier checking an n-qubit\nquantumproof(thatis, aquantumstate |ψ/an}bracketri}ht)consistsofa\nT= poly(n) length quantum circuit U=UT···U1(with\nlocalgatesUt)actingonm= poly(n) qubits, whichtakes\nthen-qubit quantum state |ψ/an}bracketri}htIas an input, together\nwithm−ninitialized ancillas, |0/an}bracketri}htA≡ |0···0/an}bracketri}htA, applies\nU, and finally measures the first qubit in the {|0/an}bracketri}ht1,|1/an}bracketri}ht1}\nbasis to return 1 (“proof accepted”) or 0 (“proof re-\njected”). Then, the class QMAcontains all problems of\nthe form: “Decide whether there exists a |ψ/an}bracketri}htsuch that\npacc(V(ψ))> a, or whether pacc(V(ψ))< bfor all|ψ/an}bracketri}ht,\nfor some chosen a−b>1/poly(n), given that one is the\ncase”. Here, the acceptance probability of a state |ψ/an}bracketri}htis\npacc(V(ψ)) :=/an}bracketle{tψ|Ω|ψ/an}bracketri}ht, with\nΩ = ( /BDI⊗/an}bracketle{t0|A)U†(|1/an}bracketri}ht/an}bracketle{t1|1⊗ /BD)U( /BDI⊗|0/an}bracketri}htA),(1)\nwhich we illustrate in Fig. 1.2\nFIG. 1: A QMAverifier consists of a sequence of Tlocal uni-\ntarygates acting onthe “quantumproof” |ψ/angbracketrightandan ancillary\nregister initialized to |0/angbracketright. The final measurement on the first\nqubit returns |1/angbracketrightor|0/angbracketrightto accept or reject the proof, respec-\ntively. Transition probabilities can be computed by doing a\n“path integral” over all intermediate configurations ( ik)k.\nThe idea behind this definition is that QMAquanti-\nfies the difficulty of computing the ground state energy\nE0(H) of a local Hamiltonian Hup to 1/poly(n) accu-\nracy. Let the verifier be a circuit estimating /an}bracketle{tψ|H|ψ/an}bracketri}ht;\nthen a black box solving QMAproblems can be used\nto compute E0(H) up to 1/poly(n) accuracy by binary\nsearch using a single QMAquery. Note also that QMAis\nthe quantum version of the class NP, where one is given\nan efficiently computable boolean function f(x)∈ {0,1}\nand one must figure out if there is an xsuch that\nf(x) = 1.\nThe class NPhas a natural counting version, known\nas#P. Here, the task is to determine the numberrather\nthan the existence of satisfying inputs, i.e., to compute\n|{x:f(x) = 1}|. We will now analogously define #BQP,\nthe countingversionof QMA. Consider the verifying map\nΩ of Eq. (1) for a QMAproblem, with the additional\npromise that Ω does not have eigenvalues between aand\nb,a−b >1/poly(n). Then the class #BQPconsists of\nall problems of the form “compute the dimension of the\nspace spanned by all eigenvectors with eigenvalues ≥a”.\nAn equivalent definition for #BQP(cf. also [6, 7]) is\nthe following: Consider a verifier Ω with the additional\npromisethatthereexistsubspaces A⊕R=C2nsuchthat\n/an}bracketle{tψ|Ω|ψ/an}bracketri}ht ≥afor all|ψ/an}bracketri}ht ∈ A, and/an}bracketle{tψ|Ω|ψ/an}bracketri}ht ≤bfor all|ψ/an}bracketri}ht ∈\nR, whereagain a−b>1/poly(n) –wecanthink of Aand\nRascontainingthe goodandbad witnesses, respectively.\n(Note that there will always be “mediocre” witnesses—\nthe question is whether there exists a decomposition into\nagoodandabadwitnessspace.) Then, #BQPconsistsof\nall problems of the form “compute dim A”. This number\nis well-defined, i.e., independent of the choice of Aand\nR, andmoreover,onecaneasilyshowthatit isequivalent\nto the definition above, cf. the Supplementary Material.\nThe gap promise we impose on the spectrum of Ω is\nnot present in the definition of QMA(though similarly\nrestricted versions of QMAwere defined in [6, 7]). Nev-\nertheless, this promise emerges naturally when consider-\ning the counting version: QMAcaptures the difficulty of\ndetermining the existence of an input state with accep-tance probability above a, up to a “grace interval” [ b,a]\nin which mistakes are tolerated (i.e., if the largest eigen-\nvalue of Ω is in [ b,a], the oracle can return either out-\ncome). Correspondingly, #BQPcaptures the difficulty\nof counting the number of eigenvalues above a, where\neivenvalues in the grace interval [ b,a] can be miscounted.\nThe reason why we choose to define #BQPwith a gap\npromise rather than with a grace interval is the same as\nforQMA, namely to have a unique outcome associated\nwith any input.\nSimilarly, the idea of the Hamiltonian formulation of\nthe problem which we will discuss below is to ask for the\nnumber of eigenstates in a certain energy interval, where\nstates which are in some small 1 /poly(n) neighborhood\nof this interval may be miscounted; again, for reasons of\nrigor we choose to consider only Hamiltonians with no\neigenstates in that interval. It should be noted, however,\nthat all of the equivalence proofs we give equally apply\nif we choose to allow for miscounting of states in those\ngrace intervals instead of requiring them to be empty, as\nthe proofs do not make use of the gap promise itself, but\nrather show that all states outside those grace intervals\nare mapped (and thus counted) correctly. Thus, while\nthe actual number returned by the grace interval formu-\nlationofthecountingproblemsmightchangeunderthose\nmappingsdue todifferent treatmentofstatesin the grace\ninterval, it will still be in the correct range.\nThe class #BQPinherits the important property from\nQMAofbeing stableunder amplification, that is, the def-\ninition of #BQPis not sensitive to the choice of aand\nb. In particular, any a−b >1/poly(n) can be ampli-\nfied (by building a new poly-size Ω′from Ω) such that\na′= 1−exp(−poly(n)),b′= exp(−poly(n)), and keep-\ning the eigenvalue gap between a′andb′, by using a\nconstruction called strong amplification , cf. Ref. [8]; as\nshown there, strong amplification acts on all eigenval-\nues independently and thus also applies to #BQP. The\ncrucial point is that strong amplification works without\nchanging the proof itself, compared to weak amplifica-\ntion which takes multiple copies of the proof as an input.\nWhile this is fine for QMA, it does change the dimension\nof the accepting subspace in an unpredictable way and is\nthus not an option for the amplification of #BQP.\nComplexity of computing the density of states.— Let\nus now show why the class #BQPis relevant for physical\napplications. In particular, we are going to show that\ncomputing the density of states of a local n-spin Hamil-\ntonianH=/summationtext\niHiwith few-body terms Hi,/bardblHi/bardbl ≤1,\nup to accuracy1 /poly(n), is a problem which is complete\nfor#BQP, i.e., it is as hard as any problem in #BQPcan\nbe. The same holds true for the (a priori weaker) prob-\nlem of counting the ground state degeneracy of a local\nHamiltonian, given a 1 /poly(n) spectral gap above (note\nthat Bravyi et al.[9] suggested this as a definition for\na quantum counting class). We can impose additional\nrestrictions on the interaction structure of our Hamilto-3\nnian, and as we will see, the hardness is preserved even\nfor 2D lattices of qubits, or 1D systems.\nThe problem dos(density of states) is defined as fol-\nlows: Given a local Hamiltonian H=/summationtext\niHi, compute\nthe number of orthogonal eigenstates with eigenvalues in\nan interval [ E1,E2] withE2−E1>1/poly(n), where we\ndo not allow for eigenvalues within a small grace interval\nof width ∆ = ( E2−E1)/poly(n) centerd around E1and\nE2; alternatively, we can allow for errorneous counts of\neigenstates in that interval. Second, the problem #lh\n(sharp local Hamiltonian) corresponds to counting the\nnumber of ground states of a local Hamiltonian which\nhas a spectral gap ∆ = 1 /poly(n) above the ground state\nsubspace, given we are told the ground state energy, and\nwhere we allow for a small splitting in the ground state\nenergies; again, we can alternatively allow to miscount\nstates in the grace interval.\nClearly,#lhis a special instance of dos, i.e., solving\n#lhcan be reduced to solving dos. In order to show\nthatdosis contained in #BQP, we can use a phase es-\ntimation circuit [10] to estimate the energy of any given\ninput|ψ/an}bracketri}htand only accept if its energy /an}bracketle{tψ|H|ψ/an}bracketri}htis in the\ninterval [E1,E2]; as the desired accuracy ∆ = 1 /poly(n),\nthis can be done efficiently. A detailed proof (using a\nmore elementary circuit) is given in the Supplementary\nMaterial.\nLet us now conversely show that #lhis a hard prob-\nlem for#BQP, that is, any problem in #BQPcan be\nreduced to counting the ground states of some gapped\nlocal Hamiltonian [22]. As in turn #lhcan be reduced\ntodos, which is contained in #BQP, this proves that\nboth#lhanddosare complete problems for #BQP,\ni.e., they capture the full difficulty of this class. To this\nend, we need to start from an arbitrary verifier circuit\nU=UT···U1and construct a Hamiltonian which has as\nmany ground states as the circuit has accepting inputs\n(corresponding to the outcome |1/an}bracketri}ht1on the first qubit).\nLetAandRbe the eigenspacesof Ω [Eq. (1)] with eigen-\nvalues≥a= 1−2−poly(n)and≤b= 2−poly(n), respec-\ntively, and define U[R] :={U|ψ/an}bracketri}htI|0/an}bracketri}htA:|ψ/an}bracketri}htI∈ R}.\nWe will follow Kitaev’s original construction for a\nHamiltonian encoding a QMAverifier circuit [4, 5], which\nfor any proof |ψ/an}bracketri}htI∈ Ahas the “proof history” |Φ/an}bracketri}ht=/summationtextT\nt=0Ut···U1|ψ/an}bracketri}htI|0/an}bracketri}htA|t/an}bracketri}htTas its ground state, where\nthe third register is used as a clock. The Hamiltonian\nH=Hinit+/summationtextT\nt=1Hevol(t) +Hfinalhas three types of\nterms:Hinit= /BD⊗( /BD−|0/an}bracketri}ht/an}bracketle{t0|A)⊗|0/an}bracketri}ht/an}bracketle{t0|Tmakes sure the\nancilla is initialized, Hevol(t) =−Ut⊗ |t/an}bracketri}ht/an}bracketle{tt−1|T+ h.c.\nensures proper evolution from t−1 tot, andHfinal=\nΠU[R]⊗ |T/an}bracketri}ht/an}bracketle{tT|Tgives an energy penalty to states |Φ/an}bracketri}ht\nbuilt from proofs |ψ/an}bracketri}htI∈ R. Note that our Hfinaldiffers\nfrom the usual choice |0/an}bracketri}ht/an}bracketle{t0|1⊗ /BD⊗|T/an}bracketri}ht/an}bracketle{tT|Tand is in fact\nnon-local; as we show in the Supplementary Material,\nthis does not significantly change the relevant spectral\nproperties (in particular, we keep the 1 /poly(n) gap, and\nthe ground state subspace is split at most exponentially).With this choice of Hfinal,Hacts independently on the\nsubspaces spanned by {Ut···U1|ψ/an}bracketri}htI|x/an}bracketri}htA|t/an}bracketri}htT}t=0,...,Tfor\nany|ψ/an}bracketri}ht ∈ Aor|ψ/an}bracketri}ht ∈ R, and|x/an}bracketri}htAthe computational\nbasis, and the restriction of Hto any of these subspaces\ndescribes a random walk which is characterized by the\nchoice of |ψ/an}bracketri}htIand the number of 1’s in |x/an}bracketri}htA. These cases\ncan be analyzed independently (see Supplementary Ma-\nterial), and it follows that Hhas a dim A-fold degenerate\nground state space with a 1 /poly(n) gap above, proving\n#BQP-hardness of #lh.\nThis shows that #lhis#BQP-hard for a Hamiltonian\nwhich is a sum of log T-local terms (i.e., each term acts\non logTsites), as the clock register is of size log T. In\norder to obtain a k-body Hamiltonian, Kitaev suggested\nto use a unary encoding of the clock (i.e., |t/an}bracketri}htTis encoded\nas|1···10···0/an}bracketri}ht, witht1’s), so that each Hamiltonian\nterm only acts on three qubits of the clock. However,\nthis makes the Hilbert space of the clock too big, and\nterms need to be added to the Hamiltonian to penal-\nize illegal clock configurations. These terms divide the\nHilbert spaceinto twoparts, Hlegal⊕Hillegal. Here,Hlegal\ncontains only legal clock states, whereas Hillegalcontains\nonly configurations with illegal clock states [4, 5]. Since\nno Hamiltonian term couples these two subspaces, the\nHamiltonian can be analyzed independently on the two\nsubspaces. It turns out that its restriction to Hillegalhas\nan at least 1 /poly(n) higher energy, while on Hlegal, the\nHamiltonian is exactly the same as before. Thus, one\nfinds that the new Hamiltonian still has the right num-\nber of ground states, and a 1 /poly(n) spectral gap. The\nvery same argument applies in the case of 1D Hamiltoni-\nans, using the QMA-construction of Ref. [2]: Again, the\nHamiltonianactsindependently on a“legal”and an“ille-\ngal” subspace, where the latter has a polynomially larger\nenergy, and the former reproduces the (low-energy) spec-\ntrum of the original Hamiltonian [11].\nAn alternative way to prove QMA-hardness on re-\nstricted interactiongraphsis to use so-called perturbation\ngadgets, which yield the Hamiltonian of the Kitaev con-\nstruction above from a perturbative expansion; in partic-\nular, this way one can show QMA-hardness of Pauli-type\nnearest-neighbor Hamiltonians on a 2D square lattice of\nqubits [1]. As shown in Ref. [12], such gadgets do in fact\napproximately preserve the whole low-energy part of the\nspectrum, and thus, our #BQP–hardness proof for #lh\nstill applies to these classes of Hamiltonians. It should\nbe noted, however, that since the eigenvalues are only\npreserved up to an error 1 /poly(n), the splitting of the\nground state space will now be of order 1 /poly(n); how-\never, it can still be chosen to be polynomially smaller\nthan the spectral gap.\nQuantum vs. classical counting complexity.— As we\nhave seen, the quantum counting class #BQPexactly\ncaptures the difficulty of counting the degeneracy of\ngroundstatesandcomputingthedensityofstatesoflocal\nquantum Hamiltonians. In the following, we will relate4\n#BQPto classicalcountingclassesand provethat #BQP\nis equal to #P, counting the number of satisfying inputs\nto a boolean function [23]. In physical terms, this shows\nthatcountingthenumberofgroundstatesordetermining\nthe density of states for a quantum Hamiltonian is not\nharder than either problem is for a classical Hamiltonian.\nClearly, #Pis contained in #BQP, as the latter in-\ncludes classical verifiers. It remains to be shown that\nany#BQPproblem can be solved by computing a #P\nfunction. We start from a verifier operator Ω, Eq. (1),\nand wish to show that the dimension of its accepting\nsubspace, i.e., the subspace Awith eigenvalues ≥a, can\nbe computed by counting satisfying inputs to some ef-\nficiently computable boolean function. Using amplifica-\ntion, we can ensure that |dimA−trΩ| ≤1\n4, i.e., we need\nto compute trΩ to accuracy1\n4. This can be done using\na “path integral” method, which has been used previ-\nously to show containments of quantum classes in the\nclassical classes PPand#P(see e.g. [13]). We rewrite\ntrΩ =/summationtext\nζf(ζ)asasumoverproductsoftransitionprob-\nabilities along a path ζ≡(i0,...,iN,j1,...,j N), where\nf(ζ) =/an}bracketle{ti0|I/an}bracketle{t0|AU†\n1|j1/an}bracketri}ht/an}bracketle{tj1|U†\n1···U†\nT|jT/an}bracketri}ht× (2)\n/an}bracketle{tiT|/bracketleftbig\n|0/an}bracketri}ht/an}bracketle{t0|1⊗ /BD/bracketrightbig\n|iT/an}bracketri}ht/an}bracketle{tiT|UT···U1|i0/an}bracketri}htI|0/an}bracketri}htA\n(cf. Fig. 1). Since any such sum over an efficiently com-\nputablef(ζ) can be determined by counting the satis-\nfying inputs to some boolean formula (see the Supple-\nmentary Material for details), it follows that Ω can be\ncomputed using a single query to a black box solving #P\nproblems.\nSummary and discussion.— In this work, we consid-\nered two problems: Computing the density of states and\ncomputing the ground state degeneracy of a local Hamil-\ntonian of a spin system. In order to capture the com-\nputational difficulty of these problems we introduced the\nquantum complexity class #BQP, the counting version\nof the class QMA. We proved that this complexity class\nexactly captures the difficulty of our two problems, even\nwhen restricting to local Hamiltonians on 2D lattices of\nqubits or to 1D chains, since all these problems are com-\nplete problems for the class #BQP[24].\nWe have further shown that #BQPis no harder than\nits classical counterpart #P. In particular this implies\nthat computing the density of states is no harder for\nquantumHamiltoniansthanit isforclassicalones. While\nthis quantum-classical equivalence might seem surpris-\ning at the Hamiltonian level, it should be noted that\nthe classes #PandPPquite often form natural “upper\nbounds” for many quantum andclassical problems.\nWhat about the problem of computing the density of\nstates for fermionic systems, such as many-electron sys-\ntems? On the one hand, this problem will be still in\n#BQPand thus #P, since any local fermionic Hamilto-\nnian can be mapped via the Jordan-Wigner transform to\na(non-local)Hamiltonianonaspinsystem,whoseenergycanstillbeestimatedefficientlybyaquantumcircuit[14].\nOn the other hand, hardness of the problem for #BQP\ncan be shown e.g. by using the #BQP-hardness of #lh,\nand encoding each spin using one fermion in two modes,\nsimilar to [14]. Thus, computing the density of states for\nfermionic systems is a #BQP-complete problem as well.\nAcknowledgments.— We thankS.Aaronson,D. Gottes-\nman, D. Gross, Y. Shi, M. van den Nest, J. Watrous\nand S. Zhang for helpful discussions and comments. Re-\nsearch at PI is supported by the Government of Canada\nthroughIndustry Canadaand by the Provinceof Ontario\nthrough the Ministry of Research and Innovation. NS is\nsupported by the Gordon and Betty Moore Foundation\nthrough Caltech’s Center for the Physics of Information\nand NSF Grant No. PHY-0803371.\nNote added.— After completion of this work, we\nlearned that Shi and Zhang [15] have independently de-\nfined#BQPand shown its relation to #Pusing the same\ntechnique.\n[1] R. Oliveira and B. M. Terhal, Quant. Inf. Comput. 8,\n900 (2009), quant-ph/0504050.\n[2] D. Aharonov, D. Gottesman, S. Irani, and J. Kempe,\nCommun. Math. Phys. 287, 41 (2009), arXiv:0705.4077.\n[3] N. Schuch and F. Verstraete, Nature Physics 5, 732\n(2009), arXiv:0712.0483.\n[4] A.Y.Kitaev, A.H.Shen, andM.N.Vyalyi, Classical and\nquantum computation (American Mathematical Society,\nProvidence, Rhode Island, 2002).\n[5] D. Aharonov and T. Naveh, (2002), quant-ph/0210077.\n[6] D.Aharonov, M. Ben-Or, F. G.Brandao, andO.Sattath,\n(2008), arXiv:0810.4840.\n[7] R. Jain, I. Kerenidis, M. Santha, and S. Zhang, (2009),\narXiv:0906.4425.\n[8] C. Marriott and J. Watrous, Comput. Complex. 14, 122\n(2005), cs/0506068.\n[9] S. Bravyi, C. Moore, and A. Russell, (2009),\narXiv:0907.1297.\n[10] R.Cleve, A.Ekert, C.Macchiavello, andM.Mosca, Proc.\nR. Soc. Lond. 454, 339 (1998), quant-ph/9708016.\n[11] N. Schuch, I. Cirac, and F. Verstraete, Phys. Rev. Lett.\n100, 250501 (2008), arXiv:0802.3351.\n[12] J. Kempe, A. Kitaev, and O. Regev, SIAM Journal of\nComputing 35, 1070 (2006), quant-ph/0406180.\n[13] L. Adleman, J. DeMarrais, and M.-D. Huang, SIAM J.\nComput. 26, 1524 (1997).\n[14] Y.-K. Liu, M. Christandl, and F. Verstraete, Phys. Rev.\nLett.98, 110503 (2007), quant-ph/0609125.\n[15] Y. Shi and S. Zhang, private communication;\nnotes available at http://www.cse.cuhk.edu.hk/\n~syzhang/papers/SharpBQP.pdf .\n[16] R. Bhatia, Matrix Analysis (Springer, 1997).\n[17] C. Jordan, Bulletin de la Soci´ et´ e Math´ ematique de\nFrance3, 103 (1875).\n[18] P. Halmos, Trans. Amer. Math. Soc. 144, 381 (1969).\n[19] E. Bernstein and U. Vazirani, Siam J. of Comp. 26, 1411\n(1997), quant-ph/9701001.\n[20] F.G.S.L. Brandao, PhD thesis (2008), arXiv:0810.0026 .5\n[21] T. J. Osborne, (2006), cond-mat/0605194.\n[22] Note that the connection between QMAwith a unique\n“good witness”, such as in our #BQPdefinition, and\nlocal Hamiltonians with unique ground state and a\n1/poly(n) gap has been shown in [6].\n[23] Formally speaking, the reduction from #BQPto#Pis\nweakly parsimonious , i.e., for any function f∈#BQP\nthere exist polynomial-time computable functions αandβ, and a function g∈#P, such that f=α◦g◦β. This\ndiffers from Karp reductions where no postprocessing is\nallowed,α= Id, but it still only requires a single call to\na#Poracle.\n[24] Note that computing the density of states to multiplica -\ntive accuracy is less difficult, see Refs. [20, 21].\nSUPPLEMENTARY MATERIAL\nQuantum Complexity Classes\nWe start with the relevant definitions. Let xbe a bi-\nnary string. Then, we denote by the verifier V≡Vxa\nquantum circuit of length T= poly(|x|),U=UT···U1\n(with local gates Ut) acting on m= poly(|x|) qubits,\nwhich is generated uniformly from x. The verifier takes\nann= poly(|x|) qubit quantum state |ψ/an}bracketri}htIas an in-\nput (we will express everything in terms of ninstead\nof|x|in the following), together with m−ninitialized\nancillas, |0/an}bracketri}htA≡ |0···0/an}bracketri}htA, appliesU, and finally mea-\nsures the first qubit in the {|0/an}bracketri}ht1,|1/an}bracketri}ht1}basis to return\n1 (“proof accepted”) or 0 (“proof rejected”). The ac-\nceptance probability for a proof |ψ/an}bracketri}htis then given by\npacc(V(ψ)) :=/an}bracketle{tψ|Ω|ψ/an}bracketri}ht, with\nΩ = ( /BDI⊗/an}bracketle{t0|A)U†(|1/an}bracketri}ht/an}bracketle{t1|1⊗ /BD)U( /BDI⊗|0/an}bracketri}htA).(3)\nDefinition 1 (QMA).Let0≤a,b≤1s.th.a−b>1\np(n)\nfor some polynomial p(n). A language Lis inQMA(a,b)\nif there exists a verifier s.th.\nx∈L⇒λmax(Ω)>a\nx /∈L⇒λmax(Ω)<b .\nDefinition 2 (#BQP).Let0≤a,b≤1s.th.a−b >\n1/poly(n), and letΩbe a verifier map with noeigenvalues\nbetweenaandb. Then, the class #BQP(a,b)consists of\nall problems of the form “compute the dimension of the\nspace spanned by all eigenvectors of Ωwith eigenvalues\n≥a” .\nAn alternative definition for #BQPis the following:\nDefinition 3 (#BQP, alternate definition) .Consider a\nverifierΩwith the property that there exist subspaces A⊕\nR=CN(N= 2n) such that /an}bracketle{tψ|Ω|ψ/an}bracketri}ht ≥afor all|ψ/an}bracketri}ht ∈ A,\nand/an}bracketle{tψ|Ω|ψ/an}bracketri}ht ≤bfor all|ψ/an}bracketri}ht ∈ R, where again a−b >\n1/poly(n). Then#BQP(a,b)consists of all problems of\nthe form “compute dimA”.\nNote that dim Ais well-defined: Consider two decom-\npositions CN=A⊕RandCN=A′⊕R′. Without lossof generality, if we assume dim A>dimA′, it follows\ndimA+dimR′>N, and thus there exists a non-trivial\n|µ/an}bracketri}ht ∈ A∩R′, which contradicts the definition.\nTheorem 4. Definition 2 and Definition 3 are equiva-\nlent.\nProof.ToshowthatDefinition2impliesDefinition3, let\nAbe spanned by the eigenvectors with eigenvalues ≥a.\nTo show the converse, we use the minimax principle for\neigenvalues [16], which states that the kth largest eigen-\nvalueλkof a Hermitian operator Ω in an N-dimensional\nHilbert space can be obtained from either of the equiva-\nlent optimizations\nλk(Ω) = max\nMkmin\n|x/angbracketright∈Mk/an}bracketle{tx|Ω|x/an}bracketri}ht (4)\n= min\nMN−k+1max\n|x/angbracketright∈MN−k+1/an}bracketle{tx|Ω|x/an}bracketri}ht,(5)\nwhereMkis a subspace of dimension k, and|x/an}bracketri}htis a unit\nvector. Now notice that Def. 3 implies that\nmin\n|x/angbracketright∈A/an}bracketle{tx|Ω|x/an}bracketri}ht ≥aand max\n|x/angbracketright∈R/an}bracketle{tx|Ω|x/an}bracketri}ht ≤b.(6)\nNext, consider the minimax theorem for k= dimA.\nFrom Eq. (4) we have\nλdimA= max\nMdimAmin\n|x/angbracketright∈MdimA/an}bracketle{tx|Ω|x/an}bracketri}ht\n≥min\n|x/angbracketright∈A/an}bracketle{tx|Ω|x/an}bracketri}ht\n≥a.\nNow consider the case that k= dimA+1. From Eq. (5),\nusing the fact that N−(dimA+1)+1 = dim R, we have\nλdimA+1= min\nMdimRmax\n|x/angbracketright∈MdimR/an}bracketle{tx|Ω|x/an}bracketri}ht\n≤max\n|x/angbracketright∈R/an}bracketle{tx|Ω|x/an}bracketri}ht\n≤b.\nThus we have\nλdimA≥a>b≥λdimA+1, (7)6\nsincea−b≥1/poly(n). It follows that λdimAis the\nsmallest eigenvalue of Ω which is still larger than a, and\ntherefore the span of the first dim Aeigenvectors of Ω\nis equal to the span of all eigenvectors with eigenvalue\n≥a. The equivalence follows. /square\nThe class #BQP(a,b) inherits the useful property of\nstrong error reduction from QMA: the thresholds ( a,b)\ncan be amplified to (1 −2−r,2−r),r= poly(n) while\nkeeping the size of the proof:\nTheorem 5. Let#BQPn(a,b)denote#BQPwith an\nnqubit witness. Then #BQPn(a,b)⊂#BQPn(1−\n2−r,2−r)for everyr∈poly(n).\nThis followsdirectly from the strongamplification pro-\ncedure presented in [8], which describes a procedure to\namplify any verifier map Ω such that any eigenvalue\nabovea(belowb) is shifted above 1 −2��r(below 2−r) at\nan overhead polynomial in r.\nUsing Thm. 5, we will write #BQPinstead of\n#BQP(a,b) from now on, where a−b >poly(n), and\na,bcan be exponentially close to 1 and 0, respectively.\nThe Complexity of the Density of States\nWe now use the class #BQPto characterize the com-\nplexity of the density of states problem and the problem\nof counting the number of ground states of a local Hamil-\ntonian. We start by defining these problems, as well as\nthe notion oflocal Hamiltonian, and then showthat both\nproblems are #BQP-complete.\nDefinition 6 (k-local Hamiltonian) .Given a set of\npoly(n)quantum spins each with dimension bounded by\na constant, a Hamiltonian Hfor the system is said to\nbek-local ifH=/summationtext\niHiis a sum of at most poly(n)\nHermitian operators Hi,/bardblHi/bardbl ≤1, each of which acts\nnontrivially on at most kspins.\nNote thatk-local does not imply any geometric local-\nity, only that each spin interacts with at most k−1 other\nspins for any given interaction term. However, we re-\nstrict ourselves to k=O/parenleftbig\nlog(n)/parenrightbig\nso that each Hican be\nspecified by an efficient classical description.\nDefinition 7 (dos).LetE2−E1>1/poly(n),∆ =\n(E2−E1)/poly(n), andletH=/summationtext\niHibe ak-local Hamil-\ntonian such that Hhas no eigenvalues in the intervals\n[E1−∆\n2,E1+∆\n2]and[E2−∆\n2,E2+∆\n2]. Then, the prob-\nlemdos(density of states) is to compute the number\nof orthogonal eigenstates with eigenvalues in the interval\n[E1,E2].\nDefinition 8 (#lh).LetE0≤E1<E2, withE2−E1>\n1/poly(n), and letH=/summationtext\niHibe ak-local Hamiltonian\ns.th.H≥E0, andHhas no eigenvalues between E1andE2. Then, the problem #lh≡#lh(E1−E0)(sharp\nlocal Hamiltonian) is to compute the dimension of the\neigenspace with eigenvalues ≤E1.\nNote that #lhdepends on the “energy splitting”\nE1−E0of the low-energy subspace. In particular, for\nE1−E0= 0,#lh(0) corresponds to computing the de-\ngeneracy of the ground state subspace. As we will see\nin what follows, the class #lh(σ) is the same for any\nsplitting exp( −poly(n))≤σ≤1/poly(n).\nWe now show that #lhanddosare both #BQP-\ncomplete. We do so by giving reductions from\n#lh(1/poly(n)) todos, fromdosto#BQP, and from\n#BQPto#lh/parenleftbig\nexp(−poly(n))/parenrightbig\n; this will at the same\ntime prove the claimed independence of #lh(σ) of the\nsplittingσ.\nTheorem 9. #lh(1/poly(n))reduces to dos.\nProof.If we denote the parameters of the #lhproblem\nby˜E0,˜E1,˜E2, then we can simply relate them to the\nparameters E1,E2,∆ of adosproblem by ∆ = ˜E2−˜E1,\nE1=˜E0−1\n2∆ andE2=˜E1+1\n2∆, and the result follows\ndirectly. /square\nTheorem 10. dosis contained in #BQP.\nProof. We start with a k-local Hamiltonian Has in\nDef. 7. Now define a new Hamiltonian\nH′:=ν(H2−(E1+E2)H+E1E2).\nH′is a 2k-local Hamiltonian; here, ν= 1/poly(n) is\nchosen such that each term in H′is subnormalized. Any\neigenvalueof Hin the interval[ E1+∆\n2;E2−∆\n2]translates\ninto an eigenvalue of H′which is below\nA:=−ν∆\n2(E2−E1−∆\n2)≤ −1/poly(n),\nwhereas any eigenvalue outside [ E1−∆\n2;E2+∆\n2] trans-\nlates into an eigenvalues of H′above\nB:=ν∆\n2(E2−E1+∆\n2)≥1/poly(n).\nThe original dosproblem now translates into counting\nthe number of eigenstates of H′with negative energy,\ngiven a spectral gap in a 1 /poly(n) sized interval [ A;B]\naroundzero. Wenowusethe circuitwhichwasusedin[5]\nto prove that log-local Hamiltonian is in QMA; it accepts\nany input state |ψ/an}bracketri}htwith probability\np(|ψ/an}bracketri}ht) =1\n2−/an}bracketle{tψ|H′|ψ/an}bracketri}ht\n2m,\nwherem= poly(n) is the number of terms in H′. [The\nidea is to randomly pick one term H′\niin the Hamiltonian\nand toss a coin with probability (1 − /an}bracketle{tψ|H′\ni|ψ/an}bracketri}ht)/2.]\nComputing the answer to the original dosproblem\nthen translates to counting the number of states with7\nacceptance probability ≥a=1\n2+A\n2m, given that there\nare no eigenstates between aandb=1\n2+B\n2m, and since\na−b= (A−B)/2m≥1/poly(n), this shows that this\nnumber can be computed in #BQP./square\nTheorem 11. #lh/parenleftbig\nexp(−poly(n))/parenrightbig\nis#BQP-hard.\nProof. To show #BQP-hardness of #lh, we need\nto start with an arbitrary QMAverifier circuit U=\nUT...U1and construct a Hamiltonian with as many\nground states as the circuit has accepting inputs. By\namplification, we can assume that the acceptance and re-\njection thresholds for the verifier are a= 1−ǫandb=ǫ,\nwhere we can choose ǫ=O(exp(−p(n))) for any polyno-\nmialp(n). As before, let AandRbe the eigenspaces of\nΩ with eigenvalues ≥aand≤b, respectively. Define\nU[R] :={U|ψ/an}bracketri}htI|0/an}bracketri}htA:|ψ/an}bracketri}htI∈ R} (8)\nand denote the projector onto this space by Π U[R]. No-\ntice that for any state |χ/an}bracketri}ht ∈U[R], due to our rejection\nthresholdb=ǫ, we have\n/an}bracketle{tχ|/parenleftbig\n|1/an}bracketri}ht/an}bracketle{t1|1⊗ /BD/parenrightbig\n|χ/an}bracketri}ht ≤ǫ. (9)\nWe now followKitaev’soriginalconstructionto encode\naQMAverifier circuit into a Hamiltonian which has a\n“proof history” as its ground state for any proof |φ/an}bracketri}htI∈\nA[4, 5]. That is, the ground states of the Hamiltonian\nare given by\n|Φ/an}bracketri}ht=T/summationdisplay\nt=0Ut...U1|φ/an}bracketri}htI|0/an}bracketri}htA|t/an}bracketri}htT (10)\nwhere the third register is used as a “clock”. The Hamil-\ntonian has the form\nH=Hinit+T/summationdisplay\nt=1Hevol(t)+Hfinal (11)\nwhere\n•Hinit= /BDI⊗( /BD− |0/an}bracketri}ht/an}bracketle{t0|A)⊗ |0/an}bracketri}ht/an}bracketle{t0|Tchecks that\nthe ancilla is property initialized, penalizing states\nwithout properly initialized ancillas;\n•Hevol(t) =−1\n2Ut⊗|t/an}bracketri}ht/an}bracketle{tt−1|T−1\n2U†\nt⊗|t−1/an}bracketri}ht/an}bracketle{tt|T\n+1\n2\n/BD⊗|t/an}bracketri}ht/an}bracketle{tt|T+1\n2\n/BD⊗|t−1/an}bracketri}ht/an}bracketle{tt−1|T\nchecks that the propagation from time t−1 totis\ncorrect, penalizing states with erroneous propaga-\ntion;\n•Hfinal= ΠU[R]⊗|T/an}bracketri}ht/an}bracketle{tT|Tcauses each state |φ/an}bracketri}htbuilt\nfrom an input |ψ/an}bracketri}htI∈ R(but which nonetheless has\na correctly initialized ancilla) to receive an energy\npenalty.As we show in Lemma 12, the total Hamiltonian Hhas\na spectral gap 1 /poly(n) above the dim A–dimensional\nground state subspace. However, Hfinalis not a local\nHamiltonian, but as we argue in the following, it can be\nreplaced by the usual term Hstd\nfinal=|0/an}bracketri}ht/an}bracketle{t0|1⊗ /BD⊗|T/an}bracketri}ht/an}bracketle{tT|T\nwhile keeping the ground space dimension (up to small\nsplitting in energies) and the 1 /poly(n) spectral gap. As\nwe prove in Lemma 13,\nHstd\nfinal≥Hfinal−√ǫ /BD. (12)\nThus, replacing HfinalbyHstd\nfinalwill decrease the energy\nof any excited state by at most√ǫ=O(exp(−p(n)/2)).\n(The energy of the ground states is already minimal and\ncannot decrease.) On the other hand, the energy of any\nproper proof history for Hcannot increase by more than\n/an}bracketle{tχ|Hstd\nfinal|χ/an}bracketri}ht=/an}bracketle{tχ||0/an}bracketri}ht/an}bracketle{t0|1⊗ /BD|χ/an}bracketri}ht ≤ǫ=O(exp(−p(n)),\ndue to our choice of acceptance threshold, i.e., the low\nenergy subspace has dimension dim A. We thus obtain\na Hamiltonian with a dim Adimensional ground state\nsubspace with energy splitting ≤ǫ= exp(−p(n)) and a\n1/poly(n) spectral gap above. /square\nThe following two lemmas are used in the preceding\nproof of Theorem 11.\nLemma 12. Hhas a spectral gap of size 1/poly(n).\nProof. Our proof follows closely the discussion in\nRef. [4], cf. also [11]. We can block diagonalize Hby\nthe (conjugate) action of the following unitary operator,\nW=T/summationdisplay\nj=0Uj···U1⊗|j/an}bracketri}ht/an}bracketle{tj|T, (13)\nwhich maps H→H′=W†HW. As this has no effect\non the spectrum, we can work with the simpler H′from\nnow on.\nLet us explicitly write the effect of conjugation by W\non the terms of H. The first term is unaffected, H′\ninit=\nHinit. The final term becomes H′\nfinal= ΠR⊗|0/an}bracketri}ht/an}bracketle{t0|A⊗\n|T/an}bracketri}ht/an}bracketle{tT|T, where Π Ris simply the projectorontothe space\nR, and the ancillas are in the correct initial state.\nWe can now conjugate each of the terms in Hevol(t)\nseparately. For example, the first term gives\nW†(Ut⊗|t/an}bracketri}ht/an}bracketle{tt−1|)W= /BD⊗|t/an}bracketri}ht/an}bracketle{tt−1|.(14)\nThe other terms are exactly analogous, and we find that\nH′\nevol(t) = /BD⊗1\n2/bracketleftbig\n|t−1/an}bracketri}ht/an}bracketle{tt−1|+|t/an}bracketri}ht/an}bracketle{tt|\n−|t/an}bracketri}ht/an}bracketle{tt−1|−|t−1/an}bracketri}ht/an}bracketle{tt|/bracketrightbig\n.\nThe total evolution Hamiltonian is then block-diagonal\na matrix which looks like a hopping Hamiltonian in the\nclock register,\n/summationdisplay\ntH′\nevol(t) = /BD⊗E, (15)8\nwhere the ( T+ 1)-by-(T+ 1)–dimensional tri-diagonal\nmatrixEis given by\nE=\n1\n2−1\n2\n−1\n21−1\n2\n−1\n2...\n1−1\n2\n−1\n21\n2\n.(16)\nTo discuss the spectrum of H′, let\nS1=A⊗|0/an}bracketri}ht/an}bracketle{t0|A⊗CT+1,\nS2=R⊗|0/an}bracketri}ht/an}bracketle{t0|A⊗CT+1,\nandS3the orthogonal complement of S1⊕ S2; i.e.,S1\ncorresponds to evolutions starting from good proofs, S2\nto those starting from wrong proofs, and S3to evolutions\nwith wrongly initialized ancillas. Note that H′=H|S1⊕\nH|S2⊕H|S3, thus, we can analyze the spectrum for H|Sp\nseparately. Note that the restriction to Spdoes not affect\nH′\nevol(t). Since we expect the ground state subspace to\noccur on S1, we need to compute ground state energy\nand gap of H′|S1, as well as lower bound the ground\nstate energies of H′|S2andH′\nS3.\nFirst,H′\ninit|S=H′\nfinal|S= 0, i.e., the spectrum of H′|S\nequalsthe spectrum of E, which canbe straightforwardly\ndetermined to be 1 −cosωn, withωn=nπ/(T+1), and\neigenfunctions (cos1\n2ωn,cos3\n2ωn,...); the ground state\ndegeneracy is indeed dim Aas desired.\nOn the other hand, H′\nfinal|S2= 1, andH′\ninit|S3≥1, i.e.,\nin both cases the ground state energy of H′|Spis lower\nbounded by the ground state energy of\nE′=\n1\n2−1\n2\n−1\n21−1\n2\n−1\n2...\n1−1\n2\n−1\n23\n2\n,\nwhichhaseigenvalues1 −cosϑn, withϑn= (n+1\n2)π/(T+\n3\n2), and eigenfunctions (cos1\n2ϑn,cos3\n2ϑn,...).\nIt follows that H′(and thus H) has a ground\nstate energy of 1 −cosω0= 0, and a gap\n1−cosπ\n2T+3=O(1/T2) =O(1/poly(n)) above.\n/square\nIt remains to prove Eq. (12), which follows from the\nfollowingLemmabychoosing P=|0/an}bracketri}ht/an}bracketle{t0|1⊗ /BD,Q= ΠU[R],\nand using Eq. (9).\nLemma 13. LetPandQbe projectors such that /bardblQ( /BD−\nP)Q/bardbl∞≤ǫ. Then\nP−Q≥ −√ǫ /BD. (17)Proof. We begin by recalling the result due to Jor-\ndan [17] (see Ref. [18] for a more modern treatment)\nfor the simultaneous canonical form of two projectors.\nIn the subspace where PandQcommute, both opera-\ntors are diagonal in a common basis and the spectrum\nis either (0,0),(0,1),(1,0), or (1,1), and direct sums of\nthose terms. In the subspace where they don’t commute,\nthe problem decomposes into a direct sum of two-by-two\nblocks given by\nPj=/parenleftbigg\n1 0\n0 0/parenrightbigg\n, Qj=/parenleftbigg\nc2cs\ncs s2/parenrightbigg\n,(18)\nwheres= sin(θj) for some angle θj,c2+s2= 1, and the\nsubscriptjjust labels a generic block.\nIn fact, this two-by-two block form is completely gen-\neral if we allow embedding our projectors into a larger\nspace while preserving their rank. The rank-preserving\ncondition guarantees that our bound is unchanged, since\nwe are only appending blocks of zeros, and so we will\nconsider this two-by-two form without loss of generality.\nThe constraint that /bardblQ( /BD−P)Q/bardbl∞≤ǫimplies con-\nstraintsonthe values that sin( θj) can take. In particular,\nwecan directly compute this operatornormin eachblock\nseparately, and we find that for all j\n/bardblQj( /BD−Pj)Qj/bardbl∞= sin2(θj)≤ǫ. (19)\nWe can also directly compute in each block that\nPj−Qj=/parenleftbigg\n1−c21−cs\n−cs−s2/parenrightbigg\n. (20)\nThe spectrum of this operator is easily computed to\nbe±|sin(θj)|. Thus, the least eigenvalue of P−Qis\nbounded from below by −√ǫ./square\nQuantum vs. Classical Counting Complexity\nWe finally want to relate #BQPto the classical count-\ning class #P. It is clear that any #Pproblem can be\nsolved in #BQPby using a classical circuit. We will\nnow show that conversely, #BQPcan be reduced to #P\nunderweakly parsimonious reductions : That is, for any\nfunctionf∈#BQPthere exist polynomial-time com-\nputable functions αandβ, and a function g∈#P, such\nthatf=α◦g◦β. This differs from Karp reductions\nwhere no postprocessing is allowed, α= Id, but still only\nrequiresa singlecall to a #Poracle, in contrastto Turing\nreductions.\nTheorem 14. There exists a weakly parsimonious re-\nduction from #BQPto#P.\nProof.We start from a verifier operator Ω for a #BQP\nproblem. First, we use strong error reduction to let a=9\n1−2−(n−2)andb= 2−(n+2). It follows that\n|dimA−trΩ| ≤2n2−(n+2)=1\n4(21)\nand thus we need to compute trΩ to accuracy 1 /4. This\ncan be done using the “path integral” method previously\nused to show containments of quantum classes inside PP\nand#P[13]. We rewrite trΩ =/summationtext\nζf(ζ), where the sum\nis over products of transition probabilities along a path,\nwhich we label\nζ≡(i0,...,iN,j1,...,j N), (22)\nso that\nf(ζ) =/an}bracketle{ti0|I/an}bracketle{t0|AU†\n1|j1/an}bracketri}ht/an}bracketle{tj1|U†\n1···U†\nT|jT/an}bracketri}ht× (23)\n/an}bracketle{tiT|/bracketleftbig\n|0/an}bracketri}ht/an}bracketle{t0|1⊗ /BD/bracketrightbig\n|iT/an}bracketri}ht/an}bracketle{tiT|UT···U1|i0/an}bracketri}htI|0/an}bracketri}htA.\n(cf. Fig. 1 in the main manuscript for an illustration).\nSinceanyquantumcircuitcanberecastintermsofreal\ngates at the cost of doubling the number of qubits [19],\nwe can simplify the proof by assuming f(ζ) to be real.\nTo achieve the desired accuracy it is sufficient to approx-\nimatefup to|ζ|+ 2 digits, where |ζ|= poly(n) is the\nnumber of bits in ζ. Now define\ng(ζ) := round/bracketleftbig\n2|ζ|+2(f(ζ)+1)/bracketrightbig\n(24)and note that g(ζ) is a positive and integer-valued func-\ntion satisfying\n/vextendsingle/vextendsingle/vextendsingle/bracketleftbig\n2−|ζ|−2/summationdisplay\nζg(ζ)−1/bracketrightbig\n−/summationdisplay\nζf(ζ)/vextendsingle/vextendsingle/vextendsingle≤1\n4.(25)\nFinally, by defining a boolean indicator function,\nh(ζ,ξ) =/braceleftigg\n1 if 0≤ξ<g(ζ)\n0 otherwise\nwe can write g(ζ) =/summationtext\nξ≥0h(ζ,ξ), and thus,\n/summationdisplay\nζg(ζ) =/summationdisplay\nξ,ζh(ζ,ξ).\nThis shows that trΩ can be approximated to accuracy\n1\n4, and thus dim Acan be determined by counting the\nnumber of satisfying assignments of a single boolean\nfunctionh(ζ,ξ) that can be efficiently constructed from\nΩ, i.e., by a single query to a black box solving #P\nproblems, together with the efficient postprocessing\ndescribed by Eq. (25). /square"    },    {        "title": "1010.4935v1.Correlation_and_Entanglement_of_Multipartite_States.pdf",        "content": "arXiv:1010.4935v1  [quant-ph]  24 Oct 2010Correlation and Entanglement of Multipartite States\nY. B. Band and I. Osherov\nDepartments of Chemistry and Electro-Optics and the Ilse Ka tz Center for Nano-Science,\nBen-Gurion University, Beer-Sheva 84105, Israel\n(Dated: November 12, 2018)\nWe derive a classification and a measure of classical- and qua ntum-correlation of multipartite\nqubit, qutrit, and in general, n-level systems, in terms of SU( n) representations of density matrices.\nWe compare the measure for the case of bipartite correlation with concurrence and the entropy of\nentanglement. The characterization of correlation is in te rms of the number of nonzero singular\nvalues of the correlation matrix, but that of mixed state ent anglement requires additional invariant\nparameters in the density matrix. For the bipartite qubit ca se, the condition for mixed state\nentanglement is written explicitly in terms of the invarian t paramters in the density matrix. For\nidentical particle systems we analyze the effects of exchang e symmetry on classical and quantum\ncorrelation.\nPACS numbers: 03.67.-a, 03.67.Mn, 03.65.Ud\nQuantum entanglement is an information resource; it\nplays an important role in many protocols for quantum-\ninformation processing, including quantum computation\n[1], quantum cryptography [2], teleportation [3], super-\ndense coding [4], and quantum error correction protocols\n[5]. Techniques for characterizing the bipartite entangle-\nment and correlation of pure and mixed quantum states\nhaveenabledmanyadvancesinquantuminformationand\nthe study ofdecoherence[6]. Many quantum information\nprotocols use bipartite entanglement, but multipartite\nentanglement [7], also has quantum-information appli-\ncations, e.g., controlled secure direct communication [8],\nquantumerrorcorrection[9], controlledteleportation[10]\nandsecretsharing[11]. Ithasbeen shownthatanyinsep-\narable two-qubit states can be distilled to a singlet-state\nform with enough copies of the qubit-pairs [12] and algo-\nrithms for multi-copy entanglement distillation for pairs\nof qubits have been developed [13]. Moreover, multipar-\ntite entanglement offers a means of enhancing interfero-\nmetric precision beyond the standard quantum limit and\nis therefore relevant to increasing the precision of atomic\nclocksbydecreasingprojectionnoiseinspectroscopy[14].\nHere we use a representation of the density matrix for\nqubit, qutrit, and more generally, n-level systems con-\ntaining 2, 3, ..., and N-parts, in terms of the correla-\ntions between the subsystems to quantify the classical\nand quantum correlation of multipartite systems. Our\nclassification of correlation is in terms of the correlation\nmatrix and its singular values, and our classification of\nentanglement of mixed states [15] is associated with the\nPeres-Horodeckicriterion [19], which we express in terms\nof additional invariant parameters in the density matrix.\nSeparate measures of bipartite, tripartite, etc., correla-\ntion are required, since general mixed states can have\nbipartitecorrelationaswellashighersubsystem-number-\ncorrelation.\nWerner [15] defined a mixed state of an N-partite sys-\ntem asseparable , i.e.,classically-correlated , if it can bewritten as a convex sum,\nρ=/summationdisplay\nkpkρA\nkρB\nk...ρN\nk, pk>0,/summationdisplay\nkpk= 1,(1)\nwhereρA\nkis a valid density matrix of subsystem A,\netc. Otherwise, Werner defined it to be entangled , i.e.,\nquantum-correlated . Unfortunately, this definition of en-\ntanglement for mixed states is not constructive, since, in\ngeneral, it cannot be used to decide whether a given den-\nsity matrix is separable or entangled. Moreover, a quan-\ntitativemeasureofentanglementofmulti-partitesystems\nhas proven to be difficult to devise. Note that studies of\nthe best separable approximation to an arbitrary density\nhave been carried out and have led to a proposal of a\nmeasure for entanglement [16]. Furthermore, aspects of\nthe geometry of separability and entanglement based on\nSchmidt decomposition have been studied and led to an\nanalysis of the question of separability for the two-qubit\ncase [17].\nIn what follows, we categorize classically-correlated\nandquantum-correlatedstatesandcharacterizetheircor-\nrelation in terms the number of nonzero singular values\n[18],{di}, of the correlation matrix C, and characterize\nentanglementofbitpartite qubit systemsusing the Peres-\nHorodecki criterion [19] which is reformulated totally in\nterms of the parameters used in forming the density ma-\ntrix.\nFirst, let us consider a bipartite qubit system. For two\nuncorrelated qubits, call them AandB, we can write\nthe density matrix as a product, ρAB=ρAρB, where\nthe individual qubit density matrices can be written as\nρJ=1\n2(1+nJ��σJ), whereJ=A,B, theσJare Pauli\nmatrices for particle Jand the Bloch vectors are nJ=\n/an}bracketle{tσJ/an}bracketri}ht= TrσJρJ[20]. For two correlated qubits,\nρAB=1\n4/bracketleftbig\n(1+nA·σA)(1+nB·σB)+σA·CAB·σB/bracketrightbig\n,\n(2)\nwhere the tensor CABspecifies the qubit correlations,\nCAB\nij≡ /an}bracketle{tσi,Aσj,B/an}bracketri}ht−/an}bracketle{tσi,A/an}bracketri}ht/an}bracketle{tσj,B/an}bracketri}ht=/an}bracketle{tσi,Aσj,B/an}bracketri}ht−ni,Anj,B.\n(3)2\nThe density matrix ρABis a 4×4 Hermitian matrix with\ntrace unity, so 15 parameters are required to parame-\nterize it. The 3 components of nA, the 3 components\nofnB, and the 9 components Cijof the 3×3 matrix C,\nwhere we have no longer explicitly shown the subsystem\nsuperscripts, are sufficient for this purpose.\nSimilarly for the bipartite qutrit case. The 3 ×3 den-\nsity matrix of a single qutrit can be written as ρ=\n1\n3/parenleftbig\n1+3\n2/an}bracketle{tλi/an}bracketri}htλi/parenrightbig\nwhere theλiare the eight traceless Her-\nmitian Gellman matrices familiar from SU(3) [21], and\n/an}bracketle{tλi/an}bracketri}ht= Trλiρ. A bipartite qutrit density matrix can be\nparameterized in the form\nρAB=1\n9[(1+3\n2/an}bracketle{tλi,A/an}bracketri}htλi,A)(1+3\n2/an}bracketle{tλj,B/an}bracketri}htλj,B)+9\n4λi,ACijλj,B],\n(4)\nCij≡ /an}bracketle{tλi,Aλj,B/an}bracketri}ht−/an}bracketle{tλi,A/an}bracketri}ht/an}bracketle{tλj,B/an}bracketri}ht, (5)\nwhereCijspecifies the correlationbetween λi,Aandλj,B.\nHere,ρABis a 9×9 Hermitian matrix with trace unity, so\n80 parameters are required to parameterize it. The eight\ncomponents of /an}bracketle{tλi,A/an}bracketri}ht, eight components of /an}bracketle{tλi,B/an}bracketri}ht, and\n64 components Cijof the 8×8 matrix Care sufficient\nfor this purpose. The same procedure can be used for\nbipartite 4-level systems using the 15 traceless 4 ×4 Her-\nmitiangeneratormatricesforSU(4), andbipartite n-level\nsystems with the n2−1 tracelessn×nHermitian matri-\nces. Likewise, a general qubit-qutrit 6 ×6 density matrix\ntakesthe form ρAB=1\n6[(1+ni,Aσi,A)(1+3\n2/an}bracketle{tλj,B/an}bracketri}htλj,B)+\n6\n4σi,ACijλj,B] withCij≡ /an}bracketle{tσi,Aλj,B/an}bracketri}ht − /an}bracketle{tσi,A/an}bracketri}ht/an}bracketle{tλj,B/an}bracketri}ht,i=\n1,2,3 andj= 1,...,8.\nOurbipartitecorrelationmeasureforan n-levelandm-\nlevel system is based on the ( n2−1)×(m2−1) correlation\nmatrixC:\nEC≡n2\n<\n4(n2\n<−1)TrCCT=n2\n<\n4(n2\n<−1)/summationdisplay\ni,jCijCT\nji,(6)\nwheren<= min(n,m).EC=n2\n<\n4(n2\n<−1)Tr(ρAB−ρAρB)2\nis a nonnegative real number. If Cis a normal ma-\ntrix [18], Tr CCTequals to the sum of the squares of\nits eigenvalues, but Cneed not be normal. ECis basis-\nindependent; any rotation in Hilbert space leaves it un-\nchanged. The normalization factor n2\n</[4(n2\n<−1)] in (6)\nis such that the maximum possible value of ECis unity.\nECmeasures both classical- and quantum-correlation.\nThis measure of bipartite correlation was suggested in\nRef. [22] for pure states and n=m.\nThecorrelationmatrix Cquantifiesthecorrelationand\nthe entanglement of bipartite states. For pure two-qubit\nstates, the number of nonzero singular values (NSVs) of\nCiszerofornon-entangledstates( Cvanishes), andthree\nfor entangled states. For classically-correlated states\nwith two terms in the sum [see Eq. (16)], only one NSV\noccurs, two NSVs occur for three terms, three NSVs\noccur for four or more terms, and for entangled (i.e.,\nquantum-correlated) mixed states there are three NSVs.\nThese cases are summarized in Fig. 1. Entangled mixedstates can be differentiated from classically-correlated\nstates with 3 NSVs by applying the Peres-Horodecki\n(PH) partial transposition condition [19] [which corre-\nsponds to changing the sign of ny,Band the matrix el-\nementsCAB\niythat multiply σy,Bin (2), and determining\nwhether the resulting ρis still a genuine density matrix\n— if it is, the state is classically correlated, i.e., unentan-\ngled but correlated] to the density matrices with 3 NSVs.\nTheonlycategoriesthat cannot be distinguished without\nuse of the PH condition are the mixed-entangled and the\nclassically correlated states with ≥4 NSVs.\npure \nentangled uncorrelated \n(unentangled) mixed \nmixed- \nentangled classically- \ncorrelated uncorrelated \n3-terms \n1 NSVs 2 NSVs 3 NSVs 3 NSVs, PH cond. 3 NSVs 0 NSVs 0 NSVs \n2-terms \nFIG. 1: Classification of two-qubit states. Categories can b e\nexperimentally distinguished by measuring nA,nB, and using\nBell measurements [7] to determine the Cmatrix.\nSimilarly, for a two qutrit pure state, the number of\nNSVs of Cis zero for non-entangled states (the Cmatrix\nvanishes), three, if only two basis states are present in\nthe entangled state, five, if one of the qutrits contains\nonly two basis states but the other contains three, and\neight if all three basis states are present. For classically\ncorrelated qutrit states, there are 1, 2, ..., 8 NSVs for\n2, 3, ..., and 9 or more terms in the sum, etc. A similar\nclassification in terms of the number of NSVs exists for\nqubit-qutrit and n-level systems.\nA general three-qubit density matrix can be written as\nρABC=1\n8[(1+nA·σA)(1+nB·σB)(1+nC·σC)\n+σA·CAB·σB+σA·CAC·σC+σB·CBC·σC\n+/summationdisplay\nijkσi,Aσj,Bσk,CDijk], (7)\nwhereCAB,CAC, andCBCare the bipartite correla-\ntion matrices and the tensor that specifies the tripartite\ncorrelations is\nDijk≡ /an}bracketle{tσi,Aσj,Bσk,C/an}bracketri}ht−/an}bracketle{tσi,A/an}bracketri}ht/an}bracketle{tσj,B/an}bracketri}ht/an}bracketle{tσk,C/an}bracketri}ht.(8)\nA tripartite qutrit state can be similarly parameterized:\nρABC=1\n27[/productdisplay\nI(1+3\n2/summationdisplay\ni/an}bracketle{tλi,I/an}bracketri}htλi,I)+9\n4/summationdisplay\nI,J/summationdisplay\ni,jλi,ICij,IJλj,J\n+27\n8/summationdisplay\nI,J,K/summationdisplay\ni,j,kλi,Aλj,Bλk,CDijk,IJK],(9)3\nDijk,IJK≡ /an}bracketle{tλi,Iλj,Jλk,K/an}bracketri}ht−/an}bracketle{tλi,I/an}bracketri}ht/an}bracketle{tλj,J/an}bracketri}ht/an}bracketle{tλk,K/an}bracketri}ht.(10)\nOur tripartite correlation measure EDis based on the\ncorrelationmatrix D,ED≡K/summationtext\ni,j,kD2\nijk, whichcanalso\nbe written as\nED=KTr(ρABC−ρAρBρC−/summationdisplay\nI,J(I/negationslash=J)/summationdisplay\ni,jCI,J\nijσi,Iσj,J)2,\n(11)\nwhereK= 1/4 for qubits, and K= 27/160 for qutrits\nwithσs replaced by λs.EDis also a basis-independent\nnonnegative real number; any rotation in Hilbert space\nleaves it unchanged. A tripartite system may have\nbipartite- as well as tripartite-correlation. The bipar-\ntite correlation of a tripartite system is the sum of the\ncorrelation for the three bipartite pairs,\nEC≡n2\n4(n2−1)/summationdisplay\nI,J(I/negationslash=J)TrCI,J(CI,J)T,(12)\nwhereI,J=A,B,C. The density matrices of four-\nparticle and higher qubit, qutrits, and n-level system\nstates can be constructed similarly, but with increased\ncomplexity. For example, it is clear from Eq. (11) how to\ngeneralizeandobtainthefour-particlecorrelationoffour-\nparticle systems: EE≡K′/summationtext\ni,j,k,lE2\nijkl, where the four-\nparticle-correlationterm of the four-qubit density matrix\nρABCDis/summationtext\nijklσi,Aσj,Bσk,Cσl,DEijklandK′= 1/8.\nWe now present some examples of qubit and qutrit\nbipartite and tripartite correlated states. The maximally\nentangled bipartite qubit states are the Bell states,\n|Ψ±/an}bracketri}ht=1√\n2[|↑↓/an}bracketri}ht±|↓↑/an}bracketri}ht ],|Φ±/an}bracketri}ht=1√\n2[|↑↑/an}bracketri}ht±|↓↓/an}bracketri}ht ].(13)\nForallthesestates, /an}bracketle{tσA/an}bracketri}ht=/an}bracketle{tσB/an}bracketri}ht=0, i.e.,nA=nB=0.\nFor the singlet, /an}bracketle{tσi,Aσj,B/an}bracketri}ht=−δij(the spins are oppo-\nsitely polarized). The density matrices of the Bell states\nare:\nρΨ−=1\n4(1A1B−σA·σB),\nρΨ+=1\n4(1A1B+σA·σB−2σz,Aσz,B),\nρΦ+=1\n4(1A1B+σA·σB−2σy,Aσy,B),\nρΦ−=1\n4(1A1B+σA·σB−2σx,Aσx,B).(14)\nThe correlation matrices of the Bell’s states are diagonal\nand the correlation measure is EC= 1, i.e., they are\nmaximally entangled.\nLet us now consider the Rashid pure states [23],\n|φ+/an}bracketri}ht= (2cosh(2θ))−1/2(e−θ|↑↑/an}bracketri}ht+eθ|↓↓/an}bracketri}ht),(15)\nwhose density matrix is ρφ+=1\n4{[1A−\ntanh(2θ)σz,A][1B−tanh(2θ)σz,B]+sech(2θ)(σx,Aσx,B−\nσy,Aσy,B) + sech2(2θ)σz,Aσz,B}. When θ= 0,|φ+/an}bracketri}ht=|Φ+/an}bracketri}ht, and asθ→ ±∞, an unentangled state re-\nsults. The nonvanishing correlation matrix elements are:\nCxx= sech(2θ),Cyy=−sech(2θ),Czz= sech2(2θ).\nUsing (6) we obtain the correlation measure\nEC(|φ+/an}bracketri}ht) =1\n3TrCCT=1\n3(2sech2(2θ) + sech4(2θ)).\nThe concurrence C[19, 24] is\nC(|φ+/an}bracketri}ht) =/radicalbig\n2(1−Tr [ρA2]) = sech(2θ),\nsince\nρA=1\n2/parenleftbigg\n1−tanh(2θ) 0\n0 1+tanh(2 θ)/parenrightbigg\n,\nand the entanglement entropy is S≡ −Tr[ρAlog2ρA].\nThese results are graphically presented in Fig. 2. All the\nmeasuresequalunity for θ= 0anddecreaserapidlyvs. θ.\nFIG. 2: (color online) Comparison of the ECmeasure of cor-\nrelation, the concurrence Cand the entanglement entropy S\nfor the Rashid pure states.\nTwo-qubit classically-correlated states take the form\nρCC=1\n4/summationdisplay\nk≥2pk(1+nA,k·σA)(1+nB,k·σB),(16)\nwith/summationtext\nkpk= 1 andpk>0. The density matrix for the\nclassically-correlated state can be written in the form of\nEq. (2) with Bloch vectors\nnA=/summationdisplay\nkpknA,k,nB=/summationdisplay\nkpknB,k,(17)\nand correlation matrix\nCij=/summationdisplay\nkpkni,A,k/bracketleftBigg\nnj,B,k−/summationdisplay\nlplnj,B,l/bracketrightBigg\n.(18)\nForexample, forclassically-correlatedmixedstatesofthe\nformρCC= (2sech2(2θ))−1/2(e−θ|↓↑/an}bracketri}ht/an}bracketle{t↓↑|+eθ|↑↓/an}bracketri}ht/an}bracketle{t↑↓|),4\nwe find that all the correlation coefficients vanish, except\nforCzz=−sech2(2θ), the density matrix in representa-\ntion (2) isρCC=1\n4/parenleftbig\n1A1B−sech2(2θ)σz,Aσz,B/parenrightbig\n, and the\nclassical-correlation measure is ECC\nC=1\n3sech4(2θ).\nIt is elucidating to consider the Werner two-qubit den-\nsity matrix composed of a sum of a singlet state and the\nmaximally mixed state, ρW=p|Ψ−/an}bracketri}ht/an}bracketle{tΨ−|+1−p\n41, or, the\nmore general Werner two-qubit density matrix,\nρGW=p|ψ−/an}bracketri}ht/an}bracketle{tψ−|+1−p\n41, (19)\nwhere|ψ−/an}bracketri}ht= (2cosh(2θ))−1/2(e−θ|↑↓/an}bracketri}ht −eθ|↓↑/an}bracketri}ht).ρGW\nreduces toρWforθ= 0. ForρGW,\nnA=−nB=ptanh(2θ)ˆz, (20)\nand\nCGW=−p\nsech(2θ) 0 0\n0 sech(2 θ) 0\n0 0 1 −p+psech2(2θ)\n.\n(21)\nThe PH entanglement criterion [19] shows that this state\nisentangledif p[(1+2sech(2 θ)]≥1. Figure3plotsthePH\ncriterion limit and the correlation measure, EC(p,θ) =/summationtext\nid2\ni= 1−p+ (2p2+p)sech2(2θ), for the generalized\nWerner state. Note that the PH criterion is not obtain-\nable from Calone, but can be obtained using the invari-\nant parameters ξ≡/summationtext\nidi−nA·C·nB\nnA·nBandnA·nB. More\nexplicitly,p[1+ 2sech(2 θ)] =−ξ+/radicalbig\nξ2/4−nA·nB, so\nthe PH condition reads\n−ξ\n2+−ξ+/radicalbig\nξ2−4nA·nB\n2≥1,(22)\nwhich can be written as the condition: the largest root of\nthe quadraticequation, ( x+ξ/2)2+ξ(x+ξ/2)+nA·nB=\n0, is greater than unity. Thus, mixed state entanglement\nis determined not only by Cbut by additional invariant\ncharacteristics of the density matrix, i.e., invariant char-\nacteristics composed of the parameters C,nAandnB\nused to form the density matrix (whereas the correlation\nis determined only in terms of C). The physical signif-\nicance of the scalar product nA·nBas the projection\nof the expectation value of the spin of one qubit on the\nother, is clear, as is the physical significance of ξas a\nspecific projection of the singular values of the correla-\ntion matrix that depends on the average spins nAand\nnB[25]. However, the physical significance of the PH\nentanglement criterion is not yet clear; i.e., the physical\ninterpretation of Eq. (22) [or the quadratic equation] re-\nmains to be uncovered. But at least the PH condition is\nnow expressed only in terms of the physical parameters\nappearing in the density matrix, rather than by the par-\ntial transposition condition, which is more removed from\nphysical interpretation.\nAsanexampleofatripartitepurequtritstate,consider\n|ψE3/an}bracketri}ht=eθ1eθ2|v1v1v1/an}bracketri}ht+e−θ1|v2v2v2/an}bracketri}ht+e−θ2|v3v3v3/an}bracketri}ht√\ne2θ1e2θ2+e−2θ1+e−2θ2,\n(23)\nFIG. 3: (color online) EC(p,θ) versus pandθfor the gener-\nalized Werner density matrix ρGW, and the Peres-Horodecki\nentanglement criterion limit, p[1+2sech(2 θ)] = 1, drawn on\nthep-θplane and projected onto the ECsurface.\nwhere|v1/an}bracketri}ht= (1,0,0)T,|v2/an}bracketri}ht= (0,1,0)T,|v3/an}bracketri}ht= (0,0,1)T.\nThe qutrit bipartite correlation ECand tripartite corre-\nlationEDare plotted in Fig. 4. The maximum of EDis at\nθ1=θ2= 0, where ED= 1, but the bipartite correlation\ndips there. Three ridges of high bipartite and tripartite\ncorrelation occur, one at θ1=θ2= negative, and two\nothers at 120 degrees rotation from the first.\n(a)\n(b)\nFIG. 4: (color online) The bipartite and tripartite correla tion\nmeasures, ECandEDfor the tripartite-qutrit pure state (23).5\nIn most quantum information systems, qubits are dis-\ntinguishable, so there is no need to account for exchange\nsymmetry, but for identical bosonic or fermionic systems,\nthe density matrix must be properly symmetrized, e.g.,\nρsym\nAB=SρABS. For two qubits, the symmetrization op-\nerator is S=1\n2(1+PAB) =3\n4+1\n4σA·σBand the anti-\nsymmetrization operator is A=1\n2(1−PAB) =1\n4−1\n4σA·\nσB, hence,ρsym\nAB=/parenleftbig3\n4+1\n4σA·σB/parenrightbig\nρAB/parenleftbig3\n4+1\n4σA·σB/parenrightbig\n,\nρanti\nAB=/parenleftbig1\n4−1\n4σA·σB/parenrightbig\nρAB/parenleftbig1\n4−1\n4σA·σB/parenrightbig\n. If the qubits\nare antisymmetric, ρAB=ρanti\nAB=AρABA, the state\nmust be pure singlet, ρanti\nAB=1\n4(1−σA·σB); it can-\nnot be a mixed state, as opposed to ρsym\nABwhich can be\nmixed. If spatial degrees of freedom need to be included\nin the description, in addition to the internal degrees of\nfreedom, a bipartite density matrix can always be writ-\nten as a product of an internal (i.e., spin) part and an\nexternal (i.e., space) part. Hence, a symmetric density\nmatrixρsym\nABfor the internal degrees of freedom must be\nmultiplied by a symmetric [antisymmetric] spatial den-\nsity matrix for the spatial degrees of freedom {rA,rB}for bosons [fermions], and ρanti\nABmust be multiplied by\nan antisymmetric [symmetric] spatial density matrix for\nbosons [fermions], so that the full density matrix has the\nright exchange symmetry. We show elsewhere that this\nhas relevance to collisional shifts in atomic clocks [26].\nIn summary, we have developed a classification of cor-\nrelation for multipartite N-level quantum systems by\nwriting their density matrices in terms of SU( N) gen-\nerators, and we defined a measure of correlation for such\nsystems, based upon their correlation matrices. The en-\ntanglement involves not just the correlation matrix but\nalso other invariant parameters in the density matrix for\nthe system. This formulation can now be used for a vari-\nety of applications, e.g., in the optimization of quantum\ngates and in the calculation of collisional clock shifts.\nThis work was supported in part by grants from the\nU.S.-IsraelBinationalScience Foundation (No. 2006212),\ntheIsraelScienceFoundation(No. 29/07),andtheJames\nFranck German-Israel Binational Program.\n[1] P. W. Shor, SIAM J. Comput. 26, 1484 (1997); L. K.\nGrover, Phys. Rev. Lett. 79, 325 (1997).\n[2] A. K. Ekert, Phys. Rev. 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Rev.\nA67, 022310 (2003).\n[14] D. J. Wineland, J. J. Bollinger, W. M. Itano and D. J.Heinzen, Phys. Rev. A 50, 67 (1994).\n[15] R. F. Werner, Phys. Rev. A 40, 4277 (1989).\n[16] M. Lewenstein andA.Sanpera, Phys.Rev.Lett. 80, 2261\n(1998); S. Karnas and M. Lewenstein, J. Phys. A 34,\n6919 (2001); J. Sperling and W. Vogel, Phys. Rev. A 79,\n052313 (2009).\n[17] J. M. Leinaas, J. Myrheim and E. Ovrum, Phys. Rev.\nA74, 012313 (2006).\n[18]C=U dV†whereUandVare orthogonal and dis\ndiagonal. G. Strang, Linear Algebra and Its Applications ,\n(Brooks Cole, 2005).\n[19] A.Peres, Phys.Rev.Lett. 77, 1413(1996); M.Horodecki,\nP. Horodecki, R. Horodecki, Phys. Lett. A 223,1 (1996);\nR. Horodecki, P. Horodecki, M. Horodecki, and K.\nHorodecki, Rev. Mod. Phys. 81, 865 (2009).\n[20] U. Fano, Rev. Mod. Phys. 55, 855(1983).\n[21] H. Georgi, Lie Algebras in Particle Physics , (Westview\nPress, 1999).\n[22] J. Schlienz and G. Mahler, Phys. Rev. A 52, 4396 (1995).\n[23] M. A. Rashid, J. Math. Phys. 19, 1391 (1978).\n[24] W. K. Wootters, Phil. Trans. R. Soc. Lond A 356, 1117-\n1731 (1998).\n[25] In the case of the generalized Werner state, ξprojects\nout the sum of two nonzero singular values −psech(2θ).\n[26] Y. B. Band and I. Osherov, “Collisionally Induced\nAtomicClockShiftsandCorrelations”, (tobepublished)."    },    {        "title": "1010.5708v1.Novel_Ground_State_Crystals_with_Controlled_Vacancy_Concentrations__From_Kagomé_to_Honeycomb_to_Stripes.pdf",        "content": "arXiv:1010.5708v1  [cond-mat.mtrl-sci]  27 Oct 2010Novel Ground-State Crystals with Controlled Vacancy\nConcentrations: From Kagom´ e to Honeycomb to Stripes\nRobert D. Batten,1David A. Huse,2Frank H. Stillinger,3and Salvatore Torquato2,3,4,5,6,7, ∗\n1Department of Chemical Engineering,\nPrinceton University, Princeton, NJ 08544, USA\n2Department of Physics, Princeton University, Princeton, N J, 08544, USA\n3Department of Chemistry, Princeton University, Princeton , NJ, 08544, USA\n4Princeton Institute for the Science and Technology of Mater ials,\nPrinceton University, Princeton, NJ, 08540, USA\n5Program in Applied and Computational Mathematics,\nPrinceton University, Princeton, NJ, 08544, USA\n6Princeton Center for Theoretical Science,\nPrinceton University, Princeton, NJ, 08644, USA\n7School of Natural Sciences, Institute for Advanced Study, P rinceton, NJ, 08544 USA\n(Dated: August 13, 2021)\n1Abstract\nWeintroduceaone-parameter family, 0 ≤H≤1, ofpairpotential functionswithasinglerelative\nenergy minimum that stabilize a range of vacancy-riddled cr ystals as ground states. The “quintic\npotential” is a short-ranged, nonnegative pair potential w ith a single local minimum of height H\nat unit distance and vanishes cubically at a distance of√\n3. We have developed this potential\nto produce ground states with the symmetry of the triangular lattice while favoring the presence\nof vacancies. After an exhaustive search using various opti mization and simulation methods, we\nbelieve that we have determined the ground states for all pre ssures, densities, and 0 ≤H≤1. For\nspecific areas below 3√\n3/2, the ground states of the “quintic potential” include high -density and\nlow-density triangular lattices, kagom´ e and honeycomb cr ystals, and stripes. We find that these\nground states are mechanically stable but are difficult to sel f-assemble in computer simulations\nwithout defects. For specific areas above 3√\n3/2, these systems have a ground-state phase diagram\nthat corresponds to hard disks with radius√\n3. For the special case of H= 0, a broad range\nof ground states is available. Analysis of this case suggest s that among many ground states, a\nhigh-density triangular lattice, low-density triangular lattice, and striped phases have the highest\nentropy for certain densities. The simplicity of this poten tial makes it an attractive candidate for\nexperimental realization with application to the developm ent of novel colloidal crystals or photonic\nmaterials.\nPACS numbers:\n2I. INTRODUCTION\nThefield ofself-assembly provides numerous examples ofusing atom s, molecules, colloids,\nand polymers as building blocks in the development of self-organizing f unctional materials.\nFor example, researchers have used nanoparticles to create sta cked rings and spirals us-\ning DNA linkers,1to construct photonic crystals using binary systems of colloids,2and to\nprototype a self-organzing colloidal battery.3\nRecently, there have been significant efforts to design pair potent ials that yield self-\nassembly of targeted ground-state (potential energy minimizing) structures using “inverse\nmethods.”4,5With an inverse approach, one chooses a targeted structure or p roperty and\nuses optimization methods to develop a robust pair potential funct ion. We envision that\ncolloids will be the experimental testbed for the optimized interactio ns because colloids\noffer a significant range of repulsive and attractive interactions th at can be tailored in the\nlaboratory.6For these models, the many-body interactions of a system are red uced to pair\ninteractions. For a pair potential v(r), where ris the distance between two particles, the\npotential energy per particle uof a many-body configuration rNofNparticles reduces to\nu(rN) =1\nN/summationtext\ni<jv(rij).\nInverse approaches have been applied to produce low-coordinate d ground states such as\nhoneycomb,7square,8simple cubic,9diamond and wurtzite crystals10as well as many-body\nconfigurationsonasphere.5Inadditiontotargetingmaterialstructures, theinverseapproa ch\nhasbeenappliedtomaterialpropertiesincluding negativePoisson’s r atio,11negativethermal\nexpansion,12and scattering characteristics of many-particle systems.13The ability to control\nvacancy concentrations and arrangements in ground-state str uctures is one property yet\nto be achieved via inverse methods. We expect the design of vacanc y-riddled crystals to\nhave important technological applications and may provide fundame ntal insight into certain\nphysical phenomena. The scattering of light,14ionic conductivity,15transport processes in\nheterogeneous materials,16and possibly supersolid behavior in quantum systems17,18are\ndirectly related to vacancies in a material structure.\nIn this paper, we introduce the “quintic potential” as a pair interact ion function that\nyields vacancy-riddled lattices and low-coordinated crystals as gro und states at high density\nand striped patterns as ground states at low-density. Although n ot a result of a true inverse\nmethod, this family of potential functions arose through a selectio n process so that vacancy-\n30.5 0.75 11.25 1.5 1.75 2r0123v(r)H = 0\nH = 0.25\nH = 0.50\nH = 0.75\nH = 1\nH\nFIG. 1: (Color online) Quintic potential, Eq. 6 for various Hvalues. The scales of energy and\nlength are dimensionless.\nriddled and low-coordinated lattices are energetically degenerate o r possibly favorable to the\ntriangular lattice for certain densities. Each quintic potential func tion curve, illustrated in\nFig. 1, is steeply repulsive for small r, has a local minimum at r= 1 so that v(1) =H, is\nnonnegative, and is smoothly truncated to be zero at r=√\n3 and beyond. The algebraic\nform of the potential is given in Sec. II. The understanding of the p hase behavior associated\nwith the quintic potential is the first step toward developing a more c omprehensive inverse\napproach to controlling vacancies in ground-state structures.\nWe find that ground states can vary from a high-density triangular lattice, the kagom´ e\nand honeycomb crystals, a striped phase, a low-density triangular lattice, and a hard-disk-\nlike fluid as the pressure decreases. The positivity of the local relat ive minimum is a key\nfeature that allows vacancies to be stabilized in the ground state. C onsider a system of\nparticles arranged in a triangular lattice at a specific area (area per particle) a=√\n3/2 so\nthat each of the particle’s first neighbors lie at r= 1 and second neighbors at r=√\n3.\nBecause the local minimum in v(r) is at positive energy, the removal of a particle will lower\nthe total potential energy of the system. However, to maintain t he constraint on a, the\nlattice spacing must decrease, which increases the total potentia l energy. This family of\npotentials is designed so that the creation of a vacancy and the sub sequent reduction in\nlattice spacing yields a net decrease in the potential energy per par ticle.\n4However, thegeneralcontrolofvacanciesisachallengingtaskdu etolong-rangedvacancy-\nvacancy interactions. For the Lennard-Jones and many short-r anged repulsive potentials,\nvacancies arise in equilibrium solids at positive temperature due to the ir entropic contribu-\ntion to the free energy.19AtT= 0, they can be “frozen in” during cooling, though they\nare not present in the equilibrium ground states of these systems. For typical potentials\nwhose ground states are ordered, some pair potentials, such as t he Lennard-Jones interac-\ntion, vacancies cost energy due to missing “bonds” of negative ene rgy, while with repulsive\npotentials and the electron crystal, vacancies cost energy due to strain energy in the crystal.\nIf a vacancy is introduced into an otherwise perfect crystal at po sitive pressure, strains\nwill arise so that the system can reduce its potential energy and ac hieve mechanical stabil-\nity. In typical materials such as Lennard-Jones systems, these r elaxations cannot reduce the\nenergy below that of the ground state. When more than one vacan cy is present, the strain\nfields interact and subsequently mediate long-ranged vacancy-va cancy interactions. These\ninteractions are highly nontrivial even in the case of two vacancies in a crystal. In two\ndimensions, several studies have examined these effective interac tion energies between two\nvacancies as a function of separation distance for a number of pot entials including the elec-\ntron crystal (Coulombic potential with positive background charg e),20–22Lennard-Jones,23\nGaussian core,24and a repulsive r−3potential.25\nIn the two-dimensional electron crystal, vacancy-vacancy inter actions are attractive and\nfall off as r−3\nv,21wherervis the separation distance between two vacancies, which agrees\nwith continuum elasticity theory.20For the Lennard-Jones potential, vacancy-vacancy inter-\nactions are also attractive at least for short distances.23These effective vacancy interaction\npotentials are not necessarily monotonic as a function of lattice spa cing.22,23The displace-\nment fields can be highly anisotropic near a vacancy.24,26In the vicinity of the vacancy, the\natomistic details account for the discrepancy between numerical r esults for the displacement\nfields and those predicted by continuum elasticity theory. Beyond fi fteen lattice spacings,\nthis continuum theory accurately reproduced the results of nume rical studies under some\nboundary conditions.24\nThe arrangement of low concentrations of vacancies may not simply be controllable be-\ncause of the nontrivial coupling between the pair potential functio n, the local deformations\nthat it produces when a vacancy is introduced into a perfect cryst al, and the effective\nvacancy-vacancy interactions. However, a number of pair poten tial functions have been\n5found to give rise to ground states with high concentrations of vac ancies. For example, the\n“honeycomb potential” was developed via inverse methods to yield th e honeycomb crys-\ntal as a robust ground-state structure.7,8At positive temperatures, the honeycomb struc-\nture remained stable for a large temperature and pressure region of the phase diagram.27\nThis double-well potential was generalized as the Lennard-Jones- Gaussian potential, which\nshowed a number of complex crystal structures including pentago nal, hexagonal, nonago-\nnal, and decagonal unit cells as the depths and locations of the ener gy wells were varied.28,29\nColloidalparticleswiththree“patches,” orattractiveinteractions itesonthesurfaceofapar-\nticle, have recently been shown to yield a honeycomb structure for some pressures at T= 0\nin addition to other low-coordinated crystal structures.30In addition, the square-shoulder\npotential, ahard-corepotential witha softcorona ofvariableleng th, shares someof thesame\nground-statefeatures asthe quintic potential. In particular, low -coordinated structures such\nas a honeycomb-like trivalent configuration and lanes are ground-s tate features31,32that are\nshared with the quintic potential. Such a potential can also form lame llar and micellar\nphases for relatively long-ranged coronas as was shown theoretic ally.33\nLastly, we note that several potentials qualitatively similar to the qu intic potential have\nbeen studied in other contexts. For example, a potential that give s rise to quasiperiodic\norder and the square lattice in two dimensions has one local minimum an d local maximum\nand finite cutoff,34as does the Dzugutov potential35which gives rise to a number of com-\nplex local clusters in three dimensions.36However, both of these potentials have a local\nattractive minimum ( i.e.negative potential energy). A piecewise linear potential with a\nhard core, single local minimum, and single local maximum showed the fo rmation of chains\nand labyrinths at positive temperature, though the ground state is not fully characterized.37\nHowever, in contrast to the above potentials, the quintic potentia l is short-ranged, positive,\nisotropic, continuous and differentiable, which may be beneficial to e xperimentalists hoping\nto realize this potential in a laboratory setting. We find that with this potential, a number\nof low-coordinated structures arise as ground states including th e honeycomb and kagom´ e\ncrystals as well as a “striped” phase. This represents a significant achievement and a first\nstep toward controlling vacancies in ground-state structures.\nThe remainder of this paper is as follows. The quintic potential is defin ed and detailed in\nSec. II while our methodology is detailed in Sec. III. In Sec. IV, we de fine the phase diagram\nin the specific area- Hplane and in the pressure- Hplane. We discuss the characteristics of\n6the phases, the results of simulation, and their mechanical stability . In Sec. V, we focus on\nthe special case where H= 0. This case yields anomalous behavior because the potential\nenergy vanishes inside and outside the energy barrier. Lastly, we d iscuss the implications of\nthis potential and identify extensions to this work in Sec. VI.\nII. QUINTIC POTENTIAL\nThe generalized quintic potential consists of a fifth-order polynom ial that is truncated\nbeyond√\n3 to be zero. It has the form\nf(r) =/bracketleftbigg\nL(m)/parenleftBig\nr−√\n3/parenrightBig5\n+K(m)/parenleftBig\nr−√\n3/parenrightBig4\n−/parenleftBig\nr−√\n3/parenrightBig3/bracketrightbigg\n, r≤√\n3 (1)\nand zero otherwise, where mis an unscaled height of the local minimum. The coefficients\nL(m) andK(m) are chosen so that r= 1 is a local minimum and f(1) =m, and, as a\nfunction of m, are given respectively as\nL(m) = 4m/parenleftBig√\n3−1/parenrightBig−5\n−/parenleftBig√\n3−1/parenrightBig−2\n, (2)\nK(m) = 5m/parenleftBig√\n3−1/parenrightBig−4\n−2/parenleftBig√\n3−1/parenrightBig−1\n. (3)\nThe condition that f′′(1)>0 ensures that the stationary point is a relative minimum, and\ntherefore the parameter mmust be constrainted to\nm </parenleftbig√\n3−1/parenrightbig3\n10≈0.0392304845 . (4)\nThe position of the energy barrier (the relative maximum) occurs at\nrb=√\n3−3/parenleftbig√\n3−1/parenrightbig\n5/bracketleftBig\n1−4m/parenleftbig√\n3−1/parenrightbig−3/bracketrightBig. (5)\nThe generalized pair potential is rescaled so that v(rb) = 1 to set a uniform energy scale,\nv(r) =f(r)/f(rb). (6)\nUpon rescaling, the height of the relative minimum His defined so that v(1) =Hand\nvaries from zero to unity. When H= 1, the potential is monotonically decreasing and is flat\natr= 1. In this case, there is no local minimum or maximum. These soft pot entials are\nrepulsive for small r, and the first and second derivatives vanish at r=√\n3. We construct\n70 0.01 0.02 0.03 0.04m00.20.40.60.81H\nFIG. 2: The relationship between mandHfor the quintic potential.\nthe potential such that the first and second derivatives vanish at r=√\n3 so that the second-\norder expansions ofthe potentialenergy required forstability ca lculations ( e.g.phonons) are\nwell-defined. Examples from the family of quintic potentials are shown in Fig. 1. Because\ndeveloping a simple expression relating mandHis difficult, a table relating mandHwas\nused. The relation between mandHis shown in Fig. 2.\nIII. METHODS\nAtT= 0, the free energy per particle gis identical to the enthalpy per particle and\nis related to the average potential energy per particle uviag=u+pa. To map the\nground-state phase diagram as a function of specific area a, pressure p, andH, we utilized\nthe double-tangent construction method to find the lowest free- energy structure among\ncandidate ground-state configurations. The double-tangent co nstruction is demonstrated in\nFig. 3.\nWe considered several crystal structures as candidate ground states including the trian-\ngular lattice (TRI), square lattice (SQ), honeycomb crystal (HC) , kagom´ e crystal (KAG),\nand its close relative, the “anti-kagom´ e” crystal (AKG). Wherea s the kagom´ e crystal has\nvacancies located on a triangular lattice separated by two nearest neighbor spacings, the\nanti-kagom´ e crystal has vacancies located on a rectangular latt ice. For all densities, the\nanti-kagom´ e has a potential energy greater than or equal to th e kagom´ e lattice due to the\ndiffering local coordination numbers. We considered these crystal structures because they\n8are relative simple crystal structures with varying degrees of loca l coordination. The con-\nstruction of the potential, with a well at r= 1 and vanishing energy at r=√\n3 would\nintuitively favor such geometries. We use other techniques to syst ematically explore other\ncrystals with a small number of particles in the unit cell. With these cry stal structures, we\nperformed lattice sums over the relevant range of specific area.\nWe also performed an exhaustive search for energy-minimizing n-particle crystals, peri-\nodic configurations containing nparticles per unit cell, while allowing for shape deformation\nof the unit cell. For example, consider a two-particle crystal with lat tice vectors [ d1,0] and\n[d2,d3] and a basis where particle 0is at (0,0) and particle 1is located at ( x1,y1). For this\nsystem, the potential energy per particle is given as\nu=v(r01)+1\n2/summationdisplay\ni[v(r00′)+v(r11′)+v(r01′)+v(r10′)], (7)\nwhere the summation is over all lattice sites except the origin site and 0′represents particle\n0in a cell other than the origin cell (and likewise for particle 1). Minimizing ufor a fixed\narea per particle aand eliminating d3=na/d1casts this as an unconstrained minimization\nthat is function of d1,d2,x1andy1. The function is minimized using the conjugate gradient\nalgorithm. It is straightforward to generalize this to a larger numbe r of particles in the unit\ncell.\nTo ensure that we obtain globally optimal solutions, we use a large num ber of initial\nconditions slowly and systematically varying d1,d2,x1andy1from zero to n√\n3. After each\nminimization, we ensure that the unit cell is not significantly sheared. A highly sheared\nunit cell requires summation over a large number of unit cells to ensur e that all interactions\nare accounted for and therefore we discard those solutions with a ngles less than 10◦. This\nprocedure is repeated for nearly two thousand values of aon the range 0 .7≤a≤3√\n3/2 so\nthat a smooth uversusacurve is generated.\nThese minimizations are a function of 2 nvariables, and therefore many minimizations\nat a fixed area per particle can be performed easily. However, as th e number of particles\nper unit cell grows, the number of initial conditions required to ensu re global optimality\ngrows as 2n. We performed these minimizations systematically for up to four par ticles per\nunit cell, beyond which it becomes too intense to systematically explor e thousands of initial\nconditions for thousands of specific areas.\nWealso used slow cooling via molecular dynamics (MD) to obtaincandidat e ground-state\n9structures for larger and possibly disordered systems. With a sys tem of 800 particles in a pe-\nriodic box, the system was simulated using the Verlet algorithm and An dersen thermostat.38\nThe system was initialized as a high-temperature liquid and slowly cooled according to a lin-\near temperature schedule from T= 0.4 to 0.025 over the course of 1.6 ×107time steps. After\nthe simulation was terminated, the system was quenched to a poten tial energy minimum\nusing the conjugate gradient algorithm.\nLastly, we performed lattice Monte Carlo (MC) optimizations using se veral hundred\nlattice sites. In these optimizations, the specific area was fixed and the lattice sites were\neither occupied or unoccupied by particles. The optimization variable s were the occupation\nstate of the lattice sites, the angle between the lattice vectors an d the length of one lattice\nvector. The length of the other lattice vector was constrained by the area, number of\nparticles, angle between the latticevectors, andthe length of the first lattice vector. Possible\nMonte Carlo moves included inserting a particle to a vacant site, remo ving a particle from\nan occupied site, swapping a particle froman occupied site to an unoc cupied site, perturbing\nthe angle of the lattice, and perturbing the length of one of the latt ice vectors. The system\nwas thenoptimized via simulated annealing. It wasgiven aninitial “temp erature,” orenergy\nscale, andmoves wereacceptedorrejectbasedupontheMetrop olisalgorithm. Tenthousand\nMC trials were attempted in each cycle and several hundred cycles w ere performed for each\noptimization.\nIt is important to note that although many interesting structural characteristics arose\nfrom the results of molecular dynamics simulation and Monte Carlo opt imization, no fi-\nnal configurations were identified as ground-state structures. The lattice sums and crystal\noptimization methods always provided the lowest free-energy stru ctures.\nThe double-tangent construction requires care and precision sinc e other phases can come\nvery close to the coexistence free energy. The p-gandu-acurves are discretized over several\nthousand data points and linearized between between data points. The identification of\ncoexisting phases can be done by using the lower, concave envelope of thep-gcurve. The\ncoexistence points and ranges of stability were then calculated usin g a tabulation of u,a,p\nandgand linear interpolation.\n100.5 1 1.5 2 2.5 3a00.511.52uOpt 3 Part\nOpt 2 Part\nOpt Brav\nHC\nTri\nKag\nMD + Quench\nMC\nFIG. 3: (Color online) Potential energy per particle uas a function of specific area afor selected\ncrystals and the optimal 2- and 3-particle crystals for H= 0.5. The dense triangular, honeycomb,\nstriped, and open triangular crystals are stable ground sta tes forH= 0.5. The square and anti-\nkagom´ e structures are omitted for clarity. Dashed lines re present the double tangent construction.\nIV. RESULTS\nA. Phase Diagram\nThedouble-tangentconstructionrevealedtwodifferent typesof phasebehaviordepending\nonthe value of H. Inno case did the configurations resulting fromMD or MC yield a grou nd\nstate. For 0 < H <0.762902, there exist five distinct structures. Figure 3 illustrates t heu-a\ncurves for H= 0.5. In the figure, the optimal n-particle crystals are not visible when they\nare identical to the triangular, honeycomb, or kagom´ e crystals. As shown in in Fig. 3, at\nhighest pressure, a dense triangular lattice (TRI), where neighbo ring particles lie inside the\npotential energy barrier of other neighboring particles, is the gro und state. At a reduction\nof pressure to p= 1.84432, the dense triangular lattice is in coexistence with the honeyc omb\ncrystal (HC), which is illustrated with the dashed tangent lines. In F ig. 3, the curve for the\nkagom´ e lattice appears possibly to touch the coexistence line due t o the finite thickness of\nthe line. However, plotting pagainstgreveals an intersection between the curves for the\nTRI and HC structures. The kagom´ e lattice has a higher free ener gy than both structures\nin this density range.\n11Further reduction of the pressure to 0.75263 yields a “striped” (S T) or lane phase coex-\nisting with the honeycomb phase. This phase is an affinely-stretched triangular lattice. The\nfirst few coordination shells contain two, four, and two particles. T he curves representing\nthe lowest-energy crystals with a two- and three-particle basis eit her the triangular lattice,\nhoneycomb crystal, or kagom´ e crystals generally for a <1.3 with a corresponding basis.\nFor larger a, these curves appear to come close to the coexistence line betwee n the HC and\nST phases or the ST and open TRI phases. The p-gcurves reveal that the free energy is\nalways above that of the coexisting phases. These near-ground s tates are generally crystals\nthat nearly mimic the coexistence. For example, the optimal three- particle crystal between\nthe HC and ST phases is an alternating sequence of lines with honeyco mb-like coordination\nand striped coordination. These are suboptimal structures beca use the specific neighbor\nlengths do not match exactly. At sufficiently low pressure, for H= 0.50 andp= 0.58275,\nthe ST phase coexists with an “open” triangular lattice where the ne ighbor particles lie on\nthe outside of the potential energy barrier.\nLastly, at vanishing pressure, any configuration in which particles a re separated by at\nleast√\n3 is a ground state since the free-energy vanishes. This occurs fo ra≥3√\n3/2. In\nthis region, the ground states behave like hard disks (HD) of radius√\n3. There would be\na hard-disk crystal and liquid regime with specific area scaled to the h ard-disk equation of\nstate, which has been well characterized.39\nFor 0.762902≥H≥1, the kagom´ e crystal emerges as a ground state structure fo r a nar-\nrow range of stability. This is an especially important result. In previo us work, researchers\nused an inverse methodology to design pair interactions for target ed ground states.8Al-\nthough they were successful in engineering potential for the hon eycomb and square crystals,\nthey were unable to do so for the kagom´ e. The area and pressure ranges of stability increase\nwithH. The emergence of the kagom´ e crystal as a ground state is show n in the p-acurve\nforH= 1.00 in Fig. 4. As the pressure is reduced, the sequence of stable pha ses is the dense\nTRI, KAG, HC, ST, open TRI, and HD phases.\nThe full phase diagrams in the p-Hplane and the a-hplane are shown in Figs. 5 and 6,\nrespectively. Figure 5 shows that the coexistence lines are nearly lin ear for small H. The\npressure range of stability for the open TRI, ST, and KAG phases w idens as Hincreases.\nWhenHis sufficiently large to include the KAG phase, the pressure range of s tability for\nthe HC phase narrows. As Hincreases, the area range of stability increases for most phases,\n120.5 1 1.5 2 2.5a01234 uOpt 3 Part\nOpt 2 Part\nOpt Brav\nHC\nTri\nKag\nMD + Quench\nMC \nFIG. 4: (Color online) Potential energy per particle uas a function of specific area afor selected\ncrystals and the optimal 2- and 3-particle crystals for H= 1.00. The dense triangular, kagom´ e,\nhoneycomb, striped, and open triangular crystals are stabl e ground states for H= 0.5 while for\nH= 1.00, thekagom´ e alsoemergesasastablegroundstate. Thesqu areandanti-kagom´ e structures\nare omitted for clarity. Dashed lines represent the double t angent construction.\nexcluding the dense TRI phase, which is evident in Fig. 6.\nNearH= 1.00, the slopes of the curves change dramatically. This is due to the s oftening\nof the pair potential function as Happroaches unity and the fact that the relative minimum\nand maximum come together much more rapidly as Happroaches unity. Fig. 1 shows that\ntheH= 1.00 potential is significantly softer near r= 1 than other potentials.\nThe softening ofthe potential andthe rise of therelative minimum ar eimportant features\nfor the inclusion of the KAG phase. In comparing Fig. 3 to Fig. 4, one c an see that the curve\nfor the kagom´ e crystal initially begins above the double tangent co nnecting the TRI and HC\nphases. As the potential softens and the relative minimum increase s, the first minimum in\neach curve increases proportionally to H. However, the softening of the potential bends the\nleft-side of the curves in such a way that the KAG curve lies at lower f ree energy than that\nTRI-HC coexistence line. The combination of the height of the relativ e minimum and the\nsoftening are directly related to the potential energy and pressu re of the system, the two\ncomponents of the T= 0 free energy.\nThe KAG phase emerges at a triple point at H= 0.762909 and p= 2.938, where the\n130 1 2 3 4 5p00.20.40.60.81H\nHDTRIKAGHCTRI\nST\nFIG. 5: (Color online) Phase diagram in the p-Hplane as calculated by double tangent method.\nThe hard-disk-like state (HD) occurs for p= 0 andH >0 and the filled triangle on the lower curve\nrepresents the triple point.\n0.5 1 1.5 2 2.5 3a00.20.40.60.81HTRITRI\nHard \nDiskST-TRI\nTRI-HCHC-STTRI-KAGKAG KAG-HC\nHC\nVRLST\nFIG. 6: Phase diagram in the a-Hplane as calculated by the double tangent method.\n140.811.2 1.4 1.6 1.8 2 2.2r-1-0.500.511.52v(r); -v'(r) /2v(r)\n-v'(r)/2\nTri 6,6\nHC 3,6\nKag 4,4\nFIG. 7: (Color online) Comparison of the location and coordi nation number of the triangular\nlattice, honeycomb crystal and kagom´ e crystal for a= 0.82443, 1.18745, and 1.06445, respectively,\nforH= 0.762902. At this Handp= 2.938, the TRI, HC, an KAG phases form a triple point and\nare in coexistence as ground-state structures. The delicat e balance between potential energy and\nforces allow for coexistence.\nTRI, HON, and KAG phases form a triple point. We have examined the s ubtle interplay\nbetween the shape of the pair potential function and its derivative at this coexistence point\nin Fig. 7. The figure shows v(r) and−v′(r)/2 forH= 0.762902. The plot also shows the\nlocation of the nearest and next-nearest neighbors for the crys tals. The HC phase has the\nclosest neighbors while the TRI phase has the smallest forces.\nThe coexistence of these structures requires that a complex sys tem of equations be sat-\n15isfied. For the coexistence pressure p∗and free energy g∗, the system of equations becomes\ng∗= 3/bracketleftBigg\nv(dt)+v(dt√\n3)−dt\n2v′(dt)−dt√\n3\n2v′(dt√\n3)/bracketrightBigg\n(8)\np∗=−√\n3\ndt/bracketleftBig\nv′(dt)+√\n3v′(dt√\n3)+/bracketrightBig\n(9)\ng∗= 3/bracketleftBigg\n1\n2v(dh)+v(dh√\n3)−dh\n4v′(dh)−dh√\n3\n2v′(dh√\n3)/bracketrightBigg\n(10)\np∗=−2\ndh√\n3/bracketleftbigg1\n2v′(dh)+√\n3v′(dh√\n3)/bracketrightbigg\n(11)\ng∗= 2v(dk)+2v(dk√\n3)−dkv′(dk)−dk√\n3v′(dk√\n3) (12)\np∗=−√\n3\n2dk/bracketleftBig\nv′(dk)+√\n3v′(dk√\n3)/bracketrightBig\n, (13)\nwherediis the nearest-neighbor distance for each structure and is neces sarily less than unity\nfor the quintic potential. Evidently, the quintic potential for H= 0.762902 satisfies these\nequations. The complexity of the coexistence equations is clear in th at each structure weighs\ncertain parts differently. The design of other potentials that stab ilize these structures may\nutilize this system of equations within an appropriate optimization fra mework.\nB. Stability\nThe positivity of the squared frequency of all phonons ensures me chanical stability of a\ncrystal. We have examined the phonon spectra for the KAG, HC, an d ST phases. These\nphases are confirmed to be mechanically stable within the relevant de nsity and Hranges.\nFigure 8 illustrates the phonon spectra for certain points in the red uced first Brillouin zone\nfor the KAG, HC, and ST structures for H= 0.875. THE HC and KAG structures have one\nparticularly soft acoustic branch, labeled in the figures. Using the r atioω2\nmax/ω2\nminatM\npoint a simple metric for the extent of mechanical stability, this ratio stays relatively fixed\nfor the KAG lattice as Hincreases. This marks an achievement since previous attempts\nto stabilize the kagom´ e crystal had been unsuccessful.8For the HC structure, this ratio is\nhigher for H= 1.00 than for that which is plotted in Fig. 8, indicating that the crystal is\nmore stable as Hincreases. In general, the quintic potential and the honeycomb po tential8\nyield similar ratios. The striped phase whose lattice vectors for a= 1.51 andH= 0.875 are\n[1.65029,0] and [0.27688,0.91490] is also mechanically stable. The phonon spectra from the\n160100200300400 ω2\nK Γ M*\n(a)050100150200250 ω2\nK Γ M*\n(b)050100150200 ω2\nΓ Γ 1 2 3 2123\nΓ\nkykx\n(c)\nFIG. 8: (Color online) Phonon spectra for (a) the kagom´ e cry stal ata= 1.035 and (b) the\nhoneycomb crystal at a= 1.2, and (c) the striped phase at a= 1.51 for wave vectors in the\nreduced first Brillouin zone for H= 0.875. The first Brillouin zone for the striped phase is shown\nin inset in (c). These crystals are mechanically stable at th ese densities with the quintic potential.\nThe branches containing soft acoustic modes are denoted wit h a *.\nother “quadrants” are identical to the quadrant displayed in Fig. 8 (c) due to the symmetry\nof the Brillouin zone.\nC. Low-Energy Structures\nAlthough those structure obtained by MD and MC methods were sub optimal, Figures 3\nand 4 show that the free-energy differences between these stru ctures and the ground states\nare small. Several structural motifs emerge for this potential an d are shown in Fig. 9 for\nH= 0.5. These metastable states may have technological value if the fre ezing kinetics are\nsufficiently slow. Preliminary simulations suggest the freezing behavio r varies with density.\nForexample, with a <1.2, systems typically freeze into atriangularlatticewithvacancies\nrandomly distributed as in Fig. 9(a). Although this is not a ground-st ate structure, it yields\na vacancy-riddled lattice. The vacancies appear to have no particu lar order. The freezing\ntransition, the temperature at which the system changes from a h igh-density liquid to an\nordered phase, appears to be first order. The drop in potential e nergy as the temperature\nis reduced is sharp in this density range.\nSystems at densities where coexistence between the HC and ST pha ses are the ground\nstates tend to exhibit a freezing from the liquid phase to a rigid, stru ctured phase. However,\n17thestructured phase, which we believe tobemetastable, ischarac terized byringsandstrings\nas in Fig. 9(b). The six-particle ring is a characteristic of the HC stru cture and the strings\nare characteristics of the ST phase. These metastable states ar e disordered due to the fluid\nnature of the strings. The drop in potential with temperature is no t as sharp for these\ndensities compared to those at higher densities.\nLastly, at lower densities, as with those associated with the ST and o pen TRI phases, the\nmetastable, low-energy states adopt labyrinthine characteristic s, Fig. 9(c), and eventually\ncolloidal polymers and monomers, 9(d). In this density range, the d rop in potential energy\nassociated with freezing is weak. The phenomena and characterist ics of low-energy states\nare common to all H. However, as Hincreases so does the propensity to form labyrinthine\ncharacteristics. This is due to the lower energy barrier and the dec rease inrb, the location\nof the energy barrier. A qualitatively similar, but piecewise linear, pot ential also yields\na labyrinth phase at positive temperature.37Although the positive temperature behavior\nhas not been fully explored in this paper, we anticipate that the equa tion of state for this\nfamily of potentials will have interesting behavior. Manipulation of the kinetics to achieve\nsuch unusual structures represents one path to achieving kinet ically stable materials with\ncontrolled vacancy concentrations.\nV. SPECIAL CASES: H= 0\nTheH= 0 pair potential has interactions that vanish at a pair distance of u nity and\nbeyond√\n3. At positive pressure, the ground state is the dense triangular la ttice. However,\nat zero pressure and a≥√\n3/2, ground-state configurations will have a vanishing poten-\ntial energy and pressure (enthalpy vanishes). Thus, the type of available ground states is\ndependent on the area.\nA number of interesting structures arise as ground states. For a >√\n3/2, dilutions of the\ntriangular lattice with unit neighbor spacing are ground states, inclu ding the honeycomb\nand kagom´ e crystals and other lattice gases. For a=√\n11/2, the striped phase, a Bravais\nlattice with lattice vectors of [1 ,0] and [1 /2,√\n11/2], becomes available as a ground state.\nEach particle has two neighbors at unit separation and four neighbo rs at separation of√\n3.\nFora >√\n11/2, dilutions of this lattice are also ground states. In addition, any sm all\nexpansion in the direction normal to the stripes, will allow each stripe to gain some fluidity.\n18(a)a= 0.9 (b)a= 1.4\n(c)a= 1.7 (d)a= 2.3\nFIG. 9: (Color online) Metastable configurations obtained v ia molecular dynamics for H= 0.5.\nIn an attempt to cool to a liquid to the ground state, the syste m has difficulty crystallizing\nand/or phase separating into the appropriate ground-state structures. As density is decreased,\nthe metastable states go from ordered structures with vacan cies to a labyrinthine network to a\nmonomeric liquid.\nComplex liquids of strings have found as ground states using molecula r dynamics for at least\na≥2.15. However, the geometric problem is highly nontrivial since for sev eral runs with\na >2.15, ground states could not always be obtained. For a= 3√\n3/2, an open triangular\nlattice with neighbor spacing of√\n3 is a ground state structure.\nThe question remains as to which type of configurations are entrop ically (thermodynami-\n19cally) favorable, orratherwhichtypeofsystem makesupthelarge stfractionofconfiguration\nspace. We first make the distinction between discrete ( e.g.lattice gas) and continuous en-\ntropy (e.g.hard-disk crystal). In the classical case, continuous entropy is u ncountable while\ndiscrete entropy is not. Making the analogy to hard disks and hard s pheres, we believe that\nthe highest entropy phases for a specified area are those that ha ve the availability for contin-\nuous entropy. Configurations withthe highest-dimensional config urationspace dominate the\nT= 0 entropy. We estimate that, for a certain a, configurations with the fewest constrained\ndegrees of freedom will have the highest dimensional configuration space, and therefore the\nhighest entropy. For a≤√\n3/2, all degrees of freedom per particle are constrained. For\na≥3√\n3/2, no degrees of freedom per particle are constrained.\nIn order to gain continuous entropy, the system must have the ab ility to expand the\n√\n3 “bonds,” which can be either first-neighbor or second-neighbor c onnections between\nparticles. Any slight expansion of the dense triangular lattice is not a ground state because\neach neighbor must be constrained to unit separation. However, a n expansion of the open\ntriangular lattice is still a ground state because the√\n3 bonds are not constrained from\nabove. The problem can be cast as a tiling problem of four triangular t iles. The possible\ntriangular tiles are those with side lengths of unity or√\n3. These are depicted in Table I\nalong with their shorthand notations.\nBecause there are no constraints beyond distances of√\n3, the edges with lengths of√\n3\ncan be considered “elastic.” For simplicity, we first consider the tiling p roblem where these\nedge lengths are fixed. The rules of the tiling problem are as follows:\n•The plane must be tiled with no gaps.\n•For pairs of tiles that share an edge, the edges must be of the same length so that the\nvertices also match.\n•The(1,1,r)and(1,r,r)tilescannotsharea√\n3edgesincethisviolatesasecondneighbor\nconstraint ( i.e.two vertices are separated by a length between unity and√\n3).\nThe internal angle of the (1,r,r) tiles are such that they cannot inte grate well with the\nother tiles. Therefore, in any tiling that includes (1,r,r) tiles, these t iles must “phase sepa-\nrate” from the others. We consider two types of tilings - those with (1,r,r) tiles and those\nwithout. First, we consider those without (1,r,r) tiles. An example of such a tiling is shown\n20TABLE I: Available triangular tiles for the H= 0 tiling problem.\nName Shape Angles (◦) Area\n(1,1,1) 60, 60, 60√\n3/4\n(r,r,r) 60, 60, 60 3√\n3/4\n(1,1,r) 30, 30, 120√\n3/4\n(1,r,r) 33.56, 73.22, 73.22√\n11/4\nin Fig. 10(a) where each vertex is decorated with a particle. These t ilings are simply di-\nlutions of the dense triangular lattice, or coexistence between a de nse and open triangular\nlattice. The (1,1,r) tiles have two roles. They can act as intermediarie s between domains\nof the dense triangular lattice and the open triangular lattice, and t hey can dimerize to\nform a rhombus making a domain of the dense triangular lattice. This c oexistence between\nthe dense and open triangular lattices is represented as a double-t angent line connecting\nthe fully constrained dense triangular lattice to the unconstrained open triangular lattice as\nshown in Fig. 11.\nNext we consider those tilings that include (1,r,r) tiles. Two periodic tilin gs with a\nspecific area of a=√\n11/2≈1.658312 exist and are shown in Figs. 10(b) and 10(c). We\ndeem these thestripe and“zig-zag”tilingsrespectively. Inthese p hases, any small expansion\nperpendicular to the stripe or zig-zag will allow the stripes to have flu idity. On average,\nthese configurations will constrain one degree of freedom per par ticle, as plotted in Fig. 11.\nFor any tiling consisting of all tiles, the (1,r,r) tiles must segregate so that the the other\ntiles can fully tile the plane. Dimerizing the (1,r,r) tiles along the√\n3 edges creates a wide\nstripe. Building off the wide stripe requires (1,1,1) tiles or (1,1,r) tiles an d forms a local\ncoexistence with the dense triangular lattice as shown in Fig. 10(d). Dimerizing the (1,r,r)\ntiles along the unit length edge creates a narrow stripe. The expose d edges have length√\n3,\nrequiring (r,r,r) tiles to be directly adjacent to the stripe. This form s a coexistence between\n21(a)\n (b)\n (c)\n(d)\n (e)\nFIG. 10: (Color online) (a) Tiling excluding (1,r,r) tiles, (b) striped phase using only (1,r,r) tiles,\n(c) zig-zag phase using only (1,r,r) tiles (d) coexistence b etween dense triangular lattice and the\nstriped phase, and (e) coexistence between the open triangu lar lattice and the striped phase.\nthe the striped phase and the open triangular lattice. The double ta ngent construction of\nthesecoexistences isdisplayed asthesolidredlineinFig.11. These co existences between the\nstriped phase and the dense triangular lattice and the striped phas e and the open triangular\nlattice maximize the number of unconstrained degrees of freedom a nd likely maximize the\nentropy of the ground state. In addition, these system can take on discrete entropy by way\nof stacking variants of striped phase and the lattice phase. Using “ S” and “T” to denote a\nstripe and a triangular lattice layer, a STTTTSSS system is degenera te with a STSTSTST\nsystem.\nThe zig-zag phase can only form local coexistence with the open tria ngular lattice. To do\nso, the(1,r,r)tilesmustformatrimerwhichexposesthe√\n3edgesoneithersideofthetrimer.\nSince they are in trimers, their stacking entropy would be less that t he stacking entropy\navailable to the coexistence between the stripe and open triangular lattice. Therefore, we\nbelieve the stripe phase has a higher entropy than the zig-zag phas e.\n220.5 11.5 22.5 3\na = A/N00.511.52Config. Space Dim. /N Dense/Open Tri Coex\nDense/ST, ST/Open Coex\nFIG. 11: (Color online) Unconstrained degrees of freedom pe r particle for candidate ground-state\nstructuresfor H= 0potential. Configurationswiththemaximumnumberofunco nstraineddegrees\nof freedom are presumed to be the ground state (red circles).\nVI. DISCUSSION AND CONCLUDING REMARKS\nIn this paper, we have developed the ground-state phase diagram of a new potential\nthat gives rise to a number of novel phases that include low-coordin ated crystal structures.\nGiven the unusual nature of the ground state, we expect that th e equation of state and full\nphase diagram for these systems to exhibit other unusual behavio rs at positive temperature.\nFor example, in the global phase diagram of the the Lennard-Jones -Gauss, or honeycomb\npotential, there was no gas-liquid coexistence in the low-temperatu re, low-density part of\nthe phase diagram, nor was there a liquid-liquid phase coexistence27We expect most of\nthe solid-solid transitions that we find at zero temperature will rema in at small nonzero\ntemperature. Our preliminary calculations suggest that strings, o r polymers, may arise in\nequilibrium at low densities.\nIn addition, we have interest in the mobility of vacancies, particularly in the cases where\nvacancy concentrations are dilute, ( e.g.H= 0 and a≈√\n3/2), due to the possible relation\nto supersolid behavior.17,18Forajust above√\n3/2 andH= 0, we expect there to be a small\nnumber of vacancies in the system at low-temperature. We have es timated the potential\n23energy required for a particle to “hop” from a lattice site to a vacan t site. By initializing a\nsystem with one vacancy with a “hopping” particle along the transitio n to the vacant site,\nin an otherwise undistorted lattice, we minimized F(rN) =|∇u(rN)|2, whererNrepresents\nparticle coordinates, using conjugate gradient minimization. The re sulting configuration\nrepresents a saddle point in the energy landscape. The difference in potential energy of the\nsaddle point and the ground state is the energy barrier required fo r the particle and vacancy\nto swap positions. This barrier sets an activation energy for classic al thermal motion. For\nthe quantum case, it would also enter in the tunneling rate. For H= 0 potential, there is a\nsingle saddle point in the vicinity of the numerous initial conditions in the energy landscape\nwith a total energy barrier height of 5.569015. The diffusing particle is midway between the\norigin and destination sites while the bracing particles are displaced off the lattice line to\naccommodate the jump. Using molecular dynamics, we expect to rela te this saddle point\nenergy to a vacancy diffusion coefficient.\nThe quintic potential can further be generalized by varying the dist ance at which the\nfunction is truncated. We set this cutoff distance to be√\n3. However, allowing this cutoff\ndistance to vary introduces a larger class of potentials. A systema tic study of the cutoff\nradius on the robustness of the ground states is necessary for e xperimental realization. The\nhard-core plus square shoulder potential has ground states tha t vary significantly depending\non the relative lengthscale of the hard-core distance and the squa re-shoulder distance31,32\nIt is expected that the ground states of the generalized quintic po tential would be sensitive\nto the location of the minimum and the cutoff distance, though it is cur rently unknown\nhow sensitive the ground-state phase diagram is to these paramet ers. Understanding this\nsensitivity is important to experimentalists who want a simple, robust potential. Developing\nan optimization procedure to make self-assembly more robust would be particularly useful.\nIn addition, an extension to three-dimensions may provide additiona l fundamental insight.\nAs mentioned earlier, inverse optimization techniques have been effe ctive in develop-\ning potentials for targeted material properties. We intend to deve lop a general and broad\ninverse optimization technique to target specific vacancy arrange ments by accounting for\nand/or manipulating long-ranged vacancy-vacancy interactions. For example, one might de-\nvelop an objective function whose variables include the strength, s ign (attractive/repulsive),\nand angular dependence of vacancy-vacancy interactions. Alter natively, one might consider\na two-component system system of a heavy particle and a light part icle and apply an in-\n24verse optimization technique to this system. Using a broad family of p otentials, one could\nthen optimize over the available parameters to achieve a dilute conce ntration of effectively\nrepulsive vacancies.\nVII. ACKNOWLEDGMENTS\nWe thank Lawrence Cheuk for initial work on a related model. S.T. tha nks the Institute\nfor Advanced Study for its hospitality during his stay there. This wo rk was supported by the\nOffice of Basic Energy Sciences, U.S. Department of Energy, Grant DE-FG02-04-ER46108..\n∗Corresponding author; Electronic address: torquato@elec tron.princeton.edu\n1J. Sharma, R. Chhabra, A. Cheng, J. Brownell, Y. Liuand H. Yan ,Science, 2009,323, 112–116.\n2A. P. Hynninen, J. H. J. Thijssen, E. C. M. Vermolen, M. Dijkst ra and A. van Blaaderen,\nNature Materials , 2007,6, 202–205.\n3Y. K. Cho, R. Wartena, S. M. Tobias and Y. M. Chiang, Adv. Func. Mat. , 2007,17, 379–389.\n4S. Torquato, Soft Matter , 2009,5, 1157–1173.\n5H. Cohn and A. Kumar, Proc. Nat. Acad. Sci. , 2009,106, 9570–9575.\n6W. B. Russel, D. A. 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Rev. , 1962,127, 359–361.\n26"    },    {        "title": "1011.4564v1.Fractional_occupation_in_Kohn_Sham_density_functional_theory_and_the_treatment_of_non_pure_state_v_representable_densities.pdf",        "content": "Fractional occupation in Kohn-Sham density-function al theory and the \ntreatment of non-pure-state v-representable densities. \n \nEli Kraisler \nRaymond and Beverley Sackler Faculty of Exact Scien ces, School of Physics and Astronomy, Tel Aviv Univ ersity, \nTel Aviv 69978, Israel and Physics Department, NRCN , P.O. Box 9001, Beer Sheva 84190, Israel   \n \nGuy Makov \nPhysics Department, Kings College, London, The Stra nd, London WC2R 2LS, UK and \nPhysics Department, NRCN, P.O. Box 9001, Beer Sheva  84190, Israel   \n \nNathan Argaman \nPhysics Department, NRCN, P.O. Box 9001, Beer Sheva  84190, Israel   \n \nItzhak Kelson \nRaymond and Beverley Sackler Faculty of Exact Scien ces, School of Physics and Astronomy, Tel Aviv Univ ersity, \nTel Aviv 69978, Israel \n \n  \nIn the framework of Kohn-Sham density-functional th eory, systems with ground-state densities that are \nnot pure-state v-representable in the non-interacting reference sys tem (PSVR) occur frequently. In the \npresent contribution, a new algorithm, which allows  the solution of such systems, is proposed. It is \nshown that the use of densities which do not corres pond to a ground state of their non-interacting \nreference system is forbidden. As a consequence, th e proposed algorithm considers only non-interacting  \nensemble v-representable densities. The Fe atom, a well-known  non-PSVR system, is used as an \nillustration. Finally, the problem is analyzed with in finite temperature density-functional theory, wh ere \nthe physical significance of fractional occupations  is exposed and the question of why degenerate stat es \ncan be unequally occupied is resolved. \n \n31.15.E-, 31.15.A-, 31.15.-p  I. INTRODUCTION. \n \nDensity-Functional Theory (DFT) is the leading theo retical framework for studying the electronic \nproperties of matter [1, 2, 3]. Most practical appl ications rely on the Kohn-Sham (KS) formulation [4] , \nwhich was originally formulated assuming that the g round state density of an interacting N-electron \nsystem in an external potential ( ) v r /arrowrightnosp can be represented by the ground state density of a non-degenerate \nnon-interacting N-electron system in a reference potential ( ) eff v r /arrowrightnosp. This assumption is known as pure-\nstate v-representability of the density in the non-interac ting system, and is denoted PSVR below.  \nFor PSVR systems [4], the N-electron wave function 1( ,..., ) N r r Ψ/arrowrightnosp /arrowrightnosp of the reference system can be written \nas a single Slater determinant of the one-electron wave functions ( ) irψ/arrowrightnosp, which are solutions of the non-\ninteracting Schrödinger equation ( )2 2 ( / 2 ) ( ) e eff i i i m v r ψ εψ − ∇ + = /arrowrightnosp/planckover2pi . The index i refers to the relevant set \nof quantum numbers, e.g. { , , , } i n l m σ =  for spherical atoms. The density is given by 2( ) ( ) N\ni\nin r r ψ=∑/arrowrightnosp /arrowrightnosp, \nwhere the sum is over the N lowest-energy eigenstates of the system. The effec tive potential \n[ ] ( ) [ ] [ ] eff H xc v n v r v n v n = + + /arrowrightnosp is a functional of the density, where 2 3 [ ] ( ) / Hv n e n r r r d r ′ ′ ′ = − ∫/arrowrightnosp /arrowrightnosp /arrowrightnosp and \n[ ] / xc xc v n E n δ δ =  are the Hartree and the exchange-correlation poten tials, respectively. The density and \nthe total energy 3[ ] [ ] ( ) ( ) [ ] [ ] s H xc E n T n n r v r d r E n E n = + + + ∫/arrowrightnosp /arrowrightnosp of the interacting system are found by \nsolving self-consistently the equations above. Here  2 2 [ ] /(2 ) N\ns e i i \niT n m ψ ψ =− ∇ ∑/planckover2pi  is the kinetic \nenergy and [ ] HE n  and [ ] xc E n  are the Hartree and exchange-correlation energy te rms. It was assumed, \nand later shown to be true [5, 6], that for ground state densities the exchange-correlation energy \nderivative /xc E n δ δ  exists.  Note, however, that the self-consistent s olution of the Kohn-Sham \nequations requires the existence of the exchange-co rrelation energy derivative /xc E n δ δ  for any density \nconsidered in the iterative process, not only for t he ultimate density.  \nTwo questions arise immediately from this formulati on, the first of which has been considered \nextensively and the second far less so. (i) Are all  ground state densities PSVR? (ii) For which densit ies \ndoes [ ] / xc xc v n E n δ δ =  exist and how can the self-consistent solution pro cess ensure that only such \ndensities are considered? From early on it was know n [7, 8] that there are physical systems (e.g., Fe and Co atoms in the spherical approximation), for which  the density was not represented by a pure state. F or \nthese atoms, no self-consistent result was found fo r which integrally occupied orbitals were also the \nlowest in energy. The absence of a self-consistent solution expressed itself in the so-called ‘Fermi \nstatistics’ problem [9]: the energy levels of the e ffective potential cross repeatedly during the iter ations, \nand being occupied according to the ground-state ru le, yield radically different orbital densities, so  that \nthe self-consistent cycle cannot converge.  \nA possible stopgap measure is to suspend the requir ement of ground state occupation and to allow the \nsystem to relax while freezing the energy level occ upation, resulting in convergence to a solution in \nwhich the representing non-interacting system is in  an excited state. In this spirit, Janak [8] propos ed an \nextension of the Kohn-Sham scheme to include occupa tion numbers { } ig (termed the electronic \nconfiguration) as parameters in the definitions of the density 2( ) ( ) i i \nin r g r ψ=∑/arrowrightnosp /arrowrightnosp and the kinetic energy \n2\n2\n2s i i i \ni eT g mψ ψ = − ∇ ∑/planckover2pi/tildenosp . To determine the occupation numbers, Janak minimi zed the total energy \n3[{ }] ( ) ( ) [ ] [ ] i s H xc E g T n r v r d r E n E n = + + + ∫/arrowrightnosp /arrowrightnosp/tildenosp /tildenosp  with respect to { } ig. It was shown  [8] that the minimum \nof the total energy E/tildenosp is achieved only for a \"proper\" electronic configu ration, i.e. one in which the \nenergy levels are occupied without gaps. This inclu des not only the PSVR systems treated by Kohn and \nSham, but also cases where the representing non-int eracting N-electron system possesses a ground-state \ndegeneracy, with the highest electron orbitals dege nerate and partially occupied. For such a \nconfiguration the density is ensemble v-representable in the non-interacting reference sys tem [2], and \nthis is denoted by NI-EVR below. However, Janak que stioned the validity of his results in the non-\nPSVR cases, especially in situations with unequal o ccupations of the degenerate orbitals. \nAddressing the problem of v-representability, a recent study by Ullrich and Ko hn [10] on topologies of \nthe v and n spaces examined for lattice systems showed that po tentials which generate NI-EVR densities \nare not isolated points in v-space, because not every perturbation potential im posed on the system will \nlift the degeneracy of its energy levels. Even more  significant is that NI-EVR densities occupy \nmanifolds of finite volume in n-space. Thus, systems with NI-EVR densities are not  rare, but form a \nsignificant physical class that should be treated w ith proper care. \nFew examples of EVR densities have been studied in detail.  Some of these studies are analytical, and \nthus relate to the exact (but unknown) exchange-cor relation functional, while others involve direct computations using available approximate functional s. Recently, two analytical examples were \nintroduced [10, 11]: a finite lattice problem, whic h produced either a PSVR or an NI -EVR solution, \ndepending on the value of the potential on the diff erent lattice points, and the Be ion series. \nUpon application of the Janak formalism, NI-EVR den sities have been identified in several physical \nsystems, such as Fe and Co atoms [7, 8]. The occupa tions of the 3d↓ and 4s↓ degenerate levels are \nfound to be fractional and unequal, contrary to wha t one might expect from elementary statistical \nmechanics considerations. This surprising result ma de the authors [7, 8] question their findings and \nspeculate that they might be a consequence of the s pherical symmetry approximation or the local-densit y \napproximation employed.  \nOther authors have rejected fractional occupation a ltogether. Averill and Painter [12] devised a \nnumerical scheme in which the minimization over the  occupations in Janak’s functional is performed as \npart of the iterative process, but took the view th at the physical representation of the system would \nnecessarily involve an electronic configuration wit h integral occupation numbers, even if it is \n“improper”. Alternatively, in a well-known comprehe nsive study in atomic physics performed by \nKotochigova et al. [13], the experimental electroni c configurations were used throughout the \ncalculations. Although for some atoms the occupatio ns thus obtained are \"improper\", the validity of su ch \na choice of electronic configuration for different atoms/ions was not discussed. \nThe Kohn-Sham scheme requires differentiability of both the kinetic and the exchange-correlation \nfunctionals [5, 6]. After the generalization of den sity-functional theory in the constrained-search \nformalism to all N-representable densities [14, 15], it was found [5, 6, 16] that differentiability exists \nonly for densities which are both interacting ensem ble v-representable (I-EVR) and non-interacting \nensemble v-representable (NI-EVR) (see Sec. II for the defini tions). However, during the iterative \nconvergence of the Janak-Kohn-Sham equations, the d ensities generated are not restricted to be EVR. \nThis is unlike the situation in the self-consistent  solution of the Kohn-Sham equations, where if the \ndensities are generated from ground states of the t rial effective potentials, then they are NI-EVR by \nconstruction. Therefore, the Janak scheme is expect ed to be invalid for the exact form of [ ] xc E n  and is \nformally applicable only for approximations, which are differentiable at all densities, such as the lo cal-\ndensity approximation (LDA). \nTherefore, in consequence of the extension of the D FT-KS formalism to include NI-EVR densities, it is \ndesirable to have an algorithm which will guarantee  that only EVR densities are considered in the self -\nconsistent solution of the Kohn-Sham equations. To the best of our knowledge, such an algorithm has not yet been suggested. We note that it was proved [5, 17] that for lattice systems all the densities are \nEVR. Therefore, in computational implementations on  a grid all densities are implicitly EVR. However, \nthis result has not been extended to the continuum,  and this indicates the benefit of an algorithm whi ch \nis restricted to explicitly EVR densities. \nSurprisingly, in all of these developments the fini te-temperature version of DFT (T-DFT) is scarcely \nmentioned. The finite-temperature theory can be rep resented as an extension of the standard theory of \nthermal ensembles to non-uniform systems [3]. It is  well-known that finite-temperature DFT does not \nsuffer from the v-representability problem, and connects continuousl y to the ground-state theory in the \nlimit 0→T  [18]. This has previously allowed the resolution o r reinterpretation of several theoretical \nissues in DFT. \nIn the present work we present three contributions to the ongoing discussion.  First, we show that \ndensities generated by \"improperly\" occupying the o rbitals in the reference system cannot be ground \nstate densities of the given external potential, co mplementing the known result that the ground state can \nbe described by properly occupying the reference sy stem. Consequently, we present a new algorithm \nwhich allows the solution of Kohn-Sham systems with  ground-state densities that are not PSVR, while \nconsidering only EVR densities in the process and o vercoming the “Fermi statistics” convergence \ndifficulties [9]. We apply the algorithm to the wel l-known case of Fe and show that the results obtain ed \nare converged, stable numerically and the emergence  of NI-EVR density is insensitive to the choice of \nthe exchange-correlation functional approximation. We also demonstrate that the subspace of external \npotentials which yield NI-EVR densities is finite, by considering variations of the nuclear charge abo ut \n26 Z= .  In passing, we discuss the solution of the Be io n series in the LDA.  Finally, our third \ncontribution is an analysis of the EVR problem with in T-DFT, where we expose the physical \nsignificance of the fractional occupations and reso lve the question of why degenerate states can be \nunequally occupied. \n \n \nII. THEORY. \n \nThe extension of the Kohn-Sham formalism to EVR den sities is briefly presented below for \ncompleteness. An interacting ensemble- v-representable (I-EVR) density is defined [5] as an  integrable \nnon-negative function ( ) ( ) ( ) k\nkkn r n r λ=∑/arrowrightnosp /arrowrightnosp, with 0kλ≥, 1kkλ=∑ , where 2( ) ( ) \n2 2 ( ) ... ( , ,..., ) ... k k \nN N n r N r r r dr dr = Ψ ∫ ∫ /arrowrightnosp /arrowrightnosp/arrowrightnosp /arrowrightnosp /arrowrightnosp /arrowrightnosp and ( ) kΨ are orthonormal degenerate ground states of the \nHamiltonian of an interacting many-electron system : \n  ˆ ˆ ˆ ˆ H T V W = + +  (1) \nHere 2 2 ˆ /(2 ) e i iT m =− ∇ ∑/planckover2pi  is the kinetic energy operator, ˆ ( ) i iV v r =∑/arrowrightnosp is the external potential operator \nand 2 ˆ( / 2) 1/ i j i j W e r r ≠= − ∑/arrowrightnosp /arrowrightnosp is the electron-electron repulsion operator.  The pure-state case is a \nparticular case with only one 1kλ=, and the others are zero. A non-interacting ensemb le-v-representable \n(NI-EVR) density is defined similarly, with a non-i nteracting Hamiltonian 0ˆ ˆ ˆ \neff H T V = +  ( ˆ0 W=).  \nAccording to the constrained search formulation of Hohenberg-Kohn theorems, the ground state energy \nis the minimum value of the functional of the densi ty 3[ ] ( ) ( ) [ ] v L E n v r n r d r F n = + ∫/arrowrightnosp /arrowrightnosp, for a given \nexternal potential ( ) v r /arrowrightnosp. [ ] LF n  is the Lieb functional ([2], p. 14 and references therein, [13], [16]): \n  ˆ ˆ [ ] min \ni i iL i i i \nd n n iF n d T W \n=  = Φ + Φ     ∑∑ . (2) \nwith 0id≥, 1iid=∑ , 2\n2 2 ( ) ... ( , ,..., ) ... i i N N n r N r r r dr dr = Φ ∫ ∫ /arrowrightnosp /arrowrightnosp/arrowrightnosp /arrowrightnosp /arrowrightnosp /arrowrightnosp, where 1 2 ( , ,..., ) i N r r r Φ/arrowrightnosp /arrowrightnosp /arrowrightnosp are all anti-\nsymmetric normalized N-particle wave functions.  \nSimilarly to the pure-state case, the density is re presented through a reference system of non-interac ting \nelectrons, defined by ( ) eff v r /arrowrightnosp. Then, the energy [ ] vE n  can be written as \n3\n, [ ] ( ) ( ) [ ] [ ] [ ] v L H xc L E n v r n r d r T n E n E n = + + + ∫/arrowrightnosp /arrowrightnosp, where \n  ˆ [ ] min \ni i iL i i i \nd n n iT n d T \n=  = Φ Φ     ∑∑ , (3) \n[ ] HE n  is the Hartree energy term and the exchange-correl ation term is defined as \n,[ ] [ ] [ ] [ ] xc L L H L E n F n E n T n = − − .   \nTo obtain the density from the reference system, th e effective potential is determined, up to a consta nt, \nvia the differential Euler relations for the intera cting and the non-interacting systems : \n  , / ( ) [ ] / L H xc L T n v r v n E n const δ δ δ δ + + + = /arrowrightnosp, (4) \n  / ( ) L eff T n v r const δ δ + = /arrowrightnosp. It follows that , ( ) ( ) [ ] [ ] eff H xc L v r v r v n v n = + + /arrowrightnosp /arrowrightnosp, only if both /LT n δ δ  and , , / [ ] xc L xc L E n v n δ δ =  exist. \nOtherwise, no relation between the interacting syst em and the reference system can be established. \nThe differentiability conditions for both [ ] LF n  and [ ] LT n  were formulated by Englisch and Englisch [5]: \n(1) the functional [ ] LT n  is differentiable at all NI-EVR densities, and onl y there; (2) the functional \n,[ ] xc L E n  is differentiable at densities, which are both NI- EVR and I-EVR, and only there. While the \nequivalence of these two sets of densities has the status of a conjecture, it can be proved that for a ny I-\nEVR density there exists a NI-EVR density arbitrari ly close. Therefore, a Kohn-Sham scheme can \nalways be set up [16]. Regarding condition (2), not e that for explicit approximations employed for \n,[ ] xc L E n , such as the LDA, mathematical differentiability i s always assured, even if for the given density \nthe exact functional would not be differentiable. \nAs to condition (1), we note that the ground state wave-functions iΦ of a non-interacting system must \nobviously be constructed as Slater determinants of the N lowest eigenfunctions ( ) irψ/arrowrightnosp that emerge from \nthe Schrödinger equation ( )2 2 ( / 2 ) ( ) e eff i i i m v r ψ εψ − ∇ + = /arrowrightnosp/planckover2pi . It can be shown that the kinetic energy \nfunctional [ ] LT n  takes the form  \n  \n22\n2[ ] 2min \ni i iL i i i \ni g n eT n g mψψ ψ \n=  = − ∇     ∑∑/planckover2pi (5) \nand the density is given by  \n  2( ) ( ) i i \nin r g r ψ=∑/arrowrightnosp /arrowrightnosp,  (6)  \nwhere the occupation numbers ig are linearly related to the coefficients id and are restricted to be : \n   :\n:\n0 : i i F \ni i i F \ni F D\ng x ε ε \nε ε \nε ε < \n= = \n> . (7) \nHere [0, ] i i x D ∈ , Fε is the highest occupied energy level and iD is the degeneracy of the i-th level, \nwhich can now be greater than 1, depending on the s pecial symmetries the problem may have, e.g. \nspherical symmetry for atoms. The restriction (7) i s a straightforward consequence of the fact that on ly \nground states of a reference system are considered.  We denote occupations consistent with (7) as \n“proper”. It can be seen from (7) that fractional occupation can occur only if for more than one energy level \ni F ε ε =. The physical interpretation of such a situation i s discussed by us in Sec. IV.  \nNow we consider the use of \"improper\" densities, i. e. those which are excited states of their own \neffective potential. As frequently happens in Janak -type calculations [7, 8, 12], a trial density ( ) n r /arrowrightnosp is \nrepresented in the form (6) with an “improper” (exc ited) occupation of levels of a non-interacting \nsystem, defined by ( ) eff v r /arrowrightnosp, which obeys ( ) ( ) [ ] [ ] eff H xc v r v r v n v n = + + /arrowrightnosp /arrowrightnosp. It is not evident that in such a \ncase the derivative /LT n δ δ  must exist [5]. However, if it exists, then there exists another non-\ninteracting system defined by [ ] ( ) [ ] [ ] eff H xc v n v r v n v n = + + /arrowrightnosp/tildenosp /tildenosp , for which ( ) n r /arrowrightnosp is the ground-state density \nor a linear combination of ground-state densities. Thus, ( ) n r /arrowrightnosp can be represented in terms of the ~( ) -\nsystem as 2\ni i in g ψ=∑/tildenosp/tildenosp  with \"proper\" occupation. Then, however, ( ) n r /arrowrightnosp cannot be the ground state of \nthe original system of interest because Euler’s rel ations (4) cannot be satisfied for [ ] eff v n  if they were \nsatisfied for [ ] eff v n /tildenosp . Therefore, we arrive at the conclusion that an “i mproper” configuration cannot be \nused in (6) to construct the density, even if /LT n δ δ  exists, as could be misunderstood from Ref. 19. As  \na corollary, an algorithm that relies on the genera lized form [ ] LT n  must be restricted to “proper” \ndensities only. \nConsidering the validity of Janak’s theorem questio ned by Valiev and Fernando [19], we comment that \nJanak’s theorem is valid, but only for Janak’s E/tildenosp, which is an extension of the energy functional to  \ninclude systems with variable and fractional number  of electrons, which are beyond the scope of the \npresent work. In the domain of systems with integer  occupation, the ground state densities are restric ted \nto be \"proper\", as in (7) and the { } ig are determined uniquely by the self-consistency re quirement. In \nthis formalism, Janak’s theorem, in the sense of a derivative of the energy functional with respect to  \noccupation numbers,  is inapplicable and this, we b elieve, should be the context of Ref. 19.  \nThe above formalism can be extended [2] to spin-pol arized cases via representing a spin-polarized \nsystem as two distinct (but coupled) electronic sub systems, denoted by the index {},σ= ↑ ↓ . The \ngeneralized kinetic energy functional [ , ] [ ,0] [ ,0] L L L T n n T n T n ↑ ↓ ↑ ↓ = + , where \n \n22\n2[ ,0] 2min \ni i iL i i i \ni g n eT n g m\nσ σ σ σ σ σ σ \nψψ ψ \n== − ∇ \n∑∑/planckover2pi. Each of the subsystems is restricted to be “properl y” occupied by (7), but there can be gaps in the \noccupation of the whole system if the first vacant level of one of the sub-systems lies below the last  \noccupied level of the another one. We call such a s ystem “proper in a broad sense”. This situation ari ses \nbecause we are constrained from having partial occu pation of spin states by the physical requirement \nthat the total spin in the reference system is half -integer, i.e. that the spin occupations remain int egral. \nTo summarize, the only densities which can be consi dered as candidate densities in the constrained \nsearch formalism are those with “proper” sets of oc cupation numbers.  In the spin-polarized case, each  \nof the spin subsystems must be properly occupied, e ven though the combined system may be \n“improperly” occupied due to the constraint of inte gral spin occupations. It remains to be shown in Se c. \nIII how to restrict the self-consistent search for the density to NI-EVR densities only. \n \n \nIII. NUMERICAL ALGORITHM AND APPLICATIONS. \n \nIn this section we present an algorithm to solve th e Kohn-Sham equations that is restricted to searchi ng \nonly over explicitly NI-EVR densities in accordance  with the results of Sec. II. We have shown above \nthat Janak’s approach involves minimizing the energ y over a set of densities which are not necessarily  \n“proper”, and therefore the existence of [ ] / xc xc v n E n δ δ =  is not assured. We apply our algorithm to the \nspherical Fe atom which is a simple, well-establish ed [7, 8, 12] non-PSVR system, which is NI-EVR. \n As an example of a NI-EVR system, we investigate s ome of its properties.  In passing we show that the  \nBe ion series which has been shown to be a NI-EVR s ystem [10, 11, 20] is instead found to be PSVR in \nthe LDA. \nIn the case of the spherical Fe atom, the use of in tegral occupation numbers does not lead  to a “prop er” \nsolution of the Kohn-Sham equations, a fact that wa s troubling from early days [7, 8]. Allowing \nfractional occupation numbers determined by the min imization of the total energy in Janak’s approach \nyields a NI-EVR density for the spin 2S=. E.g., using the Vosko, Wilk and Nusair (VWN) [21]  \nparameterization of the LSDA for the exchange-corre lation functional and a spherical approximation for  \nthe density yields -1261.229308 hartree E=  as the total energy for Fe, with an electronic con figuration \nof 5 1 \n1.398 0.602 [Ar]3 4 d s  [22].  This result is in agreement with earlier pu blications on Fe [7, 8, 12]. Our alternative algorithm for treating NI-EVR densi ties is to start with a reasonable guess for ,( ) eff v r σ/arrowrightnosp \nand occupy the energy levels of the reference syste ms “properly” given the total spin value S. Therefore, \nwe are assured that the density generated by the re ference system in each iteration is NI-EVR and by t he \nEVR conjecture [16] is also I-EVR. This allows us t o calculate the effective potential from this densi ty. \n If we employ this potential directly to calculate a new density, then as expected [9], the process st arts \nalternating between two electronic configurations, e.g. for 2S= between 5 1 \n1 1 [Fe] [Ar]3 4 d s =  and \n5 1 \n2 0 [Fe] [Ar]3 4 d s = .  \nInstead, as the first new step in the algorithm, we  reduce the linear mixing coefficient for the effec tive \npotential each time the electronic configuration ch anges. The reduction factor can vary around 0.5. By  \nrepeating this procedure, the eigenvalues eventuall y stop crossing and slowly approach each other. It is \npossible then to find an effective potential, with energy levels (3dε↓ and 4sε↓ in this example) that \ncoincide, to any required accuracy ( 610  hartree − was required for Fe) with the original integer \nconfiguration. However, this solution is not necess arily a self-consistent one, and additional iterati ons in \nthe domain of NI-EVR densities are required to obta in the final result.  \nThe second new step in the algorithm continues the search in the subspace of potentials that generate NI-\nEVR densities while relaxing the constraint on inte ger occupation. Once an effective potential with tw o \ndegenerate levels is found, it can produce a range of legitimate, “proper” NI-EVR densities by \nfractionally occupying the degenerate levels. Utili zing the fact that potentials generating NI-EVR \ndensities are not, in general, points in eff v-space, we choose the occupation that will generate  a new \npotential which preserves the degeneracy of the ene rgy levels for the next iteration. Assuming the \npotential change between two consecutive iterations  (denoted k and k+1) ( 1) ( 1) ( ) [ ({ })] k k k \neff i eff w v n g v + + = −  is \nsmall, it can be shown by means of perturbation the ory [23] that the degeneracy of levels a and b will be \npreserved if  \n  ( ) ( ) ( ) ( ) 0k k k k \na a b b w w ψ ψ ψ ψ − = .  (8) \nHere ( ) k\neff v, ( ) kn and ( ) k\nmψ are the effective potential, the density and the w avefunction of the m-th level, in \nthe k-th iteration, respectively, and { } ig symbolizes the dependence of the density on fracti onal \noccupations. Note that the presence of a Fermi-stat istics problem implies that equality (8) is fulfill ed \nwithin the available range, as the left-hand side c hanges sign as a function of the occupations, with the \ndegeneracy of the levels a and b lifted in one direction for one extreme occupation , and in the other direction for the other extreme. Numerically, equal ity (8) is enforced by searching for the electronic  \nconfiguration { } ig that, once employed to obtain a new density ( 1) ({ }) k\ni n g +, and consequently \n( 1) ( 1) [ ({ })] k k \neff i v n g + + , will nullify the left-hand side of eq. (8). The s earch is made using Newton-Raphson’s \nmethod, which converges within 3-5 steps for Fe, wh ere we have required a convergence of 610 − for the \noccupation numbers. This search is repeated for eac h iteration in this part of the convergence process , \naffording us to preserve the degeneracy in the refe rence system. \nContinuing the iterations described above, while pr eserving the degeneracy of the energy levels (3dε↓ \nand 4sε↓ in Fe), but not constraining their value, we event ually converge to a self-consistent result, \nwhich is confirmed by increasing the mixing coeffic ient. This result for Fe coincides, within the \nconvergence accuracy of 610  hartree E−∆ = , with the result reached by Janak’s approach.  \nA schematic description of convergence of Janak’s a lgorithm and the algorithm proposed above in the \neff v n − plane is shown in Fig. 1. Janak’s algorithm passes  through “improper” densities. A full \nminimization of occupation numbers brings it, howev er, to a “proper” fractionally occupied \nconfiguration. The algorithm proposed here uses onl y “proper” configurations, and by reduction of the \nmixing constant, enters the NI-EVR range of potenti als, where it converges to the desired density. \nThe process above is performed for all values of th e total spin S. In the example of Fe, the energy levels \nfor 0,1 S=  are occupied “properly in a broad sense” and posse ss higher total energies, as shown in \nTable I. As mentioned above, the stability of the r esults obtained was verified by increasing the mixi ng \ncoefficient once the iterations converged. No separ ation of the degenerate energy levels was observed.  \nThe influence of the initial guess for the effectiv e potential on the final result was also checked. I t was \nfound that the eigenvalue at which the energy level s first became degenerate depends significantly on \nthe guess. However, during the subsequent iterative  process within the degenerate subspace, the \neigenvalues converge to the expected value, along w ith the total energy and other quantities. Therefor e, \nwe conclude that the initial condition affects only  the number of iterations needed to reach self-\nconsistency, and not the converged self-consistent values. \nIn order to demonstrate that an emergence of a NI-E VR solution for Fe is not an artifact related to th e \nlocal-density approximation for the exchange-correl ation functional as speculated in Refs. 7 and 8, we  \nrepeated the calculation adding the PBE-GGA [24] to  the VWN-LSDA. We obtained a NI-EVR density \nin this approximation as well, with -1263.446126 hartree E=  for the total energy and 5 1 \n1.349 0.651 [Ar]3 4 d s  for the electronic configuration, as can be seen fr om Table I. Moreover, we have applied our algorithm  \nsuccessfully to additional atoms and ions using the  same procedures described here.  The details of th ese \ncalculations are omitted for brevity. \nIt was emphasized recently [10] that NI-EVR densiti es are not points in v-space, i.e. given an external \npotential ( ) v r /arrowrightnosp that produces a NI-EVR density, there are potentia ls in its environment that also produce \nNI-EVR densities. We use the Fe atom to demonstrate  such a situation. Changing the nucleus’ charge Z, \nwe obtain a region 25.82 26.25 Z< <  for which NI-EVR densities occur, whereas for 25.82 Z<  or \n26.25 Z>  PSVR solutions are obtained. The energy levels of the reference system and the occupation \nnumbers are plotted in Fig. 2 and 3, as a function of Z. \nFinally, we comment on the Be ion series that has b een previously presented as an example of a NI-EVR \nsystem. It was proposed [10, 11] to solve the Be io n series in the limit of large Z analytically, by \nmapping it onto a four-electron system with an elec tron-electron interaction, which becomes negligible  \nin the limit Z→∞ . The mapping is performed by a scaling / 4 r Zr =/tildenosp  for the distances and 2 2 4 / e e Z =/tildenosp  \nfor the electron charge, in the Hamiltonian (1). Wh en Z→∞ , the electron-electron interaction in the \n(~) -system vanishes and the energy levels 2 s and 2 p become degenerate, with a first-order correction \nproportional to 1/ Z. If the perturbation applied is of the exact form 2\n1 2 /e r r −/arrowrightnosp /arrowrightnosp, then the correct zeroth \norder wave functions mix the 2(2 ) s and 2(2 ) p configurations and generate the NI-EVR density as \nreported by Ullrich and Kohn [10] and in agreement with the CI-based procedure of Morrison [20].  \nHowever, Kohn-Sham calculations do not support this  result, for reasons unknown [20]. We propose \nthat this disagreement arises from symmetry conside rations. If a Hartree type approximation modified \nby the LDA (or an extension) is the perturbation, t hen its spherical character will cause the relevant  \nwave functions to separate into the 2(2 ) s and 2(2 ) p configurations, thus leading to a PSVR type \ndensity. Such a separation can be seen in Fig. 4, w hich illustrates the difference between the energy \nlevels 2sε and 2pε of the Be ion series calculated for 4, 40,100,200 Z=  with VWN-LSDA.   \n \n \nIV . DISCUSSION. T-DFT ANALYSIS. \n A better understanding of ensemble v-representable systems can be achieved by consideri ng the finite-\ntemperature approach to DFT [3, 10, 25], in which g round-state DFT corresponds to the limit of \nvanishing temperature [18]. \nIt can be shown (see [3] and references therein) th at at finite temperatures the grand canonical poten tial  \n  ˆ ˆ ˆ[ ( )] ln Tr exp B\nBT W v v r k T k T ρ       + + Ω =− −                 /arrowrightnosp (9)  \nis convex with respect to ( ) v r /arrowrightnosp and is always differentiable (for definitions see Sec. II). Here \n1ˆ ( ) N\ni ir r ρ δ == − ∑/arrowrightnosp /arrowrightnosp is the density operator, whose statistical average  yields the electron density \nˆ ( ) n r ρ=/arrowrightnosp. The first functional derivative of Ω equals the density: / ( ) ( ) v r n r δ δ Ω = /arrowrightnosp /arrowrightnosp. As a result, the \nHohenberg-Kohn free-energy functional  3[ ( )] [ ( )] ( ) ( ) HK F n r v r n r v r d r =Ω − ∫/arrowrightnosp /arrowrightnosp /arrowrightnosp /arrowrightnosp is differentiable with \nrespect to the density, and its first derivative is  / ( ) ( ) HK F n r v r δ δ =− /arrowrightnosp /arrowrightnosp. Since for a non-interacting \nelectron system, [ ( )] HK F n r /arrowrightnosp reduces to the kinetic energy functional, this las t functional is differentiable \nas well. Therefore, for electronic systems at finit e temperatures the v-representability question does not \nexist.  \nIn T-DFT, a many-electron interacting system at a t emperature T is described by reference to a non-\ninteracting system with the same temperature [26]. It follows then that the occupation numbers in the \nreference system are determined by the Fermi-Dirac distribution and the entropy is given by \n( ) ln( ) (1 )ln(1 ) B i i i i i \niS k D f f f f =− + − − ∑ , where the sum is over all energy levels and /i i i f g D = .  The \neffective potential preserves the form ( ) ( ) [ ] [ ] eff H xc v r v r v n v n = + + /arrowrightnosp /arrowrightnosp, but now the exchange-correlation \npotential depends also on T.  \nAt finite temperature the occupancy 1(1 exp[( ) / ]) i i i B g D k T ε µ −= + −  is determined by the combination \n( ) / i B k T ε µ − , which would lead us to expect that the fractional  occupations of degenerate states will be \nequal.  However, closer observation shows that in t he limit of zero temperature the Fermi-Dirac \ndistribution becomes multi-valued for iε µ =, taking on all values 0i i g D ≤ ≤ .  \nFor cases with several degenerate states at the Fer mi level having different occupations, the degenera cy \nis lifted linearly with temperature, with the combi nations ( ) / i B k T ε µ −  left essentially unchanged. This \ncan be seen in detail from the following relations.  Expanding the energy and the chemical potential fo r low temperatures, we obtain (0) (1) (2) 2 1( ) 2i i i i T T T ε ε ε ε = + + ,  (0) (1) (2) 2 1( ) 2T T T µ µ µ µ = + +  up to terms \n3( ) O T , where ( ) \n0 ( / ) n n n \ni i T d dT ε ε = =  and ( ) \n0 ( / ) n n n \nT d dT µ µ = = . If at 0T= there are two or more \ndegenerate partially-occupied levels, whose energy is (0) ε, then obviously (0) (0) µ ε = . Then, the partial \noccupation numbers of the i-th level at zero temperature are determined by the  relation: \n   (0) 1 (1) (1) 1 \n0lim (1 exp[( ) / ]) (1 exp[( ) / ]) i i i B i i B Tg D k T D k ε µ ε µ − − \n→  = + − = + −   . (10) \nThis exposes the physical significance of the magni tude of the partial occupation as indicating the \ntemperature dependence of the eigenstate relative t o that of the chemical potential. States with \noccupation less than / 2 iD  will be more temperature-dependent than the chemic al potential and vice \nversa. Only states which are both degenerate (i.e. have the same value of (0) \niε) and physically equivalent \n(i.e. have the same value of (1) \niε) will be equally occupied, whereas if the states a re accidentally \ndegenerate then the occupation numbers will be uneq ual, thus resolving Janak’s conundrum [8]. \nExpanding the occupation numbers ig, we obtain the following relation: \n  ( )(0) \n(0) (0) (2) (2) 11 ... 2i\ni i i i \nigg g g T Dε µ   = − − − +   \n   (11) \nIt can be seen that the occupation depends linearly  on the temperature through the second derivatives of \nthe energy and the chemical potential, therefore va rying significantly only at high temperatures.  \nWe have conducted some finite-temperature calculati ons for Fe, which support the above explanation. \nWe approximated the temperature-dependent exchange- correlation functional by its zero-temperature \nlimit 2[ ]( ) [ ] ( ) xc xc F n T E n O T = +  [25], and used the PBE-GGA approximation, as above . We \nconcentrated on the 2S= case only. Figure 5 shows the dependence of the ei genvalues 3dε↓ and 4sε↓ \nand the chemical potential as a function of the tem perature. Figure 6 demonstrates the dependence of t he \nnormalized occupation numbers if on the temperature. The numerical results agree wi th the expansions \nabove. In particular, it can be seen that for a van ishing temperature the results for \n(0) (0) (0) \n3 4 0.154624 hartree d s ε ε µ ↓ ↓ ↓ = = =− , 3 3 5 1.349 d d g f ↓ ↓ = =  and 4 4 0.651 s s g f ↓ ↓ = =  reproduce the zero-\ntemperature DFT results presented in Sec. III. In a ddition, from fig. 5 we deduce \n(1) \n3/ 3.0869 hartree/K B dkε↓= , (1) \n4/ 1.4659 hartree/K B skε↓=  and (1) / 2.0909 hartree/K Bkµ↓= . Substitution \ninto equation (10) yields (0) \n31.349 dg↓=  and (0) \n40.651 sg↓= , in full agreement with the results from Sec. III.    \n \nV . CONCLUSIONS. \nIn conclusion, in the present contribution we intro duce a new algorithm for the self-consistent soluti on \nof the Kohn-Sham equations in cases of non-PSVR den sities. Since we have shown that densities which \nare constructed from excited states of the effectiv e potential cannot be ground state densities of the  \ninteracting system, our algorithm solves for non-PS VR systems while considering NI-EVR densities \nonly.  It is perhaps surprising that such algorithm s have hitherto scarcely been discussed.  Avenues f or \nfuture research include further development of such  algorithms, e.g., based on studies of the efficien cy \nof their performance. \nThe Fe atom was used as an illustration. By conside ring variations in the nuclear charge Z in Fe, we \nhave also provided a demonstration that the subspac e of external potentials which yields NI-EVR \ndensities in Fe is finite. In addition, the fact th at NI-EVR densities appear in Fe was found to be \ninsensitive to the particular approximation of the exchange-correlation functional.   \nLastly, the physical meaning of fractional occupati on numbers, and in particular the unequal occupatio n \nof degenerate levels, has been clarified through th e finite-temperature density-functional formalism. As \nthe occurrence of ensembles in this formalism is na tural, it may be expected that it will aid progress  on \nadditional issues as well, including the abovementi oned issue of seeking improved algorithms. \n \n \nACKNOWLEDGEMENT. \nOne of us (E.K.) acknowledges fruitful discussions with Mr. Nir Sapir.  \nConfiguration S XC (3 ) dε↑ \n(hartree) (3 ) dε↓ \n(hartree) (4 ) sε↑ \n(hartree) (4 ) sε↓ \n(hartree) E (hartree) \"proper\"  \n[Ar]3d 5\n1.3974s 1\n0.603 2 LSDA  -0.275997 -0.161152 -0.196220 -0.161151 -1261.229308 yes  \n[Ar]3d 5\n24s 0\n1 1  LSDA  -0.187671 -0.107214 -0.155805 -0.162969 -1261.197603 b. s. \n[Ar]3d3\n44s 1\n0 0  LSDA  -0.140353 -0.163057 -0.168303 -0.147304 -1261.145890 b. s. \n[Ar]3d 5\n1.349 4s 1\n0.651  2 GGA  -0.277168 -0.154624 -0.191431 -0.154623 -1263.446126 yes  \n[Ar]3d 5\n24s 0\n1 1  GGA  -0.181669 -0.096826 -0.147664 -0.158220 -1263.411856 b. s. \n[Ar]3d 3\n44s 1\n0 0  GGA  -0.131012 -0.154871 -0.163296 -0.137957 -1263.357365 b. s. \n \nTABLE I: The electronic configuration, energy level s and the total energy for Fe at different spin \nvalues solved within LSDA and GGA (see definitions in text). The cases are identified as “proper” \nor “proper in a broad sense” (b.s.). \n \n FIG. 1. Schematic representation of the numerical c onvergence with various algorithms in the veff  \n– n plane. Each solid curve represents an integral ele ctronic configuration. The dashed curve on \npanel (c) corresponds to a fractional electronic co nfiguration. “Proper” occupations are denoted \nby circles, “improper”- by asterisks. Converged res ults (“proper” and “improper”) are marked by \nsquares. The region of NI-EVR effective potentials is denoted by a gray frame.  \n (a) A PSVR example. (b) A NI-EVR example. No conve rgence due to the ‘Fermi statistics’ \nproblem. (c) A NI-EVR example. Janak’s algorithm. ( d) A NI-EVR example. The proposed \nalgorithm.  \n \n00.5 11.5 2\n25.4 25.6 25.8 26.0 26.2 26.4 26.6 \nZgi\n \nFIG. 2. The occupations of the 3d↓(rhombi) and 4s↓ (squares) energy levels as a function of Z.  -0.4 -0.3 -0.2 -0.1 0\n25.4 25.6 25.8 26.0 26.2 26.4 26.6 \nZεi (hartree) \n \nFIG. 3. The energy levels 3dε↓ (rhombi) and 4sε↓ (squares) as a function of Z. \n \n-20 -15 -10 -5 0\n0 40 80 120 160 200 240 \nZε2s-ε2p (hartree) \n \nFIG. 4. The difference between the occupied 2s and the vacant 2p energy levels in Be ion series \ngrows proportionally to the atomic number Z. \n \n -0.160 -0.155 -0.150 -0.145 -0.140 -0.135 -0.130 \n0.000 0.002 0.004 0.006 0.008 0.010 \nkBT (hartree) εi (hartree) \n \nFIG. 5. The dependence of the chemical potential (c ircles), the 3d↓(rhombi) and the 4s↓ (squares) \nenergy levels of Fe on the temperature. The horizon tal marks are 1Bk T  above and below the \nchemical potential. \n \n00.1 0.2 0.3 0.4 0.5 0.6 0.7 \n0.000 0.002 0.004 0.006 0.008 0.010 \nkBT (hartree) fi\n \nFIG. 6. The normalized occupation numbers of 3d↓(rhombi) and the 4s↓ (squares) energy levels \nof Fe as a function of temperature.  \n                                                 \n[1]  R G Parr and W Yang, Density-Functional Theory of Atoms and Molecules  (Oxford University \nPress, New York, 1989). \n[2]  R M Dreizler and E K U Gross, Density Functional Theory  (Springer, Berlin, 1990). \n[3]  N Argaman and G Makov, Am. J. Phys. 68 , 69 (2000).  \n[4]  W Kohn and L J Sham, Phys. Rev. 140 , A1133 (1965).  \n[5]  H Englisch and R Englisch,  Phys. Status Solid i B 123 , 711 (1984);  ibid.,  124 , 373 (1984). \n[6]  R van Leeuwen, Ph.D. thesis, Vrije Universitei t, Amsterdam, The Netherlands, 1994.   \n[7]  V L Moruzzi, J F Janak, and A R Williams, Calculated Electronic Properties of Metals  (Pergamon, \n1978). \n[8] J F Janak, Phys. Rev. B 18 , 7165 (1978). \n[9]  J Harris, Phys. Rev. A 29 , 1648 (1984).  \n[10]  C A Ullrich and W Kohn, Phys. Rev. Lett. 89 , 156401 (2002).  \n[11]  C A Ullrich and W Kohn,  Phys. Rev. Lett. 87 , 093001 (2001).  \n[12]  F W Averill and G S Painter,  Phys. Rev. B 46 , 2498 (1992).  \n[13]  S Kotochigova, Z H Levine, E L Shirley, M D S tiles, and C W Clark, Phys. Rev. A 55 , 191 (1997).  \n[14]  M Levy, Phys. Rev. A 26 , 1200 (1982). \n[15]  E H Lieb, Int. J. Quantum Chem. 24 , 243 (1983).  \n[16]  R van Leeuwen,  Adv. Quantum Chem. 43 , 24 (2003).   \n[17]  J T Chayes, L Chayes, and M B Ruskai,  J. Sta t. Phys. 38 , 497 (1985).   \n[18]  N Argaman and G Makov,  Phys. Rev. B 66 , 052413 (2002).  \n[19]  M M Valiev and G W Fernando, Phys. Rev. B 52 , 10697 (1995).   \n[20]  R C Morrison, J. Chem. Phys. 117 , 10506 (2002).                                                                                                                                                                            \n[21]  S H Vosko, L Wilk, and M Nusair,  Can. J. Phy s. 58 , 1200 (1980). \n[22] In this notation, the upper number near each o rbital denotes the amount of electrons with spin up , \nthe lower number – the amount of electrons with spi n down. \n[23]  L D Landau and E M Lifshitz, Quantum Mechanics (Non-Relativistic Theory)  (Pergamon, 1991), \npp. 138-142 . \n[24]  J P Perdew, K Burke, and M Ernzerhof, Phys. R ev. Lett. 77 , 3865 (1996). \n[25]  N David Mermin, Phys. Rev. 137 , A1441 (1965). \n[26]  W Kohn and P Vashishta, “General Density Func tional Theory” in The Inhomogeneous Electron \nGas  edited by S Lundqvist and N H March (Plenum, New Y ork, 1989). \n "    },    {        "title": "1012.0921v1.Photonic_density_of_states_maps_for_design_of_photonic_crystal_devices.pdf",        "content": "Author's personal cop y\nMicroelectronics Journa l 39 (2008) 685–689\nPhotonic density of stat es maps for design of p hotonic crystal devices\nI.A. Sukhoivanova,�, I.V. Guryevb, J.A. Andrade Lucioa, E. Alvarado Mendeza,\nM. Trejo-Durana, M. Torres-Cisnerosa\naUniversidaddeGuanaju ato,Prol.Tampico912, Salamanca,GTO36730 ,Mexico\nbNationalUniversityofR adioElectronics,Lenin Avenue14,Kharkov611 66,Ukraine\nAvailable online 28 Au gust 2007\nAbstract\nIn this work, it has been investigated whether ph otonic density of states maps can be applied to the design of photonic c rystal-based\ndevices. For this reason , comparison between photonic density of sta tes maps and transmitt ance maps was carried out. Results of\ncomparison show full c orrespondence between these characteristics. Ph otonic density of states maps appear to be pre ferable for the\ndesign of photonic crys tal devices, than photon ic band gap maps prese nted earlier and than tr ansmittance maps show n in the paper.\nr2007 Elsevier Ltd. All rights reserved.\nKeywords:Photonic crystal; Photo nic density of states ma p; Photonic crystal devi ce; Transmittance map\n1. Introduction\nAt present, photonic c rystal (PhC) devices h ave wide\nrange of applications in telecommunications and lasers as,\nfor instance, PhC fiber s[1], laser resonators [2], passive\nelements, such as high- efficiency waveguides [3], couplers\n[4], and multiplexers [5]. One of the crucial mom ents in the\ndesign stage of such devices is quick and accurate\nparameters selection of PhC elements. One of t he possible\nways to select param eters is the investigat ion of full\nphotonic band gaps (P BG) maps. PBG maps represent\nthe so-called reduced band structures that give the\ninformation about full PBG at different PhC p arameters.\nSuch maps were propo sed by Joannopoulos e t al.[6]and\ntheir application for th e design of the demult iplexer was\nshown in [7].However,suchPBGm apsprovide incomplete\ninformation about trans mittance characteristics of the PhC\nbecause they contain i nformation only about full PBGs\nand do not indicates the behavior of the PhC ou tside PBG.\nWe propose to use pho tonic density of states ( PhDOS)\nmaps instead of full P BG maps analysis. Su ch PhDOS\nmaps consist of PhDOS taken at different PhC p arametersand contain full inform ation about transmittan ce of finite-\nsize PhC. Peculiar pr operties of PhDOS o f periodic\nstructures were used to modify spontaneous em ission of\nactive elements embedd ed to the periodic struct ure, and we\npropose to apply them for the design of PhC d evices.\nIn our work, we have c omputed the PhDOS m ap by the\nplane wave expansion m ethod. Such maps are r epresented\nby the PhDOS compu ted for different radiu s values of\nrods. The transmittanc e spectra of the finite structure at\ndifferent rod radii fo r the same structure were also\ncomputed. Comparison with PhDOS shows full adequate-\nness of transmittance to PhDOS.\n2. Photonic density of s tates computation for 2D PhC\nPhDOS is computed ba sed on the band structu re of PhC\n[8]. Photonic band struc ture can be obtained by the\nsolution of eigenvalue problem for stationary Helmholtz\nequation given in follow ing form:\n�q\nqx1\n�ð~rjjÞq\nqxþq\nqy1\n�ð~rjjÞq\nqy� �\nHzð~rjjÞ ¼o2\nc2Hzð~rjjÞ, (1)\nwherexandyare coordinates in plan e of 2D PhC, ~rjjis the\nradius-vector in plane o f 2D PhC,�ð ~rjjÞis the 2D dielectric\nfunction of the PhC,ois the light frequency ,cis the\nvelocity oflightinvacuu m,andHzð~rjjÞis theeigenfunction.ARTICLE IN PRESS\nwww.elsevier.com/locat e/mejo\n0026-2692/$-see front m atterr2007 Elsevier Ltd. All rights reserved.\ndoi:10.1016/j.mejo.2007.07. 091�Corresponding author. Tel.: +524646880911x2 21;\nfax: +524646472400.\nE-mailaddresses: i.sukhoivanov@ieee.org ,\nsukhoivanov@salamanc a.ugto.mx (I.A. Sukhoi vanov).Author's personal cop yThe most common m ethod for the solution of such\nproblems is the plane w aves expansion (PWE) method. It\nconsists of solution of eigenvalue problem for the\nHelmholtz equation, c arried out for an infin ite strictly\nperiodic structure by means of representatio n of wave\nfunctions according to Bloch–Floquet theorem , as the\nproduct of a plane wa ve and a periodic fun ction with\nlattice period:\n~Hð~rjjÞ ¼ ~H~kjjnð~rjjÞ ¼ ~H0\n~kjjnð~rjjÞei�~kjj�~rjj, (2 )\nwhere ~H0\n~kjjnð~rjjÞis a periodic function w ith lattice period, ~kjj\nis the wave vector,nis the number of eigen states.\nDue to the periodicity, it is convenient to repr esent the\nwave function as well a s inverted dielectric fun ction as a\nFourier series in follow ing form:\n~H~kjjnð~rjjÞ ¼X\n~GjjH0\n~kjjnð~GjjÞexpðið ~kjjþ~GjjÞ �rjjÞ,\n1\n�ð~rjjÞ¼X\nGjjwð~GjjÞexpði ~Gjj�~rjjÞ,ð3Þ\nwhere ~Gjjis the reciprocal lattice vector in plane of 2D\nPhC,wð ~GjjÞare Fourier expansion coefficients of inverted\ndielectric function of Ph C.\nAs a consequence of sim plification consisting tru ncation\nof infinite Fourier serie s, system of linear equa tions which\nis solved numerically is obtained. In case of 2D PhC, the\nsystem takes following form[8]:\nX\n~Gjjwð~Gjj�~G0\njjÞð~kjjþ~GjjÞð~kjjþ~G0\njjÞH0\nz;~kjjnð~G0\njjÞ\n¼oðHÞ2\n~kjjn\nc2H0\nz;~kjjnð~GjjÞ, ð4Þ\nwhereH0\nz;~kjjnð~G0\njjÞare wave functions in representation of\nwave vectors. Searchin g for eigenvalues of m atrix com-\nposed of coefficients in wave functions gives us the set of\nPhC eigenfrequencies for some specific valu e of wave\nvector.\nIn order to obtain t he PhDOS, it is nec essary to\nobtain the set of eigen frequencies for each w ave vector\nvalue within the first B rilluoin zone (see Fig. 1). In case\nof 2D PhC, the lowe st bands of the band structure\nobtained in this way will be represented by a number\nof surfaces for one zone each. The band structure\nrepresentedthus givesfu lldescription ofspectral properties\nof PhC and can be used as a basis for t he PhDOS\ncomputation.\nHaving the band struct ure corresponding to th e specific\ntype of PhC, the PhDO SN(o) is defined by ‘‘countin g’’ all\nallowedstateswithinag ivenfrequencyo,i.e.bythesumof\nall bands and the integ ral over the first BZ of a Dirac-d\nfunction [9]\nNðoÞ ¼X\nnZ\nBZd2kdðo�o nð~kÞÞ. (5)InFig. 2, the band structure co mputed using the PWE\nmethod for the PhC re presented by a periodic system of\ndielectric rods with re fractive indexn¼3.5 and radius\nr¼0.2a(ais a distance between nearest rods’ centers—\npitch) separated by air as well as PhDOS of such PhC\ncomputed using (5) is depicted. Comparing th e obtained\ncharacteristics, it can b e mentioned that absen ce or low\nPhDOS corresponds to full photonic band ga p or low\nnumber of states in the frequency domain . On the\nother hand, peaks of P hDOS fall at the dom ains where\nplane zones exist or t he maximum number of states is\nconcentrated.\nFig. 2shows the PhDOS for one value ofr/aonly.\nHowever, the solution of device design tasks where it is\nnecessary to determin e the PhC parameters from the\ntransmission characteri stics requires the infor mation on\nthe PhDOS taken is so me range of parameter —PhDOS\nmap. In our work, we computed the PhDOS m ap forr/a\nparameterrange0.1–0.4 . TheexampleofsuchP hDOS map\nis shown in Fig. 3. In the obtained PhDO S diagram, light\nspots (PhDOS values i n range 500–1700) cor respond toARTICLE IN PRESS\nFig. 1. Band structure of PhC computed for a ll wave vectors inside 1 st\nBrilluoin zone.\nFig. 2. Band structure and PhDOS of PhC wit hn¼3.5 andr/a¼0.2.I.A.Sukhoivanovetal. /MicroelectronicsJourn al39(2008)685–689 686Author's personal cop y\nhigh PhDOS and dark spots (PhDOS values in range\n150–500) correspond to low PhDOS. Black spo ts (PhDOS\nvalues less than 150) co rrespond to the absence of PhDOS,\ni.e. full PBG. Itcan bes een inFig. 3that there exist regions\nwhere PhDOS values ar e extremely low. Howev er, they are\nnot equal to zero. This m eans that they will not b e depicted\nin full PBG maps, as that can cause a mist ake while\ndesigning the PhC-base d device using such ma ps.\nThus, the PWE method has been used for the co mputa-\ntion of PhDOS. The co mputation was carried o ut with the\nstep ofr/aequal to 0.002 within the limits of 0.1–0.4. A t\nthis, the computation time on the IBM PC with Intel\nPentium 2.7GHz proce ssor amounted 7h.\n3. Transmittance maps computation\nInordertoprovetheap plicabilityofPhDOSma pstothe\ndesign of PhC-based devices, we compared them with\ntransmittance maps of t he finite size PhC. In ou r work, we\nusedthefinitedifference stimedomain(FDTD) method for\nthe computation of t ransmittance maps. T he method\nconsists in digitization of Maxwell’s equations . At that,\ndifferential operators a re replaced by differenc es. For 2D\nPhC in case of TM pola rization[6]the system of Maxwell’s\nequations takes followin g form:\nHnþ1=2\nxði;jÞ¼Hn�1=2\nxði;jÞ�Dt\nmDyðEn\nzði;jÞ�En\nzði;j�1Þ Þ,\nHnþ1=2\nyði;jÞ¼Hn�1=2\nyði;jÞþDt\nmDxðEn\nzði;jÞ�En\nzði�1;jÞ Þ,\nEnþ1\nzði;jÞ¼En\nzði;jÞþDt\n�DxðHnþ1=2\nyðiþ1;jÞ�Hnþ1=2\nyði;jÞÞ\n�Dt\n�DyðHnþ1=2\nxði;jþ1Þ�Hnþ1=2\nxði;jÞÞ,ð6Þ\nwhereHn\nxði;jÞ,Hn\nyði;jÞandEn\nzði;jÞare field components at nodes\nof computation mesh,Dtis time step,Dx,Dydetermine the\nsize of mesh element ,eandmare permittivity and\npermeability of the me dium. The system was solved by\nassigning initial and bo undary conditions. The harmonicsignal at one of the bo undaries of computatio n area was\nused as initial conditio ns. The cross-section o f the signal\nwas a Gaussian funct ion with maximum va lue in the\nmiddle of computation area. Boundary conditi ons used in\nthe work were represen ted by the perfectly ma tched layer\n(PLM)[10]. The layer is characte rized by high value of\nwave impedance hence high-radiation absorpti on is emu-\nlated. For TM polariza tionEzfield component is split ted\ninto two partsE zxandEzyfor the purpose of intro duction\nof impedance compone nts for different propag ation direc-\ntion. Thus, we obtain th e system of four equatio ns in terms\nof finite differences:\nHnþ1=2\nxði;jÞ¼Hn�1=2\nxði;jÞþDt\nm�1\nDyðEn\nzði;jÞ�En\nzði;j�1Þ Þ �s�\nyHn�1=2\nxði;jÞ� �\n;\nHnþ1=2\nyði;jÞ¼Hn�1=2\nyði;jÞþDt\nm1\nDxðEn\nzði;jÞ�En\nzði�1;jÞ Þ �s�\nxHn�1=2\nyði;jÞ� �\n;\nEnþ1\nzxði;jÞ¼En\nzxði;jÞ þDt\n�1\nDxðHnþ1=2\nyðiþ1;jÞ�Hnþ1=2\nyði;jÞÞ �s xEn\nzxði;jÞ� �\n;\nEnþ1\nzyði;jÞ¼En\nzyði;jÞ þDt\n��1\nDyðHnþ1=2\nxði;jþ1Þ�Hnþ1=2\nxði;jÞÞ �s yEn\nzyði;jÞ� �\n;\n(7)\nwheres xands yarex- andy-components of electric field\nimpedance, respectively ,s�\nxands�\nyarex- andy-compo-\nnents of magnetic field impedance, respectively . Computa-\ntion area and PML reg ions are shown in Fig. 4. In this\nwork, the graded imp edance was used with quadratic\ndependence on the c oordinate to avoid n on-physical\nreflection from the PM L[11]. At this, the impeda nce\nequals to zero at the inn er PML boundary and achieves its\nmaximum value at the outer boundary.\nThe transmittance spect rum was obtained comp uting the\nfield distribution at ea ch of the wavelengths within the\nrequired range. The tra nsmittance map was o btained by\ncomputing the transmi ttance spectrum for al l values of\nPhC elements paramete r in range.ARTICLE IN PRESS\nFig. 3. PhDOS of the P hC made of rods withn¼3.5 separated by air.\nFig. 4. Computation a rea for the FDTD met hod with PML bounda ry\nconditions.I.A.Sukhoivanovetal. /MicroelectronicsJourn al39(2008)685–689 687Author's personal cop yThe transmittance map computation was carrie d out for\ntwo differentr/asteps. In first case, its v alue was equal to\n0.01 and in second case it was 0.002. Moreover , transmit-\ntance spectra in first cas e were computed with 1 .5 times less\naccuracy than in secon d case. Computation t ime on the\nIBM PC with Intel Pen tium 2.7GHz processor amounted\n7h in first case and 12 0h in second case. The significant\ngrowth of computation time compared to PhD OS maps\ncomputation was cause d by the fact that com putation of\ntransmittance maps by FDTD method require s computa-\ntion of field distribut ion at each waveleng th at each\nmoment starting from the introduction of ra diation to\nthe structure. Hence, in creasing the computatio n accuracy\nyields significant growth of computation time.\n4. Results and discussio n\nResults of transmittance map computation for tw o cases\ndescribed above are dep icted inFig. 5. The computations\nwere carried out for the PhC represented by diel ectric rods\nwith refractive indexn¼3.5 separated by air. Th e range of\nvariation ofr/aparameter is 0.1–0.4. A t the transmittancemap light spots corres pond to high transmit tance, dark\nspots correspond to lo w transmittance and b lack spots\ncorrespond to zero tran smittance of the radiati on through\nthe finite-size PhC. The comparison between tra nsmittance\nmap (Fig. 5) and PhDOS map ( Fig. 3) demonstrates their\nfull correspondence, i.e. high PhDOS (from 500 to 1700 in\nFig. 3) corresponds to high tr ansmittance (from 0.5 t o 1 in\nFig. 5) and full photonic band gap corresponds to abse nce\nof transmittance (0 in Fig. 5). From the physical po int of\nview,suchcorresponden cecanbeexplainedbyt hefactthat\nhigh PhDOS correspon ds to high number of w ave vectors\nallowed for the propag ation at the specific wa velength in\nthe structure, i.e. high r adiation flux through th e PhC.\nIn the work [7], it was shown the app lication of PBG\nmaps to the design o f the 2D PhC-based wavelength\ndivision demultiplexer. The design method c onsists in\nobtaining geometric pa rameters of PhC eleme nts forming\nthe input and output w aveguide channels as w ell as PhCs\nforming band pass fil ters in the output cha nnels. The\nscheme of the paramete rs selection by the analy sis of PBG\nmaps is depicted in Fig. 6a. The selection of param eters of\nthe PhC is based on th e information about th e placementARTICLE IN PRESS\nFig. 5. Transmittance m aps of 2D PhC with (a) low resolution and (b) h igh resolution.\n0.1 0.15 0.2 0.25 0.3 0.35 0.40.20.250.30.350.40.450.5\nr/a0.10 0.15 0.20 0.25 0.30 0.35 0.40\nr/aωa/2πcω2\nω1\n0.200.250.300.350.400.450.50ω a/2πcr1r2r3\nr1r2r3\nω2\nω1\nFig. 6. PBG map (a) an d PhDOS map (b) with an example of PhC par ameters selection.I.A.Sukhoivanovetal. /MicroelectronicsJourn al39(2008)685–689 688Author's personal cop yof PBG edges. It is a ssumed that the trans mittance is\nuniform outside the PB G. At this, in Fig. 6a,r1,r2andr3\nare radii of PhC elemen ts forming the first pass band filter,\nmain waveguide chann el and the second pass band filter,\nrespectively. In Fig. 6bit is depicted the PhDO S map with\nmarks pointingoperatin gfrequencies(horizonta l lines) and\nradii of PhC elements (vertical ones) both ob tained by\nanalysis of PBG map, s hown on the Fig. 6a. It can be seen\nfrom the PhDOS map that the cross-point of liner1and\nlineo 1lies not in maximum o f the PhDOS and, hen ce,\nit does not correspon d to maximum of tra nsmittance\nspectrum. Moreover, th e cross-point of linesr 3ando 2falls\nat low PhDOS value. C onsequently, one of the operating\nfrequencies (o 1) will not correspond to maximum of\ntransmittance and anot her one (o 2) will fall at minimum\nof transmittance of pas s band filter instead of maximum\nexpected from PBG ma ps.\nThus, PhDOS maps pr ovide more detailed inf ormation\nabout the radiation b ehavior in periodic st ructures in\ncontrast to PBG maps b ecause of the capability to describe\ntransmittance of the PhC outside the PBG making\nthem more preferable f or the design of the dev ices on the\nbasis of PhCs.\n5. Conclusion\nIn this work, compari son between PhDOS m aps and\ntransmittance maps fr om the point of view of their\napplicability for designi ng the PhC-based devic es has been\ncarriedout.The present edresults showfullcorr espondence\nof the shape of PhDOS and transmittance spect rum. Local\nextremums of the tran smittance spectrum cor respond to\nlocal extremums of Ph DOS in frequency as well as in\nmagnitude. At this, the computation time for th e PhDOS is\n17 times less than the ti me required for the com putation of\nthe transmittance map.\nThus, utilization of Ph DOS maps appears to be more\neffective for the design o f PhC-based devices tha n the PBGmaps because PhDOS m aps provide the informa tion about\nthe transmittance of the structure outside the PB G. On the\nother hand, they are m ore preferable than tra nsmittance\nmaps because they can be obtained much mor e quickly\nbut provide the same in formation about the tra nsmission\ncharacteristics of PhC.\nReferences\n[1] T.A. Birks, J.C. K night, P. St. J. Russel l, Endlessly single-mod e\nphotonic crystal fiber, O pt. Lett. 22 (13) (1997) 961–963.\n[2] S.G.Romanov,D.N .Chigrin,V.G.Solovyev ,T. Maka,N.Gaponik,\nA. Eychmu ¨ller, A.L. Rogach, C .M. Sotomayor Torre s, Light\nemission in a directiona l photonic bandgap, Ph ys. Stat. Sol. (a) 197\n(3) (2003) 662–672.\n[3] S.Y. Lin, E. Chow, S.G. Johnson, J.D. Joa nnopoulos, Demonstra-\ntion of highly efficient w aveguiding in a photon ic crystal slab at the\n1.5-mm wavelength,, O pt. Lett. 25 (17) (2000) 1297–1299.\n[4] P. Bienstman, S. Assefa, S.G. Johnson , J.D. Joannopoulos,\nG.S. Petrich, L.A. Kol odziejski, Taper structu res for coupling into\nphotonic crystal slab w aveguides, J. Opt. Soc. Am. B 20 (9) (2003)\n1817–1821.\n[5] B.-S. Song, T. Asa no, Y. Akahane, Y. T anaka, S. Noda, Mult i-\nchannel add/drop filter based on in-plane heter o photonic crystals,\nJ. Lightwave Technol. 2 3 (3) (2005) 1449–1455.\n[6] J.D. Joannopoulos, R.D. Meade, J.N. Win n, Molding the Flow o f\nLight, Princeton Univer sity Press, Princeton, N J, 1995.\n[7] I.A. Sukhoivanov, I.V. Guryev, O.V. S hulika, A.V. Kublyk,\nO.V.Mashoshina,E.Al varado-Me ´ndez,J.A.Andrade-Luc io,Design\nof the photonic crystal demultiplexers for ultra -short optical pulses\nusing the gap-maps an alysis, J. Optoelectron. Adv. Mater. 8 (4)\n(2006) 1626–1630.\n[8] K. Sakoda, Optical Properties of Photonic Crystals, Springer, 2001 .\n[9] K. Busch, M. Fr ank, A. Garcia-Marti n, D. Hermann, S.F .\nMingaleev, M. Schilling er, L. Tkeshelashvili, A solid state theoretical\napproach to the optica l properties of photonic crystals, Phys. Stat.\nSol. (a) 197 (3) (2003) 6 37–647.\n[10] J.-P. Berenger, Th ree-dimensional perfect ly matched layer for t he\nabsorption of electrom agnetic waves, J. Comp ut. Phys. 127 (1996)\n363–379.\n[11] D.M. Hockanson, Perfectly matched lay ers used as absorbing\nboundaries in a three-d imensional FDTD Cod e, Technical Report,\nUmr Emc Laboratory, pp. 1–10.ARTICLE IN PRESS\nI.A.Sukhoivanovetal. /MicroelectronicsJourn al39(2008)685–689 689"    },    {        "title": "1012.2964v1.Theorems_on_ground_state_phase_transitions_in_Kohn_Sham_models_given_by_the_Coulomb_density_functional.pdf",        "content": "arXiv:1012.2964v1  [cond-mat.stat-mech]  14 Dec 2010Theorems on ground-state phase transitions in\nKohn-Sham models given by the Coulomb density\nfunctional\nKoichi Kusakabe, Isao Maruyama\nGraduate School of Engineering Science, Osaka University, 1-3 Ma chikaneyama-cho,\nToyonaka, Osaka 560-8531, Japan\nE-mail:kabe@mp.es.osaka-u.ac.jp\nAbstract. Some theorems on derivatives of the Coulomb density functional wit h\nrespect to the coupling constant λare given. Consider an electron density nGS(r)\ngiven by a ground state. A model Fermion system with the reduced c oupling constant,\nλ <1, is defined to reproduce nGS(r) and the ground state energy. Fixing the charge\ndensity, possible phase transitions as level crossings detected in a value of the reduced\ndensity functional happen only at discrete points along the λaxis. If the density is\nv-representable also for λ <1, accumulation of phase transition points is forbidden\nwhenλ→1. Relevance of the theorems for the multi-reference density fun ctional\ntheory is discussed.\nPACS numbers: 02.30.Sa, 31.15.ec, 71.15.Mb\nSubmitted to: J. Phys. A: Math. Gen.Theorems on ground-state phase transitions in Kohn-Sham mo dels 2\n1. Introduction\nThe density functional theory (DFT)[1, 2] is one of successful fr ameworks in the theory\nof electron systems. For a present standard scheme of this theo ry, the universal energy\ndensity functional F[n] defined by the constrained minimization method[3, 4, 5, 6] is\nused as a key quantity. Lieb has analyzed F[n], which is often cited as the Levy-Lieb\nfunctional.[7] This functional connects a Fermion density n(r) to a variational energy\nof an electron wave function Ψ showing the same density. Correspo ndence from n(r) to\nthis minimizing state vector |Ψ/an}b∇acket∇i}htis shown to exist,[7] which is a basic principle in the\ndensity functional theory.\nA continuous modification of the Coulomb operator appearing in the d efinition\nofF[n] is one of relevant techniques to analyze this functional.[8, 9, 5, 6] T he idea\nis similar to the ordinal perturbation theory of electron systems.[1 0, 11] To have an\nexact expression of the so-called exchange-correlation function al, people fixed thecharge\ndensity and considered continuous reduction of the coupling const ant by multiplying a\nfactorλin the energy density functional.[8, 9]\nWe re-analyze this modified functional, which is called Fλ[n] in this article, to\nconsider phase transition points appearing in a generalized Kohn-Sh am scheme.[12] We\nhave two main purposes for this analysis. The first purpose is to stu dy a parameter\ndifferentiability of this λ-modifiedF[n]. Fixing a density at an N-representable density\ninIN,[7]theparameterderivativeof Fλ[n]isshowntobewell-defined. Namely, existence\nof state vectors giving Dini’s derivatives is shown. As the second pur pose, a condition\non phase transitions around the v-representable density is derived from the theorems\nonFλ[n]. Existence of level-crossing points, i.e.phase transition points, is known. We\naddress existence of an ε-vicinity around the true solution of the many-electron system.\nIn this finite region, no level crossing point is found provided that th e density of the\nCoulomb system is also v-representable with a model given by Fλ[n]. The definition of\nthe model is given in section 6. Relevance of this ε-vicinity for a general Kohn-Sham\nscheme is discussed in section 8.\n2. The setup of the problem\nWe consider a static state of a material. The number of electrons Nis fixed to be a\nfinite integer. We apply the Born-Oppenheimer approximation (BOA) , in which motion\nof nuclei is separated from the motion of electrons. To find a stable state, we consider\na classical state of nuclei and fix the coordinates of nuclei, whose n umber isM. Motion\nof the electron system is determined for this classical configuratio n of nuclei, which give\na static electric potential for electrons.\nLet|0/an}b∇acket∇i}htbe the electron vacuum. The electronic state in an external scalar potential\nis symbolically written by a state vector |Ψ/an}b∇acket∇i}ht. This electron state is described by Fermion\nfield operators, ψ†\nσ(r) andψσ(r), satisfying the canonical anti-commutation relations.\n/bracketleftbig\nψ†\nσ(r),ψσ′(r′)/bracketrightbig\n+=δ(r−r′)δσ,σ′.Theorems on ground-state phase transitions in Kohn-Sham mo dels 3\nConsidering a position basis |{ri,σi}/an}b∇acket∇i}ht=/producttextN\ni=1ψ†\nσi(ri)|0/an}b∇acket∇i}ht, we have a wave function as an\ninner product between this position vector and |Ψ/an}b∇acket∇i}ht.\nΨ({ri,σi}) =/an}b∇acketle{t{ri,σi}|Ψ/an}b∇acket∇i}ht. (1)\nThe number operator ˆ n(r) probing existence of an electron at a point ris defined\nas\nˆn(r)≡/summationdisplay\nσψ†\nσ(r)ψσ(r). (2)\nThe kinetic energy operator ˆTfor electrons is assumed to be,\nˆT=−/planckover2pi12\n2m/integraldisplay\nd3r/summationdisplay\nσlim\nr′→rψ†\nσ(r′)∆rψσ(r), (3)\nwith the electron mass m.\nOwing to BOA, we have an external scalar potential vext(r) given by the charge of\nnuclei. Using position vectors RI(I= 1,···,M) of fixed nuclei, the potential vext(r) is\ngiven as,\nvext(r) =−M/summationdisplay\nI=1ZIe2\n|RI−r|. (4)\nHere, the charge of the I-th nucleus is ZIe. The potential term for the Hamiltonian of\nelectrons is given by the next operator.\nˆVext=/integraldisplay\nd3rvext(r)ˆn(r). (5)\nIn the following discussion, we omit a constant Coulomb energy coming from the ion-ion\ninteraction,\nVii=/summationdisplay\n/angbracketleftI,J/angbracketrightZIZJe2\n|RI−RJ|, (6)\nalthough this term is important for the charge neutrality condition. Hereafter, the\nsymbol/an}b∇acketle{tI,J/an}b∇acket∇i}htdenotes a pair of two different integers ranging from 1 to a finite inte ger\n(here it isM), and the summation with respect to these pairs are written as/summationtext\n/angbracketleftI,J/angbracketright.\nIf we have more than two electrons, the electron-electron intera ction always takes\nplace. In a static state, the inter-electron interaction is describe d by the next operator.\nˆVee=1\n2/integraldisplay\nd3rd3r′e2\n|r−r′|/summationdisplay\nσ,σ′ψ†\nσ(r)ψ†\nσ′(r′)ψσ′(r′)ψσ(r)\n=1\n2/integraldisplay\nd3rd3r′e2\n|r−r′|: ˆn(r)ˆn(r′) :. (7)\nThe symbol : O: for an operator Odenotes the normal ordering of the field operators.\nIn :O:, the order of ψandψ†is reorganized so that the creation operators come to the\nleft of the annihilation operators by interchanging operators. If in terchange of two field\noperators is made mtimes, the sign of ( −1)mis multiplied to the reordered operator.Theorems on ground-state phase transitions in Kohn-Sham mo dels 4\nWe consider an isolated electron system having N≥2 in an equilibrium.\nCompetition between the Coulomb interaction ˆVeeand other single-particle parts,\nˆT+ˆVext, causes various phase transitions in the electron systems. We sea rch for the\nminimum energy allowed for the electron system. Thus any state vec tor, which has an\nanti-symmetric property for the spin-1/2 Fermion system, has to be normalizable, and\nhas to have a finite kinetic energy. The class of wavefunctions for t he allowed state\nvectors isH1(R3N).\nFor any state vector |Ψ/an}b∇acket∇i}htof anNelectron state with N≥2, we have |Ψ′/an}b∇acket∇i}ht ≡\nψσ′(r′)ψσ(r)|Ψ/an}b∇acket∇i}ht, and the quantum state |Ψ′/an}b∇acket∇i}hthas a positive semidefinite norm, /an}b∇acketle{tΨ|:\nˆn(r)ˆn(r′) :|Ψ/an}b∇acket∇i}ht=/an}b∇acketle{tΨ′|Ψ′/an}b∇acket∇i}ht ≥0. If we assume that /an}b∇acketle{tΨ|ˆVee|Ψ/an}b∇acket∇i}ht= 0, positivity of the\nCoulomb kernel requires that ∀rand∀r′,/an}b∇acketle{tΨ|: ˆn(r)ˆn(r′) :|Ψ/an}b∇acket∇i}ht= 0. But, then we have\nzero of the integral\n/integraldisplay\nd3rd3r′/an}b∇acketle{tΨ|: ˆn(r)ˆn(r′) :|Ψ/an}b∇acket∇i}ht=N(N−1)\n2= 0,\nwhich yields N= 0 orN= 1.\nThe expectation value of ˆVeeby Ψ∈H1(R3N) is known to be finite.[7] Actually,\n/an}b∇acketle{tΨ|ˆVee|Ψ/an}b∇acket∇i}ht=/summationdisplay\n/angbracketlefti,j/angbracketright/integraldisplayN/productdisplay\nl=1drle2\n|ri−rj|Ψ∗({rk,σk})Ψ({rk′,σk′})\n=/summationdisplay\n/angbracketlefti,j/angbracketright/integraldisplayN/productdisplay\nl=1drle2θ(|rij|)+e2[1−θ(|rij|)]\n|rij||Ψ({rk,σk})|2\n≤/summationdisplay\n/angbracketlefti,j/angbracketright/braceleftBigg/integraldisplayN/productdisplay\nl=1drle2θ(|rij|)\n|rij|/integraldisplayN/productdisplay\nl=1drl|Ψ({rk,σk})|2\n+/integraldisplayN/productdisplay\nl=1drle2[1−θ(|rij|)]|Ψ({rk,σk})|2/bracerightBigg\n<∞. (8)\nHere, we wrote ri−rj=rijandθ(x) = 1 if |x| ≤1,θ(x) = 0 if |x|>1. Thus, for\nN≥2, we have,\n0</an}b∇acketle{tΨ|ˆVee|Ψ/an}b∇acket∇i}ht<∞. (9)\n3. The charge density being the primary order parameter\nThe electron system is characterized by the electron charge dens itynΨ(r)≡ /an}b∇acketle{tΨ|ˆn(r)|Ψ/an}b∇acket∇i}ht.\nThis physical quantity is observable by X-ray diffraction measureme nts. Since the wave\nfunction Ψ( {ri,σi}) is given as a function in the space L2of integrable functions, and\nsince we assume that the kinetic energy is finite for Ψ, nΨ(r)1/2and∇nΨ(r)1/2is inL2.\nThen,nΨ(r) is known to be also in L3.[7] The space L3/2+L∞is a dual of the L1+L3\nspace. Since the Coulomb potential is in L3/2+L∞,vext(r)nΨ(r) is integrable.\nThe Hamiltonian of this electron system is given as\nˆH=ˆT+ˆVee+ˆVext. (10)Theorems on ground-state phase transitions in Kohn-Sham mo dels 5\nThe total energy E[Ψ,vext] of this electron system is given by,\nE[Ψ,vext]≡ /an}b∇acketle{tΨ|ˆH|Ψ/an}b∇acket∇i}ht=/an}b∇acketle{tΨ|ˆT+ˆVee|Ψ/an}b∇acket∇i}ht+/integraldisplay\nd3rvext(r)nΨ(r). (11)\nThe lowest steady stateis given by minimizing this energy inthe space o f wavefunctions.\nE0[vext] = min\nΨE[Ψ,vext] = min\nΨ/bracketleftbigg\n/an}b∇acketle{tΨ|ˆT+ˆVee|Ψ/an}b∇acket∇i}ht+/integraldisplay\nd3rvext(r)nΨ(r)/bracketrightbigg\n.(12)\nA definition of the order parameter is given by a derivative of the tot al energy with\nrespect to the external field. For the electron system, this exte rnal degrees of freedom\nis given by vext(r). Let’s consider a stable electronic state |Ψ/an}b∇acket∇i}htwithout degeneracy. This\nstate has an energy E0[vext] =/an}b∇acketle{tΨ|ˆH|Ψ/an}b∇acket∇i}ht. If an infinitesimal variation of vext(r) is given\ninL3/2+L∞, the wavefunction minimizing E[Ψ,vext+δvext] exists. This minimizing\nwavefunction may be given as\n|Ψ+δΨ/an}b∇acket∇i}ht=eiθ(|Ψ/an}b∇acket∇i}ht+|δΨ/an}b∇acket∇i}ht).\nAcomplex phase factor eiθrepresents a gaugedegree offreedom. Wealso use anotation,\nδˆH=/integraldisplay\nd3rδvext(r)ˆn(r). (13)\nWhen||δvext||=δ→0, the norm of |δΨ/an}b∇acket∇i}htgoes to zero, and the state |δΨ/an}b∇acket∇i}htis orthogonal\nto|Ψ/an}b∇acket∇i}ht. Then, we have a derivative of E0[vext].\nE0[vext+δvext]−E0[vext]\n=/an}b∇acketle{tΨ|ˆH+δˆH|Ψ+δΨ/an}b∇acket∇i}ht\n/an}b∇acketle{tΨ|Ψ+δΨ/an}b∇acket∇i}ht−/an}b∇acketle{tΨ+δΨ|ˆH|Ψ/an}b∇acket∇i}ht\n/an}b∇acketle{tΨ+δΨ|Ψ/an}b∇acket∇i}ht=/an}b∇acketle{tΨ+δΨ|δˆH|Ψ/an}b∇acket∇i}ht\n/an}b∇acketle{tΨ+δΨ|Ψ/an}b∇acket∇i}ht\n=/parenleftBig\n/an}b∇acketle{tΨ|δˆH|Ψ/an}b∇acket∇i}ht+/an}b∇acketle{tδΨ|δˆH|Ψ/an}b∇acket∇i}ht/parenrightBig/parenleftbig\n1−/an}b∇acketle{tδΨ|Ψ/an}b∇acket∇i}ht+O(δ2)/parenrightbig\n=/an}b∇acketle{tΨ|δˆH|Ψ/an}b∇acket∇i}ht+O(δ2)\n=/integraldisplay\nd3rδvext(r)nΨ(r) =/integraldisplay\nd3rδE0[vext]\nδvext(r)δvext(r). (14)\nThis derivation follows a proof of the force theorem by Parr.[13]\nIn a steady state, since the state has to be stable against any per turbation,\nthe functional derivative should exist. The coefficients of the func tional derivative\nδE0[vext]\nδvext(r)=nΨ(r) is the primary order parameter of the electron system. If we hav e a\nfirst-order phase transition by introducing δvext(r), a level crossing in the lowest energy\nstate|Ψ/an}b∇acket∇i}htoccurs. Then, a jump in directed derivatives could be found. In suc h a special\npoint, the derivative becomes ill-defined. However, this jump is dete cted as a jump in\nthe charge density. In this sense, the electron charge density sh ould be the primary\norder parameter.\nIn section 5, we will define another phase transition without a jump in n(r). This\ntransition can happen in electron systems owing to internal degree s of freedom, e.g.the\nelectron spin. The transition without change in n(r) can happen much frequently, when\nthe interaction strength or the form of the inter-particle interac tion are modified. Once\nthe interaction strength is shifted from that of the Coulomb intera ction, the resulting\nHamiltonian describes a model system. The model can be identical to the Kohn-ShamTheorems on ground-state phase transitions in Kohn-Sham mo dels 6\nmodel.[2] Thus, for the analysis of the Kohn-Sham scheme, it is import ant to analyze\nthis second-type phase transitions in an abstract model space.\n4. The density functional theory as a Landau theory\nTo define the universal energy density functional, a Sobolev space INof functions in\nR3was introduced in the density functional theory. For a function n(r) in the set IN,\nn(r)≥0 andn1/2(r) is square integrable. Its gradient ∇n1/2(r) is also square integrable.\nIn addition, n(r) satisfies,\n/integraldisplay\nd3rn(r) =N. (15)\nWe use the theorem 3.3 of Ref. [7] stating that, if n(r)∈ IN,F[n] given by the next\ndefinition exists.\nF[n] = min\nΨ→n/an}b∇acketle{tΨ|ˆT+ˆVee|Ψ/an}b∇acket∇i}ht. (16)\nThe symbol Ψ →nrepresents that a minimizing state |Ψ/an}b∇acket∇i}htis searched with a constraint\n/an}b∇acketle{tΨ|ˆn(r)|Ψ/an}b∇acket∇i}ht=n(r). Then, the existence of the minimizing Ψ is relevant.\nWe have the next constrained minimization process.\nE0[vext] = min\nnmin\nΨ→nE[Ψ,vext] = min\nn/bracketleftbigg\nF[n]+/integraldisplay\nd3rvext(r)n(r)/bracketrightbigg\n.(17)\nHere,F[n] behaves as a free energy of the electron system.\nIf we have a well-defined description of the free energy of the syst em as a functional\nof the primary order parameter, we have a complete expression of the Landau free\nenergy. The density functional theory actually gives an example. I n anN-representable\nform of the energy density functional, the Landau free energy of the system is given as\na summation of the so-called universal energy density functional a nd the energy of the\nexternal scalar potential. The latter,/integraldisplay\nd3rvext(r)nΨ(r), islinear inthe orderparameter.\n5. Theλmodified functional\nLetλbe a real parameter in [0 ,1]. We now consider a reduced energy density functional\nFλ[n] defined by,\nFλ[n] = min\nΨ′→n/an}b∇acketle{tΨ′|ˆT+λˆVee|Ψ′/an}b∇acket∇i}ht. (18)\nExistence of the minimizing Ψ′is given also by the theorem 3.3 of Ref. [7]. We would\nlike to address a next statement.\nTheorem 1 If0≤λ≤1, and ifn(r)∈ IN,Fλ[n]is a monotone increasing continuous\nfunction of λ.Fλ[n]is concave as a function of λ.Theorems on ground-state phase transitions in Kohn-Sham mo dels 7\nFirst, choose 0 ≤λ1<λ2≤1. Assume that Fλ1[n]≥Fλ2[n]. Choose a minimizing\nstate|Ψ0/an}b∇acket∇i}ht →nof the expectation value /an}b∇acketle{tΨ′|ˆT+λ2ˆVee|Ψ′/an}b∇acket∇i}ht. Then, we have,\nFλ1[n] = min\nΨ′→n/an}b∇acketle{tΨ′|ˆT+λ1ˆVee|Ψ′/an}b∇acket∇i}ht\n≥min\nΨ′→n/an}b∇acketle{tΨ′|ˆT+λ2ˆVee|Ψ′/an}b∇acket∇i}ht=/an}b∇acketle{tΨ0|ˆT+λ2ˆVee|Ψ0/an}b∇acket∇i}ht\n=/an}b∇acketle{tΨ0|ˆT+λ1ˆVee|Ψ0/an}b∇acket∇i}ht+(λ2−λ1)×/an}b∇acketle{tΨ0|ˆVee|Ψ0/an}b∇acket∇i}ht\n>/an}b∇acketle{tΨ0|ˆT+λ1ˆVee|Ψ0/an}b∇acket∇i}ht.\nThis inequality contradicts the definition of Fλ1[n]. Thus, Fλ[n] is a monotone\nincreasing function of λ. Next, assume that Fλ[n] is not continuous when λ=\nλ0≥0. This is equivalent to a statement that ∃ε >0,∀δ >0,∃λ >0,\ns.t.{|λ−λ0|<δ&|Fλ[n]−Fλ0[n]|>ε.}For simplicity, let us further assume that\n0< λ−λ0< δand thenFλ[n]−Fλ0[n]> ε. Ifn(r)∈ IN, for any minimizing state\nvector|Ψ0/an}b∇acket∇i}htofFλ0[n], the wave function of Ψ 0→n(r) is inH1, and\n0</an}b∇acketle{tΨ0|ˆVee|Ψ0/an}b∇acket∇i}ht<∃C0<∞. (19)\nIf we letδ=ε/(2C0), we have,\nmin\nΨ′→n/an}b∇acketle{tΨ′|ˆT+λˆVee|Ψ′/an}b∇acket∇i}ht\n>min\nΨ′→n/an}b∇acketle{tΨ′|ˆT+λ0ˆVee|Ψ′/an}b∇acket∇i}ht+ε\n=/an}b∇acketle{tΨ0|ˆT+λ0ˆVee|Ψ0/an}b∇acket∇i}ht+ε\n≥ /an}b∇acketle{tΨ0|ˆT+λˆVee|Ψ0/an}b∇acket∇i}ht+(λ0−λ)C0+ε\n>/an}b∇acketle{tΨ0|ˆT+λˆVee|Ψ0/an}b∇acket∇i}ht−δC0+ε\n=/an}b∇acketle{tΨ0|ˆT+λˆVee|Ψ0/an}b∇acket∇i}ht+ε\n2. (20)\nThis inequality contradicts the definition of the minimum.\nConsider 0 ≤λ1< λ2≤1, 0< ξ <1, andλξ≡ξλ1+(1−ξ)λ2. We call a state\nΨλξ→n(r), which minimizes /an}b∇acketle{tΨ′|ˆT+λξˆVee|Ψ′/an}b∇acket∇i}ht. Then, we have\nFξλ1+(1−ξ)λ2[n]\n=ξ/an}b∇acketle{tΨλξ|ˆT+λ1ˆVee|Ψλξ/an}b∇acket∇i}ht+(1−ξ)/an}b∇acketle{tΨλξ|ˆT+λ2ˆVee|Ψλξ/an}b∇acket∇i}ht\n≥ξFλ1[n]+(1−ξ)Fλ2[n]. (21)\nThis inequality ensures concavity of Fλ[n] as a function of λ./squaresolid\nFollowing knowledge on the monotone increasing continuous function s and the\nconvex (concave) functions, we immediately obtainresults onderiv atives andan integral\nof the derivative.[14] Let’s define Dini’s derivatives of Fλ[n],\nD−(λ) = lim\ny→0−sup\ny<h<0Fλ+h[n]−Fλ[n]\nh, (22)\nD+(λ) = lim\ny→0+inf\n0<h<yFλ+h[n]−Fλ[n]\nh. (23)\nWhenλ= 1, we formally define D+(λ) by considering Fλ[n] forλ >1. In practical\nsimulation, however, D+(λ) is not required at λ= 1.Theorems on ground-state phase transitions in Kohn-Sham mo dels 8\nCorollary 2 Forn(r)∈ IN, the monotone increasing concave function Fλ[n]ofλhas\na directed derivative at ∀λ∈(0,1]andD−(λ)≥D+(λ)\nWhenD−(λ) =D+(λ), wehavethederivative,d\ndλFλ[n], ofthemonotoneincreasing\nfunction of Fλ[n]. The differentiability of a monotone increasing function is given by H.\nLebesgue. Besides, the next statement holds.\nCorollary 3 If0≤λ≤1, forn(r)∈ IN,Fλ[n]is differentiable a.e. The derivative\nd\ndλFλ[n]is integrable in [0,1].\nNext, consider the case with N≥2. A set of state vectors |Ψ/an}b∇acket∇i}htreproducing n(r)\nis denoted as Qn. Fixλ∈(0,1]. We consider a subset of |Ψ/an}b∇acket∇i}ht ∈ Q nminimizing\n/an}b∇acketle{tΨ|ˆT+λˆVee|Ψ/an}b∇acket∇i}htand call it Pn(λ). For any |Ψ/an}b∇acket∇i}ht ∈ P n(λ), 0</an}b∇acketle{tΨ|ˆVee|Ψ/an}b∇acket∇i}ht<∞. Thus\nwe have a finite range including values of /an}b∇acketle{tΨ|ˆVee|Ψ/an}b∇acket∇i}htfor|Ψ/an}b∇acket∇i}ht ∈ P n(λ). The maximum\nof this range is given by a state vector in Pn(λ), and also the minimum is given by\nanother vector. They might be different with each other. Thus the y are denoted as\n|Ψ−\nλ/an}b∇acket∇i}ht ∈ Pn(λ) and|Ψ+\nλ/an}b∇acket∇i}ht ∈ Pn(λ). The very definitions of Pn(λ) and these vectors ensure\nthat\nFλ[n] =/an}b∇acketle{tΨ−\nλ|ˆT+λˆVee|Ψ−\nλ/an}b∇acket∇i}ht=/an}b∇acketle{tΨ+\nλ|ˆT+λˆVee|Ψ+\nλ/an}b∇acket∇i}ht, (24)\nand that\n/an}b∇acketle{tΨ+\nλ|ˆVee|Ψ+\nλ/an}b∇acket∇i}ht ≤ /an}b∇acketle{tΨ−\nλ|ˆVee|Ψ−\nλ/an}b∇acket∇i}ht. (25)\nNow we prove existence of a vector giving the directed derivative of Fλ[n].\nLemma 4 Forλ∈(0,1], we haveD+(λ) =/an}b∇acketle{tΨ+\nλ|ˆVee|Ψ+\nλ/an}b∇acket∇i}htandD−(λ) =/an}b∇acketle{tΨ−\nλ|ˆVee|Ψ−\nλ/an}b∇acket∇i}ht.\nFor 0<∀δ <λ, we have 0 <λ′=λ−δ <λand\nFλ′[n]≤ /an}b∇acketle{tΨ−\nλ|ˆT+λ′ˆVee|Ψ−\nλ/an}b∇acket∇i}ht=/an}b∇acketle{tΨ−\nλ|ˆT+λˆVee|Ψ−\nλ/an}b∇acket∇i}ht−δ/an}b∇acketle{tΨ−\nλ|ˆVee|Ψ−\nλ/an}b∇acket∇i}ht\n≤ /an}b∇acketle{tΨ+\nλ|ˆT+λˆVee|Ψ+\nλ/an}b∇acket∇i}ht−δ/an}b∇acketle{tΨ+\nλ|ˆVee|Ψ+\nλ/an}b∇acket∇i}ht=/an}b∇acketle{tΨ+\nλ|ˆT+λ′ˆVee|Ψ+\nλ/an}b∇acket∇i}ht.(26)\nThus,\nD−(λ) = lim\ny→0+sup\n0<h<yFλ[n]−Fλ−h[n]\nh\n≥lim\ny→0+sup\n0<h<yFλ[n]−/an}b∇acketle{tΨ−\nλ|ˆT+λˆVee|Ψ−\nλ/an}b∇acket∇i}ht+h/an}b∇acketle{tΨ−\nλ|ˆVee|Ψ−\nλ/an}b∇acket∇i}ht\nh\n=/an}b∇acketle{tΨ−\nλ|ˆVee|Ψ−\nλ/an}b∇acket∇i}ht. (27)\nSimilarly, we have,\nD+(λ)≤ /an}b∇acketle{tΨ+\nλ|ˆVee|Ψ+\nλ/an}b∇acket∇i}ht. (28)\nThese inequalities together with Eq. (25) give just another proof o f Corollary 2.Theorems on ground-state phase transitions in Kohn-Sham mo dels 9\nLet’s assume that D−(λ)>/an}b∇acketle{tΨ−\nλ|ˆVee|Ψ−\nλ/an}b∇acket∇i}ht. Then, we have\n0<lim\ny→0+sup\n0<h<yFλ[n]−Fλ−h[n]−h/an}b∇acketle{tΨ−\nλ|ˆVee|Ψ−\nλ/an}b∇acket∇i}ht\nh\n= lim\ny→0+sup\n0<h<y/an}b∇acketle{tΨ−\nλ|ˆT+(λ−h)ˆVee|Ψ−\nλ/an}b∇acket∇i}ht−Fλ−h[n]\nh. (29)\nIndependently, we have a next inequality by the definition of Fλ−h[n].\n/an}b∇acketle{tΨ−\nλ|ˆT+(λ−h)ˆVee|Ψ−\nλ/an}b∇acket∇i}ht ≥Fλ−h[n], (30)\nwhich yields, ∀h∈[0,y],\nf(h)≡ /an}b∇acketle{tΨ−\nλ|ˆT+(λ−h)ˆVee|Ψ−\nλ/an}b∇acket∇i}ht−Fλ−h[n]≥0. (31)\nBy Eq. (24), f(0) = 0.\nWe show that f(h)/his also a monotoneincreasing function. Actually, if we assume\nthat 0<∃h0<y, and thatf(h0)\nh0>f(y)\ny, we have a next inequality.\n/parenleftbigg\n1−h0\ny/parenrightbigg\nFλ[n]+h0\nyFλ−y[n]>Fλ−h0[n].\nThis contradicts to the concavity of Fλ[n] given by Eq. (21). Thus, Eq. (29) tells that\n0<lim\ny→0+/an}b∇acketle{tΨ−\nλ|ˆT+(λ−y)ˆVee|Ψ−\nλ/an}b∇acket∇i}ht−Fλ−y[n]\ny. (32)\nWe call a state vector, which is in Pn(λ−y) and maximizes the expectation value of\nˆVee,|Ψ−\nλ−y/an}b∇acket∇i}ht. Then the above expression yields,\nlim\ny→0+/an}b∇acketle{tΨ−\nλ−y|ˆVee|Ψ−\nλ−y/an}b∇acket∇i}ht\n>/an}b∇acketle{tΨ−\nλ|ˆVee|Ψ−\nλ/an}b∇acket∇i}ht+ lim\ny→0+1\ny/bracketleftBig\n/an}b∇acketle{tΨ−\nλ−y|ˆT+λˆVee|Ψ−\nλ−y/an}b∇acket∇i}ht−/an}b∇acketle{tΨ−\nλ|ˆT+λˆVee|Ψ−\nλ/an}b∇acket∇i}ht/bracketrightBig\n≥ /an}b∇acketle{tΨ−\nλ|ˆVee|Ψ−\nλ/an}b∇acket∇i}ht. (33)\nWe note that lim y→0+|Ψ−\nλ−y/an}b∇acket∇i}htexists. The continuity of Fλ[n] ensures that\nlimy→0+|Ψ−\nλ−y/an}b∇acket∇i}ht ∈ Pn(λ). The inequality, Eq. (33), contradicts to the definition of |Ψ−\nλ/an}b∇acket∇i}ht.\nThus we conclude that D−(λ) =/an}b∇acketle{tΨ−\nλ|ˆVee|Ψ−\nλ/an}b∇acket∇i}ht. Similarly, D+(λ) =/an}b∇acketle{tΨ+\nλ|ˆVee|Ψ+\nλ/an}b∇acket∇i}ht./squaresolid\nFor a fixed n(r), we now analyze the number of discontinuous points, where\nD(λ)≡D−(λ)−D+(λ)>0. Lemma 4 tells that, when D(λ)>0,Pn(λ) has multiple\nelements which are distinguished by difference in the expectation valu e ofˆVee, and\nalso in the expectation value of ˆT. At this discontinuous point, we have a change in\nthe minimizing state from |Ψ+\nλ/an}b∇acket∇i}htto|Ψ−\nλ/an}b∇acket∇i}htby reducing λ. Therefore we have the next\ndefinition.\nDefinition 5 A point where D(λ)>0is called a transition point in the model space.\nWe should note that a priori|Ψ+\nλ/an}b∇acket∇i}htand|Ψ−\nλ/an}b∇acket∇i}htare defined independently with each\nother. Lemma 4 implies that, for a point with D(λ) = 0, we can identify |Ψ+\nλ/an}b∇acket∇i}htwith|Ψ−\nλ/an}b∇acket∇i}ht\nand replace one with the other. Thus the point with D(λ) = 0 is not a level crossing\npoint.Theorems on ground-state phase transitions in Kohn-Sham mo dels 10\nTheorem 6 i) In[0,1], we have at most a finite number of transition points, where\nFλ[n]has a finite discontinuity, which is greater than an arbitrar y small number ε.\nii) At an accumulation point of the discontinuous points D(λi)(i= 1,2,···,∞) with\nλi∈(0,1], andλi<λi+1, we haveD(λi)→0fori→ ∞.\ni) For any finite number ε>0, we have a set Λ( ε) of discontinuous points, at which\nD(λ)> ε. Assume that the number of elements of Λ( ε) is more than the countable\ninfinite. We can select a countable infinite subset of Λ( ε) and name it ¯Λ(ε). Points in\n¯Λ(ε) are in [0,1] and are to be ordered as λi<λi+1. This is due to the selection axiom.\nLet’s number the points in ¯Λ(ε) asλi(i= 1,2,···). We have,\nD+(λj)<D−(λj)≤D+(λj−1)<D−(λj−1),\nD+(λj)<D−(λj)−ε<D−(λj−1)−2ε<D−(λ1)−jε.\nSince the set ¯Λ(ε) is infinite, we have an integer J >D−(λ1)/εfor which the derivative\nD+(λJ) becomes strictly negative. This result contradicts to the increas ing property of\nFλ[n]. Thus the set Λ( ε) has to be a finite set.\nii) We consider inf iD(λi) fori= 1,2,···. Owing to i), a finite non-zero infimum is\ndenied. Now consider D= limN→∞supi>ND(λi). Assume that D >0. Then, we can\nfind an integer l1, for which D(λl1)≥D. But, we can also find another integer l2>l1,\nfor whichD(λl2)≥D, because of the definition of D. Then we have an infinite series\nofλli(i= 1,2,···) withD(λli)≥D>0, which contradicts to i). Thus, D= 0./squaresolid\nLemma 7 Forn(r)∈ IN, the functiond\ndλFλ[n]ofλis Lipschitz continuous in [0,1]\nand we have,\n/integraldisplay1\n0dλd\ndλFλ[nΨ′] =Fλ=1[n]−Fλ=0[n]. (34)\nFor∀Ψ∈H1, we have a finite Rsatisfying /an}b∇acketle{tΨ|ˆHee|Ψ/an}b∇acket∇i}ht ≤R. Choose Rsuch\nthatD+(0)≤R. SinceFλ[n] is continuous monotone increasing and concave, for\n0≤ ∀λ1<∀λ2≤1, we have,\n0≤Fλ2[n]−Fλ1[n]≤D+(λ1)(λ2−λ1)≤D+(0)(λ2−λ1)≤R(λ2−λ1).(35)\nThus, we have Eq. (34). /squaresolid\n6. The Kohn-Sham minimization scheme\nTo discuss relevance ofour theorems for thediscussion ofground -statephase transitions,\nwe introduce a Kohn-Sham minimization scheme.[16, 12] For an extern al potential\nvext(r), wehaveagroundstateoftheCoulombsystem, |ΨGS/an}b∇acket∇i}ht, whichgivesaground-state\ndensity,nGS(r). First, we have a next equality.\nE0[vext]\n=/an}b∇acketle{tΨGS|ˆT+ˆVee|ΨGS/an}b∇acket∇i}ht+/integraldisplay\nd3rvext(r)nGS(r)Theorems on ground-state phase transitions in Kohn-Sham mo dels 11\n= min\nn/bracketleftbigg\nF[n]+/integraldisplay\nd3rvext(r)n(r)/bracketrightbigg\n= min\nn/bracketleftbigg\nmin\nΨ′→n/an}b∇acketle{tΨ′|ˆT|Ψ′/an}b∇acket∇i}ht+Fλ=1[n]−Fλ=0[n]+/integraldisplay\nd3rvext(r)n(r)/bracketrightbigg\n= min\nn/bracketleftbigg\nmin\nΨ′→n/braceleftbigg\n/an}b∇acketle{tΨ′|ˆT|Ψ′/an}b∇acket∇i}ht+/integraldisplay1\n0dλd\ndλFλ[nΨ′]+/integraldisplay\nd3rvext(r)nΨ′(r)/bracerightbigg/bracketrightbigg\n= min\nΨ′/bracketleftbigg\n/an}b∇acketle{tΨ′|ˆT|Ψ′/an}b∇acket∇i}ht+/integraldisplay1\n0dλd\ndλFλ[nΨ′]+/integraldisplay\nd3rvext(r)nΨ′(r)/bracketrightbigg\n= min\nΨ′G0,vext[Ψ′]. (36)\nHere, we used a notation for the charge density given by |Ψ/an}b∇acket∇i}htas,\nnΨ(r)≡ /an}b∇acketle{tΨ|ˆn(r)|Ψ/an}b∇acket∇i}ht. (37)\nWe can show two conditions on the minimizing Ψ of G0,vext[Ψ]. Note that,\nG0,vext[Ψ] =/an}b∇acketle{tΨ|ˆT|Ψ/an}b∇acket∇i}ht+Fλ=1[nΨ]−Fλ=0[nΨ]+/integraldisplay\nd3rvext(r)nΨ(r).(38)\nThen we have,\nG0,vext[Ψ]≥min\nΨ′→nΨ/an}b∇acketle{tΨ′|ˆT|Ψ′/an}b∇acket∇i}ht+Fλ=1[nΨ]−Fλ=0[nΨ]+/integraldisplay\nd3rvext(r)nΨ(r)\n=Fλ=1[nΨ]+/integraldisplay\nd3rvext(r)nΨ(r)\n≥Fλ=1[nGS]+/integraldisplay\nd3rvext(r)nGS(r). (39)\nWe see that equalities in Eq. (39) are satisfied,\n(i) if Ψ is identical to a states Ψ′, which minimizes /an}b∇acketle{tΨ′|ˆT|Ψ′/an}b∇acket∇i}ht, with the constraint that\nΨ′→nΨ,\n(ii) and ifnΨis identical to the true ground-state charge density nGS(r).\nThus, the minimizing process of G0,vext[Ψ] gives us a state that reproduces nGS(r) and\nminimizes /an}b∇acketle{tΨ|ˆT|Ψ/an}b∇acket∇i}ht. Two fundamental statements are addressed here.\n(i) A minimizing state Ψ of G0,vext[Ψ] is the state which is searched in the constrained\nminimization of Fλ=0[nGS]. Thus Ψ is the state motivated to be searched in the\nKohn-Sham scheme.\n(ii) We do not need to have a secular equation to define the minimization process of\nG0,vext[Ψ] for our discussion.\nThe definition of G0,vext[Ψ] suggests us that we have a plenty of models for electron\nsystems. Actually, we have another equality.\nE0[vext] = min\nΨGλ,vext[Ψ], (40)\nGλ,vext[Ψ]≡ /an}b∇acketle{tΨ|ˆT+λVee|Ψ/an}b∇acket∇i}ht+/integraldisplay1\nλdλ′d\ndλ′Fλ′[nΨ]+/integraldisplay\nd3rvext(r)nΨ(r).(41)Theorems on ground-state phase transitions in Kohn-Sham mo dels 12\nThe wave-function funcional, Gλ,vext[Ψ], determines |Ψλ/an}b∇acket∇i}ht, which may appear as a\nminimizing state in the definition of Fλ[nGS]. Therefore, when we fix vext(r),\nminimization of Gλ,vext[Ψ] withλ∈[0,1] produces a set of states {|Ψλ/an}b∇acket∇i}ht}and thus\nthe function Fλ[nGS] ofλas\nFλ[nGS] =/an}b∇acketle{tΨλ|ˆT+λˆVee|Ψλ/an}b∇acket∇i}ht. (42)\nWemayintroducetheHartreetermtoformulateamuchfamiliarform inthedensity\nfunctional theory.\nG0,vext[Ψ]\n=/an}b∇acketle{tΨ|ˆT|Ψ/an}b∇acket∇i}ht+1\n2/integraldisplay\nd3rd3r′e2\n|r−r′|nΨ(r)nΨ(r′)+/integraldisplay\nd3rvext(r)nΨ(r)\n+/integraldisplay1\n0/braceleftbigg\ndλd\ndλmin\nΨ′→nΨ/an}b∇acketle{tΨ|ˆT+λˆVee|Ψ/an}b∇acket∇i}ht−1\n2/integraldisplay\nd3rd3r′e2\n|r−r′|nΨ(r)nΨ(r′)/bracerightbigg\n=/an}b∇acketle{tΨ|ˆT|Ψ/an}b∇acket∇i}ht+1\n2/integraldisplay\nd3rd3r′e2\n|r−r′|nΨ(r)nΨ(r′)+/integraldisplay\nd3rvext(r)nΨ(r)\n+/integraldisplay1\n0dλ1\n2/integraldisplay\nd3rd3r′e2\n|r−r′|{/an}b∇acketle{tΨλ|: ˆn(r)ˆn(r′) :|Ψλ/an}b∇acket∇i}ht−nΨ(r)nΨ(r′)}\n=/an}b∇acketle{tΨ|ˆT|Ψ/an}b∇acket∇i}ht+1\n2/integraldisplay\nd3rd3r′e2\n|r−r′|nΨ(r)nΨ(r′)+/integraldisplay\nd3rvext(r)nΨ(r)\n+Exc[nΨ]. (43)\nSimilarly, we have\nGλ,vext[Ψ]\n=/an}b∇acketle{tΨ|ˆT+λˆVee|Ψ/an}b∇acket∇i}ht+1−λ\n2/integraldisplay\nd3rd3r′e2\n|r−r′|nΨ(r)nΨ(r′)+/integraldisplay\nd3rvext(r)nΨ(r)\n+/integraldisplay1\nλdλ′1\n2/integraldisplay\nd3rd3r′e2\n|r−r′|{/an}b∇acketle{tΨλ′|: ˆn(r)ˆn(r′) :|Ψλ′/an}b∇acket∇i}ht−nΨ(r)nΨ(r′)}\n=/an}b∇acketle{tΨ|ˆT+λˆVee|Ψ/an}b∇acket∇i}ht+1−λ\n2/integraldisplay\nd3rd3r′e2\n|r−r′|nΨ(r)nΨ(r′)+/integraldisplay\nd3rvext(r)nΨ(r)\n+Exc,λ[nΨ]. (44)\nWe callGλ,vext[Ψ] theλ-parametrized model energy functional. All of these models can\ndetermineE0[vext] as shown by Eq. (40) via a determination of |Ψλ/an}b∇acket∇i}ht. The charge density\nof|Ψλ/an}b∇acket∇i}htsatisfies,nΨλ(r) =nGS(r). When 0 < λ <1, since a reduced-interaction λˆVee\nappears in the definition of Gλ,vext[Ψ], its minimizing state, |Ψλ/an}b∇acket∇i}ht, is represented by a\nsummation of Slater determinants.\n7. Existence of εvicinity\nFrom now on, we consider possible level crossings in Fλ[nGS] given in Eq. (42). A level\ncrossing occurs, when two or more states appear as minimizing stat es|Ψλ/an}b∇acket∇i}htofˆT+λˆVee\nand when a jump in Dini’s derivatives, D(λ)>0, happens.Theorems on ground-state phase transitions in Kohn-Sham mo dels 13\nExistence ofthecrossing pointsisexemplified byaphasetransitionf romthenormal\nstate to a ferromagnetic state recognized in the uniform electron gas system.[10, 19] At\nthephaseboundary, wehavetwouniformelectrongasgroundsta tes,i.e.aparamagnetic\nstate without spin polarization and a partially ferromagnetic state w ith a finite total\nspin. In general, we need to assume that crossing points appear at some ofλ∈[0,1].\nOnce there appears a crossing point, we can count the number of c rossing points on the\nλaxis.\nFirst, we note that the level crossings happen at discretized point s on theλaxis.\nTwo statements of Theorem 6 deny a possibility to have crossing poin ts withD(λ)>0\ncontinuously or densely on the λaxis. Next we need to consider a case with countable\ninfinite numbers of crossing points in a finite region of λ. This case causes appearance\nof accumulation points of the crossing points on the λaxis.\nThus we can choose a finite interval of λsatisfying one of two possible conditions\nforD(λ): i)∀λ∈(λa,λb),D(λ) = 0, or ii) one of the boundaries λl(l=aorb) is an\naccumulation point of D(λi)>0 and lim i→∞D(λi) = 0. Our concern is whether the\ncase ii) happens at λb= 1 for av-representable ground state density or not.\nConsider auniquegroundstate |ΨGS/an}b∇acket∇i}htofanelectronsystem intheexternal potential\nvext(r), which has an electron charge density nGS(r). Simbolically, we write the v-\nrepresentability of nGS(r) asnGS(r)∈ AN.[7] The ground state is supposed to be stable\nagainst small perturbation. Let’s assume that the point of λ= 1 is an accumulation\npoint of the level crossings along the λaxis, and that we have the case ii).\nCrossing points are labeled as λi(i= 1,2,···,∞) andλi< λi+1for the present\ndiscussion. At any level crossing point λi, owing to Lemma 4 there are at least two\nminimizing states, |Ψ+\nλi/an}b∇acket∇i}htand|Ψ−\nλi/an}b∇acket∇i}ht. The expectation values /an}b∇acketle{tˆT+λiˆVee/an}b∇acket∇i}htby these states\nare the same at the crossing point. However, |Ψ+\nλi/an}b∇acket∇i}htand|Ψ−\nλi/an}b∇acket∇i}htare distinguished by /an}b∇acketle{tˆVee/an}b∇acket∇i}ht\nand/an}b∇acketle{tˆT/an}b∇acket∇i}ht. Thus, they are linearly independent as state vectors. We can also show that\n/an}b∇acketle{tΨ+\nλi|ˆVee|Ψ+\nλi/an}b∇acket∇i}htdecreases, when iincreases, as exemplified in the proof of Theorem 6.\nIn between two neighboring crossing points, we have a continuous c hange in |Ψλ/an}b∇acket∇i}ht\nandD(λ) = 0. The state, |Ψ+\nλi/an}b∇acket∇i}htis connected to |Ψ−\nλi+1/an}b∇acket∇i}ht. We may select |Ψ+\nλi/an}b∇acket∇i}htas a\nrepresentative state for this finite range [ λi,λi+1].\nExistence of a big number of level crossing points at a close vicinity of this\naccumulation point requires existence of a plenty number of nearly d egenerate states,\n|Ψ+\nλi/an}b∇acket∇i}htwithi= 1,2,···,∞. Here, the state, |Ψ+\nλi/an}b∇acket∇i}ht, might not be an eigen state of\nany potential problem, but minimizes just the expectation value of /an}b∇acketle{tˆT+λˆVee/an}b∇acket∇i}htkeeping\nthe density. They are not distinguished by the primary order param eter,n(r), nor by\ndifference in any external symmetry breaking observed by the ext ernal potential vext(r).\nOnly internal degrees of freedom distinguish these infinite numbers of|Ψ+\nλi/an}b∇acket∇i}ht.\nWe should note that only to have these states does not directly mea n existence\nof infinite numbers of degenerate eigen states at λ= 1. This is because we have a\npossibility that all of |Ψ+\nλi/an}b∇acket∇i}htexcept for |ΨGS/an}b∇acket∇i}htbecome non-eigen-states of ˆH, but are just\nvariational states. Furthermore, if continuous change in |Ψλ/an}b∇acket∇i}htis allowed in a finite range\nofλ, only one minimizing state appears as a unique minimum for /an}b∇acketle{tˆT+λˆVee/an}b∇acket∇i}htin theTheorems on ground-state phase transitions in Kohn-Sham mo dels 14\nrange.\nHowever, the state |Ψ+\nλi/an}b∇acket∇i}htexists and may be used as variational states at any\nλ∈[0,1]. By a simple inspection, we can see that at λ= 1, the variational energies of\n/an}b∇acketle{tΨ+\nλi|ˆT+ˆVee|Ψ+\nλi/an}b∇acket∇i}htare separated by a finite gap with each other. However, when λ= 1\nis an accumulation point, we have |/an}b∇acketle{tΨ+\nλi|ˆT+ˆVee|Ψ+\nλi/an}b∇acket∇i}ht −/an}b∇acketle{tΨ+\nλi+1|ˆT+ˆVee|Ψ+\nλi+1/an}b∇acket∇i}ht| →0 for\ni→ ∞. Thus the gap for i≫Nshould be much smaller than any energy separation in\nthe energy spectrum of the finite size system.\nTo discuss the accumulation points further, we now analyze existen ce or non-\nexistence of a potential vλ(r) which gives a secular equation determining a state vector\n|Φλ/an}b∇acket∇i}ht. Namely, for λ= 1−δλwith 0< δλ≪1, we formulate a quantum mechanical\nproblem, whose solution |Φλ/an}b∇acket∇i}htsatisfies||/an}b∇acketle{tΦλ|ˆn(r)|Φλ/an}b∇acket∇i}ht−nGS(r)||∞= 0, and |Φλ/an}b∇acket∇i}ht → |ΨGS/an}b∇acket∇i}ht\nforλ→1. Simbolically, we write this statement on another v-representability of nGS(r)\nasnGS(r)∈ Aλ,N.\nFor simplicity, we consider a compact space by introducing a cube with volume\nL3under the periodic boundary condition. Then, we can introduce a Fo urier series\nexpansion for the potential vλ(r), which is used as the Lagrange multiplier to fix the\ncharge density of |Φ/an}b∇acket∇i}ht, as,\nvλ(r) =/summationdisplay\nGvλ,Gexp(iG·r). (45)\nConsider an Nelectron state |Φ/an}b∇acket∇i}ht. We introduce a functional Q[|Φ/an}b∇acket∇i}ht,vλ,G,E:δλ] as,\nQ[|Φ/an}b∇acket∇i}ht,vλ,G,E:δλ]\n=/an}b∇acketle{tΦ|ˆT+(1−δλ)ˆVee+/integraldisplay\nd3rvext(r)ˆn(r)|Φ/an}b∇acket∇i}ht\n+/integraldisplay\nd3rvλ(r)[/an}b∇acketle{tΦ|ˆn(r)|Φ/an}b∇acket∇i}ht−nGS(r)]−E[/an}b∇acketle{tΦ|Φ/an}b∇acket∇i}ht−1]\n=/an}b∇acketle{tΦ|ˆT+ˆVee+/integraldisplay\nd3rvext(r)ˆn(r)|Φ/an}b∇acket∇i}ht\n+/an}b∇acketle{tΦ|/braceleftBigg/summationdisplay\nG/negationslash=0/summationdisplay\nG′/summationdisplay\nσvλ,Gc†\nG′−G,σcG′,σ−δλˆVee/bracerightBigg\n|Φ/an}b∇acket∇i}ht\n−/summationdisplay\nG/negationslash=0vλ,GnGS(−G)−¯E[/an}b∇acketle{tΦ|Φ/an}b∇acket∇i}ht−1]. (46)\nHere,nGS(G) is the Fourier component of nGS(r),cG,σare electron annihilation\noperators with the spin σ, and¯E=E−Nvλ,0. By making derivatives of Qwith\nrespect to variables except for a parameter δλ, we have next secular equations.\n/braceleftBig\nˆH+ˆHδλ/bracerightBig\n|Φ/an}b∇acket∇i}ht=¯E|Φ/an}b∇acket∇i}ht, (47)\nˆHδλ=/summationdisplay\nG/negationslash=0/summationdisplay\nG′/summationdisplay\nσvλ,Gc†\nG′−G,σcG′,σ−δλˆVee (48)\n/an}b∇acketle{tΦ|Φ/an}b∇acket∇i}ht= 1, (49)/summationdisplay\nG′/summationdisplay\nσ/an}b∇acketle{tΦ|c†\nG′+G,σcG′,σ|Φ/an}b∇acket∇i}ht=nGS(G). (50)Theorems on ground-state phase transitions in Kohn-Sham mo dels 15\nAtλ= 1,δλ= 0, thesolutionofEqs. (47), (49), and(50)isgivenbythenormaliz ed\nstate|ΨGS/an}b∇acket∇i}htwithvλ,G= 0 for all G. For anyvλ(r)∈L3/2+L∞, or equivalently for any\nset ofvλ,G, withδλ>0, we have a normalized eigen state |Φ/an}b∇acket∇i}htof Eqs. (47) and (49). So\nthe construction of vλ,Gto meet Eq. (50) is the problem. If vλ,Gexists,nGS(r)∈ Aλ,N.\nLemma 8 When a unique ground state of ˆHgives the charge density nGS(r), and if\nthere happens accumulation of points λi(i= 1,2,···,∞) satisfying 0< λi< λi+1<1\nandD(λi)>0,nGS(r)is not in Aλi,N. We have a series of potentials vλi(r), which\ngives a set of pure states |˜Ψλi/an}b∇acket∇i}ht, and||/an}b∇acketle{t˜Φλi|ˆn(r)|˜Φλi/an}b∇acket∇i}ht−nGS(r)||∞→0asi→ ∞.\nTo show this lemma, we classify possible conditions and deny possibilities except\nfor a case that nGS(r) is not in Aλi,N, which is the case 4 below.\ncase 1There exists vλi(r), and|Ψ+\nλi/an}b∇acket∇i}htand|Ψ−\nλi/an}b∇acket∇i}htare eigen states of ˆH+ˆHδλ=\nˆT+λiˆVee+/integraldisplay\nd3rvλi(r)ˆn(r).\ncase 2There exist v+\nλi(r) andv−\nλi(r), which are different from each other more than\na constant. Two states, |Ψ±\nλi/an}b∇acket∇i}ht, are eigen states of ˆT+λˆVee+/integraldisplay\nd3rv±\nλi(r)ˆn(r),\nrespectively.\ncase 3One of|Ψ+\nλi/an}b∇acket∇i}htand|Ψ−\nλi/an}b∇acket∇i}htis an eigen state of ˆT+λˆVee+/integraldisplay\nd3rvλi(r)ˆn(r), and the\nother is not.\ncase 4Both of minimizing states, |Ψ+\nλi/an}b∇acket∇i}htand|Ψ−\nλi/an}b∇acket∇i}ht, are not eigen states of any potential\nproblem given as ˆT+λˆVee+/integraldisplay\nd3rvλi(r)ˆn(r).\nWe first deny the case 1. Let’s assume that the solution vλ,Gexists fornGS(r). It\nmeans that we have a minimum of Q, which is given by an eigen state |Φ/an}b∇acket∇i}htof Eq. (47).\nIf we insert |Ψλ/an}b∇acket∇i}htin the functional Q, we have\nQ[|Ψλ/an}b∇acket∇i}ht,vλ,G,E:δλ]\n=/an}b∇acketle{tΨλ|ˆT+λˆVee|Ψλ/an}b∇acket∇i}ht+/integraldisplay\nd3rvext(r)nGS(r)\n≤ /an}b∇acketle{tΦ|ˆT+λˆVee|Φ/an}b∇acket∇i}ht+/integraldisplay\nd3rvext(r)nGS(r)\n=Q[|Φ/an}b∇acket∇i}ht,vλ,G,E:δλ]. (51)\nThus, by the variational principle of the quantum mechanics, we see that|Ψλ/an}b∇acket∇i}htis also an\neigen state of Eq. (47). If we choose λ=λi, we have two degenerate eigen states, |Ψ+\nλi/an}b∇acket∇i}ht\nand|Ψ−\nλi/an}b∇acket∇i}ht. They are orthogonal with each other. By taking the limit i→ ∞, we have\ntwo limiting states, |Ψ+\nλ∞/an}b∇acket∇i}htand|Ψ−\nλ∞/an}b∇acket∇i}ht, since these state vectors are in a Banach space\nand existence of the weak limit is ensured by /an}b∇acketle{tΨ±\nλi|ˆVee|Ψ±\nλi/an}b∇acket∇i}ht → /an}b∇acketle{tΨGS|ˆVee|ΨGS/an}b∇acket∇i}htowing to\nTheorem 6. Orthogonality between |Ψ+\nλ∞/an}b∇acket∇i}htand|Ψ−\nλ∞/an}b∇acket∇i}htholds. Of course λ∞= 1. These\nstates minimize Q[|Ψ/an}b∇acket∇i}ht,vλ=1,G,E:δλ= 0]. This means that the ground state has to be\ndegenerate, which contradicts to the uniqueness of the ground s tate providing nGS(r).Theorems on ground-state phase transitions in Kohn-Sham mo dels 16\nIf we have a case 2, we have two independent potential v+\nλi(r) andv−\nλi(r), both\nof which give the same density nGS(r). This case contradicts the Hohenberg-Kohn\ntheorem[1] and thus it is denied at any i.\nIf we have a case 3, we again have a difficulty. Both of |Ψ+\nλi/an}b∇acket∇i}htand|Ψ−\nλi/an}b∇acket∇i}htgive the\nsame variational energy for ˆT+λˆVee+/integraldisplay\nd3rvλi(r)ˆn(r). The variational principle tells\nthat a state having the variational energy of the lowest eigen stat e gives a degenerate\neigen state. This fact contradicts an assumption that one of thes e two states is not an\neigen state of any potential problem.\nSo, we conclude the case 4, which says that vλi,Gdoes not exist, when λ= 1 is\nan accumulation point. Since nGS(r)∈ IN, we know existence of an N-particle density\nmatrix Γ, which gives the infimum of the Lieb functional.[7] Γ has to be th at for a mixed\nstate,\nΓ =/summationdisplay\nlCl\nλi|Ψl\nλi/an}b∇acket∇i}ht/an}b∇acketle{tΨl\nλi|. (52)\n|Ψl\nλi/an}b∇acket∇i}htis given as an eigen state of a potential problem and these states ar e orthogonal\nwith each other. Since the limit of λi→1 is given as a pure state, when i→ ∞,\ncoefficients Cl\nλiand state vectors satisfy,\nCl\nλi→δl,l0, (53)\n|Ψl0\nλi/an}b∇acket∇i}ht → |ΨGS/an}b∇acket∇i}ht. (54)\nThus the second statement of the lemma holds. /squaresolid\nThemeaningofthislemmaissomewhatredundant. Whenwehaveanac cumulation\npoint atλ= 1, we have no potential series, vλi(r), whose pure ground state reproduces\nexactlynGS(r). Thus the case 4, if it is found, gives an example of an N-representable\ndensity which is in ANbut not in Aλ,Nforλ<1. This density is apparently not pure-\nstatev-representable in any Kohn-Sham scheme. The accumulation of cro ssing points\nfor a ground state density contradicts a picture of the ordinal Ko hn-Sham scheme,\nwhich assumes that a unique ground state of the Coulomb problem is r eproduced by a\nnon-interacting system with an optimized potential.\nIn this case, we should follow the present lemma to know existence of a converging\nseries of quantum mechanical models. Actually, the potential serie s ofvλi(r), whose\nground state density is slightly different from the final solution, can be used to find a\nCauchy sequence of nλi(r) converging to nGS(r). This converging series is found in the\nmodel space of the multi-reference generalization of the Kohn-Sh am scheme. We may\ncall the regionof the model space as an ε-vicinity, in which a convergence of a simulation\nis guaranteed.\nConversely, ifadensity nGS(r)ispure-state v-representable inaKohn-Shamscheme\nincluding a multi-reference generalization, there is no accumulation o f crossing points\natλ= 1. In this case, we have a well-defined ε-vicinity around the true ground state\nin the model space, in which no level crossing is found. More precisely , we have a next\nstatement. If the density nGS(r) isv-representable also for λ <1, accumulation ofTheorems on ground-state phase transitions in Kohn-Sham mo dels 17\nphase transition points is forbidden when λ→1. Finding a convergence in the density\nsearched inanoptimization process of model quantum systems may allow us to conclude\nno remaining level-crossing point along a line approaching in the true Co ulomb system\ngiven by ˆH. Therefore, we conclude the existence of an εvicinity around the Coulomb\nsystem in the model space, where no essential phase transition oc curs in the direction\nto reduce the interaction strength by introducing λ<1, and keeping the charge density\nn(r) unchanged.\nNow we re-analyze the potential, vλ,G, in a pertabative argument. Although this\nproblem was treated more elegantly by Kohn,[20] we want to show a su btle problem on\nexistence of vλ,G. We have the limiting solution of |ΨGS/an}b∇acket∇i}htwithvλ,G= 0, which is an\neigen solution of Eq. (47) at least as a stationary state. The const ruction ofQis based\non the quantum mechanical variational principle, and the limit of Qforλ→1 should\nbehave regularly. In order to inspect on the v-representability of the normal solution\nnGS(r) forλ <1, we argue a perturbative construction method of vλ,G. When the\nspectrum of ˆHis normal, and when the excited states are written as |Ψi/an}b∇acket∇i}ht, the solution\nof Eq. (47) is represented by the ordinal perturbation theory as ,\n|Φ/an}b∇acket∇i}ht=1\nC/braceleftBigg\n|ΨGS/an}b∇acket∇i}ht+/summationdisplay\ni1\nE0−Ei|Ψi/an}b∇acket∇i}ht/an}b∇acketle{tΨi|ˆHδλ|ΨGS/an}b∇acket∇i}ht+o(δλ)/bracerightBigg\n. (55)\nThe expectation value in Eq. (50) gives,\n/summationdisplay\nG′/summationdisplay\nσ/an}b∇acketle{tΦ|c†\nG′+G,σcG′,σ|Φ/an}b∇acket∇i}ht−nGS(G)\n=1\nC2/braceleftBigg/summationdisplay\nP/negationslash=0/summationdisplay\ni1\nE0−Ei/an}b∇acketle{tΨGS|/summationdisplay\nG′/summationdisplay\nσc†\nG′+G,σcG′,σ|Ψi/an}b∇acket∇i}ht\n×/an}b∇acketle{tΨi|/summationdisplay\nG′′/summationdisplay\nσc†\nG′′−P,σcG′′,σ|ΨGS/an}b∇acket∇i}htvλ,P\n−/summationdisplay\ni1\nE0−Ei/an}b∇acketle{tΨGS|/summationdisplay\nG′/summationdisplay\nσc†\nG′+G,σcG′,σ|Ψi/an}b∇acket∇i}ht/an}b∇acketle{tΨi|δλˆVee|ΨGS/an}b∇acket∇i}ht+c.c.+o(δλ)/bracerightBigg\n=/summationdisplay\nP/negationslash=0AG,Pvλ,P+δλBG+c.c.+o(δλ). (56)\nHere,Cisanormalizationconstant. Ingeneral, wehaveanon-zerovector BG. Sincethe\nmatrixAG,Pis an Hermite matrix, we have a solution vλ,Pof Eq. (50) in o(δλ). Thus,\nwe may utilize the determination method of vλ,Pto analyze the Kohn-Sham method, in\nwhich the density nGS(r) is reproduced by another artificial model system.\nHowever, we find a difficulty in Eq. (56). The required conditions amou nt to the\nsamenumberas vλ,P. So,onceweconsiderhigher-orderconditionsfor λnasindependent,\nthe number of the conditions is over the number of vλ,P. This suggests that the solution\ncould befound ina non-localpotential. Even if so, the number of con ditions is toolarge.\nIfwe consider alltheexpansions inEq. (56)asfunctions of λandvλ,P, thedetermination\nequations form simultaneous equations of vλ,P. The structure is not trivial for higherTheorems on ground-state phase transitions in Kohn-Sham mo dels 18\norder terms. Since they are non-linear determination equations, t he solution is expected\nto exist only when all the expressions are derived exactly.\nBefore closing this section, we summarize conditions for Lemma 4 and Theorem 6.\nThe proofs of these statements tell that the expectation value o f the relevant interaction\nterm has to be positive and finite, i.e., 0</an}b∇acketle{tˆVee/an}b∇acket∇i}ht<∞. The convexity (or precisely the\nconcavity nature of Fλ[n]) comes from linear nature of the λ-modifiedFλ[n] with respect\ntotheparameter λ. Theconstrainedminimizationallowsustoconcludebothcontinuous\nnature of the functional and existence of minimizing states, which g ive Dini’s derivative\nof the parametrized energy density functional.\n8. Summary and conclusions\nIn section 5, we consider the λmodification of F[n], while the primary order parameter\nn(r)doesnotchange. Weconsider aspaceofmodels, Gλ,vext[Ψ], withλ∈[0,1]insection\n6. In the model space, at a discontinuous point of Fλ[n] with fixed n(r), the minimizing\nstates,|Ψ+\nλ/an}b∇acket∇i}htand|Ψ−\nλ/an}b∇acket∇i}ht, exist. Existence of discontinuous points may be detected by\nd\ndλFλ[n], ord2\ndλ2Fλ[n]. The latter behaves as a delta function at the discontinuous\npoints. We can use these functions as an indicator for discontinuou s transition points\nin a set of states giving Ψ →n(r).\nOwing to the above discussion, we have a physical conclusion on the u niversal\nenergy density functional. Consider a true density nGS(r), which is given by a ground\nstate in an external potential vext(r). If the state is non degenerate (or at least finitely\ndegenerated) and if it is stable, we have no accumulation point of the level crossing\npoints forFλ[nGS] atλ= 1 in the searching process utilizing a potential problem with\nthe reduced interaction strength. Thus, when we move away from λ= 1 on the λaxis,\nwe have an εvicinity, where no level crossing happens in the model space, which is to be\ndefined by a secular equation of the many-Fermion system with the r educed interaction.\nIn a multi-reference density functional theory (MR-DFT), one of the authors\nconsidered general modification of the universal energy density f unctional.[12, 17] In\naddition to the Kohn-Sham description of the true density,[2] we hav e plenty of effective\ndescriptions using model Fermion systems. The models include partia lly correlated\nsystems, whose ground state is obtained in a multiple Slater determin ants. We can\nintroduce distance between two modelsgiven by thenormof thecha rgedensity andthen\nthe set of models with the charge distance becomes a space of mode ls. An advantage of\nthis generalization of DFT is that we can search for an optimized mode l in a space of\nmodels including non-interacting Fermion models and interacting Ferm ion models.\nIn an optimization process in the model space, the density is search ed in a\nv-representable subset. Thus, in a realization of MR-DFT, we never meet the\naccumulation of unreachable crossing points, when we see a conver gence of the density\nin a searching step. The existence of an ε-vicinity in the model space suggests that we\nhave a physically converged model, which may continuously connect t o the true electronTheorems on ground-state phase transitions in Kohn-Sham mo dels 19\nsystem. EvenwhentheoriginalKohn-Shammodel isseparatedfro mtheelectronsystem\nby crossing points, the generalized Kohn-Sham model in MR-DFT can be settled in the\nε-vicinity.\nAcknowledgement\nThis work was supported by the Global COE Program (Core Researc h and Engineering\nofAdvancedMaterial-InterdisciplinaryEducationCenter forMate rialsScience), MEXT,\nJapan, Grand Challenges in next-generation integrated nanoscien ce, Grant-in-Aid for\nScientific Research in Priority Areas (No. 17064006, No. 19051016 )anda Grants-in-Aid\nfor Scientific Research (No. 19310094).\nReferences\n[1] Hohenberg P and Kohn W 1964 Inhomogeneous electron gas Phys. Rev. 136B864\n[2] Kohn W and Sham L J 1965 Self-consistent equations including exch ange and correlation effects\nPhys. Rev. 140A1133\n[3] Levy M 1979 Universal variational functionals of electron densit ies, first-order density matrices,\nand natural spin-orbitals and solution of the v-representability problem Proc. Natl. Acad. Sci.\n(U.S.A.) 766062\n[4] Levy M 1982 Electron densities in search of Hamiltonians Phys. Rev. A 261200\n[5] ParrRGandYangW1989 Density-Functional Theory of Atoms and Molecules (NewYork: Oxford\nUniv. Press)\n[6] Driezler R M and Gross E K U 1990 Density Functional Theory (Berlin: Springer)\n[7] Lieb E 1983 Density functionals for Coulomb systems Int. J. Quantum. Chem. 24243\n[8] Gunnarsson O and Lundqvist B I 1976 Exchange and correlation in atoms, molecuels and solids\nby the spin-density-functional formalism Phys. Rev. B 134274\n[9] LangrethDC andPerdewJP1977Exchange-correlationenergy ofa metallicsurface: Wave-vector\nanalysis Phys. Rev. B 152884\n[10] Pines D 1964 The Many-Body Problem (New York: W.A. Benjamin, Inc.)\n[11] FetterALandWaleckaJD1971 QuantumTheory of Many-Particle Systems (NewYork: McGraw-\nHill, Inc.)\n[12] Kusakabe K 2001 A Rigorous Extension of the Kohn-Sham Equat ion for Strongly Correlated\nElectron Systems J. Phys. Soc. Jpn. 702038\n[13] Parr R G 1964 Theorem governing changes in molecular conforma tionJ. Chem. Phys. 403726\n[14] Rockafellar R T 1970 Convex Analysis (New Jersey: Princeton Univ. Press.)\n[15] Lieb E H and Loss M 1997 Analysis (USA: American Math. Soc.)\n[16] Hadjisavvas N and Theophilou A 1984 Rigorous formulation of the Kohn and Sham theory Phys.\nRev. A302183\n[17] Kusakabe K, Suzuki N, Yamanaka S, and Yamaguchi K 2007 A se lf-consistent first-principles\ncalculation scheme for correlated electron systems J. Phys.: Condens. Matter 19445009\n[18] Kusakabe K 2009 Pair-Hopping Mechanism for Layered Superco nductors J. Phys. Soc. Jpn. 78\n114716\n[19] Ceperley D M and Alder B J 1980 Ground state of the electron gas by a stochastic method Phys.\nRev. Lett. 45566\n[20] Kohn W 1983 v-Representability and density functional theory Phys. Rev. Lett. 511596"    },    {        "title": "1101.1327v1.Non_Universality_of_Density_and_Disorder_in_Jammed_Sphere_Packings.pdf",        "content": "arXiv:1101.1327v1  [cond-mat.stat-mech]  6 Jan 2011Non-Universality of Density and Disorder in Jammed Sphere\nPackings\nYang Jiao\nDepartment of Mechanical and Aerospace Engineering,\nPrinceton University, Princeton New Jersey 08544, USA\nFrank H. Stillinger\nDepartment of Chemistry, Princeton University,\nPrinceton New Jersey 08544, USA\nSalvatore Torquato∗\nDepartment of Chemistry, Princeton University,\nPrinceton New Jersey 08544, USA\nDepartment of Physics, Princeton University,\nPrinceton New Jersey 08544, USA\nPrinceton Center for Theoretical Science,\nPrinceton University, Princeton New Jersey 08544, USA and\nProgram in Applied and Computational Mathematics,\nPrinceton University, Princeton New Jersey 08544, USA\n(Dated: March 16, 2022)\n1Abstract\nWe show for the first time that collectively jammed disordere d packings of three-dimensional\nmonodisperse frictionless hard spheres can be produced and tuned using a novel numerical proto-\ncol with packing density φas low as 0.6. This is well below the value of 0.64 associated w ith the\nmaximally random jammed state and entirely unrelated to the ill-defined “random loose packing”\nstate density. Specifically, collectively jammed packings are generated with a very narrow distri-\nbution centered at any density φover a wide density range φ∈[0.6,0.74048...] with variable\ndisorder. Our results support the view that there is no unive rsal jamming point that is distin-\nguishable based on the packing density and frequency of occu rence. Our jammed packings are\nmapped onto a density-order-metric plane, which provides a broader characterization of packings\nthan density alone. Other packing characteristics, such as the pair correlation function, average\ncontact number and fraction of rattlers are quantified and di scussed.\nPACS numbers: 05.20.Jj, 45.70.-n, 61.50.Ah\n2I. INTRODUCTION\nThe fundamental study of disordered jammed packings has a long h istory dating back at\nleast to the pioneering work of Bernal1,2on the so-called random-close-packing (RCP) state.\nThatwastraditionallythoughttocorrespondtothedensest “ran dom”packing withaunique\npacking fraction (or density) φ≈0.64 for frictionless monodisperse three-dimensional (3D)\nspheres. That work spawned substantial research on disordere d jammed packings.3–7,9–13\nAbout a decade ago,7it was argued that the RCP state is ill-defined for various reasons,\nincluding the fact that “randomness” was never quantified and the idea that a “random”\npacking can ever achieve a maximal density is not meaningful becaus e one can infinites-\nimally increase the density of the putative RCP state, which indeed is d ependent on the\npacking protocol and container boundaries, with imperceptible cha nge in the amorphous\npair correlation function. It was suggested that the RCP state sh ould be replaced by the\nmaximally random jammed (MRJ) state,7which is the one that minimizes a scalar order\nmetricψsubject to the condition of the degree of jamming.13,14Studies of different order\nmetrics for 3D frictionless spheres have consistently led to a minimum at approximately the\nsame density φ≈0.64,7–9for collectively and strictly jammed packings in the φ-ψplane\n(i.e., the “order map”).14This consistency among the different order metrics speaks to the\nutility of the order-metric concept, even if a perfect order metric has not yet been identified.\nThe fact that this jammed state is epitomized by maximal disorder ha s been advocated by\nother groups.10Bernal originally studied such jammed packings to understand the s tructure\nof liquids, but it is now known that 3D MRJ monodisperse sphere packin gs possess quasi-\nlong-range pair correlations.11This property is markedly different from typical liquids with\nshort-range interactions, which possess pair correlations decay ing exponentially fast. Thus,\nMRJ sphere packings can be regarded to be prototypical glasses in that they possess the\nhighest degree of disorder among all jammed packings with diverging elastic moduli.15\nThe definition of the MRJ state implies that the density φalone is not sufficient to\ncharacterize jammed packings, since any packing configuration, j ammed or not, is a point\nin theφ-ψplane. This two-parameter description is but a very small subset of the rele-\nvant parameters that are necessary to fully characterize a confi guration, but it nonetheless\nenables one to draw important conclusions.13The frequency of occurrence of a particular\nconfiguration is irrelevant insofar as the order map is concerned, i.e ., the order map em-\n3phasizes a “geometric-structure” approach to analyze packings by characterizing individual\nconfigurations, regardless of their occurrence probability.13\nOne might argue that the maximum of an appropriate “entropic” met ric (based on the\nfrequency of occurrence of the packings) would be an ideal way to characterize the ran-\ndomness of a packing and therefore the MRJ state. However, as p ointed out by Ref. 9,\na substantial hurdle to overcome in implementing such an order metr ic is the necessity to\ngenerate all possible jammed states in an unbiased fashion using a “u niversal” protocol in\nthe large-system limit, which is an intractable problem. Even if such a u niversal protocol\ncould be developed, the issue of what weights to assign the resulting configurations remains.\nMoreover, there are other fundamental problems with entropic m easures. It is well known\nthat the lack of “frustration”13in two-dimensional analogs of three-dimensional computa-\ntional and experimental protocols that lead to putative MRJ state s result in packings of\nmonodisperse circular disks that are highly crystalline, forming rath er large triangular co-\nordination domains. Because such highly ordered packings are the m ost probable outcomes\nfor these typical protocols, “entropic measures” of disorder wo uld identify these as the most\ndisordered, which is clearly a misleading conclusion.\nOn the other hand, an appropriate geometric-structure order m etric is capable of iden-\ntifying a particular configuration not an ensemble of configurations of considerably lower\ndensity. For example, a jammed vacancy-diluted triangular lattice p acking (and its multido-\nmain variant) or a jammed packing containing many small crystalline re gions and “grain”\nboundaries that is consistent with our intuitive notions of maximal dis order possesses small\nscalar order metrics. However, typical packing protocols would alm ost never generate such\ndisordered disk configurations because of their inherent implicit bias toward undiluted crys-\ntallization. The readers are referred to Ref. 13 for further discu ssion. Thus we seek to\ndevise order metrics that can be applied to single jammed configurat ions, as prescribed by\nthe geometric-structure point of view. The geometric-structur e approach incorporates not\nonly maximally dense packings (e.g., Kepler’s conjecture) and random “Bernal” packings,\nbut an infinite class of other significant jammed states not previous ly recognized, including\n“tunneled” crystals that are putatively at the jamming threshold w ithφ≈0.49365....\nFurthermore, the geometric-structure approach naturally inco rporates the algorithmic\nvariability of different packing protocols that leads to a diversity of d ensity and disorder\nin jammed sphere packings.13In a typical numerical packing protocol, either the particle\n4growth or system compression leads to an increase of φ,3,4,16,17which makes the particle-pair\nnonoverlap constraints consume larger and larger portions of the configuration space.13Fur-\nther increase of φcauses the available configuration space to fracture, generating isolated\n“islands” that each eventually collapse into final jammed states with distinct density and\nstructure. Presumably, anyprotocol would sample thedisconnec ted regions ofthe configura-\ntionspacewithfixedprobability, leadingtowell-definedaverageofan ypacking characteristic\nof interest.13Unless chosen to be highly restrictive, a typical packing protocol a pplied to a\nsystem with Nparticles could produce a largenumber of geometrically distinguishab le pack-\nings with some dispersion in their φandψvalues. A narrowing of the distributions of the\npacking characteristics with increasing Ncan be expected due to operation of a central limit\ntheorem. Theparticular valuestowhich thedistributions individually c onvergeareprotocol-\nspecific, meaning that these values can be controlled by choosing ap propriate protocols or\ntuning the parameters of a protocol. Indeed, jammed packings wit h a diversity of density\nand disorder have been produced via a variety of protocols,4,7–9,15,18,19includingφ≈0.64\nwhich is empirically the outcome of a considerable large number of labor atory experiments\nand numerical simulations for identical frictionless spheres. Since t he jammed packings with\nφ≈0.64 are apparently the “most probable states”, significance has be en attached to the\nso-called unique J (jamming) point, i.e., φ≈0.64, which is suggested to correspond to the\nonset of collective jamming in soft-sphere systems.10However, the wide spectrum of density\nvalues [e.g., φ∈(0.63,0.74048...)] that has been achieved clearly suggests that conclusions\ndrawn from any particular protocol are highly specific rather than general and thus claims\nof uniqueness of packing states based on their frequency of occu rrence overlook the wide\nvariability of packing algorithms and the distribution of configuration s that they generate.\nWe will elaborate on this issue in the Conclusions and Discussion (Sec. I V).\nIn this paper, we show explicitly that exploring algorithmic variability of packing pro-\ntocols can lead to a diversity of density and disorder of jammed sphe re packings, which is\nconsistent with the geometric-structure approach. In particula r, we demonstrate that collec-\ntively jammed packings can be generated with a narrow distribution c entered at any density\nφover a wide range φ∈[0.6,0.74048...]. A novel sequential linear programming (SLP)\npacking algorithm20is used to produce jammed disordered packings with φas low as 0.6\nfor the first time, i.e., the onsetof disordered jamming occurs well below the MRJ density\nof about 0.64. The Lubachevsky-Stillinger (LS) packing algorithm3is employed to produce\n5jammed packings with φspanning continuously from that of MRJ state ( φ≈0.64) all the\nway up to the face-centered-cubic (fcc) close-packed density ( φ= 0.74048...).7–9However,\nthe standard LS algorithm as well as all previously used numerical pr otocols do not pro-\nduce disordered collectively jammed states with φwell below 0.64. We control the jamming\ndensity by tuning certain parameters of the packing protocols. Th e jammed packings with\na diversity of disorder are mapped onto a density-order-metric pla ne. Our results strongly\nsupport the view that there is no universal jamming point that is dist inguishable based on\nthe packing density and its frequency of occurence. Other packin g characteristics, such as\nthe pair correlation function g2, average contact number Z, and fraction of rattlers frare\nquantified and discussed.\nII. PACKING PROTOCOLS AND CONTROL PARAMETERS\nA. Lubachevsky-Stillinger Algorithm\nThe LS algorithm is an event-driven molecular dynamics in which particle s can grow in\nsize at a certain expansion rate γ=1\n2dD/dt(Dis the diameter of the sphere) in addition to\ntheir thermal motion.3Note thatγis a key parameter that controls the jamming density.7,11\nSufficiently small γenables the system almost to be in equilibrium during the densification,\nwhich will finally crystallize in three dimensions into a fcc packing. For lar geγ, the system\nwill quickly fall out of equilibrium and reaches a jammed state with anam orphous structure.\nIntermediate γwill result in various degrees of partial crystallization in the system, which\nleads to a continuous range of jamming densities. It is noteworthy t hat a very small value\nofγshould be used toward the jamming limit such that a true particle cont act network can\nbe formed for both crystalline and disordered packings.13,21We use a modified version of the\nLS alogorithm22to generate jammed packings for φ≥0.64.\nB. Sequential-Linear-Programming Algorithm\nA recently devised SLP algorithm20is used here to produce disordered jammed sphere\npackings with φ<0.64. This algorithm solves an optimization problem called the adaptive\nshrinking cell (ASC) scheme:16,17jammed particle packings are generated by maximizing\nthe packing density subject to interparticle nonoverlapping const raints. The optimization\n6variables include particle positions as well as shape and size of the simu lation box. Starting\nfrom an initial configuration, a new configuration is obtained by (loca lly) maximizing the\ndensity via both individual particle motions and collective motions induc ed by the defor-\nmation/shrinkage of the simulation box. For spheres, the objectiv e function and constraints\ncan be linearized for a given packing configuration and the SLP metho d is used to solve\nthe optimization problem. The final packings produced by the SLP ar e at least collectively\njammed due to its incorporation of inherent collective particle motion s. Details of this al-\ngorithm and its applications for generating both disordered and max imally dense packings\nin high dimensional Euclidean space are given in Ref. 20.\nByremoving spheres fromthefccpacking anditsstacking variants without destroying the\nrigidity of the contact network, Torquato and Stillinger15have constructed strictly jammed\n“tunneled crystal” packings with φas low as 0.49365.... Similar removal procedures can be\nappliedtoMRJpackings. However thecontactnetwork oftherema ining packing isgenerally\nnot rigid anymore. The removed particles are required to have a larg e coordination number\nandto bemutually separated by at least a few sphere diameters. Co mpressing the remaining\npacking using the SLP algorithm leads to a re-jammed configuration, with a minimum\ndegree of structural relaxation (i.e., significant particle reconfigu rations are localized around\nthe cavities).20The re-jammed configurations generally possess φ <0.64 and the removal-\ncompression procedure can be repeated several times until a lowe r limit ofφis reached.\nThe control parameters of the SLP algorithm include the initial MRJ c onfigurations and the\nnumber of removed particles.\nIII. JAMMED PACKINGS WITH VARIABLE DENSITY AND DISORDER\nA. Histograms for Jammed Packings with φ >0.64\nWe employ the LS algorithm to generate a large number of jammed pac kings of monodis-\nperse spheres in three dimensions for φ≥0.64 withN= 250,500,1000,2500,5000, and\n10000, and a wide range of initial expansion rate γ∈[10−2,10−6]. Results are verified to be\natleast collectively jammed,21using either alinearprogramming protocol21orbymonitoring\nthe instantaneous pressure of the systems for long time periods;11and rattlers are included\nto compute the reported densities. By tuning γ, the density at which the systems jam can\n7be controlled. Figure 1(b), (c) and (d) contrast distributions of φfor two distinctly different\nsystem sizes ( N= 250 and 2500) converging onto φ≈0.64,0.68 and 0.72, respectively. (The\ncase ofφ≈0.60 shown in Fig. 1(a) will be discussed separately below.) It is clear tha t asN\nincreases the φdistributions narrow for all three mean density values. We also find t hat such\nnarrowing becomes even more significant for N= 5000 and 10000. Specifically, Fig. 2 shows\nhow the standard deviation σof the density distribution varies with system size Nfor the\ncase in which the mean φ≈0.66. Observe that σis a monotonically decreasing function of\nNandσapproximately scales N−1/2for largeN. A similar narrowing of the φdistribution\noccurs forφ≈0.60 shown in Fig. 1(a), and this case will be discussed separately below .\nThus, onecan expect that in the “thermodynamic” limit (i.e., N→ ∞) the jamming density\nwill converge to a well-defined value anywhere over the interval φ∈[0.60,0.74048...]. These\nresults imply that any inclination to select a specific φvalue as uniquely significant (e.g.,\nφ≈0.64) is primarily based on inadequate sampling of the full range of algor ithmic richness\nand diversity that is available at least in the underlying mathematical t heory of jamming.\nB. Histograms for Jammed Packings with φ <0.64\nWe produce the majority of jammed packings with φ <0.64 using the SLP algorithm.\nMRJ packings generated via the LS algorithm are used as initial config urations. Each time\napproximately fs= 0.1%−2.5% of the spheres are removed from the initial packing and\nthe remaining spheres are compressed to a jammed state using the SLP algorithm. Such a\nprocedure is repeated nr= 5−10 times before a lower limit on φis reached. The fraction\nfsof removed spheres is decreased as the limit is approached. We stre ss that our packings\nwithφ <0.64 are not so-called random loose packings ,23which are not even collectively\njammed.21A few packings with φ≈0.62 are generated using the LS algorithm with open\nsimple-cubic lattice packings as initial configurations.9\nFigure 1(a) contrasts φdistributions for two distinctly different system sizes ( N= 216\nand 2235) converging onto φ≈0.60. It can be seen that as Nincreases the φdistributions\nnarrow. Such narrowing is also observed for other convergence d ensities within the interval\n[0.6,0.64]. We note that it is very difficult to produce jammed packings with φsignificantly\nlower than 0.6 using the SLP algorithm (though removing the rattlers in the packing results\nin a slightly lower density ∼0.595), while it has been rigorously shown that strictly jammed\n8sphere packings (e.g., tunneled crystals and the associated stack ing variants) can possess φ\nas low as√\n2π\n9= 0.49365....15However, jammed packings that combine the tunneled crystals\nandMRJpackingscanbeconstructed. Inparticular, stackingsof layersofhoneycomb-lattice\npackings of spheres stabilized with triangular-lattice layers on top a nd bottom are inserted\ninto the MRJ packings. These “layered” packings are then compres sed to jamming using\nthe SLP algorithm. This construction enables one to obtain jammed p ackings with variable\ndisorder within the density range φ∈(0.49,0.64).\nC. Order Metrics and Other Packing Characteristics for Jammed Packing with\nφ∈[0.49,0.74]\nIt has been established that a variety of different useful order me trics are positively\ncorrelated7–9and hence any one of them can be used to characterize the packing s. To\nquantify the order of the packings, we compute the translational order metric T,7defined as\nT=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleNs/summationdisplay\ni(ni−nideal\ni)/Ns/summationdisplay\ni(nFCC\ni−nideal\ni)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (1)\nwhereniis the average occupation number for the shell icentered at a distance from a\nreference sphere that equals the ith nearest-neighbor separation for the open FCC lattice at\nthat density and Nsis the total number of shells for the summation ( Ns= 45 is used here\nforN∼2500);nideal\niandnFCC\niare the corresponding shell occupation numbers for an ideal\ngas (spatially uncorrelated spheres) and the open FCC lattice. For a completely disordered\nsystem (e.g., a Poisson distribution of points) T= 0, whereas T= 1 for the FCC lattice.\nFigure 3 shows the φ-Tplane on which representative jammed packings with N∼2500\nandφ∈[0.49,0.74] are mapped. The packings generated using the LS algorithm are\nshown as black circles. Note that for these packings each φis associated with a range of T\nvalues. Moreover, increasing φfrom the MRJ state can be achieved at the cost of decreasing\nthe degree of disorder, as pointed out in Ref. 7. Further increase ofφis associated with\npartial crystallization of the packings, as indicated by the sharp pe aks of the pair correlation\nfunctiong2of the packings (see Fig. 4). The fraction of “rattlers” (i.e., locally u njammed\nindividual particles that can move freely within cages of jammed neigh bors) decreases (from\n∼2.8% withφ≈0.64 to 0% with φ= 0.74048...) and the average contact number per\nparticleZincreases (from6 to 12)as thedensity increases due to the(part ial) crystallization\n9of the packings, and we have Z≈6.00,6.45 and 7.96 respectively for densities φ≈0.64,0.68\nand 0.72 (rattlers are excluded when computing Z).\nSeveral representative packings obtained via the SLP algorithm ar e also mapped onto the\nφ-Tplane (red squares in Fig. 3).24It can be seen that jamming at φ<0.64 is necessarily\nassociated with an increase in the degree of order, which is also indica ted by the increase\nof average contact number per particle (i.e., Z≈6.37 forφ≈0.6) and the decrease of\nthe fraction of rattlers (i.e., ∼1.1% forφ≈0.6). The pair correlation function g2of a\nrepresentative packing configuration (with N= 2235) is shown in Fig. 4.\nThe “tunneled crystal” packings are mapped onto the φ−Tplane (green up-triangles).\nThe FCC tunneled crystal possesses the highest translational or der metric T, the ”disor-\ndered” Barlow tunneled crystal (a random stacking of layers of ho neycomb-lattice sphere\npackings)possessesthelowest T,andthe“zig-zag”tunneledcrystal(analogofthehexagonal-\nclose sphere packing)15is in between. The dashed blue lines show the spectrum of packings\ngeneratedbyrandomlyfillingthevacanciesinthecorresponding“tu nneledcrystal”packings,\nleading to perfect FCC, HCP and ”disordered” Barlow packings at th e maximal density.13\nSeveral representative layered packings are also mapped onto th eφ-Tplane (purple dia-\nmonds) in Fig. 3. The simple-cubic ( φ= 0.5235..., blue left-triangle) and body-centered-\ncubic (φ= 0.6801..., orange right-triangle) packings are also shown in Fig. 3. Note the S C\nand BCC packings are not collectively jammed.\nIV. CONCLUSIONS AND DISCUSSION\nIn summary, we have shown that jammed sphere packings can be pr oduced with a narrow\ndistribution centered at any φover a wide range φ∈[0.6,0.74048...] with a diversity\nof disorder, by exploring the algorithmic variability of packing protoc ols. This suggests\nthat any temptation to select a specific φvalue as uniquely significant or universal (e.g.,\nφ≈0.64)based on its frequence of occurrence is primarily due to an inadequate sampling\nof the full range of algorithmic richness and variability of packing pro tocols. Our packings\nare characterized by a “geometric-structure” approach, i.e., th ey are mapped onto the φ-T\nplane; and we have shown that moving away from the MRJ density in bo th directions (i.e.,\nlower or higher φ) leads to a higher degree of order and a larger average contact nu mber.\nIt is claimed that the protocol (e.g., the athermal relaxation metho d used in Ref. 10)\n10in which the jammed states of soft spheres are weighted by the volu me of their basins of\nattraction has a clear interpretation in terms of the energy landsc ape, e.g., an ensemble of\njammed states is considered. In fact, the conceptual framewor k of such a protocol (e.g.,\nthe energy landscape picture and the ensemble of inherent struct ures) was introduced by\none of us over 40 years ago.25–27Although this approach is well-defined and theoretically\npossible, it is inevitable that all protocols sample the energy landscap e in their own biased\nfashion. Therefore, there is no compelling mathematical criterion f or selecting one protocol\nover the others. For example, from completely random initial config urations (i.e., Poisson\ndistributionofpoints), itisfoundthejammingdensity sharplypeaks atφ≈0.64.10However,\nit was shown nearly a decade ago that a much wider density range can be achieved for both\nmonodisperse and polydisperse sphere packings using the Lubache vsky-Stillinger packing\nalgorithm;7–9,28and recently similar results for polydisperse spheres have been obt ained18\nusing the same athermal protocol as in Ref. 10. More recently, we have devised a novel\n(athermal) linear-programming packing protocol that enables one to obtain a spectrum\nof inherent structures ranging from disordered jammed packings up to the maximal density\npackings starting from random initial configurations .20All of these results suggest that there\nis no “universal” protocol that can generate all possible jammed st ates in an unbiased\nfashion.\nMoreover, it isnot clear howthe jammedstates should beweighted in anensemble. It has\nbeen suggested that the volume of the basins of attraction for sy stems starting from Poisson\ndistributions of interacting points10should be used for the weighting. However, in reality\nvirtually no jammed systems are experimentally produced using a Pois son distribution as\nan initial configuration. In addition, the jammed packings produced either numerically or\nexperimentally always contain a small but different fraction of rattle rs. Strictly speaking,\nthese jammed states are not single points on the energy landscape but bounded regions with\ndimensions equal to the number of degrees of freedom of the ratt lers. The dimension of\nthe jamming basins are different, and therefore it is ambiguous to co nsider their volumes\nfor weighting and to characterize them on the same footing. Ideally , each jamming basin\nshould be characterized individually as emphasized in the “geometric- structure” approach.\nRemoving the rattlers will not affect the jamming nature of the pack ings, but leads to the\nlower jamming density. This further adds ambiguity to a density-alon e characterization of\njammed packings, which is employed in the “ensemble” approach.\n11We also would like to note that as exercised the “ensemble” approach has focused on\nprotocols that disproportionately generate disordered packings , presumably because such\npackings have a high frequency of occurrence in typical experimen ts or simulations. The\nspecific protocols employed discriminate against the ordered struc tures such as the FCC\npacking (and its stacking variants) and the tunneled crystals15and result in an inappropri-\nate narrow vision of the set of all possible collectively jammed packing s. Thus, the physical\nrelevance of jammed packings should not be determined based on th eir frequency of occur-\nrence in experiments or simulations, which again are protocol depen dent.\nFromourpointofview, the“ensemble” and“geometric-structure ” approachesdonotcon-\nflict with each other but rather are complementary. For example, t he “geometric-structure”\napproachcharacterizes individual packing configurationsdrawnf romwell-defined ensembles.\nHowever, in our experience, the “geometric-structure” approa ch can always provide nontriv-\nial solutions when the “ensemble” approach breaks down, such as t he possible identification\nof disordered jammed two-dimensional packings discussed in the In troduction. Therefore,\nwe believe that the “geometric-structure” approach, when utilize d in the broad context of a\nfull set of available protocols, has a capability to discover, integrat e and characterize packing\nstructures that go well beyond the disordered set. We emphasize that all of the aforemen-\ntioned issues have been addressed in the literature.7–9,13,29In this paper, we have examined\nthe applications of the “geometric-structure” approach in a broa der context by amplifying\nand quantifying some of these issues.\nAn interesting open question that naturally arises is whether φ≈0.60 is the lower\nlimit on the density of jammed amorphous sphere packings? We note t hat the distribution\nof density for N∼2500 shown in Fig. 1(a) can be fitted with a Gaussian with mean\nφ= 0.602 and standard deviation σ= 0.002. This strongly indicates that there is a\nsmall but finite probability of finding jammed packings with even lower d ensities. Given\nenough number of trials, such packings could be obtained in principle. It is noteworthy\nthat the density φ=√\n2π\n9= 0.49365...associated with the “tunneled crystals” is likely to\nbe the threshold (i.e., lowest possible) density for strictly jammed sp here packings in three\ndimensions.15However, neither the LS nor the SLP algorithms as normally implement ed\nis able to produce such packings, presumably because both algorith ms tend to densify the\npacking and jamming is a consequence of their “compression” natur e. Lowerφis attainable\nviatheSLPalgorithmbecausejammingisachieved throughlocalreco nfigurations. However,\n12itiscurrently notdesignedtofindhighlyunsaturatedjammedpackin gs, such asthetunneled\ncrystals. Therefore, we see no reason that jammed amorphous s phere packings with even\nlower density cannot be produced via carefully designed protocols.\nA limitation of current protocols designed to produce jammed packin gs is that they in-\nevitably lead to packings with high densities. It is highly desirable to dev ise protocols that\nexplicitly take into account the requirement of jamming as well as oth er packing charac-\nteristics (e.g., φandZ). One possible approach is to delineate the conditions for jamming\n(and other characteristics) in a quantitative way and to include the m as constraints of an\noptimization problem. We have suggested possible solutions to the pr oblem of producing\nlow-density jammed amorphous packings in Ref. 20. In future rese arch, we will focus on the\ndevelopment of such packing protocols, which is a highly nontrivial ch allenge.\nAcknowledgments\nThisworkwassupportedbytheNationalScience Foundationunder GrantsDMS-0804431\nand DMR-0820341.\n∗Electronic address: torquato@electron.princeton.edu\n1J. D. Bernal and J. Mason, Nature 188, 910 (1960).\n2J. D. Bernal, in Liquids: Structure, Properties, Solid Interactions . Ed. Hughel TJ (Elsevier,\nNew York, 1965) pp. 25-50.\n3B. D. Lubachevsky and F. H. Stillinger, J. Stat. Phys. 60, 561 (1990).\n4R. J. Speedy, J. Chem. Phys. 100, 6684 (1994).\n5S. F. Edwards, in Granular Matter . Ed. Mehta A (Springer-Verlag, New York, 1994).\n6A. J. Liu and S. R. Nagel, Nature 396, 21 (1998).\n7S. Torquato, T. M. Truskett and P. G. Debenedetti, Phys. Rev. Lett.84, 2064 (2000).\n8T. M. Truskett, S. Torquato and P. G. Debenedetti, Phys. Rev. E62, 993 (2000).\n9A. R. Kansal, S. Torquato and F. H. Stillinger, Phys. Rev. E 66, 041109 (2002).\n10C. S. O’Hern, S. A. Langer, A. J. Liu and S. R. Nagel, Phys. Rev. E68, 011306 (2003).\n11A. Donev, F. H. Stillinger and S. Torquato, Phys. Rev. Lett. 95, 090604 (2005).\n1312G. F. Parisi and F. Zamponi, Rev. Mod. Phys. 82, 789 (2010).\n13S. Torquto and F. H. Stillinger, Rev. Mod. Phys. bf 82, 2633 (2 010).\n14S. Torquato and F. H. Stillinger, J. Phys. Chem. B 105, 11849 (2001).\n15S. Torquato and F. H. Stillinger, J. Appl. Phys. 102, 093511 (2007).\n16S. Torquato and Y. Jiao, Nature 460, 876 (2009).\n17S. Torquato and Y. Jiao, Phys. Rev. E 80, 041104 (2009).\n18P.Chaudhuri,L.BerthierandS.Sastry, Phys.Rev.Lett. 104, 165701 (2010). Wenotethatthese\nauthors have explicitly shown that disordered three dimens ional binary soft-sphere packings can\njam within a density range (0 .646,0.662), by tuning the density of initial configurations used\nto produce the jammed packings. This is a relatively narrow d ensity range, but is consistent\nwith our results.\n19M. Hermes and M. Dijkstra, Europhys. Lett 89, 38005 (2010).\n20S. Torquato and Y. Jiao, Phys. Rev. E 82, 061302 (2010).\n21A. Donev, S. Torquato, F. H. Stillinger and R. Connelly, J. Ap pl. Phys. 95, 989 (2004).\n22A. Donev, S. Torquato and F. H. Stillinger, J. Comput. Phys. 202, 737 (2005).\n23G. Y. Onoda and E. G. Liniger, Phys. Rev. Lett. 64, 2727 (1990).\n24Note that we have also computed the “crystal-independent” t ranslational order metric T∗de-\nfined in Ref. 8 for typical packings for 0 .6≤φ≤0.64. As expected, these results for T∗are\npositively correlated with those for the aforementioned “c rystal-dependent” translational order\nmetricTand hence these two translational order metrics are consist ent with one another.\n25F. H. Stillinger, E. A. DiMarzio, and R. L. Kornegay, J. Chem. Phys.40, 1564 (1964).\n26F. H. Stillinger and T. A. Weber, Phys. Rev. A 25, 978 (1982); F. H. Stillinger and T. A. We-\nber, Science 225, 983 (1984). In these papers, the concept of inherent struct ures was formalized\nfor many-particle system interacting via continuous soft p otentials. The set of configurational\npoints that map to the same mechanically stable local energy minimum via a steepest-descent\ntrajectory define uniquely a basin associated with the local miminum called an inherent struc-\nture.\n27F. H. Stillinger and T. A. Weber, J. Chem. Phys. 83, 4767 (1985). In this paper, the concept\nof inherent structures was rigorously generalized to hard- sphere systems by considering the\ninfinite-nlimit of purely repulsive power-law pair potentials 1 /rn.\n28A. R. Kansal, S. Torquato and F. H. Stillinger, J. Chem. Phys. 117, 8212 (2002).\n1429A. Donev, S. Torquato, F. H. Stillinger and R. Connelly, Phys . Rev. E 70, 043301 (2004).\n15FIG. 1: Histograms of jammed sphere packings that are center ed around different mean densities.\nPanel (a) shows packings generated using the SLP algorithm w ithφ≈0.6. Panels (b), (c) and (d)\nshow packings generated using the LS algorithm with φ≈0.64, 0.68 and 0 .72, respectively. The\ndistributions become narrower as the system size increases .\nFIG. 2: The standard deviation σof the density distribution with mean φ≈0.66 as a function of\nsystem size N. System sizes N= 500,1000,2500,5000 and 10000 are used here. The linear fit for\nlnσvs. lnNgives a slope k≈ −0.533, which indicates that σapproximately scales as N−1/2for\nlarge N. Such a σ-Nrelation is also observed for density distributions with th e other mean values\nstudies here.\nFIG. 3: Ordermap for jammed spherepackings with N∼2500: translational order metric Tversus\npacking density φ. Thedashedbluelines(which areconsistent withthequalit ative trendsindicated\nin the order maps discussed in Ref. 13) show the spectrum of pa ckings generated by randomly\nfilling the vacancies in the corresponding tunneled crystal pckings, which leads to perfect FCC,\nHCP and Barlow packings at the maximal density. Thesimple-c ubic (SC) and body-centered-cubic\n(BCC) packings (not jammed) are also shown.\nFIG. 4: Pair correlation function g2andaverage contact number Zofrepresentative jammedsphere\npackings at different densities, where Dis the sphere diameter.\n16(a) (b)\n(c) (d)\nFIG. 1: Jiao, Stillinger, Torquato\n176 7 8 9\nln(N)-5.5-5-4.5-4-3.5-3ln(σ)Standard Deviation Data\nLinear Fit\nFIG. 2: Jiao, Stillinger, Torquato\n180.5 0.6 0.7\nPacking Density00.20.40.60.81Translational Order Metric Tunneled Crystals\nSLP Packings\nLS Packings\nLayered PackingsSC Packing\nBCC Packing\nFIG. 3: Jiao, Stillinger, Torquato\n19123\n φ = 0.639, Z = 6.0\n123g2(r)φ = 0.663, Z = 6.1\n123\nφ = 0.681, Z = 6.4\n0 1 2 3 4\nr/D-1123\nφ = 0.723, Z = 8.1123\n φ = 0.602 , Z = 6.3\nFIG. 4: Jiao, Stillinger, Torquato\n20"    },    {        "title": "1101.2148v1.On_the_Furstenberg_measure_and_density_of_states_for_the_Anderson_Bernoulli_model_at_small_disorder.pdf",        "content": "arXiv:1101.2148v1  [math.AP]  11 Jan 2011ON THE FURSTENBERG MEASURE AND DENSITY OF STATES\nFOR THE ANDERSON-BERNOULLI MODEL AT SMALL DISORDER\nJ. Bourgain\nAbstract. We establish new results on the dimension of the Furstenberg mea sure and\nthe regularity of the integrated density of states for the Anders on-Bernoulli model at\nsmall disorder.\n§0. Summary\nLetH= ∆ +λVwhere ∆ is the lattice Laplacian on ZandV= (Vn)n∈Zare\nindependent random variables in {1,−1}. We assume |λ|small and restrict the energy\nEoutside a fixed neighborhood of {0,2,−2}. It is shown that the Furstenberg measure\nνEofthecorresponding SL2(R)-cocycle/parenleftbigg\nE−λVn−1\n1 0/parenrightbigg\nhasdimensionatleast γ(λ),\nwhereγ(λ)λ→0−→1. As a consequence, we derive that the integrated density of states\n(IDS)N(E) is H¨ older-regular with exponent at least s(λ)λ→0−→1.\nThe spectral theory of the Anderson-Bernoulli (A-B) model h as been studied by\nvarious authors. It was shown by Halperin (see [S-T]) that fo r fixedλ >0,N(E) is\nnot H¨ older continuous of any order αlarger than\nα0=2log2\nArc cosh (1+ λ). (0.1)\nH¨ older regularity for some α>0 has been established in several papers. In [Ca-K-M],\nlePage’smethodisused. Differentapproaches(includingon eusingthesuper-symmetric\nformalism) appear in the important paper [S-V-W] that relie s on harmonic analysis\nprinciples around the uncertainty principle. In [B1], the a uthor proved H¨ older regu-\nlarity of the IDS using the Figotin-Pastur expansion of the L yapounov exponent and\nmartingale theory. We note that in both [S-V-W] and [B1], the H¨ older exponent α\nremains uniform for λ→0 (in fact, [B1] gives an explicit exponent α(λ)>1\n5+εfor\nλ→0).\nTypeset by AMS-TEX\n1Thus the result in this Note just falls short of establishing the conjectured Lipschitz\nregularity of IDS of the A-B model for small λ. Related is the question whether the\nFurstenberg measure on projective space is absolutely cont inuous when λis small (or\neven better). As pointed out at the end of the paper, a natural approach to these\nproblems is through certain spectral gap properties that do not depend on hyperbol-\nicity. There have been recent advances (cf. [BG1], [BG2], [B 2]), that are based on\nmethods from arithmetic combinatorics. But presently, thi s theory seems to restrictive\nfor an application to A-B-cocycles. It does apply however fo r Schr¨ odinger operators\nwith single site distribution given by a measure of positive dimension.\n§1 Probabilistic inequalities on the Boolean cube\nThe following statement is a consequence of Sperner’s combi natorial Lemma.∗\nLemma 1. Letf=f(ε1,...,ε n)be a real valued function on {1,−1}nand denote\nIj=f|εj=1−f|εj=−1 (1.1)\nthej-influence, which is a function of εj′,j′/\\e}atio\\slash=j.\nAssume that for all j= 1,...,n\nIj≥0 (1.2)\n(i.e.fis monotone increasing) and moreover\nIj≥κ>0onΩj∩Ω′\nj (1.3)\nwhereΩj(resp.Ω′\nj) are subsets of {1,−1}ndepending only on the variables ε1,...,ε j−1\n(resp.εj+1,...,ε n). Then, for any t∈R, we have\nmes[|f−t|<κ]≤1√n+/summationdisplay\nj(2−mesΩj−mesΩ′\nj). (1.4)\nProof.\nDenote\n˜Ω =/intersectiondisplay\n1≤j≤n(Ωj∩Ω′\nj) (1.5)\n∗It was also used in [B1] and [B-K] in the context of the Anderson -Bernoulli model.\n2for which\n1−mes˜Ω≤/summationdisplay\nj(2−mesΩj−mesΩ′\nj). (1.6)\nWe claim that the set [ |f−t|<κ]∩˜Ω does not contain a pair of distinct comparable\nelementsε= (εj)1≤j≤nandε′= (ε′\nj)1≤j≤n. Assume otherwise and ε<ε′, i.e.εj≤ε′\nj\nfor eachj. Then\nf(ε′)−f(ε) =/summationdisplay\n1≤j≤n/parenleftbig\nf(ε1,...,ε j−1,ε′\nj,...,ε′\nn)−f(ε1,...,ε j,ε′\nj+1,...,ε′\nn)/parenrightbig\n=/summationdisplay\n1≤j≤n\nεj/ne}ationslash=ε′\njIj(ε1,...,ε j,ε′\nj+1,...,ε′\nn). (1.7)\nSinceε∈Ωj,ε′∈Ω′\nj, it follows from our assumption on Ω j,Ωj′that\n(ε1,...,ε j,ε′\nj+1,...,ε′\nn)∈Ωj∩Ω′\nj\nand henceIj(ε1,...,ε j,ε′\nj+1,...ε′\nn)≥κby (1.3). In particular, since ε/\\e}atio\\slash=ε′,\n(1.7)≥#{1≤j≤n;εj/\\e}atio\\slash=ε′\nj}κ≥κ\nwhich is however impossible if |f(ε)−t| ≤κand|f(ε′)−t| ≤κ. This establishes the\nclaim.\nTherefore, by Sperner’s lemma on the maximal size of subsets of{1,−1}nnot\ncontaining any pair of distinct comparable elements, we get\nmes(˜Ω∩[|f−t|<κ])/lessorsimilar1√n(1.8)\nand (1.4) follows from (1.6), (1.8). This proves Lemma 1.\nWe will use the following corollary of Lemma 1.\nLemma 2. LetfandIjbe as in Lemma 1 and assume each Ij≥0.\nAssume further κ,δ>0and for each 1≤j <n\nf|εj=1,εj+1=1−f|εj=−1,εj+1=−1≥κforε∈Ωj (1.9)\nwhereΩj⊂ {1,−1}nis a set only depending on the variables εj+2,...,ε nand such\nthat\nmesΩj>1−δ. (1.10)\n3Then, for all t∈R\nmes[|f−t|<κ]/lessorsimilar1√n+nδ. (1.11)\nProof.Assumen= 2meven and write ω= (ε1,ε′\n1,...,ε m,ε′\nm) for the {1,−1}n-\nvariable. With this notation, let Ω jrefer to the set Ω 2j−1.\nPartition\n{1,−1}2m=/uniondisplay\nS⊂{1,...,m}VS\nwith\nVS={ω;εj=ε′\njifj∈Sandεj/\\e}atio\\slash=ε′\njifj/\\e}atio\\slash∈S}. (1.12)\nThus\nmes[|f−t|<κ] =/summationdisplay\nS⊂{1,...,m}mes[VS∩|f−t|<κ]. (1.13)\nFixS⊂ {1,...,m}.\nWe consider fonVSas a function of ( εj)j∈Swith the other variables ( εj,ε′\nj)j∈S\nfixed. Denoting g=g(εj;j∈S)}this function on {1,−1}|S|, we have by our assump-\ntion (1.9), for j∈S\nIj(g)(εj,j∈S) =f(ε1,ε′\n1,...,ε j−1,ε′\nj−1,1,1,εj+1,ε′\nj+1,...,ε n,ε′\nn)\n−f(ε1,ε′\n1,...,ε j−1,ε′\nj−1,−1,−1,εj+1,...,ε′\nn)\n≥κ\nprovided\n(εk)k∈S∈Ω′\nj={(εk)k∈S;/parenleftbig\n(εk,εk)k∈S,(εk,ε′\nk)k/ne}ationslash∈S)∈Ωj}\n= (Ωj∩VS) (εk,ε′\nk;k/\\e}atio\\slash∈S)⊂ {1,−1}|S|\nhence only depending on ( εk)k∈S,k>j, (recall that we fixed the variables outside S).\nApplying Lemma 1 to the function g(with Ω j={1,−1}|S|for allj∈S), we obtain\n#[ω∈VS;|f(ω)−t|<κ]≤#VS\n|S|1/2+/summationdisplay\nj∈S/summationdisplay\nεk/ne}ationslash=ε′\nk,k/ne}ationslash∈S/parenleftbig\n2|S|−#(Ωj∩VS)(εk,ε′\nk;k/\\e}atio\\slash∈S)/parenrightbig\n=#VS\n|S|1\n2+/summationdisplay\nj∈S#(VS\\Ωj). (1.14)\n4Summing over S⊂ {1,...,m}gives\n(1.13)≤2−m/summationdisplay\nS⊂{1,...,m}1\n|S|1\n2+n/summationdisplay\nj=1mes(Ω\\Ωj)\n/lessorsimilar/parenleftBigm\n2/parenrightBig−1\n2+nδ (1.15)\nand hence (1.11).\n§2. Application to the Anderson-Bernoulli model\nConsider the projective action of SL2(R) onP1(R)≃T=R/Z, defined for g=/parenleftbigg\na b\nc d/parenrightbigg\n∈SL2(R) by\neiτg(θ)=(acosθ+bsinθ)+i(ccosθ+dsinθ)\n[(acosθ+bsinθ)2+(ccosθ+dsinθ)2]1/2. (2.1)\nHence\n(τg)′(θ) =sin2τg(θ)\n(ccosθ+dsinθ)2=1\n[(acosθ+bsinθ)2+(ccosθ+dsinθ)2]1/2(2.2)\nand\n/bardblg/bardbl2≥(τg)′≥1\n/bardblg/bardbl2. (2.3)\nConsider the Anderson-Bernoulli model (A-B model)\nHλ(ε) =λεnδnn′+∆ (2.4)\nwithε= (εn)n∈Z∈ {1,−1}Zat small disorder λ >0 (∆ stands for the usual lattice\nLaplacian).\nThe corresponding transfer operators MN(E)∈SL2(R) are given by\nMN=MN(E;ε) =/parenleftbigg\nE−λεn−1\n1 0/parenrightbigg/parenleftbigg\nE−λεN−1−1\n1 0/parenrightbigg\n···/parenleftbigg\nE−λε1−1\n1 0/parenrightbigg\n=1/productdisplay\nNgE(εj). (2.5)\n5Considering εj(1≤j≤N) as a continuous variable on [ −1,1], and∂jthe correspond-\ning partial derivative, we have for the projective action\nτMN=τgE(εN)···gE(εj+1)oτgE(εj)oτgE(εj−1)···gE(ε1)\nand\n(∂jτMN)(θ) =τ′\ngE(εN)···gE(εj+1)(τgE(εj)···gE(ε1)(θ)).(∂jτgE)(τgE(εj−1)···gE(ε1)).(2.6)\nSince\ncotgτgE(ε)(θ) = (E−λε)−sinθ\ncosθ(2.7)\nwe have\n(∂ετgE)(θ) =λ.sin2τgE(ε)(θ) =λcos2θ\ncos2θ+((E−λε)cosθ−sinθ)2∼λcos2θ.(2.8)\nFrom (2.3), (2.6), (2.8)\n(∂jτMN)(θ)/greaterorsimilarλ\n/bardblgE(εN)···gE(εj+1)/bardbl2cos2τgE(εj−1)···gE(ε1)(θ)\n=λ\n/bardblMN−j(E;εj+1,...,ε N)/bardbl2cos2τMj−1(E,ε)(θ). (2.9)\nIn order to deal with the issue of cos τMj−1(E,ε)(θ) being small, note that by (2.7), for\nallθ,ε\n|cosθ|+|cosτgE(ε)(θ)|>c. (2.10)\nHence (2.9) implies\n(∂jτMN)(θ)+(∂j+1τMN)(θ)/greaterorsimilarλ\n/bardblMN−j(E;εj+1,...,εN)/bardbl2(2.11)\n(for allθ).\nIn order to fulfill condition (1.9), we need an upperbound on /bardblMn(E;ε1,...,ε n)/bardbl.\nThis function can be analyzed using the Figotin-Pastur expa nsion.\nDenote\nE= 2cosκ(0≤κ≤π) (2.12)\nVn=−εn\nsinκ(2.13)\n6where we assume δ0<|E|<2−δ0and henceκstays away from 0 ,π\n2,π(hereδ0will\nbe a fixed constant independent of λ).\nThe Figotin-Pastur formula gives\n1\nNlog/bardblMN(E,ε)/bardbl=1\n2NN/summationdisplay\n1log/parenleftbig\n1+λVnsin2(ϕn+κ)+λ2V2\nnsin2(ϕn+κ)/parenrightbig\n(2.14)\nwith\nζn=e2iϕn(2.15)\nrecursively given by\nζn+1=µζn+iλ\n2Vn(µζn−1)2\n1−iλ\n2Vn(µζn−1)(2.16)\nand\nµ=e2iκ. (2.17)\nNote that by (2.13), (2.16), ζnonly depends on εn′forn′≤n−1.\nExpanding (2.14), we obtain\n(2.14) =λ2\n8NN/summationdisplay\n1V2\nn (2.18)\n+λ\n2NN/summationdisplay\n1Vnsin(ϕn+κ) (2.19)\n−λ2\n4NN/summationdisplay\n1V2\nncos2(ϕn+κ) (2.20)\n+λ2\n8NN/summationdisplay\n1V2\nncos4(ϕn+κ) (2.21)\n+0(λ3)\nand\n(2.18) =λ2\n8sin2κ=λ2\n2(4−E2). (2.22)\nBy (2.16), (2.17)\n|1−µ|/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\n1ζn/vextendsingle/vextendsingle/vextendsingle<1+0(λN)\n/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\n1ξn/vextendsingle/vextendsingle/vextendsingle<0(λN)\nsin2κ(2.23)\n7and similarly\n/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\n1ζ2\nn/vextendsingle/vextendsingle/vextendsingle<0(λN)\nsin22��<0(λN). (2.24)\nWriting cos2( ϕn+κ) =1\n2(µζn+¯µ¯ζn),cos4(ϕn+κ) =1\n2(µ2ζ2\nn+¯µ2¯ζ2\nn), (2.23), (2.24)\nimply\n(2.20),(2.21)= 0(λ3). (2.25)\nWrite\n(2.19) =−λ\n2NsinκN/summationdisplay\n1εnsin(ϕn+κ)\n=−λ\n2NsinκN/summationdisplay\n1εndn(εn′;n′<n) (2.26)\nwhich is a martingale difference sequence, with\nN/summationdisplay\n1|dn|2=N/summationdisplay\n1sin22(ϕn+κ)<N\n2+1\n2/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\n1µ2ζ2\nn+ ¯µ2¯ζ2\nn/vextendsingle/vextendsingle/vextendsingle\n</parenleftBig1\n2+0(λ)/parenrightBig\nN. (2.27)\nIn conclusion\n1\nNlog/bardblMN(E;ε)/bardbl=λ2\n8sin2κ−λ\n2NsinκN/summationdisplay\n1εndn+0(λ3) (2.28)\nand the Lyapounov exponent\nL(E) =λ2\n8sin2κ+0(λ3). (2.29)\nFrom martingale theory and (2.28), we get for a>0 the large deviation inequality\nmes/bracketleftBig\nε/vextendsingle/vextendsingle/vextendsingle1\nNlog/bardblMN(E;ε)/bardbl−L(E)/vextendsingle/vextendsingle/vextendsingle>aL(E)/bracketrightBig\n<e−/parenleftbig\na2λ2\n16sin2κ+0(λ3)/parenrightbig\nN\n<e−/parenleftbig\na2\n2L(E)+0(λ3)/parenrightbig\nN.\n(2.30)\n8In particular, taking a>2 (andλsmall)\nmes[ε|log/bardblMN(E;ε)/bardbl>aNλ2]<e−ca2λ2N. (2.31)\nReturning to (2.11), take\nN∼λ−2. (2.32)\nFor 1≤j≤N, it follows from (2.31) that\nmes[(εj+1,...,ε N);/bardblMN−j(E;εj+1,...,ε N)/bardbl>eC1(logN)1\n2]≤\nexp{[−cC2\n1λ2logN\n(λ2(N−j))2+0(λ3)](N−j)}<e[−cC2\n1logN+0(λ)]<N−C1.\n(2.33)\nRecalling (2.11), we see that Lemma 2 may be applied to the fun ction\nf=τMN(E;ε)(θ) ofε∈ {1,−1}Nwithκ∼λe−2C1(logN)1/2andδ<N−C1<N−10,\nfor a suitable choice of the constant C1.\nHence, we proved\nLemma 3. Forλsmall andN∼λ−2, we have for fixed δ0<|E|<2−δ0andθ∈T\narbitrary, the distributional inequality\nmes[ε;|τMN(ε)(θ)−t|<λe−C|logλ|1\n2]≤Cλ (2.34)\nfor allt(whereCis some constant).\n§3. Dimension of the Furstenberg measure\nFixingEas above, denote νE=νthe Furstenberg measure on Tfor the random\nwalk associated with the probability measure on SL2(R)\nµ=1\n2δ/parenleftBiggE−λ−1\n1 0/parenrightBigg+1\n2δ/parenleftBiggE+λ−1\n1 0/parenrightBigg. (3.1)\nThus for all N\n/integraldisplay\nSL2(R)ϕ(g)µ(N)(dg) =/integraldisplay\n{1,−1}Nϕ/parenleftbig\nMN(E,ε)/parenrightbig\ndε. (3.2)\n9The measure νisµ-stationary i.e.\nν=/integraldisplay\n(τg)∗[ν]µ(dg) (3.3)\nand\n/a\\}bracketle{tν,f/a\\}bracketri}ht= lim\nN→∞/integraldisplay\nf/parenleftbig\nτMN(ε)(θ)/parenrightbig\ndε (3.4)\nfor allf∈C(T) andθ∈T.\nOur goal is to show that for small λ, the dimension of νEis close to 1.\nThe main inequality is the following\nLemma 4. Leth∈SL2(R)be arbitrary such that\n/bardblh/bardbl ∼λ−1\n10.\nLetN∼λ−1andI⊂Tan arbitrary interval of size |I|<λ. Then\n/integraldisplay\n(τMN(ε)h)∗[ν](I)dε≤\neC|logλ|1\n2/braceleftbig\nmax\n|J|<λ1/10|I|ν(J)+λ1\n30max\n|J|≤|I|ν(J)+ max\nλ−1\n10<D<λ−1\n51\nDmax\n|J|<D.|I|ν(J)/bracerightbig\n(3.5)\nwithJdenoting an interval.\nProof.Write\n/integraldisplay/parenleftbig\nMN(ε)h/parenrightbig\n∗[ν](I)dε=/summationdisplay\n0≤k/lessorsimilarN/integraldisplay\n[/bardblMN(ε)/bardbl∼2k]ν(τh−1τM(ε)−1(I)/parenrightbig\ndε.(3.6)\nFrom (2.31)\nmes[/bardblMN(ε)/bardbl ∼2k]<e−ck2(3.7)\nand, if/bardblMN(ε)/bardbl ∼2k,τh−1τMN(ε)−1(I) is contained in an interval J∈Tof size at\nmost/bardblh/bardbl24k|I|. Thus the kthsummands in (3.6) is certainly bounded by\ne−ck2max\n|J|<4k/bardblh/bardbl2|I|ν(J). (3.8)\nNext, restrict k/lessorsimilar(logN)1\n2andεto [/bardblMN(ε)/bardbl ∼2k].\n10LetR1,...,R Mbe a partition of Tin intervals of size1\nM∼λ. Estimate\n/integraldisplay\n[/bardblMN(ε)/bardbl∼2k]ν/parenleftbig\nτh−1τMN(ε)−1(I)/parenrightbig\ndε≤\nM/summationdisplay\nm=1/integraldisplay\n[/bardblMN(ε)/bardbl∼2k]ν/parenleftbig\nτh−1(τMN(ε)−1(I)∩RM)/parenrightbig\ndε\n≤M/summationdisplay\nm=1mes[ε;/bardblMN(ε)/bardbl ∼2kandτMN(ε)(Rm)∩I/\\e}atio\\slash=φ]/bracketleftbigg\nmaxν(J)\n|J| ≤4kDm|I|/bracketrightbigg\n(3.9)\ndenoting\nDm= max\nθ∈Rm|τ′\nh−1(θ)|. (3.10)\nFixing some θm∈Rmandψ∈I,τMN(ε)(Rm) is contained in an 4k1\nM-neighborhood\nofτMN(ε)(θm) and hence\n|τMN(ε)(θm)−ψ|/lessorsimilar4k\nM+|I|/lessorsimilar4k\nM(3.11)\nsinceτMN(ε)(Rm)∩I/\\e}atio\\slash=φ. In view of Lemma 3\nmes[ε;(3.11)]<1\nM4keC(logN)1/2\nby (2.34) and a suitable partition of the interval [ ψ−4k\nM,ψ+4k\nM]. Hence, for kas above\nmes[ε;/bardblMN(ε)/bardbl ∼2kandτMN(ε)(Rm)∩I/\\e}atio\\slash=φ]<eC(logN)12λ. (3.12)\nLeth−1=/parenleftbigg\na b\nc d/parenrightbigg\n. By (2.2)\nτ′\nh−1(θ) =1\n(acosθ+bsinθ)2+(ccosθ+dsinθ)2\nand1\n/bardblh/bardbl2/lessorsimilarτ′\nh−1(θ)/lessorsimilarmin/parenleftBig1\n/bardblh/bardbl2/bardblθ−θh/bardbl2,/bardblh/bardbl2/parenrightBig\n(3.13)\nfor someθh∈T. Thus\n1\n/bardblh/bardbl2/lessorsimilarDm/lessorsimilarmin/bracketleftBig1\n/bardblh/bardbl2/bardblθm−θh/bardbl2,/bardblh/bardbl2/bracketrightBig\n. (3.14)\n11Hence, given D>0\n#{1≤m≤M;Dm∼D}/lessorsimilar1+M\n/bardblh/bardblD1/2. (3.15)\nFrom (3.12), (3.15), we obtain the following estimate on (3. 9)\n(3.9)<eC(logN)1/2λ(logN)/braceleftBig\nmax\n/bardblh/bardbl−2<D</bardblh/bardbl2M\n/bardblh/bardblD1/2/parenleftbig\nmax\n|J|<4kD|I|ν(J)/parenrightbig/bracerightBig\n<eC′(logN)1/2/parenleftbig\nmax\n/bardblh/bardbl−2<D</bardblh/bardbl21\nD1\n2/bardblh/bardblmax\n|J|<D|I|ν(J)/parenrightbig\n(3.16)\nsincek/lessorsimilar(logN)1/2and writing Jas a union of 4kintervals of size at most D.|I|.\nWe distinguish several contributions\n(i). ForD</bardblh/bardbl−1, estimate (3.16) by\neC′(logN)1\n2max\n|J|<|I|\n/bardblh/bardblν(J)<eC′|logλ|1\n2max\n|J|<λ1\n10|I|ν(J). (3.17)\n(ii) For 1> D >/bardblh/bardbl−1, we haveD1/2/bardblh/bardbl>/bardblh/bardbl1/2/greaterorsimilarλ−1\n20and we may bound\n(3.16) by\nλ1\n30max\n|J|≤|I|ν(J). (3.18)\n(iii) For 1 ≤D≤ /bardblh/bardbl, bound (3.16) by\neC′|logλ|1/2D1\n2\n/bardblh/bardblmax\n|J|≤|I|ν(J)≤λ1\n30max\n|J|≤|I|ν(J). (3.19)\n(iv) For/bardblh/bardbl<D</bardblh/bardbl2, estimate by\neC|logλ|1/2\nDmax\n|J|<D|I|ν(J). (3.20)\nCollecting the contributions (3.17) - (3.20) gives (3.5). T his proves Lemma 4.\nNext,returningto(3.3),writing µ=1\n2δg1+1\n2δg2wemakethefollowingconstruction.\nAssume\nν=/integraldisplay\n(τg)∗[ν]µ1(dg) (3.21)\n12whereµ1is some discrete probability measure SL2(R), such that\n/bardblg/bardbl<2λ−1\n10forg∈suppµ1. (3.22)\nIfg∈suppµ1and/bardblg/bardbl<λ−1\n10, write by (3.3)\n(τg)∗[ν] =1\n2(τgg1)∗[ν]+1\n2(τgg2)∗[ν].\nDefine then\nµ2=/summationdisplay\n/bardblg/bardbl≥λ−1\n10µ1(g)δg+1\n2/summationdisplay\n/bardblg/bardbl<λ−1\n10µ1(g)(δgg1+δgg2)\nstill satisfying (3.21).\nFrom the positivity of the Lyapounov exponent, an iteration of this process will\nclearly produce a discrete probability measure ˜ µonSL2(R) s.t.\nν=/integraldisplay\n(τg)∗[ν]˜µ(dg) (3.23)\nand\nλ−1\n10</bardblg/bardbl<2λ−1\n10forg∈supp˜µ. (3.24)\nTakingN∼λ−2, and since also by (3.2)\nν=/integraldisplay\n(τMN(ε))∗[ν]dε\n(3.23) implies\nν=/integraldisplay/bracketleftBig/integraldisplay\n(τMN(ε)h)∗[ν]dε/bracketrightBig\n˜µ(dh). (3.25)\nFrom(3.25) and Lemma 4, we conclude the following inequalit y.\nLemma 5. ForI⊂Tan interval of size at most λ, we have\nν(I)≤eC|logλ|1\n2/bracketleftBig\nmax\n|J|<λ1\n10|I|ν(J)+ max\nλ−1\n10<D<λ−1\n51\nDmax\n|J|<D|I|ν(J)/bracketrightBig\n.(3.26)\nIf we iterate (3.26) r-times, assuming λ−r\n5|I|<λ, we obtain\nν(I)≤2reC|logλ|1\n2r1\nD1ν(J) (3.27)\n13for some interval Jof size|J|<D1δ1|I|whereD1>1,0<δ1<1 andD1δ−1\n1<λ−r\n10.\nFrom random matrix product theory it is known that the Furste nberg measure ν\nhas some positive dimension α>0. Hence the right side of (3.27) is at most\n/lessorsimilarC|logλ|1\n2r1\nD1|J|α<C|logλ|1\n2rδα\n1\nD1−α\n1|I|α<C|logλ|1\n2rλr\n20min(α,1−α)|I|α.(3.28)\nAssumingγ >0 some constant (independent of λ=o(1)) satisfying\nγ <α< 1−γ (3.29)\n(3.28) and the restriction on rwould imply for λ<λ(γ)\nν(I)<(C|logλ|1\n2λγ\n20)r|I|α<λγr\n30|I|α\nand\nν(I)</parenleftBig|I|\nλ/parenrightBigγ\n6|I|α/lessorsimilar|I|α+γ\n6. (3.30)\nBut (3.30) would give that νhas dimension at least α+γ\n6, a contradiction.\nThus in order to prove\nTheorem 1. Assumingδ0<|E|<2−δ0, the dimension of the Furstenberg measure\nν(λ)\nEfor the A-B model is at least α(λ)λ→0→1.\nIt will suffice to have a uniform lower bound in λfor dimν(λ)\nE.\nThis is what we establish next.\nLemma 6. Under the assumption of Theorem 1, dimν(λ)\nE>γ >0withγindependent\nofλ.\nProof.We make the following observation. Write M=MN(E;ε) as\nM=(v⊥\n−⊗v+)λ++(v⊥\n+⊗v−)λ−1\n+\n/a\\}bracketle{tv+,v⊥\n−/a\\}bracketri}ht(3.31)\nwithv+(resp.v−) the expanding (resp. contracting) direction. Hence\n/bardblM/bardbl ∼|λ+|\n|v+∧v−|. (3.32)\n14For unit vectors u,w∈R2, we deduce from (3.31) that\n/bardblMu/bardbl\n/bardblM/bardbl=/bardbl/a\\}bracketle{tv⊥\n−,u/a\\}bracketri}htv++λ−2\n+/a\\}bracketle{tv⊥\n+,u/a\\}bracketri}htv−/bardbl=\n(1+λ−2\n+)|/a\\}bracketle{tv⊥\n−,u/a\\}bracketri}ht|+0/parenleftBig|v+∧v−|\nλ2\n+/parenrightBig(3.32)\n≤(1+λ−2\n+)|/a\\}bracketle{tv⊥\n−,u/a\\}bracketri}ht|+0/parenleftBig1\n/bardblM/bardbl/parenrightBig\n(3.33)\nand\n|/a\\}bracketle{tMu,w/a\\}bracketri}ht|\n/bardblM/bardbl=|/a\\}bracketle{tv⊥\n−,u/a\\}bracketri}ht/a\\}bracketle{tv+,w/a\\}bracketri}ht+λ−2/a\\}bracketle{tv⊥\n+,u/a\\}bracketri}ht/a\\}bracketle{tv−,w/a\\}bracketri}ht|\n≥(1+λ−2)|/a\\}bracketle{tv⊥\n−,u/a\\}bracketri}ht| |/a\\}bracketle{tv+,w/a\\}bracketri}ht|−2λ−2\n+|v+∧v−|\n≥ |/a\\}bracketle{tv⊥\n−,u/a\\}bracketri}ht| |/a\\}bracketle{tv+,w/a\\}bracketri}ht|+0/parenleftBig1\n/bardblM/bardbl/parenrightBig\n. (3.34)\nHence, given an arc Iof sizeηcentered at ν\nP/bracketleftBig\nε;v−∈Iwherev−is contracting direction of MN(ε)/bracketrightBig\n≤(3.33)\nP/bracketleftBig\nε;/bardblMN(ε)u/bardbl\n/bardblMN(ε)/bardbl<2η+0/parenleftBig1\n/bardblMN(ε)/bardbl/parenrightBig/bracketrightBig\n≤\nP/bracketleftbig\nε;/bardblMN(ε)/bardbl<eλ2\n20N/bracketrightBig\n+P/bracketleftBig\nε;/bardblMN(ε)u/bardbl\n/bardblMN(ε)/bardbl<3η/bracketrightBig\n=\n(3.35)+(3.36)\nprovided\nη>e−λ2\n20N. (3.37)\nRecalling (2.29), (2.30), we have\nmes/bracketleftBig\nε/vextendsingle/vextendsingle/vextendsingle1\nNlog/bardblMN(ε)/bardbl\nL(E)−1/vextendsingle/vextendsingle/vextendsingle>a/bracketrightBig\n<e/parenleftbig\n−a2\n2L(E)+0(λ3)/parenrightbig\nN(3.38)\nwithλ2\n8<L(E)<0(λ2).\nHence,\n(3.35)<e−(1\n50λ2+0(λ3))N<e−1\n60λ2N(3.37)\n< η1\n3 (3.39)\nforλsmall enough.\n15Next, we point out that in the analysis (2.14)-(2.28), the fo rmula (2.28) is equally\nvalid for1\nNlog/bardblMN(E;ε)(u)/bardbl, withu∈S1arbitrary (as a consequence of the argu-\nment). Thus we can write\n1\nNlog/bardblMN(ε)/bardbl=L(E)−λ\n2NsinκN/summationdisplay\n1εndn+0(λ3) (3.40)\nand\n1\nNlog/bardblMN(ε)(u)/bardbl=L(E)−λ\n2NsinκN/summationdisplay\n1εnd′\nn+0(λ3) (3.41)\nso that\nlog/bardblMN(ε)/bardbl\n/bardblMN(ε)(u)/bardbl=λ\n2sinκN/summationdisplay\n1εn(d′\nn−dn)+0(Nλ3) (3.42)\nwheredn,d′\nndepend onε1,...,ε n−1.\nLetting 1>t>0 be a parameter, write\n(3.36)<(3η)t/integraldisplay/parenleftBig/bardblMN(ε)/bardbl\n/bardblMN(ε)(u)/bardbl/parenrightBigt\ndε<(3η)te0(Nλ3t)/integraldisplay\neλt\n2sinκ/summationtextN\n1εn(d′\nn−dn)dε\n<(3η)te0(Nλ3t)eCλ2t2N(3.43)\nwhere the constant Conly depends on E.\nChoosingNs.t.\nη∼e−λ2\n103N(3.44)\nwe satisfy (3.37), and it follows from (3.43) and appropriat e choice of t, that\n(3.36)<(3η)t−C(λt+t2)<ηc1\n(again forλsmall enough) and with c1>0 independent of λ.\nHence, we showed that with Nsatisfying (3.44)\nmes[ε;v−∈Iwherev−is contracting vector of MN(ε)]<ηc1. (3.45)\nSincev+is the contracting vector of MN(ε)−1, we obtain a similar statement for the\nexpanding vector. Therefore, given any pair of η-intervalsI+,I−inS1, we proved that\nmes[ε;v+∈I+,v−∈I−\nwithv+(resp.v−) expanding (resp. contracting) direction of MN(ε)]<2ηc1\n(3.46)\n16forNsatisfying (3.44).\nReturning to (3.34), we have\nP[ε;|/a\\}bracketle{tMN(ε)u,w/a\\}bracketri}ht|\n/bardblMN(ε)/bardbl<η1]≤\nP[ε;/bardblMN(ε)/bardbl<1/η1]+P[ε;|/a\\}bracketle{tv⊥\n−,u/a\\}bracketri}ht|/lessorsimilar√η1]+P[ε;|/a\\}bracketle{tv+,w/a\\}bracketri}ht|/lessorsimilar√η1].\n(3.47)\nTakingη=η1\n2\n1andNas in (3.44), the last 2 terms in (3.37) are at most 0( η1\n2c1\n1) by\n(3.46), while the first term is bounded by mes[ ε;/bardblMN(ε)/bardbl<e1\n500λ2N]<e−1\n60λ2N<η1\nby (3.38).\nHence\n(3.47)/lessorsimilarη1\n2c1\n1withη1∼e−λ2\n500N. (3.48)\nReturning to the Furstenberg measure ν=ν(λ)\nE, we have for I⊂Ta small arc of size\nη1, by (3.4)\nν(I) = lim\nN′→∞P/bracketleftBig\nε|MN′(ε)e1\n/bardblMN′(ε)e1/bardbl∈I/bracketrightBig\n.\nTakeNas in (3.48) and N′>N. Ifwdenotes the center of I, then\n|/a\\}bracketle{tMN′e1,w⊥/a\\}bracketri}ht|\n/bardblMN′e1/bardbl<η1. (3.49)\nFixε1,...,ε N′−Nand letu=MN′−N(ε1,...,εN′−N)(e1)\n/bardblMN′−N1,...,(εN′−N)(e1)/bardbl. We have\nMN′e1\n/bardblMN′e1/bardbl=MN(εN′−N+1,...,ε N′)(u)\n/bardblMN(εN′−N+1,...,ε N′)u/bardbl.\nThus (3.49) implies\n|/a\\}bracketle{tMN(···)u,w⊥/a\\}bracketri}ht|\n/bardblMN(···)/bardbl<η1 (3.50)\nfor which the measure in εN′−N+1,...,ε N′is at mostη1\n2c1\n1by(3.48). Therefore\nν(I)/lessorsimilar|I|1\n2c1. (3.51)\nThis proves that dim ν≥1\n2c1, uniformly in λ. Hence we establish Lemma 6.\nThis also completes the proof of Theorem 1.\n17§4. Density of states\nLetu,w∈S1,η>0 small. It follows from (3.34) that\nlim\nN→∞mes/bracketleftBig\nve;|/a\\}bracketle{tMN(ε)u,w/a\\}bracketri}ht|\n/bardblMN(ε)/bardbl<η/bracketrightBig\n≤\nlim\nN→∞mes/bracketleftBig\nε;|/a\\}bracketle{tv+,w/a\\}bracketri}ht|.|/a\\}bracketle{tv⊥\n−,u/a\\}bracketri}ht|<ηwithv+,v−the eigenvectors of MN(ε)/bracketrightBig\n=\nlim\nN→∞mes/bracketleftBig\n(ε,ε′);|/a\\}bracketle{tv+,w/a\\}bracketri}ht|.|/a\\}bracketle{tv′\n+,u⊥/a\\}bracketri}ht|<ηwithv+(respv′\n+) expanding direction of\nMN(ε), (respMN(ε′))/bracketrightBig\n/lessorsimilarlog1\nη.max\nη1.η2=ηνE(Iη1(w⊥)).νE(Iη2(u))≪ηγ. (4.1)\nwhere we used (3.4) and the independence of v+,v−forN→ ∞as functions of ε.\nHereγ <dimν(λ)\nEandγ=γ(λ)→1 forλ→0.\nIt is easily seen that (4.1) implies that for given K >1 and taking Nlarge enough\n(depending on K)\nmax\nu,w∈S1E/bracketleftBig/bardblMN/bardbl\n|/a\\}bracketle{tMNu,w/a\\}bracketri}ht|∧K/bracketrightBig\n<K1−γ. (4.2)\nHereMN=MN(E) and (4.2) remains clearly valid replacing Ebyz=E+iywith\n0<y<y Nsmall enough (depending on N) and taking for u,wunit vectors in C2.\nNext, take N′>Nand consider\n/bardblM[0,N′](z;ε)/bardbl./bardblM]1N′,2N′](z;ε)/bardbl\n/bardblM[0,2N′](z,ε)/bardbl. (4.3)\nFixingεN′+1,...,ε 2N′, we obtain a unit vector ζ∈C2(depending on these variables)\nsuch that\n(4.3) =/bardblM[0,N′](z,ε)/bardbl\n/bardblM[0,N′](z,ε)(ζ)/bardbl(4.4)\nand\n(4.4)/lessorsimilar/summationdisplay\ni,j=1,2|/a\\}bracketle{tM[0,N′](z,ε)ei,ej/a\\}bracketri}ht|\n|/a\\}bracketle{tM[0,N′′](z,ε)ζ,ej/a\\}bracketri}ht|.\nFixalsoε1,...,ε N′−Nandletζ1beaunitvectorin C2withζ1paralleltoM∗\n[0,N′−N](z,ε)ej.\nHence\n|/a\\}bracketle{tM[0,N′](z,ε)ei,ej/a\\}bracketri}ht|\n|/a\\}bracketle{tM[0,N′](z,ε)ζ,ej/a\\}bracketri}ht|≤/bardblM[N′−N+1,N′](z,ε)/bardbl\n|/a\\}bracketle{tM[N′−N+1,N′](z,ε)ζ,ζ1/a\\}bracketri}ht|(4.5)\n18where the vectors ζ,ζ1do not depend on εN′−N+1,...,ε N′.\nThus\nmin/parenleftbig\n(4.3),K/parenrightbig\n/lessorsimilarmin/parenleftbig\n(4.5),K/parenrightbig\n. (4.6)\nTaking expectation of (4.6) in εN′−N+1,...,ε N′(with other variables fixed), (4.2) and\nsubsequent remark, give an estimate\nEεN′−N+1,...,εN′[(4.6)]/lessorsimilarK1−γ. (4.7)\nHence, also\nE[min/parenleftbig\n(4.3),K/parenrightbig\n]/lessorsimilarK1−γ(4.8)\nvalid forz=E+iywithy>0 small enough (depending on K) andN′>N′(K).\nDenoting Nthe IDS, recall that\n¯∂N(z) =E[G(0,0,z)]z=E+iy\nwhereG(z) = (H−z)−1is the Green’s function and N(z) the harmonic extension of\nNto Imz>0.\nFixz,Imz >0. Then from the resolvent identity and positivity of the Lya pounov\nexponent, we obtain\nG(0,0,z) = lim\nΛ=[−a,b]\na,b→∞GΛ(0,0,z) a.s\nand, by Cramer’s rule\n|G(0,0,z)| ≤limN′→∞/bardblM[−N′,0](z,ε)/bardbl /bardblM[0,N′](z,ε)/bardbl\n/bardblM[−N′,N′](z;ε)/bardbl. (4.9)\nHence by (4.8)\nE[|G(0,0,z)|∧K]≤limN′→∞E[···∧K]/lessorsimilarK1−γ(4.10)\nify>0 is small enough (depending on K). Lettingy→0 we get\nE[|G(0,0,E+io)|∧K]<K1−γ. (4.11)\nIt follows from (4.11) that for 0 <γ1<γ\nE[|G(0,0,E+io)|γ1]R<C. (4.12)\nRecall that we assumed δ0<|E|<2−δ0. Using the subharmonicity of |G(0,0,z)|γ1\non Imz>0, we deduce from (4.12) that for fixed z=E+iy,y>0\n|¯∂N(z)| ≤E[|G(0,0,z)|]<1\ny1−γ1E[|G(0,0,z)|γ1]<C\ny1−γ1.(4.13)\nHenceNisγ1-H¨ older for all γ1<γ.\nThis proves\n19Theorem 2. Forδ0<|E|<2−δ0, the IDS of the A-B model with λ-disorder is\ns-H¨ older regular, with s→1forλ→0.\n§5. Further comments\nIf one aims at going further and prove the Lipschitz regulari ty of the IDS, it seems\nreasonable to prove that the Furstenberg measures on the pro jective line P1(R)≃T\nareatleastabsolutelycontinuous. Thisisfar from anobvio usissue. Infact, itwascon-\njectured in [K-L] that if νis a finitely supported probability measure on SL2(R), then\nits Furstenberg measure on P1(R) is always singular. This conjecture was disproved in\n[BPS] using a probabilistic construction reminiscent of ra ndom Bernoulli-convolutions.\nAnexplicit examplewas givenrecently in[B2], based on a con struction from [B3]/parenleftbig\nthat\nrelies on an extension of the spectral gap theory from [BG1] f orSU(2) toSL2(R)/parenrightbig\n. A\nrough description is as follows. One produces a finite subset G ⊂SL2(R)∩Mat2×2(q),\nqa fixed large integer, such that log(# G)∼logq,Ggenerates freely the free group on\n#Ggenerators and moreover Gis contained in a small neighborhood of the identity\n(depending on q). Denoting\nν=1\n(#G)/summationdisplay\ng∈Gδg (5.1)\nthe probability measure on SL2(R), it is shown that there is a spectral gap for the\nprojective representation ρ, in the following sense. Let f∈L2(T),/bardblf/bardbl2= 1 and\nassumeˆf(n) = 0 for |n|>K, whereK=K(q) is a sufficiently large constant. Then\n1\n(#G)/vextenddouble/vextenddouble/vextenddouble/summationdisplay\ng∈Gρgf/vextenddouble/vextenddouble/vextenddouble\n2<1\n2(5.2)\nwhereρgf= (τ′\ng)−1\n2(f◦τg) andτgthe action on Tdefined for g=/parenleftbigg\na b\nc d/parenrightbigg\nby\neiτg(θ)=(acosθ+bsinθ)+i(ccosθ+dsinθ)\n[(acosθ+bsinθ)2+(ccosθ+dsinθ)2]1\n2. (5.3)\nSinceg∈ Gare close to identity, (5.2) clearly implies that for fas above\n1\n(#G)/vextenddouble/vextenddouble/vextenddouble/summationdisplay\ng∈G(f◦τg)/vextenddouble/vextenddouble/vextenddouble\n2<3\n4. (5.4)\nFrom (5.4), one may then derive easily that νhas an a.c. Furstenberg measure with\nCk-density, where kcan be made arbitrarily large.\n20Itshouldbepointedoutthatthecontractiveproperties(5. 2),(5.4)arenotexploiting\nhyperbolicity (at least in the usual sense), as the Lyapouno v exponent of the random\nmatrix product corresponding to νis small.\nReturning to the A-B-model with small λ, denote\nνλ,E=1\n2δ/parenleftBiggE+λ−1\n1 0/parenrightBigg+1\n2δ/parenleftBiggE−λ−1\n1 0/parenrightBigg (5.5)\nandν(ℓ)\nλ,Eitsℓ-fold convolution. It seems reasonable to believe that\n/vextenddouble/vextenddouble/vextenddouble/summationdisplay\ngν(ℓ)\nλ,E(g)(f◦τg)/vextenddouble/vextenddouble/vextenddouble\n2≤1\n2/bardblf/bardbl2 (5.6)\nforf∈L2(T),ˆf(n) = 0 for |n|>K(λ) and where ℓis some positive integer indepen-\ndent ofλ, or at least ℓ=o(λ−2). Such property would then again imply a.c. and\na certain smoothness of the Furstenberg measure. Unfortuna tely, available technol-\nogy to establish spectral gaps (as developed in [BG1]) so far requires algebraic matrix\nelements of bounded height and hence does not apply to (5.5).\nOne may however combine the methods from [BG1] with those of [ S-T] to prove\nthe following result, which seems new (compare also with the results from [K-S]).\nTheorem 3. Consider a random Schr¨ odinger operator H= ∆+VonZwhereV=\n(Vn)n∈Zarei.i.d’s with distribution given by a compactly supported measure βonR\nof positive dimension. Thus there is κ>0s.t.\nβ(I)/lessorsimilar|I|κforI⊂Ran interval. (5.7)\nThenHhasC∞density of states.\nWe sketch the argument.\nForfixedE,letµEbetheprobabilitymeasureon SL2(R)obtainedasimagemeasure\nofβunder the map\nv/ma√sto→/parenleftbigg\nE−v−1\n1 0/parenrightbigg\n. (5.8)\nFollowing [S-T], it will suffice to show that, for some fixed con volution power ℓ, the\nmeasureµ1=µ(ℓ)\nEonSL2(R) gives a smoothing convolution operator on P1(R). Thus\nthere is some α>0 s.t. forf∈Hs(T),s≥0\n/vextenddouble/vextenddouble/vextenddouble/integraldisplay\n(f◦τg)µ1(dg)/vextenddouble/vextenddouble/vextenddouble\nHs+α/lessorsimilar/bardblf/bardblHs (5.9)\n21(whereH′denotestheusualSobolevspacewithnorm /bardblf/bardblHs=/parenleftbig/summationtext(1+|n|)2s|ˆf(n)|2/parenrightbig1\n2/parenrightbig\n.\nDenotingx=E−v, one has\n/parenleftbigg\nx−1\n1 0/parenrightbigg/parenleftbigg\ny−1\n1 0/parenrightbigg/parenleftbigg\nz−1\n1 0/parenrightbigg\n=/parenleftbigg\nxyz−x−z1−xy\nyz−1−y/parenrightbigg\n(5.10)\nand recalling (5.7), one sees that µ(3)\nEcertainly has the property that\nµ(3)\nE(Sδ)/lessorsimilarδκ′for allδ>0 (5.11)\nifSis a proper algebraic subvariety of SL2(R) of bounded degree and Sδdenotes a\nδ-neighborhood of G. Hereκ′>0 depends on the degree bound.\nLetPδ,δ >0, denote an approximate identity on SL2(R). Using (5.11), an exten-\nsionofthe‘flatteningLemma’from[BG1]to SL2(R)(notethat,uptocomplexification,\nSU(2) andSL2(R) have the same Lie-algebra and our analysis is local), permi ts us to\nconclude the following.\nLemma 7. Fix some 0<ε<1. There isℓ=ℓ(ε)∈Z+s.t. for allδ>0, we have\n/bardblµ(3ℓ)\nE∗Pδ/bardbl∞<δ−ε(5.12)\n(in particular, µ(3ℓ)\nEhas dimension at least 3−ε).\nThis is the crucial step, depending on ‘arithmetic combinat orics’ in groups (see\n[BG1] and related refs for more details).\nTakingε= 10−3andℓ=ℓ(ε) given by Lemma 7, we can now prove that µ1=µ(3ℓ)\nE\nsatisfies (5.9). This will clearly be a consequence of the fol lowing statement.\nLemma 8. Letf∈L2(T),/bardblf/bardbl2= 1and supp ˆf⊂[2k,2k+1]∪[−2k+1,−2k]withk\nsufficiently large. Then/vextenddouble/vextenddouble/vextenddouble/integraldisplay\n(f◦τg)µ1(dg)/vextenddouble/vextenddouble\n2<2−kκ. (5.13)\nfor someκ>0.\nProof of Lemma 8.\nWe summarize the argument from [B2].\n22DenoteG=SL2(R) and takeδ= 4−k, so that, by assumption on f, we may replace\nthe left side of (5.13) by/vextenddouble/vextenddouble/vextenddouble/integraldisplay\n(f◦τg)(µ1∗Pδ)(dg)/vextenddouble/vextenddouble/vextenddouble\n2. (5.14)\nUsing (5.12), one gets\n(5.14)2/lessorsimilarδ−2ε/integraldisplay/integraldisplay\nG×G|/a\\}bracketle{tf◦τg1,f◦τg2/a\\}bracketri}ht|Ω(g1)Ω(g2)dg1dg2\nwith 0≤Ω≤1 a suitable compactly supported function on G(depending on the\nsupport ofβ). Next, by Cauchy-Schwarz\n(5.14)4/lessorsimilarδ−4ε/integraldisplay/integraldisplay/integraldisplay/integraldisplay\nG×G×T×Tf(τg1x)¯f(τg2x)f(τg1y)f(τg2y)Ω(g1)Ω(g2)dg1dg2dxdy.\n(5.15)\nTo estimate (5.15), proceed as follows. Fix x,y∈Tandg1∈Gand consider the\nintegral ing2/integraldisplay\n¯f(τgx)f(τgy)Ω(g)dg. (5.16)\nThe point here is that if one specifies τgx∈T, there remains an average in τgyto\nbe exploited, when integrating in g(unlessxandyare very close). More precisely, if\n/bardblx−y/bardbl<2−h/10, then\n|(5.16)|<2−k/bardblf/bardbl2\n1\nand the contribution in (5.15) is at most\n≤δ−4ε2−k/bardblf/bardbl4\n1<2−k/2.\nThe contribution of /bardblx−y/bardbl<2−k/10in (5.15) is easily estimated by\n/integraldisplay/integraldisplay\n/bardblx−y/bardbl<2−k/10/bracketleftBig/integraldisplay\nG|f(τgx)| |f(τgy)|Ω(g)dg/bracketrightBig2\ndxdy≤\n/integraldisplay/integraldisplay\n/bardblx−y/bardbl<2−k/10/bracketleftBig/integraldisplay\nG|f(τgx)|2Ω(g)dg/bracketrightBig/bracketleftBig/integraldisplay\nG|f(τgy)|2Ω(g)dg/bracketrightBig\ndxdy\n/lessorsimilar2−k/10/bardblf/bardbl4\n2\nand (5.13) follows.\n23References\n[B1]. J. Bourgain, On localization for lattice Schr¨ odinger operators involv ing Bernoulli\nvariables , LNM, 1850, 77–99 (2004).\n[B2]. J. Bourgain, Finitely supported measures on SL2(R)which are absolutely contin-\nuous at infinity , preprint 2010.\n[B3]. J. Bourgain, Expanders and dimensional expansion , C.R. Math. Acad. Sci. Paris\n347 (2009), no 7–8, 357–362.\n[BG1]. J. Bourgain, A. Gamburd, On the spectral gap for finitely-generated subgroups of\nSU(2), Inv. Math. 171 (2008), no 1, 83–121.\n[BG1]. J. Bourgain, A. Gamburd, Spectral gap in SU(d), C.R. Math. Acad. Sci. Paris 348\n(2010), no 11–12, 609–611.\n[B-K]. J. Bourgain, C. Kenig, On localization in the continuous Anderson-Bernoulli\nmodel in higher dimension , Inv. Math. 161 (2005), no 2, 389–426.\n[B-P-S]. B. Barany, M. Pollicott, K. Simon, Stationary measures for projective transforma-\ntions: the Blackwell and Furstenberg measures , preprint 2010.\n[C-K-M]. R. Carmona, A. Klein, F. Martinelli, Anderson localization for Bernoulli and other\nsingular potentials , CMP 108, 41–66 (1987).\n[K-L]. V. Kaimanovich, V. Le Prince, Matrix random products with singular harmonic\nmeasure , Geom. Ded. (2010).\n[K-S]. A. Klein, A. Speis, Regularity of the invariant measure and the density of state s\nin the one-dimensional Anderson model , JFA 88 (1990), no 1, 211–227.\n[S-T]. B. Simon, M. Taylor, Harmonic analysis on SL2(R)and smoothness of the density\nof states in the one-dimensional Anderson model , CMP, 101, no 1 (1985), 1–10.\n[S-V-W]. C. Shubin, T. Vakilian, T. Wolff, Some harmonic analysis questions suggested by\nAnderson-Bernoulli models , GAFA 8 (1988), 932–964.\nInstitute for Advanced Study, Princeton, NJ 08540\nE-mail address :bourgain@ias.edu\n24"    },    {        "title": "1104.0845v1.Determination_of_electron_density_and_filling_factor_for_soft_X_ray_flare_kernels.pdf",        "content": "arXiv:1104.0845v1  [astro-ph.SR]  5 Apr 2011Determination of electron density and filling factor for sof t\nX-ray flare kernels\nJ. Jakimiec, U. B¸ ak-St¸ e´ slicka\nAstronomical Institute, University of Wroc/suppress law, ul. Koper nika 11, 51-622 Wroc/suppress law,\nPoland email: jjakim,bak@astro.uni.wroc.pl\nAbstract. In a standard method of determining electron density for sof t X-ray (SXR)\nflare kernels it is necessary to assume what is the extension o f a kernel along the line of\nsight. This is a source of significant uncertainty of the obta ined densities.\nIn our previous paper (B¸ ak-St¸ e´ slicka and Jakimiec, 2005 ) we have worked out another\nmethod of deriving electron density, in which it is not neces sary to assume what is the\nextension of a kernel along the line of sight. The point is tha t many flares, during their\ndecay phase, evolve along the sequence of steady-state mode ls [quasi-steady-state (QSS)\nevolution] and then the scaling law, derived for steady-sta te models, can be used to\ndetermine the electron density.\nThe aim of the present paper is: (1) to improve the two methods of density determi-\nnation, (2) to compare the densities obtained with the two me thods. We have selected a\nnumber of flares which showed QSS evolution during the decay p hase. For these flares the\nelectron density, N, has been derived by means of standard method and with our QSS\nmethod. Comparison of the Nvalues obtained with the two different methods allowed\nus: (1) to test the obtained densities, (2) to evaluate the vo lume filling factor of the SXR\nemitting plasma.\nGenerally, we have found good agreement (no large systemati c difference) between the\nvalues of electron density obtained with the two methods, bu t for some cases the values\ncan differ by a factor up to 2. For most flare kernels estimated fi lling factor turned out to\nbe about 1, near the flare maximum.\nKeywords: Sun: corona - flares - X-rays\n1. Introduction\nLoop–top flare kernels are the main sources of Soft X-ray (SXR ) emission\nnear flares maximum and during their decay. In order to invest igate energy\nbalance of the kernels it is necessary to have reliable estim ates of plasma\ndensity inside the kernels. A ”standard‘ method of determin ing the electron\nnumber density, N, consists in a few steps:\n(1) Usually, the temperature, T, is estimated from the ratio of SXR fluxes\nmeasured with two different filters:\nf1(¯T)\nf2(¯T)=F1\nF2(1)\nwhereF1andF2are measured X–ray fluxes, fi(T) are response functions\nfor measurements with two filters (i=1,2) and ¯Tis the estimated value\nc/circlecopyrt2018 Springer Science + Business Media. Printed in the USA.\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.12\nof the temperature. In the case of observations with the Reuven Ramaty\nHigh Energy Solar Spectroscopic Imager (RHESSI, Lin et al. , 2002) the\ntemperature is estimated from the slope of SXR spectrum.\nFlare kernels are multithermal (non-isothermal), i.e.\nFi=/integraldisplay\nϕ(T)fi(T)dT (2)\nwhereϕ(T) is differential emission measure (DEM). Therefore obtained\nvalues of ¯Tdepend on instrumental response functions fi(T) and we ob-\ntain different values of ¯Tfrom measurements with different instruments\n(see Table I).\n(2) Next the emission measure is calculated from the formula :\nEM =Fi/fi(¯T) (3)\nIn many cases the response function, fi(T), steeply depends on tempera-\nture. Then random errors of determination of mean temperatu re,¯T, may\ncause significant errors of EM. To improve this standard method of EM\ndetermination, we recommend that observations having flat r esponse\nfunctions (low |dfi/dT|in the investigated range of temperatures) should\nbe used to determine EM from Equation (3) (see Section 2). Then we\nhave:\nFi=/integraldisplay\nϕ(T)fi(T)dT≈¯fi/integraldisplay\nϕ(T)dT=¯fiEM, (4)\ni.e.we will obtain reliable EM from Equation (3), despite that the\nemitting plasma is multithermal.\n(3) The electron number density is estimated as:\nN=/radicalbig\nEM/V, (5)\nwhereVis the volume of emitting plasma. In order to estimate the\nvolumeVit is necessary to assume what is the extension of emitting\nplasma along the line of sight. This introduces significant u ncertainty\ninto estimates of electron density from Equation (5). The un certainty is\nhigh when the shape of a flare kernel is elongated (elliptical ) or irregular,\nsince then it is most difficult to guess what is the extension of emitting\nplasma along the line of sight.\nAdditional drawback of standard method of density determin ation is due\nto the fact that it gives values averaged over whole volume of a kernel. In\nreality, the emitting plasma can have higher density if it oc cupies only\na part of kernel volume, i.e.if its filling factor is smaller than 1.\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.23\nIn our previous paper (B¸ ak-St¸ e´ slicka and Jakimiec (2005 ), hereafter Paper\nI) we have proposed a new method of electron density determin ation in\nwhich (1) it is not necessary to assume what is the extension o f the emitting\nplasma along the line of sight and (2) which gives actual valu es of the electron\ndensity of emitting plasma.\nRosner, Tucker and Vaiana (1978) investigated models of ste ady-state coro-\nnal loops and they have found a “scaling law” of these models w hich can be\nwritten in the form:\nNSS= 2.1×106T2/L (6)\nwhereTis the temperature at the top of a loop, Lis the semilength of the\nloop and NSSis the electron number density within the loop.\nThose authors have found that Equation (6) is also valid for h ot “post-\nflare loops” having temperature up to 10 MK (see their Figure 9 b). This\nlast finding has been explained by Jakimiec et al. (1992) who carried out\nhydrodynamic modeling of the evolution of hot flaring loops. They have\nfound that if the heating rate, EH(t), of a flaring loop decreases slowly during\ndecay phase, the loop evolves along a sequence of steady-sta te models [quasi-\nsteady-state (QSS) evolution] and then Equation (6) is vali d during this\nevolution. Therefore this Equation (6) can be used to determ ine densities N\nduring the QSS evolution. The aim of the present paper is: (1) to improve\nthe two methods of density determination, (2) to compare the densities\nobtained with the two methods. We have selected a number of fla res which\nshowed QSS evolution during the decay phase. For these flares the electron\ndensity,N, has been derived by means of standard method and with our\nQSS method. Comparison of the Nvalues obtained with the two different\nmethods allowed us: (1) to test the obtained densities, (2) t o evaluate the\nvolume filling factor of the SXR emitting plasma.\nObservations and their analysis are described in Section 2. Section 3\ncontains summary and conclusions.\n2. Observations and their analysis\nLong time ago it has been shown that it is useful to display flar e evolution\nin temperature-density diagnostic diagrams (Jakimiec et al. (1987, 1992)).\nIn such diagrams the QSS branch of flare evolution is a straigh t line of\ninclination ζ=dlogT/dlogN ≈0.5. Soft X-ray images show that size of\nloop-top sources does not change significantly during decay phase of many\nflares (i.etheir volume V≈const ). Therefore in Paper I we have found\nthat simplified diagnostic diagrams (where logN is replaced by log√\nEM)\nderived from GOES ( Geostationary Operational Environmental Satellites )\nobservations, are very useful for detection of the QSS evolu tion (see Fig-\nure 1). Correct inclination, ζ≈0.5, of the QSS branch in these diagrams\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.34\nindicates that temperature, TGOES , derived from GOES observations, ad-\nequately represents mean temperature of emitting plasma du ring the QSS\nflare evolution and therefore we should use this temperature in Equation (6).\nIn Paper I we have also found that temperatures TSXT are usually lower\nthanTGOES (SXT is the Yohkoh /Soft X-ray Telescope - see Tsuneta et al. ,\n1991). This is caused by the fact that SXT has narrower spectr al range of\nsensitivity (1–5 keV for Be + Al12 filters; 1 .6–25 keV for GOES). Low values\nofTSXT result in low inclination ( ζ <0.5) of the QSS branch derived with\nthis temperature (see Ko/suppress loma´ nski et al. (2002) and our Paper I).\n2.1.Comparison of the electron densities obtained with two\nindependent methods\nUsingYohkoh /SXT and GOES observations we have selected 20 flares which\nare limb or near-the-limb flares and their GOES diagnostic di agrams show\nclear QSS branch. For the beginning of the QSS branch (near po int E in\nFigure 1) we have determined temperature, TSXT andTGOES , and densities,\nNSXT andNSS, using the SXT images and GOES data. The flaring loop\nsemilength, L, has been calculated as L=π\n2h, wherehis the loop height\nmeasured in the SXR image. Obtained values of the temperatur es and den-\nsities are given in Table I. Comparison of NSXT andNSSis shown in Figure\n2. We see good agreement between NSXT andNSSvalues. The regression\nline is:NSXT = 0.803NSS+ 3.13 [this line is NSXT = 0.784NSS+ 2.21 if\nwe neglect the point (30,54) which shows large deviation – se e discussion\nbelow]. The correlation coefficient is r= 0.844.\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.45\nFigure 1. Examples of diagnostic diagrams obtained from the GOES data . Flare evolution\nproceeds from A to F. EF is the QSS branch. Straight lines have inclination ζ= 0.5.\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.56\nFigure 2. Comparison of NSXT andNSSfor 20 selected flares. Estimates of random errors\nare given in Table 1.\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.67Table I. List of selected flares\nGOES GOES Beginning of the QSS evolution\nDate class max. StartTSXTTGOESNSXT ∆NSXTNSS ∆NSSk\n(UT) (UT) (MK) (MK)\n1. 1991 Nov. 04 M3.4 23:23 23:47 8.0 11.5 7.1 1.4 12.6 2.5 0.6\n2. 1991 Nov. 19 C8.5 09:32 09:36 8.0 11.3 9.6 1.9 10.4 2.1 0.9\n3. 1991 Dec. 03 X2.2 16:39 16:40 10.4 19.1 75.6 15.1 84.7 16.9 0.9\n4. 1992 Jan. 13 M2.0 17:34 17:41 8.2 12.8 11.9 2.4 15.5 3.1 0.8\n5. 1992 Sep. 09 M3.1 02:13 02:15 9.2 14.0 19.6 3.9 27.3 5.5 0.7\n6. 1992 Oct. 04 M2.4 22:27 22:31 8.1 12.5 14.9 3.0 20.3 4.1 0.7\n7. 1992 Nov. 22 M1.6 23:10 23:11 9.0 13.3 20.1 4.0 39.2 7.8 0.5\n8. 1993 Jan. 02 C8.4 23:52 23:59 7.7 9.1 7.2 1.4 9.0 1.8 0.8\n9. 1993 Mar. 23 M2.3 01:53 02:11 8.3 10.7 4.4 0.9 4.2 0.8 1.0\n10. 1994 Jan. 16 M6.1 23:25 23:35 9.0 12.3 30.4 6.1 20.4 4.1 1.5\n11. 1998 Aug. 18 X2.8 08:24 08:31 10.9 15.6 40.5 8.1 30.1 6.0 1.3\n12. 1998 Aug. 18 X4.5 22:15 22:27 11.1 17.1 54.1 10.8 30.0 6.0 1.8\n13. 1998 Aug. 19 M3.0 14:26 14:50 8.0 10.3 14.9 3.0 6.8 1.4 2.2\n14. 1998 Nov. 22 X3.7 06:42 06:45 11.2 16.5 38.1 7.6 52.7 10.5 0.7\n15. 1998 Nov. 22 X2.5 16:23 16:28 12.3 16.3 35.5 7.1 34.5 6.9 1.0\n16. 1998 Nov. 23 X2.2 06:44 06:57 11.3 15.3 27.4 5.5 26.5 7.3 1.0\n17. 1999 Aug. 04 M6.0 05:57 06:09 8.5 12.9 17.4 3.5 22.0 4.4 0.8\n18. 2000 Jan. 12 M1.1 20:49 20:57 8.1 10.6 8.1 1.6 11.4 2.3 0.7\n19. 2000 Jun. 23 M3.0 14:31 14:42 9.0 12.3 14.0 2.8 10.1 2.0 1.4\n20. 2001 Oct. 29 M1.3 01:59 02:01 8.3 13.2 18.5 3.7 38.4 7.7 0.5\nValues of NSXT andNSSare in 1010cm−3,k=NSXT/NSS, ∆Nare 20% random errors (see text)\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.78\nFigure 3. EM for various values of temperature. The calculations have be en carried out\nfor a constant value of the SXR flux, Fi, measured with the Al12 filter ( Fi= 1.5×106\nDN).\nThe good agreement between NSXT andNSSvalues shows that both\nmethods of density determination give reliable values of th e density (no\nlarge systematic errors in their calibration).\nIt is surprising that we obtain agreement between the densit ies despite of sig-\nnificant systematic differences between the temperatures TSXT andTGOES .\nExplanation of this paradox is the following: The response f unction, fi(T)\nfor theYohkoh /SXT with Al12 filter changes only a little in wide range of\ntemperature, T≈8−40 MK [see curve fin Figure 9 in Tsuneta et al. (1991)].\nTherefore, the emission measure, calculated according to E quation (3), only\nweakly depends on temperature ¯T. To illustrate this effect, we have assumed\na constant value of the X-ray flux, Fi, and calculated EM for various values\nof temperature (Figure 3). For Al12 filter the dependence of EM onTis\nweak in the range of flare temperatures. Dependence of the ele ctron density,\nNSXT, on temperature is even weaker, since NSXT∼√\nEM. This explains\nwhy we obtain correct values of NSXT, despite of the fact that TSXT and\n|dTSXT/dt|are systematically too low.\nFigure 3 displays function 1 .5×106[DN]/fi(T) for SXT observations with\nAl12 filter. We have calculated values of this function for TSXT values which\nare presented in Table 1. From these values of the function we obtained\nits mean value and random ( r.m.s. ) error. This gives the following form of\nEquation (3):\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.89\nEM[1048cm−3] = (6.48±0.27)×10−6Fi[DN] (7)\nwhich does not depend on estimated value of mean temperature ¯T, provided\nthat ¯T≥8 MK (DN is Digital Number in Yohkoh /SXT observations).\nHowever, the digital coefficient in Equation (7) can slowly ch ange with time\ndue to slow change in calibration of the instrument.\nIn Table I we see that most flares in our sample were strong ones (GOES\nclass M and X). Electron number density, measured near SXR fla re maxi-\nmum, for most of the flares falls within the limits of (4 −50)×1010cm−3.\nBut for one flare [No. 3 in Table I, point (85,76) in Figure 2] th e density was\nextremely high, N≈8×1011cm−3. This high density has been obtained\nwith both methods of density determination. SXR image of thi s flare is\nshown in Figure 4. We see that it was a low flaring loop with very bright,\ncompact loop-top kernel (at the beginning of the QSS evoluti on the diameter\nof the kernel was about 6000 km). This flare was also very stron g in HXRs\nduring its impulsive phase (about 1300 counts/s/SC in the Yohkoh /HXT\n23-33 keV light curves, SC is a subcollimator).\nMain source of random errors (deviations from the regressio n line in Fig-\nure 2) in the standard method of density determination from S XR images,\nis due to the fact that we should assume what is extension of a S XR kernel\nalong the line of sight. Main source of random errors in the QS S method\nof density determination is due to uncertainty in estimates of the flaring-\nloop semilength, L. Estimates of Lare usually based on assumption that\nthe shape of the loop is semicircular and (in the case of limb fl ares) that\nthe loop lies in the plane of image. Deviations from these ass umptions are\nthe main source of random errors in the QSS method of electron density\ndetermination.\nNext we have calculated values of the ratio k=NSXT/NSS. The values of\nthe ratio kare given in Table I and their distribution is shown in Figure 5. We\nsee that for two flares (No. 12 and 13) the values of kare highest ( k= 1.8 and\n2.2). SXR images of these two flares are shown in Figure 6. We see t hat their\nSXR kernels are elongated. The volumes, V, of such kernels were calculated\nunder assumption that they have ”sausage-like” shape i.e.that they are\nellipsoids with semiaxes a×b×b. The high values of the ratio ksuggest that\nactual volume of the SXR emitting plasma is significantly lar ger,i. e. the\nthird semiaxis is larger than b. This indicates that estimates of NSXT for\nsuch elongated kernels can be loaded with highest random err ors. Therefore,\nwe have excluded these two large values of kfrom calculation of the mean\nvalue ¯k. We have obtained:\n¯k= 0.88, s =±0.28 (8)\nwheresis random (r. m. s.) error of single estimate of k.\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.910\nFigure 4. GOES light curve, diagnostic diagram and sample SXT(Be119) image (inverse\nintensity scale in colour) of December 3, 1991 flare (solid li ne shows solar limb).\nIn Figure 5 we see that k-values are concentrated around this mean value.\nThe value of ¯kwhich is close to one, confirms the good agreement between\nNSXT andNSSvalues. We see that random ( r.m.s. ) error,s, ofk-values is\nabout 30%. Assuming that contribution of random errors of NSXT andNSS\nto this random error of kare similar in size, we conclude that random error\nof a single determination of NSXT orNSSis about 30% /√\n2≈20%.\nThe values of NSXT, obtained with the standard method, are mean\nelectron number densities averaged over the SXR-kernel vol ume. But the\nQSS method of density determination provides actual densit y of the emit-\nting plasma. Therefore, the ratio, k=NSXT/NSS, provides an estimate\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.1011\n 0 2 4 6 8 10 12\n 0.3  0.7  1.1  1.5  1.9  2.3number of flares\nk\nFigure 5. Histogram of the ratio k=NSXT/NSSfor 20 analysed flares\nof the fraction of kernel’s volume which is filled with the SXR emitting\nplasma (”filling factor”). In Figure 5 we see that k-values are concentrated\naroundk≈0.9 which means that, typically, near the flare maximum the\nkernel’s volume is nearly entirely filled with the hot SXR emi tting plasma.\nFigure 6. Sample SXT(Be119) images (inverse intensity scale in colou r) of August 18,\n1998 ( left) and August 19, 1998 ( right) flares\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.1112\nRemaining, (1 −k), fraction of the volume can be filled with plasma of\nlower temperature, T <4 MK, whose contribution to the investigated SXR\nemission is marginal. Values k >1 are due to random errors in both methods\nof density determination.\n3. Summary and conclusions\nWe have found that simplified diagnostic diagrams ( Tvs.√\nEM), derived\nfrom GOES observations, are adequate to identify quasi-ste ady-state (QSS)\nphase of flare evolution. Correct inclination, ζ≈0.5, of the QSS branches in\nthe diagnostic diagrams shows that the temperature, TGOES , derived from\nthe GOES observations adequately represents mean temperat ure of the hot,\nSXR emitting plasma.\nFor the QSS branch of flare evolution we were able to determine electron\nnumber density, N, for SXR loop-top flare kernels, with two independent\nmethods which are described in Sections 1 and 2. Good agreeme nt of the\nN-values obtained with the two methods confirms reliability o f the densities\nderived from Yohkoh /SXT images. In Section 2 we have explained why we\nobtain correct NSXTelectron densities, despite of the fact that Yohkoh /SXT\ntemperatures are systematically too low.\nWe have obtained that electron density near the SXR flare maxi mum falls\nwithin the limits of (4 −50)×1010cm−3, but in one case of strong, compact\nflare it was extremely high, N≈8×1011cm−3.\nDensities, NSXT, derived from SXR images, are mean electron densities\naveraged over SXR flare kernels. But the densities, NSS, obtained with the\nsecond (QSS) method, provide actual densities of emitting p lasma. Therefore\nthe ratio NSXT/NSSprovides an estimate of the filling factor of the kernel\nwith hot, SXR emitting plasma. Our analysis of obtained kvalues shows\nthat, typically, flare kernels near SXR flare maximum are near ly entirely\nfilled with the hot, SXR emitting plasma.\nAcknowledgements\nTheYohkoh satellite is a project of the Institute of Space and Astronau tical\nScience of Japan. GOES is a satellite of National Oceanic and Atmospheric\nAdministration (NOAA), USA. We thank the anonymous referee for useful\ncomments and suggestions. This investigation has been supp orted by grant\nN203 1937 33 from the Polish Ministry of Science and High Educ ation.\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.1213\nReferences\nB¸ ak-St¸ e´ slicka, U., Jakimiec, J., 2005, Solar Phys .,231, 95.\nDonnelly, R. F., Grubb, R. N., Cowley, F. C., 1977, NOAA Tech. Memo. ERL SEL-48.\nJakimiec, J., Sylwester, B., Sylwester, J., Lemen, J. R., Me we, R. et al.: 1987, in: So-\nlar Maximum Analysis , Stepanov, V. E., Obridko,V. N. (Eds.), VNU Science Press,\nUtrecht, p. 91\nJakimiec, J., Sylwester, B., Sylwester, J., Serio, S., Pere s, G. et al.: 1992, A&A253, 269.\nKo/suppress loma´ nski, S., Jakimiec, J., Tomczak, M., Falewicz, R.: 2002, Adv. Space Res. 30, No.\n3, 665.\nLin, R.P., Dennis, B.R., Hurford, G.J., Smith, D.M., Zehnde r, A., et al.: 2002, Solar Phys .\n210, 3\nRosner, R., Tucker, W., Vaiana, G.: 1978, Astrophys. J. 220, 643.\nTsuneta, S., Acton, L., Bruner, M., Lemen, J., Brown, W. et al .: 1991, Solar Phys. 136,\n37.\nJakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.13Jakimiec_Bak_rev.tex; 20/02/2018; 10:44; p.14"    },    {        "title": "1104.4140v1.Alpha_cluster_structure_and_density_wave_in_oblate_nuclei.pdf",        "content": "arXiv:1104.4140v1  [nucl-th]  20 Apr 2011KUNS-2333, RIKEN-MP-19\nAlpha-cluster structure and density wave in oblate nuclei\nYoshiko Kanada-En’yo\nDepartment of Physics, Kyoto University, Kyoto 606-8502, J apan\nYoshimasa Hidaka\nMathematical Physics Lab., RIKEN Nishina Center, Saitama 3 51-0198, Japan\nPentagon and triangle shapes in28Si and12C are discussed in relation with nuclear density\nwave. In the antisymmetrized molecular dynamics calculati ons, the Kπ= 5−band in28Si and\ntheKπ= 3−band in12C are described by the pentagon and triangle shapes, respect ively. These\nnegative-parity bands can be interpreted as the parity part ners of the Kπ= 0+ground bands and\nthey are constructed from the parity-asymmetric-intrinsi c states. The pentagon and the triangle\nshapes originate in 7 αand 3αcluster structures, respectively. In a mean-field picture, they are\ndescribed also by the static one-dimensional density wave a t the edge of the oblate states. In\nanalysis with ideal αcluster models using Brink-Bloch cluster wave functions an d that with a\nsimplified model, we show that the static edge density wave fo r the pentagon and triangle shapes\ncan be understood by spontaneous breaking of axial symmetry , i.e., the instability of the oblate\nstates with respect to the edge density wave. The density wav e is enhanced in the Z=Nnuclei\ndue to the proton-neutron coherent density waves, while it i s suppressed in Z/negationslash=Nnuclei.\nI. INTRODUCTION\nIn light nuclei, some of negative-parity rotational bands with high Kquanta are discussed in relation with specific\nsymmetry of intrinsic states. One of the famous examples is the 3−state at 9 .64 MeV in12C, which has been\ndiscussed for a long time in connection to an equilateral triangle confi guration of 3 αcluster structure. The 3−state is\nthe lowest negative-parity state, and its spin is contradict to the n aive expectation from shell model calculations. The\nreason for the low-lying 3−state is understood by the point group D3hsymmetry of the equilateral triangle 3 αcluster\nstructure [1–3] [see Fig. 1(a)], which is characterizedby the n-fold symmetry with n= 3 of the intrinsic structure. The\n3−state is interpreted as the band head of the Kπ= 3−band constructed by the parity and total-angular-momentum\nprojection from the D3hsymmetry of the intrinsic state. Although the 4−state in the Kπ= 3−band has not yet\nconfirmed, a possible assignment 4−for the level at 13 .35 MeV was suggested [4]. In microscopic 3 αcluster models,\ntheKπ= 3−band is considered to form a parity doublet with the Kπ= 0+ground band [5].\nIn28Si, theKπ= 5−rotationalband startingfromthe 5−\n1state at9 .70MeVwasreportedin γ-raymeasurementsby\nGlatz et al. [6], and it was discussed with a 7 αcluster structure with a pentagon shape [7, 8]. In the 7 αcluster model\nwith Brink-Bloch (BB) α-cluster wave functions [1], an oblate solution for negative parity sh ows theD5hsymmetry\n[see Fig. 1(b)]. Since a Kπ= 5−band can be constructed from the D5hsymmetry of the intrinsic state, the existing\nKπ= 5−band might be an indirect evidence of the pentagon shape and may be regarded as the parity partner of\ntheKπ= 0+ground state.\nIn spite of the reasonable description of the Kπ= 5−band in28Si with the 7 αcluster model, the BB α-cluster\nmodel is too simple to quantitatively describe low-lying energy spectr a of28Si [8]. Moreover, the validity of the ansatz\nthat28Si consists of seven αclusters is not obvious but it should be checked with frameworks with out any cluster\nassumptions because αclusters might be dissociated or melted down due to the spin-orbit fo rce insd-shell nuclei [9].\nRecently, more sophisticated calculations of28Si were performed by one of the authors with antisymmetrized\nmolecular dynamics (AMD) [10, 11], which is a framework free from clus ter assumptions. The calculations reproduce\nwell low-lying positive-parity levels of the oblate ground band and the excited prolate band by incorporating the\nspin-orbit force with a proper strength. Interestingly, the Kπ= 5−band is constructed from the oblate state with a\npentagon shape even though existence of any clusters is not assu med in the calculations [11]. This means that the\npentagon shape can be induced by αcluster correlation in the oblate intrinsic state of28Si.\nFrom the viewpoint of the symmetry breaking, the pentagon shape in the intrinsic state is regarded as the sponta-\nneous breaking of the axial symmetry. The surface density is oscilla ting along the edge of the oblate shape, that is,\nthe spontaneous symmetry breaking (SSB) occurs in the rotation al invariance around the symmetric axis. In relation\nto the SSB concerning density, this structure is associated with de nsity wave (DW) in a nuclear matter, in which the\nSSB of the translational invariance occurs. The DW in nuclear matte r has been discussed for a long time [12–14], and\nit was suggested that the one-dimensional DW could be stable in a low d ensity nuclear matter [14]. The nonuniform\nnuclear matter with density oscillation has also been investigated with cluster models as the α-cluster matter [15–17].\nIn analogy to the nuclear matter DW, the pentagon shape can be int erpreted as the static one-dimensional DW at2\nthe edge of the oblate state. Our aim is to give description of the pen tagon and triangle shapes in terms of the DW\nwith the wave number five and three to discuss the relation of the 7 αand 3αcluster structures with the static edge\nDW, i.e., the spontaneous axial symmetry breaking of the oblate sta tes.\nIn this paper, we report the AMD calculations of28Si while focusing on the pentagon shape in the oblate states.\nIn analysis of single-particle wave functions of the obtained AMD wav e functions, we show that the pentagon shape\nis expressed by one-particle and one-hole excitations having the wa ve number five on the oblate state and it can be\ninterpreted as the one-dimensional DW. To see the instability of the axial symmetry with respect to the pentagon\nshape, analyses of ideal cluster models using BB wave functions are performed. Similarly, focusing on the triangle\nshape of the 3 αcluster structure, structures of12C are also discussed. We introduce a simplified model for the\none-dimensional DW in the oblate state by truncating active orbits f or particle and hole states, and show that the\nproton-neutroncoherentDWsin Z=Nnucleipromotetheinstabilityoftheoblatestateswithrespecttot hepentagon\nand triangle shapes. The suppression mechanism of cluster struct ures in neutron-rich nuclei is discussed from the\nviewpoint of proton DW with excess neutrons.\nThe paper is organized as follows: In the next section, we explain the AMD calculations for28Si and12C. Analyses\nwith ideal cluster models using BB α-cluster wave functions are given in Sec. III, and those using exte nded BB wave\nfunctions in Sec. IV. Discussions with a simplified model for the one-d imensional DW are given in Sec. V. Finally, in\nSec. VI, a summary and an outlook are given.\n1\n23\n1\n2345\n67(b) (a)\nFIG. 1: The schematic figures for spatial configurations of cl uster centers of (a) a triangle structure consisting of thre eαclusters\nin12C and (b) a pentagon structure of seven αclusters in28Si.\nII. AMD CALCULATIONS OF28SI AND12C\nA. Method of AMD calculations\nThe AMD method has been applied for various nuclei and succeeded t o describe shell-model structures and cluster\nstructures of ground and excited states in a light-mass region [18– 20]. Here we briefly describe a simple version of the\nAMD method and its application to28Si and12C [10, 18] focusing on cluster features of the oblate bands. The de tails\nof the previous AMD calculations for28Si are described in Refs. [10, 11], in which the energy levels of the Kπ= 0+\n1,\nKπ= 0+\n2,Kπ= 3−and,Kπ= 5−bands are well reproduced.\nAn AMD wave function for an A-nucleon system is given by a Slater determinant of Gaussian wave pa ckets,\nΦAMD(Z) =1√\nA!A{ϕ1,ϕ2,...,ϕA}, (1)\nwhere the i-th single-particle wave function is written as\nϕi=φZiXi, (2)\nφZi(rj)∝exp/bracketleftig\n−ν/parenleftig\nrj−Zi√ν/parenrightig2/bracketrightig\n. (3)\nXiis the spin-isospin function and fixed to be p↑, p↓,n↑, orn↓. The spatial part is represented by complex\nvariational parameters, Z xi, Zyi, Zzi, which indicate the centers of the i-th Gaussian wave packet. The parameter νis\nchosen to be ν= 0.15 fm−2andν= 0.175 fm−2for28Si and12C so as to minimize the energy of the positive-parity\nstate.\nIn the AMD model, all the centers {Z1,Z2,···,ZA}of single-nucleon Gaussians are treated independently as\ncomplex variational parameters. Thus, the AMD method is based co mpletely on single nucleons and therefore it\nis free from such assumptions as cluster existence or axial symmet ry. Nevertheless, if a cluster structure is favored\nin a system, the cluster structure can be described as an optimum s olution of AMD wave functions because BB3\ncluster wave functions are included in the AMD model space. For inst ance,αcluster formation is expressed by the\nconcentration of Gaussian centers for four nucleons, p↑,p↓,n↑, andn↓at a certain position.\nBy using an effective Hamiltonian,\nHeff=/summationdisplay\niTi+/summationdisplay\ni<jvij+/summationdisplay\ni<j<kvijk (4)\nconsisting of kinetic terms and two-body and three-body interact ion terms as effective nuclear forces, the energy\nvariationis performedwithin the AMD modelspaceto obtainthe optim umsolutions, whichcorrespondto the intrinsic\nwave functions for low-lying states. As in Refs. [10, 11, 18], the en ergy variation is done after parity projection by\noperating (1 ±Pr) on the AMD wave function. After the energy variation for (1 ±Pr)ΦAMD(Z) with respect to Zthe\noptimized intrinsic wave functions, Φ AMD(Z(+)) and Φ AMD(Z(−)), are obtained for the positive- and negative-parity\nstates, respectively. Then, the total-angular-momentum proje ction,PJ\nMK, is operated on the obtained AMD wave\nfunctions, PJ\nMK(1±Pr)ΦAMD, to calculate expectation values of parity and angular-momentum e igenstates, J±.\nThe adopted effective nuclear forces are the same as those in Refs . [10, 11] with which AMD calculations reproduce\nthe energy levels of28Si. Namely, the MV1 force (case 1) [21], which consists of finite-rang e two-body and zero-range\nthree-body forces, with a parameter set ( b=h= 0,m= 0.62) is used for the central force. As for the spin-orbit force,\nthe spin-orbit term of the G3RS force [22] with the strengths uI=−uII= 2800 MeV is adopted. Coulomb force is\napproximated by seven-range Gaussians.\nB. AMD results of28Si\nIn the present work, we focus only on the oblate rotational bands ,Kπ= 0+\n1,Kπ= 3−\n1, andKπ= 5−\n1, though\nthe prolate excited Kπ= 0+\n2band exists in28Si [10]. We adopt the oblate solutions of the intrinsic wave func-\ntions Φ AMD(Z(+)) and Φ AMD(Z(−)) obtained by the energy variation after positive- and negative-pa rity projections.\nΦAMD(Z(+)) and Φ AMD(Z(−)) correspond to the intrinsic states of the lowest positive- and neg ative-parity bands.\nDensity distributions of these AMD wave functions are shown in Fig. 2 . Interestingly, the wave function Φ AMD(Z(+)),\nwhich correspondsto the Kπ= 0+ground band, shows the pentagon shape due to the 7 α-like structure even though α\nclusters are not a prioriassumed in the framework. We should comment that αclusters in Φ AMD(Z(+)) are somehow\ndissociated due to the spin-orbit force as discussed in Ref. [10]. Fro m these two intrinsic states Φ AMD(Z(+)) and\nΦAMD(Z(−)), we calculate the J±states by performing the parity and angular-momentum projectio ns and diagonal-\nizing Hamiltonian and norm matrices with respect to PJ\nMK(1±Pr)ΦAMD(Z(+)) andPJ\nMK(1±Pr)ΦAMD(Z(−)). Here,\nstates with each parity are described by a linear combination of the p arity and angular-momentum eigenstates pro-\njected from both of Φ AMD(Z(+)) and Φ AMD(Z(−)). The calculated energy levels of28Si are shown in Fig. 3 compared\nwith the experimental data of the members in the oblate bands, Kπ= 0+\n1,Kπ= 3−\n1, andKπ= 5−\n1. The experi-\nmental energy levels are reproduced rather well by the calculation s. The calculated in-band E2 transition strengths\nalso reproduce well the experimental data (see Table. I). The gro und band, Kπ= 0+\n1, and the lowest negative-parity\nband,Kπ= 3−\n1, are dominantly constructed from Φ AMD(Z(+)) and Φ AMD(Z(−)), respectively. The Kπ= 5−\n1band\nis mainly constructed from Φ AMD(Z(+)) having the pentagon shape though the Kπ= 5−\n1band member states have\nsignificant mixing with the Kπ= 3−\n1band members. This means that the Kπ= 0+\n1and the Kπ= 5−\n1bands can\nbe interpreted as the parity partners constructed from the par ity-asymmetric-intrinsic state Φ AMD(Z(+)) with the\npentagon shape.\nNext we analyzethe single-particlewavefunctions in the pentagonin trinsicstate. The single-particlewavefunctions\nϕiin Eq. (2) written by Gaussian wave packets are non-orthogonal t o each other. We can make linear transformation\nfromthe set {ϕi}toanorthonormalbasisset {ϕ′\ni}keepingthe Slaterdeterminantunchangedexcept fornormalizatio n,\ndet{ϕ′\ni} ∝det{ϕi}= ΦAMD(Z). From this orthonormal basis, we construct the Hartree-Fock (HF) single-particles\n{ϕHF\ni}, which diagonalize the HF single-particle Hamiltonian as described in Ref . [23]. Analysis of {ϕHF\ni}is helpful\nto discuss intrinsic states in a mean-field picture.\nAmong the HF single-particle wave functions in the intrinsic wave func tion, Φ AMD(Z(+)), we find single-particle\norbits with pentagon density distributions (see Fig. 4). The pentag on orbits show the parity asymmetry indicating\nmixing of positive-parity and negative-parity components. We extr act each parity component from the pentagon\norbits, and find that about 5% negative-parity component is mixed in the dominant positive-parity component in\neach orbit. As shown in Fig. 4, both the positive- and negative-parit y components show donut shapes in their density\ndistributions, and the pentagon orbits can be roughly described by a linear combination cφ(0,0,±2)+c′φ(0,0,∓3)with\n|c′|2∼0.05 for|c|2+|c′|2= 1, in terms of harmonic oscillator (H.O.) single-particle orbits labeled b y quantum\nnumbers ( nz,nρ,ml) in the cylinder coordinates (see Appendix A for the expressions of φ(0,0,±2)andφ(0,0,±3)). In4\n28Si(+)28Si(−)\n12C(+)(b)\nC(−)12(c) (d)(a)\nFIG. 2: Density distributions of the AMD wave functions for t he positive- and negative-parity states in28Si and12C.\n 0 5 10 15 20\nSi28\n0+2+4+\n0+2+4+6+8+5−6−7−8−\n4−6−\n5−\n3−\nπ+K  =0π+K  =06−\n4−5−\n3−6−\n5−7−\n7−Excitation energy (MeV) EXP AMDK  =3−ππ− π−π−\nK  =3K  =5\nK  =58−\n 0 5 10 15 20\n0+4+\n2+3−\n4+\n2+\n0+\nK  =0π+ K  =0π+K  =3π−C12\n3−− −(2 ,4  )Excitation energy (MeV) AMD EXP4−−5\n−\nFIG. 3: (a) Energy levels calculated with AMD calculations a nd the experimental levels of the Kπ= 0+\n1,Kπ= 3−\n1, and\nKπ= 5−\n1bands in28Si. (b) Those of the Kπ= 0+\n1andKπ= 3−\n1bands in12C. The experimental data are taken from\nRefs. [4, 6].\nthe pentagon state of28Si, totally eight pentagon orbits are found corresponding to cφ(0,0,±2)+c′φ(0,0,∓3)occupied\nby four species of nucleons, p↑,p↓,n↑, andn↓.\nC. AMD results of12C\nThe calculations of12C with the AMD are described in Ref. [19], where the Kπ= 0+\n1andKπ= 3−\n1bands are\nconstructed from triangle states with oblate deformations. In th e present work, we use the same effective interactions5\nTABLE I: The calculated and the experimental values of E2 transition strengths in28Si. The values in Weisskopf unit,\nW.u.=5.05 e2fm−4are listed. The B(E2) values are calculated with the AMD method. The experiment al data are taken from\nRef. [6].\ninitial final B(E2)\nJ±\nfJ±\ni exp. cal.\nKπ= 0+\n1→0+\n1\n2+\n10+112.7(+0.4,−0.3) 10.6\n4+\n12+\n113.6(+1.4,−1.2) 15.1\n6+\n14+\n19.4(+3.6,−2.0) 16.2\nKπ= 3−\n1→3−\n1\n4−\n13−\n132.4(+9.6,−6.4) 22.4\n5−\n23−\n1 >3.4 2.0\n5−\n24−\n1 13.7\n6−\n24−\n1 >6.1 2.5\n6−\n25−\n2 >12 16.9\n7−\n25−\n2 5.2\n7−\n26−\n2 16.9\n8−\n26−\n2 7.8\nKπ= 5−\n1→5−\n1\n6−\n15−\n1 17(+8,−4) 14.4\n7−\n15−\n1 >2.5 7.7\n7−\n16−\n1 >16.5 13.8\n8−\n16−\n1 10.1\nKπ= 5−\n1→3−\n1\n5−\n13−\n10.034(+0.01,−0.01) 4.0\n5−\n14−\n1 2(+0.2,−0.2) 7.1\n7−\n15−\n2 >2.3 2.8\n6−\n14−\n1 7.7\nas those for28Si.\nBoth of the intrinsic wave functions Φ AMD(Z(+)) and Φ AMD(Z(−)) for positive and negative parity show equilateral\ntriangle shapes because of the 3 αstructure. As seen in density distributions shown in Figs. 2(c) and 2 (d), the\ndevelopment of the 3 α-cluster structure is more remarkable in the negative-parity intrin sic state, Φ AMD(Z(−)), than\nthe positive-parity intrinsic state, Φ AMD(Z(+)). To calculate energy levels, we perform parity and angular-momen tum\nprojections and obtain the rotational Kπ= 0+\n1andKπ= 3−\n1bands constructed from the triangle intrinsic states,\nΦAMD(Z(+)) and Φ AMD(Z(−)), respectively. If we tolerate the difference of the degree of the cluster development\nbetween positive- and negative-parity states, the Kπ= 0+\n1andKπ= 3−\n1bands are roughly interpreted as the parity\npartners of the parity-asymmetric-intrinsic state with the triang le shape as argued in Ref. [5]. Then, we can say that\nthe triangle shape of12C has an analogy to the pentagon shape of28Si constructing the parity partner Kπ= 0+\n1and\nKπ= 5−\n1bands.\nIn a similar way to28Si, we analyze the HF single-particles {ϕHF\ni}in the Φ AMD(Z(+)) and find the parity-mixing\nsingle-particle orbits with the triangle shape. The density distributio n of aϕHF\nifor the triangle shape is shown in\nFig. 4(d), and those of the positive- and negative-parity compone nts extracted from this triangle orbit are shown in\nFigs. 4(e) and 4(f). Both the positive- and negative-parity compo nents show donut shape densities, and the triangle\norbit can be roughly interpreted as a linear combination cφ(0,0,±1)+c′φ(0,0,∓2)with|c′|2∼0.06 for|c|2+|c′|2= 1, in\nterms of H.O. single-particle orbits expressed by cylinder coordinat es (see Appendix A). In the triangle structure of\n12C, totally eight triangle orbits are found corresponding to cφ(0,0,±1)+c′φ(0,0,∓2)occupied by p↑,p↓,n↑, andn↓.6\n28Si(+)n  (1+P )+ rφkφkφk\n12C(+)n  (1+P )+ rφkφkφk(a) (b) (c)− rn  (1−P )\n(d) (e) (f)− rn  (1−P )\nFIG. 4: (a) Density of the highest single-particle orbit ϕHF\nkin ΦAMD(Z(+)) of28Si. (b) and (c) Density for the positive- and\nnegative-parity components of the highest orbit. Each comp onent is normalized to be 1 by multiplying npmwith 1/n2\n±=\n/angbracketleftϕHF\nk|(1±Pr)2|ϕHF\nk/angbracketright. (d) Density of the highest single-particle orbit in Φ AMD(Z(+)) for12C. (e) and (f) Density for the positive\nand negative-parity components of the highest orbit. Each c omponent is normalized to be 1.\nIII. ANALYSIS WITH BRINK-BLOCH αCLUSTER MODELS\nAs described in the previous section, the pentagon and triangle sha pes are found in the AMD results of28Si and\n12C, respectively. The AMD calculations show that these shapes origin ate in the 7 α- and 3α-cluster features in which\ntheαclusters are somehow dissociated because of the spin-orbit force . The fact that the 7 α- and 3α-like structures\nare actually formed in the AMD without a priori assuming any clusters indicates that these cluster structures ar e\nfavored in28Si and12C.\nThe pentagon and triangle shapes are interpreted as spontaneou s breaking of axial symmetry of the oblate28Si and\n12C. In this section, to understand the mechanism of the symmetry b reaking, we investigate properties of ideal 7 α-\nand 3α-cluster states by using BB α-cluster wave functions [1].\nA. Brink-Bloch α-cluster wave functions\nBBα-cluster wave functions ΦBB\nXαfor even-even Z=Nnuclei with the mass number A= 4Xare described by\nXα-cluster wave functions consisting of (0 s)4αclusters [1, 3]. The i-thαcluster is located around a certain position\nSi, and ΦBB\nXαis characterized by a spatial configuration of center positions of Xαclusters, {S1,···,SX}.\nA BBα-cluster wave function ΦBB\nXαfor aXαstate can be expressed also by an AMD wave function with a\nspecific configuration of Gaussian centers {Z}. When Xiis chosen to be p↑,p↓,n↑, andn↓fori={1···X},\ni={X+1,···,2X},i={2X+1,···,3X}, andi={3X+1,···,4X}, respectively, and Gaussian centers for four\nnucleons ( p↑,p↓,n↑,n↓) are common and real values, Zi=Zi+X=Zi+2X=Zi+3X=Si/√ν(i= 1···X), the\nAMD wave function is equivalent to the corresponding BB α-cluster wave function for X αclusters localizing at the\npositions S1,S2,···,SX.\nB. Pentagon 7αBB wave function\nLet us consider the pentagon structure of a 7 αsystem. The α-cluster centers Siofαclusters are taken to have the\npentagon configuration illustrated in Fig. 1(b) as\nSi=/parenleftig\nd√νcos/parenleftig2π\n5i/parenrightig\n,d√νsin/parenleftig2π\n5i/parenrightig\n,0/parenrightig\n(5)\nfori= 1,···,5, and\nSi= (0,0,±d′√ν) (6)\nfori= 6,7.dis the dimensionless pentagon size. Thus, the defined ΦBB\n7αis determined by three parameters ν,d,d′,\nand hence, we denote the pentagon 7 αwave functions by ΦBB\n7α(ν,d,d′).7\nNext we explain the relation between ΦBB\n7αand shell model wave functions by transforming the single-particle wave\nfunctions of ΦBB\n7αin the expansion with respect to the pentagon size d. In general, when α-cluster centers {Si}are\nlocated around the origin, the BB wave function can be connected t o a H.O. shell model wave function by using\ninvariance of a Slater determinant det {φi(rj)}=n0det{φ′\ni(rj)}under a linear transformation φi(r)→φ′\ni(r). Heren0\nis a normalization factor.\nFor oblate systems, we use H.O. single-particle wave functions φ(nz,nρ,ml)in the expression of cylinder coordinates\ndescribed in Appendix A. In the small d′limit, the spatial wave functions for spin-up protons in ΦBB\n7α(ν,d,d′) can be\ntransformed to det {φ′\ni(rj)}, which is given by the Taylor expansion with respect to the pentagon sizedas follows:\ndet{φZ1,φZ2,···,φZ7}=n0det{φ′\n1,φ′\n2,···,φ′\n7},\nφ′\n1=φ(0,0,0)+O(d2),\nφ′\n2=φ(1,0,0)+O(d2),\nφ′\n3=φ(0,0,+1)+O(d2),\nφ′\n4=φ(0,0,−1)+O(d2),\nφ′\n5=φ(0,1,0)+O(d2),\nφ′\n6=φ(0,0,−2)−d√\n6φ(0,0,+3)+O(d2),\nφ′\n7=φ(0,0,+2)+d√\n6φ(0,0,−3)+O(d2), (7)\nwheren0has the order O(d9,d′). We define φ′(0)\n1,2,3,4,5,6,7≡φ(0,0,0),φ(1,0,0),φ(0,0,+1),φ(0,0,−1),φ(0,1,0),φ(0,0,+2),\nφ(0,0,−2). In the small dlimit, single-particle orbits φ′\niapproach φ′(0)\niand the 7 αwave function becomes equivalent\nto the following 0¯ hωshell model wave function with the s2\nπs2\nνp6\nπp6\nν(sd)6\nπ(sd)6\nνconfiguration,\nΦBB\n7α(ν,d′,d)→Φ(0¯hω)\n7α≡n4\n0/productdisplay\nτσdet{φ′(0)\n1Xτσ,···,φ′(0)\n7Xτσ}, (8)\nwhereτ={p,n}andσ={↑,↓}. In this limit, Φ(0¯hω)\n7αhas the axial symmetric oblate shape. Here after we consider\nonly a small d′limit and investigate properties of ΦBB\n7α(ν,d,d′) as functions of νandd, ΦBB\n7α(ν,d). In particular, the\nsymmetry breaking of the oblate shape caused by the finite dis discussed.\nLe us consider the pentagon shape described by ΦBB\n7αwith a finite pentagon size d. As the pentagon size dincreases,\nthe pentagon shape develops and the axial symmetry breaking of t he oblate state enlarges. The leading terms of the\ndeviation from Φ(0¯hω)\n7αare contained in φ′\n6andφ′\n7. The orbits φ′\n6andφ′\n7are the parity-mixed orbits, and they show\ndensity with the pentagon shape as expressed in the following explicit form of the density,\nφ′∗\n6(r)φ′\n6(r) =φ′∗\n7(r)φ′\n7(r)\n=1\n2(πb2)3/2/parenleftigρ\nb/parenrightig4\ne−r2/b2/parenleftbigg\n1+d\n3√\n2ρ\nbcos(5φ)+O(d2)/parenrightbigg\n. (9)\nThe second term cos(5 φ) gives the density oscillation with the wave number five along the edge of the oblate shape\nand show the pentagon feature. The orbits φ′\n6andφ′\n7are given by linear combinations of φ(0,0,∓2)andφ(0,0,±3).\nDue to the mixing of φ(0,0,±3)inφ(0,0,∓2)of the amplitude d2/6, the density of these orbits changes from the axial\nsymmetric density to the oscillating density. This is nothing but the sy mmetry breaking of the rotational invariance\naround the zaxis. If we associate the rotational invariance with the translation al invariance of a uniform matter,\nand the z-component of the angular momentum mlwith the momentum k, then we find a good correspondence of\ntheφ′\n6andφ′\n7orbits with the single-particle wave functions of the nuclear matter DW proposed by Overhauser [12].\nIn other words, the pentagon shape can be interpreted as the st atic DW at the edge of the oblate state. As shown\nbelow, ΦBB\n7α(ν,d) is expressed by coherent particle-hole configurations from Φ(0¯hω)\n7α.\nWe show the particle-hole representation of ΦBB\n7α(ν,d) below. We assume that Φ(0¯hω)\n7αis the Hartree-Fock vacuum\n|0∝an}bracketri}htF, andφ(0,0,±3)Xτσandφ(0,0,±2)Xτσare the levels above and below the Fermi level, respectively. We defin e the\nparticle and hole operators as\na†\n±k,τσ=c†\n±k,τσ,\nb†\n±q,τσ=c∓q,τ−σ, (10)8\nwhere the labels k≡3 andq≡2 indicate mlfor particles and holes. When higher order terms, O(d2), in the\nsingle-particle wave functions φ′are ignored, ΦBB\n7α(ν,d) can be approximated to be\nΦBB\n7α(ν,d)≈/productdisplay\nχ/parenleftbigg\n1+d√\n6a†\n−k,χb†\n−q,−χ/parenrightbigg/parenleftbigg\n1−d√\n6a†\n+k,χb†\n+q,−χ/parenrightbigg\n|0∝an}bracketri}htF, (11)\nχ=τσand−χ=τ−σ. As clearly seen, the product of the particle and hole operators, a†\n±k,χb†\n±q,−χ, brings quanta\nK=±5.\nC. Triangle 3αBB wave function\nIn a similar way to the pentagon 7 αstate ΦBB\n7α(ν,d), the equilateral triangle 3 αstate is related to axial symmetry\nbreaking of the oblate state in p-shell, and triangle shape is described by parity-mixed orbits. In the BBα-cluster\nwave function ΦBB\n3αfor the 3αstructure, the parameters Si(i= 1,···,3) with a triangle configuration are written as\nSi=/parenleftig\nd√νcos/parenleftig2π\n3i/parenrightig\n,d√νsin/parenleftig2π\n3i/parenrightig\n,0/parenrightig\n. (12)\nΦBB\n3αis specified by the parameters ν,dasΦBB\n3α(ν,d).dis the dimensionless trianglesize. With a propertransformation\nφZi(r)→φ′\ni(r) and the Taylor expansion of the transformed single-particle orbit sφ′\ni(r) with respect to the triangle\nsized, we can rewrite the spatial wave function for three identical nucle ons,\ndet{φZ1,φZ2,φZ3}=n0det{φ′\n1,φ′\n2,φ′\n3}, (13)\nwith\nφ′\n1=φ′(0)\n1,\nφ′\n2=φ′(0)\n2+d\n2φ(0,0,+2)+O(d2),\nφ′\n3=φ′(0)\n3−d\n2φ(0,0,−2)+O(d2), (14)\nwhere\nφ′(0)\n1≡φ(0,0,0),\nφ′(0)\n2≡φ(0,0,−1),\nφ′(0)\n3≡φ(0,0,+1). (15)\nIn the small dlimit, ΦBB\n3α(ν,d) becomes equivalent to the 0¯ hωshell model wave function,\nΦBB\n3α(ν,d)→n4\n0Φ(0¯hω)\n3α,\nΦ(0¯hω)\n3α≡/productdisplay\nτσdet{φ′(0)\n1Xτσ,···,φ′(0)\n3Xτσ}. (16)\nThe orbits φ′\n2andφ′\n3are the parity-mixed orbits, and their density show a triangle shape with the form\nφ′∗\n6(r)φ′\n6(r) =φ′∗\n7(r)φ′\n7(r)\n=1\n(πb2)3/2/parenleftigr\nb/parenrightig2\ne−r2/b2/parenleftbigg\n1+d\n2√\n2r\nbcos(3φ)+O(d2)/parenrightbigg\n. (17)\nSimilarly to the particle-hole representation of the 7 αwave function, ΦBB\n3α(ν,d) of the order dcan be written in the\nparticle-hole representation by using alternative definitions k≡2,q≡1, and|0∝an}bracketri}htF≡Φ(0¯hω)\n3α(ν,d),\nΦBB\n3α(ν,d)≈/productdisplay\nχ/parenleftbigg\n1−d\n2a†\n−k,χb†\n−q,−χ/parenrightbigg/parenleftbigg\n1+d\n2a†\n+k,χb†\n+q,−χ/parenrightbigg\n|0∝an}bracketri}htF. (18)9\nD. Development of the 7αand3αstates\nWe calculate energies of the 7 αpentagon state ΦBB\n7α(ν,d) and the 3 αtriangle state ΦBB\n3α(ν,d) as functions of νand\nd. The energies are evaluated by calculating expectation values of th e Hamiltonian Heffgiven in IIA with respect\nto the intrinsic state ΦBB\n7α(ν,d), and also the positive-parity projected state (1 + Pr)ΦBB\n7α(ν,d), the negative-parity\nprojected state (1 −Pr)ΦBB\n7α(ν,d), and the Kπ= 0+projected one PK=0(1 +Pr)ΦBB\n7α(ν,d). The parity projection\ncorresponds to the restoration of the broken parity symmetry o f the intrinsic state, and the K= 0 projection restores\nthe broken axial symmetry. The energy minimum state of PK=0(1+Pr)ΦBB\n7α(ν,d) may relate to the structure of the\nKπ= 0+\n1band in28Si, while that of (1 −Pr)ΦBB\n7α(ν,d) corresponds to the Kπ= 5−\n1band because (1 −Pr)ΦBB\n7α(ν,d) is\nequivalent to Kπ= 5−projected state at least in case of small d. Note that the projected states may contain higher\ncorrelations beyond a single Slater determinant.\nThe contour plots of energy surfaces are shown in Fig. 5 as functio ns ofνandd2/6. It is found that the energy\nminimum of the energy surface for ΦBB\n7α(ν,d) with no projection locates at d2/6∼0.1. The finite pentagon size\ndof the energy minimum indicates the development of pentagon shape , namely, the spontaneous breaking of axial\nsymmetry in the intrinsic structure. As seen in the minima of the ener gy surfaces shown in Fig. 5(d) and (c), the\ndevelopment of the pentagon shape enhances in the Kπ= 0+projected state, and it is largest in the negative-parity\nprojected state.\nThe value d2/6 indicates approximately the mixing amplitude of the negative-parity component in the pentagon\norbits,φ′\n6andφ′\n7, as given in Eq. (7). The value d2/6∼0.1 at the energy minimum of the positive-parity projected\nstate indicates ∼10% mixing which is comparable to 5% mixing of the negative-parity comp onent in the pentagon\norbits in the AMD wave function Φ AMD(Z+) for28Si. The main reason for the smaller mixing in the AMD result\nthan that in the ideal 7 αcluster model may be αcluster dissociation effects in the AMD calculations.\nWe also calculate energy of the equilateral triangle 3 αstate ΦBB\n3α(ν,d) as functions of νandd. Energies are\ncalculated with respect to the intrinsic state ΦBB\n3α(ν,d), and the positive-parity projected state (1 + Pr)ΦBB\n3α(ν,d),\nthe negative-parity projected state (1 −Pr)ΦBB\n3α(ν,d), and the Kπ= 0+projected one PK=0(1+Pr)ΦBB\n3α(ν,d). The\nenergy minimum state of PK=0(1+Pr)ΦBB\n3α(ν,d) describes the structure of the Kπ= 0+\n1band in12C, while that of\n(1−Pr)ΦBB\n3α(ν,d) corresponds to the Kπ= 3−\n1band.\nThe contour plots of the energy surfaces are shown in Fig. 6 as fun ctions of νandd2/4.d2/4 is approximately\nthe mixing amplitude of φ0,0,∓2inφ0,0,±1as described in Eq. (13). The finite d2/4 value of the energy minimum\nindicates the development of 3 αcluster structure. The energy minimum of the energy surface for ΦBB\n3α(ν,d) with no\nprojection locates at d2/4∼0.1. The cluster development slightly enhances in the Kπ= 0+projected state, and it is\nmost remarkable in the negative-parity projected state. This is co nsistent with the AMD calculations of12C shown\nin Fig. 2. One of the interesting features of ΦBB\n3α(ν,d) is that the energy surface is quite shallow against the large\ntriangle size datν∼0.20. This corresponds to three αcluster break up.\nE. Roles of the parity and Kπprojections in a mean-field picture\nAs mentioned before, in the particle-hole representation based on |Φ(0¯hω\n7α∝an}bracketri}ht=|0∝an}bracketri}htF, ΦBB\n7α(ν,d) can be approximately\nexpressed as Eq. (11). At least in the oder d, the negative-parity projected state is nothing but a linear combin ation\nof 1p-1hstates,\n(1−Pr)|ΦBB\n7α(ν,d)∝an}bracketri}ht ≈d√\n6/summationdisplay\nχ/parenleftig\na†\n−k,χb†\n−q,−χ−a†\n+k,χb†\n+q,−χ/parenrightig\n|0∝an}bracketri}htF. (19)\nThus, (1 −Pr)ΦBB\n7α(ν,d) is described by the coherent sum of the 1 p-1hstates,a†\n−k,χb†\n−q,−χ|0∝an}bracketri}htFanda†\n+k,χb†\n+q,−χ|0∝an}bracketri}htF. It\nindicates that the negative-parity state can be described by the Kπ= 5−vibration mode on the oblate shape. This is\nan alternative interpretation of the Kπ= 5−band. However, as already discussed before, since the axial symm etry of\nthe oblate state is already broken in the intrinsic state before the n egative-parity projection (see Fig. 5), the Kπ= 5−\nband is regarded as the static pentagon “shape” instead of the Kπ= 5−vibration on the oblate state.\nThe positive-parity projected state corresponds to the mixing of 2p-2hstates having K= 0 and K=±10 into the\ndominant Φ(0¯hω)\n7αstate. The high Kcomponents do not affect to the Kπ= 0+ground band, and actually, they are\ndropped off in the K= 0 projection. Consequently, the Kπ= 0+projected state contains only the 2 p-2hstates with\nK= 0,\nPK=0(1+Pr)|ΦBB\n7α(ν,d)∝an}bracketri}ht ≈ |0∝an}bracketri}htF−d2\n6/summationdisplay\nχ/summationdisplay\nχ′a†\n−k,χb†\n−q,−χa†\n+k,χ′b†\n+q,−χ′|0∝an}bracketri}htF. (20)10\n (fm   )−2ν  (fm   )−2ν  (fm   )−2ν  (fm   )−2ν\n2d  /6\n2d  /6\n2d  /62d  /6\n−195MeV\n−165(a)no projection\n(b)parity +\n(c)parity −\n(d)K=0\nFIG. 5: (a) Energy expectation values of ΦBB\n7α(ν,d) plotted as functions of d2/6 andν. (b), (c), and (d) Those of the positive-\nparity state, the negative-parity state, and the K= 0 state projected from the intrinsic wave function ΦBB\n7α(ν,d). The parameter\nd′is taken to be a small value, d′/√ν= 0.1 fm.\nThe contained 2 p-2hstates are kand−kparticle pairs and qand−qhole pairs, and they are associated with Cooper\npairs in BCS theory [24]. Let us remind the reader that the normal pa irings in nuclear systems are considered to\nbe neutron-neutron pairing and proton-proton one in the spin S= 0 channel. However, the 2 p-2hterms in Eq. (20)\nhave not only spin-zero nnandpppairs but also spin-one nppairs. Namely, a ( S,T) = (1,0) (spin-one isoscalar)\nparticle-particle pair and a ( S,T) = (1,0) hole-hole pair couple to be totally S= 0 and T= 0, while a ( S,T) = (0,1)\n(spin-zeroisovector)particle-particlepairanda( S,T) = (0,1)hole-holepaircouple tobe S= 0andT= 0. Becauseof\nthe large number of coherent pairs, the Kπ= 0+state projected from the 7 αcluster state may gain much correlation\nenergy.\nIn these analyses, we can say that, in both the negative-parity an dKπ= 0+states, the coherent particle-hole\nconfigurations due to the coherent edge DWs of four kinds χ=p↑,p↓,n↑, andn↓play an important role in the\ndevelopment of the pentagon shape. We will show the importance of the coherence for the SSB in the later sections.\nIV. EXTENSION OF BB α-CLUSTER MODELS\nAs discussed in the previous section, the coherent proton and neu tron edge DWs are essential to develop the\npentagon and triangle shapes. In this section, we investigate the d evelopment of the pentagon and triangle shapes\nwithout the proton-neutron coherence by considering a pentago n or triangle neutron structure with a frozen proton\nstructure. For this aim, we extend the BB α-cluster wave functions for the pentagon 7 α- and the triangle 3 α-cluster11\n (fm   )−2ν  (fm   )−2ν  (fm   )−2ν  (fm   )−2ν\n2d  /4\n2d  /4\n2d  /42d  /4\n−75MeV\n−45(a)no projection\n(b)parity +\n(c)parity −\n(d)K=0\nFIG. 6: (a) Energy expectation values of ΦBB\n3α(ν,d) plotted as functions of d2/4 andν. (b), (c), and (d) Those of the positive-\nparity state, the negative-parity state, and the K= 0 state projected from the intrinsic wave function ΦBB\n3α(ν,d). The parameter\nd′is taken to be a small value, d′/√ν= 0.1 fm.\nstates as follows: We assume the pentagon configurations of prot on and neutron structures for a Z=N= 14 system\nbut take the pentagon size din Eq. (5) independently for protons and neutrons. We take an eno ugh small value of\nthe pentagon size dpfor protons and vary the size dnfor neutrons. Thus, the defined wave function can be written in\nthe expansion of the order dnas\nΦ7α-n(ν,dn)≈n′\n0/productdisplay\nσdet{φ′(0)\n1Xpσ,···,φ′(0)\n7Xpσ}\n×/productdisplay\nσdet{φ′(0)\n1Xnσ,···,φ′(0)\n5Xnσ,φ′\n6Xnσ,φ′\n7Xnσ},\nφ′\n6=φ(0,0,−2)−dn√\n6φ(0,0,+3)+O(d2\nn),\nφ′\n7=φ(0,0,+2)+dn√\n6φ(0,0,−3)+O(d2\nn). (21)\nIn a similar way, we also assume the triangle configurations of proton and neutron structures for a Z=N= 6\nsystem by taking the triangle size din Eq. (12) independently for protons and neutrons. We take an en ough small\nvalue of the triangle size dpfor protons and vary the size dnfor neutrons. Then the wave function can be written in12\nthe expansion of the order dnas\nΦ3α-n(ν,dn)≈n′\n0/productdisplay\nσdet{φ′(0)\n1Xpσφ′(0)\n2Xpσφ′(0)\n3Xpσ}\n×/productdisplay\nσdet{φ′(0)\n1φ′\n2Xnσφ′\n3Xnσ},\nφ′\n2=φ′(0)\n2+dn\n2φ(0,0,+2)+O(d2\nn),\nφ′\n3=φ′(0)\n3−dn\n2φ(0,0,−2)+O(d2\nn). (22)\nMoreover, we consider the triangle proton structure in a Z= 6 and N= 14 system to study the proton edge DW\nin a neutron-rich system. For the frozen neutron structure, we adopt a pentagon configuration of the neutron part\nwith an enough small pentagon size dn. The proton structure is assumed to be a triangle structure with t he triangle\nsizedp, which is a variational parameter. The wave function can be written in the expansion of the order dpas\nΦ20C-p(ν,dp)≈n′\n0/productdisplay\nσ={↑,↓}det{φ′(0)\n1Xpσ,φ′\n2Xpσ,φ′\n3Xpσ}\n×/productdisplay\nσ={↑,↓}det{φ′(0)\n1Xnσ,···,φ′(0)\n7Xnσ},\nφ′\n2=φ′(0)\n2+dp\n2φ(0,0,+2)+O(d2\np),\nφ′\n3=φ′(0)\n3−dp\n2φ(0,0,−2)+O(d2\np). (23)\nThis model corresponds to a 3 αcore structure in20C system.\nWe calculate energies of Φ 7α-n(ν,dn), Φ3α-n(ν,dn), and Φ 20C-p(ν,dp) states and compare the results with ΦBB\n7α(ν,d)\nand ΦBB\n3α(ν,d). The energies are evaluated by calculating expectation values of t he effective Hamiltonian Hefffor these\nstates with no projection, the positive- and negative-parity proj ected states, and the Kπ= 0+projected states.\nAt first, we compare the pentagon size dependence of the energie s of Φ7α-n(ν,dn) having the frozen proton structure\nwith that of ΦBB\n7α(ν,d) having the proton-neutron coherent pentagon shapes. The en ergy curves are shown in Fig. 7.\nIn each system, νis fixed to be the optimum value at the energy minimum solution in the ν-dplane for the positive-\nparity projected state. As already discussed in the previous sect ion, ΦBB\n7α(ν,d) shows the deep energy pocket around\nthe energy minimum at the finite dvalue [see Fig. 7(a)]. This indicates the development of the pentagon shape, which\ncorresponds to the spontaneous breaking of the axial symmetry of the oblate state Φ(0¯hω)\n7α. The potential pockets\nare deeper in the projected states than the intrinsic state with no projection. In contrast to the energy curve for\nΦBB\n7α(ν,d), the energy curve for Φ 7α-n(ν,dn) with no projection has the minimum around dn= 0, which corresponds\nto the axial symmetric oblate state Φ(0¯hω)\n7α. Even in the projected states, there is no deep pocket in a finite dnregion,\nand the pentagon shape of the neutron structure is suppressed in the frozen proton structure. This means that the\nneutron edge DW on the oblate state Φ(0¯hω)\n7αdoes not occur without the coherent proton edge DW. It may lead t o\nsuppression of pentagon shape in Z∝ne}ationslash=Nnuclei.\nNext we discuss the triangle structures in Z= 6 nuclei. In a similar way to the pentagon structure, we compare\ntriangle size dependence of the energies of Φ 3α-n(ν,dn) having the frozen proton structure with that of ΦBB\n3α(ν,d)\nhaving the proton-neutron coherent triangle shapes in Figs. 8(a) and 8(b). Again we find that each energy curve\nfor Φ3α-n(ν,dn) has the minimum around dn= 0, which corresponds to Φ(0¯hω)\n3α. It is in contrast to the features of\nΦBB\n3α(ν,d), which shows the deep energy pocket at the finite dindicating to the developed triangle shape. We also\nconsider the proton triangle shape in the Z= 6,N= 14 system, Φ 20C-p(ν,dp). The finite dpcorresponds to the\ndevelopment of the 3 αcore structure in20C system with the oblate proton and neutron structures. The ene rgy of\nΦ20C-p(ν,dp) is the smallest around d= 0 and increases as the triangle size dpbecomes large. Compared with the\nenergy curve of Φ 3α-n(ν,dn), the triangle proton structure is significantly unfavored in the Z= 6,N= 14 system.\nThen we conclude that the proton-neutron coherence is essentia l in development of the pentagon and triangle\nstructures of the oblate states in Z=Nnuclei. Needless to say, this is consistent with the cluster aspect of Z=N\nnuclei. In the oblate state of neutron-rich C, the triangle cluster s tructure is suppressed. The first reason for the\nquenchingofcluster structureis lackofthe proton-neutroncoh erence. The second reasonis the expanded levelspacing\nof proton orbits in neutron-rich nuclei because protons are deep ly bound due to excess neutrons. Since the energy cost\nfor 1p-1hproton excitations increases in the neutron-rich system, the cor relation energy due to the triangle structure\nmay not be able to overcome the cost. We shall discuss the details in t he next section.13\n-195-190-185-180-175-170-165\n 0 0.1 0.2 0.3 0.4Energy (MeV)\nd2/6  ν=0.150(a)28Si-pnno proj.\nparity +\nparity -\nKπ=0+\n-190-185-180-175-170-165-160\n 0 0.1 0.2 0.3 0.4Energy (MeV)\ndn2/6  ν=0.135(b)28Si-n\nno proj.\nparity +\nparity -\nKπ=0+\nFIG. 7: (a) The energy of the 7 αstate ΦBB\n7α(ν,d) as a function of d2/6.dis the pentagon size for protons and neutrons. (b)\nThe energy of the Z=N= 14 state Φ 7α-n(ν,dn) with the frozen proton structure as a function of d2\nn/6.dnis the pentagon\nsize for neutrons. In each system, νis fixed to be the optimum value at the energy minimum solution in theν-dplane for the\npositive-parity projected states as (a) ν= 0.15 fm−2and (b) ν= 0.135 fm−2. The parameter d′is chosen to be d′/√ν= 0.1\nfm.\nV. EDGE DENSITY WAVE AND SPONTANEOUS SYMMETRY BREAKING\nAs already mentioned, the pentagon and triangle structures can b e interpreted as the static edge DWs at the\nsurface of the oblate states, which is expected to connect with th e spontaneous symmetry breaking (SSB) of rotational\ninvariance around the symmetric axis.\nInsystemswithstronginteraction,weknowvariousSSBphenomen asuchasnuclearBCS,chiralsymmetrybreaking,\nand color superconductivities. These SSB occur both homogeneou sly and inhomogeneously. In particular, inhomoge-\nneous SSB phases are discussed in the framework of nuclear DWs, c hiral DWs, and Fulde-Ferrel-Larkin-Ovchinnikov\nstate in color superconducting phase [12, 25–39], whose phase bre aks translational invariance and the corresponding\ncondensation operator depends on the spatial coordinates. In m ore general, the SSB resulting in a spatially nonuni-\nform vacuum relates to condensation operators with finite moment a. When we understand nuclear matter DW as the\ninstability of the Fermi surface, the condensation operator is give n in the form a†\nkFb†\nkF, which has the momentum 2 kF\n(kFis the Fermi momentum). In condensed matter physics, inhomogen eous phases with DW are discussed as charge\nDWs and spin DWs [40, 41].\nThese phenomena in infinite systems are discussed in terms of the or der parameters, which are characterized by\nnon-zero expectation values of certain operators. In finite syst ems, however, the symmetry cannot be broken in the\nenergy eigenstates, because the symmetry is restored even if it is broken in the intrinsic state. Nevertheless, it is useful\nto discuss the SSB in the state before projection or restoration b y analyzing expectation values of specific operators\nresemble to the condensation operatorsas done for the BCS phen omena in finite nuclei. For the pentagon and triangle\nstructures, the expressions in Eqs. (11) and (18) are the form s imilar to the matter DW operators a†\nkFb†\nkF.\nIn this section, we describe the SSB for edge DWs by introducing a sim plified model in Appendix C, and discuss\nthe development and suppression of the pentagon and triangle str uctures from the viewpoint of the edge DW. In14\n-75-70-65-60-55-50-45\n 0 0.1 0.2 0.3 0.4Energy (MeV)\nd2/4  ν=0.175(a)12C-pnno proj.\nparity +\nparity -\nKπ=0+\n-75-70-65-60-55-50-45\n 0 0.1 0.2 0.3 0.4Energy (MeV)\ndn2/4  ν=0.160(b)12C-n\nno proj.\nparity +\nparity -\nKπ=0+\n-90-85-80-75-70-65-60\n 0 0.1 0.2 0.3 0.4Energy (MeV)\ndp2/4  ν=0.135(c)20C-n\nno proj.\nparity +\nparity -\nKπ=0+\nFIG. 8: (a) The energy of the 3 αstate ΦBB\n3α(ν,d) as a function of d2/4, where dis the pentagon size for protons and neutrons.\n(b) The energy of the Z=N= 6 state Φ 3α-n(ν,dn) with the frozen proton structure as a function of d2\nn/4.dnis the pentagon\nsize for neutrons. (c) The energy of the Z= 6,N= 14 system Φ 20C-p(ν,dp) with the frozen proton structure. The pentagon\nsizednfor the frozen neutron structure is taken to be d2\nn= 0.025. In each system, νis fixed to be the optimum value at the\nenergy minimum solution in the ν-dplane for the positive-parity projected states as (a) ν= 0.175 fm−2, (b)ν= 0.160 fm−2,\nand (c)ν= 0.135 fm−2.\nthis model, Φ(0¯hω)\n7αis assumed to be the Hartree-Fock vacuum |0∝an}bracketri}htF, and the orbits |φ(0,0,±q)Xτσ∝an}bracketri}htand|φ(0,0,±k)Xτσ∝an}bracketri}ht\nare considered to be active Hartree-Fock single-particle states. It means that the model space is truncated within\nφ(0,0,±k)Xτσfor particle states and φ(0,0,±q)Xτσfor hole states. This is equivalent to the model of 8 particles for 16\nstates, which is a kind of half filled models. As for the residual interac tion, we assume a contact interaction and adopt\nHDWdefined in Eq. (C7). Note that this model is applicable also to the 3 αoblate state by replacing q= 2 and k= 315\nfor the 7 αstate with q= 1 and k= 2. Then, the Hamiltonian in the particle-hole representation can be written as\nH=H0+H1+HDW,\nH0=F∝an}bracketle{t0|H|0∝an}bracketri}htF,\nH1=/summationdisplay\nχEk,τa†\n+k,χa+k,χ+/summationdisplay\nχEk,τa†\n−k,χa−k,χ\n−/summationdisplay\nχEq,τb†\n+q,χb+q,χ−/summationdisplay\nχEk,τb†\n−q,χb−q,χ,\nHDW= 2/summationdisplay\nχ,χ′G(ph)\nχ,χ′\n×/bracketleftig\na†\n+k,χb†\n+q,−χb+q,−χ′a+k,χ′\n+a†\n−k,χb†\n−q,−χb−q,−χ′a−k,χ′/bracketrightig\n. (24)\nHereχ=τσand−χ=τ−σ.\nWe use an ansatz for the new vacuum of the edge DWs with the axial-s ymmetry breaking as\n|Ψ∝an}bracketri}ht=/productdisplay\nχ/parenleftig\nvτ+uτa†\n+k,χb†\n+q,−χ/parenrightig/productdisplay\nχ/parenleftig\nv∗\nτ−u∗\nτa†\n−k,χb†\n−q,−χ/parenrightig\n|0∝an}bracketri}htF, (25)\nwith\n|vτ|2+|uτ|2= 1, (26)\nwherevτanduτare variational parameters determined by the energy variation, a nd time reversal invariance is taken\ninto account. Our ansatz Eq. (25) has the same form as that obta ined in the approximation that the quantum\nfluctuation of the particle-hole operators such as a†\n±k,χb†\n±q,−χare omitted as shown in Appendix D. It is clear that\nΨ is equivalent to ΦBB\n7α(ν,d) withuτ=−d/√\n6 in the order dapproximation given by Eq. (11) (or ΦBB\n3α(ν,d) with\nuτ=d/2). In general, the coefficients uτandvτare complex. The phase φ0ofuτ/vτcorresponds to the constant shift\nof the rotation angle φ→φ+φ0in the density oscillation cos(5 φ) in Eq. (9). Since the phase φ0for the lowest-energy\nsolution is isospin independent; hereafter, uτandvτare taken to be real quantities. The expectation values for this\nvacuum|Ψ∝an}bracketri}htare\n/angbracketleftbig\na†\n±k,χa±k,χ/angbracketrightbig\n=/angbracketleftbig\nb†\n±q,−χb±q,−χ/angbracketrightbig\n=uτuτ,\n/angbracketleftbig\na†\n±k,χb†\n±q,−χ/angbracketrightbig\n=/angbracketleftbig\nb±q,−χa±k,χ/angbracketrightbig\n=±uτvτ. (27)\nThe normal state |0∝an}bracketri}htFhasvτ= 1 and uτ= 0, while the SSB vacuum has a finite ∝an}bracketle{ta†\n±k,χb†\n±q,−χ∝an}bracketri}ht, i.e., the finite value\nofuτvτ.\nThe values vτanduτare determined by minimizing the expectation value ∝an}bracketle{tΨ|H|Ψ∝an}bracketri}ht,\nE=∝an}bracketle{tΨ|H|Ψ∝an}bracketri}ht=H0+2Ecorr,\nEcorr=/summationdisplay\nτσ(Ek,τ−Eq,τ)u2\nτ+2/summationdisplay\nχ,χ′G(ph)\nχ,χ′uτvτuτ′vτ′. (28)\nForZ=Nsystems, when isospin dependences of single-particle energies Ek,τandEq,τare ignored, vτanduτdo\nnot depend on the isospin τ, and the energy correction Ecorrfrom the energy H0is\nEcorr=/summationdisplay\nτσ(Ek,τ−Eq,τ)u2\nτ+2/summationdisplay\nχ/ne}ationslash=χ′g(ph)uτvτuτ′vτ′\n= 4{(Ek−Eq)u2+6g(ph)u2v2}. (29)\nThe stationary condition with respect to variations of uandvwith the constraint uδu+vδv= 0 leads to the equation\n(Ek−Eq)u−6g(ph)u(u2−v2) = 0. (30)16\nFor a non-zero u,uandvare solved,\nu2=1\n2/parenleftbigg\n1+Ek−Eq\n6g(ph)/parenrightbigg\n,\nv2=1\n2/parenleftbigg\n1−Ek−Eq\n6g(ph)/parenrightbigg\n,\nuv=1\n2/radicaligg\n1−/parenleftbiggEk−Eq\n6g(ph)/parenrightbigg2\n. (31)\nIt turns out that, to obtain a non-zero uvwith real uandvvalues for the SSB vacuum the following condition must\nbe satisfied:\nEk−Eq<−6g(ph). (32)\nThis indicates that the SSB occurs provided that the strength −g(ph)of the attraction is large enough so as to satisfy\nthe above condition. In other words, the static edge DWs can exist if the correlation energy −6g(ph)overcomes the\nenergy cost Ek−Eqof a 1p-1hexcitation. For N=Zsystems, one can regard the correlation of 1 p-1has that of four\nparticles, because 1 hcorresponds to the three particles state in the particle picture.\nLet us consider the role of the proton-neutron coherence in the S SB. In the case that there is no proton-neutron\ninteraction, the coupling G(ph)\nχ,χ′is taken to be G(ph)\nχ,χ′=g(ph)δττ′(1−δσσ′). Protons and neutrons are decoupled in the\nHamiltonian, and the energy correction\nEcorr= 4{(Ek−Eq)u2+2g(ph)u2v2} (33)\nleads to the condition for the SSB,\nEk−Eq<−2g(ph). (34)\nThis condition is more difficult to be satisfied than Eq. (32). This is the r eason why the proton and neutron coherent\nedge DWs can be stable, while the incoherent neutron or proton edg e DW is unfavored in the oblate 7 αand 3αstates.\nThe reason for the three times smaller interaction term, i.e., the cor relation energy in Eq. (33) than that in Eq. (29)\nis that, in the particle picture, 1 hcorresponds to the one particle state in case with no proton-neut ron interaction,\ninstead of the 1 hstate corresponding to the three particles state in case with prot on-neutron interactions.\nWe also consider the further unfavored proton edge DW in a neutro n-rich system discussed in the previous section.\nProtons are deeply bound in a neutron-rich system, and therefor e the energy cost Ek−Eqfor the 1p−1hexcitation\nbecomes large in general. As a result, the condition Ek−Eq<−2g(ph)becomes severe, and the proton edge DW is\nsuppressed largely in neutron-rich nuclei.\nVI. SUMMARY AND OUTLOOK\nPentagon and triangle shapes in28Si and12C were discussed in the relation with DW at the edge of the oblate\nstates. In the AMD calculations, the Kπ= 5−band in28Si and the Kπ= 3−band in12C are described by the\npentagon and triangle shapes, respectively. These negative-par ity bands can be interpreted as the parity partners of\ntheKπ= 0+ground bands and they are constructed from the parity-asymme tric-intrinsic states. The pentagon and\ntriangle shapes originate in the 7 αand 3αcluster structures.\nWe performed analysis of ideal cluster model wave functions using B Bα-cluster wave functions and also extended\nBB wave functions, and investigated the development of the penta gon and triangle shapes. It was found that the\nproton-neutron coherence is essential in development of the pen tagon and triangle structures of the oblate states\ninZ=Nnuclei. Without the proton-neutron coherent density oscillation, t he pentagon and triangle shapes are\nsuppressed. Needless to say, this is consistent with the features of lightZ=Nnuclei, in which cluster structures\nare favored because of α-cluster formation. In the oblate state of neutron-rich C, the tr iangle cluster structure is\nsuppressed.\nIn analysis of single-particle orbits of the AMD wave functions and BB αcluster wave functions, the pentagon and\ntriangle shapes are regarded as the static one-dimensional DWs at the edge of the oblate states. The edge DWs can\nbe described by nonuniform orbits with parity mixing, which give densit y oscillation with the wave number five and\nthree at the surface of the 0¯ hωoblate states.17\nThe static edge DWs of the oblate Z=Nnuclei are understood by the spontaneous symmetric breaking (S SB)\nof rotational invariance around the symmetric axis of the oblate st ates. In other words, the development of the 7 α\nand 3αcluster structures is interpreted as the instability of axial symmet ry with respect to the pentagon and the\ntriangle shapes. We introduced a simplified model and discussed the S SB for the edge DWs. The development and\nthe suppression of the pentagon and triangle structures are des cribed by the SSB inducing static edge DWs.\nIn the simplified model, the 0¯ hωoblate states are assumed to be the Hartree-Fock vacuums |0∝an}bracketri}htF. The model space\nfor particle and hole states are truncated so that only φ(0,0,±3)Xτσandφ(0,0,±2)Xτσare active. Assuming a contact\ninteraction, we adopted the DW term HDWas the residual interaction. For the proton-neutron coherent e dge DWs in\nZ=Nsystems, the SSB occurs, when the condition Ek−Eq<−6g(ph)is satisfied. If there is no coupling between\nprotons and neutrons, the condition for the SSB is Ek−Eq<−2g(ph), which is more severe condition than the\nproton-neutron coherent case. This means that the proton and neutron coherent edge DWs are favored, while an\nincoherent neutron or proton edge DW is unfavored.\nConsidering the condition Ek−Eq<−2g(ph)for an incoherent edge DW, we explained the reasons why the triang le\ncluster structure is suppressed in the oblate state of neutron-r ich C. Since protons are deeply bound in neutron-rich\nnuclei, the level spacing of proton orbits becomes large. It increas es the energy cost Ek−Eqfor a 1p-1hexcitation,\nand hence, the correlation energy −2g(ph)due to the triangle structure is not able to overcome the cost Ek−Eq.\nThe scenariofor the suppression of the proton DW in neutron-rich systems could be extended also to infinite matter\nproblems. Let us mention about the instability with respect to proto n density oscillation in a neutron-rich matter,\na symmetric nuclear matter, and a pure proton matter with the sam e Fermi momentum of protons ignoring the\nCoulomb force. The proton density wave should be most unfavored in the neutron-rich matter among these three\ncases, while that might be favored with the coherent neutron DW in t he symmetric nuclear matter. It turns out that\nthe possibility of α-cluster crystallization in the neutron-rich matter may be suspiciou s. Alternatively, we can say that\ntheα-cluster crystallization may be suppressed in the neutron-rich mat ter because of the quenched effective mass of\nprotons.\nIn the present simplified model, in which active orbits are limited to be a s mall number, DW may be superior to\nBCS-type pairing in Z=Nsystems. We adopted the ansatz of the residual interaction H2=HDWand discuss the\nedge DWs in relation to the SSB. This ansatz may be applicable only to th e case that the level density is enough low,\nactive orbits are restricted in almost one dimension, and the spin-or bit force can be ignored. The oblate12C may\nsatisfy this condition and the oblate28Si would do probably. However, we should comment that, in normal n uclei,\nstatic surface DWs may yield to the BCS pairing. The spin-orbit force may also weaken static DWs. Moreover, when\nthe number of active orbits are large enough, the BCS pairing overc omes to DWs. Therefore, in heavy-mass nuclei,\nespecially, in spherical nuclei, the BCS pairing can be predominant as w ell known. In fact, various phenomena due\nto the BCS pairing has been observed in heavy-mass nuclei and are s uccessfully described by the BCS theory in the\nj-jcoupling scheme.\nAppendix A: H.O. single-particle states\nTo see the relation between BB cluster wave functions and shell mod el wave functions, it is convenient to expand a\nBB wave function with LS-coupling shell model wave functions which are described in terms of single-particle orbits\nin the spherical H.O. potential. For instance, an αcluster located at the origin is expressed by four nucleons, p↑,\np↓,n↑, andn↓occupying the 0 sorbit in the H.O. potential with the frequencies ωx=ωy=ωz=ω≡¯h/mb2. Here\nthe parameter bis related to νof cluster wave functions as ν= 1/2b2. For oblate systems, it is convenient to use the\nexpression of single-particle orbits with cylinder coordinates, ρ=/radicalbig\nx2+y2,z,φ, wherezis the symmetry axis. Then,\nH.O. single-particle orbits are characterized by quantum numbers nz,nρ,ml. Herenzandnρare the node numbers\nwith respect to zandρcoordinates, mlis the eigenvalue for the z-component of the orbital angular momentum. The\ntotal quantum number is N=nz+2nρ+|ml|.\nThe explicit forms of the H.O. single-particle orbits φ(nz,nρ,ml)for (nz,nρ,ml)=(0,0,±1), (0,0,±2), and (0 ,0,±3)\nare\nφ(0,0,±1)(r) =∓1\n(πb2)3/4ρ\nbe±iφe−r2/2b2,\nφ(0,0,±2)(r) =1√\n2(πb2)3/4/parenleftigρ\nb/parenrightig2\ne±2iφe−r2/2b2,\nφ(0,0,±3)(r) =∓1√\n6(πb2)3/4/parenleftigρ\nb/parenrightig3\ne±3iφe−r2/2b2. (A1)18\nAppendix B: Particle and hole representation\nIn this appendix, we summarize the notations of the particle and hole representation. The creation and annihilation\noperators, c†\nαandcαfor a state |α∝an}bracketri}htare defined as\nc†\nα|−∝an}bracketri}ht=|α∝an}bracketri}ht,\ncα|α∝an}bracketri}ht=|−∝an}bracketri}ht,\nc†\nα|α∝an}bracketri}ht= 0,\ncα|−∝an}bracketri}ht= 0, (B1)\nwhere|−∝an}bracketri}htis the no-particle state, and αdenotes the index of all degrees of freedom of the single-particle s tate such\nas momentum, spin, and isospin. cαandc†\nβsatisfies{cα,c†\nβ}=δα,β, and other anticommutation relations are zero.\nTo describe particle-hole excitations on the Hartree-Fock (HF) va cuum, we define the HF vacuum state as\n|0∝an}bracketri}htF≡/productdisplay\nα<Fc†\nα|−∝an}bracketri}ht, (B2)\nand the particle and hole operators as\na†\nα=c†\nα forα > F,\nb†\nα=S−αc−αforα < F,\naα=cα forα > F,\nbα=S−αc†\n−αforα < F. (B3)\nHereα < Fandα > Fmeans the states below and above the Fermi surface, respective ly. The time reversal state of\n|α∝an}bracketri}htis defined as S−α|−α∝an}bracketri}ht.\nFor an infinite matter of spin-1/2 fermions, single-particle states c an be characterized by momentum kand spin\nsz=σ. In a usual convention, |−α∝an}bracketri}ht=|−k,−σ∝an}bracketri}htfor|α∝an}bracketri}ht=|k,σ∝an}bracketri}htand\nSα≡(−1)1\n2−σα. (B4)\nForasphericallysymmetricsystem, single-particlestatescanbech aracterizedbythequantumnumbers |α∝an}bracketri}ht ≡ |nlsjm j∝an}bracketri}ht\nin thej-jcoupling picture, and the corresponding |−α∝an}bracketri}htand the phase convention are\n|−α∝an}bracketri}ht=|nlsj−mj∝an}bracketri}ht,\nSα≡(−1)j−mj. (B5)\nIn an axial symmetric system in the l-scoupling scheme such as the present 7 αand 3αmodels for28Si and12C,\nwe use the notation |α∝an}bracketri}ht=|nznρmlσ∝an}bracketri}htspecified by the quantum numbers in the cylinder coordinates and ad opt the\nfollowing conventions:\n|−α∝an}bracketri}ht=|nznρml−σ∝an}bracketri}ht,\nSα≡(−1)1\n2−σ−ml. (B6)\nThe operator b†\nαcreates a hole carrying the z-component of angular momentum mland the spin sz=σ.\nWe consider the Hamiltonian including the two-body interaction\nH=/summationdisplay\nαβ∝an}bracketle{tα|T|β∝an}bracketri}htc†\nαcβ+1\n2/summationdisplay\nαβγδVα,β,γ,δc†\nαc†\nβcγcδ,\nVα,β,γ,δ≡1\n2{∝an}bracketle{tαβ|v|γδ∝an}bracketri}ht−∝an}bracketle{tαβ|v|δγ∝an}bracketri}ht}. (B7)\nWe rewrite the Hamiltonian in normal-ordered form with respect to ne w particle and hole operators assuming that\nthe single-particle states, α, are solutions of Hartree-Focksingle-particle equations, which dia gonalize the Hamiltonian\nmatrix\n∝an}bracketle{tβ|T|γ∝an}bracketri}ht=/summationdisplay\nα<F[∝an}bracketle{tαβ|v|αδ∝an}bracketri}ht −∝an}bracketle{tαβ|v|δα∝an}bracketri}ht] =Eβδβδ. (B8)19\nThen the Hamiltonian takes the form,\nH=H0+H1+H2, (B9)\nwith\nH0=/summationdisplay\nα<F∝an}bracketle{tα|T|α∝an}bracketri}ht+1\n2/summationdisplay\nα<F/summationdisplay\nβ<F[∝an}bracketle{tαβ|v|αβ∝an}bracketri}ht −∝an}bracketle{tαβ|v|βα∝an}bracketri}ht],\nH1=/summationdisplay\nα>FEαa†\nαaα−/summationdisplay\nα<FEαb†\nαbα,\nH2=1\n2/summationdisplay\nαβγδVα,β,γ,δN(c†\nαc†\nβcδcγ), (B10)\nwhereN( ) is the normal-orderedproduct with respect to the particle and h ole operators defined before. The residual\ninteraction H2contains the particle-particle, hole-hole, and particle-hole scatte ring,\nHpp=1\n2/summationdisplay\nα,β,γ,δ>FVα,β,γ,δa†\nαa†\nβaδaγ,\nHhh=1\n2/summationdisplay\nα,β,γ,δ<FVα,β,γ,δb†\nαb†\nβbδbγ,\nHph= 2/summationdisplay\nα,γ>F/summationdisplay\nβ,δ<FVα,−δ,−β,γS−βS−δa†\nαb†\nβbδaγ. (B11)\nAppendix C: Hamiltonian of the simplified model\nWe introduce a simplified model for the oblate state Φ(0¯hω)\n7α. In this model, Φ(0¯hω)\n7αis assumed to be the Hartree-\nFock (HF) vacuum |0∝an}bracketri}htF, and possible particle-hole excitations are restricted within the HF s ingle-particle states\nof|φ(0,0,±k)Xτσ∝an}bracketri}htand|φ(0,0,±q)Xτσ∝an}bracketri}ht. This means that the model space is truncated so that active orbit s are only\n|φ(0,0,±k)Xτσ∝an}bracketri}htfor particle states and |φ(0,0,±q)Xτσ∝an}bracketri}htfor hole states with k= 3 and q= 2 (or k= 2 and q= 1 for\nthe Φ(0¯hω)\n3α). We use the labels α=±k,τσandα=±q,τσfor these active single-particle states and also adopt the\nnotations χ≡τσand−χ≡τ−σ. In this paper, we define the particle and hole operators as\na†\n±k,χ=c†\n±k,χ,\nb†\n±q,χ=c∓q,−χ. (C1)\nHere, for convenience, we adopt the definition of the hole operato rs without the phase convention instead of Eq. (B3).\nIn this model, we assume a contact two-body attraction v(r) =gδ(r) withg <0 for the residual interaction in\ntheH2term. The matrix element Vα,β,γ,δis not zero only when kα+kβ=kγ+kδandχα=χγ∝ne}ationslash=χβ=χδ(or\nχα=χδ∝ne}ationslash=χβ=χγ) are satisfied and calculated to be\nG(pp)\nχχ′≡Vkχ,−kχ′,kχ,−kχ′=g(pp)(1−δχχ′),\ng(pp)≡g\n2∝an}bracketle{tk,−k|δ(r)|k,−k∝an}bracketri}ht,\nG(hh)\nχχ′≡Vqχ,−qχ′,qχ,−qχ′=g(hh)(1−δχχ′),\ng(hh)≡g\n2∝an}bracketle{tq,−q|δ(r)|q,−q∝an}bracketri}ht,\nG(ph)\nχχ′≡Vkχ,−qχ′,kχ,−qχ′=g(ph)(1−δχχ′),\ng(ph)≡g\n2∝an}bracketle{tk,−q|δ(r)|k,−q∝an}bracketri}ht. (C2)\nUsing the symmetry (or antisymmetry) of the matrix elements Vα,β,γ,δwith respect to indexes, the Hamiltonian20\nEq. (B10) in the particle-hole representation can be rewritten in th e explicit form,\nH=H0+H1+H2,\nH0=F∝an}bracketle{t0|H|0∝an}bracketri}htF,\nH1=/summationdisplay\nχEk,χa†\n+k,χa+k,χ+/summationdisplay\nχEk,χa†\n−k,χa−k,χ\n−/summationdisplay\nχEq,χb†\n+q,χb+q,χ−/summationdisplay\nχEk,χb†\n−q,χb−q,χ,\nH2=Hph+Hpp+Hhh, (C3)\nwith\nHph= 2/summationdisplay\nχ,χ′G(ph)\nχ,χ′/bracketleftig\na†\n+k,χb†\n+q,−χb+q,−χ′a+k,χ′\n+a†\n−k,χb†\n−q,−χb−q,−χ′a−k,χ′\n+a†\n+k,χb†\n−q,−χb−q,−χ′a+k,χ′\n+a†\n−k,χb†\n+q,−χb+q,−χ′a−k,χ′/bracketrightig\n−2/summationdisplay\nχ,χ′G(ph)\nχ,χ′/bracketleftig\na†\n+k,χb†\n+q,−χ′b+q,−χ′a+k,χ\n+a†\n−k,χb†\n−q,−χ′b−q,−χ′a−k,χ\n+a†\n+k,χb†\n−q,−χ′b−q,−χ′a+k,χ\n+a†\n−k,χb†\n+q,−χ′b+q,−χ′a−k,χ/bracketrightig\n, (C4)\nHpp=/summationdisplay\nχ,χ′G(pp)\nχ,χ′/bracketleftig\na†\n+k,χa†\n+k,χ′a+k,χ′a+k,χ\n+a†\n−k,χa†\n−k,χ′a−k,χ′a−k,χ/bracketrightig\n+2/summationdisplay\nχ,χ′G(pp)\nχ,χ′/bracketleftig\na†\n+k,χa†\n−k,χ′a−k,χ′a+k,χ\n+a†\n+k,χa†\n−k,χ′a+k,χ′a−k,χ/bracketrightig\n, (C5)\nHhh=/summationdisplay\nχ,χ′G(hh)\nχ,χ′/bracketleftig\nb†\n+q,χb†\n+q,χ′b+q,χ′b+q,χ\n+b†\n−q,χb†\n−q,χ′b−q,χ′b−q,χ/bracketrightig\n+2/summationdisplay\nχ,χ′G(hh)\nχ,χ′/bracketleftig\nb†\n+q,χb†\n−q,χ′b−q,χ′b+q,χ\n+b†\n+q,χb†\n−q,χ′b+q,χ′b−q,χ/bracketrightig\n, (C6)\nwhere we omit 0 →4 and 1→3 and their inverse processes. In the particle-hole interaction ter mHph, the DW term\nHDW≡2/summationdisplay\nχ,χ′G(ph)\nχ,χ′/bracketleftig\na†\n+k,χb†\n+q,−χb+q,−χ′a+k,χ′\n+a†\n−k,χb†\n−q,−χb−q,−χ′a−k,χ′/bracketrightig\n(C7)\nmayinduce the edgeDWs havingthe wavenumber ±(k+q), which givesthe non-zeroexpectation value ∝an}bracketle{ta†\n±k,χb†\n±q,−χ∝an}bracketri}ht.\nThe terms of a†\n±k,χb†\n∓q,−χb∓q,−χ′a±k,χ′inHphmay induce the exciton mode having the wave number ±1. They\ncontains the spurious mode of translational motion and are of no int erest in finite systems. Other terms in Hphhave\nthe opposite sign and they do not give coherent effects to the corr elation energy.21\nInHppandHhh, the interactions which may induce the BCS pairing are the following te rms:\nHpp\nBCS= 2/summationdisplay\nχ,χ′G(pp)\nχ,χ′/bracketleftig\na†\n+k,χa†\n−k,χ′a−k,χ′a+k,χ (C8)\n+a†\n+k,χa†\n−k,χ′a+k,χ′a−k,χ/bracketrightig\n,\nHhh\nBCS= 2/summationdisplay\nχ,χ′G(hh)\nχ,χ′/bracketleftig\nb†\n+q,χb†\n−q,χ′b−q,χ′b+q,χ\n+b†\n+q,χb†\n−q,χ′b+q,χ′b−q,χ/bracketrightig\n. (C9)\nIn the case of Z=Nnuclei, only two types of BCS pairing, for instance, a†\n+k,p↑a†\n−k,p↓anda†\n+k,n↑a†\n−k,n↓, are usually\nconsidered among four species of nucleons, χ=p↑,p↓,n↑,n↓. This is different from the DWs induced by HDW\nwhere four types of particle-hope combination, ∝an}bracketle{ta†\n±k,χb†\n±q,−χ∝an}bracketri}ht, can be non zero simultaneously and they can give\ncoherent effects to the correlation energy. Considering that the coupling constants, G(ph),G(pp), andG(hh)are the\nsame order, the DW may be superior to the BCS-type pairing in Z=Nsystems in the present simplified model with\na limited number of active orbits. We consider HDWto be the dominant term, and adopt the ansatz of H2=HDW\nand discuss the edge DWs in the Hamiltonian H=H0+H1+HDWin relation to the SSB.\nAppendix D: Alternative method to solve DW Hamiltonian\nIn this section, we solve Eq. (24) in the approximation omitting the qu antum fluctuation of the product of the\nparticle and hole operators. It is a kind of the mean-field approache s in the field theory. We show that the same\nresult as that in Sec. V are obtained in this approximation.\nLet us consider the H=H0+H1+HDWin Eq. (24). By decomposing HDWinto mean fields and their fluctuations,\nwe can rewrite the second and third terms H1+HDWas\nH1+HDW=Hmf+Hquasi+Hfluc,\nHmf≡ −2/summationdisplay\nχ,χ′G(ph)\nχ,χ′/bracketleftig\n∝an}bracketle{ta†\n+k,χb†\n+q,−χ∝an}bracketri}ht∝an}bracketle{tb+q,−χ′a+k,χ′∝an}bracketri}ht+∝an}bracketle{ta†\n−k,χb†\n−q,−χ∝an}bracketri}ht∝an}bracketle{tb−q,−χ′a−k,χ′∝an}bracketri}ht/bracketrightig\n,\nHquasi≡H1+2/summationdisplay\nχ,χ′G(ph)\nχ,χ′/bracketleftig\na†\n+k,χb†\n+q,−χ∝an}bracketle{tb+q,−χ′a+k,χ′∝an}bracketri}ht+∝an}bracketle{ta†\n+k,χb†\n+q,−χ∝an}bracketri}htb+q,−χ′a+k,χ′\n+a†\n−k,χb†\n−q,−χ∝an}bracketle{tb−q,−χ′a−k,χ′∝an}bracketri}ht+∝an}bracketle{ta†\n−k,χb†\n−q,−χ∝an}bracketri}htb−q,−χ′a−k,χ′/bracketrightig\n,\nHfluc≡2/summationdisplay\nχ,χ′G(ph)\nχ,χ′/bracketleftig/parenleftbig\na†\n+k,χb†\n+q,−χ−∝an}bracketle{ta†\n+k,χb†\n+q,−χ∝an}bracketri}ht/parenrightbig/parenleftbig\nb+q,−χ′a+k,χ′−∝an}bracketle{tb+q,−χ′a+k,χ′∝an}bracketri}ht/parenrightbig\n+/parenleftbig\na†\n−k,χb†\n−q,−χ−∝an}bracketle{ta†\n−k,χb†\n−q,−χ∝an}bracketri}ht/parenrightbig/parenleftbig\nb−q,−χ′a−k,χ′−∝an}bracketle{tb−q,−χ′a−k,χ′∝an}bracketri}ht/parenrightbig/bracketrightig\n,(D1)\nwhereHmf,Hquasi, andHintare the mean-field energy term, the sum of H1and the interaction term between particles\nand the mean field, and the fluctuation term of the mean field, respe ctively. In the mean-field approximation, Hfluc\nis assumed to be negligible, so we drop the Hfluc. We also assume that the ground state does not break time-rever sal\nsymmetry, so that the mean fields satisfy\n∝an}bracketle{tb+q,−χa+k,χ∝an}bracketri}ht=−∝an}bracketle{ta†\n−k,−χb†\n−q,χ∝an}bracketri}ht. (D2)\nThe Hamiltonian of the quasiparticle Hquasican be written as\nHquasi=/summationdisplay\nχ/parenleftig\na†\n+k,χb+q,χa†\n−k,χb−q,χ/parenrightig\nEk,τ∆χ0 0\n∆∗\nχEq,τ0 0\n0 0 Ek,τ−∆∗\n−χ\n0 0 −∆−χEq,τ\n\na+k,χ\nb†\n+q,χ\na−k,χ\nb†\n−q,χ\n−2/summationdisplay\nχEq,τ,(D3)\nwhere the last term in Eq. (D3) comes from the anticommutation rela tion:b†\n±q,χb±q,χ=−b±q,χb†\n±q,χ+1. The gap is\ndefined by\n∆χ≡2/summationdisplay\nχ′Gχ,χ′∝an}bracketle{tbq,−χ′ak,χ′∝an}bracketri}ht. (D4)22\nSinceHquasidepends on ∆ χ, the right handed side in Eq. (D4) also depends on ∆ χthrough the expectation value;\nthus, Eq. (D4) can be regarded as a self-consistency equation. T he Hamiltonian of the quasiparticle has the quadratic\nform, so that it can be diagonalized by the following unitary transfor mation or Bogoliubov transformation:\n/parenleftigg\n˜a+k,χ\n˜b†\n+q,χ/parenrightigg\n=/parenleftigg\nvχ−uχ\nu∗\nχv∗\nχ/parenrightigg/parenleftigg\na+k,χ\nb†\n+k,χ/parenrightigg\n,\n/parenleftigg\n˜a−k,χ\n˜b†\n−q,χ/parenrightigg\n=/parenleftigg\nv∗\nχu∗\nχ\n−uχvχ/parenrightigg/parenleftigg\na−k,χ\nb†\n−k,χ/parenrightigg\n, (D5)\nwith\nuχ\nvχ=−2∆χ\nEk,τ−Eq,τ+/radicalbig\n(Ek,τ−Eq,τ)2+4|∆χ|2, (D6)\nwhich satisfies |vχ|2+|uχ|2= 1. The eigenvalues of Hquasicorresponding to the energies of the quasiparticles are\n˜Ek,χ(∆χ) =1\n2/parenleftig\nEk,τ+Eq,τ+/radicalig\n(Ek,τ−Eq,τ)2+4|∆χ|2/parenrightig\n,\n˜Eq,χ(∆χ) =1\n2/parenleftig\nEk,τ+Eq,τ−/radicalig\n(Ek,τ−Eq,τ)2+4|∆χ|2/parenrightig\n. (D7)\nThe new vacuum is defined as the state vanished by the annihilation op erators, ˜a±k,χand˜b±q,χ:\n˜a±k,χ|Ψ∝an}bracketri}ht=˜b±q,χ|Ψ∝an}bracketri}ht= 0. (D8)\nThe solution of Eq. (D8) is given by\n|Ψ(∆χ)∝an}bracketri}ht=/productdisplay\nχ(vχ+uχa†\n+k,χb†\n+k,χ)/productdisplay\nχ(v∗\nχ−u∗\nχa†\n−k,χb†\n−k,χ)|0∝an}bracketri}htF. (D9)\nThis has the same form as Eq. (25); however, they are different, b ecause the vacuum in Eq. (D9) is a function of ∆ χ,\nwhile that in Eq. (25) is a function of uτ(andvτ), which is a variational parameter. Their vacua coincide at the\nsolutions of the self-consistent equation and the variational equa tion.\nThe expectation value of Hquasibecomes\nEquasi(∆χ) =∝an}bracketle{tHquasi∝an}bracketri}ht= 2/summationdisplay\nχ/parenleftbig˜Eq,χ(∆χ)−Eq,χ/parenrightbig\n, (D10)\nwhereEquasi(∆χ)≤0 and the equality is only satisfied, when |∆χ|= 0. The mean-field term can be rewritten by ∆ χ\nas\nHmf=−/summationdisplay\nχ,χ′1−3δχ,χ′\n3g(ph)∆∗\nχ∆χ′, (D11)\nwhere we used the explicit form of the interaction G(ph)\nχχ′=g(ph)(1−δχ,χ′). Using Eqs. (D10) and (D11), we obtain\nthe correlation energy as\n2Ecorr(∆χ) =Equasi+Hmf\n= 2/summationdisplay\nχ/parenleftig\n˜Eq,χ(∆χ)−Eq,χ−/summationdisplay\nχ′1−3δχ,χ′\n6g(ph)∆∗\nχ∆χ′/parenrightig\n. (D12)\nIn the mean-field approximation, ∆ χis obtained by the stationary condition:\n∂\n∂∆∗χEcorr(∆χ) =−∆χ/radicalbig\n(Ek,τ−Eq,τ)2+4|∆χ|2+∆χ\n2g(ph)−/summationtext\nχ′∆χ′\n6g(ph)= 0. (D13)\nNotice that Eq. (D13) is the equivalent to the consistency condition Eq. (D4), which can be check by inserting\n∝an}bracketle{tbq,−χak,χ∝an}bracketri}ht=uχvχ.\nForZ=Nsystems, when Ek,τandEq,τare independent of the isospin, ∆ χis independent of χ. The solution is\n∆χ= 3g/radicaligg\n1−(Ek−Eq)2\n(6g)2. (D14)\nInserting Eq. (D14) into Eqs. (D6) and (D12), one finds that the vτanduτcoincide with Eq. (31) and also the Ecorr\ndoes with Eq. (29).23\nAcknowledgments\nThe computational calculations of this work were performed by usin g the supercomputers at YITP and done in\nSupercomputer Projects of High Energy Accelerator Research O rganization (KEK). This work was supported by\nGrant-in-Aid for Scientific Research from Japan Society for the Pr omotion of Science (JSPS). It was also supported\nby the Grant-in-Aid for the Global COE Program “The Next Generat ion of Physics, Spun from Universality and\nEmergence” from the Ministry of Education, Culture, Sports, Scie nce and Technology (MEXT) of Japan. Discussions\nduring the YITP workshops and the YIPQS long-term workshops he ld at YITP are helpful to complete this work.\nReferences\n[1] D. M. Brink, International School of Physics “Enrico Fer mi”, XXXVI, p. 247 (1966).\n[2] T. Yukawa and S. Yoshida, Phys. Lett. B 33, 334 (1970).\n[3] H. Horiuchi and K. 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Dautry and E. M. Nyman, Nucl. Phys. A 319, 323 (1979).\n[29] D. V. Deryagin, D. Y. Grigoriev and V. A. Rubakov, Int. J. Mod. Phys. A 7, 659 (1992).\n[30] E. Shuster and D. T. Son, Nucl. Phys. B 573, 434 (2000).\n[31] B. Y. Park, M. Rho, A. Wirzba and I. Zahed, Phys. Rev. D 62, 034015 (2000).\n[32] M. G. Alford, J. A. Bowers and K. Rajagopal, Phys. Rev. D 63, 074016 (2001).\n[33] E. Nakano and T. Tatsumi, Phys. Rev. D 71, 114006 (2005).\n[34] I. Giannakis and H. C. Ren, Phys. Lett. B 611, 137 (2005).\n[35] K. Fukushima, Phys. Rev. D 73, 094016 (2006).\n[36] D. Nickel, Phys. Rev. Lett. 103, 072301 (2009); Phys. Rev. D 80, 074025 (2009).\n[37] T. Kojo, Y. Hidaka, L. McLerran and R. D. Pisarski, Nucl. Phys. A 843, 37 (2010).\n[38] S. Carignano, D. Nickel and M. Buballa, Phys. Rev. D 82, 054009 (2010).\n[39] K. Fukushima, T. Hatsuda, Rept. Prog. Phys. 74, 014001 (2011).\n[40] G. Gruner, Rev. Mod. Phys. 60, 1129 (1988).\n[41] G. Gruner, Rev. Mod. Phys. 66, 1 (1994)."    },    {        "title": "1104.5053v2.Topological_density_wave_states_of_non_zero_angular_momentum.pdf",        "content": "Topological density wave states of non-zero angular momentum\nChen-Hsuan Hsu,1S. Raghu,2and Sudip Chakravarty1\n1Department of Physics and Astronomy, University of California Los Angeles\nLos Angeles, California 90095-1547\n2Department of Physics and Astronomy, Rice University, Houston, Texas 77005\n(Dated: July 15, 2022)\nThe pseudogap state of high temperature superconductors is a profound mystery. It has tantalizing evidence\nof a number of broken symmetry states, not necessarily conventional charge and spin density waves. Here we\nexplore a class of more exotic density wave states characterized by topological properties observed in recently\ndiscovered topological insulators. We suggest that these rich topological density wave states deserve closer\nattention in not only high temperature superconductors but in other correlated electron states.\nI. INTRODUCTION\nIn a paper in 2000 Nayak1provided an elegant classifica-\ntion of density wave states of non-zero angular momentum.\nThe surprise is that given the roster of multitude of such states,\nso few are experimentally observed. Of these, the angular mo-\nmentum`= 2, spin-singlet has taken on a special significance\nin the context of pseudogaps in cuprate high temperature su-\nperconductors,2It breaks translational symmetry, giving rise\nto a momentum dependent dx2\u0000y2(DDW) gap, without mod-\nulating charge or spin, but alternating circulating charge cur-\nrents from a plaquette to plaquette much like an antiferromag-\nnet. In its pristine form, in the half filled limit, that is, for\none electron per site, the Fermi surface of DDW consists of\nfour Dirac points and is therefore a semimetal. This broken\nsymmetry state has inspired much effort in characterizing the\npseudogap as a phase with an order parameter distinct from\nfluctuating superconducting order parameter.\nPresently, it appears from many experiments that the pseu-\ndogap may be susceptible to a host of possible competing or-\nders. Thus it is important and interesting to explore an order\nparameter closely related to the singlet DDW, which retains\nmany of its primary signatures such as the broken translational\nsymmetry or a particle-hole condensate of higher angular mo-\nmentum. In particular we consider a density wave of non-\nzero angular momentum of mixed singlet and triplet variety\nsuch that in the half-filled limit, it is a gapped insulator. Un-\nlike the semimetallic DDW, it has a non-vanishing quantized\nspin Hall effect for a range of values of the chemical poten-\ntial. This is in fact a topological Mott insulator3because it\nis the electron-electron interaction that is necessary for it to\nbe realized. Further addition of charge carriers, doping, leads\nto Lifshitz transitions destroying the quantization but not the\nvery existence of the spin Hall effect.\nIt is remarkable that such an unconventional broken sym-\nmetry, possibly relevant to high temperature superconductors,\nbelongs to the same class of currently discussed novel state\nof matter known as topological insulators; in fact, our work is\nto some extent motivated by these recent developments.4We\nwish to emphasize that the undoped parent compounds of high\ntemperature superconductors are proven to be antiferromag-\nnets with sizeable moments and the spin density wave trans-\nforms according to `= 0.5The proposed topological density\nwave should therefore be relevant at larger doping that per-haps originates from a nearby insulating state. In no way is\nthis different from the original suggestion of DDW.\nIt has been known that triplet i\u001bdx2\u0000y2order parameter\ncorresponds to staggered circulating spin currents around a\nsquare plaquette.6wherein the oppositely aligned spins circu-\nlate in opposite directions, as shown in Fig. 1. This reminds\nus of topological band insulators where oppositely aligned\nedge-spins travel in opposite directions. However, there is no\ntopological protection because the bulk is not gapped, but is\na semimetal instead. A more interesting case is the order pa-\nrameter (i\u001bdx2\u0000y2+dxy), where\u001b=\u00061for up and down\nspins, with the quantization axis along ^z. Such a state not\nonly satisfies time reversal invariance but is also fully gapped,\nanalogous to time reversal invariant band insulators discov-\nered recently. Singlet chiral (idx2\u0000y2+dxy)density wave that\nbreaks macroscopic time reversal symmetry was employed to\ndeduce possible polar Kerr effect and anomalous Nernst ef-\nfect7in the pseudogap phase of the cuprates. Another topo-\nlogical state with a different symmetry of the order parameter\nwas discussed in Ref. 8\nAs to topological properties of superfluids, we refer the\nreader to the book by V olovik.9Superconductors are particle-\nparticle condensates, and, as such, the orbital wave function\nconstrains the spin wave function because of the exchange\nsymmetry. What we are discussing here are particle-hole con-\ndensates, and there is no exchange requirement between a par-\nticle and a hole. Thus, orbital wave function cannot constrain\nthe spin wave function. Thus an orbital singlet can come in\nboth spin singlet and triplet varieties.\nThe plan of the paper is as follows: Section II is divided\ninto three parts. Part A discusses the topological aspects in\nthe absence of magnetic field, while Part B contains results\nfor a perpendicular magnetic field. The Part C consists of a\nthorough discussion of the bulk-edge correspondence that fol-\nlows from topological considerations. In section III we dis-\ncuss Fermi surface reconstruction via a Lifshitz transition as\nthe system is doped. In section IV possible experimental de-\ntection schemes are suggested. The symmetry of the order\nparameter that we have introduced is such that the necessary\nexperimental techniques are more subtle than the detection of\nmore common broken symmetries, such as spin or charge den-\nsity waves.arXiv:1104.5053v2  [cond-mat.supr-con]  13 Oct 20112\n!\n!\n!!\n!\n! !!\nFIG. 1. (Color online) Triplet i\u001bdx2\u0000y2density wave in the absence\nof an external magnetic field. The current pattern of each spin species\non an elementary plaquette is shown. The state is a semimetal. On\nthe other hand i\u001bdx2\u0000y2+dxycan be fully gapped for a range of\nchemical potential. An example is shown in Fig. 2.\nII. ORDER PARAMETER TOPOLOGY\nA. Zero external magnetic field\nThe order parameter that we consider is\nhcy\nk+Q;\u001bck;\u001b0i= (\b\u0016(k)\u001c\u0016)\u001b\u001b0; (1)\nwherecy\nk;\u001b(ck;\u001b) is the Fermion creation(annihilation) opera-\ntor with momentum kand spin component \u001b;\u0016= 0;\u0001\u0001\u00013,\u001c1,\n\u001c2, and\u001c3are the standard Pauli matrices and \u001c0= 1. The\nnesting vector ~Q= (\u0019=a;\u0019=a ). We choose the components\nof the order parameter to be\n\b3(k)/iW0\n2(coskx\u0000cosky)\u0011iWk (2)\n\b0(k)/\u00010sinkxsinky\u0011\u0001k: (3)\nand the remaining components are set to zero. The right hand\nside is written in terms of the gap parameters and the con-\nversion involves suitable coupling constants, which we do not\nneed to specify in a non-selfconsistent Hartree-Fock theory.\nThe lattice spacing ais set to unity.\nIn the absence of an external magnetic field, the triplet d\u0006id\nHamiltonian is\nHd\u0006id\u0000\u0016N=X\nk\ty\nkAk\tk; (4)\nwhere the summation is over the reduced Brilloin Zone (RBZ)\nbounded by ky\u0006kx=\u0006\u0019, and the spinor, \ty\nk, is defined\nas(cy\nk;\";cy\nk+Q;\";cy\nk;#;cy\nk+Q;#). The chemical potential is sub-\ntracted for convenience, Nbeing the number of particles.The\nmatrixAkis\nAk=0\nB@\u000fk\u0000\u0016\u0001k+iWk 0 0\n\u0001k\u0000iWk\u000fk+Q\u0000\u0016 0 0\n0 0 \u000fk\u0000\u0016\u0001k\u0000iWk\n0 0 \u0001 k+iWk\u000fk+Q\u0000\u00161\nCA;\n(5)with a generic set of band parameters,\n\u000fk=\u000f1k+\u000f2k (6)\n\u000f1k=\u00002t(coskx+ cosky); \u000f2k= 4t0coskxcosky:(7)\nWe may choose t= 0:15eV, renormalized by about a fac-\ntor of 2 from band calculations and t0= 0:3t, andW0\u0018\n\u0000\u00010\u0018t\u0018J, whereJis the antiferromagnetic exchange\nconstant in high temperature superconductors, for the purpose\nof illustration. Each of the two 2\u00022blocks can be written\nin terms of two component spinors,  k;\u001b= (ck;\u001b;ck+Q;\u001b)T,\n\u001b=\u00061\u0011(\";#); for example, for the up spin block we have\nH\"=X\nk y\nk;\"\u0002\n1(\u000f2k\u0000\u0016) +\u000f1k\u001c3+ \u0001k\u001c1\u0000Wk\u001c2\u0003\n k;\"\n(8)\nThe eigenvalues (\u0006refers to the upper and the lower bands\nrespectively)\n\u0015k;\u0006=\u000f2k\u0000\u0016\u0006Ek; Ek=q\n\u000f2\n1k+W2\nk+ \u00012\nk: (9)\nare plotted in Fig. 2. Since up and down spin components\nare decoupled, the Chern number for each component can be\ncomputed separately. After diagonalizing the Hamiltonian,\nwe can obtain the eigenvectors\n\b\u001b;\u0006(k) = (u\u0006ei\u001b\u0012k=2;v\u0006e\u0000i\u001b\u0012k=2)T; (10)\nwhere\nu2\n\u0006=1\n2(1\u0006\u000f1k\nEk); (11)\nv2\n\u0006=1\n2(1\u0007\u000f1k\nEk); (12)\n\u0012k= arctan(Wk\n\u0001k) +\u0019\u0002(\u0000\u0001k): (13)\nTo compute the Berry phase of the eigenstates, we define\nthe Berry curvature, ~\n\u001b;\u0006as\n~\n\u001b;\u0006\u0011i~5k\u0002h\by\n\u001b;\u0006(k)j~5kj\b\u001b;\u0006(k)i (14)\nSubstituting the eigenstates into the above equation, the\nBerry curvature can be written as\n~\n\u001b;\u0006=i~5k\u0002[(u2\n\u0006\u0000v2\n\u0006)~5k(i\u001b\u0012k\n2)]: (15)\nSinceu\u0006,v\u0006, and\u0012konly depend on kxandky, only the z\ncomponent, \n\u001b;\u0006, is non-zero, which is given by\n\n\u001b;\u0006=\u0007\u001b\n2[@\n@kx(\u000f1k\nEk)@\u0012k\n@ky\u0000@\n@ky(\u000f1k\nEk)@\u0012k\n@kx]\n=\u0006\u001b1\n2E3\nk\f\f\f\f\f\f\f\f\f\f\u0001kWk\u000f1k\n@\u0001k\n@kx@Wk\n@kx@\u000f1k\n@kx\n@\u0001k\n@ky@Wk\n@ky@\u000f1k\n@ky\f\f\f\f\f\f\f\f\f\f: (16)\nFrom the above determinant, we can see that the Berry cur-\nvature will be zero if one of \u0001kandWkis zero, so we need3\na mixing of dx2\u0000y2anddxyto have a non-trivial topological\ninvariant.\nIf we define the unit vector ^n\u001b\u0011~h\u001b=j~h\u001bj, where~h\u001b=\n(\u0001k;\u0000\u001bWk;\u000f1), the Berry curvature can be written as\n\n\u001b;\u0006=\u00071\n2^n\u001b\u0001(@^n\u001b\n@kx\u0002@^n\u001b\n@ky): (17)\nMore explicitly, the Chern numbers are\nN\u001b;\u0006=Z\nRBZd2k\n2\u0019\n\u001b;\u0006\n=\u0006\u001bZ\nRBZd2k\n2\u0019tW0\u00010\nE3\nk(sin2ky+ sin2kxcos2ky)\n=\u0006\u001b:\n(18)\nWe can focus on the lower band as long as there is a gap be-\ntween the upper and the lower bands. Then,\nN=N\";\u0000+N#;\u0000= 0 (19)\nNspin=N\";\u0000\u0000N#;\u0000= (\u00001)\u00001 =\u00002 (20)\nirrespective of the dimensionful parameters. Note, however,\nthat the Chern numbers vanish unless both \u00010andW0are\nnon-vanishing. The quantization holds for a range of chemical\npotential\u0016, as can be seen from Fig 2.\nFIG. 2. (Color online) Energy spectra, \u0015k;\u0006+\u0016, corresponding to\n(i\u001bdx2\u0000y2+dxy)density wave. Here, for illustration, we have cho-\nsenW0=tand\u00010=\u0000tand the band parameters, as described\nin the text. The chemical potential, \u0016, anywhere within the spectral\ngap, the lower band is exactly a half-filled and the system is a Mott\ninsulator, unlike the semimetallic DDW at half-filling.\nFor the fully gapped case, there will be a quantized spin\nHall conductance associated with the eigenstates. The ratio of\nthe dimensions of the quantized spin Hall conductance to the\nquantized Hall conductance should be the same as the ratio of\nthe spin to the charge carried by a particle, since in two dimen-\nsions for both quantities the scale dependence Ld\u00002cancels,\nthat is,\n[\u001bspin\nxy]\n[\u001bxy]=~\n2\ne: (21)So, the quantized spin Hall conductance will be\n\u001bspin\nxy=\u0000e2\nh~\n2eNspin=e\n2\u0019(22)\nThe eigenstates,j\t\u001b;\u0006(k)i, are also the eigenstates of S2and\nSzwith eigenvalues S2=3\n4andSz=\u0000\u001b\n2. Since the spin\nSU(2)is broken by the triplet DDW, one might wonder if\nthe Goldstone modes not contained in the Hartree-Fock pic-\nture may not ruin the quantization. If SU(2)is broken down\ntoU(1), then there is still a quantum number corresponding\nto, saySz, which is transported by the edge currents in the\nsystem. More succinctly, as long as time-reversal symme-\ntry is preserved, we will still have Kramers degeneracy in our\nHartree-Fock state, and therefore the edge modes will remain\nprotected.\nB. Non-zero magnetic field\nIn an infinitesimal external magnetic field, ~H, there will\nbe a spin flop transition in the absence of explicit spin-orbit\ncoupling, as shown in Fig. 3. We can assume ~H=H^zand\nthe spins quantized along the ^xdirection without any loss of\ngenerality. Then the Hamiltonian now becomes\nHd\u0006id=X\nk\ty\nkAk\tk (23)\nAs before, the summation is over the RBZ, and the spinor is\nthe same. The matrix Akis now\nAk=0\nB@\u000fk;\" 0 0 \u0001 k+iWk\n0\u000fk+Q;\"\u0000\u0001k\u0000iWk 0\n0\u0000\u0001k+iWk\u000fk;# 0\n\u0001k\u0000iWk 0 0 \u000fk+Q;#1\nCA;\nwhere\u000fk;\u001b=\u000fk+\u001bg\u0016BH\n2=\u000fk+\u001b\r. Although the spin up\nand down components are coupled, particles with momentum\nkand spin up only couple to holes with momentum k+Q\nand spin down, and vice versa. Therefore, by redefining the\nspinor, \t0y\nk\u0011(cy\nk;\";cy\nk+Q;#;cy\nk;#;cy\nk+Q;\"), the Hamiltonian\ncan still be expressed as a block diagonal matrix: Hd\u0006id=P\nk\t0y\nkA0\nk\t0\nk. The Chern numbers for each subblocks, i=\n1;2, can be calculated as before. Therefore, defining \u0011i= + 1\nor -1 fori= 1 or2, we obtainEk;i= [(\u000f1+\u0011i\r)2+W2\nk+\n\u00012\nk]1=2, and the Berry curvature\n\ni;\u0006=\u00071\n2E3\nk;i~hi\u0001(@~hi\n@kx\u0002@~hi\n@ky); (24)4\nH\nFIG. 3. (Color online) Spins are flopped perpendicular to the applied\nmagnetic field H. Contrast with Fig. 1.\nwhere~hi= (\u0011i\u0001k;\u0000Wk;\u000f1+\u0011i\r). Performing a surface\nintegration of the Berry curvature we get\nNi;\u0006=Z\nRBZd2k\n2\u0019\ni;\u0006\n=\u0006\u0011itW0\u00010Z\nRBZd2k\n2\u0019E3\nk;ih\nsin2ky+ sin2kxcos2ky\n\u0000\u0011i\r\n4t(coskxsin2ky+ sin2kxcosky)i\n(25)\nThe\u0006refers to the upper and the lower band respectively. The\nintegral does not depend on the external field, nor on magni-\ntude of the parameters t,W0, and \u00010. The Chern numbers\nare\nNi;\u0006=\u0006\u0011i; Ntotal=N1;\u0000+N2;\u0000= 0;\nNspin=N1;\u0000\u0000N2;\u0000=\u00002;(26)Once again the spin Hall conductance is quantized, but the\ncharge quantum Hall effect vanishes. The flopped spins carry\nthe same current as before. The corresponding spin Hall con-\nductance, as long as the gap survives, is\n\u001bspin\nxy=e\n2\u0019: (27)\nThe eigenstates,j\bi;\u0006(k)i, are the eigenstates of S2with\neigenvalues S2=3\n4, but not eigenstates of Szbecause of the\nmixing of up and down spins.\nC. Bulk-edge correspondence\nFor the (i\u001bdx2\u0000y2+dxy)order the bulk-edge correspon-\ndence can be studied by open boundary condition in the x-\ndirection but periodic boundary condition in the y-direction,\nthat is, by cutting open the torus. The edge modes if they exist\nwill reside on the ends of the cylinders. The cut then leads to\na Hamiltonian\nH=X\nky;i;j\ty\ni;kyAij(ky)\tj;ky; (28)\nwhere the spinor is \ti;ky= (ci;ky\"ci;ky+\u0019\";ci;ky#ci;ky+\u0019#)T,\nandAij(ky)is a4N\u00024Nmatrix parametrized by the wave\nvectorky, which is given by\nAij(ky) =0\nBB@Tij(ky)Sij;\"(ky) 0 0\nSy\nij;\"(ky)Tij(ky+\u0019) 0 0\n0 0 Tij(ky)Sij;#(ky)\n0 0 Sy\nij;#(ky)Tij(ky+\u0019)1\nCCA;\nwhereTij(ky)andSij;\u001b(ky)areN\u0002Nmatrices:\nTij(ky) =0\nBBBB@\u0000\u0016\u00002tcosky\u0000t+ 2t0cosky 0\u0001\u0001\u0001 \u0001\u0001\u0001\n\u0000t+ 2t0cosky\u0000\u0016\u00002tcosky\u0000t+ 2t0cosky\u0001\u0001\u0001 \u0001\u0001\u0001\n0\u0000t+ 2t0cosky\u0000\u0016\u00002tcosky\u0000t+ 2t0cosky\u0001\u0001\u0001\n............\u0000t+ 2t0cosky\n\u0000t+ 2t0cosky\u0000\u0016\u00002tcosky1\nCCCCA;\nSij;\u001b(ky) =i\u001bW0\n40\nBBBB@\u00002 cosky\u00001 0\u0001\u0001\u0001\n1 2 cos ky 1\u0001\u0001\u0001\n0\u00001\u00002 cosky\u00001\u0001\u0001\u0001\n............ (\u00001)N\u00001\n(\u00001)N(\u00001)N2 cosky1\nCCCCA+i\u00010\n2sinky0\nBBBB@0 1 0\u0001\u0001\u0001\n1 0\u00001\u0001\u0001\u0001\n0\u00001 0 1\u0001\u0001\u0001\n............(\u00001)N\n(\u00001)N01\nCCCCA:\nThe corresponding one dimensional system with Nsites de-\npends on the band structure and the order parameters defined\nabove.The eigenvalue spectra are shown in Fig 4. The spectra,\ndegenerate for up and down spins, are plotted in the range\n0\u0014ky\u0014\u0019(ky<0can be obtained by reflection). To find\nthe edge states we choose the chemical potential in the gap.5\nIn Fig. 4, we put \u0016=\u00000:075eVfor the purpose of illustra-\ntion. There are two edge states with positive group velocity,\none with up spin and the other with down spin. Let them be\n >;\"and >;#, respectively. There are also two edge modes\nwith negative group velocity denoted as  <;\"and <;#for up\nspin and down spin, respectively. By explicitly computing the\nsupport of each of these wave functions, we have verified that\nelectrons in states  >;#and <;\"are localized near the left\nedge of the system whereas those in states  <;#and >;\"are\nlocalized near the right edge. The localization length of these\nstates is essentially a lattice spacing; an example is shown in\nFig 4.\n(a)\n(b)20 40 60 80 1000.10.20.30.40.50.60.7P(x)\nxLRky/!(     )E\n-\n-RLLR0.20.40.60.81.00.40.20.00.20.40.60.8kyeV\nFIG. 4. (a) Spectrum of the triplet ( d\u0006id)-density wave on a cylinder.\nParameters are t= 0:15eV;t0= 0:3t;\u0016=\u00000:075eV;W 0=t, and\n\u00010=\u0000t. The subscripts LandRto the spins correspond to left and\nright modes. (b) The probability density for positive group velocity\nforLandRspins for a lattice of N= 100 sites.\nIt is interesting to see how this spectra compare with the one\nwhere periodic boundary conditions are applied in both xand\nydirections. After diagonalizing the Hamiltonian, we plot the\nspectra for a fixed value of kyfor all values of the energies.\nThe results are shown in Fig. 5, which are essentially identical\nto Fig. 4, except that the edge states are missing.\n0.2 0.4 0.6 0.8 1.0ky/Slash1Π/Minus0.4/Minus0.20.00.20.40.60.8/CapEpsilonky/LParen1eV/RParen1FIG. 5. The bulk spectra for fixed values of kywith the same param-\neters, as in Fig. 4.\nIII. FERMI POCKETS AND LIFSHITZ TRANSITION\nIt is interesting to track the evolution of successive Lifshitz\ntransitions as we change the parameters. At first, when we\nlower the chemical potential, four hole pockets will open up\nin the full Brillouin zone, as shown in Fig. 6 and the corre-\nsponding spin Hall effect will lose its quantization but not the\neffect itself. But in mean field theory this cannot continue\nindefinitely with the nodal or the antinodal gaps fixed. So\nthe parameters W0and\u00010will also decrease and will lead\nto a further opening of two electron pockets in the full Bril-\nlouin zone, as shown in Fig. 6. Ultimately, when the doping\nis increased further, the large Fermi surface will emerge as a\nfurther Lifshitz transition. There is good evidence that such\nLifshitz transitions indeed occur in high temperature super-\nconductors.\n/Minus3/Minus2/Minus10 1 2 3/Minus3/Minus2/Minus10123\nakxaky\n/Minus3/Minus2/Minus10 1 2 3/Minus3/Minus2/Minus10123\nakxaky\nFIG. 6. (Color online) (Left) Region plot, \u0016=\u00000:16eV. Here for\nillustration, we have chosen W0=tand\u00010=\u0000tas before. The\nhole pockets open up. (Right) W0= 0:05tand\u00010=\u00000:5tillus-\ntrating the opening of the electron pockets at (\u0019;0)and symmetry\nrelated points with enlarged hole pockets.6\nIV . EXPERIMENTAL DETECTION\nWhile there are many speculations about the nature of the\npseudogap, they largely fall into two categories: 1) it is a\ncrossover between a Mott insulator and a Fermi liquid, with-\nout any sharp, coherent excitations, and 2) it reflects a broken\nsymmetry, with quasiparticles due to reconstructed Fermi sur-\nface that, despite strong correlations in the system, can behave\nin many ways as weakly interacting particles. The resolution\nof this dichotomy will ultimately be settled by experiments,\nwhich, to date, have shown some support for both. In the ab-\nsence of a definitive evidence one way or the other, we have\nadopted the second perspective (to some extent motivated by\nrecent quantum oscillation experiments) to see what conse-\nquences there may be of having a broken symmetry phase\nwith sufficiently hidden order, in particular one that has strik-\ning similarities to topological insulators.\nA prime characteristic of a broken symmetry is that deep\nin the broken symmetry phase, an effective mean-field, or a\nHartree-Fock Hamiltonian, suffices in discussing the proper-\nties of matter, and the symmetries alone determine the ex-\ncitation spectra and the collective modes. It is only in the\nproximity of quantum critical points that such a description\nbreaks down but that is not the subject of discussion here.\nMoreover, those properties that are determined by symmetries\nalone should be robust and can be understood in the weak cou-\npling limit, simplifying our task of exploring correlated elec-\ntron system.\nThe mixed triplet-singlet order parameters considered here\nis even more hidden than the corresponding singlet DDW. Not\nonly do they not modulate charge or spin, but so long as spin-\norbit coupling is absent, they are also invisible to elastic neu-\ntron scattering because there is no associated staggered mag-\nnetic field, as in a singlet DDW.\nInelastic neutron scattering can detect its signature in terms\nof a spin gap at low energies in the longitudinal susceptibil-\nity and signatures in the transverse susceptibility of quasi-\nGoldstone modes, and even onset of a finite frequency res-\nonance mode. Recall that at any finite temperatures SU(2)\nsymmetry cannot be spontaneously broken in two dimen-\nsions; interlayer coupling is necessary to stabilize it. Thus\nthe scale of symmetry breaking must be considerably smaller\nthant\u0018J, and the signature must be sought at higher ener-\ngies. It could be a challenge to disentangle the signal from\ninelastic spin density wave excitations. On the other hand\nsince the quasiparticle excitations are essentially identical to\nthe singlet DDW, the quantum oscillation properties will be\nsimilar,10except perhaps those in a tilted field,11which is cur-\nrently being explored. The essence of this order parameters is\nmodulation of spin current and kinetic energy. So, it will re-\nquire probes that can detect higher order correlation functions,\nsuch as the two-magnon Raman scattering. In the presence of\nmodest spin-orbit coupling, it may be possible to find small\nshifts of nuclear quadrupolar frequency (NQR). The modula-tion of the kinetic energy arising from the dxycomponent, in\nparticular staggered modulation of t0, may lead to anomalies\nin the propagation of ultrasound12at a temperature where such\nan order is formed, presumably at the pseudogap temperature\nT\u0003. The detection of the unique features of the proposed or-\nder parameter, the spin Hall effect and edge currents would be\neven more challenging.\nThe effects of non-magnetic impurities on the mixed triplet-\nsinglet phase studied here are rather subtle. We expect such\ndisorder to couple only weakly to spin currents. Generically,\ndisorder will couple differently to the i\u001bdx2\u0000y2anddxycom-\nponents since each breaks a different symmetry. However, by\nbreaking both the point group and lattice translation symme-\ntries, disorder can enable mixing with (generally incommen-\nsurate) density wave states in other angular momentum chan-\nnels. For example, at the level of Landau theory, we expect\nterms in the free energy proportional to product of quadratic\npowers of the component order parameters, which would be\nproportional to the impurity concentration, thus inducing spin\nor charge density waves. So long as spin rotational symmetry\nis preserved in the normal state, the phase transition into the\ni\u001bdx2\u0000y2state can remain sharp.\nFrom the standpoint of topological order at zero tempera-\nture, the effects of weak disorder are somewhat simpler. Since\nthe density wave phase considered here is a gapped phase with\ntopological order that is protected by time-reversal symmetry,\nit remains robust against weak non-magnetic disorder. Thus,\nthe phase can still be described in terms of its topology at zero\ntemperature, a feature which it shares with topological band\ninsulators.\nLastly, we remark that in the presence of magnetic impu-\nrities, the phase is not sharply defined - either as a broken\nsymmetry or in terms of it’s underlying topology.\nIn terms of microscopic models beyond the phenemenol-\nogy discussed here, it is almost certain that correlated hopping\nprocesses will play a key role,13Finally, since dx2\u0000y2anddxy\nare two distinct irreducible representations on a square lattice,\ngenerically they will each have their own transition tempera-\ntures, as dictated by Landau theory. The development of the\ndxyorder parameter would be at a higher temperature com-\npared to the triplet component which breaks SU(2)and there-\nfore requires interlayer coupling. Thus it follows that when\napplied to cuprates there must be two transitions in the pseu-\ndogap regime. Since the topological phase studied here arises\nfrom spontaneous symmetry breaking, it can support charged\nskyrmion textures in analogy with.14The properties of such\ntextures and their transport signatures shall be the topic of a\nforthcoming publication.\nACKNOWLEDGMENTS\nThis work is supported by NSF under the Grant DMR-\n1004520. We thank Liang Fu, Pallab Goswami, and Chetan\nNayak for discussion.7\n1We shall depart slightly from the notation of C. Nayak, Phys. Rev.\nB,62, 4880 (2000).\n2S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Phys.\nRev. B, 63, 094503 (2001).\n3S. Raghu, X.-L. Qi, C. Honerkamp, and S.-C. Zhang, Phys. Rev.\nLett., 100, 156401 (2008).\n4M. Z. Hasan and C. L. Kane, Rev. Mod. Phys., 82, 3045 (2010);\nX. Qi and S. Zhang, ArXiv e-prints (2010), arXiv:1008.2026\n[cond-mat.mes-hall].\n5S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. Rev. B,\n39, 2344 (1989).\n6A. A. Nersesyan, G. I. Japaridze, and I. G. Kimeridze, J. Phys. C,\n3, 3353 (1991).\n7S. Tewari, C. Zhang, V . M. Yakovenko, and S. Das Sarma,\nPhys. Rev. Lett., 100, 217004 (2008); C. Zhang, S. Tewari, and\nS. Das Sarma, Phys. Rev. B, 79, 245424 (2009); P. Kotetes and\nG. Varelogiannis, ibid.,78, 220509 (2008); Phys. Rev. Lett., 104,106404 (2010).\n8Y . Ran, A. Vishwanath, and D. Lee, ArXiv e-prints (2008),\narXiv:0806.2321 [cond-mat.str-el].\n9G. E. V olovik, The Universe in a Helium Droplet (Oxford Univer-\nsity Press, NY , 2003) and references therein.\n10S. Chakravarty and H.-Y . Kee, Proc. Natl. Acad. Sci. USA, 105,\n8835 (2008); N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Lev-\nallois, J.-B. Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy,\nand L. Taillefer, Nature, 447, 565 (2007).\n11D. Garcia-Aldea and S. Chakravarty, Phys. Rev. B, 82, 184526\n(2010); B. J. Ramshaw, B. Vignolle, J. Day, R. Liang, W. N.\nHardy, C. Proust, and D. A. Bonn, Nat Phys, 7, 234 (2011).\n12A. Shekhter, personal communication; S. Bhattacharya, M. J. Hig-\ngins, D. C. Johnston, A. J. Jacobson, J. P. Stokes, D. P. Goshorn,\nand J. T. Lewandowski, Phys. Rev. Lett., 60, 1181 (1988).\n13C. Nayak and E. Pivovarov, Phys. Rev. B, 66, 064508 (2002).\n14T. Grover and T. Senthil, Phys. Rev. Lett., 100, 156804 (2008)."    },    {        "title": "1105.3196v3.Local_density_of_states_of_a_quarter_filled_one_dimensional_Mott_insulator_with_a_boundary.pdf",        "content": "arXiv:1105.3196v3  [cond-mat.str-el]  30 May 2012Local density of states of a quarter-filled one-dimensional\nMott insulator with a boundary\nDirk Schuricht1,2\n1Institute for Theory of Statistical Physics, RWTH Aachen, 5 2056 Aachen, Germany\n2JARA-Fundamentals of Future Information Technology\n(Dated: July 7, 2018)\nWe study the low-energy limit of a quarter-filled one-dimens ional Mott insulator. We analytically\ndetermine the local density of states in the presence of a str ong impurity potential, which is modeled\nby a boundary. To this end we calculate the Green function usi ng field theoretical methods. The\nFourier transform of the local density of states shows signa tures of a pinning of the spin-density\nwave at the impurity as well as several dispersing features a t frequencies above the charge gap.\nThese features can be interpreted as propagating spin and ch arge degrees of freedom. Their rela-\ntive strength can be attributed to the “quasi-fermionic” be havior of charge excitations with equal\nmomenta. Furthermore, we discuss the effect of bound states l ocalized at the impurity. Finally,\nwe give an overview of the local density of states in various o ne-dimensional systems and discuss\nimplications for scanning tunneling microscopy experimen ts.\nPACS numbers: 68.37.Ef, 71.10.Pm, 72.80.Sk\nI. INTRODUCTION\nOver the past decades scanning tunneling microscopy\n(STM) and spectroscopy techniques have been estab-\nlished as an important experimental tool to study\nstrongly correlated electron systems with a spatial res-\nolution down to the atomic scale. In these experiments\nthe tunneling current between the sample and the STM\ntip is measured as a function of its position and the ap-\nplied voltage. The tunneling current is directly related1\nto the local density of states (LDOS) in the sample; thus\nSTM experiments provide a tool to investigate the spa-\ntial dependence of the LDOS, which may arise for exam-\nple due to impurities. These spatial modulations can be\nanalyzed in terms of the Fourier transform of the tun-\nneling conductance which is directly proportional to the\nFourier transform of the LDOS. Using this method of\nanalyzing STM data one can infer informations about\nthe bulk state of matter like the properties of its quasi-\nparticle excitations. For example, this has been used to\nstudy high-temperature superconductors2as well as car-\nbon nanotubes3.\nTheoretical studies of the LDOS and STM in the pres-\nence of boundaries havebeen carriedout in particular for\none-dimensional systems including Luttinger liquids4–8,\nopenHubbardchains5,9,andcharge-densitywave(CDW)\nstates10,11. In these systems the coupling to impurities\nis relevant11–13, which leads, at sufficiently low energies,\nto an effective cutting of the system into disconnected\npieces. In this sense the effect of an impurity can be\nmodeled by a boundary condition. Previous studies of\nthe Fourier transform of the LDOS in both gapless6–8\nand gapped10,11systems revealed a pinning of the CDW\nat the impurity as well as individually propagating spin\nand charge degrees of freedom.\nHere we study a quarter-filled one-dimensional Mott\ninsulator in the presence of a strong impurity poten-tial. It has been established13,14that for sufficiently\nstrong interactions double Umklapp processes are rele-\nvant and generate a gap in the charge sector. As a mi-\ncroscopic realization one may think of a quarter-filled\nHubbard model extended by including nearest-neighbor\ninteractions, which possesses15an insulating phase for\nsufficiently strong repulsions. Experimentally these sys-\ntems arerelevantforthe descriptionoforganicquasi-one-\ndimensional conductors, i.e. the Bechgaard and Fabre\nsalts16.\nIn this article we study the low-energy behavior of\nthe LDOS by employing the field theoretical description\nof the Mott insulator. The spectrum consists of mass-\nless spin excitations (spinons) carrying spin ±1/2 but\nno charge and massive charge excitations (solitons and\nantisolitons) carrying charge ±e/2 but no spin. In this\nframeworkthe impurityis modeledbyaboundaryforthe\ncollective excitations. In particular, we will focus on sig-\nnatures of dispersing quasiparticles in the spatial Fourier\ntransform of the LDOS, as they have been clearly ob-\nserved in similar studies of Luttinger liquids6–8and half-\nfilled Mott insulators as well as CDW states10,11. The\nspectralfunctionofthetranslationallyinvariant,quarter-\nfilled Mott insulator has been previously derived17by\nEssler and Tsvelik. From the quantum numbers stated\nabove it is clear that an electron will fractionalize into\nat least one spinon and two antisolitons. Thus the spec-\ntral function exhibits a featureless scattering continuum\nof at least three excitations, which is also consistent with\nexperimental results18from angle-resolved photoelectron\nspectroscopy on Bechgaard salts.\nThis article is organized as follows. In Sec. II we\nwill briefly discuss the field theoretical description of the\nquarter-filled Mott insulator. In Sec. III we present our\nresults for the single-particle Green function. This will\nthen be used to derive an analytic expression for the\nLDOS, which we will discuss in detail in Sec. IV. Our\nmain result is presented in Eq. (15) as well as Figs. 1and 2, which show the Fourier transform of the LDOS in\nthepresenceofanimpurity. Surprisingly,wefindtwodis-\npersingfeatureswhichfollowthedispersionrelations(19)\nand (20) respectively. These features can be interpreted\nas arising from an antisoliton or a spinon-antisoliton pair\npropagating between the position of the STM tip and\nthe impurity. In addition, we observe a non-dispersing\nsingularity at momentum Q= 2kFwhich is indicative of\na pinning of a spin-density wave (SDW) at the bound-\nary. A detailed analysis further reveals dispersing fea-\ntures originating in propagating spinons and antisoliton\npairs. The suppression of these features is attributed to\nthe “quasi-fermionic” behavior of antisolitons with equal\nmomenta. Finally we explain the different findings in\nthe spectral function and the LDOS. In Sec. V we dis-\ncuss the effect of possible boundary bound states on the\nLDOS and in Sec. VI we compare our results to the half-\nfilled Mott insulator studied10,11previously. Finally, in\nSec. VII we set our results in the context of other one-\ndimensional systems and discuss implications for STM\nexperiments on quasi-one-dimensional materials. Read-\ners mainly interested in the qualitative features of the\nLDOS may start with this section and study the detailed\nresults afterwards. Technical details have been moved to\nthe appendix.\nII. THE MODEL\nAs the underlying microscopic model for the quarter-\nfilled Mott insulator one may start with the extended\nHubbard model13\nHHubbard=−t/summationdisplay\nj,σ/bracketleftBig\nc†\nj,σcj+1,σ+c†\nj+1,σcj,σ/bracketrightBig\n+U/summationdisplay\njnj,↑nj,↓+V/summationdisplay\njnjnj+1,(1)\nwherec†\nj,σandcj,σare electron creation and annihilation\noperators at site jwith spin σ=↑,↓,nj,σ=c†\nj,σcj,σ, and\nnj=nj,↑+nj,↓. At quarter filling and for sufficiently\nlarge on-site and nearest-neighbor repulsions U,V≥4t\nthis model possessesa Mott insulating phase15which can\nbe thought of as a microscopic realization of the field-\ntheoretical system we will study below. The extended\nHubbard model (1) can be used as an effective model\nfor the description of Bechgaard salts16; its spectral and\ntransport properties have been investigated in detail us-\ning various techniques13,19.\nInstead of working with the microscopic model (1) we\nwill study here the correspondinglow-energyfield theory.\nThe continuum description is obtained by focusing on\nthe degrees of freedom around the Fermi points ±kF.\nWe introduceslowlyvaryingright-and left-movingFermi\nfields as\ncj,σ√a0→Ψσ(x) =eikFxRσ(x)+e−ikFxLσ(x),(2)and similarly for the electron creation operators. Here\na0denotes the lattice spacing, x=ja0, and the Fermi\nmomentumis givenby kF=π/4a0atquarterfilling. The\nimpurity potential is assumed to be strong such that we\ncan model it by a boundary condition on the continuum\nelectron field\nΨσ(x= 0) = 0 . (3)\nUsingstandardbosonizationoftheright-andleft-moving\nFermi fields it was shown13,14that double Umklapp pro-\ncesses result in a cosine term in the charge sector, which\nis relevant at low energies and generates a gap. Thus the\neffective low-energy Hamiltonian is given by20\nH=Hc+Hs, (4)\nHc=vc\n16π/integraldisplay0\n−∞dx/bracketleftBig/parenleftbig\n∂xΦc/parenrightbig2+/parenleftbig\n∂xΘc/parenrightbig2/bracketrightBig\n−gc\n(2π)2/integraldisplay0\n−∞dxcos/parenleftbig\nβΦc), (5)\nHs=vs\n16π/integraldisplay0\n−∞dx/bracketleftBig/parenleftbig\n∂xΦs/parenrightbig2+/parenleftbig\n∂xΘs/parenrightbig2/bracketrightBig\n,(6)\nwhere, as a consequence of (3), the canonical Bose fields\nΦc,sare subject to the hard-wall boundary conditions\nΦc,s(x= 0) = 0 . (7)\nThe fields Θ c,sare dual to Φ c,s. The charge and spin ve-\nlocitiesvc,s, the Luttinger parameter β, and the coupling\nconstant gcare functions of the hopping and interactions\ninanunderlyingmicroscopicmodel. Within thefieldthe-\nory they can be viewed as phenomenological parameters.\nExperimental estimates21for the Luttinger parameter in\nthe Bechgaard salts yield β2≈0.9.\nIn the bosonization procedure leading to the effective\nlow-energy Hamiltonian (4) the explicit relation between\nthe Fermi and the Bose fields is given by17,22\nR†\nσ=ησ√\n2πexp/bracketleftbiggi\n4/parenleftbiggβ\n2Φc+2\nβΘc/parenrightbigg/bracketrightbigg\n×exp/bracketleftbiggi\n4fσ/parenleftbig\nΦs+Θs/parenrightbig/bracketrightbigg\n,(8)\nL†\nσ=ησ√\n2πexp/bracketleftbigg\n−i\n4/parenleftbiggβ\n2Φc−2\nβΘc/parenrightbigg/bracketrightbigg\n×exp/bracketleftbigg\n−i\n4fσ/parenleftbig\nΦs−Θs/parenrightbig/bracketrightbigg\n,(9)\nwhere the Klein factors ησsatisfy the anticommutation\nrelations {ησ,η′\nσ}= 2δσσ′andf↑= 1 =−f↓.\nThe charge excitations are described by the sine-\nGordon model on the half-line (5). For β <1 the cosine\nterm is relevant and generates a gap. The excitations\nare massive solitons and antisolitons which carry charges\n±e/2 respectively. They possess a relativistic dispersion\nrelation; we parametrize their energy and momentum in\nthe usual way using the rapidity θby\nE= ∆cosh θ, P=∆\nvcsinhθ. (10)\n2The soliton mass ∆ is a function23of the bare parame-\ntersgcandβbut will be viewed here as a phenomenolog-\nical parameter replacing the coupling constant gc. In the\nregime1/2≤β2solitonsand antisolitonsarethe only ex-\ncitations in the charge sector; for β2<1/2 propagating\nbreather (soliton-antisoliton) bound states exist as well.\nAt the Luther-Emery point (LEP) β2= 1/2 the charge\nsector is equivalent to a free massive Dirac theory24. The\nsine-Gordon model (5) with the boundary condition (7)\nisintegrable25,26, which wewill usebelowtocalculatethe\nnecessary correlation functions of right- and left-moving\nFermi fields. The spin sector (6) describes massless rela-\ntivistic spinon excitations which propagate with velocity\nvsand carry spin ±1/2. We have already assumed spin\nrotational invariance, i.e. a possible Luttinger parameter\nin the spin sector is set to one.\nWe note that the half-filled Mott insulator studied\npreviously10,11possesses the same effective low-energy\nHamiltonian (4)–(6). The difference to the quarter-filled\ncase studied in this article is given22by the bosonization\nrelations (8) and (9) for the right- and left-moving Fermi\nfields. This leads to differing Green functions as well as\nLDOS. We will proceed with the presentation of the re-\nsults for the quarter-filled Mott insulator and present a\ndetailed comparison to the half-filled system in Sec. VI.\nIII. GREEN FUNCTION\nIn order to derive the LDOS below we calculate the\ntime-ordered Green function in Euclidean space,\nGσσ′(τ,x1,x2) =−∝an}bracketle{t0b|TτΨσ(τ,x1)Ψ†\nσ′(0,x2)|0b∝an}bracketri}ht,\n(11)\nwhere|0b∝an}bracketri}htis the ground state of (4) in the presence of\nthe boundary and τ= itdenotes imaginary time. At low\nenergies we can linearize around the Fermi points ±kF,\nwhich results in\nGσσ′=eikF(x1−x2)GRR\nσσ′+e−ikF(x1−x2)GLL\nσσ′\n+eikF(x1+x2)GRL\nσσ′+e−ikF(x1+x2)GLR\nσσ′,(12)\nwhere e.g. GRL\nσσ′=−∝an}bracketle{t0b|TτRσ(τ,x1)L†\nσ′(0,x2)|0b∝an}bracketri}ht. For\nthe calculation of the LDOS we have to set x1=x2be-\nlow. In particular, we will focus on the spatial Fourier\ntransform of the LDOS as physical properties can be\nmore easily identified. The four terms in the decomposi-\ntion (12) then contribute in different regions in momen-\ntum space, i.e. GRL\nσσ′andGLR\nσσ′contribute for Q≈ ±2kF\nwhileGRR\nσσ′andGLL\nσσ′contributefor Q≈0. Inthetransla-\ntionally invariant system right- and left-moving fields in\nthe spin sector decouple which results in GRL\nσσ′=GLR\nσσ′=\n0. The presence of a boundary couples right and left\nsectors and concomitantly the Fourier transform of the\nGreen function (12) acquires a non-zero component at\nQ≈ ±2kF. We will focus on the 2 kF-part of the Green\nfunction and the LDOS in the following, since this pro-\nvides a particularly clean way of investigating boundaryeffects. We note that the other components of the Green\nfunction (12) can be calculated in the same way.\nDue to the spin-charge separated Hamiltonian (4) the\nGreen function GRL\nσσ′factorizes into a product of correla-\ntionfunctionsinthespinandchargesectors. Thecorrela-\ntion functions in the spin sector can be straightforwardly\nderived using standard methods like boundary conformal\nfield theory27. On the other hand, the integrability ofthe\nsine-Gordon model on the half-line (6) enables us to cal-\nculate correlationfunctions in the chargesector using the\nboundary state formalism25together with a form-factor\nexpansion22,28,29. This yields the following result for the\n2kF-component of the Green function\nGRL\nσσ′(τ,x1,x2)\n=−1\n2πδσσ′/radicalbig\nvsτ−i(x1+x2)∞/summationdisplay\nk=0Gk(τ,x1,x2).(13)\nThe infinite seriesoriginatesfrom the applied form-factor\nexpansion, which constitutes an expansion of correlation\nfunctions in the number of contributing solitons and an-\ntisolitons. In App. A we derive explicit expressions for\nthe first three terms Gkwhich correspond to two-particle\nprocesses in the charge sector. We note that (13) is valid\nat zero temperature. The effect of small temperatures\nT≪∆ on the gapless spin sector can be incorporated\nusing conformal field theory27as discussed for the half-\nfilled Mott insulator in Ref. 11. In the next section we\nuse the result (13) to derive the Fourier transform of the\nLDOS.\nIV. LOCAL DENSITY OF STATES\nThe LDOS can now be calculated directly from the\nGreen function. In order to analyze the physical prop-\nerties it is useful to consider the spatial Fourier trans-\nform of the LDOS as features like dispersing quasipar-\nticles can be more easily identified. This technique was\npreviouslyapplied to the LDOS ofLuttinger liquids6–8as\nwell as CDW states and half-filled Mott insulators10,11.\nThe Fourier transform of the LDOS for positive energies\nE >0 is given by30\nN(E,Q) =−1\n2π/integraldisplay0\n−∞dx/integraldisplay∞\n−∞dtei(Et−Qx)\n×Gσσ(τ >0,x,x)/vextendsingle/vextendsingle/vextendsingle\nτ→it+δ.(14)\nHere the Green function has been analytically continued\nto real times and we have taken the limit x1→x2≡x.\nWe note that due to the assumed spin rotational invari-\nance the LDOS does not depend on σ. As mentioned\nbefore, we will concentrate on the 2 kF-component as it\nvanishes in the absence of the boundary and hence of-\nfers a particularly clean way of investigating boundary\neffects. For Q≈2kFonlyGRLcontributes and starting\n3from (13) we arrive at our main result ( |q| ≪2kF)\nN(E,2kF+q) =−Θ(E−2∆)2/summationdisplay\nk=0Nk(E,2kF+q),(15)\nNk(E,2kF+q) =Z2e−iπ/4√vs\n2π3/2\n×/integraldisplaydθ1dθ2\n(2π)2Θ(E−∆/summationtext\nicoshθi)/radicalbig\nE−∆/summationtext\nicoshθi\n×hk(θ1,θ2)\nvsqk−2(E−∆/summationtext\nicoshθi)+iδ,(16)\nwhere\nh0(θ1,θ2) =1\n2|G(θ1−θ2)|2,\nh1(θ1,θ2) =K(θ1+iπ\n2)eθ1/4G(θ2−θ1)G(θ1+θ2)∗,\nh2(θ1,θ2) =−1\n2e(θ1+θ2)/42/productdisplay\ni=1K(θi+iπ\n2)\n×sinhθ1+θ2+iπ\nξ\nsinhθ1+θ2\nξS0(θ1+θ2+iπ)|G(θ1−θ2)|2,\nas well as q0=q,q1=q−2∆\nvcsinhθ1, andq2=\nq−2∆\nvc/summationtext\nisinhθi. Explicitintegralrepresentationsforthe\nscattering matrix S0(θ), the boundary reflection matrix\nK(θ), and the function G(θ) are given in App. B. Phys-\nicallyS0(θ) describes the scattering of two antisolitons\nwith relative momentum correspondingto the rapidity θ,\nK(θ) is the reflection amplitude ofan antisoliton with ra-\npidityθofftheboundary[andthusincorporatestheinfor-\nmationonthe boundarycondition(7)], and G(θ) appears\nin the matrix element [i.e. form factor] of the charge part\nof (8) and (9) with two-antisoliton states [see (A4)]. The\nparameter ξis defined as ξ=β2/(1−β2). The normal-\nization constant31Z2has dimension Z2∝∆β2/16+1/β2;\nthus in order to obtain dimensionless quantities we mul-\ntiply the LDOS by ∆3/2−β2/16−1/β2/√vsin all figures.\nThe singularity of the integrands in (16) is smeared out\nby taking δsmall but finite; we choose δ= 0.01 unless\nstated otherwise. This mimics the broadening of singu-\nlarities in experiments due to the instrumental resolution\nand the finite temperature. The LDOS (15) vanishes for\nE <2∆ as the fractionalization of an electron creates at\nleast two antisolitons.\nThe three terms (16) constitute the leading contribu-\ntion fromthe form-factorexpansion. Ascan be seen from\nthe double integralsthey all originatein two-particlepro-\ncesses in the charge sector. The number of boundary\nreflection matrices K(θ) indicates that Nkinvolveskin-\nteractions with the boundary, i.e. the terms (16) contain\nno, one, ortwo reflectionsofchargeexcitations. The sub-\nleading contributions involve at least three particles in\nthe charge sector which either come from a higher num-\nber of particles in the intermediate state or from higher-\norder processes due to the boundary. Similar terms were\n|N(E,2kF+q)|\n02π4π\nArg N(E,2kF+q)\n-5-4 -3 -2 -1 0 1 2 3 4 5\nvsq/∆|N(E,2kF+q)|\n02π4π\nArg N(E,2kF+q)|N|\nArg Nβ2=0.9\nβ2=0.7\nFIG.1: Constantenergyscanfor E= 3∆:|N(E,2kF+q)|and\nArgN(E,2kF+q) forvc= 1.2vsandβ2= 0.9 (upper panel)\nandβ2= 0.7 (lower panel). Both plots are shown on the same\nscale. We observe a peak at q= 0 and dispersing features at\nq=±2/radicalbig\n(E−∆)2−∆2/vcas well as q=±2√\nE2−4∆2/vc.\nForq <0 the dispersing features are strongly suppressed.\nthoroughlyanalyzedforthe LDOS in aCDWstate at the\nLEP11as well as the spectral function in the Ising model\nwith a boundary magnetic field32; they were found to be\nnegligible in both cases. We further note that the sup-\npression of sub-leading terms in the form-factor expan-\nsion for bulk two-point functions is a well-known feature\nof massive integrable field theories22,33. We thus expect\nthe higher-order corrections to (15) to be negligible in\nthe low-energy regime we consider here.\nThe constant energy scan of the Fourier transform of\nthe LDOS (15) is shown in Fig. 1. We observe a singu-\nlarity at q= 0, i.e. Q= 2kF, which is indicative of the\npinning ofthe 2 kF-SDW at the boundary (see Sec. IVA).\nFurthermore there is a peak at q >0. The constant\nmomentum scans shown in Fig. 2 reveal that this peak\nshows dispersing behavior and splits above a critical mo-\nmentum q > q0[q0will be defined in Eq. (22)]. More pre-\ncisely these features follow the dispersion relations (19)\nand (20) respectively (see Sec. IVB). The existence of\ndispersing features is at first surprising since the spectral\nfunction17of the translationally invariant system does\nnot show any dispersing quasiparticles. The processes\ncontributing to the LDOS (15) can be thought of as aris-\ningfromcreatinganelectronataposition x, thefraction-\nalization of this electron into at least one spinon and two\nantisolitons, the propagation of these elementary excita-\ntions through the system, and their subsequent recombi-\nnation at the position x. This allows a natural interpre-\ntation of the leading terms in the form-factor expansion\n(16) in terms of propagating antisolitons and spinons. In\nthe next sub-sections we will analyze the different terms\n(16) separately. Afterwards we comment on the different\nfindings in the spectral function and the LDOS.\n42 3 4 5\nE/∆-202vsq/∆=5\n4\n3\n1\n-1|N(E,2kF+q)|\nFIG. 2: |N(E,2kF+q)|forβ2= 0.9 andvc= 1.2vs. The\ncurves are constant q-scans which have been offset along the\ny-axis by a constant with respect to one another. The LDOS\nis dominated by the peak at q= 0. We further observe dis-\npersing features at E1c= ∆ +/radicalbig\n∆2+(vcq/2)2as well as\nE1cs= ∆+∆/radicalbig\n1−(vs/vc)2+vsq/2 forq > q0.\nA. No reflections in the charge sector:\nN0(E,2kF+q)\nLet us start with the analysis of the first term in the\nform-factor expansion (15), which is shown in Fig. 3. We\nfirst note that this term is dominated by a singularity\natq= 0, i.e. Q= 2kF. This singularity is due to the\npinningofthe2 kF-SDWattheboundary. As kF=π/4a0\nwe find a spatial modulation with periodicity 2 π/2kF=\n4a0as sketched13,34in Fig. 3.b. Close to q= 0 the first\nterm in the form-factor expansion and thus the whole\nLDOS behaves as35\nN(E,2kF+q)∼N0(E,2kF+q)∼1√vsq.(17)\nWe note that the exponent is independent of the Lut-\ntinger parameter β(see Fig. 3.a). This singularity is\nsimilar to the ones observed in the LDOS of Luttinger\nliquids6,8as well as CDW states10,11where, however, the\nexponents depend on the interaction strength and thus\nthe Luttinger parameter.\nFurthermore, N0possesses a weak dispersing feature\nfollowing (see36Fig. 3.c)\nEs(q) = 2∆+vsq\n2, q >0. (18)\nAs already discussed, the created electron fractionalizes\ninto two antisolitons and (at least) one spinon which can\npropagate through the system. In this way of thinking\nthe dispersing feature at Esarises from two antisolitons\nwith zeromomentum (solid linesin Fig.3.d) contributing\nan energy 2∆ and a massless soliton with momentum q\n(dashed line in Fig. 3.d). The appearance of vs/2 inEs\nis due to the fact that the spinon has to propagate to\nthe boundary and back, thus covering the distance 2 xin\ntimet.2 3 4 5\nE/∆00.1|N0(E,2kF+q)|vsq/∆=2\nvsq/∆=3\nvsq/∆=4\nvsq/∆=5-2 -1 0\nlog10(vsq/∆)-101log10|N0(E,2kF+q)|β2=0.5\nβ2=0.7\nβ2=0.9b)\nd)a)\nc)~1/2kF\nFIG. 3: (Color online) N0(E,2kF+q) forvc= 1.6vs. a)\nLogarithmic plot close to the singularity at q= 0 for δ=\n0.001. The dotted line is a guide to the eye; it has gradient\n−1/2. b)Sketch13,34ofthe2kF-SDWpinnedat theboundary.\nc)|N0(E,2kF+q)|forβ2= 0.9 and different momenta. We\nobserve a dispersing feature at Es= 2∆ + vsq/2 indicated\nby small arrows. d) Sketch of the process giving rise to the\npropagating feature. The electron decomposes into two zero -\nmomentum antisolitons (solid lines) and one spinon (dashed\nline) reflected at the boundary.\nB. One reflection in the charge sector:\nN1(E,2kF+q)\nThe next term in the form-factor expansion (15) con-\ntains one boundary reflection matrix K(θ). Thus it is\nnatural to interpret it as arising from processes involv-\ning the reflection of one antisoliton at the boundary. In\nFig. 4.a we show N1for different momenta. We clearly\nobservea dispersing feature (indicated by up-arrows) fol-\nlowing\nE1c(q) = ∆+/radicalbigg\n∆2+/parenleftBigvcq\n2/parenrightBig2\n. (19)\nThis feature originates in the process where one anti-\nsoliton propagates with momentum qwhile the other\nantisoliton and the spinon possess zero momentum (see\nFig. 4.b). The first term in E1cis the rest mass of the\nzero-momentum antisoliton while the second term is the\nenergy of the propagating antisoliton.\nFurthermore, when qexceeds the critical value q0a\nsecond dispersing feature appears (indicated by down-\narrows in Fig. 4.a) at\nE1cs(q) = ∆+∆/radicalBigg\n1−/parenleftbiggvs\nvc/parenrightbigg2\n+vsq\n2.(20)\nThe critical momentum q0can be determined from the\ncondition that the spin and charge excitations have the\n52 3 4 5 6\nE/∆00.10.20.3|N1(E,2kF+q)|vsq/∆=2\nvsq/∆=3\nvsq/∆=4\nvsq/∆=5\nb) c)a)\nFIG. 4: (Color online) a) |N1(E,2kF+q)|forβ2= 0.9,vc=\n1.6vs, and different momenta. We observe dispersing features\natE1c= ∆ +/radicalbig\n∆2+(vcq/2)2indicated by up-arrows and\nE1cs= ∆+∆/radicalbig\n1−(vs/vc)2+vsq/2indicatedbydown-arrows.\nb) Process resulting in E1c. The electron decomposes into one\nantisoliton and spinon with zero momentum as well as one\nantisoliton reflected at the boundary. c) Process resulting in\nE1cs.\nsame group velocity\n∂E1c\n∂q/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq=q0!=∂Es\n∂q=vs\n2, (21)\nwhich leads to\nq0=2∆vs\nvc/radicalbig\nv2c−v2s. (22)\nThus this feature can be thought of as arising from an\nantisolitonwith momentum q0and a spinon carryingmo-\nmentum q−q0, while the second antisoliton stays at rest\n(see Fig. 4.c). The fact that the propagating antisoliton\nand the spinon have the same group velocity results in\nan increased probability that both simultaneously arrive\nat position xafter reflection at the boundary.\nThe observedsplitting ofthe quasiparticlepeakis rem-\niniscent of what is found for the single-particle spectral\nfunction in the translationally invariant, half-filled Mott\ninsulator37as well as the LDOS of a CDW state in the\npresence of a boundary in the regime of attractive inter-\nactions10,11. The peak splitting is a consequence of the\ncurvature of the antisoliton dispersion (and thus of the\nchargegap) togetherwith the condition vc> vs. Hence it\ncannot be observed in the Luttinger liquid case6–8where\nboth sectors are massless.\nC. Two reflections in the charge sector:\nN2(E,2kF+q)\nThe leading term in second order in the boundary re-\nflection matrix is N2. As in the previous term we find2 3 4 5\nE/∆00.03 |N2(E,2kF+q)|vsq/∆=2\nvsq/∆=3\nvsq/∆=4\nvsq/∆=5\nb) c)a)\nFIG. 5: (Color online) a) |N2(E,2kF+q)|forβ2= 0.9,\nvc= 1.6vs, and different momenta. We observe dispers-\ning features at E2c= 2/radicalbig\n∆2+(vcq/4)2(up-arrows) and\nE2cs= 2∆/radicalbig\n1−(vs/vc)2+vsq/2 (down-arrows). Processes\nresulting in (b) E2cand in (c) E2cs.\ntwo dispersing features. First, indicated by up-arrows in\nFig. 5.a, there is a feature following\nE2c(q) = 2/radicalbigg\n∆2+/parenleftBigvcq\n4/parenrightBig2\n. (23)\nThis feature can be thought of as arising from the prop-\nagation of both antisolitons with momenta q/2 and thus\nthe same velocity, while the spinon has zero momentum\n(see Fig. 5.b). The second feature is found at\nE2cs(q) = 2∆/radicalBigg\n1−/parenleftbiggvs\nvc/parenrightbigg2\n+vsq\n2,(24)\nprovided qexceeds the critical momentum 2 q0. This fea-\nture is indicated by down-arrows in Fig. 5.a. The in-\nterpretation is that both antisolitons carry momentum\nq0while the excess momentum q−2q0is carried by the\nspinon (see Fig. 5.c).\nWe note that at finite momentum q >0 the term N1\ndominates the LDOS as can be seen by comparing the\nscales in Figs. 3.c, 4.a, and 5.a, respectively. More im-\nportantly, the dispersing features originating in N1are\nmuch more pronouncedthan those resulting from N0and\nN2. We attribute this to the fact that, at least in the\nnaive interpretation of Figs. 4.b and c, the antisolitons\ninvolved in processes contributing to N1possess different\nmomenta as one of them is at rest while the other propa-\ngatesthroughthe system. In contrast, the dispersingfea-\ntures inN0andN2originate from processes in which the\ntwo antisolitons possess equal momenta (see Figs. 3.d as\nwellas 5.b andc, respectively). Thustheir relativerapid-\nity vanishes and the scattering matrix is S0(θ= 0) =−1\n(see App. B). The resulting “quasi-fermionic” behavior\nleads to a strong suppression of the dispersing features\ninN0andN2via the Pauli principle.\n6Finally let us comment on the different findings in the\nspectral function17of the translationally invariant sys-\ntem and the LDOS (15) in the presence of a bound-\nary. The spectral function is obtained from the Green\nfunction G(τ,x,0) via Fourier transformation with re-\nspect to t=−iτandx. The contributing processes\nrequire the propagation of all excitations, and in par-\nticular both antisolitons, from (0 ,0) to (t,x). Thus one\nmayexpectadispersingfeatureat E= 2/radicalbig\n∆2+(vcq/2)2\nfrom the antisolitons propagating with the same mo-\nmentum q/2. In addition, above the critical momentum\n2q0= 2∆vs/vc/radicalbig\nv2c−v2sa linearly dispersing feature fol-\nlowingE= 2∆/radicalbig\n1−(vs/vc)2+vsqis expected. Indeed\nthese are exactly the conditions17for the threshold be-\nlow which the spectral function vanishes. Above the\nthreshold the spectral function is rather featureless, in\nparticular there are no dispersing peaks associated with\nthe propagation of charge or spin degrees of freedom.\nThe absence of dispersing features may be attributed to\nthe “quasi-fermionic” behavior of antisolitons, as all pro-\ncesses contributing to the spectral function will contain\nantisolitons with equal momenta. In contrast, the pro-\ncesses contributing to the LDOS (14) are given by the\nfractionalization of an electron at position x, the propa-\ngation of the two antisolitons and the spinon through the\nsystem, and the subsequent recombinationat the original\npositionx. In particular, processeswherethe momentum\nqis solely carried by one of the antisolitons or a spinon-\nantisolitonpairwillcontribute,resultinginthedispersing\nfeatures at E1candE1cs, respectively. The observation\nof these features thus underlines that the spectral func-\ntion and the LDOS probe rather different physical pro-\ncesses. While the spectral function requires the propaga-\ntion of all excitations between two points in the system,\nthe LDOS also probes processes where only one or two\nexcitations propagate to the impurity and back.\nV. LOCAL DENSITY OF STATES IN THE\nPRESENCE OF BOUNDARY BOUND STATES\nUp to now we have concentrated on the boundary con-\nditions (3) for the electron field or equivalently (7) for\nthe canonical Bose fields. In this section we will consider\nmore general phase shifts for the electron fields which\nlead to arbitrary Dirichlet boundary conditions for the\nBose fields. As is well known, such boundary condi-\ntionscanresultinthe existenceofboundaryboundstates\n(BBS’s), which are expected10,11to manifest themselves\nin the LDOS as features within the spectral gap. Here\nwe will focus on general Dirichlet boundary conditions in\nthe charge sector\nΦc(x= 0) = Φ0\nc,−π\nβ≤Φ0\nc≤π\nβ,(25)\nwhile in the spin sector we still assume Φ s(x= 0) = 0.\nIn the boundary state formalism25applied here the infor-\nmation about the boundary condition (25) is encoded in1 2 3 4 5\nE/∆vsq/∆=5\n4\n3\n2\n1\n0\n-1\n-2|N(E,2kF+q)|\nFIG.6:|N(E,2kF+q)|forβ2= 0.5, Φ0\nc=−4, andvc= 1.2vs.\nThe curves are constant q-scans which have been offset along\nthe y-axis by a constant with respect to one another. The\nexistence of a BBS results in spectral weight within the two-\nsoliton gap E <2∆.\nthe boundary reflection matrix K(θ). In particular, the\nexistence of a BBS will show up as a pole in the physical\nstrip 0≤Imθ≤π/2. As was shown in Ref. 38 this will\nbe the case provided\n−π\nβ<Φ0\nc<−βπ; (26)\nthe pole at θ= iγcorresponds to a BBS with energy\nEbbs= ∆sinγ, γ=π+βΦ0\nc\n2−2β2. (27)\nWhen calculating the Green function the pole of the\nboundary reflection matrix will lead to two additional\nterms,G3andG4, in the form-factorexpansion(13). This\nin turn yields two additional terms in the LDOS (15).\nTo be explicit, let us consider the LEP β2= 1/2 in\nthe following. The three leading terms in the form-factor\nexpansion of the LDOS are still given by (16) where the\nboundary reflection matrix now depends explicitly on Φ0\ns\nK(θ) =sin/parenleftbig\niθ\n2−Φ0\nc√\n8/parenrightbig\ncos/parenleftbig\niθ\n2+Φ0\nc√\n8/parenrightbig. (28)\nThis obviously possesses a pole in the physical strip 0 ≤\nImθ≤π/2 provided Φ0\nc<−π/√\n2 corresponding to the\nexistence of a BBS. The resulting two additional terms\nin the LDOS read ( k= 3,4)\nNk(E,2kF+q) =Z2√vscosΦ0\nc√\n2e−i(π−√\n2Φ0\nc)/8\nπ3/2\n×/integraldisplaydθ\n2πΘ(E−∆coshθ−Ebbs)√E−∆coshθ−Ebbs\n×hk(θ)\nvsqk−2(E−∆coshθ−Ebbs)+ivsκbbs,(29)\n7where\nh3(θ) =/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinhθ+i\n2(π+√\n2Φ0\nc)\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n,\nh4(θ) =−K(θ+iπ\n2)eθ/4sinh2θ−i\n2(π+√\n2Φ0\nc)\n2,\nEbbs=−∆sinΦ0\nc√\n2, κ bbs=−2∆\nvccosΦ0\nc√\n2,\nas well as q3=qandq4=q−2∆\nvcsinhθ. We note that\nκbbs>0 has the dimension of an inverse length. The\nspatial dependence of the terms G3andG4(see App. A)\nis dominated by ∼eκbbsx(x <0), thus it is natural to\ninterpret 1 /κbbsas the width of the BBS.\nThe terms (29) possess a threshold at E= ∆ +Ebbs\ncorresponding to the creation of the boundary bound\nstate and an additional antisoliton. Interestingly, al-\nthoughN3andN4separately possess jumps at the\nthreshold, the sum of both terms is continuous. This is\nreminiscent of the BBS contributions in the Ising model\nwith a boundary magnetic field32, where a cancellation\nof singularities yields a smooth spectral function at the\nthreshold.\nThe LDOS in the presence of a BBS is shown in Fig. 6.\nWe clearly observe a contribution within the gap E <\n2∆. In contrast to BBS’s in the CDW state10,11there is,\nhowever, no singularity at E=Ebbsas the underlying\nprocesses involve the creation of a spinon, an antisoliton,\nand the BBS. As before the LDOS is dominated by a\nsingularity at q= 0 and we observe dispersing features\nfollowing (19) and (20) respectively.\nVI. COMPARISON TO HALF-FILLED MOTT\nINSULATOR\nThe LDOS of a half-filled Mott insulator in the pres-\nence of a boundary was analyzed in detail39in Ref. 11.\nAs already mentioned, the effective low-energy Hamilto-\nnian is also given by (4)–(6) but the expressions for the\nright- and left-moving Fermi fields in terms of the Bose\nfields differ from (8) and (9). From the physical point\nof view right- and left-moving Fermi fields create and\nannihilate individual antisolitons in the half-filled case\nwhereas they create and annihilate antisoliton pairs in\nthe quarter-filled system. Thus the gap in the LDOS is\ngiven by ∆ in the former and 2∆ in the latter system.\nApart from this we find the following similarities and dif-\nferences.\nIn both cases the LDOS possesses a singularity at\nq= 0 which behavesas N(E,2kF+q)∼1/√vsqindepen-\ndently40of the Luttinger parameter β. This singularity\nis due to the pinning of the 2 kF-SDW at the boundary,\nwhich possesses a wave length 2 a0or 4a0respectively.\nBeside this singularity the LDOS of the half-filled\nMott insulator shows dispersing features at E=/radicalbig\n∆2+(vcq/2)2, atE= ∆ +vsq/2, and, for q > q 0,\natE= ∆/radicalbig\n1−(vs/vc)2+vsq/2. An interpretation isobtained by noting that the electron decomposes into\none spinon and one antisoliton, which then propagate\nthroughthesystem. Astheleadingprocessesinvolveonly\nindividual antisolitons, the ”quasi-fermionic”behavior of\nthe antisolitonsdiscussedat theend ofSec.IVC doesnot\naffect the LDOS. In contrast, in the quarter-filled system\nwe observe only two dispersing modes following (19) and\n(20) respectively (see Fig. 2). All other dispersing fea-\ntures discussed above are strongly suppressed due to the\n”quasi-fermionic” behavior of the antisolitons.\nFinally, in the presence of a BBS with energy Ebbs<\n∆ the LDOS of the half-filled Mott insulator pos-\nsesses a non-dispersing singularity with N(E,2kF+q)∼\n1/√E−Ebbswhich originates from the creation of a\nspinon and the BBS. In contrast, in the quarter-filled\ncase the LDOS is smooth at the threshold E= ∆+Ebbs\nwhich originates from the simultaneous creation of one\nantisoliton and the BBS.\nVII. DISCUSSION AND EXPERIMENTAL\nSIGNATURES\nIn this section we compare qualitative aspects of the\nLDOSofvariousone-dimensionalsystemsin thepresence\nof a boundary or strong impurity potential. We discuss\nthe spatial Fourier transform N(E,Q) defined in (14).\nSpecifically we consider Luttinger liquids6, half-filled\nMott insulators (or CDW states)10,11, and the quarter-\nfilled Mott insulator investigated in Sec. IV. We focus on\ntheQ≈2kF-component of the LDOS as it vanishes in\nthe translationally invariant systems and thus provides\na particularly clean way to investigate boundary effects.\n(We note that the LDOS in the small-momentum regime\nQ≈0 behaves qualitatively similar.) The first feature,\nwhich all three systems have in common, is a singularity\nof the LDOS N(E,2kF+q) atq= 0, which originates\nin the pinning of a charge or spin density wave at the\nimpurity. In the case of a Luttinger liquid the strength\nof this singularity depends on the Luttinger parameter\nKcand thus on the interactions between the electrons,\nwhereas in both the half-filled and quarter-filled Mott in-\nsulator the LDOS behaves as40N(E,2kF+q)∼1/√q.\nFurthermore, as was pointed out in Ref. 22, the gap of\nthe LDOS in the half-filled Mott insulator is given by the\nsoliton mass ∆, which equals the thermal activation gap\nand is half of the gap observed in optical measurements.\nIn contrast, in the quarter-filled system the LDOS pos-\nsesses a gap of 2∆, which is equal to the optical gap but\ntwice the thermal gap.\nIn addition, the LDOS possesses various dispersing\nfeatures (see Fig. 7). In the Luttinger liquid case one\nobserves two linearly dispersing modes corresponding to\nindividually propagating spin and charge degrees of free-\ndom respectively (see Fig. 7.a). In the half-filled Mott in-\nsulator an electron decomposes into massless spin excita-\ntions and one charge excitation with mass ∆. Propagat-\ningspinexcitationsgiverisetoalinearlydispersingmode\n8q0 q0E E E\nq q q∆2∆a) b) c)\nLuttinger liquid half−filled MI quarter−filled MI\nFIG. 7: Schematic picture of the dispersing features observ -\nable in the spatial Fourier transform N(E,2kF+q) of the\nLDOS in different one-dimensional systems. a) In a Luttinger\nliquid one finds two linearly dispersing modes. b) In a half-\nfilled Mott insulator (MI) there exists a gap ∆. One observes\none linear and one curved dispersion. From the latter a third\nfeature splits off at q=q0. c) The quarter-filled Mott insu-\nlator possesses a gap 2∆. One further observes one curved\ndispersion from which a linear mode splits off at q=q0. In\nboth (b) and (c) the thermal activation gap is given by ∆.\nwith velocity vs, while the propagating charge excitation\nresults in a curved dispersion relation. In addition, there\nexists a critical momentum q=q0at which the group ve-\nlocity of the charge excitation equals vs. At this momen-\ntum a third linearly dispersing feature splits away from\nthe charge mode, which has its origin in the collective\npropagation of spin and charge degrees of freedom with\nequal velocities (see Fig. 7.b). We note that the qualita-\ntive behavior discussed for the half-filled Mott insulator\nis also expected in other systems possessing gapless exci-\ntationsinonesectorandgappedonesintheotherlike, for\ninstance, CDW states. Finally, in the quarter-filled Mott\ninsulator an electron decomposes into massless spin exci-\ntations and two charge excitations with masses ∆. This\nresults in a curved dispersion originating in the propaga-\ntion of one charge excitation while the second one stays\nat the position of the STM tip. This dispersion splits at\nthe critical momentum q0due to the collective propaga-\ntion of one charge excitation and additional spin degrees\nof freedom (see Fig. 7.c). We note that one does not\nobserve individually propagating spin excitations. The\nrelevant processes would require the two charge excita-\ntions to possess equal momenta, which is forbidden by\ntheir ”quasi-fermionic” behavior (see Sec. IVC).\nTo sum up, at sufficiently large momenta qwe observe\ntwo linearly dispersing modes in a Luttinger liquid, two\nlinear and one curved dispersion relation in a half-filled\nMott insulator, and one linear and one curved dispersion\nin a quarter-filled Mott insulator. This clearly demon-\nstrates that characteristic properties of the bulk state of\nmatter and its electronic excitations can be infered from\nstudying spatial modulations of the LDOS in the pres-\nence of an impurity.\nThe LDOS discussed in this article is directly related\nto the tunneling current measured in STM experiments.\nIf the density of states in the STM tip is approximately\nindependent of energy, the tunneling conductance will be\ngiven by the thermally smeared LDOS of the sample atthe position of the tip1\ndI(V,x)\ndV∝/integraldisplay\ndEf′(E−eV)N(E,x),(30)\nwheref(E) denotes the Fermi function. N(E,Q) can\nthen easily be obtained via spatial Fourier transforma-\ntion. In this context the models discussed above ap-\nply to quasi-one-dimensional materials at energies above\nthe cross-over scale to three-dimensional behavior. This\nsituation might be experimentally realized for example\nin stripe phases of high-temperature superconductors1,6,\ncarbon nanotubes3,41, Bechgaard and Fabre salts16, self-\norganized atomic gold chains42, and two-leg ladder ma-\nterials43.\nAs discussed above already the knowledge of qualita-\ntive properties such as the presence of a spectral gap and\nits relation to the thermal activation gap as well as the\nnumber and positions of singularities can reveal substan-\ntial information about the electronic properties and ex-\ncitations of quasi-one-dimensional materials. A more de-\ntailedanalysisofquantitativefeatureslikethestrengthof\nsingularities or the precise positions of dispersing modes\ncan even lead to estimates for the Luttinger parameter\nand the velocities of charge and spin excitations. There\nare, of course, experimental limitations. Obviously the\nexperimental resolution and thermal broadenings have\nto be sufficiently small to clearly distinguish the differ-\nent singularities and dispersing modes. Moreover, the\nsimplyfied relation (30) does not contain1the matrix el-\nements for the tunneling from the sample into the tip,\nwhich may introduce a non-trivial spatial dependence of\nthe tunneling conductance on the LDOS and thus ham-\nper a natural interpretation in terms of propagating ex-\ncitations.\nVIII. CONCLUSIONS\nIn conclusion, we have studied a quarter-filledMott in-\nsulatorinthepresenceofanimpurity, whichwasmodeled\nby a boundary condition for the electron fields. In this\nsystem we calculated the 2 kF-component of the spatial\nFourier transform of the LDOS using the boundary state\nformalism together with a form-factor expansion. The\nLDOS is dominated by a singularity at Q= 2kF, which\nis indicative of a pinning of a SDW at the impurity. Fur-\nthermore, we observed several dispersing features above\nthe two-soliton gap E <2∆. We identified their physi-\ncal origin as spin and charge degrees of freedom, which\npropagatebetweentheSTMtipandtheimpurity, andat-\ntributed their relative strength to the “quasi-fermionic”\nbehavior of chargeexcitations with equal momenta. This\nalso explains the absence of dispersing features in the\nspectral function17of the translationally invariant sys-\ntem and underlines that the spectral function and the\nLDOS probe rather different physical processes. We fur-\nther showed that a BBS in the charge sector leads to\n9a non-vanishing LDOS within the two-soliton gap. Fi-\nnally, we discussed our results in the context of other\none-dimensional systems, i.e. Luttinger liquids and half-\nfilled Mott insulators, and argued that the measurement\nof the spatial modulations in the LDOS can be used to\nextract detailed informations about the electronic states\nof quasi-one-dimensional systems.\nAcknowledgments\nI would like to thank Fabian Essler, Markus Garst,\nMarkus Morgenstern, Alexei Tsvelik, and particularly\nVolker Meden for valuable discussions and comments.\nThis work was supported by the German Research Foun-\ndation (DFG) through the Emmy Noether Program.\nAppendix A: Calculation of the Green function\nIn this appendix we derive the leading terms Gkin the\nform-factor expansion of the charge part of the Green\nfunction (13), i.e. we determine the correlation function\n/angbracketleftBig\nOR(τ,x1)O†\nL(0,x2)/angbracketrightBig\n(A1)in the sine-Gordon model (5) on the half line. Here the\noperators O†\nRandO†\nLare given by the charge parts of\n(8) and (9) respectively. We calculate (A1) using the\nboundary state formalism25together with a form-factor\nexpansion22,28,29. The sametechniquewaspreviouslyap-\nplied10,11to determine the Green function of a CDW\nstate in the presence of an impurity. Here we restrict\nourselves to the main steps of the derivation and con-\ncentrate on the differences as compared to Ref. 11, to\nwhich we refer implicitly for all details and definitions\nnot discussed here.\nBy performing a rotation in Euclidean space we map25\nthe problem onto the sine-Gordonmodel on the real axis.\nThe boundary condition is translated into an initial con-\nditionencoded inaboundarystate. Expandingthe latter\nin powersof the boundary reflection matrix and inserting\na resolution of the identity between the operatorsin (A1)\nwe find\n/angbracketleftBig\nOR(τ,x1)O†\nL(0,x2)/angbracketrightBig\n=∞/summationdisplay\nn=0∞/summationdisplay\nm=0Cn2m(τ,x1,x2),(A2)\nwhere we have defined the auxiliary functions\nCn2m(τ,x1,x2) =1\n2m1\nm!1\nn!/integraldisplay∞\n−∞dθ′\n1...dθ′\nm\n(2π)m/integraldisplay∞\n−∞dθ1...dθn\n(2π)nKa1b1(θ′\n1)...Kambm(θ′\nm)\n×∝an}bracketle{t0|OR(τ,x1)|θn,...,θ 1∝an}bracketri}htcn,...,c1c1,...,cn∝an}bracketle{tθ1,...,θ n|O†\nL(0,x2)|−θ′\n1,θ′\n1,...,−θ′\nm,θ′\nm∝an}bracketri}hta1,b1,...,am,bm.(A3)\nHere the states |θn,...,θ 1∝an}bracketri}htcn,...,c1(and so on) are scatter-\ning states of solitons and antisolitons, the indices ai,bi,ci\ntake the values ∓1 and label the corresponding U(1)\ncharges, and energy and momentum of solitons and an-\ntisolitons are given in terms of the rapidities θiandθ′\ni\nby (10). We note that (A2) constitutes an expansion in\ntwo parameters: (i) the number of particles (with rapidi-\ntiesθi) in the intermediate state and (ii) the number of\nreflections of particles at the boundary [the number of\nboundary reflection matrices K(θ′\ni)].\nThe scattering of solitons or antisolitons with rapidi-\ntiesθ1andθ2is encoded in the scattering matrix44\n|θ1,θ2∝an}bracketri}htab=Scd\nab(θ1−θ2)|θ2,θ1∝an}bracketri}htdc. The boundary reflec-\ntion matrix25Kab(θ) similarly describes the reflection at\nthe boundary, |θ∝an}bracketri}hta=K¯ab(iπ\n2−θ)|−θ∝an}bracketri}htb, where ¯a=±for\na=∓. For Dirichlet boundary conditions one has25,38\nK++(θ) =K−−(θ) = 0.\nFurthermore, O(τ,x) =e−xHe−iPτOeiPτexH, where\nHandPare the Hamiltonian and the total momentum\nof the sine-Gordon model on the infinite line. The oper-\natorOR(O†\nL) creates (annihilates) two antisolitons, i.e.\nincreases(decreases)theU(1)chargeby2. Thenecessaryform factors were derived by Lukyanov and Zamolod-\nchikov31. In our conventions they are given by\n∝an}bracketle{t0|OR/L|θ1,θ2∝an}bracketri}ht−−\n=/radicalbig\nZ2e±iπ/8e±(θ1+θ2)/8G(θ1−θ2),(A4)\nwhere the normalization constant Z2and the auxiliary\nfunction G(θ) are stated in App. B. The form factors\nsatisfy the form-factor axioms as stated in Ref. 11 with\nLorentz spin s(OR/L) =±1/4 and semi-locality factor\nl−(OR/L) =e±iπ/4. In the following we will evaluate the\ntermsC20,C22, andC24which result in G0,G1, andG2,\nrespectively.\n1. Derivation of G0\nThe first term to be evaluated is C20. It does not con-\ntain any information about the boundary in the charge\nsector, i.e. it contains no boundary reflection matrices.\nA similar term was calculated in Ref. 17 in the deriva-\ntion of the spectral function in the translationally invari-\n10ant system. After shifting the contour of integration,\nθ1,2→θ1,2+iπ/2, we directly obtain\nG0=C20=Z2\n2eiπ/4/integraldisplay∞\n−∞dθ1dθ2\n(2π)2/vextendsingle/vextendsingleG(θ1−θ2)/vextendsingle/vextendsingle2\n×ei∆\nvcr/summationtext\nisinhθie−∆τ/summationtext\nicoshθi,\nwhere the center-of-mass coordinates are defined by x=\n(x1+x2)/2<0 andr=x1−x2<0. Analytical contin-\nuationτ→itto real times, taking the limit r→0, and\nFourier transformation (14) with respect to tandxyield\nthe first term N0(E,2kF+q) in the form-factorexpansion\nof the LDOS (15).\n2. Derivation of G1\nWe start with C22defined in (A3). As the operator\nORcreates two antisolitons we deduce c1=c2=−;\nK++(θ′) =K−−(θ′) = 0 yields b= ¯a. The matrix el-\nement of O†\nLcontains incoming and outgoing particles\nand thus possesses kinematical poles. We deal with these\nterms following Smirnov28and analytically continue the\nform factor as (see Ref. 11 for a detailed discussion of\nthis procedure)\n−−∝an}bracketle{tθ1,θ2|O†\nL|−θ′,θ′∝an}bracketri}hta¯a\n=−−∝an}bracketle{tθ1+i0,θ2+i0|O†\nL|−θ′,θ′∝an}bracketri}hta¯a\n+2πδ(θ1−θ′)δ−¯a−∝an}bracketle{tθ2|O†\nL|−θ′∝an}bracketri}hta(A5)\n+2πδ(θ1+θ′)δ−bSbc\na¯a(−2θ′)−∝an}bracketle{tθ2|O†\nL|θ′∝an}bracketri}htc\n+2πδ(θ2−θ′)δe¯aSde\n−−(θ1−θ2)d∝an}bracketle{tθ1|O†\nL|−θ′∝an}bracketri}hta\n+2πδ(θ2+θ′)δbeSbc\na¯a(−2θ′)\n×Sde\n−−(θ1−θ2)d∝an}bracketle{tθ1|O†\nL|θ′∝an}bracketri}htc.\nHere we have already omitted the positive imaginary\nparts of the rapidities in the form factors in the 2nd to\n5th lines as they do not possess kinematical poles. In-\nserting (A5) into C22yields to different contributions:\nThe first line leads to a term containing three integra-\ntions over rapidities and is thus a sub-leading correc-\ntion; we will not evaluate it here. The 2nd to 5th\nlines yield using the boundary cross-unitarity condi-\ntion25Kab(θ) =Sab\ncd(2θ)Kdc(−θ) as well as unitarity44\nSc1c2a1a2(θ)Sb1b2c1c2(−θ) =δb1a1δb2a2\n/integraldisplay∞\n−∞dθ′\n2πdθ\n2πK+−(θ′)∝an}bracketle{t0|OR|θ,θ′∝an}bracketri}ht−−−∝an}bracketle{tθ|O†\nL|−θ′∝an}bracketri}ht+\n×ei∆τ(sinhθ+sinhθ′)e∆\nvc(rcoshθ+2xcoshθ′).\n(A6)\nIn order to proceed let us first assume Φ0\nc= 0. This im-\npliesthattheboundaryreflectionmatrixdoesnotdepend\non the U(1) charge, i.e. K+−(θ′) =K−+(θ′) =K(θ′),\nwhereK(θ) is explicitly stated in App. B. In particular,\nK(θ′) is analytic in the physical strip 0 ≤Imθ′≤π/2.Now we shift the contours of integration θ,θ′→θ,θ′+\niπ/2(wenotethatthecontributionoriginatinginthefirst\nterm of (A5) does not possess any poles in the physical\nstrip 0≤Imθi≤π/2) and apply the crossing relation in\nthe second form factor,\n−/angbracketleftbig\nθ+iπ\n2/vextendsingle/vextendsingleO†\nL/vextendsingle/vextendsingle−θ′−iπ\n2/angbracketrightbig\n+=+/angbracketleftbig\n−θ′+iπ\n2/vextendsingle/vextendsingleOL/vextendsingle/vextendsingleθ−iπ\n2/angbracketrightbig∗\n−\n=e−iπ/4∝an}bracketle{t0|OL/vextendsingle/vextendsingle−θ′+i3π\n2,θ−iπ\n2/angbracketrightbig∗\n−−,\nwhere the phase factor e−iπ/4is due to the semi-\nlocality22,28,29of the operator OLwith respect to the\nfundamental fields creating the excitations. Finally we\nuse (A4) to obtain\nG1=Z2eiπ/4/integraldisplay∞\n−∞dθ′\n2πdθ\n2πK(θ′+iπ\n2)G(θ−θ′)G(θ+θ′)∗\n×eθ′/4ei∆\nvc(rsinhθ+2xsinhθ′)e−∆τ(coshθ+coshθ′).\nFourier transformation (14) directly yields the second\ntermN1(E,2kF+q) of the LDOS.\nOn the other hand, if Φ0\nc<−βπ, the boundary re-\nflection matrix K+−(θ′) possesses a pole at θ′= iγ. At\nthe LEP the residue is given by 2icos(Φ0\nc/√\n2) (see (28)).\nThus when shifting the contours of integration in (A6)\nwe obtain the additional BBS contribution\nG3=−2iZ2cosΦ0\nc√\n2e−i(π−√\n2Φ0\nc)/8e−Ebbsτeκbbsx\n×/integraldisplay∞\n−∞dθ\n2π/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinhθ+i\n2(π+√\n2Φ0\nc)\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nei∆\nvcrsinhθe−∆τcoshθ.\nFourier transformation yields N3(E,2kF+q).\n3. Derivation of G2\nFinally we calculate the leading term involving the re-\nflection of two antisolitons, which originates from C24.\nThe analytic continuation of the form factor of O†\nLreads\n−−∝an}bracketle{tθ1,θ2|O†\nL|−θ′\n1,θ′\n1,−θ′\n2,θ′\n2∝an}bracketri}hta¯ab¯b\n=−−∝an}bracketle{tθ1+i0,θ2+i0|O†\nL|−θ′\n1,θ′\n1,−θ′\n2,θ′\n2∝an}bracketri}hta¯ab¯b\n+S−+\n−+(θ′\n1+θ′\n2)δ+aδ+b (A7)\n×−−∝an}bracketle{tθ1,θ2|θ′\n1,θ′\n2∝an}bracketri}ht−−∝an}bracketle{t0|O†\nL|−θ′\n1,−θ′\n2∝an}bracketri}ht++\n+S−+\na¯a(−2θ′\n1)S−+\n−+(−θ′\n1+θ′\n2)δ+b\n×−−∝an}bracketle{tθ1,θ2|−θ′\n1,θ′\n2∝an}bracketri}ht−−∝an}bracketle{t0|O†\nL|θ′\n1,−θ′\n2∝an}bracketri}ht++\n+S−+\nb¯b(−2θ′\n2)S−+\n−+(θ′\n1−θ′\n2)δ+a\n×−−∝an}bracketle{tθ1,θ2|θ′\n1,−θ′\n2∝an}bracketri}ht−−∝an}bracketle{t0|O†\nL|−θ′\n1,θ′\n2∝an}bracketri}ht++\n+S−+\na¯a(−2θ′\n1)S−+\nb¯b(−2θ′\n2)S−+\n−+(−θ′\n1−θ′\n2)\n×−−∝an}bracketle{tθ1,θ2|−θ′\n1,−θ′\n2∝an}bracketri}ht−−∝an}bracketle{t0|O†\nL|θ′\n1,θ′\n2∝an}bracketri}ht++,\nwhere the scalar products are evaluated using the\nFaddeev-Zamolodchikov algebra44,45\n−−∝an}bracketle{tθ1,θ2|θ′\n1,θ′\n2∝an}bracketri}ht−−=(2π)2/bracketleftbig\nδ(θ1−θ′\n2)δ(θ2−θ′\n1)\n+S−−\n−−(θ′\n1−θ2)δ(θ1−θ′\n1)δ(θ2−θ′\n2)/bracketrightbig\n.\n11Therefore the 2nd to 5th line in (A7) yield (we also use\nthe scattering axiom for the first form factor)\n1\n2/integraldisplay∞\n−∞dθ′\n1dθ′\n2\n(2π)2K+−(θ′\n1)K+−(θ′\n2)S−+\n−+(θ′\n1+θ′\n2)\n×∝an}bracketle{t0|OR|θ′\n1,θ′\n2∝an}bracketri}ht−−∝an}bracketle{t0|O†\nL|−θ′\n1,−θ′\n2∝an}bracketri}ht++\n×ei∆τ/summationtext\nisinhθ′\nie2∆\nvcx/summationtext\nicoshθ′\ni.(A8)\nLet us again first assume Φ0\nc= 0. If we shift the contours\nof integration θ′\ni→θ′\ni+iπ/2 and use the crossingrelation\nas well as a Lorentz transformation,\n∝an}bracketle{t0|O†\nL/vextendsingle/vextendsingle−θ′\n1−iπ\n2,−θ′\n2−iπ\n2/angbracketrightbig\n++\n=e−iπ/8∝an}bracketle{t0|OL|−θ′\n2,−θ′\n1∝an}bracketri}ht∗\n−−,\nwe obtain\nG2=Z2\n2eiπ/4/integraldisplay∞\n−∞dθ′\n1dθ′\n2\n(2π)2K(θ′\n1+iπ\n2)K(θ′\n2+iπ\n2)\n×S−+\n−+(θ′\n1+θ′\n2+iπ)/vextendsingle/vextendsingleG(θ′\n1−θ′\n2)/vextendsingle/vextendsingle2\n×e(θ′\n1+θ′\n2)/4e2i∆\nvcx/summationtext\nisinhθ′\nie−∆τ/summationtext\nicoshθ′\ni.\nN2(E,2kF+q) is now readily obtained via Fourier trans-\nformation.\nStarting from (A8) the second term N4(E,2kF+q)\nrelated to the BBS is obtained by picking up the poles of\nK+−(θ′\n1) andK+−(θ′\n2) atθ′\n1,2= iγrespectively. At the\nLEP the result is\nG4=2iZ2cosΦ0\nc√\n2e−i(π−√\n2Φ0\nc)/8e−Ebbsτeκbbsx\n×/integraldisplay∞\n−∞dθ′\n2πK+−(θ′+iπ\n2) sinh2θ′−i\n2(π+√\n2Φ0\nc)\n2\n×eθ′/4ei∆\nvcrsinhθ′e−∆τcoshθ′.\nAppendix B: Explicit expressions for form factors\nand scattering matrices\nIn this appendix we collect some formulas on the sine-\nGordontheoryusedin the presentarticle. Thescatteringmatrix between solitons and antisolitons possesses the\nintegral representation22\nS0(θ) =−exp/bracketleftBigg\ni/integraldisplay∞\n0dt\ntsinθt\nπξsinh/parenleftbigt\n2(1−1\nξ)/parenrightbig\nsinht\n2cosht\n2ξ/bracketrightBigg\n,\nwhereξ=β2/(1−β2). A similar integral representation\nfor the boundary reflection matrix reads46(for Dirichlet\nboundary conditions with Φ0\nc= 0)\nK(θ)=−cos/parenleftbiggπ\n2ξ+iθ\nξ/parenrightbigg\nR0/parenleftbig\niπ\n2−θ/parenrightbig\nσ/parenleftbig\niπ\n2−θ/parenrightbig\n,\nR0(θ)=exp/bracketleftBigg\n2i/integraldisplay∞\n0dt\ntsinθt\nπξsinh3t\n4ξsinh/parenleftbigt\n4(1−1\nξ)/parenrightbig\nsinht\n4sinht\nξ/bracketrightBigg\n,\nσ(θ)=exp/bracketleftBigg\n2i/integraldisplay∞\n0dt\ntsinθt\nπξsinh/parenleftbigt\nξ(1+iθ\nπ)/parenrightbig\nsinhtcosht\nξ/bracketrightBigg\n.\nTheauxiliaryfunction G(θ) appearingintheformfactors\n(A4) is given by31\nG(θ) =−iC1sinhθ\n2\n×exp/bracketleftBigg/integraldisplay∞\n0dt\ntsinh2/parenleftbig\nt(1+iθ\nπ)/parenrightbig\nsinh/parenleftbig\nt(ξ−1)/parenrightbig\nsinh(2t) coshtsinh(tξ)/bracketrightBigg\n,\nC1= exp/bracketleftBigg\n−/integraldisplay∞\n0dt\ntsinh2t\n2sinh/parenleftbig\nt(ξ−1)/parenrightbig\nsinh(2t) coshtsinh(tξ)/bracketrightBigg\n,\nwhile for the normalization factor Z2one finds numeri-\ncally from the integral repesentation derived in Ref. 31\nZ2≈7.7320∆1681/1440forβ2= 0.9,\nZ2≈20.1857∆1649/1120forβ2= 0.7.\nThe expressionsat the LEPareobtained by setting ξ= 1\n(β2= 1/2), i.e.S0(θ) =−1,K(θ) = itanhθ\n2,G(θ) =\n−isinhθ\n2, andZ2≈38.5519∆65/32.\n1O. 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Despite the absence of a simple Landau criti cal velocity, a superfluid response sur-\nvives, but dissipation reduces the superfluid fraction. I al so suggest an idea for how the superfluid\ndensity of an example of such a system, i.e. microcavity pola ritons, might be measured.\nPACS numbers: 03.75.Kk,47.37.+q,71.36.+c,\nTheobservationofsuperfluidityisoneofthemostcom-\npelling signatures of quantum coherence in systems such\nas Helium and cold atomic gases [1]. Recently, there\nhas been much activity exploring condensation of mixed\nmatter-light excitations, i.e. semiconductor microcavity\npolaritons (see [2] for a review) as well as recent exper-\niments on photons in dye filled cavities [3]. Microcavity\npolaritons consist of superpositions of excitons confined\nin quantum wells, and photons confined in semiconduc-\ntor microcavities. Because of the photonic component of\ntheseparticles, theyhaveafinitelifetime, andsoanycon-\ndensatewill beanon-equilibriumsteadystate, whereloss\nis balanced by injection of new particles. As well as these\nmatter-light systems, recent experiments on continuous\nloading of cold atoms into traps [4] suggest that simi-\nlar questions of superfluid properties in non-equilibrium\nsteady states may soon be accessible in cold atoms.\nFinite particle lifetime prompts questions about the\nmeaning of superfluidity when the particles involved in\nthe condensate are continually being replaced. To see\nwhy such questions arise, one may observe that finite\nparticle lifetime changes the excitation spectrum from\nthe linearly dispersing Bogoliubov sound mode to a dif-\nfusive mode [5, 6], ξ(k) =−iσ+√\nc2k2−σ2. The form\nof the spectrum of long wavelength modes is commonly\ninvoked in explaining the existence of superfluidity, i.e. if\nthe low energy excitations are linearly dispersing Bogoli-\nubov sound modes, then there exists a critical velocity,\nsuch that for a fluid flowing below this velocity, it is not\nenergeticallyfavorabletocreatequasiparticleexcitations,\nand thus the flowing superfluid remains stable [1]. If one\nnaively defines a critical velocity by the dispersion of the\nreal part of ω(k), one finds that for the non-equilibrium\nspectrum, such a critical velocity vanishes. As has been\npointed out elsewhere [7, 8], a more careful argument\nregarding the response to a static defect indicates that\nthere may still be a particular velocity at which there\nis a sharp onset of drag. Nonetheless, the above illus-\ntrates why non-equilibrium condensates require one to\nre-examine questions of superfluidity.\nExperimentally, aspects of superfluid behavior have\nbeen studied in microcavity polariton systems, and fea-\ntures such as quantized vortices [9], suppression of scat-\ntering off disorder [10] and metastability of induced vor-tices [11] have been observed. While these various ex-\nperiments show that aspects of superfluid behavior can\nbe seen in such non-equilibrium condensates, they leave\nopen an important question, namely what is the super-\nfluid fraction. Even in superfluid Helium, at non-zero\ntemperaturestherearedragforcesduetothenormalfluid\ncomponent. However, the normal and superfluid compo-\nnents can be clearly distinguished by their response to a\nslowrotation[1]: Atlowangularvelocities, thesuperfluid\ncannot rotate, so only the normal component rotates,\nthus reducing the effective moment of inertia. Determin-\ning the normal and superfluid densities thus gives a fuller\ndescription of the superfluid properties of a system than\ndoes a binary distinction between superfluid and non-\nsuperfluid systems. As such, the aim of this letter is to\ndiscuss the normal and superfluid densities of an open\ndissipative condensate, and to suggest how this might be\nexplored in the microcavity polariton systems.\nThe same superfluid density as introduced above can\nbe found from the current-current response function\nχij(q). This response function gives the particle current\nJi(q) =/summationtext\nkψ†\nk+qγi(2k+q)ψk(where the current vertex\nγi(k) =ki/2mand/planckover2pi1= 1 throughout) due to a pertur-\nbationH→H−/summationtext\nqfi(q)Ji(q), i.e.Ji(q) =χij(q)fj(q).\nIn an isotropic system, one may then define ρs,ρnas [1]\nmχij(q→0,ω= 0) =ρsqiqj\nq2+ρnδij.(1)\nThe superfluid component picks out and responds only\nto the irrotational (non-transverse) part of the applied\nforce. The advantage of this approach is that it allows\nanexplicitcalculationfortheopendissipativesystem, us-\ning the Schwinger-Keldysh approach for non-equilibrium\nsystems (see e.g. [12]).\nIn the following, I will consider the simplest model of\na weakly interacting dilute Bose gas:\nH=/summationdisplay\nkǫkψ†\nkψk+/summationdisplay\nk,k′,qU\n2ψ†\nk+qψ†\nk′−qψk′ψk(2)\nwhereǫk=k2/2m. In addition, one must include pump\nand decay processes such that the bare inverse retarded\nGreen’s function [ DR\n(0)]−1=ω−ǫk+iκ���ip(ω) where\nthe pump has the form p(ω) =γ−ηωandκdescribes2\nχij(q) = + +\n+ + +\nFIG. 1. Types of Feynman diagram required for the re-\nsponse function to one-loop order. Straight lines indicate\nnon-condensate excitations. Filled symbols involve the co n-\ndensate, arising from either interactions (circles) or cou pling\nto currents (squares). Wavy lines indicate source fields cou -\npling to the current vertices.\ndecay[13]. This form of pumping is motivated by recent\nworks by Wouters and Carusotto [7] (as well as related\nmodels [14]), and simplifies the calculation of χijcom-\npared to models with density-dependent pump processes.\nWhen condensed, such a model has the diffusive spec-\ntrumζ(k) =−iσ+√\nc2k2−σ2withσ=ηµ/(1+η2),c=/radicalbig\nµ/[m(1+η2)]. Thus, finding a non-zero superfluid\ndensity in such a model addresses how the diffusive spec-\ntrum affects superfluidity.\nTo correctly find the superfluid response function [15]\nrequires vertex corrections in the response function. In\nequilibrium, this can be avoided by using sum rules that\nresult from conservation of density, however with finite\nparticle lifetimes, this is not necessarily a-priori justified.\nIwillpostponeuntillaterthediscussionofhowthevertex\ncorrectionsare to be determined and next summarize the\nphysical reason that a superfluid density can survive.\nFig. 1 illustrates the classes of Feynman diagram (in-\ncluding vertex corrections) that result at one loop order.\nThe first five diagrams contribute to the superfluid den-\nsity, while the last gives the normal density. This can\nbe seen by noting that the first five diagrams all have\nthe current vertex scatter a particle out of the conden-\nsate, and thus involve a factor γi(q)∝qi, hence they all\ncontribute to χij∝qiqj. In order that the superfluid\ndensity does not vanish, it is crucial that the fluctua-\ntion propagator DR(q,ω= 0) that also appears in these\nfive diagrams behave as 1 /q2atq→0 so that overall\nχij∝qiqj/q2remains finite. The existence of superfluid\ndensity therefore depends on how the denominator of the\nGreen’s function behaves.\nIn thermal equilibrium, the Green’s function behaves\nasDR(q,ω)∝[(ω+i0)2−c2q2]−1and so the correct\nscaling ofDR(q,ω= 0) is dependent on having the a\nlinear spectrum, hence the relation of the Landau crit-\nical velocity and superfluid density. However, despite\nthe changed spectrum of the open dissipative system,\none hasDR(q,ω)∝[ω2+ 2iσω−c2q2]−1and so the\nGreen’s function at ω= 0 still scales as DR∝1/q2,\nyielding a non-vanishing superfluid density. Such behav-\nior of the Green’s function has also been seen to exist\nin several other models of non-equilibrium polariton con-\ndensates[5,6]. ThefactthatthisstructureoftheGreen’s0.000.050.100.15\n 0  0.2  0.4  0.6  0.8  1ρN/(mµ)\nT/µγ=2κ; η=γ/10\nκ/µ=0.02\nκ/µ=0.5\nκ/µ=1.0\nFIG. 2. Normal density vs temperature for a variety of pump\nand decay rates and obeying the equilibrium fluctuation dis-\nsipation theorem. Relative pump and decay rates are scaled\nso all three curves correspond to the same condensate densit y,\nwhile the influence of pump and decay varies.\nfunction leads to a superfluid density, despite the modi-\nfied spectrum, is the first main result of this letter.\nA second result is the effect of finite particle lifetime\non the normal density. In an equilibrium single compo-\nnent system, the normal density vanishes at zero tem-\nperature [1]. The normal density of the non-equilibrium\nsystem can be straightforwardly calculated since, just as\nin the thermal equilibrium case, there are no vertex cor-\nrections at one loop order [15], so one finds (in 2D):\nρn\nm=−/integraldisplay/integraldisplaydǫk\n2πdω\n2πǫki\n4Tr/bracketleftbig\nσ3DK\nkσ3(DR\nk+DA\nk)/bracketrightbig\n(3)\nwhere the Green’s functions and Pauli matrices σiare\nwritten in Nambu space, i.e. DR\nk(t,t′) =−iθ(t−\nt′)∝an}b∇acketle{t[Ψk(t),Ψ†\nk(t′)]∝an}b∇acket∇i}ht,Ψ†\nk= (ψ†\nk,ψ−k). Even for a ther-\nmalised case, using the equilibrium fluctuation dissipa-\ntion theorem, DK\nk(ω) = (2nB(ω)+1)[DR\nk(ω)−DA\nk(ω)],\none finds that the presence of pump and decay terms af-\nfect the normal density. As shown in Fig. 2, the normal\ndensity does not vanish at zero temperature.\nHaving shown that superfluid density need not vanish\nin a dissipative condensate, but is reduced by finite life-\ntime, one may then ask how the superfluid and normal\ndensities could be measured in such a system. As an\nillustration, the following suggests a method that uses\nthe polariton polarization degree of freedom [16] in or-\nder to apply ideas that have only recently been proposed\nfor how one might measure of superfluid density in cold\natom systems [17, 18]. A number of alternative meth-\nods likely also exist, such as adaptations of proposals to\ncreate gauge fields in coupled photonic cavities[19]. To\nadapt the approach in [18], one first considers the effec-\ntive Hamiltonian HBdue to an inhomogeneous real mag-\nnetic field, in the space of polariton polarization states.\nIf the splitting of polarization states is always large, one\nmay restrict to the adiabatic ground state |ψ∝an}b∇acket∇i}ht. Because\nthe polarization composition of this ground state varies\nin space, there can be a non-trivial synthetic gauge field\nin this subspace qAsynth=i∝an}b∇acketle{tψ|∇ψ∝an}b∇acket∇i}ht. Thus, a real mag-3\n(a)\nµc\n 0 0.1 0.2 0.3\n 0 1 2 3 4 0 0.1 0.2 0.3 0.4mℓ2ω\nqAφℓ = mvℓ\nr/ℓ(b)\nFIG. 3. Panel (a), microcavity sample ( µc) placed between\n(imbalanced) anti-Helmholtz coils to induce a field Breal.\nPanel (b), velocity and angular velocity vs radius for Asynth\nas discussed in the text.\nnetic field acting on polarization degrees of freedom can\ninduce an artificial vector potential acting on the (neu-\ntral) polaritons. This artificial gauge field can mimic a\nrotating frame, thereby allowing one to distinguish the\nsuperfluid and normal response to rotation [1].\nIn order to illustrate how this might work, one may\nconsider the real magnetic field produced by an im-\nbalanced anti-Helmholtz configuration, as illustrated in\nFig. 3(a), so that the magnetic field in the microcav-\nity has the form B= (βx,βy,B z) for small in-plane\ncoordinates x,ywith constant Bz. Microcavity polari-\ntons typically involve heavy-hole excitons, so that po-\nlariton polarizations ±1 imply electron and hole spins\n(∓1/2,±3/2). As such, while a B field perpendicular to\nthe microcavity simply splits these polarization states,\nthe in-plane field is more complicated, as it mixes the\npolariton states with non-radiative excitons with spins\n(±1/2,±3/2). Furthermore, depending on the crystal\nsymmetry and quantum well growth direction, the lead-\ning order coupling between ±3/2 hole states may either\nbe linear or cubic in magnetic field [20]. In order to illus-\ntrate the basic idea, I will avoid these complications, and\nconsider the simplest situation, where a linear coupling\nexists[21]. Adiabatically eliminating the non-radiative\nexcitons, the effective Hamiltonian for the polariton po-\nlarization is H=λ[ℓ2σz+r2(e2iφσ−+e−2iφσ+)] where\nreiφ=x+iyandthelength ℓencodestheratioofin-plane\nand perpendicular fields. One may then find the ground\nstate|ψ∝an}b∇acket∇i}ht, ensuring this is smooth as r→0, and thus find\nqA=i∝an}b∇acketle{tψ|∇ψ∝an}b∇acket∇i}ht= (ˆφ/r)(1−l2/√\nr4+l4). Thisgaugefield\nis equivalent to a rotating frame with qAsynth=mω×r\nhenceω(r) =qAφ/mr. This is shown in Fig. 3(b).\nDetection of the response to this rotating frame could\npotentially be done by imaging the momentum and en-\nergy distribution of the polaritons [2]. If the condensate\nis concentrated around r≃ℓ, one has the maximum ve-\nlocity, corresponding to an energy shift for the normal\ncomponent of ∆ Emax= (1/2)mv2\nmax= 0.08/mℓ2. For\nℓ≃0.5µm, andm= 10−4melectronthis corresponds to0.2meV. Observing the differential shift of luminescence\nas one varies ℓby varying Bzcould allow one to extract\nthe superfluid fraction.\nI now turn to discuss in more detail how the super-\nfluid density can be calculated in the Schwinger-Keldysh\napproach, and to explain the origin of the vertex cor-\nrections in Fig. 1 and the form of the normal density\nin Eq. (3). Following the path integral approach to the\nSchwinger-Keldysh formalism [12], the response function\ncan be written in terms of a generating functional as:\nχij(q) =−i\n2d2Z[f,θ]\ndfi(q)dθj(−q),Z=/integraldisplay\nD(¯ψ,ψ)ei(S+δS).(4)\nHere,Sis the Keldysh action [12] due to the Hamil-\ntonian in Eq. (2) and the pump and decay terms S=/summationtext¯ψkD−1\n0ψk′−/integraltext/integraltext\ndtd2rU\n2[¯ψq¯ψcl(ψ2\ncl+ψ2\nq)+H.c.] where\nthe fieldsψare written in the Keldysh space for clas-\nsical/quantum fields [12], and the retarded, advanced\nand Keldysh components of the inverse Green’s func-\ntionD−1\n0are as discussed above. The source term\nδS=/summationtext¯ψk+q[τ1fi(q)+(τ3+iτ2)θi(q)]γi(2k+q)ψk, where\nPauli matrices τiare in the Keldysh space. The presence\nof two separate fields is necessary due to need to calcu-\nlate a normal ordered current (derivative with respect to\nθ) linearly dependent on a classical force (field f).\nIn the non-condensed state, one may immediately de-\ntermine Zat leading order by neglecting interactions,\nand performing the (Gaussian) integration over fields\n(¯ψ,ψ)cl,q. This yields mχij=ρnδij, withρngiven by\nEq. (3), where the Nambu structure in the normal state\nis trivial. When condensed, vertex corrections become\nimportant. These vertex corrections can be found by\nuse of an “honest saddle point” of the partition func-\ntion, i.e. determine the saddle point in the presence of\nthe source terms f,θ, and then integrate out fluctua-\ntions about this new saddle point. One thus finds a form\nZ ∝exp{iS0[f,θ]−1\n2Trln(1 + DA[f,θ])}whereS0is\nthe saddle point action, Dare the Green’s functions (for\nf=θ= 0) andAis the self energy due to the fields\nf,θ. BothS0andAinvolve terms arising from the shift\nof the saddle point field ψin the presence of the source\ntermsf,θ, and thus both S0andAhave higher order\ncontributions of f,θ. One may then expand these terms\nto quadratic order in f,θand evaluate χijvia Eq. (4).\nTaking derivatives with respect to f,θ(indicated by\nprimes and subscripts), one finds χij=1\n2S′′\n0,fi,θj+\ni\n4Tr(DA′′\nfiθj)−i\n4Tr(DA′\nfiDA′\nθj). Comparing the three\nterms in this expression to the diagrams in Fig. 1 the\nfirst diagram arises from the first term, the second two\ndiagrams arise from the second term, and the last three\ndiagramsfrom the third term. After explicitly evaluating\nthe shifts to the saddle point, and the self energy A, one\nfinds the explicit form:4\nmχij(q) =qiqj\nq2/braceleftbiggΛ\nU+m/integraldisplaydǫk\n2π/bracketleftbigg\n1−i\n4/integraldisplaydω\n2π/parenleftbigg\nTr/bracketleftbig\n(2σ0+σ1)DK\nk/bracketrightbig\n−2Λ\nǫqTr(DK\nkσ1)/parenrightbigg/bracketrightbigg/bracerightbigg\n−m/integraldisplaydǫk\n2π/bracketleftbiggqiqj\nq22Λ2\nǫqN22−iqi(qj+2kj)\nq2ΛN23+i(qi+2ki)qj\nq2ΛN32+(qi+2ki)(qj+2kj)\n4mN33/bracketrightbigg\n(5)\nwhereNab=/integraltextidω\n8πTr(DR\nk+qσaDK\nkσb+DK\nk+qσaDA\nkσb) in\nterms of Pauli matrices σiin the Nambu space and Λ =\nU|ψ0|2. The terms in Eq. (5) are arranged in the same\norder as the corresponding diagrams in Fig. 1.\nA number of technical issues regarding regularization\nare worth noting. Firstly, as is known elsewhere (see\ne.g. [22] and refs. therein), there can be cases where it\nis necessary to return to the discrete time coherent state\npath integral in order to correctly incorporate causality\nin performing integrals. The current problem is such a\ncase, and the term arising from this is the 1 on the first\nline of Eq. (5). Secondly, in order to give an ultravio-\nlet finite expression, it is necessary to perform the stan-\ndard T-matrix regularization of the contact interaction:\nU−1→U−1\neff−m/integraltextdǫk\n2π[2(ǫ+µ)]−1. Thirdly, the appar-\nently singular terms involving qiqj/q2ǫqin fact cancel,\nleaving only finite contributions as q→0. One may ver-\nify that Eq. (5) recovers the expected equilibrium result\nin the absence of pumping and decay.\nOne may note that the expression in Eq. (5) does not\nexplicitly involve details of the pumping. This is because\nthe only nonlinearity included being the interaction term\nU. This means that Eq. (5) survives for general models\nof pumping and decay, and so is more generic than the\nparticular model of pumping used to derive it.\nIn conclusion, the superfluid density of a non-\nequilibrium open dissipative condensate need not vanish,\ndespite the non-existence of a Landau critical velocity.\nThis is because the poles of the response function, which\ngive the spectrum, do not uniquely determine the form\nof the response function at zero frequency, which is the\nquantity that defines the superfluid density. Such a su-\nperfluid density could potentially be measured in a po-\nlariton system by using real magnetic fields to engineer\nan effective rotating frame. The current-currentresponse\nfunction can be explicitly calculated using the “honest\nsaddle point” approach. Such an approach would also\nallow calculation of dynamical response functions, allow-\ning a more nuanced understanding of the distinctions be-\ntween static and dynamic superfluid phenomena in open\ndissipative condensates.\nI acknowledge helpful discussions with Austen\nLamacraft and with Nigel Cooper, and funding from EP-\nSRC grant EP/G004714/2\n[1] A. J. Leggett, Quantum Liquids (Oxford University\nPress, 2006).[2] H. Deng, H. Haug, and Y. Yamamoto,\nRev. Mod. Phys. 82, 1489 (2010).\n[3] J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz,\nNature468, 545 (2010).\n[4] M. Falkenau et al., arXiv:1102.0928.\n[5] M. H. Szyma´ nska, J. Keeling, and P. B. Littlewood,\nPhys. Rev. Lett. 96, 230602 (2006).\n[6] M. Wouters and I. Carusotto,\nPhys. Rev. B 74, 245316 (2006).\n[7] M. Wouters and I. Carusotto,\nPhys. Rev. Lett. 105, 020602 (2010).\n[8] M. Wouters and V. Savona,\nPhys. Rev. B 81, 054508 (2010).\n[9] K. G. Lagoudakis et al., Nature Phys. 4, 706 (2008).\n[10] A. Amo et al. , Nature 457, 291 (2009);\nNature Phys. 5, 805 (2009).\n[11] D. Sanvitto et al., Nature Phys. 6, 527 (2010).\n[12] A. Altland and B. D. Simons, Condensed Matter Field\nTheory, 2nd ed. (Cambridge University Press, 2010);\nA. Kamenev, in Nanophysics: Coherence and transport ,\nLes Houches, Vol. LXXXI, edited by H. Bouchiat et al.\n(Elsevier, Amsterdam, 2005) p. 177.\n[13] To avoid ultraviolet divergence, a regularization [ κ−\nip(ω)]→[κ−ip(ω)]iΓ/(ω+iΓ) is also required. Γ is\nassumed large compared to all other energy scales.\n[14] M. Wouters and V. Savona, arXiv:1007.5453.\n[15] A. Griffin, Excitations in a Bose-Condensed Liquid\n(Cambridge University Press, Cambridge, 1994).\n[16] I.A.Shelykh et al.,Semicond. Sci. Technol. 25, 013001 (2010).\n[17] N. R. Cooper and Z. Hadzibabic,\nPhys. Rev. Lett. 104, 030401 (2010).\n[18] J. Dalibard, F. Gerbier, G. Juzeli˜ unas, and\nP.¨Ohberg, arXiv:1008.5378; N. R. Cooper,\nPhys. Rev. Lett. 106, 175301 (2011).\n[19] M. Hafezi, A. S¨ orensen, E. Demler, and\nM. Lukin, Phys. Rev. A 76, 023613 (2007);\nJ. Koch, A. Houck, K. Le Hur, and S. Girvin,\nPhys. Rev. A 82, 043811 (2010); R. O. Umucalilar and\nI. Carusotto, arXiv:1104.4071.\n[20] R. Winkler, Spin-Orbit Coupling Effects in Two-\nDimensional Electron and Hole Systems , Springer Tracts\nin Modern Physics No. 191 (Springer, Berlin, 2003).\n[21] An alternative would be to use stress to produce an in-\nplane field [23].\n[22] J. H. Wilson and V. Galitski,\nPhys. Rev. Lett. 106, 110401 (2011).\n[23] R. Balili, B. Nelsen, D. W. Snoke, L. Pfeiffer, and\nK. West, Phys. Rev. B 81, 125311 (2010)."    },    {        "title": "1106.3911v1.Density_Functional_Resonance_Theory_of_Unbound_Electronic_Systems.pdf",        "content": "Density Functional Resonance Theory of Unbound Electronic Systems\nDaniel L. Whitenack\u0003\nDepartment of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907, USA\nAdam Wassermany\nDepartment of Chemistry, Purdue University, 560 Oval Drive, West Lafayette, IN 47907, USA and\nDepartment of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907, USA\n(Dated: October 16, 2021)\nDensity Functional Resonance Theory (DFRT) is a complex-scaled version of ground-state Den-\nsity Functional Theory (DFT) that allows one to calculate the resonance energies and lifetimes of\nmetastable anions. In this formalism, the exact energy and lifetime of the lowest-energy resonance of\nunbound systems is encoded into a complex \\density\" that can be obtained via complex-coordinate\nscaling. This complex density is used as the primary variable in a DFRT calculation just as the\nground-state density would be used as the primary variable in DFT. As in DFT, there exists a\nmapping of the N-electron interacting system to a Kohn-Sham system of Nnon-interacting par-\nticles in DFRT. This mapping facilitates self consistent calculations with an initial guess for the\ncomplex density, as illustrated with an exactly-solvable model system. Whereas DFRT yields in\nprinciple the exact resonance energy and lifetime of the interacting system, we \fnd that neglect-\ning the complex-correlation contribution leads to errors of similar magnitude to those of standard\nscattering close-coupling calculations under the bound-state approximation.\nDensity Functional Theory (DFT) [1{3] provides one\nof the most accurate and reliable methods to calculate\nthe ground-state electronic properties of molecules, clus-\nters, and materials from \frst principles. It is one of\nthe workhorses of computational quantum chemistry [4].\nIn addition, DFT's time-dependent extension (TDDFT)\n[5] can now be applied to a wealth of excited-state and\ntime-dependent properties in both linear and non-linear\nregimes [6]. When the N-electron system of interest\nhas no bound ground state, however, neither DFT nor\nTDDFT can be applied in a straightforward way. A cor-\nrect DFT calculation converges to the true ground state\nby ionizing the system, thus leaving no reliable starting\npoint for a subsequent TDDFT calculation on the N-\nelectron system. In practice, a \fnite simulation box or\nbasis set can make the system arti\fcially bound [7, 8],\nbut information about the relevant lifetimes is lost in the\nprocess.\nWe address here this fundamental limitation of ground-\nstate DFT, and propose a solution.\nConsider a system of Ninteracting electrons in an\nexternal potential ~ v(r), with ground-state density ~ n(r).\nThe potential is set to be everywhere positive and go to\na positive constant Casjrj!1 . The ground-state en-\nergy is ~E > 0. We start by asking how the gound state\ndensity changes when a smooth step is added to ~ v(r) at\na radiusjRjthat is larger than the range of ~ v(r). The\nstep is such that the new potential v(r) coincides with\n~v(r) forjrj<jRjbut goes to zero at in\fnity. Since ~ v(r)\nis everywhere positive, all Nelectrons tunnel out and\nv(r) supports no bound states. The correct ground state\nenergy is now E= 0, and the new density n(r) is delocal-\nized through all space. In practical calculations, however,\nv(r) and ~v(r) cannot be distinguished if jRjis beyond\nthe size of the simulation box. The result provided byground-state DFT using the exact exchange-correlation\nfunctional is not E, but ~E > 0, and the density obtained\nis ~n(r) as if the system were bound. Even when the sim-\nulation box is large enough to include the steps, use of a\n\fnite basis-set of localized functions will arti\fcially bind\nall electrons. Clearly, such calculations do not provide\napproximations to the true ground-state energy and den-\nsity ofv(r), but to those of its lowest-energy resonance\n(LER).\nThe purpose of this letter is to establish an analog\nof KS-DFT that provides the in-principle exact LER-\ndensity along with its energy and lifetime for any \fnite\njRj. AsjRj!1 , the results coincide with those of stan-\ndard KS-DFT. For higher-energy resonances, TDDFT is\nneeded as a matter of principle [9, 10].\nFirst, we note that as jRj!1 , the complex density\nn\u0012(r) associated with the LER of\n^Hv=^T+^Vee+Z\ndr^n(r)v(r); (1)\nbecomes equal to the complex density ~ n\u0012(r) associated\nto ~v(rei\u0012). In Eq. 1, ^T=\u00001\n2PN\ni=1r2\niis the kinetic\nenergy operator, ^Vee=PN\ni;j6=ijri\u0000rjj\u00001is the electron-\nelectron interaction, and ^ n(r) =PN\ni=1\u000e(r\u0000^ri) is the\ndensity operator. (Atomic units are used throughout).\nTo \fndn\u0012(r), we complex-scale ^Hvby multiplying all\nelectron coordinates by the phase factor ei\u0012, diagonalize\nthe resulting non-hermitian operator ^H\u0012\nv, and calculate\nthe bi-expectation value of ^ n(r) as:\nn\u0012(r) =h\tL\n\u0012j^n(r)j\tR\n\u0012i; (2)\nwherej\tR\n\u0012iandh\tL\n\u0012jare the right and left eigenstates\ncorresponding to the complex eigenvalue of ^H\u0012\nvthat hasarXiv:1106.3911v1  [quant-ph]  20 Jun 20112\nthe smallest positive real part among all eigenvalues in\nthe non-rotating spectrum of ^H\u0012\nv. For a detailed review\nof this technique and related methods in non-hermitian\nQuantum Mechanics, see ref. [11, 12]. The computational\ncost of this prescription scales exponentially with the\nnumber of particles. Since n\u0012(r)!~n\u0012(r) asjRj!1 ,\nand since there is a one-to-one correspondence between\nn\u0012(r) andv(rei\u0012) [13, 14], the complex energy of the LER\nE\u0012[n\u0012] goes to ~E(notE), asjRj!1 . Its lifetimeLis\ngiven by (\u00002Im(E\u0012))\u00001, and for any \fnite jRj,\nE\u0012[n\u0012] =E[n\u0012]\u0000i\n2L\u00001[n\u0012]; (3)\nwhere the resonance energy Etends to ~EasjRj!1 .\nTo build a complex analog of Kohn-Sham DFT us-\ningn\u0012(r) as the basic variable, we \frst map the sys-\ntem of interacting electrons whose LER density is n\u0012(r)\nto one ofNparticles moving independently in a com-\nplex \\Kohn-Sham\" potential v\u0012\ns(r) de\fned such that its\nNoccupied complex orbitals f\u001e\u0012\ni(r)gyield the interact-\ning LER-density via n\u0012(r) =PN\ni=1h\u001e\u0012;L\nij^n(r)j\u001e\u0012;R\nii. The\ncomplex Kohn-Sham equations are:\n\u0012^h1\u0000\"i\u0000^h2\u00002\u001c\u00001\ni^h2+ 2\u001c\u00001\ni^h1\u0000\"i\u0013\u0012Re(\u001e\u0012\ni)\nIm(\u001e\u0012\ni)\u0013\n= 0;(4)\nwhere ^h1=\u00001\n2cos(2\u0012)r2+ Re(v\u0012\ns(r)), and ^h2=\n1\n2sin(2\u0012)r2+ Im(v\u0012\ns(r)). The set off\"igandf\u001cigpro-\nvide the orbital resonance energies and lifetimes of the\nKohn-Sham particles.\nSecond, we write E\u0012[n\u0012] as:\nE\u0012[n\u0012] =T\u0012\ns[n\u0012] +Z\ndrn\u0012(r)v(rei\u0012)\n+E\u0012\nH[n\u0012] +E\u0012\nXC[n\u0012] (5)\nin analogy to standard KS-DFT, and require: T\u0012\ns[n\u0012] =\ne\u00002i\u0012Ts[n\u0012] andE\u0012\nH[n\u0012] =e\u0000i\u0012EH[n\u0012], whereTs[n\u0012] and\nEH[n\u0012] are the standard non-interacting kinetic energy\nand Hartree functionals evaluated at the complex densi-\nties. Eq. 5 then de\fnes E\u0012\nXC[n\u0012]. The complex variational\nprinciple [12] along with the assumption that the orbitals\nused to construct the density can be expanded in an or-\nthonormal basis leads to the Euler-Lagrange equation:\n\u000eE\u0012[n\u0012]\n\u000en\u0012\u0000\u0016Z\ndrn\u0012(r) = 0: (6)\nPerforming the variarion in Eq. 5 and comparing with\nEq. 4 leads to an expression for the Kohn-Sham potential\nthat is again analogous to that of standard KS-DFT:\nv\u0012\ns(r) =v(rei\u0012) +e\u0000i\u0012vH[n\u0012](r) +v\u0012\nXC[n\u0012](r);(7)\nwherev\u0012\nXC[n\u0012](r) =\u000eE\u0012\nXC[n\u0012]=\u000en\u0012(r)jLER.\nThe simplest case where all essential aspects of this for-\nmalism can be illustrated is a system of two interactingelectrons moving in a one-dimensional potential such as\nthe one depicted in the inset of Fig. 1. We study a Hamil-\ntonian where the electrons interact via a soft-Coulomb\npotential of strength \u0015:\n^H=2X\ni=1\u0014\n\u00001\n2d2\ndx2\ni+v(xi)\u0015\n+\u0015p\n1 + (x1\u0000x2)2;(8)\nusingv(x) =a\"\n2P\nj=1\u0010\n1 +e\u00002c(x+(\u00001)jd)\u0011\u00001\n\u0000e\u0000x2\nb#\n. Its\nparent potential ~ v(x) =a(1\u0000e\u0000x2=b) goes toaasx!\n\u00061, butv(x) goes down to zero at x\u0018\u0006d.\nExact solution via 2-electron wavefunction : The\ncomplex-scaled Hamiltonian ^H\u0012=^H(fxig!fxiei\u0012g)\nwas diagonalized with the Fourier Grid Hamiltonian\n(FGH) [15] and Finite Di\u000berence Methods. The numeri-\ncally exact n\u0012(x) was calculated via Eq. 2. The complex\ndensityn\u0012(x) depends on the value of \u0012(see Fig. 1),\nbut for a large enough number of grid points the energy\ndoes not. In the complex-scaling method the resonance\nenergies are precisely those that remain stationary as \u0012\nchanges [12]. Fig. 2 shows the energy for 0 <\u0015< 1.\nExact KS solution : Two non-interacting electrons in\nthe potential indicated by solid lines in Fig.3 have the\nsamen\u0012(x) as calculated above to one part in 106(in\nthe sense that the space integral of the square of the\ndi\u000berence between their real or imaginary parts is less\nthan 106). Whenn\u0012(x) is set to integrate to the number\nof electrons (2, here), we verify this potential is given by:\nv\u0012\ns(x) =e\u00002i\u0012r2p\nn\u0012(x)\n2p\nn\u0012(x)\u0000\"H+ 2i\u001c\u00001\nH; (9)\nwhere\"H\u00002i\u001c\u00001\nHis the highest occupied complex orbital\nenergy (in this case the only one), in exact analogy to\nreal KS-potentials for bound 2-electron systems. With-\nout an explicit expression for E\u0012\nXC[n\u0012], however, the total\nenergy cannot be calculated via Eq. 5. Related work by\nErnzerhof [13] and physical intuition suggest that bound\nground-state functionals are applicable here. They are,\nin any case, the most natural candidates.\nExchange : Borrowing knowledge from bound 2-\nelectron DFT, Eqs. 4 and 7 were solved employing\nE\u0012\nX[n\u0012] =\u00001\n2E\u0012\nH[n\u0012] =\u00001\n2e\u0000i\u0012EH[n\u0012]. The complex KS\nequations can be solved self-consistently with an initial\nguess forn\u0012. Using the non-interacting complex density,\nthe SCF calculations converged in 4-5 iterations. The re-\nsulting complex energies are plotted in Fig. 2 along with\nthe exact results. For comparison, we also plot the re-\nsults from perturbation theory to \frst order in \u0015. The\ntwo yield identical answers for the resonance energies,\nand extremely close for the lifetimes for all \u0015in the range\n0<\u0015< 1. Thus, neglecting correlation, we \fnd the av-\nerage error is\u001814% for the real part and \u001835% for the\nimaginary part of the total energy. We also compare with3\nFIG. 1. Di\u000berent exact 2-electron complex densities when\nusing di\u000berent scaling angles. The model potential, v(x), used\nin this study is also shown ( a= 4,b= 0:5,c= 4, andd= 2).\nGrid \u0012 Re(E) Im( E)\n(N = 299) 0.27 4 :99895 \u00000:0149586\n0.35 4:99933 \u00000:0144161\n0.43 4:99962 \u00000:0139792\n(N = 1299) 0.27 5 :00182 \u00000:0161045\n0.35 5:00198 \u00000:0159848\n0.43 5:00200 \u00000:0159513\nTABLE I. Two-electron resonance energy values in the model\nHamiltonian of Eq. 8 calculated via exchange-only DFRT. As\nthe grid spacing decreases numerical dependence on \u0012practi-\ncally disappears. ( a= 4,b= 0:5,c= 4,d= 2 and\u0015= 1)\nstandard scattering calculations using the close-coupling\nequations under the bound state approximation [16, 17].\nThe resonance energy is predicted by this method with\nan error of 22%, comparable to our DFRT exchange-only\nresults.\nAs in standard KS-DFT, total energies are given here\nby:\nE\u0012[n\u0012] =NX\ni=1\u0000\n\"i\u00002i\u001c\u00001\ni\u0001\n+E\u0012\nHX[n\u0012]\n\u0000Z\ndrv\u0012\nHXn\u0012(r) (10)\nWe point out that the \u0012-independence of the energy is\npreserved by the SCF procedure (see Table I). As the\ngrid-size increases the dependence on \u0012becomes negli-\ngible. This is important, because within a SCF DFRT\ncalculation one is always solving the 1-body complex KS-\nequations. For these equations, one should be able to ef-\n\fciently use a large enough basis set or a \fne enough grid\nto extinguish most of the numerical \u0012dependence. Thus,\nthis well-known drawback of the complex-scaling tech-\nnique [18{20] is outdone by the bene\ft of never having\nto deal with N-particle wavefunctions, but just 1-body\n(complex) densities.\nCorrelation Potential : It is of interest to calculate the\nexact correlation potential, which we do by subtracting\n0 0.5 1345\nλRe(Eθ)\n  \nExactSCF, λ(1)\n0 0.5 1−0.02−0.01\nλIm(Eθ)\nλ(1)Exact\nSCFFIG. 2. The real and imaginary parts of E\u0012in the model\nHamiltonian of Eq. 8 calculated exactly with complex scaling\n(thick solid), a \frst order correction to the non-interacting\nenergy (dashed), and our DFRT exchange-only self-consistent\nmethod (thin solid). ( a= 4,b= 0:5,c= 4, andd= 2)\nthe hartree-exchange contribution from the exact KS po-\ntential. The individual Hartree-exchange and correlation\npotentials are shown in Fig. 4. To interpret the features\nin these complex potentials it is useful to distinguish\nbetween two regions. As the interaction between elec-\ntrons is turned on and \u0015increases from 0 to 1, the region\naround the central well is shifted up in the real part of\nthe Kohn-Sham potential. This behavior is also seen in\nstandard KS-DFT, and serves to shift up the position\nof the non-interacting orbital energies (in that case the\nreal part of the orbital energies). However, both the real\nand imaginary part of the complex Kohn-Sham potential\nhave a second region outside the central well that shows\na dramatic oscillatory structure arising purely from the\nfact that the state is unbound. It is already known that\nthe decaying oscillations in the tails of the complex LER\nwavefunction are governed by the lifetime of the reso-\nnance [21]. These oscillations serve to produce the correct\nassymptotic behavior in the interacting complex density\nand thereby give the correct interacting lifetime when\nthis density is used in the functional.\nThe analog of Koopmans' theorem does not hold in\nDFRT. Although the ionization energy of our 2-electron\nsystem is strictly zero, it is tempting to de\fne I\u0012\u0011\nE\u0012(N= 1)\u0000E\u0012(N= 2) and check whether it equals\nthe highest occupied KS orbital energy. For the param-\neters used in Figs.1-4, E\u0012(N= 1) = 1:629\u00000:003i,\nE\u0012(N= 2) = 4:127\u00000:014i, but the exact KS eigenvalue\nis 2:065\u00000:006i. Clearly, DFRT provides an unambigu-\nous prescription for the calculation of negative electron\na\u000enities.\nWe are working on the implementation of DFRT to cal-\nculate the lifetime of molecular metastable anions. The\nmethod would also be applicable to molecules connected\nto metallic leads, as in molecular electronics. Ernzer-\nhof and co-workers have developed an approach for that\npurpose where complex absorbing potentials are added\nwithin a complex-DFT framework [13, 22]. However, we\nemphasize that the complex potentials in DFRT are the\nresult of a variational calculation, and they are obtained\nself-consistently for the N-electron system treated as iso-4\n−5 0 5024\nxRe(vsθ)\n  \n−5 0 5−10\nxIm(vsθ)\nFIG. 3. The real and imaginary part of the complex Kohn-\nSham potential for the LER of 2 soft-Coulomb interacting\nelectrons in the model potential, Eq.8. The dashed lines are\nthe real and imaginary part of the complex-scaled parent po-\ntential ~v(x). (\u0012= 0:35,a= 4,b= 0:5,c= 4,d= 2, and\n\u0015= 1).\n−5 0 500.20.4\nxRe(vHXθ)\n  \nλ=1\nλ=0.75\nλ=0.5\nλ=0.25\nλ=0\n−5 0 500.04\nxIm(vHXθ)λ=1\nλ=0.5\nλ=0.25\nλ=0λ=0.75\n−5 0 5−1−0.500.5\nxRe(vcθ)\n  \nλ=0λ=1\nλ=0.5\nλ=0.25λ=0.75\n−5 0 5−1−0.50\nxIm(vcθ)λ=0\nλ=0.25\nλ=0.5\nλ=0.75\nλ=1\nFIG. 4. The individual contributions to the Kohn-Sham po-\ntential from Hartree-exchange and correlation. ( \u0012= 0:35,\na= 4,b= 0:5,c= 4, andd= 2)\nlated, rather than added to the Hamiltonian from the\nstart to model an open system.\nIn addition, DFRT should be applicable to study shape\nand Feshbach resonances in low-energy electron scatter-\ning processes [23{25] of growing interest in biological\nsystems [26{28], atmospheric sciences, lasers, and astro-\nphysics [29{32].\nIn summary, DFRT provides an unambiguous prescrip-\ntion for calculating negative electron a\u000enities based on\na complex-scaled version of standard ground state DFT.\nThis complex-scaled version has been cast in a way that\nis analogous in practice to KS-DFT. Results on a model\nsystem suggest that the same machinery that has been\ndeveloped for KS-DFT yields accurate resonance energies\nand lifetimes in DFRT. It remains to be seen if common\napproximations to EXC[n] are able to capture the impor-\ntant e\u000bects that determine properties of real transient\nanions. A more detailed study of the complex density\nfunction and various DFRT identities is forthcoming.\nAcknowledgment is made to the Donors of the Ameri-can Chemical Society Petroleum Research Fund for sup-\nport of this research under grant No.PRF# 49599-DNI6.\n\u0003dwhitena@purdue.edu; http://www.purdue.edu/dft\nyawasser@purdue.edu\n[1] P. Hohenberg and W. Kohn, Phys. Rev., 136, B864\n(1964).\n[2] W. Kohn and L. J. Sham, Phys. Rev., 140, A1133 (1965).\n[3] R. G. Parr and W. Yang, Density-Functional Theory of\nAtoms and Molecules (Oxford University Press, Oxford,\n1994).\n[4] R. M. Martin, Electronic Structure: Basic Theory and\nPractical Methods (Cambridge University Press, London,\n2004).\n[5] E. Runge and E. K. U. Gross, Phys. Rev. Lett., 52, 997\n(1984).\n[6] M. A. L. Marques, C. A. Ullrich, F. Nogueira, A. Rubio,\nK. 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Adhikari, Variational Principles and the Numerical\nSolution of Scattering Problems (John Wiley and Sons,\nInc., New Jersey, 1998).\n[17] J. R. Taylor, Scattering Theory (Dover Publications, Inc.,\nNew York, 1972).\n[18] N. Moiseyev, P. R. Certain, and F. Weinhold, Mol.\nPhys., 36, 1613 (1978).\n[19] W. P. Reinhardt, Ann. Rev. Phys. Chem., 33, 223 (1982).\n[20] B. Simon, Ann. Math., 97, 247 (1973).\n[21] U. Peskin, N. Moiseyev, and R. Lefebvre, J. Chem.\nPhys., 92, 2902 (1990).\n[22] F. Goyer, M. Ernzerhof, and M. Zhuang, J. Chem. Phys.,\n126, 144104 (2007).\n[23] R. E. Palmer and P. J. Rous, Rev. Mod. Phys., 64, 383\n(1992).\n[24] J. Simons, J. Phys. Chem. A, 112, 6401 (2008); Annu.\nRev. Phys. Chem., 67, 107 (2011).\n[25] T. A. Field, K. Graupner, A. Mauracher, P. Scheier,\nA. Bacher, S. Deni\r, F. Zappa, and T. D. Mark, J.\nPhys.: Conf. Ser., 88, 012029 (2007).\n[26] X. Pan, P. Cloutier, D. Hunting, and L. Sanche, Phys.\nRev. Lett., 90, 208102 (2003).\n[27] X. Pan and L. Sanche, Phys. Rev. Lett., 94, 1981045\n(2005).\n[28] M. Mucke, M. Braune, S. Barth, M. Frstel, T. Lischke,\nV. Ulrich, T. Arion, U. Becker, A. Bradshaw, and\nU. Hergenhahn, Nature Physics, 6, 143 (2010).\n[29] H. S. W. Massey, Negative Ions (Cambridge University\nPress, London, 1950).[30] W. Liang1, M. P. Shores, M. Bockrath, J. R. Long, and\nH. Park, Nature, 417, 725 (2002).\n[31] V. Strelkov, Phys. Rev. Lett., 104, 123901 (2010).\n[32] R. Olivares-Amaya, M. Stopa, X. Andrade, M. A. Wat-\nson, and A. Aspuru-Guzik, J. Phys. Chem. Lett., 2, 682\n(2011)."    },    {        "title": "1107.4769v1.Doping_of_epitaxial_graphene_on_SiC_intercalated_with_hydrogen_and_its_magneto_oscillations.pdf",        "content": "Doping of epitaxial graphene on SiC intercalated with hydrogen and its\nmagneto-oscillations\nSergey Kopylov,1Vladimir I. Fal'ko,1and Thomas Seyller2\n1Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK\n2Lehrstuhl f ur Technische Physik, Universit at Erlangen-N urnberg, 91056 Erlangen, Germany\nWe study the charge transfer between a quasi-free-standing monolayer graphene, produced by\nhydrogen intercalation, and surface acceptor states. We consider two models of acceptor density of\nstates to explain the high hole densities observed in graphene and \fnd the density responsivity to\nthe gate voltage. By studying magneto-oscillations of the carrier density we provide an experimental\nway to determine the relevant model.\nAmong numerious ways of graphene fabrication [1{3],\none of the promising methods for the top-down manu-\nfacturing of electronic devices consists in the graphitiza-\ntion of Si-terminated surface of silicon carbide. Graphene\nproduced by such graphitization resides on an insulat-\ning (6p\n3\u00026p\n3) R30\u000ecarbon bu\u000ber layer, which is eas-\nily spoiled by vacancies and, typically, leads to high\ngraphene doping [4]. Without special growth proto-\ncols, monolayer graphene on Si face of SiC appears\nto be highly n-doped, with electron density at about\nne= 1\u00011013cm\u00002level [5, 6]. Also, charged sur-\nface donors induce Coulomb scattering, which limits the\nmobility of electrons in such a material. To improve\nthe transport qualities of graphene and decouple it from\nthe substrate, it has been proposed to use hydrogen in-\ntercalation [7]. Hydrogen breaks Si-C bonds and con-\nverts the bu\u000ber layer into a quasi-free-standing mono-\nlayer graphene (QFMLG), which is usually p-doped [8]\ndue to electron transfer from graphene to acceptor states\nin the H-terminated surface of SiC.\nIn this work we study the charge transfer between\ngraphene and SiC in hydrogen-intercalated epitaxial\ngraphene, including the magneto-oscillations of the hole\ndensity in QFMLG. We consider two limiting models for\nthe charge transfer, which are di\u000berent by the form of the\ndensity of states of surface acceptors, \r(\"). In the model\nI we assume that all acceptor states are due to occa-\nsional unsaturated Si bonds: vacancies in the hydrogen\nlayer with the density nv, energyEIand narrow spec-\ntral density \rI(\") =nvexp(\u0000(\"\u0000EI)2=\u00012)=(p\u0019\u0001)!\nnv\u000e(\"\u0000EI), \u0001\u001cEG\u0000EI, whereEGis the work function\nof graphene (Fig. 1, inset I). In the model II we assume\na uniform density of states of acceptor levels \r(\") =\r0\n\flled up to the energy EII(Fig. 1, inset II). For both\nmodels we \fnd the hole density in graphene and the re-\nsponsivity factor describing the e\u000bectiveness of QFMLG\ncarrier density control by a gate voltage in a \feld-e\u000bect\ntransistor. We also analyse the oscillations in the carrier\ndensity dependence on a magnetic \feld, and show that\nthis can be used to distinguish experimentally between\nthe two limiting doping models.\nThe electron transfer from QFMLG to surface acceptor\nFIG. 1: a) The dependence of the hole density nin QFMLG\non the acceptor density (see nvaxis for the model I and \r0\naxis for the model II). The insets show charge distribution\nbetween graphene and acceptor states for both models. b)\nResponsivity factor rdependence on nv. The following pa-\nrameters were used for both plots: d= 0:3 nm,AI= 0:6 eV,\n\u0001 = 5 meV, AII= 0:7 eV.\nstates is described by the following equation,\nn\u0000ng=\"(n)Z\nEmind\"\r(\");\"(n) =EG\u0000e2d\n\"0(n\u0000ng)+\"F(n);\n(1)arXiv:1107.4769v1  [cond-mat.mtrl-sci]  24 Jul 20112\nwherenis the density of holes in graphene, ng=CVg=e\n(Vgis gate voltage), dis the distance between SiC and\nQFMLG, the Fermi energy (relative to the graphene\nDirac point) is \"F(n) =\u0000~vp\u0019n, andEmin=\u00001 for\nthe model I and Emin =EIIfor the model II. Note\nthat integral in Eq. 1 is taken over the electron (rather\nthan hole) energy. Each of the models is characterized\nby one energy parameter: A=EG\u0000EIfor model I and\nA=EG\u0000EIIfor model II.\nThe value of the carrier density for non-gated struc-\ntures can be obtained by solving Eq. (1) for ng= 0.\nIn Fig. 1a we illustrate the hole density dependence in\ngraphene on the amount of acceptors on hydrogenated\nSiC. For the model I, it is\nnI= min [nv;nb] ;nb=4A2\n\u0010\n~vp\u0019+q\n~2v2\u0019+4e2dA\n\"0\u00112:\n(2)\nAt a small nv, all acceptor levels are occupied and nI=\nnv. As acceptor density nvincreases, graphene doping\nsaturates at nI\u00191\u00011013cm\u00002. Model II also shows\na saturation of the carrier density, but with a smoother\ncrossover,\nnII=4A2\n\u0012\n~vp\u0019+r\n~2v2\u0019+ 4A\u0010\n1\n\r0+e2d\n\"0\u0011\u00132:(3)\nThe e\u000bectiveness of using QFMLG transistors can be\ncharacterized by the responsivity factor, r=dn=dng.\nThen,r= 1 corresponds to the regime of the e\u000bective\ntransistor operation, while r\u001c1 indicates that it is dif-\n\fcult to change carrier density in graphene. The respon-\nsivity factors for the models I and II,\nrI= 1\u00001 + sign(nv\u0000nb)\n2p\n1 + 4e2dA=(\u0019~2v2\"0);\nrII= 1\u00001r\n1 +4A\n\u0019~2v2\u0010\n1\n\r0+e2d\n\"0\u0011;\nare compared graphically in Fig. 1b.\nDue to the charge transfer between graphene and ac-\nceptors in the hydrogen layer on SiC, carrier density\nof QFMLG change upon the variation of the magnetic\n\feldB. For high acceptor densities ( nv\u001d1012cm\u00002,\n\r0\u001d1013cm\u00002eV\u00001), Fig. 2a shows the oscillations in\nn(B) dependence, which appear due to a non-zero density\nof acceptor states at the Fermi level. Similar oscillations\nwere found in graphene on a non-hydrogenated SiC sur-\nface [9], and used to explain the presence of a wide \u0017= 2\nplateau in QHE. Both models give close results for the\nn(B) dependence, which reveals two di\u000berent regimes.\n(a) The regions with a negative slope indicating the pin-\nning of the Fermi level to one of the Landau levels in\nFIG. 2: a) The magneto-oscillations of the hole density nin\nQFMLG in the regimes of large acceptor density. b) n(B)\ndependence at small acceptor density. The energy structure\nin di\u000berent regimes is shown in the insets A-C (model II) and\ninset D (model I). The parameters used here were chosen for\nillistration purposes only.\ngraphene,EN=\u0000p\n2e~v2BN. To evaluate n(B) depen-\ndence in this regime we solved Eq. (1), with \"F=EN.\n(b) The regions with a positive slope corresponding to\nthe pinning of \flling factor \u0017=nh=(eB) = 4N+ 2, for\nwhich the Fermi level in the system lies in the acceptor\nband, between Landau levels.\nThe di\u000berence between the density oscillations in\ngraphene expected on the basis of the models I and\nII is quite pronounced in structures with low acceptor\ndensities (nv.1012cm\u00002,\r0.1013cm\u00002eV\u00001), as\nshown on Fig. 2b. In this regime no oscillations are ob-\nserved for model I, in contrast to the model II, where\npinning of \u0017= 2 takes place over magnetic \feld inter-\nval of several Tesla. In the latter case, one can iden-\ntify the following charge transfer regimes illustrated us-\ning sketches in Fig. 2b. At high magnetic \felds ( B >\nB1=hA=2e=(\r\u00001\n0+e2d=\" 0)) the carrier density remains\nconstant, since the 0th Landau level (LL) is partially oc-\ncupied and the Fermi level in graphene is pinned to the\nDirac point (inset A in Fig. 2b). At lower magnetic \felds3\n(2A2=(p\ne~v2+q\ne~v2+ 4eA=(\r\u00001\n0+e2d=\" 0)=h)2<B <\nB1) 0th LL becomes completely unoccupied and the\nFermi level sticks to acceptor levels between 0th and 1st\nLLs (Fig. 2b, inset B). This regime, observed within a\nwindow of several T, is responsible for widening the \u0017= 2\nQHE plateau. The region C in Fig. 2b refers to the pin-\nning of the Fermi level to the 1st LL E1.\nIn conclusion, we have considered two models of the\ncharge transfer between hydrogen intercalated graphene\nand acceptors in SiC surface. The \frst model describes\nthe case when all acceptor states are created by unsat-\nurated Si bonds with energies from a narrow window.\nThe opposite limit of a wide acceptor energy distribu-\ntion is covered by the second model. Both models pre-\ndict a saturation behavior of carrier density in grapheneas the acceptor density increases, however, the magnetic\n\feld dependence of charge transfer in them is very dif-\nferent, especially in structures with a low initial doping\nn.1012cm\u00002by holes. As shown in Fig. 2b, the model\nII features a wide (several T) \u0017= 2 QHE plateau, while\nthe model I reveals no n(B) dependence. One of the ways\nto detect pinning of \flling factor \u0017= 2 is to measure the\nactivation energy T\u0003and/or breakdown current of QHE\n[9]. The width of the peak in the dependence of T\u0003and\nbreakdown current on the magnetic \feld in the vicinity\nof\u0017= 2 QHE plateau determines the region of \flling fac-\ntor pinning. Alternatively, one can determine the \flling\nfactor by looking at the Landau level occupancy using\nscanning tunneling spectroscopy as in Ref. [10].\n[1] Xu Du, I. Skachko, A. Barker, and Eva Y. Andrei, Nature\nNanotechnology 3, 491 (2008).\n[2] A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic,\nM. S. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).\n[3] W. Gannett, W. Regan, K. Watanabe, T. Taniguchi, M.\nF. Crommie, and A. Zettl, Appl. Phys. Lett. 98, 242105\n(2011).\n[4] S. Kopylov, A. Tzalenchuk, S. Kubatkin, and V. I. Falko,\nAppl. Phys. Lett. 97, 112109 (2010).\n[5] J. Jobst, D. Waldmann, F. Speck, R. Hirner, D. K.\nMaude, Th. Seyller, and H. B. Weber, Phys. Rev. B 81,\n195434 (2010).\n[6] C. Coletti, C. Riedl, D. S. Lee, B. Krauss, L. Patthey,\nK. von Klitzing, J. H. Smet, and U. Starke, Phys. Rev.B81, 235401 (2010).\n[7] C. Riedl, C. Coletti, T. Iwasaki, A. A. Zakharov, and U.\nStarke, Phys. Rev. Lett. 103, 246804 (2009).\n[8] F. Speck, J. Jobst, F. Fromm, M. Ostler, D. Wald-\nmann, M. Hundhausen, H. B. Weber, Th. Seyller,\narXiv:1103.3997 (2011).\n[9] T. J. B. M. Janssen, A. Tzalenchuk, R. Yakimova, S.\nKubatkin, S. Lara-Avila, S. Kopylov, and V. I. Falko,\nPhys. Rev. B 83, 233402 (2010).\n[10] 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, Nature 467, 185\n(2010)."    },    {        "title": "1108.5999v2.Generic_phases_of_cross_linked_active_gels__Relaxation__Oscillation_and_Contractility.pdf",        "content": "epl draft\nGeneric phases of cross-linked active gels: Relaxation, Oscillation\nand Contractility\nS. Banerjee1, T.B. Liverpool2andM.C. Marchetti1;3\n1Physics Department, Syracuse University, Syracuse New York, 13244-1130, USA\n2Department of Mathematics, University of Bristol, University Walk, Clifton, Bristol BS8 1TW, U.K.\n3Syracuse Biomaterials Institute, Syracuse University, Syracuse New York, 13244-1130, USA\nPACS 87.16.Ka { Filaments, microtubules, their networks, and supramolecular assemblies\nPACS 87.10.-e { General theory and mathematical aspects\nAbstract { We study analytically and numerically a generic continuum model of an isotropic\nactive solid with internal stresses generated by non-equilibrium `active' mechano-chemical reac-\ntions. Our analysis shows that the gel can be tuned through three classes of dynamical states by\nincreasing motor activity: a constant unstrained state of homogeneous density, a state where the\nlocal density exhibits sustained oscillations, and a steady-state which is spontaneously contracted,\nwith a uniform mean density.\nIntroduction. { The mechanical properties of living\ncells are largely controlled by a variety of \flament-motor\nnetworks optimized for diverse physiological processes [1].\nIn the presence of ATP, such networks are capable of gen-\nerating controlled contractile forces and spontaneous os-\ncillations. These phenomena arise from the presence of\ngroups of motor proteins, such as myosin II, that convert\nthe chemical energy from ATP hydrolysis into mechanical\nwork via a cyclic process of attachment and detachment to\nassociated polar protein \flaments, e.g. F-actin [2]. There\nis great variation in the organization of actin, myosins,\nand other cross-linking proteins in the actomyosin struc-\ntures found in cells. Myo\fbrils in striated muscle cells are\nexamples of highly organized structures [1], composed of\nrepeated subunits of actin and myosin, known as sarcom-\neres, arranged in series. Each sarcomere consists of actin\n\flament of alternating polarity bound at their pointed end\nby large clusters of myosins, known as myosin \\thick \fla-\nments\". The periodic structure of the myo\fbril allows it\nto generate forces on large length scales due to the collec-\ntive dynamics of individual units of microscopic size, giv-\ning rise to muscle oscillation and contraction. More di\u000e-\ncult is to understand the origin of spontaneous oscillations\nand contractility in cytoskeletal \flament-motor assemblies\nthat lack such a highly organized structure [3].\nOscillations are for instance observed in many organ-\nisms during repositioning of the mitotic spindle from the\ncell center towards the cell pole when unequal cell divi-sion occurs [3]. In vitro examples of such phenomena are\nsustained cilia-like beatings in self-assembled bundles of\nmicrotubules and dyneins [4] and spontaneous contractil-\nity in isotropic reconstituted actomyosin networks with\nadditional F-actin crosslinking proteins, such as \flamin or\n\u000b-actinin [5,6].\nTheoretical models have shown that spontaneous os-\ncillations can arise from the collective action of groups\nof molecular motors coupled to a single elastic ele-\nment [7]. In these models the load dependence of the bind-\ning/unbinding kinetics of motor proteins breaks detailed\nbalance and provides the crucial nonlinearity that tunes\nthe system through a Hopf bifurcation to a spontaneously\noscillating state. This simple theoretical concept has been\nadapted to describe the beating of cilia and \ragella [8],\nmitotic spindles during asymmetric cell division [9], and\nspontaneous waves in muscle sarcomeres [10{13].\nIn a parallel development, generic continuum theories\nof active polar `gels' have been constructed by suitable\nmodi\fcation of the hydrodynamic equations of equilib-\nrium liquid crystals to incorporate the e\u000bect of activity.\nThese continuum models are capable of capturing some\nof the large-scale consequences of the internal contractile\nstresses induced by active myosin crosslinkers, including\nthe presence of propagating actin waves in cells adher-\ning to a substrate [14] and the retrograde \row in the\nlamellipodium of crawling cells [15]. In these studies the\nacto-myosin network is modeled as a Maxwell viscoelastic\np-1arXiv:1108.5999v2  [cond-mat.soft]  12 Oct 2011S. Banerjee1 T.B. Liverpool2 M.C. Marchetti1,3\n\ruid, with short-time elasticity and liquid response at long\ntimes [16]. The active viscoelastic liquid cannot, however,\nsupport sustained oscillations that require low frequency,\nlong wavelength elastic restoring forces.\nIn this letter we consider a nonlinear version of the\ngeneric continuum model of an isotropic active solid dis-\ncussed recently by two of us [17]. We go beyond the linear\nstability analysis discussed in [17] and show that the pres-\nence of nonlinearities leads to both stable contracted and\noscillatory states in di\u000berent regions of parameters. The\nacto-myosin network is modeled as an elastic continuum,\nas appropriate for a cross-linked polymer gel embedded in\na permeating viscous \ruid, with elastic response at long\ntimes and liquid-like dissipation at short times. The active\nsolid model describes the various phases of acto-myosin\nsystems as a function of motor activity, including spon-\ntaneous contractility and oscillations. It provides a uni-\n\fed description of both phenomena and a minimal model\nrelevant to many biological systems with motor-\flament\nassemblies that behave as solids at low frequencies. When\nthe active solid is isotropic, as assumed in the present\nwork, the coupling to a non-hydrodynamic mode provided\nby the binding and unbinding kinetics of motor proteins\nis essential to generate spontaneous sustained oscillations.\nThe main results are summarized in Fig. 1, where we\nsketch the steady states of the system as we tune the ac-\ntivity, de\fned as the di\u000berence \u0001 \u0016between the chemical\npotential of ATP and its hydrolysis products, and the com-\npressional modulus Bof the passive gel. For a \fxed value\nofBwe \fnd a regime where the active gel supports sus-\ntained oscillating states and a contracted steady state as\n\u0001\u0016is increased. For \fxed \u0001 \u0016spontaneous contractility\nis only observed below a critical network sti\u000bness. This is\nconsistent with experiments on isotropic acto-myosin net-\nworks with additional cross-linking \u000b-actinin where spon-\ntaneous contractility was seen only in an intermediate\nrange of\u000b-actinin concentration [5]. Our model does not,\nhowever, yield a lower bound on Bbelow which the con-\ntracted state does not exist. This may be because, in con-\ntrast to the experiments where a minimum concentration\nof\u000b-actinin is required to provide integrity to the net-\nwork, our system is always by de\fnition an elastic solid,\neven at the lowest values of B. The phase diagram result-\ning from our model also resembles the state space diagram\nof a muscle sarcomere obtained experimentally [13].\nInterestingly, we \fnd that a simpli\fed dynamical system\nobtained by a one-mode approximation to our continuum\ntheory corresponds to the half-sarcomere model proposed\nrecently by G unther and Kruse [10] for a particular set of\nparameters. Our analysis shows, however, that the phase\nbehavior described above is generic and can be expected in\na wide variety of active elastic systems, as it relies solely on\nsymmetry arguments. It provides a uni\fed description of\nboth spontaneous and oscillatory states and predicts their\nregion of stability as a function of the elastic properties of\nthe network and motor activity.\nUnstrained\nSteady StateSustained\nOscillations\nContracted  \nSteady State\n0123456701234567\nDmBFig. 1: (Color online) \\Phases\" of the nonlinear active elastic\ngel obtained by varying the compressional modulus Band the\nactivity \u0001\u0016.\nModel. { In Ref. [17] two of us formulated, on the\nbasis of symmetry arguments, a phenomenological hydro-\ndynamic model of a cross-linked gel (a network of actin \fl-\naments crosslinked by \flamins or other \"passive\" linkers)\nunder the in\ruence of active forces exerted by clusters of\ncrosslinking motor proteins (e.g., myosin II mini\flaments).\nOnly linear terms were retained in the continuum equa-\ntions in [17] where the linear modes of the system and\ntheir stability were analyzed. Here we consider a non-\nlinear continuum model and show how nonlinearities can\nstabilize oscillatory and contracted states. The active gel\nconsists of a polymer network in a permeating \ruid [17].\nOn length scales large compared to the network mesh size,\nthe gel can be described as a continuum elastic medium,\nviscously coupled to a Newtonian \ruid. The model follows\nclosely that formulated for a passive gel [18]. We focus on\ncompressional deformations and consider for simplicity a\none-dimensional model. We assume the permeating \ruid\nto be incompressible and consider the case of small volume\nfraction of the gel. In this limit the permeating \ruid sim-\nply provides a frictional drag to the polymer network. The\nhydrodynamic description is then obtained in terms of two\nconserved \felds: the density \u001a(x;t) of the gel and the one-\ndimensional displacement \feld, u(x;t). The momentum\ndensity of the gel is not conserved due to drag exerted by\nthe permeating \ruid. In addition, density variations are\ndetermined by the local strain, with \u000e\u001a=\u001a\u0000\u001a0=\u0000\u001a0@xu\nand\u001a0the equilibrium mean density. The elastic free\nenergy density of the gel can be expanded in the strain\ns=@xuabout the state s= 0. To describe the possibility\nof swelling and collapse of a gel (even a passive one) one\nneeds to keep terms up to fourth order in sin the elastic\nfree energy density (dropping a linear term that can be\neliminated by rede\fning the ground state) [19]\nfe=B\n2s2+\u000b\n3s3+\f\n4s4; (1)\nwhereBis the longitudinal compressional modulus of the\ngel and\u000b;\f > 0 are phenomenological parameters cap-\np-2Generic phases of cross-linked active gels: Relaxation, Oscillation and Contractility\nturing the e\u000bects of many-body interactions and nonlinear\nelasticity of the components [20].\nIt is straightforward to show that within mean-\feld the-\nory the free energy given in Eq. (1) yields a line of \frst\norder phase transitions at B\u0011Bc= 2\u000b2=(9\f) between\nan unstrained state with s= 0 forB >Bcand a strained\nstate with \fnite sforB < Bc. The stable strained state\nis one of higher density ( s <0) for\u000b > 0, corresponding\nto a contracted state, and one of lower density ( s>0) for\n\u000b<0, corresponding to an expanded state. The transition\nline terminates at a critical point at \u000b= 0.\nActivity is induced by the presence of a concentration\nc(x;t) of bound active cross-linkers that undergo a cyclic\nbinding/unbinding transformation fueled by ATP, exert-\ning forces on the gel. The dynamics of the active gel on\ntime scales larger than the Kelvin-Voigt viscoelastic re-\nlaxation time is described by coupled equations for u(x;t)\nandc(x;t), given by\n\u0000@tu=@x\u001be\u0000@xpa; (2)\n@tc=\u0000@x(c@tu)\u0000k(s)(c\u0000c0) (3)\nwhere\u001be=@fe\n@s=Bs+\u000bs2+\fs3is the elastic stress, \u0000 is\na friction constant describing the coupling of the polymer\nnetwork to the permeating \ruid, and pa(\u001a;c) is the active\ncontribution to the pressure, describing the isotropic part\nof the active stresses induced by myosins. Equation (3) al-\nlows for convection of bound motors by the gel at speed @tu\nand incorporates the binding/unbinding dynamics, with\nk(s) the strain-dependent motor unbinding rate and c0the\nequilibrium concentration of bound motors. Since highly\nnon-processive motor proteins such as myosins are on av-\nerage largely unbound, we neglect the dynamics of free\nmotors that provide an in\fnite motor reservoir.\nThe active pressure is taken to be linear in the rate \u0001 \u0016\nof ATP consumption, with pa= \u0001\u0016\u0010(\u001a;c). This is a rea-\nsonable approximation for weakly active systems, where\nthe number of active elements make up a small fraction\nof the total mass of the gel, as is the case in most experi-\nments.1We then expand\n\u0010(\u001a;c)'\u00100\u0000\u00101\u000e~\u001a\u0000\u00102\u001e\u0000\u00103(\u000e~\u001a)2+\u00104\u000e~\u001a\u001e\n+\u00105\u001e2+\u00106(\u000e~\u001a)3:::(4)\nwith\u000e~\u001a=\u000e\u001a=\u001a 0and\u001e= (c\u0000c0)=c0the \ructuations in the\ngel and bound motor concentrations, respectively, and all\nparameters \u0010ide\fned positive. The positive sign of \u00101and\n\u00102corresponds to a contractile acto-myosin system and de-\nscribes the reduction in the longitudinal sti\u000bness of the gel\nfrom contractile forces exerted by motor proteins. The pa-\nrameters\u00103,\u00104and\u00105describe excluded volume e\u000bects. A\npositive\u00103favors contracted over expanded states. A pos-\nitive\u00106guarantees the stability of the network in regions\nof negative e\u000bective compressional modulus.\n1We note that the e\u000bects of nonlinearities in \u0001 \u0016can taken ac-\ncount of by expanding in \u000e\u0016= \u0001 \u0016\u0000\u0001\u00160about a stationary state\nwith \fnite \u0001 \u00160.There are several sources of nonlinearities in Eqs. (2,3):\nthe nonlinear strain dependence of the elastic free energy,\nthe nonlinear terms in the active pressure, and the depen-\ndence of the motor unbinding rate kon the load force on\nbound motors, which in turn is proportional to the strain\nsof the elastic gel. We assume an exponential dependence\nof the form k(s) =k0e\u000fs[21], withk0the unbinding rate\nof unloaded motors and \u000fa dimensionless parameter de-\ntermined by microscopic properties of the motor-\flament\ninteraction. In the following we expand the unbinding\nrate for small strain as k(s)'k0h\n1 +\u000fs+\u000f2\n2s2+O(s3)i\n:\nKeeping higher order terms in \u000fdoes not change qualita-\ntively the behavior described below. Finally, the \frst term\non the rhs of Eq. (3) contains a convective nonlinearity\nthat does not a\u000bect the key features of dynamics observed\nand will be neglected in most of the following.\nThe equations for the active gel can then be written as\n\u0000@tu=@x\u001ba\ne+ \u0001\u0016@x\u0002\n\u00102\u001e+\u00104s\u001e\u0000\u00105\u001e2\u0003\n; (5a)\n@t\u001e=\u0000@x[(1 +\u001e)@tu]\u0000k0\u0014\n1 +\u000fs+\u000f2\n2s2\u0015\n\u001e; (5b)\nwhere\u001ba\ne=Bas+\u000bas2+\fas3, with renormalized elastic\nconstants,Ba=B\u0000\u00101\u0001\u0016,\u000ba=\u000b+\u00103\u0001\u0016and\fa=\n\f+\u00106\u0001\u0016.\nWe render our equations dimensionless by letting u!\nu=L andt!tk\u00001\n0. We then de\fne dimensionless param-\neters as ~B=B\n\u0000k0L2, ~\u000b=\u000b\n\u0000k0L2,~\f=\f\n\u0000k0L2, \u0001~\u0016=\u0001\u0016\n\u0000k0L3\nand ~\u0010i=\u0010iL. In the following we drop the tilde to sim-\nplify notation and all quantities should be understood as\ndimensionless.\nLinear stability analysis. { There are three steady\nstate solutions of Eqs. (5a,5b): an unstrained state, with\n(us;\u001es) = (0;0), and two strained states, with ( us;\u001es) =\n(s\u0006x;0) ands\u0006=\u0010\n\u0000\u000ba\u0006p\n\u000b2a\u00004Ba\fa\u0011\n=2\fa, provided\nBa< \u000b2\na=4\fa. For concreteness we choose \u000ba;\fa>0.\nThen ifBa>0,s\u0006<0 and\u000e~\u001a\u0006>0, so that both strained\nsolutions correspond to contracted states. If Ba<0, then\ns\u0000<0 ands+>0, corresponding to contracted and\nexpanded states, respectively.\nIn this section we examine the linear stability of each of\nthese states. Letting s=s0+\u000es, wheres0= (0;s\u0006) rep-\nresents the constant value of strain in the steady state and\n\u000es=@x\u000eu, the linearized equations for the displacement\nand motor concentration \ructuations are given by\n@t\u000eu=B@2\nx\u000eu+Z@x\u001e; (6a)\n@t\u001e=\u0000@x@t\u000eu\u0000\u0014\u001e; (6b)\nwhere\nB=Ba+ 2\u000bas0+ 3\fas2\n0; (7a)\nZ= \u0001\u0016(\u00102+\u00104s0); (7b)\n\u0014= 1 +\u000fs0+\u000fs2\n0=2: (7c)\np-3S. Banerjee1 T.B. Liverpool2 M.C. Marchetti1,3\nUnstrained\nContracted\n02468100246810\nDmB\nUnstrainedUnstable\nContracted\n02468100246810\nDmB\nFig. 2: (Color online) Phase diagram obtained by analyzing\nthe linear stability of the modes. The left frame is for \u00102=\n\u00104= 0, corresponding to \u001e= 0. The right frame is obtained\nincluding motor \ructuations, with \u00102=\u0010and\u00104=\u0011\u0010. The\nother parameter values are \u000b= 0:1,\f= 0:1,\u00101=\u0010= 1,\n\u00103=\u0011\u0010, and\u00106=\u00112\u0010, with\u0011= 0:1.\nLooking for solution of the form \u000eu;\u001e\u0018ezt+iqxwe \fnd\nthat the dynamics of \ructuations is controlled by two\neigenvalues, with dispersion relations\nzu;\u001e(q) =\u0000b(q)\n2\u00061\n2q\n[b(q)]2\u00004\u0014Bq2 (8)\nwhereb(q) =\u0014+(B\u0000Z)q2andzu(q) andz\u001e(q) correspond\nto the root with the plus and minus signs, respectively.\nIf motor \ructuations are neglected ( \u001e= 0), correspond-\ning to letting Z= 0 in Eq. (8), one \fnds zu=\u0000Bq2and\nthe linear stability of the steady states is entirely deter-\nmined by the sign of B, withB>0, corresponding to a\nlinearly stable state. When \u001e= 0 the problem is equiv-\nalent to an equilibrium problem, with q2Bthe curvature\nof a free energy of the form given in Eq. (1), albeit with\nelastic constants renormalized by activity. In this case the\nunstrained state s= 0 is the global minimum of the free\nenergy for Ba>2\u000b2\na=(9\fa), while the contracted state\ns=s\u0000is the global minimum for Ba<2\u000b2\na=(9\fa). The\nlineBa= 2\u000b2\na=(9\fa) de\fnes a line of \frst order phase\ntransitions in the ( B;\u0001\u0016) (see Fig. 2, left frame). If, on\nthe other hand, we consider the problem dynamically, the\ncondition of stability of linear \ructuations given by B>0\nis a necessary, but not su\u000ecient one to identify the stable\nsteady states of the system, as multiple \fxed points with\ndi\u000berent basins of attraction coexist in the same region\nof parameters. In particular, both the unstrained s= 0\nstate and the contracted s=s\u0000state are linearly stable\nforBa< B < \u000b2\na=(4\fa), while both the contracted and\nexpanded states ( s=s\u0007) are stable for Ba<0. If we\nassume that among these linearly stable states the system\nselects dynamically the steady state with the fastest de-\ncay rateq2B, then we recover the linear phase diagram of\nFig. 2 obtained form the equilibrium analysis.\nWhen the coupling to motor concentration \ructua-\ntions is included, we \fnd a region of parameters where\nRe[zu;\u001e]>0 for all three homogeneous solutions s=(0;s\u0006). This is the white region in Fig. 2 (right frame),\nwhere the dynamical system is linearly unstable. The in-\nstability occurs in a region where the modes are complex,\ndescribing oscillatory states, and Z\u0000\u0014=q2>B>0. The\nlinear stability diagram for the system is shown in Fig. 2\nfor the shortest wavevectors \u00181=L, withLthe system\nsize. The phase diagram is constructed again by assuming\nthat the system will relax to linearly stable states charac-\nterized by the fastest relaxation rates. To linear order, the\ncoupling to motor \ructuations yields oscillatory or propa-\ngating (as opposed to purely di\u000busive) \ructuations. Some\nof these oscillatory states are linearly unstable in a region\nof parameters, as shown in the phase diagram in Fig. 2.\nNext we consider the e\u000bect of nonlinearities by \frst ex-\namining a minimal model that only retains the longest\nwavelength mode in the Fourier expansion of the nonlin-\near equations, Eqs. (2,3), and then comparing the latter\nto the numerical solution of the full nonlinear partial dif-\nferential equations.\nOne-mode model. { We begin by incorporating\nonly the nonlinearities in the unbinding rate, while ne-\nglecting convective, elastic and pressure nonlinearities.\nWe impose boundary conditions [ @xu]x=0= [@xu]x=L=\n0 and [\u001e]x=0= [\u001e]x=L= 0, and seek a solution of\nEqs. (2,3) in the form of a Fourier series as, u(x;t) =P1\nm=0um(t) cos ( ^mx) and\u001e(x;t) =P1\nm=1\u001em(t) sin ( ^mx),\nwhere, ^m=m\u0019=L . We perform a 1-mode Galerkin trun-\ncation, and only consider the dynamics of the \frst non-\ntrivial mode, m= 1 (setting um=\u001em= 0;8m6= 1).\nThis corresponds to approximating the system by only\nits longest wavelength excitations, ignoring all the shorter\nwavelength modes.\nThe coupled equations for u1and\u001e1are given by\n_u1=\u0000Ba\u00192u1+\u00102\u0001\u0016\u0019\u001e 1; (9a)\n_\u001e1=\u0019_u1\u0000\u0012\n1\u00008\n3\u000fu1+3\n8\u00192\u000f2u2\n1\u0013\n\u001e1: (9b)\nThese equations can be recast into an e\u000bective second\norder di\u000berential equation for u1that has the structure\nof the equation for a Van der Pol oscillator [22] coupled\nto a nonlinear spring, of the form  u1+ _u1[\u0015\u0000f(u1)] +\nu1Ba[1\u0000f(u1)] = 0, with f(u1) =8\u000f\n3u1\u00003\u000f2\u00192\n8u2\n1and\nfriction\u0015=\u00192(Ba\u0000\u00102\u0001\u0016) + 1. Equations (9a) and\n(9b) admit one \fxed point ( u1;\u001e) = (0;0). Linear sta-\nbility analysis about this null \fxed point shows that the\n\fxed point is unstable (a repeller) when \u0015 < 0 and is\nstable for positive \u0015. From a global analysis [22] of Eqs.\n(9a) and (9b) with the \u000fnonlinearity, we \fnd that the\nexistence of the unstable \fxed point signals the appear-\nance a stable limit cycle as \u0015crosses zero. In other words\nthe system undergoes a supercritical Hopf bifurcation at\n\u0001\u0016= \u0001\u0016c1= (B+\u0019\u00002)=(\u00101+\u00102), with sustained oscil-\nlations for \u0001 \u0016>\u0001\u0016c1and stable spirals or nodes [22] in\nthe two-dimensional, ( u1;\u001e1) phase space, otherwise.\nWe now consider the role of pressure and elastic nonlin-\nearities, while letting \u000f= 0. Within the one-mode model\np-4Generic phases of cross-linked active gels: Relaxation, Oscillation and Contractility\n05101520-4-3-2-1012\ntu1HtL\nDmc1Dmc2\n012345-4-202\nDmu1s\nFig. 3: (color online) Left frame : Time evolution of u1(t) for\nvarious values of activity : \u0001 \u0016= 1:0 (black), corresponding\nto an unstrained steady state, \u0001 \u0016= 2:0 (red) and \u0001 \u0016= 3:5\n(green), corresponding to the region of sustained oscillations\nand \u0001\u0016= 4:5 (blue), corresponding to the contracted steady\nstate. Right frame: bifurcation diagram obtained from the 1-\nmode model. The \fgure shows a plot of the value us\n1ofu1at\nlong times ( t= 100) vs \u0001 \u0016. A supercritical Hopf bifurcation\noccurs at \u0001 \u0016= \u0001\u0016c1'1:5. As one increases \u0001 \u0016within the\nrange \u0001\u0016c1<\u0001\u0016<\u0001\u0016c2, the amplitude of oscillations grows\ncontinuously. At \u0001 \u0016= \u0001\u0016c2the limit cycle disappears and the\nsystem settles into a contracted steady state for \u0001 \u0016 > \u0001\u0016c2.\nThe dashed line indicates an unstable \fxed point. Parameters:\nB= 3:0,\u000f= 0:1. Other parameter values are same as in Fig. 2.\nwe \fnd that in this case if \fa=\u000ba= 0 andBa>0 there\nis again only one \fxed point at ( u1;\u001e) = (0;0) and the\nlinear stability analysis is identical to that of the model\nwith only rate nonlinearity. In other words, if \u000f= 0,\nthe pressure nonlinearities do not stabilize the system for\n\u0015 < 0. When\u000ba6= 0;\fa6= 0 andBa<16\u000b2\na=27\fa\u00192,\nwe have two new nonzero \fxed points ( u1;\u001e)\u0011(u\u0006;0),\ndescribing a contracted and an expanded state of the gel,\nrespectively, with\nu\u0006=8\u000ba\n9\u00192\fa\u00062\n3\fa\u00192r\n16\u000b2a\n9\u00003\u00192\faBa: (10)\nSince\u000e\u001a'\u0019\u001a0u1,u+corresponds to a contracted state\nwhereasu\u0000corresponds to an expanded state when B >\n\u0010\u0001\u0016. The contracted/expanded state is linearly unstable\nwhen\u0015\u0003\n\u0006=\u0000Ba+8\u000bau\u0006\n3\u00009\fa\u00192u2\n\u0006\n4+\u00102\u00008\u00104u\u0006\n3\u00001\n\u00192>0:\nFor\u0015\u0003\n\u0006<0 the nonzero \fxed points are unstable if\nBa>16\u000b2\na=27\fa\u00192. WhenBa<16\u000b2\na=27\fa\u00192and\n\u0015\u0003\n\u0006<0 the nonzero \fxed points are stable and the or-\nbits in the two-dimensional, ( u1;\u001e1) phase-space describe\nnodes or spirals settling at long times to ( u\u0006;0) [22].\nWhen\u0015\u0003\n\u0006>0, the contracted/expanded states are lin-\nearly unstable. The nonlinearities in the active pres-\nsure stabilize these unstable states into stable asymmetric\nlimit cycles. A supercritical Hopf bifurcation occurs when\n\u0001\u0016 > \u0001\u0016c1= (B+\u0019\u00002)=(\u00101+\u00102), which terminates to\na stable contracted steady state for \u0001 \u0016 >\u0001\u0016c2(\u0001\u0016c2is\ndetermined by the solution of \u0015\u0003\n\u0006(B;\u0001\u0016) = 0).\nIt is instructive to note that when \u000ba=\fa= 0, our\n1-mode model corresponds to that of a half-sarcomere de-rived in Ref. [10]. The bifurcation diagram for the com-\nplete 1-mode model is shown in Fig. 3, with all the nonlin-\nearities incorporated. The diagram summarizes the steady\nstate crossovers of the system as we increase \u0001 \u0016keepingB\n\fxed. Finally, the steady states of the nonlinear gel within\nthe 1-mode approximation are shown in the phase dia-\ngram in Fig. 4 (indicated by dashed lines), in the ( B;\u0001\u0016)\nplane. The unstrained state of the gel is stable when \u0015>0\nwhereas the contracted state is stable for \u0015\u0003\n+>0. When\n\u0015<0 and\u0015\u0003\n+<0 the gel exhibits sustained oscillations.\nNumerical analysis of the Continuum Nonlinear\nModel. { In this section we discuss the numerical so-\nlution of the nonlinear PDEs given in Eqs. (5a,5b) (but\nwith no convective nonlinearities) with boundary condi-\ntions [@xu(t)]x=0;L= 0 and [\u001e]x=0;L= 0 and an initial\nstrained state. We spatially discretize the PDEs using\n\fnite di\u000berence method and then integrate the resulting\ncoupled ODEs using the rkf45 method.\nThe behavior of the system as we vary Band \u0001\u0016, keep-\ning all other parameters constants, is summarized in the\nnumerically constructed phase diagram, shown in Fig. 4.\nAssuming\u00101=\u00102=\u0010, the system settles into an un-\nstrained state for B > Bc1'2\u0010\u0001\u0016and Hopf bifurcates\nto sustained oscillations for B < Bc1. We note that the\nphase boundary in the 1-mode model B= 2\u0010\u0001\u0016\u00001=\u00192,\nalthough di\u000berent from the numerical phase boundary\nB=Bc1, lies within the error bar of the numerics. For\nB <Bc2'0:85\u0010\u0001\u0016the system settles into a stable con-\ntracted state. The basin of attraction of the expanded\nstate is much smaller than that of the contracted state.\nThis is due to the choice of the sign of the coupling con-\nstants,\u000baand\u00104. The regimes predicted by our con-\ntinuum phenomenological model may be used to clas-\nsify the behaviour seen in a number of experimental sys-\ntems [5, 6, 13]. To estimate the parameter in real sys-\ntems we assume \u0000 \u0018\u0011=\u00182, with\u0011the viscosity of the\npermeating \ruid and \u0018the gel mesh size [18] and use ex-\nperimental values of B. For muscle \fbers, using B\u00182\nkPa [23],\u0011\u00182\u000210\u00003Pa s [2],\u0018= 0:5\u0016m,L\u0018100\u0016m\nandk\u00001\n0= 40ms, we estimate the dimensionless modulus\n~B=B=\u0000L2k0as~B\u00181. For isotropic cross-linked acto-\nmyosin gels only direct measurements of the low frequency\nshear modulus Gare available, with G\u00181\u000010Pa[6]. It\nhas been argued, however, that these networks may sup-\nport much higher compressional forces, of the order of\nbuckling forces on the scale of the mesh-size, yielding a\nvalue ofBcomparable to that of muscle \fbers [24]. For\nB\u00181\u0000103Paand\u0011comparable to that of water, we\nobtain ~B\u001810\u00003\u00001. The active coupling \u0010\u0001\u0016can be es-\ntimated as [10] \u0010\u0001\u0016\u0018\u0018nbkm\u0001m, wherenbis the number\ndensity of bound motors, kmis the sti\u000bness of the myosin\n\flaments and \u0001 mis the steady state stretch of the bound\nmotor tails. Using nb\u0018102\u0000104\u0016m\u00003,km= 4pN=nm\nand \u0001m\u00181nm, we obtain \u0010\u0001\u0016\u00180:1\u000010 in dimen-\nsionless units. In the phase diagram in Fig. 4, these pa-\nrameter values would put muscle \fbers in the regions U,\np-5S. Banerjee1 T.B. Liverpool2 M.C. Marchetti1,3\nUSO\nC\nDl(t)\nt010-303\n0246801234567\nDmB\nFig. 4: (Color online) Phase diagram in the ( B;\u0001\u0016) plane\nfor the continuum nonlinear model described by Eqs. (5a)\nand (5b). U : Unstrained, SO : Sustained Oscillations, C :\nContracted. The points are obtained by numerical solution\nof the full PDE's; the error bars are determined by the step\nsize used in the \u0001 \u0016increments. The dashed lines indicate the\n1-mode phase boundaries. Inset : Plot of extension \u0001 `(t) =\nu(0;t)\u0000u(L;t) forB= 3:0 and \u0001\u0016= 1:0 (black), \u0001 \u0016= 2:0\n(red), \u0001\u0016= 3:5 (green) and \u0001 \u0016= 4:5 (blue). Parameter\nvalues are same as in Fig. 3.\nSO and C respectively as we increase nb. Isotropic ac-\ntomyosin networks may have a much lower value of ~B,\npossibly corresponding to the contracted region. Indeed,\nwhile spontaneous contractility has been observed in vitro\nin reconstituted actomyosin networks [5], no experimental\nevidence has yet been put forward of spontaneous oscilla-\ntions in these systems.\nConclusion. { We have presented a generic contin-\nuum model of a cross-linked active gel which can be used\nto describe a wide variety of isotropic elastic active sys-\ntems. We \fnd an elastic active gel system can, in gen-\neral, be tuned through three classes of dynamical states\nby increasing motor activity: an unstrained steady state\nof homogeneous constant density, a state where the lo-\ncal density exhibits sustained oscillations, and a sponta-\nneously contracted steady state, with a uniform mean den-\nsity. The continuum model is not strictly hydrodynamic\ndue to presence of the fast variable \u001e, describing the dy-\nnamics of motor proteins. The motor binding/unbinding\nkinetics plays a crucial role in generating oscillating states\nat \fnite wavevectors. The one-mode model is in excel-\nlent qualitative agreement with the results described by\nthe solutions to the nonlinear continuum equations and is\ncomparable to a variety of one dimensional models for ac-\ntive elastic systems. Quantitative agreement in the phase\nboundaries between the one-mode model and the contin-\nuum model fails due to the non hydrodynamic nature ofthe model and we expect the oscillatory and/or contractile\nbehaviour to depend on system size [25].\n\u0003\u0003\u0003\nMCM and SB were supported by the National Science\nFoundation through awards DMR-0806511 and DMR-\n1004789. TBL acknowledges the support of the EPSRC\nunder grant EP/G026440/1.\nREFERENCES\n[1]Alberts B. et al. ,Molecular biology of the cell (Garland,\nNew York) 2007.\n[2]Howard J. ,Mechanics of motor proteins and the cy-\ntoskeleton (Sinauer Associates Sunderland, MA) 2001.\n[3]Albertson D. ,Dev. Biol. ,101(1984) 61; Yeh E. et al. ,\nMol. Bio. Cell ,11(2000) 3949.\n[4]Sanchez T., Welch D., Nicastro D. andDogic Z. ,\nScience ,333(2011) 456.\n[5]Bendix P. M. et al. ,Biophys. J. ,94(2008) 3126.\n[6]Koenderink G. H. et al. ,Proc. Nat. Acad. Sci. ,106\n(2009) 15192.\n[7]Julicher F. andProst J. ,Phys. Rev. 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Lett. ,63(1989) 1082.\n[20]Storm C. et al. ,Nature ,435(2005) 191; Gardel M. L.\net al. ,Science ,304(2004) 1310;\n[21]Parmeggiani A., J ulicher F., Peliti L. andProst\nJ.,Europhys. Lett. ,56(2001) 603.\n[22]Strogatz S. H. ,Nonlinear dynamics and chaos (West-\nview Press) 1994.\n[23]Magid A. andLaw D. ,Science ,230(1985) 1280.\n[24]O. Chaudhuri S. H. P. andFletcher D. A. ,Nature ,\n445(2006) 295.\n[25]Thoresen T. et al. ,Biophys. J. ,100 (2011) 446; Shi-\nmamato Y. et al. ,Biochem. Biophys. Res. Comm. ,366\n(2008) 233; Dufresne E. R., Private Comm.\np-6"    },    {        "title": "1109.3727v1.Probing_the_quantum_state_of_a_1D_Bose_gas_using_off_resonant_light_scattering.pdf",        "content": "arXiv:1109.3727v1  [cond-mat.quant-gas]  16 Sep 2011Probing the quantum state of a 1D Bose gas using off-resonant l ight scattering\nA. G. Sykes1,2and R. J. Ballagh1\n1Jack Dodd Centre for Quantum Technology, Department of Phys ics,\nUniversity of Otago, PO Box 56, Dunedin, New Zealand\n2Theoretical Division and Center for Nonlinear Studies,\nLos Alamos National Laboratory, Los Alamos, NM 87545, USA\n(Dated: October 19, 2018)\nWe present a theoretical treatment of coherent light scatte ring from an interacting 1D Bose\ngas at finite temperatures. We show how this can provide a nond estructive measurement of the\natomic system states. The equilibrium states are determine d by the temperature and interaction\nstrength, andare characterized bythespatial density-den sitycorrelation function. We showhowthis\ncorrelation function is encoded in the angular distributio n of the fluctuations of the scattered light\nintensity, thus providing a sensitive, quantitative probe of the density-density correlation function\nand therefore the quantum state of the gas.\nPACS numbers: pacs numbers\nThe remarkable progress made in cold atom physics over the past tw o decades has been facilitated by the precision\nand flexibility of the measurement tools available to experimentalists. Optical probing by lasers has proved to be\none of the most important tools, for example absorption and phase contrast imaging [1] have been extensively used\nto measure atomic density in bosonic and fermionic systems. Propos als for employing coherent light scattering from\ncold atoms are long standing [2]. More than a decade ago superradian t light scattering from a Bose condensate was\ndemonstrated, and stimulated light scattering was used to implemen t Bragg scattering of matter waves [3]. More\nrecently Mekhov et al[4] have proposed light scattering as a probe of cold atoms in an optic al lattice.\nIn this letter we present a theoretical treatment of monochroma tic light scattering appropriate for a 1D Bose\ngas. We show that for far detuned incident light, a clear signature o f various quantum states of the gas can be\nobtained from the fluctuations of the scattered light. The 1D Bose gas is an important model in many-body physics,\nproviding one of the simplest paradigms of a strongly interacting sys tem, owing to its exact integrability via the\nBethe ansatz [5]. In addition, the distinction between fermions and b osons (that are well known in 3D systems), are\nsuppressed in 1D [6]. For a 1D Bose gas with repulsive interactions, th e system state can be characterized by the two\nbody spatial density-density correlation function [7]. Possible syst em states range from a nearly ideal gas in the ‘high’\ntemperature regime (where the spatial correlation function displa ys bunching), to the Tonks-Girardeau gas in the\ninteraction dominated regime, (where the correlation function disp lays antibunching). Experimental investigations of\nthe 1D Bose gas have been pursued in recent years [8]. A method for measuring momentum correlations of a cold\natom system has been demonstrated by analysing the density fluct uations of a resonant absorption image [10]. Two\nand three body correlationswere also studied via resonant absorp tion images [9]. Impressive as these experiments are,\nthis technique cannot image submicron density fluctuations. In this work we show that a nondestructive measurement\nof the spatial nonlocal density-density correlation function of a 1 D Bose gas can be made by measuring the angular\ndistribution of the fluctuations in the scattered light intensity.\nWe consider a system of two-state bosonic atoms in a long thin cylindr ical trap, illuminated by a coherent laser\nbeam of frequency ωLand wavevector kL. Each atom has resonant frequency ω0and transition dipole moment d. We\nuse the rotating wave approximation for the radiation interaction. The laser mode is treated as a classical quantity\nELei(kL·x−ωL)t, and its interaction with the atoms is characterised by the detuning ∆L=ωL−ω0and the Rabi\nfrequency Ω L=d·EL//planckover2pi1. In the case where the detuning is large, ∆ L≫ΩL,Γ, (Γ is the spontaneous emission rate)\nthe upper internal atomic state can be adiabatically eliminated, leavin g coupled equations for the scattered radiation\nmodes ˆak(t) and the atomic field operator for the lower internal state ˆΨ(x). The equation for ˆ ak(t) can be integrated,\n(ignoring small terms and employing the Markov approximation) leadin g to\nˆak(t) =πΩLgk\n∆Le−iωLtδ(∆k)ei∆kt/2/integraldisplay\nS3d3xeiq·xˆρ(x,t). (1)\nHeregkdenotes the usual atom-radiation coupling constant, ∆ k=ωL−ωk, ˆρ=ˆΨ†ˆΨ is the 3D density of the system,\nq=kL−kandS3is the region of space where the incoming light intersects the gas. We have excluded the additive\nquantum noise term from Eq. (1) as we will only consider normally orde red averages of ˆ ak.\nFor the cylindrically symmetric trapping potential V(x) =mω2\n⊥(y2+z2)/2, the 1D regime can be achieved when\n/planckover2pi1ω⊥≪kBT,µ(the chemical potential). In addition the atomic recoil energy shou ld be much less than /planckover2pi1ω⊥. In this2\ncase the atomic field operator is well approximated by ˆΨ(x,t) = (√πl⊥)−1exp[−(y2+z2)/2l2\n⊥−iω⊥t]ˆφ(x,t), where\nˆφ(x,t) is the 1D annihilation operator of the atomic ground state and l⊥=/radicalbig\n/planckover2pi1/(mω⊥) is the transverse oscillator\nlength. Performing the transverse integration in Eq. (1) gives\nˆak(t) =A e−iωLt/integraldisplay\nS1dxˆn(x)eiqxx(2)\nwhereA=πΩLgkei∆kt/2δ(∆k)e−(q2\ny+q2\nz)l2\n⊥/4/∆L, ˆn=ˆφ†ˆφis the 1D density and S1= [−L/2,L/2] is the region where\nthe light intersects the gas. In this paper, we consider the case wh ere the incident light is a plane wave polarized\nalong the zaxis, with kLin thex-yplane and making an angle θLto theyaxis. The intensity of the scattered light\nis proportional to the total number of photons in a given mode,\n/an}bracketle{tˆa†\nkˆak/an}bracketri}ht=|A|2/bracketleftBigg\nN+2n2/integraldisplayL\n0dr g(2)(r)cos(qxr)(L−r)/bracketrightBigg\n, (3)\nwhereN=nLis the total number of atoms illuminated. The first term in the square brackets in Eq. (3) arises\nfrom the atomic shot noise, while the integral provides the effect of the atomic correlations, via the density-density\ncorrelation function of the gas, g(2)(x−x′) =/an}bracketle{t:ˆn(x) ˆn(x′):/an}bracketri}ht/n2. We have assumed that, within the illuminated region,\nthe density is constant, /an}bracketle{tˆn(x)/an}bracketri}ht=n, andg(2)is translationally invariant.\nThe function g(2)characterisesthe state of a uniform density 1D Bose gas [7]. At equ ilibrium in the thermodynamic\nlimit,g(2)(r) is completely determined by two parameters[11]: the dimensionless interaction strength γ=m2ω⊥a//planckover2pi1n\n(whereais the 3D s-wave scattering length, and mis the atomic mass) and the reduced temperature τ=T/Tdwhere\nTd=/planckover2pi12n2/(2kBm) is the temperature of quantum degeneracy. Varying these two p arameters gives rise to three\nphysically distinct regimes, each of which can be further split into two subregimes.\n(i)Nearly ideal-gas regime,γ≪min{τ2,√τ}. Herethetemperaturealwaysdominatesoverthe interactionstr ength,\nand the local density-density correlation displays thermal bunchin g, 1< g(2)(0)≤2. This regime has two subregimes\ndefined by τ≪1 (quantum decoherent gas) or τ≫1 (classical decoherent gas).\n(ii)Weakly interacting regime,τ2≪γ≪1. This regime, for which both the interaction strength and the tem per-\nature are small, realises the quasicondensate where density fluctu ations are strongly suppressed but long wavelength\nfluctuationsofthe phasestillexist[12]. Thedensity-densitycorre lationisclosetothe uncorrelatedvalueof g(2)(r)≈1,\nand fluctuations occur due to either vacuum or thermal excitation s. Two further subregimes can be defined, τ≪γ\n(dominant vacuum fluctuations) or τ≫γ(dominant thermal fluctuations).\n(iii)Strongly interacting regime (Tonks gas), γ≫max{1,√τ}. Here the interaction strength always dominates\nover temperature induced effects, and the local density-density correlation displays interaction-induced antibunching\n0≤g(2)(0)<1. This regime is usefully divided into two further subregimes defined b yτ≪1 (low temperature Tonks\ngas) and τ≫1 (high temperature Tonks gas).\nIneachoftheaboveregimes,perturbationtheorycanbeapplieda ndsixdifferentanalyticexpressionsforthedensity-\ndensity correlation function have been calculated [11]. The density –density correlation function decays from its local\nvalueg(2)(0) to the uncorrelated value g(2)= 1 over a length lc( the correlation length). For the quantum decoherent\ngas we have lc= 2/(τn) (the phase coherence length), while for the classical decoheren t gas,lc=/radicalbig\n4π/(τn2) (the\nthermal de Broglie wavelength). For the weakly interacting quasico ndensate regime, (with either vacuum or thermal\nfluctuations dominant) we have lc= 1//radicalbig\nγn2(the healing length). For the low temperature Tonks gas we have\nlc= 1/nand finally for the high temperature Tonks gas we have lc=/radicalbig\n4π/(τn2)\nThe physically relevant cases for light scattering are where L≫lc> λ. In order to distinguish the incoherent from\nthe coherent scattering it is convenient to define η(r)≡g(2)(r)−1, Eq. (3) then becomes\n/an}bracketle{tˆa†\nkˆak/an}bracketri}ht=/an}bracketle{tˆa†\nk/an}bracketri}ht/an}bracketle{tˆak/an}bracketri}ht+|A|2N/bracketleftBigg\n1+2N\nL2/integraldisplayL\n0dr η(r)cos(qxr)(L−r)/bracketrightBigg\n. (4)\nThe first term on the right of Eq. (4), /an}bracketle{tˆa†\nk/an}bracketri}ht/an}bracketle{tˆak/an}bracketri}ht, is the coherent scattering and is predominantly within angular width\n∼λ/Lof theqx= 0 direction. For λ≪L, the total intensity of coherently scattered light at a distance rfrom the\ngas is\nIc=N2/parenleftbiggΩL\n2∆L/parenrightbigg2\nΓ/planckover2pi1ωL3sin2θ\n8πr2sinc2/bracketleftbiggLΘ\n2/bracketrightbigg\n, (5)3\nwhere Θ = kLx−kLsinθcosθs, andθs= arctan(kx\nky) is thein-plane scattering angle, and θ= arctan(√\nk2x+k2y\nkz) is the\nout-of-plane scattering angle (see Fig. 4).\nThe signature for the atomic g(2)(r) function is in the fluctuations of the scattered intensity arisingfrom the integral\nterm of Eq. (4). These fluctuations are typically expressed in term s of quadratures of the radiation field operator\n(relative to a particular phase reference β), which are defined by ˆXk\nβ=1\n2/parenleftbig\nˆake−iβ+h.c./parenrightbig\n. The variances of these\nquadrature operators,\nRk\nβ=/an}bracketle{t/parenleftBig\nˆXk\nβ/parenrightBig2\n/an}bracketri}ht−/an}bracketle{tˆXk\nβ/an}bracketri}ht2\n=|A|2N/bracketleftBigg\n1\n2+N\nL2/integraldisplayL\n0dr η(r)/parenleftbigg\ncos(qxr)(L−r)+cos(2 β)sin[qx(L−r)]\nqx/parenrightbigg/bracketrightBigg\n, (6)\ncanbemeasuredbywellknowntechniques. For L≫lc> λ, thedependenceon βinEq.(6)isverytightlyconcentrated\naround the transmitted and reflected directions; i.e. |qx|/lessorsimilar1/L. A simple and prudent choice of phase reference is\nβ=π\n4. In this case the quadrature noise simplifies to Rkπ\n4≈ |A|2N/bracketleftBig\n1/2+n/integraltextL\n0dr η(r)cos(qxr)/bracketrightBig\n, which consists of the\natomic shot noise term, and a Fourier transform of η(r).\nIn order to obtain an analytic understanding of our results, we firs t take the limiting cases where γandτtake on\nvalues which lie deepwithin one of the regimes defined above. The density-density corre lation then simplifies to\nη(r)≈\n\nexp[−τnr] (a)\nexp/bracketleftbig\n−τn2r2/2/bracketrightbig\n(b)\n−exp/bracketleftbig\n−τn2r2/2/bracketrightbig\n(c)\n−sin2(πnr)/(πnr)2(d)\n(a) is deep within the decoherent quantum regime, (b) is deep within t he decoherent classical regime, (c) is deep\nwithin the high temperature Tonks regime, (d) is deep within the low te mperature Tonks regime. The integral for\nRkπ\n4can then be calculated analytically;\nRkπ\n4=|A|2N\n2\n\n1+2n2τ/(q2\nx+n2τ2) (a)\n1+2/radicalbig\nπ/(2τ)e−q2\nx/(2n2τ)(b)\n1−2/radicalbig\nπ/(2τ)e−q2\nx/(2n2τ)(c)\n1−tri[qx/(2πn)] (d)(7)\nwhere tri( r) = max(1 − |r|,0). We note that for η= 0, the case of a perfectly coherent quasicondensate, density\nfluctuations are completely suppressed, and Rkπ\n4reduces to the shot noise. For intermediate values of γandτ, we\nsolve for Rk\nβnumerically using expressions for g(2)taken from [11]. As seen in Fig. 4, the intensity fluctuations\narising from η(r) are typically scattered over a wider angle than the coherent scat tering. For example, the thermal\ncase in Eq. (7)(b), the angular spread is ∼arctan(N(λ/L)(2.3√τ/2π)). As a crude approximation, we can write\nη(r) =ηe−r2/(2l2\nc), where−1≤η≤1, and we find the incoherent scattering intensity, at a distance rfrom the gas, to\nbeIic=\nN/parenleftbiggΩL\n2∆L/parenrightbigg2\nΓ/planckover2pi1ωL3sin2θ\n8πr2/bracketleftbigg\n1+nlcη√\n2πexp/parenleftbigg−l2\ncΘ2\n2/parenrightbigg/bracketrightbigg\n. (8)\nIntensity isosurfaces of both the coherent and incoherent scat tering are shown in Fig. 4.\nThe results for Rkπ\n4plotted against the scattered angle are shown in the right-hand co lumns of Figs. 1–3, for a range\nofτandγ. In the left hand-columns we plot the corresponding g(2)(r). In Fig. 1 the sequence of plots are at a fixed\nhigh temperature, and show the transition from a decoherent clas sical gas (top row) to a high temperature Tonks\ngas, as the interaction strength γincreases from 0 to 10. For weak interaction g(2)displays a peak at r≈0; thermal\nbunching. As γincreases, the peak in g(2)changes to a dip; antibunching. The behaviour of the correspondin g\nscattered intensity fluctuations, which consist of a noise floor ( |A|2N/2), plus the Fourier transform of η=g(2)−1,\ncan be readily understood. A positive (negative) function ηproduces a peak (trough) in Rkπ\n4, with a width scaling\ninversely to lc. This is clearly illustrated in the sequence of plots for θL= 0 in Fig. 1. Atom bunching produces\nenhanced intensity fluctuations over a range of angles around θs= 0, while antibunching results in noise suppression\nin the scattered light over a similar angular range. We can interpret t he more complex structure in Fig 1 (c2) as4\n012\nγ=0τ=8g(2)(r;γ,τ)(a1)\n00.51Rβk/(|A|2N)(a2)\n01\nγ=0.5τ=8(b1)\n00.5(b2)\n01\nγ=4τ=8(c1)\n00.5(c2)\n0 1 201\nγ=10τ=8\nnr(d1)\n00.5\n−π−π/20π/2πθs(d2)\nFIG. 1: (color online). Density-density correlation funct ion (LH column) and corresponding scattered intensity quad rature\nnoise (RH column). Sequence shows transition from a decoher ent classical gas to a high temperature Tonks gas. Incident a ngle\nθL= 0 (solid blue lines); π/4 (red dashed lines); π/2 (green dot-dashed lines). Other parameters ( kL)x= 20n, andL≫lc.\n11.52\nγ=0.01\nτ=2g(2)(r;γ,τ)(a1)\n01Rβk/(|A|2N)(a2)\n11.5γ=0.01\nτ=0.3(b1)\n01(b2)\n11.5γ=0.01\nτ=0.05(c1)\n01(c2)\n0 1011.5γ=0.01\nτ=0\nnr(d1)\n01\n−π−π/20π/2π\nθs(d2)\nFIG. 2: (color online). Density-density correlation funct ion and scattered intensity quadrature noise for the transi tion from a\ndecoherent classical gas to a zero temperature quasiconden sate. Incident angle θL= 0 (solid blue lines); = π/4 (red dashed\nlines); = π/2 (green dot-dashed lines). ( kL)x= 5n, andL≫lc.\nbeing due to a remnant bunching on a long spatial scale superimposed on a more dominant (and spatially narrower)\nantibunching. For an incident angle of π/4, these features are replicated around a (broader) transmitte d peak at\nθs=−π/4 and a reflected peak at −3π/4. Fig. 2, where all plots are at weak interaction strength γ= 0.01, shows\nthe transition from a decoherent classical gas to a zero temperat ure quasicondensate as temperature is decreased to\nzero. At the higher temperatures, the scattered intensity fluct uations are enhanced in a broad angular region around\nthe transmitted (and reflected) beams. At the lowest temperatu re, where g(2)displays weak antibunching with ls\ngiven by the condensate healing length (which is long), the light fluctu ations are suppressed over a narrow angular\nrange. Fig. 2(b1) exhibits competition between bunching and antibu nching on different length scales, resulting in a\nsubtle minima in g(2)atnr≈τ/(4γ). This leads to a sharp dip in the noise features Fig. 2(b2) showing th at light\nscattering is an ideal probe of these subtle, and controversial [1 1], density-density correlations. Finally, in Fig. 3, a\nsequence of plots at constant large interaction strength γ= 8, shows the transition from a high temperature to low\ntemperature Tonks gas. In our analysis we have assumed the atom s are in an equilibrium state unperturbed by the\ninteraction with the light. This assumption is valid provided the heating per atom from the recoil energy is much less\nthan the initial energy per atom, E0. The recoil component perpendicular to the trap axis is absorbed b y the trap as\na whole (because it is very tight), and so for θL= 0, only the incoherently scattered light contributes to the heatin g.\nWe can estimate the heating rate per atom to be Υ /planckover2pi12/an}bracketle{tkx/an}bracketri}ht2/2mwhere/an}bracketle{tkx/an}bracketri}ht2is the mean square recoil momentum and\nΥ = Γ(ΩL\n2∆L)2is the rate of photon scattering [13]. The perturbation to the initial state of the gas will be negligible\nprovided Υ t≪E0. Expressions for E0can be found from [5].5\n01\nγ=8\nτ=20g(2)(r;γ,τ)(a1)\n00.4Rβk/(|A|2N)(a2)\n01\nγ=8\nτ=10(b1)\n00.4\n(b2)\n01\nγ=8\nτ=5(c1)\n00.4\n(c2)\n0 1 201\nγ=8\nτ=0\nnr(d1)\n00.4\n−π−π/20π/2π\nθs(d2)\nFIG. 3: (color online). Density-density correlation funct ion and scattered intensity quadrature noise for the transi tion from a\nhigh temperature Tonks gas to a low temperature Tonks gas. In cident angle θL= 0 (solid blue lines); = π/4 (red dashed lines);\n=π/2 (green dot-dashed lines). ( kL)x= 20n,L≫lc.\nFIG. 4: Intensity isosurfaces for coherently and incoheren tly scattered light. In each of the above figures, the blue cyl inder\nshows the orientation of the 1D Bose gas, the red cylinder sho ws the direction of the incoming laser beam, and the semi-\nopaque, purple surface shows an intensity isosurface as a fu nction of scattering angle. In (a) we show the intensity of th e\ncoherently scattered light [given by Eq. (5) in the main text ] withN2/parenleftBig\nΩL\n2∆L/parenrightBig2\nΓ/planckover2pi1ωL3\n8πr2= 1,kLx= 0.7kLandkLL≈28. In\n(b)–(e) we show the incoherent intensity [given by Eq. (8) in the main text], for the approximation η(r) =ηe−r2/(2l2\nc), with\nN/parenleftBig\nΩL\n2∆L/parenrightBig2\nΓ/planckover2pi1ωL3\n8πr2= 1,kLx= 0.7kL,nlc= 1 and kLlc= 7. In (b) we have η= 0.5; in (c), η= 0.3; in (d), η=−0.3; and in\n(e),η=−1.\nIn summary, we have shown that the quantum state of a 1D Bose ga s can be nondestructively measured using off\nresonant light scattering. The angular distribution of the quadrat ure noise in the scattered light provides a direct\nsignature of the nonlocal two-body correlations which character ize different system states of the gas. The method\nis useful across the full range of temperatures and atomic intera ction strengths, and is capable of revealing subtle\nfeatures in the density fluctuations that occur across multiple leng th scales. In addition, the method could also be\nused to monitor the dynamics of two-body correlations in the syste m, by simply solving the inverse Fourier transform\nof the intensity fluctuations.\nThis work was supported by the New Zealand Foundation for Resear ch, Science and Technology under Contract\nNo. NERF-UOOX0703. 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A 79, 043619 (2009); A. G. Sykes, et al., Phys. Rev. Lett. 100, 160406 (2008).\n[12] V. N. Popov, Functional Integrals in Quantum Field Theory and Statistic al Physics (D. Reidel Pub., Dordrecht, 1983);\nD. S. Petrov, D. M. Gangardt, and G. V. Shlyapnikov, J. Phys. I V France 1, 1 (2007).\n[13] C. Cohen-Tannoudji, in Fundamental Systems in Quantum Optics , edited by J. Dalibard, et al., Atomic motion in laser\nlight, (Elsevier Science, Amsterdam 1992)."    },    {        "title": "1109.5451v1.First_principles_calculations_of_phonon_and_thermodynamic_properties_of_AlRE__RE__Y__Gd__Pr__Yb__intermetallic_compounds.pdf",        "content": "First-principles calculations of phonon and thermodynamic properties\nof AlRE (RE =Y , Gd, Pr, Yb) intermetallic compounds\u0003\nRui Wangy, Shaofeng Wang, and Xiaozhi Wu\nDepartment of Physics and Institute for Structure and Function, Chongqing University,\nChongqing 400044, P. R. China.\nAbstract\nThe phonon and thermodynamic properties of rare-earth-aluminum intermetallics AlRE (RE =Y , Gd, Pr,\nYb) with B2-type structure are investigated by performing density functional theory and density functional\nperturbation theory within the quasiharmonic approximation. The phonon spectra and phonon density of\nstates, including the phonon partial density of states and total density of states, have been discussed. Our\nresults demonstrate that the density of states is mostly composed of Al states at the high frequency. The\ntemperature dependence of various quantities such as the thermal expansions, the heat capacities at constant\nvolume and constant pressure, the isothermal bulk modulus, and the entropy are obtained. The electronic\ncontribution to the specific heat is discussed, and the presented results show that the thermal electronic\nexcitation a \u000becting the thermal properties is inessential.\nPACS: 71.20.Lp, 63.20.D-, 65.40.De, 71.15.Mb\nKeywords: Rare-earth intermetallics; Phonon; Thermodynamic properties; First-principles calculations\n1 Introduction\nAluminum alloys are widely used for aerospace industries, aircraft automotive industries, electronic industries and\nbuildings due to light weight, good corrosion resistance, reasonably high strength and favorable economics. Addition\nof rare earth (RE) elements to Al-based metal alloys have received great attention since it can result in increasing their\nwear resistance [1], mechanical properties [2], electrochemical behavior [3] and thermal stability [4],etc. Thus, they are\nimportant for design of novel materials and further scientific and technical investigations. The development and design\n\u0003The work is supported by the National Natural Science Foundation of China (11074313) and and Project No.CDJXS11102211\nsupported by the Fundamental Research Funds for the Central Universities of China.\nyTel:+8613527528737; E-mail: rcwang@cqu.edu.cn.\n1arXiv:1109.5451v1  [cond-mat.mtrl-sci]  26 Sep 2011of new Aluminum alloys need more fundamental physical data of these alloys. Recently, some investigations focusing on\nAlRE, Al 2RE, and Al 3RE systems intensely are performed from first-principle calculations based on density functional\ntheory (DFT) [5, 6, 7, 8, 9, 10, 11, 12]. The B2-type AlY is a high-temperature stable while the temperature higher\nthan 1473K [13]. Such metastale phases are also available for AlGd, AlPr, and AlYb, etc. intermetallics with B2-type\nstructure [14, 15, 16]. Srivastava et. al. [5] investigated the electronic and thermal properties by using the tight binding\nlinear mu \u000en tin orbital (TB-LMTO). Tao et. al. [6, 7] performed the projector augmented wave (PAW) method to\ncalculate the elastic properties and predicted thermodynamical properties of B2-AlRE intermetallics below 400 Kwithin\nthe Debye-type model. However, the accurate calculations of thermodynamic properties for B2-type AlRE intermetallics\nfrom the lattice dynamics, i.e. exact phonon spectra, have received little attention.\nThermodynamical properties, such as entropy, specific heat, thermal expansion, and temperature dependent equa-\ntion of state (EOS) of AlRE intermetallics are very important to investigate their properties as well as applications. A\nsimplified method for thermal expansion calculations is in the framework of density functional theory (DFT) with a\nDebye-Gr ¨uneisen based model [17], which is based on the long-wave approximation. This model is useful for calculating\nthe thermal properties at low temperature, and thermodynamic properties of B2-type AlRE intermetallics have been car-\nried out based on this model by Tao et. al [7]. A more accurate approach has been made possible by the achievements of\ndensity functional perturbation theory (DFPT) from first principles[18], which allowed exact calculations of vibrational\nfrequencies in every point of the Brillouin Zone [19]. The vibrational free energy can be obtained using the quasiharmonic\napproximation (QHA). Moreover, QHA method lets one take into account the anharmonicity of the potential at the first\norder: vibrational properties can be understood in terms of the excitation of the noninteracting phonon. QHA based on\nDFPT provided a reasonable description of the thermodynamical properties of many bulk materials below the melting\npoint [19, 20, 21, 22, 23], and had been applied with great success to more and more complex materials such as alloys\n[NiAl 3[24]], perovskite [MgSiO 3[25]], and hexaborides [LaB 6and CeB 6[26]]. More recently, we have successfully\ninvestigated the thermodynamical properties of NiAl, YAg, and YCu [27], and MgRE intermetallics [28] with B2-type\nstructures.\nIn this paper, we apply first principles calculations within QHA based on DFPT to study the thermodynamic properties\nof AlRE (RE =Y , Gd, Pr, Yb) intermetallics with B2-type structures. The phonon spectra and phonon density of states,\nincluding the phonon partial density of states and total density of states, have been discussed. Thermal expansions,\n2temperature dependence of isothermal bulk modules, heat capacities at constant volume and constant pressure, and the\nentropy as a function of temperature are presented.\n2 Theory\nTo study the e \u000bects of changing temperature, one has to look at the Helmholtz free energy, incorporating the e \u000bects\nof thermal-electronic excitations and thermal-vibrations (phonons). The Helmholtz free energy at temperature Tand\nconstant volume Vis give by [29, 30]\nF(V;T)=E0(V)+Fel(V;T)+Fvib(V;T); (1)\nwhere E0(V) is the static contribution to the internal energy at volume V and can be easily obtained form standard DFT\ncalculations. Fel(V;T) given in Eq. (1) is the thermal electronic contribution to free energy and is given by Fel=Eel\u0000TS el.\nHere, The electronic excitation energy Eelis given by\nEel(V;T)=Z1\n0n(\";V)f(\")\"d\"\u0000Z\"F\n0n(\";V)\"d\"; (2)\nwhere n(\";V) is the electronic density of state (DOS) at energy \"and volume V,f(\") is the Fermi-Dirac distribution\nfunction, and \"Frepresents the Fermi level. The electronic entropy Selis formulated as\nSel(V;T)=\u0000kBZ1\n0n(\";V)[f(\") lnf(\")+(1\u0000f(\")) ln(1\u0000f(\"))]d\"; (3)\nwhere kBis the Boltzmann constant. Thermal electronic contribution to free energy is generally considered to be negligible\naway from the melting point of the material under consideration.\nFvib(V;T) Eq. (1) is the vibrational free energy which comes from the phonon contribution. Within the quasiharmonic\napproximation (QHA), Fvib(V;T) is given by\nFvib(V;T)=kBTX\nq;\u0015ln(\n2 sinh\u0012~!q;\u0015(V)\n2kBT\u0013)\n: (4)\nHere, the sum is over all phonon branches \u0015and over all wave vectors qin the first Brillouin zone, ~is the reduced Planck\nconstant, and !q\u0015(V) is the frequency of the phonon with wave vector qand polarization \u0015, evaluated at constant volume\nV.\nThe vibrational specific heat CVat constant volume in the QHA from the following equation\nCvib\nV=X\nq\u0015kB\u0012~!q\u0015(V)\n2kBT\u00132\ncosh2\u0012~!q\u0015(V)\nkBT\u00132\n: (5)\n3The electronic specific heat can be obtained from\nCel\nV=T\u0012@Sel\n@T\u0013\nV; (6)\nand we denote that total specific heat at constant volume is then CV=Cph\nV+Cel\nV. Due to anharmonicity, the specific\nheat at a constant pressure, Cp, is di \u000berent from the specific heat at a constant volume, CVgoes to a constant which\nis given by classical equipartition law: CV=3NkB, where Nis the number of atoms in the system while Cp, which\nis what experiments determine directly, is proportional to T. QHA lets one take into account the anharmonicity of the\npotential at first order: vibrational properties can be understood in terms of the excitation of the noninteracting phonon.\nThe equilibrium volume at temperature Tis obtained by minimizing Helmholtz Fwith respect to V, i.e., min V[F(V;T)].\nThe volume thermal expansion coe \u000ecient is given by\n\u000b(T)=1\nV\u0012@V\n@T\u0013\nP; (7)\nand the linear thermal expansion is described as\n\u000f(T)=a(T)\u0000a(Tc)\na(Tc); (8)\nwhere a(Tc) is equilibrium lattice constant a(T)=[V(T)]1=3atTc=300K.\nThen, Cpcan be obtained from CVand\u000bby\nCp=CV+\u000b2BVT; (9)\nwhere B(T)=\u00001=V@2F=@V2is the bulk modulus.\nThe vibrational contribution to the entropy of the crystal is given by\nSvib=\u0000kBX\nq\u0015\"\nln\u0012\n2 sinh~!q\u0015(V)\n2kBT\u0013\n\u0000~!q\u0015(V)\n2kBTcoth~!q\u0015(V)\n2kBT#\n: (10)\n3 Computational details\nIn present work, the static energy and the thermal electronic contribution to the Helmhotz free energy are computed\nby using the first-principles calculations in the framework of the density-functional theory (DFT). We employed the gen-\neralized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) [31, 32] exchange-correlation functional\n4as implemented in the V ASP code [33, 34, 35]. The ion-electron interaction is described by the full potential frozen-core\nprojector augmented wave (PAW) method [36, 37], with energy cuto \u000bof 600eV for plane waves. The Brillouin zones of\nthe unit cells are represented by Monkhorst-Pack special k-point scheme [38]. We used a sample as 21 \u000221\u000221k-point\nmesh in the full Brillouin zone giving 726 irreducible k-points to calculate the initial structures and electronic DOS. The\nradial cuto \u000bs of the PAW potentials of Al, Y , Gd, Pr and Yb were 1.40, 1.81, 1.58, 1.64, and 1.74 Å, respectively. The 3 s\nand 3 pelectrons for Al, the 4 s, 4p, 4dand 5 selectrons for Y , the 4 fand 6 selectrons for Gd, Pr, and Yb were treated as\nvalence and the remaining electrons were kept frozen. The thermal electronic energies and entropies are evaluated using\none-dimensional integrations from the self-consistent DFT calculations of electronic DOS using Fermi-Dirac smearing as\nshown in Eqs. (2) and (3). In order to deal with the possible convergence problems for metals, a smearing technique is\nemployed using the Methfessel-Paxton scheme [39], with a smearing with of 0.05eV .\nThe vibrational free energy were obtained from the first-principles phonon calculations by using PHONOPY [23,\n40, 41] which can support V ASP interface to calculate force constants directly in the framework of density-functional\nperturbation theory (DFPT) [42]. Phonon calculations were performed by the supercell approach. Since the chosen\nsupercell size strongly influences on the thermal properties, we compare the vibrational free energies of 3 \u00023\u00023 supercell\nwith those of 5\u00025\u00025 supercell at 300K and 1000K, and find that the energy fluctuations between 3 \u00023\u00023 and 5\u00025\u00025\nsupercells are less than 0.01%. Hence, we chose the 3 \u00023\u00023 supercell with 54 atoms to calculate phonon dispersions.\nWe carried out DFPT calculations on this 54 atoms supercell using PBE-GGA exchange-correlations e \u000bects and 7\u00027\u00027\nk-point grid meshes for Brillouin zone integrations.\nIn order to get the equilibrium lattice volume as a function of temperature, we have calculated total free energy at\ntemperature points with a step of 1K from 0 to 1200K at 13 volume points. At each temperature point, the equilibrium\nvolume V(T) and isothermal bulk moduli B(T) are obtained by minimizing free-energy with respect to Vfrom fitting the\nintegral form of the Vinet equation of state (EOS) [43] at p=0. These procedures applied for AlY are demonstrated in\nFigure 1, where the Helmholtz free energy F(T;V) as a function of unit-cell volume at temperatures are shown. While,\nCphas be calculated by polynomial fittings for CVand by numerical di \u000berentiation for @V=@Tto obtain\u000b(T).\n54 Results and discussions\n4.1 Phonon spectra and density of states\nWe calculated phonon frequency using the DFPT with force-constant method [44] with forces calculated using V ASP.\nThe calculated phonon spectra of AlY , AlGd, AlPr, and AlYb, which are calculated by using 3 \u00023\u00023 supercells, are\ndisplayed in Figures 2. Due to the crystal symmetry, the spectra curves are shown along high-symmetry direction G\u0000\nX\u0000M\u0000\u0000\u0000Rof the Brillouin zone. These dispersion curves have common framework. Since the B2-type intermetallics\ncontain two atoms per primitive cubic unit cell, there are three acoustical branches and three optical branches. Our results\nobviously show that the degenerating of two transverse-acoustical (TA) modes and two transverse-optical (TO) modes in\ntheG\u0000XandG\u0000Rdirections. At the Gpoint, the degenerating of longitudinal-optical (LO) and TO modes is obtained, so\nthere are no polarization e \u000bects in the B2-type AlRE intermetallics. The phonon DOS including the partial DOS (PDOS)\nand the total DOS (TDOS) are shown in Figures 3. Sampling a 51 \u000251\u000251 Monkhorst-Pack grid for phonon wave vectors\nqis found to be su \u000ecient in order to get the mean relative error in each channel of phonon DOS. The phonon band gap of\nAlY starts at 5 THz, while those of AlGd, AlPr, and AlYb starts at less than 4Hz. Among the four intermetallics, AlY and\nAlYb has the greatest and lowest values of the maximum value of acoustic modes, respectively, since the atomic mass of\nY is lightest and that of Yb is heaviest in the four calculated RE elements. The flat regions of phonon-dispersion curves,\nwhich correspond to the peaks in the phonon PDOS, indicate localization of the states, i.e., they behave like ”atomic\nstates” [23]. In the four intermetallics, the density of states are mostly composed of Al states above the phonon band gap\nsince its atomic mass is much lighter than those of the rare earth elements RE (RE =Y , Dy, Pr, Tb).\n4.2 Bulk properties and thermal expansion\nThe calculated results of the equilibrium lattice constants a0, the isothermal bulk modulus B0, and the pressure\nderivatives of the isothermal bulk modulus B0\n0atT=0Kfor AlRE (RE =Y , Dy, Pr, Tb) intermetallics together with\nthe previous calculated results [6] and the available experimental values [13, 15, 16] are shown in Table 1. Our calculated\nresults for the equilibrium lattice constants at T=0Kshow excellent agreements with the previous theoretical and\nexperimental results. For the isothermal bulk modulus B0and the pressure derivatives of the isothermal bulk modulus B0\n0\natT=0, which are obtained from Vinet equation of state, the present results are within 1.7% errors from the previous\ncalculated values obtained from the Rose’s equation of state [6]. The temperature dependent isothermal bulk modulus\n6B(T) are shown in Figure 4. Among the four intermetallic compounds, through the temperature range 0 \u00001200 K, lihgt\nRE AlY and heavy RE AlYb have the highest and lowest bulk moduli, respectively, i.e, AlY is the most incompressible\nand AlYb is the most compressible. With increasing temperature, the di \u000berences among the bulk moduli almost remain\nunchanged. The overall observation is that BTof the intermetallics decrease with increasing temperature, and approach\nlinearity at higher temperature and zero slope around zero temperature.\nAs shown in Figure 1, the equilibrium volume at any temperature corresponds the minimum values of the fitted\nthermodynamic functions. The thermal expansion is observed as an increase in the equilibrium volume. The linear\nthermal expansion \u000fdefined by Eq. (8) as a function of the four AlRE intermetallics are shown in Figure 5(a). The linear\nexpansions of the four compounds are found AlYb >AlY>AlGd>AlPr, and those of AlY and AlGd are nearly equivalent.\nThe coe \u000ecients of the volume thermal expansion \u000bas a function of temperature are shown in Figure 5(b). With increasing\ntemperature, the thermal expansion grows rapidly up to \u0018200K, and the slops become smaller and nearly constant at high\ntemperatures except AlYb. In addition, \u000bcan be used to estimate the anharmonic e \u000bects according to Gr ¨uneisen relation\n\u000b=\rCV=VBT, and Gr ¨uneisen parameter \r=\u0000dln!=dlnV. In QHA, the phonon frequencies at given lattice parameters\nare independent of temperature. In real crystal, it is not the case. The accuracy of QHA applied to AlRE (RE =Y , Gd, Pr,\nYb) can be verified by experiment in the future.\n4.3 Specific heat and Entropy\nOnce the phonon spectrum over the entire Brillouin zone is available, the vibrational heat capacity at constant volume\nCvib\nVcan be calculated by using Eq. (5), while the electronic contribution to heat capacity at constant volume Cel\nVcan be\nobtained from the electronic DOS by using Eq. (6). Then, the specific heat at constant pressure Cpcan be computed by\nEq. (9). As a comparison, both Cpvalues, including electronic contribution and not, are plotted. We display these results\nin Figures 6. At high temperature, Cvib\nVtends to the classical constant value 6 R(Ris molar gas constant), and Cel\nVandCp\nstill increase. Considering thermal electronic contributions to specific heat, we find that Cel\nVletsCpsu\u000ber a little shift and\ncan be negligible. This character can be understood from the electronic DOS at the Fermi level n(\"F). AlRE intermetallics\nhave low electronic DOS near the Fermi level and the Fermi levels occur at a valley in the curves of electronic DOS [6].\nObviously, Cel\nVis much smaller than the value Cp\u0000CV=\u000b2BVT . Hence, the electronic excitations a \u000becting the thermal\nproperties is inessential. Comparing with our previous study of MgRE intermetallics [28], the electronic specific heat can\n7not be negligible since the Fermi level for MgRE intermetallics occur above a peak in the electronic DOS [45] and the\nelectronic excitations a \u000becting the thermal properties is remarkable. Whether we consider the electronic contribution or\nnot, the theoretical Cpkeeps positive slopes obviously. At low temperature below \u0018200K, the discrepancy between Cp\nandCVcan be neglected, and CVvalues increase rapidly. We show temperature dependence of CVfor AlRE(RE =Y , Gd, Pr,\nand Yb) below 200K as in Figure 7. At low temperature, the specific heats CVof four AlRE intermetallics obey the law of\nT3. The values of CVare found AlY <AlGd<AlPr<AlYb, and this character can be understood from elementary excitation\nof phonon relating to the mass of unit cell. We have the order of the mass of unit cell MAlY<MAlGd<MAlPr<MAlYb.\nAt low temperature limitation T!0K, the specific heat CVis determined from the vibrations of the low frequencies.\nThe entropy as a function of temperature is an important thermodynamic quantity in thermodynamic modeling. The\ncalculation results of entropies for the fore B2-type structures AlRE are shown in Figure 8. It can be clearly seen that\nthe entropies of the four compounds increase with temperature. Through the temperature range 0 \u00001200 K, the overall\nobservation is that the order of the entropy SAlY<AlGd<AlPr<AlYb. At the temperature of 300 K, the calculated\nvalues of entropy for AlY , AlGd, AlPr, and AlYb are 67.48 Jmol\u00001K\u00001, 74.87 Jmol\u00001K\u00001, 77.93 Jmol\u00001K\u00001, and 85.03\nJmol\u00001K\u00001.\n5 Conclusions\nIn conclusion, the phonon and thermodynamic properties, such as the thermal expansions, the heat capacities at\nconstant volume and constant pressure, the isothermal bulk modulus, and the entropy as function of temperature, of\nrare-earth-aluminum intermetallics AlRE (RE =Y , Gd, Pr, Yb) with B2-type structure are investigated by DFT and DFPT\nwithin the QHA. The phonon spectra and phonon density of states, including the phonon partial density of states and\ntotal density of states, have been discussed. Our results demonstrate that the density of states are mostly composed of Al\nstates at the high frequency and composed RE (RE =Y , Gd, Pr, and Yb) states at low frequency. Through the calculated\ntemperature range 0 \u00001200 K, AlY is the most incompressible and AlYb is the most compressible. The di \u000berences of the\nisothermal bulk moduli among the four intermetallics almost remain unchanged with increasing temperature. The thermal\nexpansions of the four compounds are found AlYb >AlY>AlGd>AlPr, and those of AlY and AlGd are nearly equivalent.\nThe specific heats and entropy increase with temperature, while Cvib\nVtends to a constant and Cpkeeps positive slopes at\nhight temperature. The electronic contribution to the specific heat is discussed, and the presented results show that the\n8thermal electronic excitation a \u000becting the thermal properties is inessential.\nReferences\n[1] W.X. Shi, B. Gao, G.F. Tu, S.W. Li, J. Alloys Compd. 508 (2010) 480.\n[2] K.E. 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B 60 (2007) 75.\n11Table 1: The equilibrium lattice constants a0, the isohtermal bulk modulus B0, and the pressure derivatives of\nthe isothermal bulk modulus B0\n0atT=0Kfor AlRE (RE =Y , Gd, Pr, Yb) in our calculation in comparison with\nthe previous calculated results and the experiment.\nAlY AlGd AlPr AlYb\na0(Å) 3.606, 3.605a, 3.759b3.635, 3.634a, 3.7208c3.760, 3.759a, 3.82d3.699, 3.697a\nB0(GPa) 62.84, 63.70a62.54, 61.72a54.18, 55.11a38.72, 39.11a\nB0\n03.97, 3.99a3.79, 3.81a3.86, 3.92a4.24, 4.30a\naReference[6]\nbReference[13]\ncReference[15]\ndReference[16]\n7 \u0013.\n>H9@\n7 \u0014\u0015\u0013\u0013.\nFigure 1: (Color online) The Helmholtz free energy F(V;T) as a function of unit-cell volume of AlY . The\ncircles denote F(V;T) at the volume points at every 100K between 0 and 1200K. The solid curves show the\nfitted thermodynamic functions from Vinet EOS. The minimum values of the fitted thermodynamic functions\nat temperatures are depicted by the crosses. The dashed curve passing through the crosses is guide to the eye.\n12\u000bD\f\u0003$O<\n*\n 0\n *\n 5\n>7+]@\n;\n*\n ;\n 0\n *\n 5\n>7+]@\n\u000bE\f\u0003$O*G\n*\n ;\n 0\n *\n 5\n\u000bF\f\u0003$O3U\n>7+]@\n*\n ;\n 0\n *\n 5\n\u000bG\f\u0003$O<E\n>7+]@\nFigure 2: (Color online) Phonon-dispersion curves of (a) AlY , (b) AlGd, (c) AlPr, and (d) AlYb.\n130246800.511.522.5\n0246800.511.522.53\n0246800.511.522.5\nFrequency [THz]TDOS (PDOS) [states/THz]\nTDOS (PDOS) [states/THz]Frequency [THz] Frequency [THz]TDOS (PDOS) [states/THz]\nTDOS (PDOS) [states/THz]\n0246801234\nFrequency [THz](a) AlY (b)AlGd\n(c)AlPr (d)AlYbFigure 3: Phonon TDOS and PDOS of (a) AlY , (b) AlGd, (c) AlPr, and (d) AlYb. The solid curves, dashed\ncurves, and dotted curves represent the phonon total DOS (TDOS), partial DOS (PDOS) of RE (RE =Y , Gd, Pr,\nand Yb) states, and PDOS of Al states, respectively. The PDOS indicates that the DOS are mostly composed\nof Al states at high frequency and RE states at low frequency.\n14020040060080010001200203040506070\nT [K]BT [GPa]AlY\nAlGd\nAlPr\nAlYbFigure 4: (Color online) Isothermal bulk moduli Bas a function of temperature.\n020040060080010001200−505101520x 10−3\nT [K]εAlY\nAlGd\nAlPr\nAlYb(a)\n020040060080010001200012345678x 10−5\nT [K]α [K−1]AlY\nAlGd\nAlPr\nAlYb(b)\nFigure 5: (Color online) (a) Temperature dependence of the linear thermal expansion \u000f(T) for AlRE(RE =Y ,\nGd, Pr, and Yb); (b) The coe \u000ecients of volume thermal expansion \u000bfor AlRE(RE =Y , Gd, Pr, and Yb) as a\nfunction of temperature.\n150200400600800100012000102030405060\nT [K]Specific Heat [J mol−1 K−1](a) AlY\n0200400600800100012000102030405060\nT [K]Specific Heat [J mol−1 K−1](b) AlGd\n0200400600800100012000102030405060\nT [K]Specific Heat [J mol−1 K−1](c) AlPr\n0200 400 600 8001000 12000102030405060\nT [K]Specific Heat [J mol−1 K−1](d) AlYbFigure 6: Temperature dependence of heat capacity of (a) AlY , (b) AlGd, (c) AlPr, and (d) AlYb. Solid and\ndashed curves denote the calculated Cp, including electronic contribution and not, respectively. Doted and\ndot-dashed curves show vibrational and electronic CV, respectively.\n160 50 100 150 20001020304050\nT [K]CV  [J mol−1 K−1]AlY\nAlGd\nAlPr\nAlYbFigure 7: (Color online) The specific heat at constant volume CVfor AlRE(RE =Y , Gd, Pr, and Yb) as a function\nof temperature below 200K.\n020040060080010001200020406080100120140160\nT [K]Entropy [J mol−1 K−1]AlY\nAlGd\nAlPr\nAlYb\nFigure 8: (Color online) The entropy Sfor AlRE(RE =Y , Gd, Pr, and Yb) as a function of temperature.\n17"    },    {        "title": "1110.1962v1.Bound_States_in_Coulomb_Systems___Old_Problems_and_New_Solutions.pdf",        "content": "arXiv:1110.1962v1  [physics.plasm-ph]  10 Oct 2011Contributions toPlasmaPhysics,20 June 2018\nBound Statesin CoulombSystems\n- OldProblems andNew Solutions\nW. Ebeling1∗, W.D. Kraeft2,3∗∗,andG.R¨opke3∗∗∗\n1Institutf¨ ur Physik,Humboldt-Universit¨ at zuBerlin,Ne wtonstr. 15, D-12489 Berlin\n2Institutf¨ ur Physik,Ernst-Moritz-Arndt-Universit¨ at G reifswald, Felix-Hausdorff-Str.6,D-17487 Greifswald\n3Institutf¨ ur Physik,Universit¨ at Rostock, Universit¨ at splatz 3, D-18055 Rostock\nKeywords dense plasmas, bound states, partitionfunction.\nPACS05.60.-k,05.45.Yv,05.90.+m\nWe analyze the quantum statistical treatment of bound state s in Hydrogen considered as a system of electrons\nand protons. Within this physical picture we calculate isot herms of pressure for Hydrogen in a broad density\nregion and compare to some results from the chemical picture . First we resume in detail the two transitions\nalong isotherms :\n(i)the formationof bound states occurring byincreasing th e densityfrom low tomoderate values,\n(ii)the destruction of bound states inthehigh density regi on, modelledhere byPauli-Fockeffects.\nAvoiding chemical models we will show, why bound states acco rding to a discrete part of the spectra occur\nonly in a valley in the T-p plane. First we study virial expans ions in the canonical ensemble and then in the\ngrand canonical ensemble. Weshow that infugacityrepresen tations the population of bound statessaturates at\nhigher density and that a combination of both representatio ns provides quickly converging equations of state.\nInthe case of degenerate systems wecalculated firstthe dens ity-dependent energylevels, and findthe pressure\nin Hartree-Fock-Wigner approximation showing the promine nt role of Pauli blocking and Fock effects in the\nselfenergy.\nCopyrightlinewillbe providedbythepublisher\n1 Introduction\nIn1911,exactlyhundredyearsago,Rutherfordinventedane wmodelofmatter,whichisthebasisforthescience\ndealing with strongly coupled Coulomb systems [1]. His mode l was based on the interpretation of existing\nexperiments on the scattering of electrons on matter. In May 1911, Rutherford came forth with a model for\nthe structure of atoms which provideda first understandingw hy electron scattering can so deeplypenetrate into\nthe interior of atoms, so far unexpected experimental resul ts. Rutherford explained his scattering results as the\npassage of a high speed electron through an atom having a posi tive central charge +Ne, and surrounded by\na compensating charge of Nelectrons [1]. In Rutherford’s model the atom is nearly empt y, it is made up of\na central charge and electrons and the glue keeping the syste m together are the Coulomb forces. Soon after,\nin 1913, a first quantum-mechanical treatment of the model wa s given by Bohr and a statistical treatment by\nHerzfeld. AccordingtotheBohr-Herzfeldtheorytheenergi esandthecorrespondingpartitionfunctionare\nE∗\ns=−µe4\n2¯h2s2, µ=mem+\nme+m+, σ(T) =/summationdisplay\nss2exp(−βEs) =smax/summationdisplay\ns=1s2exp/parenleftbiggI\ns2kBT/parenrightbigg\n.(1)\nWe denote the main quantum number by sin order not to be mixed up with the density n. According to the\nBohr model there are infinitely many levels close to the serie s limits→ ∞the terms increase as s2and the\nsum diverges. This is the result for hydrogen, but the proble m remains for all elements. The problem of the\ndivergency of the atomic partition function took a long time before the mathematical background for a serious\ntreatment of this problem was available. However urgent nee d to solve the ionization problems in astrophysics\nand plasma physics forced fast solutions. For the calculati ons the problem was simplified and the partition\n∗Corresponding author: e-mail: ebeling@physik.hu-berlin.de\n∗∗e-mail:wolf-dietrich.kraeft@uni-rostock.de\n∗∗∗e-mail:gerd.roepke@uni-rostock.de\nCopyrightlinewillbeprovidedbythe publisher2 Ebeling, Kraeft,and R¨ opke: Bound States inCoulomb Syste ms- OldProblems andNew Solutions\nfunction was estimated by the first term of the sum, or by two or three terms. Ignoring the open problems with\nthepartitionfunction,EggertandSahaattackedtheurgent problemsofastrophysics,consideringonlytheground\nstate contribution.\nAn idea developed later was to cut the sum at the minimal terms |Es| ≃kBT. This way of solving the\ndivergenceproblemis due to the classical paper of Planck [2 ] published in 1924. DevelopingPlancks approach\nBrillouinproposedin 1930a moresmoothwayofremovingthe d ivergencewhichledtotheformula[3]\nσBPL(T) =∞/summationdisplay\ns=1s2[exp(βI/s2)−1−(βI/s2)]. (2)\nUsing semi-classical methods,Planck couldshow thatthe (i nfinite)interactiontermsbelowandabovetheseries\nlimit compensate each other. This procedure was justified by methods of quantum statistics only in 1960 by\nVedenov and Larkin [4, 5] and subsequently by the present aut hors in collaboration with D. Kremp by using\nquantum-statistical methods [6, 7, 8, 9, 10, 11, 12, 13, 14, 1 5]. We will show here that the Brillouin-Planck-\nLarkin procedure provides the most natural way to avoid the d ivergencies for non-degenerate systems. Further\nwestudydegeneratesystemsandshowthataconsequenttreat mentoftheidentityofelectronsintheHartree-Fock\napproximation including bound states is a key for the study o f hydrogenic bound states. The present approach\nis based on the methodof Greensfunctionsas developedin [11 , 12, 13, 14, 15] and recently surveyedin a book\ndedicated to Nevill Mott [16]. The present investigationis concentratedon the so-called physical picture which\nstayswithin theRutherfordpictureandavoidsthe introduc tionof chemicalspecieslike atoms,moleculesetc. In\nsomeoftheearlierworkthetransitiontoachemicalpicture wasmadefollowingtheprincipleofequivalencethat\nbound states are to be treated on the same footing as free part icles [14, 17]. This approachwas quite successful\nin the description of partial ionization[11, 18, 19, 20, 21, 22]. On the other hand the chemical picture may lead\nto principaldifficultiese.g. if at highdensitiesthe minim umof the freeenergyis not well defined[23, 24]. This\nisthereasonwhywestayherewithinthephysicalpicturetry ingtosumuptheimportanthigherorderterms,this\nway followingtheline developedin severalrecentworks[25 , 26,27, 28].\n2 Density andfugacity expansionsfornondegenerate system sincluding non-\nlinearring and Saha-type contributions\nAnysuccessfuldescriptionofplasmashastogobeyondtheSa hatheorywhichissomethinglikethezerothorder\napproximationforplasmas. ThederivationofSaha-typeapp roximationsfromquantumstatisticswasfirststudied\nby Planck and Brillouin [2, 3] and is since then a topic of perm anent interest due to the importance of Saha-\ntype equations for experimental work and for many technolog ical applications [6]. We use here a systematic\nquantum statistical approach to the pressure on the basis of the Greens function representations of the pressure\n[11, 12,13, 14]\np(β,µe,µi) =pid−1\n2V/integraldisplay1\n0dλ\nλ/integraldisplay\nd1d˜1V(1˜1)G2(1,˜1,1++,1+:˜t1=t+\n1) (3)\nwhere1 ={/vector p1,σ1}denotes momentum and spin variables. The Green’s function r epresentation works in the\ngrandcanonicalensemblewhichprovidesusa seriesinthefu gacities(sa-spinofparticles, a=e,i)\nza=2sa+1\nΛ3aexp(βµa) =naexp(βµex\na); Λ a=h√2πmakBT.\nNote thatµex\nais the excess part of the chemical potential (the part beyond the ideal Boltzmann term) and that\nthereforeza→nafor small densities. A representation by a density series ma y be obtain by inversion, i.e. by\nexcludingthe fugacities. We summarize the existing result s about the pressure of a hydrogenicquantumplasma\nasafunctionofthedensityinaratherspecialformwhichisa symtoticallyvalidintheregionoflowandmoderate\ntemperatures T≤I/kBT. Usingforthisregiontheasymptoticexpressionsoftheexa ctresults,thepressuremay\nbe writtenas( κ2= 8πβne2[11, 13]\nβp(β,n) =n[1+1\n8√\n2nΛ3\ne+O(n2)]+n−κ3\n24π[1−3√π\n8(κλ)+3\n10(κλ)2+...]\n−n2γ[1−βe2κ]+O(n3logn). (4)\nCopyrightlinewillbeprovidedbythe publishercpp header willbe provided by the publisher 3\nWe introducedherethe meanthermalwave length\nλ=λie=¯h\n2µkBT=Λe\n2√π;µ=memi\nme+mi;γ= Λ3\neσBPL(T) = 8π√πλ3\nieσBPL(T)\nwhereλis an effective quantum length, being in some correspondenc e to the Debye-H¨ uckel parameter in the\nclassical theory. We note as a special property of the given t emperature regime that he interaction terms in the\npressure dependonlyon reducedmass µof the electron-protoninteractionandnot on the otherredu cedmasses.\nAn equivalentformofeq.(4)whichshowsbetterthegenerals tructureisthefollowing\nβp(β,n) =n[1+1\n8√\n2nΛ3+O(n2)]+n−nγ[(1−βe2κ+O(κ2))+...]\n−κ3\n24π[1−3\n2γ+O(γ2)][1−3√π\n8(κλ)+3\n10(κλ)2+O(κ3)]+O(n3/2). (5)\nTheexpressionforthepressureofhydrogeninthelimit ofsm alldensities,givenintheformulaeeqs. (4)and(5)\nhas been drawn in Fig. 1. This is a first step to a Saha-type desc ription, since the pressure related to the ideal\npressureisdecreasingwiththedensitywithafactorpropor tionaltothedistributionfunction,howeverweseethat\nourformulafailsathigherdensitiesgoingeventonegative values. LookingatthecurvesinFig. 1weseethatthe\nscreeningeffectsandthecontributionsfromtheBrillouin -Planck-Larkinfunctionbothtendtolowerthepressure.\nThis lowering is too large at increasing densities, so we hav e to look for contributions limiting this growth. In\norder to proceed in this direction, we investigate first the s tructure of eq. 5, aiming to complete the unfinished\nseries with respect to γandκλ. The structure of eq. (5) suggests that the expressions in th e parenthesis may\n 0 0.2 0.4 0.6 0.8 1\n-14 -12 -10 -8 -6 -4 -2p / 2 n kB T:  T/I=0.1;0.15\nlg n’=n aB**3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1\n 16  18  20  22  24p / pid\nlg(nc), T =  15000;20000;50000 K \nFig. 1Left panel: Pressure related to the total classical pressur e2nkBTat constant temperatures T= 15788 K and\nT= 23682 Kindependence onthedensity(log-scale). Theresultofden sityexpansions upto2ndordervirialterms(redand\ngreen) are compared tocorresponding fugacity expansions. Right panel: Pressure related tothe full ideal pressure (in cluding\nthe Fermi pressure) calculated from an extended density rep resentation including the ring sum φ(x)and the Saha function\nA(x)for the temperatures T= 15000 K,20000 K,50000 K independence on the density(inlog-scale 15 -25).\nbe only the first terms of some infinite series with respect to nΛ3,κandγ. The first series is evidently nothing\nelse than the ideal electron pressure, which may easily be ex tended to an infinite series using the ideal Fermi\npressure. Inordertofindthehigherordersintheseriesin γandinκλwewilluseamorecomplicatedprocedure\nwhich is basedon fugacityexpansions[11, 17, 29] According to ourgeneralphilosophy,the strongdecayof the\npressureobservedinsecondorderdensityexpansions(Fig. 1)isduetothefactthatimportanthigherorderterms\ncorrespondingto some infinite series were omitted in the der ivation based on the canonical ensemble. Alastuey\net al. [25, 26] succeeded in deriving a Saha type expression i n summing up an infinite series of terms. Here we\nfollowasimilarideabutgoanalternativewaythroughfugac ityexpansions. Asshownmuchearlier,fugacity-like\nexpansions are equivalent to mass-action laws. This was use d in plasma theory by Bartsch, Ebeling and others\n[11, 17, 28, 29] andworkedoutin largedetail byRogerset al. [30]. We mentionthat Rogerset al. succeededto\nCopyrightlinewillbeprovidedbythe publisher4 Ebeling, Kraeft,and R¨ opke: Bound States inCoulomb Syste ms- OldProblems andNew Solutions\nshowthatextendedfugacityexpansionsallowanexcellentd escriptionofmeasurementsfortheoscillationmodes\nofthe sun[31].\nA systematical quantumstatistical approachtothepressur eincludingthelow aswellasthe highpressureregion\nmay be given on the basis of Green’s function representation s of the pressure [11, 12, 13, 14]. The Green’s\nfunctionrepresentationworksinthegrandcanonicalensem blewhichprovidesusaseriesinthefugacitiesinstead\nof a series in the densities. Note that µex\nais the excess part of the chemical potential (the part beyond the ideal\nBoltzmann term) and that therefore za→nafor small densities. The structure of this series is similar to the\ndensity series, we have e.g. similar screening effects, how ever there are some differences. e.g. the fugacity\nseriescontainsmorediagrams. Thecontributionsofbounds tatesofanelectronandanion(atoms)arecontained\nin the contribution of order zezi, the contribution of molecules is contained in the term of or derz2\nez2\ni. General\nexpressionsforscreenedfugacityserieswere written down first byMontrollandWard, VedenovandLarkinand\nexplicit calculations were done by De Witt and Larkin [4, 5, 3 2]. Semiclassical fugacity series were given by\nseveral workers [11, 29]. We are interested here mostly in th e bound state effect and may simplify by using\napproximations allowed for the region T≤157000K were the plasma is nondegenerate and where the bound\nstatesaredominant. Aswehaveseeninthedensityrepresent ation,thedifferencesofthemassesofelectronsand\nions (protons) do not play a role in the region T≪I kBand the plasma behaves asymptotically like a system\nwith equal masses and equal relative De Broglie radii. In the lowest approximationcorrespondingto eq. (4) we\nfindforlowtemperatureH-plasmasthefollowingresults[11 , 13,33]\nβp=ze+zi+κ3\ng\n12πf(κgλ)+8πzeziλ3exp[1+βe2κg]σBPL(T)+O(z5/2), (6)\nwherethegrand-canonicalscreeningquantityisdefinedby κ2\ng= 4πβ(ze+zi)e2andthequantum-statisticalring\nfunctionisa transcendentalfunctionhavingtheseries\nf(x) =/bracketleftbigg\n1−3√π\n16x+1\n10x2+.../bracketrightbigg\n. (7)\nThedensitiesmaybe expressedbyfugacities\nni=zi∂βp\n∂zi=zi+κ2\ng\n16∂\n∂λ(κgλf(κgλ))+4πzezi[1+βe2κg+···]λ3σBPL(T)···, (8)\nne=ze∂βp\n∂ze=ze+κ2\ng\n16∂\n∂λ(κgλf(κgλ))+4πzezi[1+βe2κg+···]λ3σBPL(T)···. (9)\nWe will showbyiterationsthat goingfromthe fugacityvaria bleto the densitythisresult occursto be equivalent\nto eq. (4). In order to proceed with this complicated system o f equations we go to the special case of non-\ndegenerate hydrogenat lower temperatures T≤I/kBand satisfying nΛ3\ne≤1. As shown above, in this region\nthe differencesof the electronion masses doesnot matter an dwe my assume ze=zi=z. In orderto represent\nthe pressure by densities we neglect now in a first step the bou nd state contributions assuming σBPL(T) = 0.\nExpressingin thisapproximationfirst thefugacitiesbyden sitieswefind\nz=nexp/bracketleftbigg\n−βe2κ\n2G(κλ)/bracketrightbigg\n;G(x) =√π\nx/bracketleftbigg\n1−exp(x2\n4)/bracketrightbigg/bracketleftBig\n1−Φ(x\n2)/bracketrightBig\n= 1−√π\n4x+1\n6x2+O(x3)···,(10)\nwhereΦ(x)is the error function. Then excluding the fugacities from th e series step by step we arrive at the\nrepresentation\nβp= 2n−κ3\n24πφ(κλ);κ2= 8πβne2. (11)\nHereφ(x)isanothertranscendentfunction,relatedto f(x)andf′(x), theso-calledpressureringfunctionwhich\nis an analogue of the Debye-H¨ uckel ring function [4, 11]. A f ull representation of this transcendent function is\ngivenbytheinfiniteseries\nφ(x) = 1−√π\n3∞/summationdisplay\nk=1(k+1)(k+3)x2k−1\n2kk!+1\n3∞/summationdisplay\nk=1(k+1)(k+3)x2k\n2k(2k−1)!!. (12)\nCopyrightlinewillbeprovidedbythe publishercpp header willbe provided by the publisher 5\nInthecalculationsweusedmoreconvenientPad´ eformulae. Thenonlinearfunctions φ(x)andG(x)convergeto\nzeroat largeargumentswhatisimportantforthehigh-densi tyasymptoticsofthepressure. InFig. 1theresultof\nfugacityexpansionsup to 2nd ordervirial terms for T= 15788 K andT= 23682 K (redand green curves)are\ncompared with corresponding density expansions. We see tha t the behaviour of truncated fugacity expansions\nseems to be much better than that of truncated density series . However this is not true in all cases and may lead\nto serious problems. We may show this on the example of the ide al Fermi gas. Truncating here the series after\nquadratictermsthe representationof the pressure leadsfo rnΛ3>√\n2to an imaginarypressure. For this reason\nwe hold the opinion that the a controlled summing up of terms i n the density series may be more reliable. The\nresults we obtained so far may be summarizedas follows. By me ans of grand-canonicalmethodswe succeeded\nalready to obtain one infinite series in the screening parame terκinstead of the first linear and quadratic terms\nonly we found in the density series, (see eq.(5)). This summi ng up a series in κwhich leads to a saturating\nfunctionwhatimprovesverymuchthebehaviorat largerdens ities.\nInourcanonicalrepresentationeq. (5)appearsalsoanothe rseriesexpansioninthedensityparameter γ. Inorder\nsumupthisseriesin γ,weproceedinasimilarway. Westudyatfirsttheboundstatec ontributions,neglectingthe\ncontributionscomingfromtheringtermsandfromthetermsd uetodegeneracy. Thisyieldsthesimplequadratic\nequation\nn=z+8πz2λ3σBPL(T). (13)\nBy theway,thisformulashowsalreadythatthefugacitiesof theelectronsandtheprotonsshouldberathersmall\nin the bound state region where the partition function is lar ge. The quadratic equation for the fugacities can be\neasily solved. Thisway wefindaftereliminationofthefugac ityz:\nz0=nA(γ);A(x) =1\n2x[√\n1+4x−1];γ= 8πnλ3σBPL(T). (14)\nIntroducingthezerothstep ofiteration z0intothepressurewefind\nβp0=n(1+A(γ)). (15)\nThisrepresentationincludesanonlinearfunction A(x)whichsaturatesatlargedensitiesandiscloselyrelatedto\ntheresultoftheSahatheory,exceptthatinourversiontheB PLpartitionfunctionappears. Thenewrepresentation\ncontains all terms in the density up to n2as well as several higher order terms in the density containe d in the\nnonlinearSahafunction A(γ). We will showthatthiskindoffunctionallowsusto reproduc eanideal Saha-type\nbehavior. We notethe followingseriesexpansionandasympt oticsofthenonlinearfunction A(x):\nA(x) = 1−x+2x2−···;A(x)≃1√x+···. (16)\nThe saturatingbehaviorof the nonlinearfunction A(x)in comparisonto its linear approximationhasthe conse-\nquencethat intheregionoflargepartitionfunctionsthefu gacitiesdisappearas\nz0=1/radicalbig\n8πnλ3σBPL(T). (17)\nThisisimportantfortheunderstanding,whythefugacityse rieshasintheregionofboundstatesa betterconver-\ngence as the density series. The better convergenceis due to the fact that the fugacities disappear in the region\nwheretheboundstatecontributionislarge. Thefugacityse riescontainsmoretermsthanthedensityseries,sowe\nmayexpectthatsomeofthedifficultiesconnectedwithdensi tyexpansions,asthestrongdecreaseofthepressure\nwith increasingdensity shown in Fig. 1 may be avoided. Indee d, a representationof the curvecorrespondingto\neq. (6)showsamorereasonablebehaviorwithincreasingden sitywhichismuchclosertotheSaha-typebehavior.\nAs the curvesshown in Fig. 1 demonstrate, the pressure accor dingto the fugacityexpansion goes to saturation.\nThisisduetothefactthatthefugacityisnotfixed,itdecrea seswithincreasingvalueofthepartitionfunctionand\nthis way limits the growth. Evidently the fugacity expansio ns provide a more correct description of the bound\nstate contributionsasdemonstratedinFig. 1. Inordertodr awsomefirst conclusions:\nDensity aswell asfugacityexpansionshavebothadvantages aswell as disadvantages: 1)Thedensityexpansion\ndescribeswell the screeningeffectsbutit failsto copewit h the divergingcontributionsfromthe screeningterms\nCopyrightlinewillbeprovidedbythe publisher6 Ebeling, Kraeft,and R¨ opke: Bound States inCoulomb Syste ms- OldProblems andNew Solutions\nandfromtheBPL-partitionfunction. 2)Thefugacityexpans ioncorrespondstoaninfinitedensityseriesincluding\nthe partition function σ. Ifσis large, then the fugacity goes to zero what guaranteesalwa ys finite contributions\nto the pressure. This is also true for the screening contribu tions; any strong increase of contributionsdamps the\nfugacities.\nWe mayexpectthat the best representationis obtainedby ext endeddensity expansionswhichcontainadditional\ncontributions corresponding to the important damping term s in the fugacity expansions and provide Saha-like\nterms. Thiskindofmixedexpansionscombinethepositivefe aturesofbothexpansionsavoidingthenegativefea-\ntures. In orderto demonstratethiswe started herefromthe d ensityexpansionsandused the fugacityexpansions\nforfindingtherightcontinuationoftheinfiniteserieswith respecttothe γ-parameterandthe κλparameter.\nThiswaywe obtainina first stepthe followingexpressionfor thefugacityandthepressure\nz=nA/parenleftbig\nexp[−βe2κG(κλ)]γ/parenrightbig\n, (18)\nβp=n−κ′3/2\n24πφ(κ′λ)+nA/parenleftbig\nexp[−βe2κG(κ′λ)]γ/parenrightbig\n, κ′=κA(γ)1/2, (19)\nwherearenormalizedvalueofthescreeningparameter κ′appears. Thisprovidesusaclosedandrelativelysimple\nformulaforthepressurewhichhoweverisrestrictedtothen ondegenerateregion. Howeversofarourresultsrefer\nto non-degenerate electrons. The good convergence of the ne w expression including nonlinear functions A(x)\nandφ(x)based on the fugacityseries is explainedby the fact that the fugacitiesof the electronsand the protons\nare rather small in the bound state region. The overall behav ior of the new representation is much better than\nthat of pure density or pure fugacity representations. We ma y conclude that the most appropriatedescription of\nCoulombicsystems is by density expansionsand includingin finite sumsrepresentingscreeningand boundstate\neffects. We note that the new theory based on eq. (19) is consi stent with the Saha-theory and in particular also\nwith the Saha-Debye-H¨ uckeltheory[11]. We underlinethat several of the termsbeyond n2andz2are basedon\nextrapolationswhich still need further confirmations. How everin the regionof low temperature T <I/k Band\nnon-degenerateplasmas our rather simple formulae give a ra ther good behavior and describe well the transition\nfromlowdensitytothevalleyofboundstates. Wenotehoweve rthatseveralphysicaleffectsase.g. plasmaphase\ntransitions[11, 18] arenotyetdescribedbythepresentapp roach. Maybethiseffectappearsonlyaftertransition\nto somechemicalpicture[11, 18].\nIn order to describe as well the transition to full ionizatio n in the degenerate region, additional effects are to be\ntaken into account, in particular the symmetry between elec trons and protons is lost. The most important new\neffects at high densities are connectedof the replacement o f the ring terms by Hartree-Fockterms and a density\ndependenceandfinaldisappearanceoftheboundstateswithi ncreasingdensity.\n3 Calculationofthepressure athighdensities inmean fielda pproximation\nThe theory presented in the previous section provides isoth erms of the pressure for the nondegenerate region.\nAfter crossing the line of degeneracy nΛ3\ne≃1the theory is no longer valid since degeneracy effects and th e\ndissymmetry of the massed are relevant. In this regionthe fo rmulaegiven aboveare just an extrapolationwhich\nis exact only in the (dominant) ideal Fermi contributions. F or very dense, strongly degenerate plasmas, bound\nstates do not exist, the plasma behaves like a degenerate non ideal Coulomb gas. Let us assume, we start from\na high density and decrease the density along an isotherm. Th e question is then, at which density bound states\nwill appear. In other wordswe have to ask the question at whic h densities the spectrumwill have discrete levels\ncorresponding to bound states of two particles, three parti cles, four particles etc. Turning the question around\nwe may ask, what is the density dependence of discrete states . This way, the most important effect to be taken\ninto accountin the degenerateregion nΛ3\ne>1is the changeof the energylevels by effectivedensity-depe ndent\nenergylevelsand the questionof mergingof discrete levels in the continuum. This stronglyaffectsthe equation\nofstate ofhydrogen[13,34].\nIn the region of strongly degenerate electrons, the thermod ynamics is in good approximation described by the\nCopyrightlinewillbeprovidedbythe publishercpp header willbe provided by the publisher 7\nHartree-Fock approximation, the protons form a kind of a Wig ner lattice [13]. Based on eq. (3) the grand-\ncanonicalpressuremaybe approximatedby[11, 13,14]\np(β,ze,zi) =pid+pHF\ne(ze)+pWi\ni(zi)+zeziΛ3\ne˜σ(T)+... (20)\n˜σ(T) =/summationdisplay/bracketleftBig\nexp(−β˜En)−1+β˜En/bracketrightBig\n(21)\nwith\nne=ze+ze∂βpHF\ne(ze)\n∂ze+zeziΛ3\ne˜σ(T), (22)\nni=zi+zi∂βpWi\ni(zi)\n∂zi+zeziΛ3\ne˜σ(T). (23)\nWe proceedasintheprevioussection. Neglectingfirst thebo undstatecontributionwegetforthefugacityofthe\nelectronsinHartree-Fockapproximation[11, 13]\nze=nexp[−βe2\nΛeI−1/2(α)];α=βµid\ne. (24)\nHere theIk(x)are the well-known Fermi integrals. We use in the following t he high density limit including\nthe first temperature correctionscalculated with the Somme rfeldmethod [11, 12, 13, 35] ( t=kBT/Ry). Since\ntheT2correction in the Sommerfeld expansion changes the sign wit h increasing temperature we used in the\ncalculationsamoreconvenientPad´ eexpressionwiththe T2−terminthedenominator\npHF\ne[Ry]≃ −0.3059n\nrs+6.030t2r5s;zHF\ne≃nexp[−1.222\ntrs+0.1216t3r5s], r s=/bracketleftbigg3\n4πn/bracketrightbigg1/3\n.(25)\nThe ionic contribution at high density is more difficult [13] . However in the limit of high density, the Wigner\nlimit, wemaygivetheresultinananalyticallyquitesimila r formwithlowest order T-corrections:\nzi≃nexp[−2.3856(1+0.05rs)\ntrs];pWi\ni[Ry]≃0.5964n(1+0.01rs)\nrs. (26)\nIndifferencetothesituationatsmalldensities,thefugac itiesareathighdensities(small rs)rathersmallincom-\nparison to the densities. For this reason the contributiono f the boundstates to the pressure is just a perturbation\nand we do not need to sum up an infinite Saha-type series, as in t he low density case. Putting now together\nall terms we arrive in the canonical ensemble at the followin g simple relation for the high density pressure of\nhydrogen(pF\ne-Fermipressureoftheelectrons)\np(β,n)≃pF\ne+pHF\ne+pWi\ni+zHF\nezWi\niΛ2\ne/summationdisplay/bracketleftBig\nexp(−β˜En)−1+β˜En/bracketrightBig\n. (27)\nThis way we found now for the limit of high density a closed exp ression for the pressure of hydrogen in the\nphysical picture. It remains to find the effective two-parti cle energy levels which are now density-dependent.\nThetwo-particleboundstateenergiesinadensesystem ˜Enmaybederivedfromaneffectivewaveequation,the\nso-calledBethe-Salpeterequation.\nIn this paper,the influenceof the medium is taken only in mean -fieldapproximation,i.e., by Pauli blockingand\nFockselfenergy. Theeffectivewaveequationreadsinthisc ase(inthefollowingthetildeisomitted)[13,34,36]\np2\n2meψn(p)+/summationdisplay\nqV(q)ψn(p+q)−Enψn(p) =/summationdisplay\nqV(q)[ψn(p+q)fe(p)−ψn(p)fe(p+q)].(28)\nHere the l.h.s. represents the standard Schr¨ odinger equat ion in momentum representation. The terms on the\nr.h.s. containing the electron distribution function fe(p)model effects connected with the Fermi character of\nthe electrons. The first of these two contributions we denote as Pauli-blocking and the second one as Fock\ncontribution. The Fock term and the Pauli blocking contribu tions have opposite signs and compensate each\nother partially. The more complete Bethe-Salpeter equatio n containsalso dynamic self-energyand dynamically\nCopyrightlinewillbeprovidedbythe publisher8 Ebeling, Kraeft,and R¨ opke: Bound States inCoulomb Syste ms- OldProblems andNew Solutions\nscreened interactioncontributions. These correctionswe re studied for hydrogenand electron-holesystems, see,\ne.g.,[13,34, 36,37, 38,39, 40,41].\nThe corresponding shifts will be neglected here since we con sider only the high density limit. In perturbation\ntheory, the sum of Pauli and Fock shifts [15, 39, 42] can be cal culated with the solutions φnof the unperturbed\nSchr¨ odingerequation.\n∆EPF\nn=−/summationdisplay\np,qφn(p)V(q)[fe(p)φn(p+q)−fe(p+q)φn(p)]. (29)\nIn the limit of strong degeneracy where the Fermi function ca n be replaced by the Heaviside step function, the\nintegrals can be performedanalytically. The result for the shift of the lowest three energy eigenvaluesis shown\nin Fig. 2. Inthelow-densitylimit wehaveforthegroundstat en={10}\nlim\nn→0∆EPF\n10=n\n2/summationdisplay\nq4πe2\nq2φ10(q)[φ10(0)−φ10(q)] = 32πn′−20πn′= 12πn′. (30)\nInanalogywe findanalyticalexpressionsforthe excited s-states. Inthe low-densitylimit we have\nlim\nn→0∆EPF\n20= 48πn′,lim\nn→0∆EPF\n30= 108πn′. (31)\nAs we see, due to compensationeffects, the sum of Pauli and Fo ck shifts is, in linear approximation,less than a\nhalf of the Pauli shift in the groundstate and is also reduced in the excited states. The evaluationof eq. (29) for\narbitrary densities leads to deviations from the linear den sity dependence as shown in Fig. 2. In contrast to the\nboundstates, there is no Pauli shift for the scattering stat es. The loweringof the continuumedge is givenby the\nFockenergyshift atzeromomentum,\n∆EFock(p= 0) =−/summationdisplay\nqV(q)fe(q) =−4pF\nπ=−4/parenleftbigg3n′\nπ/parenrightbigg1/3\n. (32)\nThe Fock shift at the Fermi momentum ∆EFock(pF)results as ∆EFock(p= 0)/2. This is in agreement with\ntheestimateoftheshiftofthecontinuumbythechemicalpot entialsgivenabovesinceinthestronglydegenerate\ncase the chemical potential of the electronsis given by the e nergyat the Fermi momentum. The lowering of the\ncontinuum edge is also shown in Fig. 2. The bound states vanis h when they meet the continuum edge. We see\nthat forn′<0.0018, only two levels survive and for n′<0.00018still 3 levels are left. Only at very small\ndensitiesmorethan3levelsareleft. Werepeatherethatthe solutionofanin-mediumSchr¨ odingermorecomplex\nthan Eq.(28) was dealtwith in [13, 34, 36, 37, 38, 39, 40, 41]. In[39] it was shownthat perturbationprocedures\nare justified evenforin-mediumSchr¨ odingerequations. So lutionsbeyondperturbationtheorycan also be found\nwithina variationalapproach[16,42].\n0 0.005 0.01 0.015 0.02\nelectron density n aB3-1-0.8-0.6-0.4-0.20bound state energies [Ryd]continuum edge\nE30(n)\nE20(n)\nE10(n)\n0 0.0001 0.0002\nelectron density n aB3-0.25-0.2-0.15-0.1-0.050Bound state energy [Ryd]continuum edge\nE30(n)\nE20(n)\nFig. 2The shifts of the ground state and two excited states in Pauli -Fock approximation in relation to the lowering of the\ncontinuum level.\nAsanapplicationwecalculatedthepressurerelatedtothec lassicalidealpressureforthecaseofhighdensities\natatemperatureof 10000−20000K,correspondingtotheHartree-FockandWignerapproximat ions,seeFig. 3.\nCopyrightlinewillbeprovidedbythe publishercpp header willbe provided by the publisher 9\nTheenergyshiftsaretakenintoaccount. Forcomparisonweh avegivenacurvefor10000KgivenbyVorberger\net al. [43] and with a result for T= 20000 K obtained by means of a Pad´ e approximation in combination\nwith a mass action law [19]. The agreement of our new, rather s imple approachwith the previousones is rather\nsatisfactory.\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 23.4  23.6  23.8  24  24.2  24.4  24.6  24.8  25p / pid \nlg(nc), T = 10000; 15000; 20000 K  0 1 2 3 4 5 6 7 8\n 23.2  23.4  23.6  23.8  24  24.2  24.4  24.6  24.8  25p / 2 n k T \nlg(nc), T =  20000 K \nFig. 3Calculations of the pressure corresponding to the Hartree- Fock approximation taking into account the energy shifts\nforthecaseofhighdensities. Theleftpanelshows p/pidforthetemperatures T= 10000,15000,20000K.Forlogn<23.5\nbegins the region of developed bound states which is not well described by this method. The magenta curve represents\nnumerical and analytical results by Vorberger et al. [43] fo rT= 10000 K. In the right panel we show the pressure related\nto the classical ideal pressure at the same temperatures and for comparison we give for T= 20000 K a curve (in magenta)\nobtained earlier using aPad´ e approximation for the pressu re incombination witha nonideal Saha equation [19].\n4 DiscussionandConclusion\nWediscussherethedivergenceofthepartitionfunctionoft heBohratomandtheBPLmethodofrenormalization,\nand present several new results within the physical picture . At small densities we discussed closed expressions\nfor the ring sums. At high densities we concentrated on the Ha rtree-Fock approximation and took into account\nthe shiftsin Hartree-Fockapproximation. We assume that th ese shiftsprovidethe most importanteffectsforthe\ndestructionofboundstatesin the high-densityregion. Inp articularwe contributehereto the theoryofhydrogen\nat high pressures in the region where a Mott transition to ful l ionization has been predicted and where recent\nexperimentshaveshownatransitionfrominsulatingbehavi ortometal-likeconductivity.\nIn order to understand this transition several effects have to be taken into account. We concentrated here on\nso-called Pauli blocking effects expressing the rule that s tates occupied by atomic electrons cannot be occupied\nbyfreeelectronswiththesamespinstate. Thisleadsathigh electrondensitiestothedestructionofatomicstates\nwhich need a relatively high amount of phase space. We calcul ated the energy shifts due to Pauli effects and\ndiscusstheMott effectssolvingeffectiveSchr¨ odingereq uationsforstronglycorrelatedsystems.\nStill a word on the plasma phase transition. In the temperatu re region 10000−20000K which we studied\nhere, no first order transitions were observed. The physical picture which we used here cannot be extended\nwithout problemsto lower temperatures, where H2molecules dominate. In order to check for phase transitions\nwe introduced the energy levels calculated here into the che mical picture developed earlier [16] which includes\nH2molecules. A resulting isotherm for T= 6000K shows a small wiggle. According to this estimate the\ntransitionoccursintherangebelow6000K,thecriticaltem peratureTcisslightlyabove. Summarizing,weshow\nhere that in the non-degenerate the most essential effects a re the screening and the BPL reduction, resulting in\nabsorbingtheordersin e2, e4andthecompensationeffectsofthestatesbelowandaboveth eserieslimit. Another\nCopyrightlinewillbeprovidedbythe publisher10 Ebeling, Kraeft,and R¨ opke: Bound States inCoulomb Syst ems- OldProblems andNew Solutions\nimportant effect is the summation of an infinite series in the parameterγ= Λ3σBPL(T)leading to a Saha-type\nbehavior. 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Theor. 42, 214033 (2009). .\n[43] J. Vorberger, D.O.Gericke, W.D.Kraeft, Equation of statefor high density hydrogen ; http://arxiv.org/abs/1108.4826.\nCopyrightlinewillbeprovidedbythe publisher"    },    {        "title": "1111.5384v1.Observation_of_reentrant_quantum_Hall_states_in_the_lowest_Landau_level.pdf",        "content": "Observation of reentrant quantum Hall states in the lowest Landau level\nYang Liu, C.G. Pappas, M. Shayegan, L.N. Pfei\u000ber, K.W. West, and K.W. Baldwin\nDepartment of Electrical Engineering, Princeton University, Princeton, New Jersey 08544\n(Dated: March 5, 2022)\nMeasurements in very low disorder two-dimensional electrons con\fned to relatively wide GaAs\nquantum well samples with tunable density reveal reentrant \u0017= 1 integer quantum Hall states in\nthe lowest Landau level near \flling factors \u0017= 4=5 and 6/5. These states are not seen at low\ndensities and become more prominent with increasing density and in wider wells. Our data suggest\na close competition between di\u000berent types of Wigner crystal states near these \fllings. We also\nobserve an intriguing disappearance and reemergence of the \u0017= 4=5 fractional quantum Hall e\u000bect\nwith increasing density.\nAt low temperatures and subjected to a strong per-\npendicular magnetic ( B), a low-disorder two-dimensional\nelectron system (2DES) displays a myriad of novel col-\nlective states [1{3] arising from the dominance of the\nCoulomb interaction energy over the kinetic energy and\nthe disorder potential. At high B, when the electrons\noccupy the lowest ( N= 0) Landau level (LL) [4], the\n2DES exhibits fractional quantum Hall e\u000bect (FQHE)\nat several series of odd-denominator LL fractional \fll-\nings\u0017=nh=eB (nis the 2DES density) as it condenses\ninto incompressible liquid states [1{5]. At even higher\nB, the last series of FQHE liquid states is terminated by\nan insulating phase which is reentrant around a FQHE\nat\u0017= 1=5 and extends to lower \u0017 < 1=5 [6{8]. This\ninsulating phase is generally believed to be an electron\nWigner crystal (WC), pinned by the small but ubiquitous\ndisorder potential [6{8].\nIn the higher ( N > 0) LLs, at the lowest temperatures\nand in the cleanest samples, other collective states com-\npete with the FQHE states. These include anisotropic\n\"stripe\" phases at half-integer \fllings for N\u00152 and\nseveral insulating phases at non-integer \fllings which ex-\nhibit reentrant integer quantum Hall e\u000bect (RIQHE) be-\nhavior [9{12]. The latter phases are most prominent in\nthe second ( N= 1) LL, and their hallmark feature is a\nHall resistance ( Rxy) which is quantized at the value of a\nnearby integer quantum Hall plateau and is often accom-\npanied by a vanishing longitudinal resistance ( Rxx) at\nthe lowest achievable temperatures. The origin of these\nRIQHE phases is not entirely clear, although they are\nwidely considered to signal the condensation of electrons\nin a partially-\flled LL into \"bubble\" phases where sev-\neral electrons are localized at one lattice site [9{13]. Such\nRIQHEs have not been seen until now in the N= 0 LL,\nconsistent with the expected the instability of the bubble\nphases in 2DESs in the lowest LL [14].\nWe report here the observation of RIQHE in the low-\nest LL in very clean, high-density 2DESs con\fned to\nrelatively wide GaAs quantum wells (QWs). Figure 1\nhighlights our main \fnding. It shows data taken in a\n42-nm-wide GaAs QW at two di\u000berent densities. At\nthe lower density, the RxxandRxytraces show what\nis normally seen in very clean 2DESs: strong QHE at\u0017= 1 and 2/3 and, between these \fllings, several FQHE\nstates at\u0017= 4/5, 7/9, 8/11, and 5/7. At the higher n,\nhowever, a RIQHE (marked by down arrows) is observed\nnear 1=\u0017= 1:20, as evidenced by an Rxxminimum and\nandRxyquantized at h=e2. Also evident is a developing\nRIQHE state between \u0017= 4=5 and 7/9, signaled by a dip\ninRxy(up arrow in Fig. 1(b)). As we detail below, the\nRIQHE phases near \u0017= 4=5, as well as similar phases\nnear\u0017= 6=5 on the low-\feld \rank of \u0017= 1, show a\nspectacular evolution with density. An examination of\nthe conditions under which we observe these reentrant\nphases suggests that they are likely WC states, similar\nto those observed near \u0017= 1=5 in high-quality 2DESs,\nand it is the larger electron layer thickness in our samples\nthat stabilizes them here in the lowest LL near \u0017= 1.\nOur samples were grown by molecular beam epitaxy,\nand each consist of a wide GaAs quantum well (QW)\nbounded on each side by undoped Al 0:24Ga0:76As spacer\nlayers and Si \u000e-doped layers. We report here data for\n0.2\n0.0Rxx (kΩ) Rxy (h/e2)\n1.01.5\nν = 4/5\n1.2 1.41.78 x 1011 cm-2\n5/7\n1.2 1.4\n1/ν5/72.46 x 1011 cm-2\n7/98/11 8/11\n7/9(a) (b)\nν = 4/5\nFIG. 1.RxxandRxyvs. 1=\u0017traces atT= 30 mK for a\n42-nm-wide GaAs QW at two densities: (a) n= 1:78, and\n(b) 2:46\u00021011cm\u00002. In (b) the RIQHE phases observed on\ntwo sides of \u0017= 4=5 are marked by arrows. Note also the two\nsharpRxxmaxima surrounding the \u0017= 4=5Rxxminimum.arXiv:1111.5384v1  [cond-mat.mes-hall]  23 Nov 20112\n0.8\n0.6\n0.4\n0.0Rxx (kΩ) Rxy (h/e2)\n1.01.5\n1.02.02.5\n÷ 2\n1.78\n1.91\n2.05\n2.17\n2.32\n2.61\n2.87\n3.14\n3.28\n3.412.46W = 42 nm\n T  30 mK3.0\nν = 4/5\n6/5\n0.8 0.9 1.2 1.4\n1/νn\n(1011 cm-2)5/7\n9/7\n1.3 1.50.8 0.9 1.2 1.4 1.3 1.5\n1.78\n1.91\n2.05\n2.17\n2.32\n2.61\n2.87\n3.14\n3.28\n3.412.46\nFIG. 2. (color online) Waterfall plots of RxxandRxyvs.\n1=\u0017for the 42-nm-wide QW as nis changed from 1 :78 to\n3:41\u00021011cm\u00002. Traces are shifted vertically for clarity. The\ndown arrows mark the development of RIQHE phases near\n1=\u0017= 1:20 asnis raised from 2 :05 to 2:61\u00021011cm\u00002, and\nnear 1=\u0017= 0:86 asnis further increased. The up arrows in\nthe top panel mark the development of similar, albeit weaker,\nRIQHE phases near 1 =\u0017= 1:28 and 0.83. The yellow band\nmarks the range 0 :85\u0014\u0017\u00141:15; see text.three samples, with QW widths W= 31, 42 and 44 nm,\nand as-grown densities of n'3.3, 2.5 and 3.8 \u00021011\ncm\u00002, respectively. The low-temperature mobilities of\nthese samples are \u0016'670, 910 and 600 m2/Vs, respec-\ntively. The samples have a van der Pauw geometry and\neach is \ftted with an evaporated Ti/Au front-gate and\nan In back-gate. We carefully control nand the charge\ndistribution symmetry in the QW by applying voltage\nbiases to these gates [15, 16]. All the data reported here\nwere taken by adjusting the front- and back-gate biases\nso that the total charge distribution is symmetric. The\nmeasurements were carried out in superconducting and\nresistive magnets with maximum \felds of 18 T and 35 T\nrespectively. We used low-frequency ( '10 Hz) lock-in\ntechniques to measure the transport coe\u000ecients.\nFigure 2 shows a series of RxxandRxytraces in the\nrange 2=3< \u0017 < 4=3 for the 42-nm-wide QW sample,\ntaken asnis changed from 1.78 to 3 :41\u00021011cm\u00002.\nThese traces reveal a remarkable evolution for the dif-\nferent reentrant phases of this 2DES. At the lowest n,\nFQHE states are seen at \u0017= 9/7, 6/5, 4/5, and 5/7.\nWhennis increased to 2 :05\u00021011cm\u00002, the\u0017= 4=5\nFQHE disappears and a minimum in Rxydevelops near\n\u0017= 4=5. Asnis further increased, the Rxyminimum\nbecomes deeper and moves towards 1 =\u0017= 1:20, and an\nRxxminimum starts to develop at the same \flling (see\ndown arrows near 1 =\u0017= 1:20 in Fig. 2). Meanwhile,\nthe\u0017= 4=5 FQHE reappears to the right of these min-\nima. As we keep increasing n, theRxyminimum deep-\nens and becomes quantized at h=e2forn\u00152:46\u00021011\ncm\u00002, and theRxxminimum gets deeper and vanishes\nforn>2:87\u00021011cm\u00002. At the highest nthese minima\nmerge into the Rxyplateau and the Rxxminimum near\n\u0017= 1.\nFigure 2 traces also show that, with increasing n, an-\notherRxyminimum starts to develop on the right side\nof\u0017= 4=5, as marked by the up arrows at 1 =\u0017= 1:28.\nThis minimum, too, becomes deeper at higher nand, as\nwe will show later, approaches h=e2. Also noteworthy are\nthe two sharp Rxxmaxima on the \ranks of \u0017= 4=5 after\nthe\u0017= 4=5 FQHE reemerges at high densities.\nThe evolution observed on the right side of \u0017= 1 is\nqualitatively seen on the left side also, but at higher n.\nThe\u0017= 6=5 FQHE, e.g., disappears at n= 2:87\u00021011\ncm\u00002and a dip in Rxxdevelops near 1 =\u0017= 0:86 (see\ndown arrows in Fig. 2), concomitant with Rxylifting up\nand eventually becoming quantized at h=e2.\nFigure 3 illustrates the T-dependence of RxxandRxy\natn= 2:46\u00021011cm\u00002. At the highest T, theRxx\nandRxytraces near \u0017= 4=5 look \"normal\": There is a\nrelatively strong FQHE at \u0017= 4=5 as signaled by a deep\nminimum in Rxxand anRxyplateau at 5 h=4e2. Away\nfrom\u0017= 4=5,Rxyfollows a nearly linear dependence\nonB. AsTis lowered, however, Rxydevelops a mini-\nmum near 1 =\u0017= 1:20 which eventually turns to a plateau\nquantized at h=e2at the lowest T. Meanwhile, a strong3\n0.00.51.5 Rxx (kΩ) Rxy (h/e2)\n1.0\n1/νW = 42 nm\n n = 2.46 x 1011 cm-2\nν = 4/55/7\n5/7 7/930 mK\n50 mK\n110 mK\n180 mK\n1.4 1.0 1.2 1.1 1.3 1.5\nFIG. 3. (color online) Evolution of RxxandRxywith tem-\nperature for the 42-nm QW at n= 2:46\u00021011cm\u00002.\nminimum develops in Rxxat 1=\u0017= 1:20. On the right\nside of\u0017= 4=5, anotherRxyminimum is developing.\nData taken on the 31- and 44-nm-wide QW samples\nreveal the generality of these phenomena. Figure 4(a)\nshows data for the 31-nm QW. Similar to the data of\nFig. 2, asnis increased, the \u0017= 4=5 FQHE is quickly\ndestroyed and Rxydevelops a deep minimum which ap-\nproaches the \u0017= 1 plateau at h=e2near the highest B\nthat we can achieve in this sample. The resemblance of\nthe traces shown in Fig. 4(a) to the top four Rxyand\nRxxtraces in Fig. 2 is clear. Note, however, that in the\nnarrower QW of Fig. 4(a) we need much higher nto\nreproduce what is seen in the wider QW of Fig. 2.\nIn Fig. 4(b) we show data for the 44-nm QW at a very\nhigh density ( n= 3:83\u00021011cm\u00002). Given the larger\nQW width and higher nof this sample compared to Fig.\n2 sample, we expect the RIQHE near 1 =\u0017= 1:20 to be\nfully developed, and the RIQHE on the high-\feld side of\n\u0017= 4=5 whose emergence is hinted at in Fig. 2 Rxytraces\n(see up arrows near 1 =\u0017= 1:28 in Fig. 2) to become more\npronounced. This is indeed seen in Fig. 4(b): The Rxy\ntrace shows a very deep minimum on the high-\feld side\nof\u0017= 4=5 at 1=\u0017= 1:28, nearly reaching the \u0017= 1\nplateau ath=e2when the smallest sample current (1 nA)\nis used to achieve the lowest electron temperature for this\nsample. Note also the appearance of a small but clearly\nvisibleRxxminimum at 1 =\u0017= 1:28, consistent with the\n1.1 1.43\n01.01.6Rxx (kΩ) Rxy (h/e2)\n1/νW = 31 nm\n3.05\n3.19\n3.33\n3.48ν = 5/7\n4/5\n1.2 1.410 nA\n1 nA\n2\n01.01.5\nW = 44 nmRxy (h/e2) Rxx (kΩ)n\n(1011 cm-2)n = 3.83\n        x 1011 cm-2\n1/ν(a) (b)\nν = 5/7\n4/5 T (mK)\n183160\n3.33\nT   25 mK T   25 mK\n1.2 1.3 1.310 nAFIG. 4. (color online) (a) Waterfall plots of RxxandRxyvs.\n1=\u0017for the 31-nm QW sample, showing how the RIQHE phase\nnear 1=\u0017= 1:20 starts to develop as nis raised from 3 :05 to\n3:48\u00021011cm\u00002. Data are shifted vertically for clarity. The\ninset shows the T-dependence of the RIQHE at n= 3:33\u00021011\ncm\u00002. (b)RxxandRxytraces for the 44-nm QW at n=\n3:83\u00021011cm\u00002displaying a RIQHE phase at 1 =\u0017= 1:28.\ndevelopment of RIQHE at this \flling [17].\nA natural interpretation of the RIQHE phases that we\nobserve near \u0017= 4=5 and 6=5 is that these are pinned\nWC phases, similar to the insulating phases seen around\nthe\u0017= 1=5 FQHE and extending to \u0017\u001c1=5 [6{8]. In\nthe present case, electrons at \u0017= 1\u0006\u0017\u0003can be con-\nsidered as a \flled LL, which is inert, plus excess elec-\ntrons/holes with \flling factor \u0017\u0003, which could conduct.\nAt su\u000eciently small values of \u0017\u0003and low temperatures,\nthese excess electrons/holes crystalize into a solid phase\nwhich is pinned by disorder and does not participate in\ntransport. Thus the magneto-transport coe\u000ecients, Rxx\nandRxy, approach those of the \u0017= 1 IQHE.\nIn high-quality 2DESs, there have been no prior re-\nports of RIQHE phases near \u0017= 1 in the lowest LL\nfrom low-frequency (essentially dc) magneto-transport\nmeasurements[18]. Our results indicate that such RIQHE\nphases set in only at high densities and in relatively wide\nQWs. We suggest that it is the large electron layer thick-\nness (spread of the electron wavefunction perpendicular\nto the 2D plane) in our wide QW samples that induces\nthe WC formation [19]. Supporting this conjecture, the-\noretical work [20] indeed indicates that in wide QWs,\nthanks to the softening of the Coulomb interaction at\nshort distances, a WC phase is favored over the FQHE\nliquid state if the QW width becomes several times larger4\nthan the magnetic length ( lB). In our samples, W=lBis\nlarge and ranges from '4.6 to'6.0 at the densities above\nwhich we start to observe the RIQHE near \u0017= 4=5, qual-\nitatively consistent with the theoretical results.\nIn a relevant study, high-frequency (microwave) reso-\nnances were observed very close to \u0017= 1 (\u0017\u0003<0:15) in\nhigh-quality 2DESs and were interpreted as signatures\nof a pinned WC [21]. We have highlighted this \flling\nrange with a yellow band in Figs. 2 and 3. This is the\nrange where we see a deep Rxxminimum and a very well\nquantizedRxy(ath=e2), signaling a strong \u0017= 1 QHE.\nIn our samples, however, we observe RIQHE phases at\nrelatively large values of \u0017\u0003, extending to \u0017\u0003>1=5. In\nparticular, 1 =\u0017'1:28 where we see the RIQHE on the\nright side of \u0017= 4=5 corresponds to \u0017\u0003'0:22. This is\nclearly outside the \u0017\u0003range where Chen et al: observed\nmicrowave resonances, indicating that the RIQHE phases\nwe are reporting here are distinct from the phase docu-\nmented in Ref. [21]. Also the RIQHE we observe at\n1=\u0017= 1:20 (\u0017\u0003'0:17) is separated from the deep QHE\nregion very near \u0017= 1 by maxima in RxxandRxy(see,\ne.g., the right-hand-side edge of the yellow band in Figs.\n2 and 3). This observation provides additional evidence\nthat two distinct insulating phases exist. If so, then these\nresistance maxima signal additional scattering at the do-\nmain walls separating these phases.\nIt is worth noting that microwave experiments at very\nsmall\u0017have revealed two distinct resonances [22]. One\nresonance (\" A\") was seen in the insulating phases reen-\ntrant around \u0017= 1=5, and another (\" B\") was domi-\nnant at very small \fllings ( \u0017 <0:15). It was suggested\nthat these resonances signal the existence of two di\u000ber-\nent types of correlated WCs, stabilized by the crystal-\nlization of composite Fermions with di\u000berent number of\n\rux quanta attached to them. Such an interpretation\nis corroborated by theoretical calculations [23, 24]. It\nis tempting to associate the RIQHE phases we observe\nreentrant around \u0017= 4=5 (\u0017\u0003= 1=5) with the type \" A\"\nWC and the resonance seen very near \u0017= 1 (\u0017\u0003<0:15)\nin Ref. [21] with the type \" B\" WC.\nA very intriguing aspect of our data is the disappear-\nance followed by a rapid reemergence of the \u0017= 4=5\n(\u0017\u0003= 1=5) FQHE when the RIQHE near \u0017= 4=5 starts\nto be seen [25]. We do not have a clear explanation\nfor this observation. We speculate that it might signal\nthe existence of multiple types of WC phases that have\nground-state energies very close to the ground-state en-\nergy of the FQHE liquid phase. Theoretical calculations\nindeed indicate that such WC phases, which are based on\ncomposite Fermion FQHE liquid wavefunctions, do exist\nand might well describe the insulating phases observed\nnear\u0017= 1=5 [23, 24]. The disappearance and reap-\npearance of the \u0017= 4=5 FQHE we observe may stem\nfrom a close competition between the FQHE and such\nWC states. Consistent with this scenario is the following\nobservation in Fig. 4(a). When the \u0017= 4=5 FQHE dis-appears,Rxyat\u0017= 4=5 immediately starts to dip down\ntowardsh=e2, suggesting a transition to a RIQHE phase.\nThis is in sharp contrast to Rxymaintaining its value of\n5h=4e2on the classical Hall line, if the \u0017= 4=5 FQHE\nstate made a transition to a compressible liquid phase.\nIn conclusion, we observe RIQHE phases near \u0017= 4=5\nand 6/5 in the lowest LL in very clean 2DESs con\fned to\nrelatively wide GaAs QWs. In a given QW, the RIQHE\nis absent at low densities and develops above a certain\ndensity which is higher for narrower QWs. Our obser-\nvations are consistent with the crystallization of excess\nelectrons/holes in the un\flled lowest LL into a WC.\nWe thank L. W. Engel and J. K. Jain for illuminat-\ning discussions. We acknowledge support through the\nMoore Foundation and the NSF (DMR-0904117 and MR-\nSEC DMR-0819860) for sample fabrication and charac-\nterization, and the DOE BES (DE-FG0200-ER45841) for\nmeasurements. This work was performed at the National\nHigh Magnetic Field Laboratory, which is supported by\nNSF Cooperative Agreement No. DMR-0654118, by the\nState of Florida, and by the DOE. We thank L. W. Engel\nfor illuminating discussions, and E. Palm, J. H. Park, T.\nP. Murphy and G. E. Jones for technical assistance.\n[1] J. K. Jain, Composite Fermions (Cambridge University\nPress, Cambridge, UK, 2007).\n[2]Perspectives in Quantum Hall E\u000bects , Edited by S. Das\nSarma and A. Pinczuk (Wiley, New York, 1997).\n[3] See, e.g., M. Shayegan, \"Flatland Electrons in High Mag-\nnetic Fields,\" in High Magnetic Fields: Science and Tech-\nnology , Vol. 3, edited by Fritz Herlach and Noboru Miura\n(World Scienti\fc, Singapore, 2006), pp. 31-60. [cond-\nmat/0505520].\n[4] This regime corresponds to \u0017 <2, accounting for the two\nN= 0 LLs, one for each spin orientation.\n[5] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys.\nRev. Lett., 48, 1559 (1982).\n[6] H. W. Jiang, R. L. Willett, H. L. Stormer, D. C. Tsui,\nL. N. Pfei\u000ber, and K. W. West, Phys. Rev. Lett., 65,\n633 (1990).\n[7] V. J. Goldman, M. Santos, M. Shayegan, and J. E. Cun-\nningham, Phys. Rev. Lett., 65, 2189 (1990).\n[8] See articles by H. A. Fertig and by M. Shayegan in Ref.\n[2].\n[9] M. P. Lilly, K. B. Cooper, J. P. Eisenstein, L. N. Pfei\u000ber,\nand K. W. West, Phys. Rev. Lett., 82, 394 (1999).\n[10] R. R. Du, D. C. Tsui, H. L. Stormer, L. N. Pfei\u000ber, K. W.\nBaldwin, and K. W. West, Solid State Communications,\n109, 389 (1999).\n[11] J. P. Eisenstein, K. B. Cooper, L. N. Pfei\u000ber, and K. W.\nWest, Phys. Rev. Lett., 88, 076801 (2002).\n[12] J. S. Xia, W. Pan, C. L. Vicente, E. D. Adams, N. S.\nSullivan, H. L. Stormer, D. C. Tsui, L. N. Pfei\u000ber, K. W.\nBaldwin, and K. W. West, Phys. Rev. Lett., 93, 176809\n(2004).\n[13] A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii,\nPhys. Rev. Lett., 76, 499 (1996); M. M. Fogler and A. A.5\nKoulakov, Phys. Rev. B, 55, 9326 (1997).\n[14] According to Ref. [13], bubble phases with more than\none electron per lattice site are not stable in the N= 0\nLL. (A bubble phase with one electron per lattice site is\nindistinguishable from a WC phase.).\n[15] Y. W. Suen, H. C. Manoharan, X. Ying, M. B. Santos,\nand M. Shayegan, Phys. Rev. Lett., 72, 3405 (1994).\n[16] Y. Liu, J. Shabani, and M. Shayegan, Phys. Rev. B, 84,\n195303 (2011).\n[17] In all samples we measured, the Rxyquantization is al-\nways more robust than the vanishing of Rxx. Similar\nbehavior is seen for the RIQHE phases observed in the\nN= 1 LL; see e.g. Ref. [11].\n[18] W. Li et al. [Phys. Rev. Lett. 105, 076803 (2010)] have\nreported a RIQHE between \flling factors \u0017= 2=3 and\n3/5 and its particle-hole symmetric state, a reentrant\ninsulating phase between \u0017= 1=3 and 2/5, in samples\nwhere 2D electrons reside in an Al xGa1\u0000xAs alloy. They\nattribute their observation to the strong short-range (al-\nloy) disorder which favors a pinned WC state in their\nsamples.\n[19] In the parameter range we study our samples, often two\nelectric subbands, which are separated in energy by at\nleast\u001840 K, are occupied. Near \u0017= 1, however, we ex-pect that all the electrons occupy the lowest LL of the\nlowest electric subband. We therefore suspect the upper\nsubband is not playing a major role, althought we cannot\nrule out the e\u000bect of subband mixing.\n[20] R. Price, X. Zhu, S. Das Sarma, and P. M. Platzman,\nPhys. Rev. B, 51, 2017 (1995).\n[21] Y. Chen, R. M. Lewis, L. W. Engel, D. C. Tsui, P. D.\nYe, L. N. Pfei\u000ber, and K. W. West, Phys. Rev. Lett.,\n91, 016801 (2003).\n[22] Y. P. Chen, R. M. Lewis, L. W. Engel, D. C. Tsui, P. D.\nYe, Z. H. Wang, L. N. Pfei\u000ber, and K. W. West, Phys.\nRev. Lett., 93, 206805 (2004).\n[23] H. Yi and H. A. Fertig, Phys. Rev. B, 58, 4019 (1998);\nR. Narevich, G. Murthy, and H. A. Fertig, ibid.,64,\n245326 (2001).\n[24] C.-C. Chang, G. S. Jeon, and J. K. Jain, Phys. Rev.\nLett., 94, 016809 (2005); C.-C. Chang, C. T oke, G. S.\nJeon, and J. K. Jain, Phys. Rev. B, 73, 155323 (2006).\n[25] We observe a qualitatively similar development near \u0017=\n6=5: The traces at n>2:61\u00021011cm\u00002in Fig. 2 show\nthe disappearance of the 6/5 FQHE as the RIQHE sets\nin, although we cannot reach high enough densities to see\na clear reemergence of the 6/5 FQHE state."    },    {        "title": "1111.6545v2.Local_density_of_states_of_the_one_dimensional_spinless_fermion_model.pdf",        "content": "arXiv:1111.6545v2  [cond-mat.str-el]  12 Jan 2012Local density of states of the one-dimensional\nspinless fermion model\nE Jeckelmann\nInstitut f¨ ur Theoretische Physik, Leibniz Universit¨ at Hannover , Appelstraße 2,\nD-30167 Hannover, Germany\nE-mail:eric.jeckelmann @itp.uni-hannover.de\nAbstract. We investigate the local density of states of the one-dimensional h alf-\nfilled spinless fermion model with nearest-neighbor hopping t >0 and interaction V\nin its Luttinger liquid phase −2t < V≤2t. The bulk density of states and the local\ndensity of states in open chains are calculated over the full band wid th∼4twith an\nenergy resolution ≤0.08tusing the dynamical density-matrix renormalization group\n(DDMRG) method. We also perform DDMRG simulations with a resolution of 0.01t\naround the Fermi energy to reveal the power-law behaviour D(ǫ)∼ |ǫ−ǫF|αpredicted\nby the Luttinger liquid theory for bulk and boundary density of stat es. The exponents\nαare determined using a finite-size scaling analysis of DDMRG data for la ttices with\nup to 3200 sites. The results agree with the exact exponents given by the Luttinger\nliquid theory combined with the Bethe Ansatz solution. The crossove r from boundary\nto bulk density of states is analyzed. We have found that boundary effects can be seen\nin the local density of states at all energies even far away from the chain edges.\nPACS numbers: 71.10.Fd,71.10.Pm,71.27.+a,73.21.Hb\nSubmitted to: J. Phys.: Condens. MatterLocal density of states of the one-dimensional spinless fer mion model 2\n1. Introduction\nOne-dimensional conductors have fascinated physicists for more than 50 years [1, 2]\nbecause they feature unusual properties which set them apart f rom ordinary metals.\nOur understanding of ordinary metals is based on the Fermi liquid par adigm [3]. In one\ndimension, however, thistheoryfails. Instead, thelow-energyph ysics ofone-dimensional\nconductors is described by the Luttinger liquid paradigm [2, 4, 5]. Th e predictions of\nthe Luttinger liquid theory differ fundamentally from those of the Fe rmi liquid theory.\nFor instance, Fermi liquids have a finite density of states D(ǫ) at the Fermi energy ǫF.\nIn contrast the bulk density of states of Luttinger liquids vanishes as a power law at the\nFermi energy\nD(ǫ)∼ |ǫ−ǫF|α(1)\nwhere the exponent α >0 depends on the system. This feature is regarded as one\nhallmark of a Luttinger liquid.\nThe density of states D(ǫ) can be measured experimentally in photoemission\nspectroscopy and in scanning tunnelling spectroscopy (STS). The STS method yields a\nspatially-resolved local density of states (LDOS) with a resolution o f a few˚A. The\nexperimental observation of vanishing densities of states has bee n reported in the\nphotoemission or STS spectrum of various quasi-one-dimensional c onductors such as\nBechgaard salts [6], the organic charge transfer salt TTF-TCNQ [7 ], large samples\nof single-walled carbon nanotubes [8], and the purple bronze Li 0.9Mo6O17[9, 10].\nConsequently, these materials are believed to be realizations of Lut tinger liquids\nalthough this interpretation remains often controversial. Only ver y recently, the power-\nlaw behaviour (1) has been observed unambiguously in the STS and ph otoemission\nspectra of gold wires deposited on semiconducting Ge(001) surfac es [11].\nFrom a field-theoretical point of view, the low-energy properties o f Luttinger\nliquids are very well understood thanks to powerful methods such as bosonization\nand renormalization group [2, 4, 5]. However, field theory only descr ibes the low-\nenergy scaling |ǫ−ǫF| →0 of various physical quantities such as the density of states.\nExperimentally, this asymptotic behaviour could only be observed in a more or less\nbroad window of excitation energies. Indeed, as real materials are three-dimensional,\nthere is always a dimensional crossover [1, 2] at low excitation energ y below which\none-dimensional physics can no longer be observed. Moreover, as field-theoretical\ninvestigations are based on the linear dispersion of excitations close to the Fermi energy,\nthere is always a high-energy limit above which the density of states s hould deviate\nfrom the power-law (1) because of the band curvature in real mat erials [12]. Therefore,\nthe theory must be extended to finite-energy scales beyond the a symptotic behaviour\ncovered by the Luttinger liquid theory to facilitate the interpretat ion of experiments in\none-dimensional conductors.\nWe know that the Luttinger liquid paradigm describes the low-energy properties\nof various one-dimensional quantum lattice models for interacting e lectrons in their\nmetallic phases, such as the Hubbard model [13] away from half filling a nd theLocal density of states of the one-dimensional spinless fer mion model 3\nspinless fermion model [2, 5]. Lattice models enable us to study the infl uence of\nfinite energy scales such as the curvature and finite width of excita tion bands in one-\ndimensional conductors. Some of these models are exactly solvable by the Bethe Ansatz\nmethod. However, it is very difficult to obtain dynamical correlation f unctions related\nto spectroscopic experiments from a Bethe Ansatz solution. (The spectral functions\nof the spinless fermion models have been investigated only very rece ntly using the\nBethe Ansatz [14] but the issue of the density of states has not be en discussed in that\nwork.) Therefore, the spectral properties of these models at fin ite excitation energy\nhave mostly been determined using numerical methods for the quan tum-many body\nproblem [15] such as exact diagonalizations, quantum Monte Carlo sim ulations and the\ndensity-matrix renormalization group (DMRG) method. Several po wer-law divergences\n(∼ |ǫ|αwithα <0) predicted by field theory have been confirmed with these numeric al\nmethods. Yet the power-law singularity (1) with an exponent α >0 has not been\nobserved univocally at finite-energy scale in quantum lattice models s o far.\nThe DMRG method is one of the most powerful numerical method for computing\nthe properties of one-dimensional quantum lattice models [15, 16]. The density\nof states of some lattice models have been investigated using the or iginal DMRG\nmethod [17, 18, 19]. In these studies DMRG was used to compute the spectral weight of\nthe lowest few eigenstates in order to verify the prediction of the b osonization approach\nin the asymptotic low-energy limit. The dynamical DMRG (DDMRG) is an e xtension\nof DMRG which makes possible the calculation of dynamical correlation functions over\ntheir full band width [20, 21]. It yields spectra which are broadened b y a Lorentzian of\nwidthηwhich sets the actual energy resolution. Over the last decade DDM RG has been\nused successfully in many studies of spectral properties in quantu m lattice models with\nenergy resolution down to a few hundredths of the bare band width [22]. Nevertheless,\nthis accuracy has not allowed for a direct observation of the power -law behaviour at\nfinite excitation energy until now. For instance, DDMRG has been us ed to calculate the\ncomplete single-particle spectral functions of the Hubbard model away from half filling\nwith a resolution of ��= 0.1t[22]. The dispersion of holon and spinon branches in the\nDDMRGspectral functionsagreeperfectlywiththeexact BetheA nsatzdispersions. Yet\nat the Fermi energy, where the power law (1) should be seen, the m omentum-integrated\nspectral weight barely shows a shallow dip.\nIn this paper we investigate the density of states of the half-filled s pinless fermion\nmodel in its Luttinger liquid phase. In this model we can study significa ntly larger\nsystems and thus reach a much better resolution than in the Hubba rd model. Moreover,\ninthehalf-filledspinlessmodelanyexponent α≥0canbeachievedwhileintheHubbard\nmodel only 0 ≤α≤1/8 is possible.\nBoth the bulk density of states and the local density of states clos e to a chain\nedge are calculated numerically using the DDMRG method. They are de termined over\nthe full band width ∼4twith a resolution of η= 0.04tor 0.08t. Using lattices with\nup to 1600 sites we have also performed DDMRG simulations with a reso lution of\nη= 0.01taround the Fermi energy to reveal the power-law behaviour (1) p redicted byLocal density of states of the one-dimensional spinless fer mion model 4\nthe Luttinger liquid theory for bulk and boundary density of states . The exponents α\nare determined using a finite-size scaling analysis of DDMRG data for la ttices with up\nto 3200 sites and compared to the predictions of the Luttinger liquid theory combined\nwith the exact Bethe Ansatz solution. Finally, we discuss the crosso ver from boundary\nto bulk density of states as one moves away from the chain edges.\nOur paper is organized as follows: In the next section we introduce t he model and\nmethod used in this work. In the third section we discuss our results for the bulk density\nof states while the LDOS close to a chain edge is analyzed in the fourth section. Finally,\nour findings are summarized in the last section.\n2. Models and method\nThe one-dimensional spinless fermion model is one of the simplest rea lizations of a one-\ncomponent Luttinger liquid in a lattice model. It can be interpreted as a system of\nspin-polarized electrons. The model is defined by the Hamiltonian\nH=−tN−1/summationdisplay\nj=1/parenleftBig\nc†\njcj+1+c†\nj+1cj/parenrightBig\n+VN−1/summationdisplay\nj=1/parenleftbigg\nnj−1\n2/parenrightbigg/parenleftbigg\nnj+1−1\n2/parenrightbigg\n(2)\nwherec†\njandcjare the creation and annihilation operators for a spinless fermion on\nsitej, and the density operator is nj=c†\njcj. The model parameters are the hopping\namplitude t >0 between nearest-neighbor sites and the Coulomb interaction Vbetween\nparticles on nearest-neighbor sites. The half-filled system corres ponds to N/2 fermions\non theN-site lattice.\nThe one-dimensional spinless fermion model isexactly solvableby the Bethe Ansatz\nmethod [2, 5]. For −2t < V≤2tits excitation spectrum is gapless and its low-energy\nproperties are described by the Luttinger liquid theory. At half filling the dispersion of\nelementary excitations [23] is given by\nǫ(k) = 2t∗|sin(ka)| (3)\nwith the renormalized “hopping term”\nt∗=πt\n2/radicalBig\n1−/parenleftbigV\n2t/parenrightbig2\narccos/parenleftbigV\n2t/parenrightbig. (4)\nThe universal properties of a one-component Luttinger liquid are d etermined by two\nparameters: The velocity of elementary excitations (renormalized Fermi velocity) vand\na dimensionless Luttinger parameter K. From the Bethe Ansatz solution we know the\nrelation between these Luttinger liquid parameters and the lattice m odel parameters at\nhalf filling,\nv=2at∗\n/planckover2pi1(5)\nwitht∗given by (4) and\nK=π\n21\nπ−arccos/parenleftbigV\n2t/parenrightbig (6)Local density of states of the one-dimensional spinless fer mion model 5\nwhereais thelattice constant. In thenon-interacting chain V= 0 this yields K= 1 and\nthe usual Fermi velocity v=vF= 2ta//planckover2pi1of the one-dimensional tight-binding model.\nThe local density of states (LDOS) is defined by\nD(j,ǫ) =\n\n/summationdisplay\nn/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\nn/vextendsingle/vextendsingle/vextendsinglec†\nj/vextendsingle/vextendsingle/vextendsingle0/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2\nδ(ǫ−En+E0) for ǫ > ǫF\n/summationdisplay\nn/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\nn/vextendsingle/vextendsingle/vextendsinglecj/vextendsingle/vextendsingle/vextendsingle0/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2\nδ(ǫ+En−E0) for ǫ < ǫF(7)\nwhere|n/angbracketrightdenotes the eigenstates of the Hamiltonian HandEntheir eigenenergies in\nthe Fock space. The ground state for the chosen number of part icles correspond to\nn= 0. The total spectral weight is\n/integraldisplay+∞\n−∞dǫ D(j,ǫ) = 1. (8)\nAs theHamiltonian(2) isinvariant under theparticle-hole transform ationc†\nj↔cj(−1)j,\nits density of states is symmetric at half filling: D(j,ǫ) =D(j,−ǫ) withǫF= 0.\nWe use the DDMRGmethod [20, 21] to compute the density of states (7). DDMRG\nsimulations yield the convolution of a Lorentzian of width η >0 with the local density\nof states on a N-site lattice\nDDMRG(j,ǫ) =1\nπ/integraldisplay+∞\n−∞dǫ′η\n(ǫ−ǫ′)2+η2D(j,ǫ′). (9)\nAs we are interested in the density of states in the thermodynamic lim it, we should,\nin principle, extrapolate our numerical data to an infinite lattice size N→ ∞and\nthen to a vanishing broadening η→0. However, the spectrum of a N-site chain\nis indistinguishable from the same spectrum in the thermodynamic limit if they are\ncompared on an energy scale (resolution) ∆ ǫ > W/N . The parameter Whas to be\ndetermined empirically but it is typically of the order of the effective ba nd width of\nthe spectrum considered (here W∼4t∗). Thus if we choose a broadening η > W/N ,\nDDMRG spectra can be regarded as the spectra of infinite systems with the same\nbroadening η, which sets the actual resolution. In this work we have systematic ally\nused\nη=16t\nN>4t∗\nN. (10)\nThe DMRG algorithm truncates the Hilbert space in an optimal way to r epresent\nsome chosen target states such as the ground state. Thus DMRG data are affected by\ntruncation errors which depend on the number of density-matrix e igenstates kept in the\ncalculation [15, 16]. In the DDMRG method the truncated Hilbert spac e is optimized\nto represent the ground state and eigenstates with a given excita tion energy ǫor lying\nin a given excitation energy range [ ǫ1,ǫ2] withǫ2−ǫ1/lessorsimilarη. The calculations are carried\nout independently for each excitation energy. We have kept up to 4 00 density-matrix\neigenstates in our simulations. Numerical errors are always very sm all in absolute values\nor when they are compared to the total spectral weight (8). Thu s the DDMRG results\npresented here can be regarded as numerically exact. This is illustra ted in the followingLocal density of states of the one-dimensional spinless fer mion model 6\n-6-5-4 -3 -2 -1 0 1 2 3 4 56\nε/t00.10.20.30.40.50.6D(ε)t\nFigure 1. Bulk density of states for V=−1.9t(- - - -),V= 0 (—·—), and V= 2t\n(——) calculated in the middle of a 400-site chain ( η= 0.04t).\nsections by comparison with exact spectra for non-interacting fe rmions. However, as the\nspectral weight approaches zero at the Fermi energy, relative e rrors become unavoidably\nlarger and can be seen as a scattering of DDMRG data close to the Fe rmi energy. In\nour numerical calculations we use the energy scale t= 1, the time scale /planckover2pi1/t= 1 and the\nlength scale a= 1.\n3. Bulk density of states\nIn principle, the bulk density of states D(ǫ) should be calculated as the average of the\nLDOS (7) over the full lattice in the thermodynamic limit. Here we ident ify the bulk\ndensity of states with the LDOS in the middle of a N-site chain, D(j=N/2,ǫ). While\nthis seems intuitively correct for insulators, in a Luttinger liquid boun dary effects could\na priori be felt as far as the chain center. First, we have verified th e validity of this\napproach for several exactly solvable non-interacting models. Fo r instance, the bulk\ndensity of states in a non-interacting chain ( V= 0) can be calculated analytically,\nD(ǫ) =1\nπ1√\n4t2−ǫ2(11)\nfor|ǫ|<2tandD(ǫ) = 0for |ǫ|>2t. Figure1showstheLDOScalculatedwithDDMRG\nin the middle of a 400-site non-interacting chain with η= 0.04t. On the scale of this\nfigure the DDMRG spectrum is indistinguishable from the exact result (11) convolved\nwith a Lorentzian of the same width η. This demonstrates the accuracy of the DDMRG\nmethod and also confirms that the scaling of the broadening (10) is a ppropriate for the\ndensity of states of the spinless fermion model.\nAdditionally, we have checked in interacting systems with DDMRG that D(j,ǫ)\ndepends negligibly onthesiteposition jclosetothechain center andthat D(j=N/2,ǫ)\nis indistinguishable from the momentum-integrated spectral funct ions [21]. Finally, we\nwill examine the crossover from boundary LDOS to bulk LDOS in more d etail in the\nnext section. All our results confirm that the DDMRG spectra calcu lated in the middleLocal density of states of the one-dimensional spinless fer mion model 7\n-2 -1 0 1 2\nV/t00.51π/2εp/2t, t*/t\nFigure 2. Position ǫp/2tof the peaks in the bulk density of states calculated with\nDDMRG in 200-sitechains ( ◦) and renormalized“hopping term” t∗/tfrom (4) (——)\nas a function of the interaction parameter V.\nof a spinless fermion chain with a broadening (10) can be regarded as its bulk density\nof states in the Luttinger liquid phase.\nFigure 1 shows the bulk density of states over the full band width fo r three different\nvalues of the interaction V. The band edge singularities of the non-interacting density\nof states (11) are seen as two peaks at ǫ=±2tbecause of the broadening η. Similar\npeaks are visible at positions ǫ=±ǫpforV/negationslash= 0. The energy ǫpincreases monotonically\nwithVstarting from 0 for V=−2t. Actually, figure 2 shows that ǫp/2tvaries exactly\nas the effective “hopping term” t∗(4) as a function of V. Thusǫp= 2t∗and the peak\npositions correspond to the edges of the elementary excitation ba nds [i.e., the maximum\nof the dispersion (3)] for all values of V.\nIn figure 1 we see some spectral weight beyond ǫpwhich is clearly due to the\nbroadening ηforV= 0 but cannot be explained by this broadening for V=−1.9t\nandV= 2t. Therefore, excitations which are made up from more than one elem entary\nexcitations contribute to the density of states (7) at energy |ǫ|> ǫp.\nAccording to the Luttinger liquid paradigm the bulk DOS should vanish a s the\npower law (1) at the Fermi energy. The Luttinger liquid theory also p redicts that the\nexponent is\nα=(K−1)2\n2K(12)\nfora one-component Luttinger liquid. Thus forthespinless fermion model this exponent\ncan be calculated exactly as a function of the interaction paramete rVusing the Bethe\nAnsatz relation for the Luttinger parameter (6). In the non-inte racting chain ( V= 0)\nthis yields α= 0, which implies that the bulk DOS does not vanish but approaches a\nconstant value at the Fermi energy, while one gets α >0 for interacting chains ( V/negationslash= 0).\nThe power-law behaviour is not visible on the scale of figure 1. The van ishing of\nthe density of states is just suggested by narrow minima which are s een atǫ= 0 for\ninteracting fermion systems. Figure 3 shows the bulk density of sta tes around the FermiLocal density of states of the one-dimensional spinless fer mion model 8\n-0.3 -0.2 -0.1 0 0.1 0.2 0.3\nε/t00.010.020.030.040.050.060.070.08D(ε)t\nFigure 3. Bulk density of states around the Fermi energy for V= 2tcalculated with\nDDMRG in a 1600-site chain ( η= 0.01t). The solid line is the power law (1) with the\nexact exponent α=1\n4.\nenergy at a higher resolution ( η= 0.01t) forV= 2t. The power law (1) is also shown\nwith the exact exponent α=1\n4forV= 2t. Clearly, the DDMRG data agree with the\npower law over the energy range shown in this figure.\nActually, we have often found an approximate power-law behaviour over a rather\nbroad energy range of the order of t∗when the exponent αis not too large. This is\nillustrated in figure 4 which shows the bulk density of states for two v alues ofVon a\ndoublelogarithmicscale. Inbothcases theDDMRGdata clearlydeviat e fromthepower\nlaw at high energy ǫ/greaterorsimilart∗. This is not surprising as the curvature of the dispersion (3)\nbecomes relevant above this energy scale. The DDMRG data can also deviate from\nthe power law at low energy ǫ/lessorsimilarη= 0.01t, where the spectrum is dominated by the\nLorentzian broadening (9), although this is not apparent in both ex amples in figure 4.\nForV= 2tthe DDMRG spectrum agree particularly well with the power law. For m ost\n10-310-210-1100\nε/t0.010.11D(ε)t\nFigure 4. Bulk density of states for V=−√\n2t(◦) andV= 2t(/square) calculated with\nDDMRG in a 1600-site chain ( η= 0.01t). The solid lines show the power law (1) with\nthe exact exponent α=1\n4for both cases.Local density of states of the one-dimensional spinless fer mion model 9\n-0.04 -0.02 0 0.02 0.04\nε/t00.010.020.030.040.050.060.07D(ε)t\nFigure 5. Bulk density of states around the Fermi energy for V=−1.9tcalculated\nwith DDMRG in a 1600-site chain ( η= 0.01t). The solid line is the power law (1) with\nthe exact exponent α≈1.574 but shifted vertically by 0.0145.\nvalues of V, however, there is only a rather rough agreement as shown by the second\nexample, V=−√\n2t.\nFor large exponents α/greaterorsimilar1 discrepancies become apparently significant. This is\nillustrated in figure 5 which shows the density of states for V=−1.9tcorresponding to\nα≈1.574. Clearly, a simple power law (1) can not describe the DDMRG data b ecause\nthey converge to a finite value for ǫ→0. This is due to the convolution (9) which\ntransfers a spectral weight DDDMRG(ǫF= 0)∼η∼1/Nwhere the original spectrum\nhas none. While this finite-size effect occurs for all values of V, it becomes relevant for\nlarge exponents αonly. Nevertheless, a simple shift is enough to offset the finite densit y\nof states at the Fermi energy and to recover an excellent agreem ent with the low-energy\npower-law behaviour as shown in figure 5.\nWe note that a similar effect is seen in the STS spectra of atomic chains [11]\nand the purple bronze [9]. Clearly, the experimental densities of sta tes shown in these\npublications are well fitted by power laws around the Fermi energy b ut they remain\nfinite at the Fermi energy. There, interchain couplings, finite temp erature or disorder\nmay be responsible for this deviation from a pure power-law behaviou r.\nFigure 4 suggests that the bulk density of states decreases appr oximately as a\npower law over a rather broad energy range. Yet we have rarely su cceeded in extracting\naccurate values for the exponent αfrom fits to such spectra. The exponent αcan be\ndetermined more accurately and with less computational effort usin g a finite-size scaling\nanalysis [20]. For this purpose we calculate the density of states D(ǫ) for several system\nsizesNat an energy ǫwhich scales as 1 /N. We know that D(ǫ∼1/N) scales as\nN−αin the thermodynamic limit and thus we can determine αusing a power-law fit.\nIn this work we have used ǫ=η= 16t/Nand up to N= 3200 sites for this scaling\nanalysis. The exponents that we have obtained in the range 0 .08< α <1.6 agree\nwithin 10% with the predictions of the Luttinger liquid theory (12) com bined with the\nBethe Ansatz solution (6). Larger exponents than α≈1.6 can be achieved in the half-Local density of states of the one-dimensional spinless fer mion model 10\nfilled spinless fermion chain for −1.9t > V > −2t. We have not explore this regime\nbecause the effective energy scale (4) becomes rapidly very small a nd thus numerical\nsimulations become increasingly difficult. A reliable determination of sma ller exponents\nthanα≈0.08 would require DDMRG computations for a larger range of system s izes\nthan in this work (i.e., N≫103). In principle, a similar approach could be used to\ndetermine whether the peaks found at ǫ=±2t∗are band edge singularities like in the\nnon-interacting system (11) or smoother structures but we hav e not carried out this\nanalysis yet.\n4. Local density of states\nWe now examine the LDOS D(j,ǫ) close to a chain end (hard wall boundary). This\nconfiguration can also be realized experimentally, for instance in gold chains on Ge\nsurfaces [11]. Previous theoretical works [2, 17, 18, 24, 25] have shown that in a\nLuttinger liquid the LDOS at chain edges differs significantly from the b ulk density\nof states. In particular, the low-energy behaviour is very differen t for spinless fermions\nwith attractive ( V <0) or repulsive ( V >0) interactions. Additionally, the LDOS\nof the spinless fermion model next to a site impurity has been investig ated using the\nfunctional renormalization group (fRG) method [26]. (In a Luttinge r liquid the LDOS\nnext to an impurity, however weak, is similar to the LDOS close to a cha in end in the\nasymptotic low-energy limit [2, 5].)\nWe first explore the boundary LDOS, i.e D(j= 1,ǫ) on the first lattice site. In a\nsemi-infinite non-interacting chain ( V= 0) this LDOS is given by\nD(j= 1,ǫ) =1\n2πt2√\n4t2−ǫ2. (13)\nThe DDMRG spectrum for V= 0 is shown in figure 6 with a broadening η= 0.04t. On\nthe scale of this figure there is no visible difference between these DD MRG data and\nthe exact result (13) with the same broadening. This confirms that the scaling of the\n-5-4 -3 -2 -1 0 1 2 3 4 5\nε/t00.10.20.3D(1,ε)t\nFigure 6. Boundary LDOS D(j= 1,ǫ) calculated with DDMRG on the first site of a\n400-site chain ( η= 0.04t) forV= 0 (—·—),V= 1t(- - - -), and V= 2t(——).Local density of states of the one-dimensional spinless fer mion model 11\n-5-4 -3 -2 -1 0 1 2 3 4 5\nε/t00.511.522.5D(1,ε)t\nFigure 7. Boundary LDOS D(j= 1,ǫ) calculated with DDMRG on the first site of a\n200-site chain ( η= 0.08t) forV= 0 (—·—),V=−1t(- - - -), and V=−1.9t(——).\nbroadening (10) is appropriate for the LDOS close to chain edges to o and demonstrates\nagain the accuracy of the DDMRG method.\nFigure 6 also shows the boundary LDOS calculated with DDMRG for two repulsive\ninteractions V. The semi-elliptic spectrum (13) of the non-interacting chain seems to\nsplit into two smaller semi-elliptic bands which move apart as Vincreases. A narrow\nminimum(butnogapupto V= 2t)appearsattheFermienergy ǫ= 0. Theonsetsofthe\nupper and lower bands correspond to the peak positions ǫ=±2t∗in the bulk density of\nstatesdiscussed intheprevioussectionandthustotheedgesoft heelementaryexcitation\nbands (3). As in the bulk density of states we observe some spectr al weight beyond 2 t∗\nwhich can be explained by the convolution (9) for V= 0 but not for interacting fermions\n(V/negationslash= 0).\nIn contrast for attractive interactions V <0 the boundary LDOS develops an\nincreasingly sharppeakattheFermienergyas Vdecreases. Figure7showstheDDMRG\nspectra for two values of V <0 as well as V= 0 for comparison. For V=−twe can still\nsee that the spectrum has apparent onsets at energies ǫ≈ ±2t∗corresponding to the\nband edges of elementary excitations (3). For larger Vwe can no longer distinguish the\nspectrum onsets but most of the spectral weight is still concentr ated within the band of\nelementary excitations −2t∗≤ǫ≤2t∗.\nTheLuttinger liquidtheorypredicts thattheboundarydensity ofs tatesalsofollows\na power law (1) but the boundary exponent is\nαB=1\nK−1 (14)\nfor a one-component Luttinger liquid [2, 17]. Previous DMRG investiga tions have\nconfirmed this expression for the boundary exponent in terms of t he Luttinger\nparameter (6) obtained from the Bethe Ansatz solution [17, 18].\nFor repulsive interactions V >0 the Luttinger parameter is K <1 according to (6)\nand thus αB> α >0. This increase of the exponent at a chain boundary has been\nobserved experimentally in gold chains on semiconducting Ge surface s [11]. The power-Local density of states of the one-dimensional spinless fer mion model 12\nlaw behaviour is not visible on the scale of figure 6 and the narrow minima at the Fermi\nenergy just hints at the presence of a pseudogap. As for the bulk density of states we\nhave examined the low-energy LDOS with resolution of η= 0.01tand obtained similar\nresults. The boundary LDOS follows an approximate power law over a n energy range\n∼t∗for small exponents α. Forα≈1 (⇔V≈2t) we have to offset some finite\nspectral weight at ǫF= 0 to recover the correct low-energy behaviour. Finally, using\na finite-size scaling analysis of our DDMRG data we obtain exponents αwhich agree\nwithin 10% with the exact results of the Luttinger liquid theory (14) c ombined with the\nBethe Ansatz solution (6).\nIn contrast for attractive interactions V <0 the Luttinger parameter is K >1 and\nthusαB<0. Therefore, the Luttinger liquid theory predicts a power-law dive rgence in\nthe boundary LDOS at the Fermi energy. As the effective band widt h∼4t∗and the\nexponent αBdecrease rapidlywithdecreasing V, thissingularityshouldbecomestronger\nand sharper. When Vapproaches −2t, the peak width must vanish according to (4)\nwhile the exponent αBconverges to -1 according to (6) and (14). Thus the power-law\nsingularity turns into a Dirac δ-peak at the Fermi energy in that limit. The predictions\nof the Luttinger liquid theory agree with the boundary LDOS calculat ed with DDMRG\nwhich are shown in figure 7. In the DDMRG spectra the Fermi-energy singularity has\nbeen smoothed into a peak by the broadening (9). Again we can perf orm a finite-size\nscaling analysis to confirm that the peak is a singularity and to determ ine the precise\nvalues of the exponent αB. This analysis is much easier for power-law singularities with\nnegative exponents (i.e., divergences) than with positive ones (pse udogaps). Thus we\nfind exponents which agree very well (within 2%) with the exact value s given by the\nLuttinger liquid exponent (14) combined with the Bethe Ansatz solut ion (6).\nWe now turn to the crossover from the boundary LDOS to the bulk d ensity of\nstates. First, it is helpful to discuss the LDOS close to the chain edg e in a semi-infinite\nnon-interacting chain ( V= 0). Numbering the sites j= 1,2,...from the chain edge we\nobtain for small j≪Nand|ǫ|<2t\nD(j,ǫ) =2\nπ1√\n4t2−ǫ2/bracketleftBig\n1−T2\nj/parenleftBigǫ\n2t/parenrightBig/bracketrightBig\n(15)\nwhereTj(x) = cos(arccos( x)j) are the Chebyshev polynomials. As for the bulk density\nof states D(j,ǫ) = 0 for |ǫ|>2t. The boundary LDOS (13) is recovered for j= 1. The\nLDOSD(j,ǫ) oscillates between 0 and twice its bulk value (11). There are exactly j\npeaks in D(j,ǫ) considered as a function of ǫ. In particular, there is a peak at the Fermi\nenergy for odd jbutD(j,ǫ) vanishes as ǫ2for even j. At low energy these oscillations\ncorrespond to two modes ( −1)jand cos( ǫj/t) = cos[2 ǫja/(/planckover2pi1vF)]. Field theoretical\nstudies have shown that these same two oscillation modes also domina te the low-energy\nLDOS of Luttinger liquids with the renormalized velocity vsubstituted for vF[24].\nAs (15) has been derived for j≪N, it is not surprising that it does not converge\ntoward the bulk density of states (11) for large j. Nevertheless, if we broaden each\nLDOSD(j,ǫ) with a Lorentzian (9) (or a Gaussian, ...) of width ∼4t/j, we find that\nthe result converges toward the bulk density of states for j→ ∞. As in the middleLocal density of states of the one-dimensional spinless fer mion model 13\n-6-4 -2 0 2 4 6\nε/t012345D(j,ε)t(a)\n-6-4 -2 0 2 4 6\nε/t012345D(x,ε)t(b)\nFigure 8. LDOSD(j,ǫ) calculated on a 200-site chain ( η= 0.08t) forV= 2t. (a)\nFrom one edge to the middle of the chain: j= 1,10,20,...,100 from top to bottom.\n(b) Close to the chain edge: j= 1,2,...,10 from top to bottom.\nof aN-site chain this broadening ∼4t/jis smaller than the one used in DDMRG\ncalculations (10), this property explains why the LDOS calculated in t he middle of a\nchain with DDMRG agrees so well with the (broadened) bulk density of states (11).\nThe crossover from boundary to bulk density of states is shown in fi gure 8 for a\nrepulsive interaction V= 2tand in figure 9 for an attractive interaction V=−1t. First,\nthe right panels of both figures reveal that the characteristic sh apes of the boundary\nLDOSD(j= 1,ǫ) in figures 6 and 7 are purely local features. The LDOS D(j >2,ǫ) on\nother sites look completely different, even on the closest sites. The refore, it is unlikely\nthat the boundary exponent (14) can be determine precisely in STS experiments which\naverage over several ˚A [11]. Second, we note that the LDOS for j >1 are quite similar\nfor attractive and repulsive interactions besides the different ban d width ≈4t∗. The\nmain reason for this likeness is that the spectra D(j,ǫ) are clearly dominate by jpeaks\nup to (at least) j= 10 for all interactions V. These oscillations are similar to those\nfound in the LDOS for non-interacting spinless fermions (15). Finally , in the left panels\nof figures 8 and 9 we see that the amplitude of these oscillations decr ease rapidly with\nincreasing distance jfrom the boundary. Already for j/greaterorsimilar50 =N/4 both LDOS do not\nappear to change on the scale of the broadening η= 0.08t. This confirms that DDMRG\ncalculations of the LDOS in the middle of a chain yield a (broadened) bulk density of\nstates.\nThe above discussion is based on our DDMRG data in chains with up to 20 0 sites\nandη≥0.08t. As seen in the study of bulk and boundary density of states, this\nresolution is not good enough to analyze the asymptotic behaviour c lose to the Fermi\nenergy. Moreover, themultipleoscillationsobservedintheLDOSinth ecrossover regime\n1< j≪Nimply that a power-law behaviour could only be observed in a significant ly\nsmaller energy range than for the bulk and boundary density of sta tes. We estimate\nthat carrying out such a study would require a computational effor t which is one order\nof magnitude higher than for the present work (which amounts to 2 ·104CPU hours).Local density of states of the one-dimensional spinless fer mion model 14\n-4 -3 -2 -1 0 1 2 3 4\nε/t0123456D(j,ε)t(a)\n-4 -3 -2 -1 0 1 2 3 4\nε/t012345D(j,ε)t(b)\nFigure 9. LDOSD(j,ǫ) calculated on a 200-site chain ( η= 0.08t) forV=−t. (a)\nFrom one edge to the middle of the chain: j= 1,10,20,...,100 from top to bottom.\n(b) Close to the chain edge: j= 1,2,...,10 from top to bottom.\nThe crossover from boundary to bulk density of states in Luttinge r liquids has\nalready been studied with field theory [24, 25]. We have not perform ed any quantitative\ncomparison between DDMRG spectra and the field-theoretical res ults or the fRG\nresults [26] because of the limited resolution of our data. Neverthe less, there is a clear\ndiscrepancy between our results and one of the main predictions of bosonization. In an\nextensionoftheTomonaga-Luttingermodelwithhardwallbounda riesithasbeenshown\nthat boundaries affect the LDOS D(j,ǫ) only below a crossover energy [24]. Above this\nenergy bulk behaviour is observed. This crossover energy is given b yv/planckover2pi1/(ja) = 2t∗/j.\nHowever, in DDMRG spectra (see figures 8 and 9) boundary effects are clearly seen as\nrapid oscillations over the full band width, even for j= 20. Thus our results indicate\nthat there is no crossover energy above which only bulk behaviour is observed in the\nspinless fermion model. This is also demonstrated by the exact LDOS o f noninteracting\nfermions (15) which can be regarded as a Luttinger liquid with K= 1. A correct\nstatement for the spinless fermion model is that boundary effects are only visible in the\nLDOSD(j,ǫ) at a resolution /lessorsimilar4t∗/j. If the LDOS is examined at a resolution larger\nthan this value (for instance, if it has been convolved with a Lorentz ian or Gaussian\ndistribution of width /greaterorsimilar4t∗/j), then boundary effects are not visible. Thus in the\nDDMRG spectra boundary effects disappear when the broadening ( 10) exceeds a value\n≈4t∗/j.\n5. Conclusion\nWe have investigated the local density of states (LDOS) D(j,ǫ) of the one-dimensional\nhalf-filled spinless fermion model in its Luttinger liquid phase. Using the DDMRG\nmethod we have determined the LDOS over its full energy range with a resolution of\nη= 0.08tor better. In all cases most of the spectral weight is concentrat ed in the band\n|ǫ|<2t∗of elementary excitations from the Bethe Ansatz solution. We have found that\nthe bulk density of states can be identified with the LDOS computed in the middleLocal density of states of the one-dimensional spinless fer mion model 15\nof a chain if we use a large enough broadening η. It is dominated by sharp peaks at\nthe band edges ǫ=±2t∗for all interactions V. For repulsive interactions ( V >0) the\nboundary LDOS (on the first lattice site) consists in two symmetric j oined bands which\nare reminiscent of Hubbard bands. For attractive interactions ( V <0) the boundary\nLDOS is dominated by a peak at the Fermi energy, which becomes incr easingly sharp\nasVdecreases and turns into a Dirac δ-peak for V→ −2t.\nWe have also examined the bulk and boundary density of states close to the Fermi\nenergy with a resolution η= 0.01ton lattices with 1600 sites. In the bulk density of\nstates for V/negationslash= 0 and in the boundary LDOS for V >0 we can observe the power-law\npseudogap D(j,ǫ)∼ |ǫ|αpredicted by the Luttinger liquid theory over an energy range\n∼t∗although we have to compensate explicitly for finite-size effects whe n the exponent\nαis large. In the boundary LDOS for V <0 we see the power-law divergence predicted\nby the Luttinger liquid theory. Using a finite-size scaling analysis of DD MRG data on\nlattices with up to 3200 sites we obtain exponents αwhich agree quantitatively with\nthe exact results given by the Luttinger liquid theory combined with t he Bethe Ansatz\nsolution.\nFinally, we have explored the crossover from the boundary LDOS D(j= 1,ǫ) to\nthe bulk density of states D(j=N/2,ǫ) as one moves away from the chain edge. The\ncharacteristic features of the boundary LDOS D(j= 1,ǫ) in Luttinger liquids appear to\nbe a purely local effect as all D(j >2,ǫ) look completely different. They are dominated\nby oscillations which are similar to those observed in the non-interact ing chain. The\ncrossover is smooth and boundary effects can be visible far away fr om the chain edge\n(j≫1) at all energies ǫbut vanish beyond a point set by the DDMRG resolution, i.e.\nwhen 4t∗/j/lessorsimilarη.\nThe LDOS of the spinless fermion model cannot be used directly to int erpret\nphotoemission or STS experiments. For this purpose we have to con sider electronic\nmodels such as the Hubbard model. The present work has been motiv ated in part by\nthe failure to observe a power law in the density of states of the Hub bard model using\nDDMRG with an energy resolution η= 0.1t. For spinless fermion models the DDMRG\nmethod allows us to study larger system sizes Nand thus to reach a better resolution\n∼1/Nthan for electronic models. Our work confirms that the power-law f eatures of\nLuttinger liquid LDOS can be studied with DDMRG. However, the resolu tion required\nfor the Hubbard model is certainly much smaller than 0 .1t. In the spinless fermion\nmodel we have to use a resolution of about 10−1t∗forα≈1 and 10−2t∗forα≈1\n4to\nobserve a power law. In the Hubbard model α <1/8 and the energy scale of the power-\nlaw behaviour is much lower than the bare band width 4 t. This scale is exponentially\nsmall∼exp(−t/U) for small U/t[18] and it is probably set by the effective exchange\ncoupling J∼t2/Ufor large U/t. Thus a resolution even smaller than 0 .01tis required\nwhich implies system sizes well in excess of 103sites.\nThe recent experimental observation of Luttinger liquid LDOS in ato mic wires\ncertainly calls for a renewed effort in the study of the LDOS in Hubbar d-type models\nfor interacting electrons in one dimension. Fortunately, larger exp onentsαare achievedLocal density of states of the one-dimensional spinless fer mion model 16\nin extended Hubbard models with a non-local Coulomb interaction [27, 28], which\nare anyway more realistic than the original Hubbard model. We think t hat in these\nelectronic models the LDOS and the power-law pseudogap could be inv estigated with a\nsufficient accuracy and a reasonable computational effort using th e DDMRG method.\nAcknowledgments\nI thank Volker Meden for helpful comments. The GotoBLAS library d eveloped by\nKazushige Goto was used to perform the DDMRG calculations. Some o f these\ncalculations were carried out on the RRZN cluster system of the Leib niz Universit¨ at\nHannover.\nReferences\n[1] Baeriswyl D and Degiorgi (eds) 2004 Strong Interactions in Low Dimensions (Dordrecht: Kluwer\nAcademic Publishers)\n[2] Giamarchi T 2003 Quantum Physics in One Dimension (Oxford: Clarendon Press)\n[3] Giuliani G F and Vignale G 2005 Quantum Theory of the Electron Liquid (Cambridge: Cambridge\nUniversity Press)\n[4] Sch¨ onhammer K 2002 J. Phys.: Condens. Matter 1412783\n[5] Sch¨ onhammerK 2004Luttinger liquids: the basic concepts Strong Interactions in Low Dimensions\ned. D Baeriswyl and L Degiorgi (Dordrecht: Kluwer Academic Publish ers)\n[6] Vescoli V et al2000Eur. Phys. J. B13503\n[7] Sing M et al2003Phys. Rev. B68125111\n[8] Ishii H et al2003Nature426540\n[9] Hager J et al2005Phys. Rev. Lett. 95186402\n[10] Wang F et al2006Phys. Rev. B74113107\n[11] Blumenstein C et al2011Nature Physics 7776\n[12] Imambekov A and Glazman L I 2009 Science323228\n[13] Essler F H L, Frahm H, G¨ ohmann F, Kl¨ umper A and Korepin V E 20 05The One-Dimensional\nHubbard Model (Cambridge: Cambridge University Press)\n[14] Kohno M, Arikawa M, Sato J and Sakai K 2010 J. Phys. Soc. Jap. 79043707\n[15] Fehske H, Schneider R and Weiße A (eds) 2008 Computational Many-Particle Physics (Berlin:\nSpringer)\n[16] Schollw¨ ock U 2005 Rev. Mod. Phys. 77259\n[17] Sh¨ onhammer K, Meden V, Metzner W, Schollw¨ ock U and Gunnar sson O 2000 Phys. Rev. B61\n4393\n[18] Meden V, Metzner W, Schollw¨ ock U, Schneider O, Stauber T and Sch¨ onhammer K 2000 Eur.\nPhys. J. B16631\n[19] Schneider I, Struck A, Bortz M and Eggert S 2008 Phys. Rev. Lett. 101206401\n[20] Jeckelmann E 2002 Phys. Rev. B66045114\n[21] Jeckelmann E and Benthien H 2008 Dynamical Density-Matrix Ren ormalization Group\nComputational Many-Particle Physics ed.HFehske, RSchneiderandAWeiße(Berlin: Springer)\n[22] Jeckelmann E 2008 Progress of Theoretical Physics Supplement 176143\n[23] Caux J-S, Konno H, Sorrell M and Weston R 2011 Phys. Rev. Lett. 106217203\n[24] Eggert S, Johannesson H and Mattsson A 1996 Phys. Rev. Lett. 761505\n[25] Schneider I and Eggert S 2010 Phys. Rev. Lett. 104036402\n[26] Andergassen S, Enss T, Meden V, Metzner W, Schollw¨ ock U and Sch¨ onhammer K 2004 Phys. Rev.\nB70075102Local density of states of the one-dimensional spinless fer mion model 17\n[27] Ejima S, Gebhard F and Nishimoto S 2005 Europhysics Letters 70492\n[28] Ejima S, Gebhard F and Nishimoto S 2006 Phys. Rev. B74245110"    },    {        "title": "1112.1159v1.Evolution_of_the_single_mode_squeezed_vacuum_state_in_amplitude_dissipative_channel.pdf",        "content": "arXiv:1112.1159v1  [quant-ph]  6 Dec 2011Evolution of the single-mode squeezed vacuum state in ampli tude dissipative channel\nHong-Yi Fan1, Shuai Wang1and Li-Yun Hu2∗\n1Department of Physics, Shanghai Jiao Tong University, Shan ghai 200240,China\n2College of Physics & Communication Electronics,\nJiangxi Normal University, Nanchang 330022, China\n∗Corresponding author. E-mail: hlyun2008@126.com.\nUsing the way of deriving infinitive sum representation of de nsity operator as a solution to the\nmaster equation describing the amplitude dissipative chan nel by virtue of the entangled state rep-\nresentation, we show manifestly how the initial density ope rator of a single-mode squeezed vac-\nuum state evolves into a definite mixed state which turns out t o be a squeezed chaotic state with\ndecreasing-squeezing. We investigate average photon numb er, photon statistics distributions for this\nstate.\nI. INTRODUCTION\nSqueezed states are such for which the noise in one\nof the chosen pair of observables is reduced below the\nvacuum or ground-state noise level, at the expense of\nincreased noise in the other observable. The squeezing\neffect indeed improves interferometric and spectroscopic\nmeasurements, so in the context of interferometric detec-\ntion and of gravitational waves the squeezed state is very\nuseful [1, 2]. In a very recently published paper, Agarwal\n[3] revealed that a vortex state of a two-mode system can\nbe generated from a squeezed vacuum by subtracting a\nphoton, such a subtracting mechanism may happen in\na quantum channel with amplitude damping. Usually,\nin nature every system is not isolated, dissipation or de-\nphasing usually happens when a system is immersed in\na thermal environment, or a signal (a quantum state)\npasses through a quantum channel which is described by\na master equation [4]. For example, when a pure state\npropagates in a medium, it inevitably interacts with it\nand evolves into a mixed state [5]. Dissipation or de-\nphasing will deteriorate the degree of nonclassicality of\nphotonfields, sophysicistspaymuchattentiontoit [6–8].\nIn this present work we investigate how an initial\nsingle-mode squeezed vacuum state evolves in an ampli-\ntude dissipative channel (ADC). When a system is de-\nscribed by its interaction with a channel with a large\nnumber of degrees of freedom, master equations are set\nup for a better understanding how quantum decoherence\nis processed to affect unitary character in the dissipation\nor gain of the system. In most cases people are inter-\nested in the evolution of the variables associated with\nthe system only. This requires us to obtain the equa-\ntions of motion for the system of interest only after trac-\ning over the reservoir variables. A quantitative measure\nof nonclassicality of quantum fields is necessary for fur-\nther investigating the system’s dynamical behavior. For\nthis channel, the associated loss mechanism in physical\nprocesses is governed by the following master equation\n[4]\ndρ(t)\ndt=κ/parenleftbig\n2aρa†−a†aρ−ρa†a/parenrightbig\n, (1)\nwhereρis the density operator of the system, and κisthe rate of decay. We have solved this problem with use\nof the thermo entangled state representation [9]. Our\nquestions are: What kind of mixed state does the initial\nsqueezedstateturnsinto? Howdoesthephotonstatistics\ndistributions varies in the ADC?\nThus solving master equations is one of the fun-\ndamental tasks in quantum optics. Usually peo-\nple use various quasi-probability representations, such\nas P-representation, Q-representation, complex P-\nrepresentation, and Wigner functions, etc. for converting\nthemasterequationsofdensityoperatorsintotheircorre-\nsponding c-number equations. Recently, a new approach\n[9, 10], using the thermal entangled state representation\n[11, 12] to convert operator master equations to their c-\nnumber equations is presented which can directly lead to\nthe corresponding Kraus operators (the infinitive repre-\nsentation of evolved density operators) in many cases.\nThe work is arranged as follows. In Sec. 2 by virtue\nof the entangled state representation we briefly review\nour way of deriving the infinitive sum representation of\ndensity operator as a solution of the master equation.\nIn Sec. 3 we show that a pure squeezed vacuum state\n(with squeezing parameter λ) will evolves into a mixed\nstate (output state), whose exact form is derived, which\nturns out to be a squeezed chaotic state. We investigate\naverage photon number, photon statistics distributions\nforthisstate. Theprobabilityoffinding nphotonsinthis\nmixed state is obtained which turns out to be a Legendre\npolynomial function relating to the squeezing parameter\nλand the decaying rate κ. In Sec. 4 we discuss the\nphoton statistics distributions of the output state. In\nSec. 5 and 6 we respectively discuss the Wigner function\nand tomogram of the output state.\nII. BRIEF REVIEW OF DEDUCING THE\nINFINITIVE SUM REPRESENTATION OF ρ(t)\nFor solving the above master equation, in a recent\nreview paper [13] we have introduced a convenient ap-\nproach in which the two-mode entangled state [11, 12]\n|η/angbracketright= exp(−1\n2|η|2+ηa†−η∗˜a†+a†˜a†)|0˜0/angbracketright,(2)2\nis employed, where ˜ a†is a fictitious mode independent\nof the real mode a†,[˜a,a†] = 0.|η= 0/angbracketrightpossesses the\nproperties\na|η= 0/angbracketright= ˜a†|η= 0/angbracketright,\na†|η= 0/angbracketright= ˜a|η= 0/angbracketright, (3)\n(a†a)n|η= 0/angbracketright= (˜a†˜a)n|η= 0/angbracketright.\nActing the both sidesofEq.(1) onthe state |η= 0/angbracketright ≡ |I/angbracketright,\nand denoting |ρ/angbracketright=ρ|I/angbracketright, we have\nd\ndt|ρ/angbracketright=κ/parenleftbig\n2aρa†−a†aρ−ρa†a/parenrightbig\n|I/angbracketright\n=κ/parenleftbig\n2a˜a−a†a−˜a†˜a/parenrightbig\n|ρ/angbracketright, (4)\nso its formal solution is\n|ρ/angbracketright= exp/bracketleftbig\nκt/parenleftbig\n2a˜a−a†a−˜a†˜a/parenrightbig/bracketrightbig\n|ρ0/angbracketright,(5)\nwhere|ρ0/angbracketright ≡ρ0|I/angbracketright, ρ0is the initial density operator.\nNoticing that the operators in Eq.(5) obey the follow-\ning commutative relation,\n/bracketleftbig\na˜a,a†a/bracketrightbig\n=/bracketleftbig\na˜a,˜a†˜a/bracketrightbig\n= ˜aa (6)\nand\n/bracketleftbigga†a+˜a†˜a\n2,a˜a/bracketrightbigg\n=−˜aa, (7)\nas well as using the operator identity [14]\neλ(A+σB)=eλAeσ(1−e−λτ)B/τ, (8)\n(which is valid for [ A,B] =τB), we have\ne−2κt/parenleftBig\na†a+˜a†˜a\n2−a˜a/parenrightBig\n=e−κt(a†a+˜a†˜a)eT′a˜a,(9)\nwhereT′= 1−e−2κt.\nThen substituting Eq.(9) into Eq.(5) yields [13]\n|ρ/angbracketright=e−κt(a†a+˜a†˜a)∞/summationdisplay\nn=0T′n\nn!an˜an|ρ0/angbracketright\n=e−κta†a∞/summationdisplay\nn=0T′n\nn!anρ0a†ne−κt˜a†˜a|I/angbracketright\n=∞/summationdisplay\nn=0T′n\nn!e−κta†aanρ0a†ne−κta†a|I/angbracketright,(10)\nwhich leads to the infinitive operator-sum representation\nofρ,\nρ=∞/summationdisplay\nn=0Mnρ0M†\nn, (11)\nwhere\nMn≡/radicalbigg\nT′n\nn!e−κta†aan. (12)We can prove\n/summationdisplay\nnM†\nnMn=/summationdisplay\nnT′n\nn!a†ne−2κta†aan\n=/summationdisplay\nnT′n\nn!e2nκt:a†nan:e−2κta†a\n= :eT′e2κta†a:e−2κta†a\n= :e(e2κt−1)a†a:e−2κta†a= 1,(13)\nwhere : : stands for the normal ordering. Thus Mnis a\nkind of Kraus operator, and ρin Eq.(11) is qualified to\nbe a density operator, i.e.,\nTr[ρ(t)] =Tr/bracketleftBigg∞/summationdisplay\nn=0Mnρ0M†\nn/bracketrightBigg\n=Trρ0.(14)\nTherefore, for any given initial state ρ0, the density op-\neratorρ(t) can be directly calculated from Eq.(11). The\nentangled state representation provides us with an ele-\ngant way of deriving the infinitive sum representation of\ndensity operator as a solution of the master equation.\nIII. EVOLVING OF AN INITIAL SINGLE-MODE\nSQUEEZED VACUUM STATE IN ADC\nIt is seen from Eq.(11) that for any given initial state\nρ0, the density operator ρ(t) can be directly calculated.\nWhenρ0is a single-mode squeezed vacuum state,\nρ0= sechλexp/parenleftbiggtanhλ\n2a†2/parenrightbigg\n|0/angbracketright/angbracketleft0|exp/parenleftbiggtanhλ\n2a2/parenrightbigg\n,\n(15)\nwe see\nρ(t) = sech λ∞/summationdisplay\nn=0T′n\nn!e−κta†aanexp/parenleftbiggtanhλ\n2a†2/parenrightbigg\n|0/angbracketright\n×/angbracketleft0|exp/parenleftbiggtanhλ\n2a2/parenrightbigg\na†ne−κta†a. (16)\nUsing the Baker-Hausdorff lemma [15],\neλˆAˆBe−λˆA=ˆB+λ/bracketleftBig\nˆA,ˆB/bracketrightBig\n+λ2\n2!/bracketleftBig\nˆA,/bracketleftBig\nˆA,ˆB/bracketrightBig/bracketrightBig\n+···.(17)\nwe have\nanexp/parenleftbiggtanhλ\n2a†2/parenrightbigg\n|0/angbracketright=etanhλ\n2a†2e−tanhλ\n2a†2anetanhλ\n2a†2|0/angbracketright\n=etanhλ\n2a†2/parenleftbig\na+a†tanhλ/parenrightbign|0/angbracketright.(18)\nFurther employing the operator identity [16]\n/parenleftbig\nµa+νa†/parenrightbigm=/parenleftbigg\n−i/radicalbiggµν\n2/parenrightbiggm\n:Hm/parenleftbigg\ni/radicalbiggµ\n2νa+i/radicalbiggν\n2µa†/parenrightbigg\n:,\n(19)3\nwhereHm(x) is the Hermite polynomial, we know\n/parenleftbig\na+a†tanhλ/parenrightbign\n=/parenleftBigg\n−i/radicalbigg\ntanhλ\n2/parenrightBiggn\n:Hn/parenleftBigg\ni/radicalbigg\n1\n2tanhλa+i/radicalbigg\ntanhλ\n2a†/parenrightBigg\n:.(20)\nFrom Eq.(18), it follows that\nanetanhλ\n2a†2|0/angbracketright=/parenleftBigg\n−i/radicalbigg\ntanhλ\n2/parenrightBiggn\netanhλ\n2a†2\n×Hn/parenleftBigg\ni/radicalbigg\ntanhλ\n2a†/parenrightBigg\n|0/angbracketright.(21)\nOn the other hand, noting e−κta†aa†eκta†a=\na†e−κt,eκta†aae−κta†a=ae−κtand the normally ordered\nform of the vacuum projector |0/angbracketright/angbracketleft0|=:e−a†a:,we have\nρ(t) = sechλ∞/summationdisplay\nn=0T′n\nn!e−κta†aanetanhλ\n2a†2|0/angbracketright\n×/angbracketleft0|etanhλ\n2a2a†ne−κta†a\n= sechλ∞/summationdisplay\nn=0(T′tanhλ)n\n2nn!ee−2κta†2tanhλ\n2\n×Hn/parenleftBigg\ni/radicalbigg\ntanhλ\n2a†e−κt/parenrightBigg\n|0/angbracketright/angbracketleft0|\n×Hn/parenleftBigg\n−i/radicalbigg\ntanhλ\n2ae−κt/parenrightBigg\nee−2κta2tanhλ\n2\n= sechλ∞/summationdisplay\nn=0(T′tanhλ)n\n2nn!:ee−2κt(a2+a†2)tanhλ\n2−a†a\n×Hn/parenleftBigg\ni/radicalbigg\ntanhλ\n2a†e−κt/parenrightBigg\nHn/parenleftBigg\n−i/radicalbigg\ntanhλ\n2ae−κt/parenrightBigg\n:\n(22)\nthen using the following identity [17]\n∞/summationdisplay\nn=0tn\n2nn!Hn(x)Hn(y)\n=/parenleftbig\n1−t2/parenrightbig−1/2exp/bracketleftBigg\nt2/parenleftbig\nx2+y2/parenrightbig\n−2txy\nt2−1/bracketrightBigg\n,(23)\nandeλa†a=:e(eλ−1)a†a:,we finally obtain the expres-\nsion of the output state\nρ(t) =Weß\n2a†2ea†aln(ßT′tanhλ)eß\n2a2,(24)\nwithT′= 1−e−2κtand\nW≡sechλ/radicalbig\n1−T′2tanh2λ,ß≡e−2κttanhλ\n1−T′2tanh2λ.(25)By comparing Eq.(15) with (23) one can see that after\ngoing through the channel the initial squeezing parame-\nter tanh λin Eq.( 15) becomes to ß ≡e−2κttanhλ\n1−T′2tanh2λ,and\n|0/angbracketright/angbracketleft0| →1√\n1−T′2tanh2λea†aln(ßT′tanhλ),a chaotic state\n(mixed state), dueto T′>0,wecanprovee−2κt\n1−T′2tanh2λ<\n1,which means a squeezing-decreasing process. When\nκt= 0, then T′= 0 and ß = tanh λ, Eq.(22) becomes the\ninitial squeezed vacuum state as expected.\nIt is important to check: if Tr ρ(t) = 1. Using Eq.(22)\nand the completeness of coherent state/integraltextd2z\nπ|z/angbracketright/angbracketleftz|= 1\nas well as the following formula [18]\n/integraldisplayd2z\nπeζ|z|2+ξz+ηz∗+fz2+gz∗2=1/radicalbig\nζ2−4fge−ζξη+fη2+gξ2\nζ2−4fg,\n(26)\nwhose convergent condition is Re( ζ±f±g)<0\nandRe/parenleftBig\nζ2−4fg\nζ±f±g/parenrightBig\n<0, we really see\nTrρ(t) =W/integraldisplayd2z\nπ/angbracketleftz|eß\n2a†2ea†aln(ßT′tanhλ)eß\n2a2|z/angbracketright\n=W/radicalBig\n(ßT′tanhλ−1)2−ß2= 1. (27)\nsoρ(t) is qualified to be a mixed state, thus we see\nan initial pure squeezed vacuum state evolves into a\nsqueezed chaotic state with decreasing-squeezing after\npassing through an amplitude dissipative channel.\nIV. AVERAGE PHOTON NUMBER\nUsing the completeness relation of coherent state\nand the normally ordering form of ρ(t) in Eq.\n(22), and using eß\n2a2a†e−ß\n2a2=a†+ßa, as well\nasea†aln(ßT′tanhλ)a†e−a†aln(ßT′tanhλ)=a†ßT′tanhλ,we\nhave\nTr/parenleftbig\nρ(t)a†a/parenrightbig\n=W/integraldisplayd2z\nπ/angbracketleftz|eß\n2a†2ea†aln(ßT′tanhλ)eß\n2a2a†a|z/angbracketright\n=W/integraldisplayd2z\nπ/angbracketleftz|eß\n2a†2ea†aln(ßT′tanhλ)eß\n2a2za†|z/angbracketright\n=W/integraldisplayd2z\nπz/angbracketleftz|eß\n2a†2ea†aln(ßT′tanhλ)/parenleftbig\na†+ßa/parenrightbig\neß\n2a2|z/angbracketright\n=Wß/integraldisplayd2z\nπzeß\n2(z∗2+z2)/angbracketleftz|/parenleftbig\na†T′tanhλ+z/parenrightbig\nea†aln(ßT′tanhλ)|z/angbracketright\n=Wß/integraldisplayd2z\nπ/parenleftbig\n|z|2T′tanhλ+z2/parenrightbig\n×exp/bracketleftbigg\n(ßT′tanhλ−1)|z|2+ß\n2/parenleftbig\nz∗2+z2/parenrightbig/bracketrightbigg\n.(28)4\n            0 1 2 3 4 50.00.20.40.60.81.01.21.4\n( ) n t N\ntN\nFIG. 1: (Color online) The average ¯ n(κt) as the function of\nκtfor different values of squeezing parameter λ(from bottom\nto topλ= 0,0.1,0.3,0.5,1.)\nIn order to perform the integration, we reform Eq.(28)\nas\nTr/parenleftbig\nρ(t)a†a/parenrightbig\n=Wß/braceleftbigg\nT′tanhλ∂\n∂f+2\nß∂\n∂s/bracerightbigg\n×/integraldisplayd2z\nπexp/bracketleftbig\n(ßT′tanhλ−1+f)|z|2\n+ß\n2/parenleftbig\nz∗2+(1+s)z2/parenrightbig/bracketrightbigg\nf=s=0\n=1−ßT′tanhλ\n(ßT′tanhλ−1)2−ß2−1 (29)\nin the last step, we have used Eq.(27). Using Eq.(29), we\npresent the time evolution of the average photon num-\nber in Fig. 1, from which we find that the average pho-\nton number of the single-mode squeezed vacuum state in\nthe amplitude damping channelreducesgraduallyto zero\nwhen decay time goes.\nV. PHOTON STATISTICS DISTRIBUTION\nNext, we shall derive the photon statistics distribu-\ntions ofρ(t). The photon number is given by p(n,t) =\n/angbracketleftn|ρ(t)|n/angbracketright. Noticing a†m|n/angbracketright=/radicalbig\n(m+n)!/n!|m+n/angbracketright\nand using the un-normalized coherent state |α/angbracketright=\nexp[αa†]|0/angbracketright, [19, 20] leading to |n/angbracketright=1√\nn!dn\ndαn|α/angbracketright|α=0,/parenleftbig\n/angbracketleftβ|α/angbracketright=eαβ∗/parenrightbig\n, as well as the normal ordering form of\nρ(t) in Eq. (22), the probability of finding nphotons in\nthe field is given by\np(n,t)\n=/angbracketleftn|ρ(t)|n/angbracketright\n=W\nn!dn\ndβ∗ndn\ndαn/angbracketleftβ|eß\n2β∗2ea†aln(ßT′tanhλ)eß\n2α2|α/angbracketright/vextendsingle/vextendsingle/vextendsingle\nα,β∗=0\n=W\nn!dn\ndβ∗ndn\ndαnexp/bracketleftbigg\nβ∗αßT′tanhλ+ß\n2β∗2+ß\n2α2/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nα,β∗=0.(30)Note that\n/bracketleftBig\neß\n2a†2ea†aln(ßT′tanhλ)eß\n2α2/bracketrightBig†\n=eß\n2a†2ea†aln(ßT′tanhλ)eß\n2α2\nso\n/angbracketleftn|ρ(t)|n/angbracketright∗=/angbracketleftn|ρ(t)†|n/angbracketright=/angbracketleftn|ρ(t)|n/angbracketright\n∂n+n\n∂tn∂t′nexp/bracketleftbig\n2xtt′−t2−t′2/bracketrightbig\nt=t′=0\n= 2nn![n/2]/summationdisplay\nm=0n!\n22m(m!)2(n−2m)!xn−2m,(31)\nwe derive the compact form for p(n,t), i.e.,\np(n,t)\n=W\nn!/parenleftbigg\n−ß\n2/parenrightbiggndn\ndβ∗ndn\ndαne−2T′tanhλβ∗α−β∗2−α2/vextendsingle/vextendsingle/vextendsingle\nα,β∗=0\n=W(ßT′tanhλ)n[n/2]/summationdisplay\nm=0n!(T′tanhλ)−2m\n22m(m!)2(n−2m)!.(32)\nUsing the newly expression of Legendre polynomials\nfound in Ref. [21]\nxn[n/2]/summationdisplay\nm=0n!\n22m(m!)2(n−2m)!/parenleftbigg\n1−1\nx2/parenrightbiggm\n=Pn(x),\n(33)\nwecanformallyrecastEq.(32)intothefollowingcompact\nform, i.e.,\np(n,t) =W/parenleftBig\ne−κt√\n−ßtanhλ/parenrightBign\nPn/parenleftBig\neκtT′√\n−ßtanhλ/parenrightBig\nnote that since√\n−ßtanhλis pure imaginary, while\np(n,t)isreal, sowemuststillusethepower-seriesexpan-\nsion on the right-hand side of Eq.(32) to depict figures of\nthe variation of p(n,t). In particular, when t= 0, Eq.(32\n) reduces to\np(n,0) = sech λ(tanhλ)nlim\nT′→0[n/2]/summationdisplay\nm=0n!(T′tanhλ)n−2m\n22m(m!)2(n−2m)!\n=/braceleftbigg(2k)!\n22kk!k!sechλtanh2kλ, n= 2k\n0 n= 2k+1,(34)\nwhich just correspond to the number distributions of the\nsqueezed vacuum state [22, 23]. From Eq.(34) it is not\ndifficult to see that the photocount distribution decreases\nas the squeezing parameter λincreases. While for κt→\n∞,we see that p(n,∞) = 0.This indicates that there is\nno photon when a system interacting with a amplitude\ndissipative channel for enough long time, as expected.\nIn Fig. 2, the photon number distribution is shown for\ndifferent κt.5\n    \n       \n0tN ) (b \u000b\fP n 0.5tN \u000b\fP n ) (a\nn n\n     \n\u000b\fP n 1.0tN \u000b\fP n ( ) c 2tN ( ) d\nn n\nFIG. 2: (Color online) Photon number distribution of the\nsqueezed vacuum state in amplitude damping channel for λ=\n1, and different κt: (a)κt= 0, (b)κt= 0.5,(c)κt= 1 and\n(d)κt= 2.\nVI. WIGNER FUNCTIONS\nIn this section, we shall use the normally ordering for\nofdensity operatorsto calculatetheanalyticalexpression\nof Wigner function. For a single-mode system, the WF\nis given by [24]\nW(α,α∗,t) =e2|α|2/integraldisplayd2β\nπ2/angbracketleft−β|ρ(t)|β/angbracketrighte−2(βα∗−β∗α),\n(35)\nwhere|β/angbracketrightis the coherent state [19, 20] . From Eq.(22) it\nis easyto see that oncethe normalorderedform of ρ(t) is\nknown, we can conveniently obtain the Wigner function\nofρ(t).\nOnsubstitutingEq.(24)intoEq.(35)weobtaintheWF\nof the single-mode squeezed state in the ADC,\nW(α,α∗,t)\n=We2|α|2/integraldisplayd2β\nπ2exp/bracketleftBig\n−(1+ßT′tanhλ)|β|2\n−2(βα∗−β∗α)+ß\n2β∗2+ß\n2β2/bracketrightbigg\n=W\nπ/radicalBig\n(1+ßT′tanhλ)2−ß2exp/bracketleftBig\n2|α|2/bracketrightBig\n×exp/bracketleftBigg\n2−2(1+ßT′tanhλ)|α|2+ß/parenleftbig\nα∗2+α2/parenrightbig\n(1+ßT′tanhλ)2−ß2/bracketrightBigg\n(36)\nIn particular, when t= 0 and t→ ∞, Eq.(36)\nreduces to W(α,α∗,0) =1\nπexp[−2|α|2cosh2λ+/parenleftbig\nα∗2+α2/parenrightbig\nsinh2λ], andW(α,α∗,∞) =1\nπexp/bracketleftBig\n−2|α|2/bracketrightBig\n,\nwhich are just the WF of the single-mode squeezed vac-\nuum state and the vacuum state, respectively. In Fig. 3,\nthe WF of the single-mode squeezed vacuum state in the\namplitude damping channel is shown for different decay\ntimeκt.    \n       \n\u000b\f*,WDD \u000b\f*,WDD 0tN 0.5tN ) (a ) (b\n\u000b\f ReD \u000b\f ReD \u000b\f ImD \u000b\f ImD\n     \n\u000b\f*,WDD\u000b\f*,WDD 2tN 1.0tN ( ) d ( ) c\n\u000b\f ReD \u000b\f ReD \u000b\f ImD \u000b\f ImD\nFIG. 3: (Color online) Wigner function of the squeezed vac-\nuumstate inamplitude dampingchannel for λ= 1.0, different\nκt: (a)κt= 0.0, (b)κt= 0.5, (c)κt= 1, and ( d)κt= 2.\nVII. TOMOGRAM\nAs we know, once the probability distributions Pθ(ˆxθ)\nofthequadratureamplitudeareobtained,onecanusethe\ninverse Radon transformation familiar in tomographic\nimaging to obtain the WF and density matrix [25]. Thus\nthe Radon transform of the WF is corresponding to the\nprobability distributions Pθ(ˆxθ). In this section we de-\nrive the tomogram of ρ(t).\nForasingle-modesystem, the RadontransformofWF,\ndenoted as Ris defined by [26]\nR(q)f,g=/integraldisplay\nδ(q−fq′−gp′)Tr[∆(β)ρ(t)]dq′dp′\n=Tr/bracketleftBig\n|q/angbracketrightf,g f,g/angbracketleftq|ρ(t)/bracketrightBig\n=f,g/angbracketleftq|ρ(t)|q/angbracketrightf,g(37)\nwhere the operator |q/angbracketrightf,g f,g/angbracketleftq|is just the Radon trans-\nform of single-mode Wigner operator ∆( β), and\n|q/angbracketrightf,g=Aexp/bracketleftBigg√\n2qa†\nB−B∗\n2Ba†2/bracketrightBigg\n|0/angbracketright,(38)\nas well as B=f−ig, A =/bracketleftbig\nπ/parenleftbig\nf2+g2/parenrightbig/bracketrightbig−1/4exp[−q2/2/parenleftbig\nf2+g2/parenrightbig\n]. Thus the\ntomogram of a quantum state ρ(t) is just the quantum\naverage of ρ(t) in|q/angbracketrightf,grepresentation (a kind of\nintermediate coordinate-momentum representation) [27].\nSubstituting Eqs.(24) and (38) into Eq.(37), and using\nthe completeness relation of coherent state, we see that\nthe Radom transform of WF of ρ(t) is given by\nR(q)f,g\n=Wf,g/angbracketleftq|eß\n2a†2ea†aln(ßT′tanhλ)eß\n2a2|q/angbracketrightf,g\n=WA2\n√\nEexp/braceleftBigg\nq2ß\nE|B|4/parenleftbig\nB2+B∗2/parenrightbig\n+2q2ß\nE|B|2/parenleftbig\nTtanhλ+ß−ßT2tanh2λ/parenrightbig/bracerightBigg\n,(39)6\nwhere we have used the formula (26) and /angbracketleftα|γ/angbracketright=\nexp[−|α|2/2−|γ|2/2+α∗γ], as well as\nE=/parenleftbigg\n1+ßB\nB∗/parenrightbigg/parenleftBigg\n1+B∗\nBß−B∗(ßT′tanhλ)2\nB∗+ßB/parenrightBigg\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+ßB\nB∗/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n−( ßT′tanhλ)2. (40)\nIn particular, when t= 0,(T= 0), then Eq.(39) reduces\nto (B\nB∗=e2iφ)\nR(q)f,g=A2sechλ\n|1+e2iφtanhλ|\n×exp\n\nq2/parenleftBig\nB2+B∗2+2|B|2tanhλ/parenrightBig\ntanhλ\n/vextendsingle/vextendsingle/vextendsingle1+e2iφ|B|4tanhλ/vextendsingle/vextendsingle/vextendsingle2\n\n,(41)\nwhich is a tomogram of single-mode squeezed vacuum\nstate; while for κt→ ∞,(T= 1), then R(q)f,g=A2,\nwhich is a Gaussian distribution corresponding to the\nvacuum state.In summary, using the way of deriving infinitive sum\nrepresentation of density operator by virtue of the entan-\ngled state representation describing, we conclude that in\nthe amplitude dissipative channel the initial density op-\nerator of a single-mode squeezed vacuum state evolves\ninto a squeezed chaotic state with decreasing-squeezing.\nWe investigate average photon number, photon statistics\ndistributions, Wigner functions and tomogram for the\noutput state.\nAcknowledgments\nThis work is supported by the National Natural Sci-\nence Foundation of China (Grant No.11175113 and\n11047133), Shandong Provincial Natural Science Foun-\ndation in China (Gant No.ZR2010AQ024), and a grant\nfrom the Key Programs Foundation of Ministry of Edu-\ncation of China (Grant No. 210115),as well as Jiangxi\nProvincial Natural Science Foundation in China (No.\n2010GQW0027).\n[1] Caves C M 1981 Phys. 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Theor. 44445306\n[17] Rainville E D 1960 Special Functions (New York:\nMacMillan Company)\n[18] Puri R R 2001 Mathematical Methods of Quantum Optics\n(Berlin/Heidelberg/New York: Springer-Verlag)\n[19] Glauber R J 1963 Phys. Rev. 1302529\n[20] Glauber R J 1963 Phys. Rev. 1312766\n[21] Fan H Y, Hu L Y and Xu X X 2009 Mod. Phys. Lett. A\n241597\n[22] Kim M S, de Oliveira F A M and Knight P L 1989 Phys.\nRev. A402494\n[23] Paulina Marian 1992 Phys. Rev. A 452044\n[24] Fan H Y andZaidi H R 1987 Phys. Lett. A 124303\n[25] Vogel K and Risken H 1989 Phys. Rev. A 402847\n[26] Fan H Y and Niu J B 2010 Opt. Commun. 2833296\n[27] Fan H Y and Hu L Y 2009 Opt. Commun. 2823734"    },    {        "title": "1112.1716v3.Bounds_on_the_density_of_states_for_Schr___odinger_operators.pdf",        "content": "BOUNDS ON THE DENSITY OF STATES FOR SCHR ODINGER\nOPERATORS\nJEAN BOURGAIN AND ABEL KLEIN\nAbstract. We establish bounds on the density of states measure for Schr odinger\noperators. These are deterministic results that do not require the existence\nof the density of states measure, or, equivalently, of the integrated density of\nstates. The results are stated in terms of a \\density of states outer-measure\"\nthat always exists, and provides an upper bound for the density of states mea-\nsure when it exists. We prove log-H older continuity for this density of states\nouter-measure in one, two, and three dimensions for Schr odinger operators,\nand in any dimension for discrete Schr odinger operators.\nContents\n1. Introduction 1\n2. Discrete and one-dimensional Schr odinger operators 5\n2.1. A lower bound for the maximal L1norm 5\n2.2. Discrete Schr odinger operators 6\n2.3. One-dimensional Schr odinger operators 8\n3. Multi-dimensional Schr odinger operators 9\n3.1. Local behavior of approximate solutions of the stationary Schr odinger\nequation 9\n3.2. A quantitative unique continuation principle for approximate solutions\nof the stationary Schr odinger equation 15\n3.3. Two and three dimensional Schr odinger operators 19\nReferences 22\n1.Introduction\nWe study the density of states of the Schr odinger operator\nH=\u0000\u0001 +Von L2(Rd); (1.1)\nwhere \u0001 is the Laplacian operator and Vis a bounded potential. The density of\nstates measure of an interval gives the \\number of states per unit volume\" with\nenergy in the interval; its cumulative distribution function is the integrated density\nof states. Finite volume density of states measures, i.e., density of states measures\nfor restrictions of the Schr odinger operator to \fnite volumes, are always well de\fned.\nThe density of states measure is given by appropriate limits of \fnite volume density\nof states measures, when such limits exist. These limits are known to exist for\nSchr odinger operators where the potential Vis in some sense uniform in space (e.g.,\nA.K. was supported in part by the NSF under grant DMS-1001509.\n1arXiv:1112.1716v3  [math-ph]  12 Oct 20122 JEAN BOURGAIN AND ABEL KLEIN\nperiodic potentials, ergodic Schr odinger operators), but not for general Schr odinger\noperators. The density of states measure and the corresponding integrated density\nof states cannot be de\fned for general Schr odinger operators. For this reason we\nintroduce the density of states outer-measure, which always exists, and provides\nan upper bound for the density of states measure, when it exists. We prove upper\nbounds on the density of states outer-measure of small intervals, establishing log-\nH older continuity in one, two, and three dimensions for Schr odinger operators, and\nin any dimension for discrete Schr odinger operators.\nWe let\n\u0003L(x) :=x+\u0003\n\u0000L\n2;L\n2\u0002d=\b\ny2Rd;jy\u0000xj1<L\n2\t\n(1.2)\ndenote the (open) box of side Lcentered at x2Rd. By a box \u0003 Lwe will mean a\nbox \u0003L(x) for somex2Rd. We writek k=k k2for 2L2(Rd) or 2L2(\u0003).\nWe setV1=kVk1, the norm of the bounded potential V. By\u001fBwe denote\nthe characteristic function of the set B. Constants such as Ca;b;::: will always be\n\fnite and depending only on the parameters or quantities a;b;::: ; they will be\nindependent of other parameters or quantities in the equation. Note that Ca;b;:::\nmay stand for di\u000berent constants in di\u000berent sides of the same inequality.\nGiven a \fnite box \u0003 \u001aRd, we letH]\n\u0003and \u0001]\n\u0003be the restriction of Hand \u0001\nto L2(\u0003) with]boundary condition, where ]=D(Dirichlet), N(Neumann), or P\n(periodic). We de\fne \fnite volume density of states measures \u0011\u0003;]on Borel subsets\nB ofRdby\n\u0011\u0003;](B) :=1\nj\u0003jtrn\n\u001fB(H]\n\u0003)o\nfor]=D;N;P; (1.3)\n\u0011\u0003;1(B) :=1\nj\u0003jtrf\u001fB(H)\u001f\u0003g:\nNote that for for all Borel subsets B\u001a]\u00001;E] we have\n\u0011\u0003;](B)\u0014Cd;V1;E<1for]=1;D;N;P: (1.4)\nMoreover, given f2Cc(R) and\u000e >0, there exists L(d;V1;\u000e;f) such that for all\nL\u0015L(d;V1;\u000e;f) andx02Rdwe have\n\f\f\u0011\u0003L(x0);]1(f)\u0000\u0011\u0003L(x0);]2(f)\f\f\u0014\u000efor]1;]2=1;D;N;P: (1.5)\n(This can be extracted from [DoIM, see Theorem 3.6, Theorem 6.2, and their\nproofs].) The \fnite volume integrated density of states are the corresponding cu-\nmulative distribution functions:\nN\u0003;](E) :=\u0011\u0003;](]\u00001;E]): (1.6)\nFor periodic and ergodic Schr odinger operators, density of states measures \u0011]\ncan be de\fned as weak limits of the \fnite volume density of states measures \u0011\u0003;]\nfor sequences of boxes \u0003 !Rdin an appropriate sense. In this case, the integrated\ndensity of states N](E) :=\u0011](]\u00001;E]) satis\fesN](E) = lim \u0003!RdN\u0003;](E) except\nfor a countable set of energies. Moreover, they all coincide, so we de\fne the density\nof states measure \u0011and the integrated density of states N(E) by\u0011(B) :=\u0011](B)\nandN(E) :=N](E) for]=1;D;N;P . (See [KM, PF, CL, DoIM, N].)\nSince in\fnite volume density of states measures and integrated density of states\ncannot be de\fned for general Schr odinger operators, we de\fne density of statesBOUNDS ON THE DENSITY OF STATES FOR SCHR ODINGER OPERATORS 3\nouter-measures on Borel subsets B of Rdby\n\u0011\u0003\nL;](B) := sup\nx2Rd\u0011\u0003L(x);](B)\n\u0011\u0003\n](B) := lim sup\nL!1\u0011\u0003\nL;](B); ] =1;D;N;P: (1.7)\nThese are always \fnite on bounded sets in view of (1.4). (They are indeed outer-\nmeasures, so we call them outer-measures for lack of a better name.) Moreover, it\nfollows from (1.5) that for all E1;E22R,E1\u0014E2, and\u000e>0 we have\n\u0011\u0003\n]1([E1;E2])\u0014\u0011\u0003\n]2([E1\u0000\u000e;E2+\u000e]) for all ]1;]2=1;D;N;P: (1.8)\nWe will say that we have continuity of the density of states outer-measure \u0011\u0003\n]if\nlim\n\"!0\u0011\u0003\n]([E\u0000\";E+\"]) = 0 for all E2R: (1.9)\nIn view of (1.8), continuity of \u0011\u0003\n]for some value of ]implies continuity of \u0011\u0003\n]for all\nvalues of], and we have\n\u0011\u0003\n1([E1;E2]) =\u0011\u0003\nD([E1;E2]) =\u0011\u0003\nN([E1;E2]) =\u0011\u0003\nP([E1;E2]) (1.10)\nfor allE1;E22R,E1\u0014E2. In this case we set\n\u0011\u0003([E1;E2]) :=\u0011\u0003\n]([E1;E2]) for]=1;D;N;P; (1.11)\nand say that we have continuity of the density of states outer-measure.\nWe are ready to state our main result. Note that if the density of states measure\n\u0011]exists, we always have\n\u0011](B)\u0014\u0011\u0003\n](B) for all Borel sets B\u001aRd; (1.12)\nand hence continuity of the density of states outer-measure implies continuity of\nthe integrated density of states\nTheorem 1.1. LetHbe a Schr odinger operator as in (1.1) , whered= 1;2;3.\nThen we have continuity of the density of states outer-measure. Moreover, given\nE02R, for allE\u0014E0and\"\u00141\n2we have\n\u0011\u0003([E;E +\"])\u0014Cd;V1;E0\u0000\nlog1\n\"\u0001\u0014d;where\u00141= 1; \u00142=1\n4; \u00143=1\n8: (1.13)\nWe also prove a similar result for discrete Schr odinger operators, i.e., for\nH=\u0000\u0001 +Von`2(Zd); (1.14)\nwhereVis a bounded potential and \u0001 is the centered discrete Laplacian,\n\u0001 (x) =X\ny2Zd;jx\u0000yj=1 (y) forx2Zd: (1.15)\n(Our results are still valid if we take \u0001 to be any translation invariant \fnite range\nself-adjoint operator on `2(Zd).) In Zdwe de\fne the box of side Lcentered at\nx2Zdby\n\u0003 = \u0003L(x) =\b\ny2Zd;jy\u0000xj1\u0014L\n2\t\n; (1.16)\nand de\fne \fnite volume operators H]\n\u0003and \u0001]\n\u0003as the restriction of Hand \u0001 to\n`2(\u0003) with]boundary condition, where ]=D(Dirichlet, i.e., simple boundary\ncondition) or P(periodic). We de\fne \fnite volume density of states measures \u0011\u0003;]\nas in (1.3) and density of states outer-measures \u0011\u0003\nL;];\u0011\u0003\n]as in (1.7) for ]=1;D;P .4 JEAN BOURGAIN AND ABEL KLEIN\nIn the discrete case it is easy to see that we also have (1.8), and hence continuity\nof\u0011\u0003\n]for some value of ]implies (1.10), in which case we de\fne \u0011\u0003as in (1.11).\nTheorem 1.2. LetHbe a discrete Schr odinger operator as in (1.14) . Then for\nalld= 1;2;::: we have continuity of the density of states outer-measure, and for\nallE2Rand\"\u00141\n2we have\n\u0011\u0003([E;E +\"])\u0014Cd;V1\nlog1\n\": (1.17)\nWe are not aware of previous results in the generality of Theorems 1.1 and 1.2.\nPublished results appear to be restricted to cases where we have existence of the\nintegrated density of states. For periodic potentials, continuity of the the integrated\ndensity of states is equivalent to the nonexistence of eigenvalues, a nontrivial result\nproved by Thomas [T]. For ergodic Schr odinger operators, continuity of the the\nintegrated density of states is equivalent to the nonexistence of energies that are\neigenvalues of in\fnite multiplicity with probability one (see [CL, Lemma V.2.1]).\nAlthough Schr odinger operators can have eigenvalues of in\fnite multiplicity (see\n[ThE]), it is hard to imagine how a \fxed energy can be an eigenvalue of in\fnite\nmultiplicity for almost all realizations of an ergodic Schr odinger operator.\nCraig and Simon proved log-H older continuity (with exponent 1) of the inte-\ngrated density of states for one-dimensional ergodic Schr odinger operators [CrS1]\nand for ergodic discrete Schr odinger operators in any dimension [CrS2]. Delyon\nand Souillard [DS] provided a simple proof of continuity of the integrated density\nof states in the discrete case. But continuity of the the integrated density of states\nfor multi-dimensional (continuous) ergodic Schr odinger operators, albeit expected,\nhas been hard to prove in full generality. It is Problem 14 in [Si2], where it was\ncalled (in 2000) a 15 year old open problem.\nFor random Schr odinger operators continuity of the integrated density of states\nfollows from a suitable Wegner estimate. The most general result is due to Combes,\nHislop and Klopp [CoHK] that proved that for the Anderson model, both contin-\nuous and discrete, we always have continuity of the integrated density of states if\nthe single-site probability distribution has no atoms. (They show that the inte-\ngrated density of states has as much regularity as the concentration function of the\nsingle-site probability distribution.) Germinet and Klein [GK2] proved log-H older\ncontinuity of the integrated density of states for the continuous Anderson model\nwith arbitrary single-site probability distribution (e.g., Bernouilli) in the region of\nlocalization. (More precisely, in the region of applicability of the multiscale analy-\nsis; the log-H older continuity of the integrated density of states is derived from the\nconclusions of the multiscale analysis.)\nThe casesd= 1 andd= 2;3 of Theorem 1.1 have separate proofs, the proof for\nd= 1 being similar to the proof of Theorem 1.2. Note that it su\u000eces to establish\n(1.13) and (1.17) with Dirichlet boundary condition ( ]=D), since we would then\nhave (1.11). Thus in the following sections we assume Dirichlet boundary condition\nand drop it from the notation.\nTheorem 1.2 and the d= 1 case of Theorem 1.1 are proved in Section 2; they\nare immediate consequences of Theorems 2.2 and 2.3, respectively.\nSection 3 is devoted to multi-dimensional Schr odinger operators. We start by\nstudying the local behavior of approximate solutions of the stationary Schr odinger\nequation in Subsection 3.1; see Theorem 3.1. Solutions of the stationary Schr odingerBOUNDS ON THE DENSITY OF STATES FOR SCHR ODINGER OPERATORS 5\nequation admit a local decomposition into a homogeneous harmonic polynomial and\na lower order term [HW, B]; in Lemma 3.2 we establish a quantitative version of\nthis decomposition with explicit estimates of the lower order term. This result\nis extended to approximate solutions in Lemma 3.3, implying Theorem 3.1. We\nthen state and prove Theorem 3.4, a version of Bourgain and Kenig's quantitative\nunique continuation principle [BoK, Lemma 3.10], in which we make explicit the\ndependence on the parameters relevant to this article. Finally, in Subsection 3.3\nwe prove Theorem 3.7, which implies the d= 2;3 cases of Theorem 1.1.\nThe restriction to d= 1;2;3 in Theorem 1.1 is due to the present form of the\nquantitative unique continuation principle (Theorem 3.4), where there is a term Q4\n3\nin the exponent on the left hand side of (3.62). If we had Q\fin (3.62), we would be\nable to prove Theorem 3.7, and hence Theorem 1.1, for dimensions d<\f\n\f\u00001. Since\n\f=4\n3, we getd <4. It is reasonable to expect that something like Theorem 3.4\nholds with\f= 1+ (there are no counterexamples for real potentials), in which case\nTheorem 1.1 would hold for all d, with\u0014d=\f\u0000d(\f\u00001)\n2\f=1\n2\u0000ford\u00152 in (1.13).\n2.Discrete and one-dimensional Schr odinger operators\nTo prove Theorem 1.2 and the d= 1 case of Theorem 1.1, we will select a\nclass of approximate eigenfunctions for which we establish a global upper bound,\nand use Lemma 2.1 to pick an approximate eigenfunction for which we have a\nlower bound for the global upper bound. In more detail: Given an energy E,\n0< \"\u00141\n2, and a box \u0003, we set P=\u001f[E;E+\"](H\u0003) and consider the linear space\nRanP. (Note that  2RanPis an approximate eigenfunction for H\u0003in the sense\nthatk(H\u0003L\u0000E) k\u0014\"k k.) We select a linear subspace of Fof RanPfor which\nthe L1-norms are uniformly bounded in terms of the L2-norms (a global upper\nbound). We then use Lemma 2.1 to pick  02F for which we have a lower bound\nfork 0k1. Comparing this lower bound with the global upper bound yields the\nbound on\u0011\u0003([E;E +\"]).\n2.1.A lower bound for the maximal L1norm.\nLemma 2.1. LetVbe a \fnite dimensional linear subspace of L1(\n;P), where\n(\n;P)is a probability space. Then there exists  2V withk k2= 1such that\nk k1\u0015p\ndimV: (2.1)\nThis lemma is known to follow immediately from the theory of absolutely sum-\nming operators, but can also be proved by a direct argument. We present both\nproofs for completeness.\nProof of Lemma 2.1 using absolutely summing operators. LetVpdenote the linear\nspaceVviewed as subspace of Lp(\n;P) and letIp;qbe the identity map from Vpto\nVq, with\u00192(Ip;q) being its 2-summing norm. (We refer to [DiJA] for the de\fnition\nand properties of the 2-summing norm.) Then \u00192(I2;2) =p\ndimV, since it is the\nsame as the Hilbert-Schmidt norm of I2;2. FactorI2;2=I2;1I1;2, so\u00192(I2;2)\u0014\r\rI2;1\r\r\u00192(I1;2) by the the ideal property [DiJA, item 2.4]. Since \u00192(I1;2)\u00141\n[DiJA, Example 2.9(d)], we have\r\rI2;1\r\r\u0015p\ndimV, and the lemma follows. \u0003\nProof of Lemma 2.1 (direct proof). Using the the Gelfand-Neumark Theorem (e.g.,\n[S, Section 73]) we can assume, without loss of generality, that \n is a compact6 JEAN BOURGAIN AND ABEL KLEIN\nHausdor\u000b space and L1(\n;P) =C(\n). ThusVis a \fnite dimensional linear\nsubspace of C(\n)\u001aL2(\n;P). LetN= dimV, and pick an orthonormal basis\nf\u001ejgN\nj=1forV. In particular,\n\u001e(x;y) :=NX\nj=1\u001ej(x)\u001ej(y)2C(\n2); (2.2)\nand we have\nN=Z\n\n\u001e(x;x)P(dx) =Z\n\n(\n\u001e(x;x)p\n\u001e(x;x))2\nP(dx)\u0014Z\n\nmax\ny2\n(\nj\u001e(x;y)jp\n\u001e(x;x))2\nP(dx):\n(2.3)\nSincePis a probability measure, there exists x02\n such that\nmax\ny2\nj\u001e(x0;y)jp\n\u001e(x0;x0)\u0015p\nN: (2.4)\nSetting\n =\u001e(x0;\u0001)p\n\u001e(x0;x0)=1p\n\u001e(x0;x0)NX\nj=1\u001ej(x0)\u001ej; (2.5)\nwe have 2V,k k2= 1, andk k1\u0015p\nN. \u0003\n2.2.Discrete Schr odinger operators. Theorem 1.2 is an immediate consequence\nof the following theorem.\nTheorem 2.2. LetHbe a discrete Schr odinger operator as in (1.14) . Then for\nall0<\"\u00141\n2and boxes \u0003 = \u0003LwithL\u0015Ld;V1log1\n\"we have\n\u0011\u0003([E;E +\"])\u0014Cd;V1\nlog1\n\"for allE2R: (2.6)\nProof. Let \u0003L= \u0003L(x0) withx02Zd,E2R, and\"2]0;1\n2]. We set P=\n\u001f[E;E+\"](H\u0003L), and note that\nk(H\u0003L\u0000E) k1\u0014k(H\u0003L\u0000E) k\u0014\"k kfor all 2RanP; (2.7)\nsince we havek k1\u0014k kfor all 2`2(\u0003).\nWe assume\n\u001a:=\u0011\u0003L([E;E +\"]) =1\nj\u0003LjtrP > 0; (2.8)\nsince otherwise there is nothing to prove.\nWe \fxR22N,R<L , to be selected later, and pick G\u001a\u0003Lsuch that\n\u0003L=[\ny2G\u0003R(y) andj\u0003Lj\nj\u0003Rj\u0014#G\u00142dj\u0003Lj\nj\u0003Rj: (2.9)\nNote that ( L\u00001)d<j\u0003Lj=\u0000\n2\u0004L\n2\u0005\n+ 1\u0001d\u0014(L+ 1)d, andj\u0003Rj= (R+ 1)d. We set\n@2\u0003R(y) =\b\nx2\u0003R(y);jx\u0000yj12\bR\n2;R\n2\u00001\t\t\n; (2.10)\nand let\n@R\u0003L=[\ny2G@2\u0003R(y): (2.11)BOUNDS ON THE DENSITY OF STATES FOR SCHR ODINGER OPERATORS 7\nWe have\nj@2\u0003R(y)j\u0014cdRd\u00001;soj@R\u0003Lj\u00142dcdRd\u00001j\u0003Lj\nj\u0003Rj\u00142dcdj\u0003Lj\nR: (2.12)\nWe take\nL>2d+1cd\n\u001a+ 2; (2.13)\nsince otherwise there is nothing to prove for large L, and pick\nR2h\n2d+1cd\n\u001a;2d+1cd\n\u001a+ 2\u0011\n\\2N: (2.14)\nWe now consider the vector space\nF=f 2RanP; (x) = 0 for all x2@R\u0003Lg: (2.15)\nSinceFis the vector subspace of Ran Pde\fned byj@R\u0003Ljlinear conditions, it\nfollows from (2.8), (2.12), and (2.14) that\ndimF\u0015\u001aj\u0003Lj\u0000j@R\u0003Lj\u00151\n2\u001aj\u0003Lj\u00151: (2.16)\nLet 2Fwithk k= 1. Ify2G, it follows from (1.14), (1.15), and (2.7) that if\nwe know thatj (x)j\u0014Cfor allxwithjx\u0000yj1=k+ 1;k+ 2, then we must have\nj (x)j\u0014CA+\"forjx\u0000yj1=k, whereA= 2d\u00001 +kV\u0000Ek1. Since (x) = 0\nifjx\u0000yj1=R\n2;R\n2\u00001, we get\nj (x)j\u0014\"PR\n2\u00002\u0000jx\u0000yj1\nj=0 Aj\u0014\"\u0000R\n2\u00001\u0001\nAR\n2\u00002for allx2\u0003R\u00004(y);(2.17)\nsoj (x)j\u0014\"\u0000R\n2\u00001\u0001\nAR\n2\u00002for allx2\u0003R(y). We conclude, using (2.9), that\nk k1\u0014\"\u0000R\n2\u00001\u0001\nAR\n2\u00002: (2.18)\nWe now use Lemma 2.1, obtaining  02F,k 0k= 1, such that\nk 0k1\u0015s\ndimF\nj\u0003Lj\u0015q\n1\n2\u001a: (2.19)\n(The volumej\u0003Ljappears because the measure in Lemma 2.1 is normalized.) Com-\nbining (2.18), (2.19), and (2.14) we get\nq\n1\n2\u001a\u0014\"\u0000R\n2\u00001\u0001\nAR\n2\u00002\u0014\"2dcd\n\u001aA2dcd\n\u001a\u00001\u0014\"2dcd\n\u001aA2dcd\n\u001a; (2.20)\nwhich implies\n\u001a\u0014Cd;kV\u0000Ek1\nlog1\n\"; (2.21)\nwhich is valid when (2.14) holds, i.e., L\u0015C0\nd;kV\u0000Ek1log1\n\".\nSince\u001b(H\u0003L)\u001a[\u00002d\u0000V1;2d+V1], we have \u0011\u0003L([E;E +\"]) = 0 unless\njEj\u00142d+V1+1\n2, so we get (2.6) if L\u0015Ld;V1log1\n\". \u00038 JEAN BOURGAIN AND ABEL KLEIN\n2.3.One-dimensional Schr odinger operators. The cased= 1 of Theorem 1.1\nis an immediate consequence of the following theorem. Note that one dimensional\nboxes are intervals.\nTheorem 2.3. LetHbe a Schr odinger operator as in (1.1) withd= 1. Given\nE02R, there exists LV1;E0such that for all 0< \"\u00141\n2, open intervals \u0003 = \u0003L\nwithL\u0015LV1;E0log1\n\", and energies E\u0014E0, we have\n\u0011\u0003([E;E +\"])\u0014CV1;E0\nlog1\n\": (2.22)\nProof. Let \u0003 = \u0003 L=]a0;a0+L[,E2R,\"2]0;1\n2]. We setP=\u001f[E;E+\"](H\u0003).\nRecall that Ran P\u001f\u0003\u001aD(\u0001\u0003)\u001aC1(\u0003) sinced= 1, and note that\nk(H\u0003\u0000E) k\u0014\"k kfor all 2RanP: (2.23)\nGiven 0<R<L , setaj=a0+jRforj= 1;2;:::;\u0006L\nR\u0007\n\u00001. We introduce the\nvector space\nFR:=\u001a\n 2RanP; (aj) = 0(aj) = 0 forj= 1;2;:::;\u0018L\nR\u0019\n\u00001\u001b\n:(2.24)\nGiven 2FRandj= 1;:::;\u0006L\nR\u0007\n\u00001, it follows from Gronwall's inequality (see\n[Ho]), (aj) = 0(aj) = 0, and (2.23) that for all x2]aj\u0000R;aj+R[\\\u0003 we have\nj (x)j\u0014eKjx\u0000ajj\f\f\f\f\fZx\naje\u0000Kjy\u0000ajjj(H\u0003\u0000E) (y)jdy\f\f\f\f\f\u0014(2K)\u00001\n2eKR\"k k;(2.25)\nwhereK= 1+kV\u0000Ek1. Since \u0003 is the union of these intervals, we conclude that\nk k1\u0014(2K)\u00001\n2eKR\"k kfor all 2FR: (2.26)\nWe now assume that\n\u001a:=\u0011\u0003L([E;E +\"]) =1\nLtrP >4\nL; (2.27)\nsince otherwise there is nothing to prove for large L.. TakingR=4\n\u001a, it follows from\n(2.27) that\ndimFR\u0015\u001aL\u00002\u0000\u0006L\nR\u0007\n\u00001\u0001\n\u0015\u001aL\u00002L\nR=1\n2\u001aL> 1: (2.28)\nApplying Lemma 2.1, we obtain  02FR, 06= 0, such that\nk 0k1\u0015r\ndimFR\nLk 0k\u0015q\n1\n2\u001ak 0k: (2.29)\nIt follows from (2.26) and (2.29) that\nq\n1\n2\u001a\u0014(2K)\u00001\n2eKR\"= (2K)\u00001\n2e4K\n\u001a\": (2.30)\nThus, we get\n\u001a\u00148K\nlog1\n\"; (2.31)\nifL>4\n\u001a\u0015log1\n\"\n2K.\nSince\u001b(H\u0003)\u001a[\u0000V1;1[, we have \u0011\u0003([E;E +\"]) = 0 unless E\u0015\u0000V1\u00001\n2.\nThus, given E02R, there exists LV1;E0such that, for all 0 <\"\u00141\n2, open intervals\n\u0003 = \u0003LwithL\u0015LV1;E0log1\n\", and energies E\u0014E0, we have (2.22). \u0003BOUNDS ON THE DENSITY OF STATES FOR SCHR ODINGER OPERATORS 9\n3.Multi-dimensional Schr odinger operators\nTo prove Theorem 1.1 for d= 2;3, we will select a class of approximate eigen-\nfunctions for which we can establish uniform local upper bounds, and pick an\napproximate eigenfunction for which we have a global lower bound for the global\nupper bound. The local upper bounds will come from the local behavior of approx-\nimate solutions of the stationary Schr odinger equation (Theorem 3.1); the global\nupper bound will come from the quantitative unique continuation principle (Theo-\nrem 3.4).\nGivenx2Rdand\u000e>0, we setB(x;\u000e) :=\b\ny2Rd;jy\u0000xj<\u000e\t\n.\n3.1.Local behavior of approximate solutions of the stationary Schr odinger\nequation.\nTheorem 3.1. Let\n\u001aRdbe an open subset, where d= 2;3;:::, and \fx a real\nvalued function W2L1(\n), LetB(x0;r0)\u001a\nfor somex02Rdandr0>0.\nSupposeFis a linear subspace of H2(\n)such that\nk(\u0000\u0001 +W) kL1(B(x0;r0))\u0014CFk kL2(\n)for all 2F: (3.1)\nThen there exist constants \rd>0and0< r1=r1(d;W1)< r0, whereW1=\nkWkL1(\n), with the property that for all N2Nthere is a linear subspace FNof\nF, with\ndimFN\u0015dimF\u0000\rdNd\u00001; (3.2)\nsuch that for all  2FNwe have\nj (x)j\u0014\u0010\nCN2\nd;W1;r1jx\u0000x0jN+1+CF\u0011\nk kL2(\n)for allx2B(x0;r1):(3.3)\nWe taked= 2;3;:::, and set N0=f0g[N. We consider sites x2Rd, partial\nderivatives @j=@\n@xjforj= 1;2;:::;d , multi-indices \u000b2Nd\n0, and set\nx\u000b=dY\nj=1x\u000bj\nj; D\u000b=dY\nj=1@\u000bj\nj;j\u000bj=dX\nj=1j\u000bjj; \u000b ! =dY\nj=1\u000bj!: (3.4)\nWe letH(d)\nm=Hm(Rd) denote the vector space of homogenous harmonic polyno-\nmials on Rdof degreem2N0, and recall that [ABR, Proposition 5.8 and exercises]\nwe have dimH(d)\n0= 1, dimH(d)\n1=d, and, form= 2;3;:::,\ndimH(d)\nm=\u0012d+m\u00001\nd\u00001\u0013\n\u0000\u0012d+m\u00003\nd\u00001\u0013\n: (3.5)\nIn particular, we have\ndimH(2)\nm= 2 and dimH(3)\nm= 2m+ 1 form= 2;3;:::; (3.6)\nand dimH(d)\nm<dimH(d)\nm+1ford>2. Moreover\nlim\nm!1dimH(d)\nm\nmd\u00002=2\n(d\u00002)!ford\u00152: (3.7)\nWe also de\fneH(d)\n\u0014N=LN\nm=0H(d)\nm, the vector space of harmonic polynomials on\nRdof degree\u0014N. It follows from (3.7) that for d= 2;3;:::there exists a constant10 JEAN BOURGAIN AND ABEL KLEIN\n\rd>0 such that\ndimH(d)\n\u0014N=NX\nm=0dimH(d)\nm\u0014\rdNd\u00001for allN2N: (3.8)\nLet\n\b(x) =\bd(x) :=(\n(d(d\u00002)!d)\u00001jxj\u0000d+2ifd= 3;4;:::\n\u00001\n2\u0019logjxj ifd= 2: (3.9)\n\b(x) is the fundamental solution to Laplace's equation; !ddenotes the volume of\nthe unit ball in Rd. In particular,\n\u0000\u0001\b(x) =\u000e(x) on Rd; (3.10)\nand\njD\u000b\b(x)j\u0014Cd;j\u000bjjxj\u0000d+2\u0000j\u000bj: (3.11)\nGiven \n = B(x;r) for somex2Rdandr >0 andW2L1(\n) real valued, we\nsetW1=kWkL1(\n), and consider the the stationary Schr odinger equation\n\u0000\u0001\u001e+W\u001e= 0 a.e. on \n : (3.12)\nWe letE0(\n) =E0(\n;W) denote the vector subspace formed by solutions \u001e2\nH2(\n). We de\fne linear subspaces\nEN(\n) =\u001a\n\u001e2E0(\n); lim sup\nx!x0j\u001e(x)j\njx\u0000x0jN<1\u001b\nforN2N: (3.13)\nNote thatE1(\n) =f\u001e2E0(\n);\u001e(x0) = 0g,EN(\n)\u001bEN+1(\n) for allN2N0, and\n\\1\nN=0EN(\n) =f0gby the unique continuation principle.\nA solution of the equation (3.12) admits a local decomposition into a homoge-\nneous harmonic polynomial and a lower order term [HW, B]. The following lemma\nis a quantitative version of this decomposition; it gives an explicit estimate of the\nlower order term.\nLemma 3.2. Let\n =B(x0;3r0)for somex02Rdandr0>0,d= 2;3;:::, and\n\fx a real valued function W2L1(\n). For allN2N0there exists a linear map\nY(\n)\nN:EN(\n)!H(d)\nNsuch that for all \u001e2EN(\n)we have\n\f\f\f\u001e(x)\u0000\u0010\nY(\n)\nN\u001e\u0011\n(x\u0000x0)\f\f\f (3.14)\n\u0014r\u0000d\n2\n0\u0010\nr\u00001\n0eCd;r2\n0W1\u0011N+1\u000016\n3\u0001(N+1)(N+2)\n2((N+ 1)!)d\u00002jx\u0000x0jN+1k\u001ekL2(\n)\nfor allx2B(x0;r0\n2). As a consequence, for all N2N0we have\nEN+1(\n) = kerY(\n)\nN and dimEN+1(\n)\u0015dimEN(\n)\u0000dimH(d)\nN: (3.15)\nIn particular, ifJis a vector subspace of E0(\n)we have\ndimJ\\EN+1(\n)\u0015dimJ\u0000\rdNd\u00001for allN2N; (3.16)\nwhere\rdis the constant in (3.8) .\nProof. We prove the lemma for \n = B(0;3); the general case then follows by\ntranslating and dilating. We set \n0=B\u0000\n0;3\n2\u0001\n, and note that \u001e2E0(En=En(\n))\nsatis\fes elliptic regularity estimates:\nk\u001ekL1(\n0)\u0014Cd;W1k\u001ekL2(\n); (3.17)BOUNDS ON THE DENSITY OF STATES FOR SCHR ODINGER OPERATORS 11\nkr\u001ekL1(B(0;1))\u0014Cd;W1k\u001ekL1(\n0): (3.18)\nThe estimate (3.17) follows immediately from [GiT, Theorem 8.17]. If we knew\n\u001e2C2(\n)\\E0, the estimate (3.18) would follow directly from [GiT, Theorem 8.32].\nTo prove (3.18) for arbitrary \u001e2E0, we \fx a molli\fer \u000b2C1(Rd) (i.e.,\u000b\u00150,R\n\u000b(x) dx= 1, supp\u000b\u001aB(0;1)), let\u000bn(x) =nd\u000b(nx) forn2N, and de\fne\n\u001en=\u000bn\u0003\u001eonRd. (We extend \u001etoRdby\u001e(x) = 0 forx =2\n.) We have\n\u001en2C1(Rd) and\u001en!\u001einH2(\n0). (See [GiT, Chapter 7].) Since \u001e2E0, we\nhave\n\u0000\u0001\u001en=\u000bn\u0003(\u0000\u0001)\u001e=\u000bn\u0003((\u0000\u0001 +W)\u001e)\u0000\u000bn\u0003(W\u001e) =\u0000\u000bn\u0003(W\u001e) on \n0:\n(3.19)\nIn addition, setting \n00=B(0;5\n4), takingn\u00154, and using Young's inequality for\nconvolutions, we have\nk\u001enkL1(\n00)\u0014k\u001ekL1(\n0); (3.20)\nk(\u0000\u0001 +W)\u001enkL1(\n00)\u0014k\u000bn\u0003(W\u001e)kL1(\n00)+kW\u001enkL1(\n00)\u00142W1k\u001ekL1(\n0):\nAppealing to [GiT, Theorem 8.32], and using (3.20), we get\nkr\u001enkL1(B(0;1))\u0014Cd;W1\u0010\nk\u001enkL1(\n00)+k(\u0000\u0001 +W)\u001enkL1(\n00)\u0011\n(3.21)\n\u0014C0\nd;W1k\u001ekL1(\n0)forn\u00154:\nSince we can \fnd a subsequence \u001enksuch thatr\u001enk!r\u001ea.e. on \n0, (3.18)\nfollows from (3.21).\nGiven\u001e2E0we consider its Newtonian potential given by\n (x) =\u0000Z\n\n0W(y)\u001e(y)\b(x\u0000y) dyforx2Rd: (3.22)\nIn view of (3.17), we have\nj (x)j\u0014W1k\u001ekL1(\n0)k\bkL1(\n)\u0014Cd;W1W1k\u001ekL2(\n)for allx2\n0:(3.23)\nIt follows from (3.10) that \u0001  =W\u001e weakly in \n0. Thus, letting h=\u001e\u0000 \nwe have \u0001h= 0 weakly in \n0, so we conclude that his a harmonic function in\n\n0\u001bB(0;1). In particular (see [ABR, Corollary 5.34 and its proof]), his real\nanalytic in \n0and\nh(x) =1X\nm=0pm(x) for all x2B(0;1); (3.24)\nwherepm2H(d)\nmfor allm= 0;1;:::, and form= 1;2;:::we have\njpm(x)j\u0014Cdmd\u00002jxjmsup\ny2@B(0;1)jh(y)jfor allx2B(0;1): (3.25)\nIn addition, it follows from the mean value property that for all y2@B(0;1) we\nhave\njh(y)j\u00141\njB(y;1\n2)jZ\nB(y;1\n2)jh(y0)jdy0\u0014Cd;W1k\u001ekL2(\n); (3.26)\nusing (3.17) and (3.23). Thus, for all m= 1;2;:::it follows from (3.25) that\njpm(x)j\u0014Cd;W1md\u00002k\u001ekL2(\n)jxjmfor allx2B(0;1): (3.27)12 JEAN BOURGAIN AND ABEL KLEIN\nSettinghN=PN\nm=0pm(x)2H(d)\n\u0014N, it follows that\njh(x)\u0000hN(x)j\u0014Cd;W1k\u001ekL2(\n)(N+ 1)d\u00002jxjN+1for allx2B\u0000\n0;1\n2\u0001\n:\n(3.28)\nFor eachy2Rdnf0gwe consider \by(x) =\b(x\u0000y), a harmonic function on\nRdnfyg. In particular, \by(x) is real analytic in B(0;jyj), so, de\fning\nJm(x;y) =X\n\u000b2Nd\n0;j\u000bj=m1\n\u000b!D\u000b\b(y)x\u000bforx2Rd; (3.29)\nwe have (see [ABR])\n\b(x\u0000y) =\by(x) =1X\nm=0Jm(x;y) for all x2B(0;jyj); (3.30)\nthe series converging absolutely and uniformly on compact subsets of B(0;jyj).\nMoreover,Jm(\u0001;y)2H(d)\nm, and for all y2Zdandm= 1;2;:::we have (see [ABR,\nCorollary 5.34 and its proof]) that\njJm(x;y)j\u0014Cdmd\u00002\u0010\n4jxj\n3jyj\u0011m\nsup\nx02@B\u0010\n0;3\n4jyj\u0011j\by(x0)j\u0014Cdmd\u00002\u0010\n4jxj\n3jyj\u0011m\n\b(y\n4);\n(3.31)\nfor allx2B\u0000\n0;3\n4jyj\u0001\n. Setting \by;N(x) =PN\nm=0Jm(x;y)2H(d)\n\u0014N, it follows that\nforx2B(0;1\n2jyj) we have\nj\by(x)\u0000\by;N(x)j\u0014Cd(N+ 1)d\u00002\u0010\n4jxj\n3jyj\u0011N+1\n\b(y\n4): (3.32)\nWe now proceed by induction. We de\fne Y0:E0!H(d)\n0byY0\u001e=\u001e(0). Given\n\u001e2E0, it follows from the mean value theorem and the elliptic regularity estimates\n(3.17) and (3.18) that\nj\u001e(x)\u0000\u001e(0)j\u0014 sup\ny2B(0;1)jr\u001e(y)jjxj\u0014Cd;W1k\u001ekL2(\n)jxjforx2B(0;1):\n(3.33)\nThus the lemma holds for N= 0.\nWe now let N2Nand suppose that the lemma is valid for N\u00001. If\u001e2EN, it\nfollows that \u001e2EN\u00001withYN\u00001\u001e= 0, so by the induction hypothesis\nj\u001e(x)j\u0014CNk\u001ekL2(\n)jxjNfor allx2B\u0000\n0;1\n2\u0001\n; (3.34)\nwhere\nCN=eCN\nd;W1\u000016\n3\u0001N(N+1)\n2(N!)d\u00002: (3.35)\nUsing (3.31) and (3.34), we de\fne\n N(x) =\u0000Z\n\n0W(y)\u001e(y)\by;N(x) dy2H(d)\n\u0014N: (3.36)\nWe \fxx2B\u0000\n0;1\n2\u0001\nand estimate\nj (x)\u0000 N(x)j\u0014W1Z\n\n0j\u001e(y)jj\by;>N(x)jdy; (3.37)BOUNDS ON THE DENSITY OF STATES FOR SCHR ODINGER OPERATORS 13\nwhere \by;>N(x) =\by(x)\u0000\by;N(x). Appealing to (3.32) and (3.34), we get\nZ\nB(0;1\n2)nB(0;2jxj)j\u001e(y)jj\by;>N(x)jdy\u0014CdCNk\u001ekL2(\n)(N+ 1)d\u00002\u00004\n3\u0001N+1jxjN+1:\n(3.38)\nIfy =2B(0;2jxj) we havejyj\u00152jxj\u00151, and hence, using (3.32),\nZ\n\n0n\u0010\nB(0;2jxj)[B(0;1\n2)\u0011j\u001e(y)jj\by;>N(x)jdy (3.39)\n\u0014Cd(N+ 1)d\u00002\u00004\n3\u0001N+1\b(1\n4)jxjN+1Z\n\n0j\u001e(y)jdy\n\u0014Cd(N+ 1)d\u00002\u00004\n3\u0001N+1jxjN+1k\u001ekL2(\n):\nUsing (3.31) and (3.34), we get\nZ\nB(0;1\n2)\\B(0;2jxj)j\u001e(y)jj\by;>N(x)jdy (3.40)\n\u0014CNk\u001ekL2(\n)Z\nB(0;1\n2)\\B(0;2jxj)jyjNj\by;>N(x)jdy\n\u0014CNk\u001ekL2(\n)Z\nB(0;1\n2)\\B(0;2jxj)jyjNj\b(x\u0000y)jdy\n+CdCNk\u001ekL2(\n)NX\nm=0md\u00002\u00004\n3jxj\u0001mZ\nB(0;1\n2)\\B(0;2jxj)jyjN\u0000m\f\f\b(y\n4)\f\fdy\n\u0014CdCNk\u001ekL2(\n)\u0010\n1 +Nd\u00002\u00004\n3\u0001N+1\u0011\njxjN+1;\nwhere we usedjx\u0000yj\u00143jxjfory2B(0;2jxj). (Note that we get jxjN+2ifd\u00153\nandjxj(N+2)\u0000ifd= 2.) Also using (3.31), we get\nZ\n\n0nB(0;1\n2)j\u001e(y)jj\by;>N(x)jdy\u0014Z\n\n0nB(0;1\n2)j\u001e(y)jj\b(x\u0000y)jdy (3.41)\n+CdNX\nm=0md\u00002\u00004\n3jxj\u0001mZ\n\n0nB(0;1\n2)j\u001e(y)jjyj\u0000m\f\f\b(y\n4)\f\fdy\n\u0014Cdk\u001ekL2(\n)\u0010\n1 +Nd\u00002\u00004\n3\u0001N+1\u0011\n;\nwhere we usedjxj\u00141\n2. Sincejxj>1\n4ify2B(0;2jxj)nB(0;1\n2), we obtain\nZ\n(\n0\\B(0;2jxj))nB(0;1\n2)j\u001e(y)jj\by;>N(x)jdy (3.42)\n\u0014Cdk\u001ekL2(\n)\u0010\n1 +Nd\u00002\u000016\n3\u0001N+1\u0011\njxjN+1:\nPutting together (3.37), (3.38), (3.39), (3.40), and (3.42), we conclude that for all\nx2B(0;1\n2) we have ( CN\u00151)\nj (x)\u0000 N(x)j\u0014CdCNW1(N+ 1)d\u00002\u000016\n3\u0001N+1jxjN+1k\u001ekL2(\n): (3.43)14 JEAN BOURGAIN AND ABEL KLEIN\nWe now de\fne YN\u001e=hN+ N2H(d)\nN. Since\u001e=h+ , for allx2B\u0000\n0;1\n2\u0001\nit\nfollows from (3.28), (3.43), and (3.35), that\nj\u001e(x)\u0000(YN\u001e) (x)j\u0014jh(x)\u0000hN(x)j+j (x)\u0000 N(x)j (3.44)\n\u0014(Cd;W1+CdW1CN) (N+ 1)d\u00002\u000016\n3\u0001N+1jxjN+1k\u001ekL2(\n)\n\u0014eCd;W1CN(N+ 1)d\u00002\u000016\n3\u0001N+1jxjN+1k\u001ekL2(\n)\n\u0014eCd;W1\u0012\neCN\nd;W1\u000016\n3\u0001N(N+1)\n2(N!)d\u00002\u0013\n(N+ 1)d\u00002\u000016\n3\u0001N+1jxjN+1k\u001ekL2(\n)\n\u0014eCN+1\nd;W1\u000016\n3\u0001(N+1)(N+2)\n2((N+ 1)!)d\u00002jxjN+1k\u001ekL2(\n);\nby choosing the constant eCd;W1in (3.35) large enough. This completes the induc-\ntion.\nThe lemma is proven, as (3.15) is an immediate consequence of (3.14), and (3.16)\nfollows from (3.15) and (3.8). \u0003\nTheorem 3.1 is an immediate consequence from the following lemma.\nLemma 3.3. Let\n\u001aRdbe an open subset, where d= 2;3;:::, and \fx a real\nvalued function W2L1(\n). LetB(x0;r1)\u001a\nfor somex02Rdandr1>0.\nSupposeFis a linear subspace of H2(\n)such that\nk(\u0000\u0001 +W) kL1(B(x0;r1))\u0014CFk kL2(\n)for all 2F: (3.45)\nThen there exists 0< r2=r2(d;W1)< r1, whereW1=kWkL1(\n), with the\nproperty that for all r2]0;r2]there is a linear map Zr:F!E 0(B(x0;r))such that\nk \u0000Zr kL1(B(x0;r))\u0014Cd;rCFk kL2(\n)where lim\nr!0Cd;r= 0: (3.46)\nAs a consequence, for all N2Nthere is a vector subspace FNofF, with\ndimFN\u0015dimF\u0000\rdNd\u00001; (3.47)\nwhere\rdis the constant in (3.8) , such that for all  2FNwe have\nj (x)j\u0014\u0010\nbCN+1\nd;W1;r1((N+ 1)!)d\u000023N2jx\u0000x0jN+1+CF\u0011\nk kL2(\n) (3.48)\n\u0014\u0010\nCN2\nd;W1;r1jx\u0000x0jN+1+CF\u0011\nk kL2(\n)\nfor allx2B(x0;r2\n4).\nProof. It su\u000eces to consider x0= 0. We set Br=B(0;r). Given 0 <r <r 1and\n 2F, we de\fne Zr 2E0(Br) as the unique solution \u001e2H2(Br) to the Dirichlet\nproblem on Brgiven by(\n\u0000\u0001\u001e+W\u001e= 0 onBr\n\u001e= on@Br: (3.49)\nThis map is well de\fned in view of [GiT, Theorem 8.3] and is clearly a linear map.\nTo prove (3.46) we will use the Green's function Gr(x;y) for the ball Br. We\nrecall that, abusing the notation by writing \b(jxj) instead of \b(x) (see [GiT, Sec-\ntion 2.5]; note that with our de\fnition \b(x) =\u0000\u0000(jxj)),\nGr(x;y) =(\n\b(jx\u0000yj)\u0000\b\u0010\njyj\nr\f\f\fx\u0000r2\njyj2y\f\f\f\u0011\nify6= 0\n\b(jxj)\u0000\b(r) if y= 0: (3.50)BOUNDS ON THE DENSITY OF STATES FOR SCHR ODINGER OPERATORS 15\nUsing Green's representation formula [GiT, Eq. (2.21)] for  andZr , for allx2Br\nwe have\n (x) =\u0000Z\n@Br (\u0010)@\u0017Gr(x;\u0010)dS(\u0010)\u0000Z\nBrW(y) (y)Gr(x;y)dy (3.51)\n+Z\nBr(\u0000\u0001 +W(y)) (y)Gr(x;y)dy;\nZr (x) =\u0000Z\n@Br (\u0010)@\u0017Gr(x;\u0010)dS(\u0010)\u0000Z\nBrW(y)Zr (y)Gr(x;y)dy; (3.52)\nwhere dSdenotes the surface measure and @\u0017is the normal derivative. Since by an\nexplicit calculation we have, with p2= 2 andpd=d\u00001\nd\u00002ford\u00153, that for all x2Br\nkGr(x;\u0001)kL1(Br)\u0014C0\ndrd(pd\u00001)\npdkGr(x;\u0001)kLpd(Br)\u0014Cdrd(pd\u00001)\npd; (3.53)\nit follows that\nk \u0000Zr kL1(Br)(3.54)\n\u0014Cdrd(pd\u00001)\npd\u0010\nW1k \u0000Zr kL1(Br)+k(\u0000\u0001 +W) kL1(Br)\u0011\n:\nSelectingr22]0;r1[ such that Cdrd(pd\u00001)\npd\n2 (1 +W1)\u00141\n2, and using (3.45), we get\n(3.46).\nNow letJ= RanZr2, a linear subspace of E0(Br2); note that\ndimJ+ dim kerZr2= dimF: (3.55)\nWe setJN=J\\EN+1(Br2) andFN=Z\u00001\nr2(JN). It follows from (3.16) and (3.55)\nthat\ndimFN= dim kerZr2+ dimJN\u0015dimF\u0000\rdNd\u00001: (3.56)\nIf 2FN, we haveZr2 2EN+1(Br2) and\nk kL1(Br2)\u0014k \u0000Zr2 kL1(Br2)+kZr2 kL1(Br2); (3.57)\nso (3.48) follows from (3.46) and (3.14). \u0003\n3.2.A quantitative unique continuation principle for approximate so-\nlutions of the stationary Schr odinger equation. We state and prove a a\nversion of Bourgain and Kenig's quantitative unique continuation principle [BoK,\nLemma 3.10], in which we make explicit the dependence on the parameters relevant\nto this article. We give a proof following [GK2, Theorem A.1].\nGiven subsets AandBofRd, and a function 'on setB, we set'A:='\u001fA\\B.\nIn particular, given x2Rdand\u000e>0 we write'x;\u000e:='B(x;\u000e).\nTheorem 3.4. Let\nbe an open subset of Rdand consider a real measurable\nfunctionVon\nwithkVk1\u0014K <1. Let 2H2(\n) be real valued and let\n\u00102L2(\n)be de\fned by\n\u0000\u0001 +V =\u0010a.e. on \n: (3.58)\nLet\u0002\u001a\nbe a bounded measurable set where k \u0002k2>0. Set\nQ(x;\u0002) := sup\ny2\u0002jy\u0000xjforx2\n: (3.59)16 JEAN BOURGAIN AND ABEL KLEIN\nConsiderx02\nn\u0002such that\nQ=Q(x0;\u0002)\u00151andB(x0;6Q+ 2)\u001a\n: (3.60)\nThen, given\n0<\u000e\u0014min\b\ndist (x0;\u0002);1\n24\t\n; (3.61)\nwe have\n\u0012\u000e\nQ\u0013m\u0010\n1+K2\n3\u0011\u0012\nQ4\n3+logk \nk2\nk \u0002k2\u0013\nk \u0002k2\n2\u0014k x0;\u000ek2\n2+\u000e2k\u0010\nk2\n2; (3.62)\nwherem> 0is a constant depending only on d.\nWe we will apply this theorem with \u000e\u001c1\u001cQ.\nThe proof of this theorem is based on the the Carleman-type inequality estimate\ngiven in [BoK, Lemma 3.15], [EV, Theorem 2], We state it as in [GK2, Lemma A.5].\nLemma 3.5. Given% >0, the function w%(x) ='(1\n%jxj)onRd, where'(s) :=\nse\u0000Rs\n01\u0000e\u0000t\ntdt, is a strictly increasing continuous function on [0;1[,C1on]0;1[,\nsatisfying\n1\nC1%jxj\u0014w%(x)\u00141\n%jxjforx2B(0;%);whereC1='(1)\u000012]2;3[:(3.63)\nMoreover, there exist positive constants C2andC3, depending only on d, such\nthat for all \u000b\u0015C2and all real valued functions f2H2(B(0;%))with suppf\u001a\nB(0;%)nf0gwe have\n\u000b3Z\nRdw\u00001\u00002\u000b\n%f2dx\u0014C3%4Z\nRdw2\u00002\u000b\n%(\u0001f)2dx: (3.64)\nProof of Theorem 3.4. Letx02\nn\u0002 satisfy (3.60), where C1is de\fned in (3.63).\nFor convenience we may assume x0= 0, in which case \u0002 \u001aB(0;2C1Q), and take\n\n =B(0;%), where%= 2C1Q+ 2.\nLet\u000ebe as in (3.61), and \fx a function \u00112C1\nc(Rd) given by \u0011(x) =\u0018(jxj),\nwhere\u0018is an evenC1function on R, 0\u0014\u0018\u00141, such that\n\u0018(s) = 1 if3\u000e\n4\u0014jsj\u00142C1Q; \u0018 (s) = 0 ifjsj\u0014\u000e\n4orjsj\u00152C1Q+ 1; (3.65)\n\f\f\f\u0018(j)(s)\f\f\f\u0014\u00004\n\u000e\u0001jifjsj\u00143\u000e\n4;\f\f\f\u0018(j)(s)\f\f\f\u00142jifjsj\u00152C1Q; j = 1;2:\nNote thatjr\u0011(x)j\u0014p\ndj\u00180(jxj)jandj\u0001\u0011(x)j\u0014dj\u001800(jxj)j.\nWe will now apply Lemma 3.5 to the function \u0011 . In what follows C1;C2;C3\nare the constants of Lemma 3.5, which depend only on d. ByCj,j= 4;5;:::, we\nwill always denote an appropriate nonzero constant depending only on d.\nGiven\u000b\u0015C2>1 (without loss of generality we take C2>1), it follows from\n(3.64) that\n\u000b3\n3C3%4Z\nRdw\u00001\u00002\u000b\n%\u00112 2dx\u00141\n3Z\nRdw2\u00002\u000b\n%(\u0001(\u0011 ))2dx\u0014Z\nRdw2\u00002\u000b\n%\u00112(\u0001 )2dx\n+ 4Z\nsuppr\u0011w2\u00002\u000b\n%jr\u0011j2jr j2dx+Z\nsuppr\u0011w2\u00002\u000b\n%(\u0001\u0011)2 2dx;(3.66)\nwhere suppr\u0011\u001a\b\u000e\n4\u0014jxj\u00143\u000e\n4\t\n[f2C1Q\u0014jxj\u00142C1Q+ 1g.BOUNDS ON THE DENSITY OF STATES FOR SCHR ODINGER OPERATORS 17\nUsing (3.58), recalling kVk1\u0014K, and noting that w%\u00141 on supp\u0011, we get\nZ\nRdw2\u00002\u000b\n%\u00112(\u0001 )2dx\u00142K2Z\nRdw\u00001\u00002\u000b\n%\u00112 2dx+ 2Z\nRdw2\u00002\u000b\n%\u00112\u00102dx:(3.67)\nWe take\n\u000b0:=\u000b\u001a\u00004\n3\u0015C4\u0010\n1 +K2\n3\u0011\n; (3.68)\nensuring\u000b>C 2and\n\u000b3\n3C3%4=\u000b3\n0\n3C3\u00156K2: (3.69)\nAs a consequence, using (3.63) and recalling (3.59), we obtain\nZ\nRdw\u00001\u00002\u000b\n%\u00112 2dx\u0015\u0012%\nQ\u00131+2\u000b\nk \u0002k2\n2\u0015(2C1)1+2\u000bk \u0002k2\n2: (3.70)\nCombining (3.66), (3.67), (3.69), and (3.70), we conclude that\n2\u000b3\n0\n9C3(2C1)1+2\u000bk \u0002k2\n2\u00144Z\nsuppr\u0011w2\u00002\u000b\n%jr\u0011j2jr j2dx (3.71)\n+Z\nsuppr\u0011w2\u00002\u000b\n%(\u0001\u0011)2 2dx+ 2Z\nsupp\u0011w2\u00002\u000b\n%\u00112\u00102dx:\nWe haveZ\nf2C1Q\u0014jxj\u00142C1Q+1gw2\u00002\u000b\n%\u0010\n4jr\u0011j2jr j2+ (\u0001\u0011)2 2\u0011\ndx (3.72)\n\u00144d2\u0012C1%\n2C1Q\u00132\u000b\u00002Z\nf2C1Q\u0014jxj\u00142C1Q+1g\u0010\n4jr j2+ 2\u0011\ndx\n\u0014C5\u00005\n4C1\u00012\u000b\u00002Z\nf2C1Q\u00001\u0014jxj\u00142C1Q+2g\u0000\n\u00102+ (1 +K) 2\u0001\ndx\n\u0014C5\u00005\n4C1\u00012\u000b\u00002\u0010\nk\u0010\nk2\n2+ (1 +K)k \nk2\n2\u0011\n;\nwhere we used an interior estimate (e.g., [GK1, Lemma A.2]). Similarly,\nZ\nf\u000e\n4\u0014jxj\u00143\u000e\n4gw2\u00002\u000b\n%\u0010\n4jr\u0011j2jr j2+ (\u0001\u0011)2 2\u0011\ndx (3.73)\n\u001416d2\u000e\u00002\u0000\n4\u000e\u00001C1%\u00012\u000b\u00002Z\nf\u000e\n4\u0014jxj\u00143\u000e\n4g\u0010\n4jr j2+ 2\u0011\ndx\n\u0014C6\u000e\u00002\u0000\n4\u000e\u00001C1%\u00012\u000b\u00002Z\nfjxj\u0014\u000eg\u0000\n\u00102+ (K+\u000e\u00002) 2\u0001\ndx\n\u0014C6\u000e\u00002\u0000\n16\u000e\u00001C2\n1Q\u00012\u000b\u00002\u0010\nk\u0010\nk2\n2+ (K+\u000e\u00002)k 0;\u000ek2\n2\u0011\n:\nIn addition,\nZ\nsupp\u0011w2\u00002\u000b\n%\u00112\u00102dx\u0014\u0000\n4\u000e\u00001C1%\u00012\u000b\u00002k\u0010\nk2\n2\u0014\u0000\n16\u000e\u00001C2\n1Q\u00012\u000b\u00002k\u0010\nk2\n2:(3.74)\nThus, if we have\n\u000b3\n0\u00008\n5\u00012\u000bk \u0002k2\n2\u0015C7(1 +K)k \nk2\n2; (3.75)18 JEAN BOURGAIN AND ABEL KLEIN\nwe obtain\nC5\u00005\n4C1\u00012\u000b\u00002(1 +K)k \nk2\n2\u00141\n22\u000b3\n0\n9C3(2C1)1+2\u000bk \u0002k2\n2; (3.76)\nso we conclude that\n\u000b3\n0\n9C3(2C1)1+2\u000bk \u0002k2\n2\u0014C8\u000e\u00002\u0000\n16\u000e\u00001C2\n1Q\u00012\u000b\u00002\u0010\n(K+\u000e\u00002)k 0;\u000ek2\n2+k\u0010\nk2\n2\u0011\n;\n(3.77)\nwhere we used (3.61). Thus,\n\u000b3\n0Q2\u0010\n(8C1Q)\u00001\u000e\u00112\u000b\nk \u0002k2\n2\u0014C9\u0010\n(K+\u000e\u00002)k 0;\u000ek2\n2+k\u0010\nk2\n2\u0011\n; (3.78)\nwhich implies\n\u000b3\n0Q4\u0012\u000e\nQ\u00134\u000b+4\nk \u0002k2\n2\u0014C10\u0010\n(1 +K)k 0;\u000ek2\n2+\u000e2k\u0010\nk2\n2\u0011\n; (3.79)\nsince\u000e\nQ\u00141\n24<1\n8C1by (3.61).\nWe now choose \u000b. Requiring (3.68), to satisfy (3.75) it su\u000eces to also require\n\u000b\u0015C11\u0012\n1 + logk \nk2\nk \u0002k2\u0013\n: (3.80)\nThus we can satisfy (3.68) and (3.75) by taking\n\u000b=C12\u0010\n1 +K2\n3\u0011\u0012\nQ4\n3+ logk \nk2\nk \u0002k2\u0013\n: (3.81)\nCombining with (3.79), and recalling Q\u00151, we get\n\u0010\n1 +K2\n3\u00113\u0012\u000e\nQ\u0013C13\u0010\n1+K2\n3\u0011\u0012\nQ4\n3+logk \nk2\nk \u0002k2\u0013\nk \u0002k2\n2(3.82)\n\u0014C14\u0010\n(1 +K)k 0;\u000ek2\n2+\u000e2k\u0010\nk2\n2\u0011\n;\nand hence,\n\u0012\u000e\nQ\u0013m\u0010\n1+K2\n3\u0011\u0012\nQ4\n3+logk \nk2\nk \u0002k2\u0013\nk \u0002k2\n2\u0014k 0;\u000ek2\n2+\u000e2k\u0010\nk2\n2; (3.83)\nwherem> 0 is a constant depending only on d. \u0003\nWe will apply Theorem 3.4 to approximate eigenfunctions of Schr odinger opera-\ntors de\fned on a box \u0003 with Dirichlet boundary condition. In this case the second\ncondition in (3.60) seems to restrict the application of Theorem 3.4 to sites x02\u0003\nsu\u000eciently far away from the boundary of \u0003. But, as noted in [GK2, Corollary A.2],\nin this case Theorem 3.4 can be extended to sites near the boundary of \u0003 as in the\nfollowing corollary.\nCorollary 3.6. Consider the Schr odinger operator H\u0003:=\u0000\u0001\u0003+VonL2(\u0003),\nwhere \u0003 = \u0003L(x0)is the open box of side L > 0centered at x02Rd,\u0001\u0003is\nthe Laplacian with either Dirichlet or periodic boundary condition on \u0003, andV\na is bounded potential on \u0003withkVk1\u0014K <1. Let 2D(\u0001\u0003)and \fx a\nbounded measurable set \u0002\u001a\u0003wherek \u0002k2>0. SetQ(x;\u0002) := supy2\u0002jy\u0000xjBOUNDS ON THE DENSITY OF STATES FOR SCHR ODINGER OPERATORS 19\nforx2\u0003, and consider x02\u0003n\u0002such thatQ=Q(x0;\u0002)\u00151. Then, given\n0<\u000e\u0014min\b\ndist (x0;\u0002);1\n24\t\nsuch thatB(x0;\u000e)\u001a\u0003, we have\n\u0012\u000e\nQ\u0013m\u0010\n1+K2\n3\u0011\u0012\nQ4\n3+logk k2\nk \u0002k2\u0013\nk \u0002k2\n2\u0014k x0;\u000ek2\n2+\u000e2kH\u0003 k2\n2; (3.84)\nwherem> 0is a constant depending only on d.\nThis corollary is proved exactly as [GK2, Corollary A.2].\n3.3.Two and three dimensional Schr odinger operators.\nTheorem 3.7. LetHbe a Schr odinger operator as in (1.1) , whered= 2;3. Given\nE02R, there exists Ld;V1;E0such that for all 0<\"\u00141\n2, open boxes \u0003 = \u0003Lwith\nL\u0015Ld;V1;E0\u0000\nlog1\n\"\u00013\n8, and energies E\u0014E0, we have\n\u0011\u0003([E;E +\"])\u0014Cd;V1;E0\u0000\nlog1\n\"\u00014\u0000d\n8: (3.85)\nProof. We \fx\"2]0;1\n2], letL\u0015L0(\"), whereL0(\")>0 will be speci\fed later, and\ntake a box \u0003 = \u0003 L. Since\u001b(H\u0003)\u001a[\u0000V1;1[, it su\u000eces to consider E0\u0015\u0000V1\u00001\nandE2[\u0000V1\u00001;E0]. We setP=\u001f[E;E+\"](H\u0003); note that Ran P\u001aD(\u0001\u0003)\u001a\nH2(\u0003) and\nk(H\u0003\u0000E) k\u0014\"k kfor all 2RanP: (3.86)\nMoreover, for  2RanPwe have\nk k1=\r\r\re\u0000(H\u0003+V1)e(H\u0003+V1) \r\r\r\n1(3.87)\n\u0014\r\r\re\u0000(H\u0003+V1)\r\r\r\nL2(\u0003)!L1(\u0003)\r\r\re(H\u0003+V1) \r\r\r\u0014CdeE0+V1+1k k;\nwhere we used that for t>0\r\r\re\u0000t(H\u0003+V1)\r\r\r\nL2(\u0003)!L1(\u0003)\u0014\r\ret\u0001\u0003\r\r\nL2(\u0003)!L1(\u0003)\u0014\r\ret\u0001\r\r\nL2(Rd)!L1(Rd)<1:\n(3.88)\nSinceP(H\u0003\u0000E) = (H\u0003\u0000E)P = (H\u0003\u0000E) for 2RanP, we conclude\nthat\nk(H\u0003\u0000E) k1\u0014\"Cd;V1;E0k kfor all 2RanP: (3.89)\nLet\n\u001a:=\u0011\u0003([E;E +\"]) =1\nLdtrP: (3.90)\nRecalling the estimate tr P\u0014Cd;V1;E0Ld(e.g., [GK1, Eq. (A.7)]), where Cd;V1;E0\u0015\n1, we obtain the uniform upper bound\n\u001a\u0014\u001aub:=Cd;V1;E0with\u001aub\u00151: (3.91)\nWe assume that\nLd>23d+1\rd\u001aub\n\u001a; (3.92)\nsince otherwise there is nothing to prove for Lsu\u000eciently large. Here \rdis the\nconstant in Theorem 3.1; we assume 2d\rd\u00151 without loss of generality. We take\nRsuch that\n2d+1\rd\u001aub\n\u001a\u0014Rd<\u0000L\n4\u0001d; (3.93)\nnote that\n2\u0014\u001aRdand 2\u0014Rd: (3.94)20 JEAN BOURGAIN AND ABEL KLEIN\nIn particular, it follows from (3.91) and (3.93) that\nN:=\u0016\u0010\n\u001a\n2d+1\rd\u00111\nd\u00001Rd\nd\u00001\u0017\n\u0015\u0016\n\u001a1\nd\u00001\nub\u0017\n\u00151: (3.95)\nWe pickG\u001a\u0003 such that\n\u0003 =[\ny2G\u0003R(y) and #G=\u0000\ndL\nRe\u0001d2h\u0000L\nR\u0001d;\u00002L\nR\u0001di\n\\N: (3.96)\nGiveny12G, we apply Theorem 3.1 with \n = \u0003 \u001bB(y1;1),W=V\u0000E, and\nF= RanP. The hypothesis (3.1) follows from (3.89). We conclude that there\nexists a vector subspace Fy1;Nof RanPandr0=r0(d;V1;E0)2(0;1), such that,\nusing also (3.95) and (3.93), we have\ndimFy1;N\u0015\u001aLd\u0000\rdNd\u00001\u00151; (3.97)\nand for all 2Fy1;Nwe have\nj (y1+x)j\u0014\u0010\nCN2\nd;V1;E0jxjN+1+\"Cd;V1;E0\u0011\nk kifjxj<r0: (3.98)\nPickingy22 G,y26=y1, and applying Theorem 3.1 with \n = \u0003 \u001bB(y2;1),\nW=V\u0000E, andF=Fy1;N, we obtain a vector vector subspace Fy1;y2;NofFy1;N,\nand hence of Ran P, such that\ndimFy1;y2;N\u0015dimFy1;N\u0000\rdNd\u00001\u0015\u001aLd\u00002\rdNd\u00001\u00151; (3.99)\nand (3.98) holds for all  2Fy1;y2;Nalso withy2substituted for y1. Repeating this\nprocedure until we exhaust the sites in G, we conclude that there exists a vector\nsubspaceFRof RanPandr0=r0(d;V1;E0)2(0;1), such that\ndimFR\u0015\u001aLd\u0000\u00002L\nR\u0001d\rdNd\u00001\u00151\n2\u001aLd\u001523d\rd\u001aub\u00151; (3.100)\nwhere we used the assumption (3.92), and for all  2FRandy2Gwe have\nj (y+x)j\u0014\u0010\nCN2\nd;V1;E0jxjN+1+\"Cd;V1;E0\u0011\nk kifjxj<r0: (3.101)\nWe letQRdenote the orthogonal projection onto FR. Since trQR= dimFR, it\nfollows from (3.100) that that we can \fnd a box \u0003 1= \u0003 1(x1)\u001a\u0003 such that\ntr\u001f\u00031QR\u001f\u00031\u0015(dLe)\u0000d1\n2\u001aLd\u00152\u0000(d+1)\u001a: (3.102)\nButQR=QRP=PQRsinceFR\u001aRanP, and hence\n2\u0000(d+1)\u001a\u0014tr\u001f\u00031QR\u001f\u00031= tr\u001f\u00031PQR\u001f\u00031\u0014k\u001f\u00031Pk1kQR\u001f\u00031k: (3.103)\nRecall that\nk\u001f\u00031Pk1=kP\u001f\u00031k1\u0014Cd;V1;E0: (3.104)\n(This is [Si1, Theorem B.9.2] when \u0003 = Rd. But by an argument similar to (3.88)\nthe crucial estimate [Si1, Eq. (B11)] holds on \fnite boxes \u0003 with constants uniform\nin \u0003, so a careful reading of the proof of [Si1, Theorem B.9.2] shows that the result\nholds on \fnite boxes \u0003 with constants uniform in \u0003.). We thus conclude that\nkQR\u001f\u00031k\u0015C0\nd;V1;E0\u001awithC0\nd;V1;E0>0; (3.105)\nso there exists  0=QR 0withk 0k= 1 such that\nk\u001f\u00031 0k\u0015\r\u001a; where\r=1\n2C0\nd;V1;E0>0: (3.106)\n(Note that \r\u001a<k 0k= 1.)BOUNDS ON THE DENSITY OF STATES FOR SCHR ODINGER OPERATORS 21\nWe picky02Gsuch that\n1\n2\u00141\n4R\u0014dist (y0;\u00031)\u00145p\nd\n2R; (3.107)\nwhich can done by our construction, and apply Corollary 3.6 with x0=y0, \u0002 = \u0003 1,\nand potential V\u0000E; note that (recall (3.59))\nR\n4+p\nd\u0014Q=Q(y0;\u00031)\u00145p\nd\n2R+p\nd\u00143p\ndR: (3.108)\nLet 0< \u000e < \u000e 0:= min\b1\n24;r0\t\n, wherer0is as in (3.101). It follows from Corol-\nlary 3.6, using (3.86), that\n\u0010\n\u000e\n3p\ndR\u0011m\u0010\n1+K2\n3\u0011\u0010\nR4\n3\u0000logk 0\u001f\u00031k2\u0011\nk 0\u001f\u00031k2\n2\u0014\r\r 0\u001fB(y0;\u000e)\r\r2\n2+\"2;(3.109)\nwith a constant m=md>0 andK=kV\u0000Ek1. Using (3.101) and (3.106), we\nget\n\u0010\n\u000e\n3p\ndR\u0011m\u0010\n1+K2\n3\u0011\u0010\nR4\n3\u0000log(\r\u001a)\u0011\n(\r\u001a)2\u0014CdCN2\nd;V1;E0\u000e2(N+1)+d+Cd;V1;E0\"2:\n(3.110)\nSince\u001a\u00152R\u0000dand\u000e\n3p\ndR<\u000e\n3p\nd<1 by (3.94), the inequality (3.110) implies\nthe existence of strictly positive constants eR=eRd;V1;E0andM=Md;V1;E0such\nthat\n\u0000\u000e\nR\u0001MR4\n3\u0014CN2\nd;V1;E0\u000e2N+Cd;V1;E0\"2forR\u0015eR: (3.111)\nWe require\nR>bR= maxn\neR;\u000e\u00001\n0o\n; (3.112)\nand choose \u000eby (noteCN\nd;V1;E0\u00151)\n\u000e=\u0000\nCN\nd;V1;E0R\u0001\u00001<\u000e0;so\u000e\nR=CN\nd;V1;E0\u000e2=\u0000\nCN\nd;V1;E0R2\u0001\u00001;(3.113)\nobtaining\n\u0000\u000e\nR\u0001MR4\n3\u0014\u0000\u000e\nR\u0001N+Cd;V1;E0\"2: (3.114)\nWe now take d= 2;3 and take Rlarge enough so that\n\u0000\u000e\nR\u0001N\u00141\n2\u0000\u000e\nR\u0001MR4\n3;i.e.,\u0000\nCN\nd;V1;E0R2\u0001N\u0000MR4\n3\u00152: (3.115)\nTo see this, note that4\n3<d\nd\u00001ford= 2;3, so\nMR4\n3<N =\u0016\u0010\n\u001a\n2d+1\rd\u00111\nd\u00001Rd\nd\u00001\u0017\nif\u001a>C0\nd;V1;E0Rd\u00004\n3; (3.116)\nand hence\n\u0000\nCN\nd;V1;E0R2\u0001N\u0000MR4\n3\u00154N\u0000MR4\n3\u00152 if\u001a>C00\nd;V1;E0Rd\u00004\n3: (3.117)\nWe now choose Rby\n\u001a=cd;V1;E0Rd\u00004\n3; (3.118)22 JEAN BOURGAIN AND ABEL KLEIN\nwhere the constant cd;V1;E0is chosen large enough to ensure that all the conditions\n(3.93), (3.112), and (3.117) (and hence (3.115)) are satis\fed. (This can be done\nusing (3.91).) It then follows from (3.114) and (3.115) that\n1\n2\u0000\u000e\nR\u0001MR4\n3\u0014Cd;V1;E0\"2;i.e.,\u0000\nCN\nd;V1;E0R2\u0001\u0000MR4\n3\u00142Cd;V1;E0\"2:(3.119)\nRecalling (3.95), and using (3.118) with a su\u000eciently large constant cd;V1;E0, we\nget from (3.119) that\ne\u0000M0R8\n3= e\u0000M0Rd\u00004\n3(d\u00001)+d\nd\u00001+4\n3\u0014Cd;V1;E0\"2; (3.120)\nwhereM0=M0\nd;V1;E0. Thus\nlog1\n\"\u0014Cd;V1;E0R8\n3=C0\nd;V1;E0\n\u001a8\n4\u0000d; (3.121)\nand hence\n\u001a\u0014Cd;V1;E0\u0000\nlog1\n\"\u0001\u00004\u0000d\n8; (3.122)\nas long asLis large enough to satisfy (3.93) with the choice of Rin (3.118), namely\nL\u0015Ld;V1;E0\u0000\nlog1\n\"\u00013\n8. \u0003\nReferences\n[ABR] Axler, S., Bourdon, P., Ramey, W.: Harmonic function theory . Second edition. Graduate\nTexts in Mathematics 137. Springer-Verlag, New York, 2001\n[B] Bers, L.: Local behavior of solutions of general linear elliptic equations. Comm. Pure\nAppl. Math. 8, 473-496 (1955)\n[BoK] Bourgain, J., Kenig, C.: On localization in the continuous Anderson-Bernoulli model in\nhigher dimension, Invent. Math. 161, 389-426 (2005)\n[CL] Carmona, R, Lacroix, J.: Spectral Theory of Random Schr odinger Operators. Boston:\nBirkha user, 1990\n[CoHK] Combes, J.M., Hislop, P.D., Klopp, F.: Optimal Wegner estimate and its application to\nthe global continuity of the integrated density of states for random Schr odinger operators.\nDuke Math. J. 140, 469-498 (2007)\n[CrS1] Craig, W., Simon, B.: Subharmonicity of the Lyaponov index. Duke Math. J. 50, 551-560\n(1983)\n[CrS2] Craig, W., Simon, B.: Log H older continuity of the integrated density of states for\nstochastic Jacobi matrices. Comm. Math. Phys. 90, 207-218 (1983)\n[DS] Delyon, F., Souillard, B.: Remark on the continuity of the density of states of ergodic\n\fnite di\u000berence operators. Comm. Math. Phys. 94, 289-291(1984)\n[DiJA] Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators . Cambridge Studies\nin Advanced Mathematics 43. Cambridge University Press, Cambridge, 1995\n[DoIM] Doi, S., Iwatsuka, A., Mine, T.: The uniqueness of the integrated density of states for\nthe Schr odinger operators with magnetic \felds. Math. Z. 237, 335-371 (2001)\n[EV] Escauriaza, L., Vessella, S.: Optimal three cylinder inequalities for solutions to parabolic\nequations with Lipschitz leading coe\u000ecients. Inverse problems: theory and applications\n(Cortona/Pisa, 2002), 79-87, Contemp. Math. 333, Amer. Math. Soc., Providence, RI,\n2003\n[GK1] Germinet, F., Klein, A.: A characterization of the Anderson metal-insulator transport\ntransition. Duke Math. J. 124, 309-351 (2004)\n[GK2] Germinet, F., Klein, A.: A comprehensive proof of localization for continuous Anderson\nmodels with singular random potentials. J. Europ. Math. Soc. To appear\n[GiT] Gilbarg, D.. Trudinger, N.: Elliptic partial di\u000berential equations of second order . Reprint\nof the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001\n[HW] Hartman, P., Wintner, A.: On the local behavior of solutions of non-parabolic partial\ndi\u000berential equations. Amer. J. Math. 75, 449-476. (1953)BOUNDS ON THE DENSITY OF STATES FOR SCHR ODINGER OPERATORS 23\n[Ho] Howard, R.: The Gronwall Inequality [Online].\nAvailable: http://www. math.sc.edu/ howard/Notes/gronwall.pdf\n[KM] Kirsch, W., Martinelli, F.: On the density of states of Schrdinger operators with a random\npotential. J. Phys. A 15, 2139-2156 (1982)\n[N] Nakamura, S.: A remark on the Dirichlet-Neumann decoupling and the integrated density\nof states. J. Funct. Anal. 179, 136-152 (2001)\n[PF] Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Heidelberg:\nSpringer-Verlag, 1992\n[S] Simmons, G. F.: Introduction to topology and modern analysis . McGraw-Hill Book Co.,\nInc., New York-San Francisco, Calif.-Toronto-London, 1963\n[Si1] Simon, B.: Schr odinger semi-groups. Bull. Amer. Math. Soc. 7, 447-526 (1982)\n[Si2] Simon, B.: Schr odinger operators in the twenty-\frst century. Mathematical Physics 2000,\n283-288, Imp. Coll. Press, London, 2000\n[T] Thomas, L.: Time dependent approach to scattering from impurities in a crystal. Comm.\nMath. Phys. 33, 335-343 (1973)\n[ThE] Thurlow, C., Eastham, M.: The existence of eigenvalues of in\fnite multiplicity for the\nSchr odinger operator. Proc. Roy. Soc. Edinburgh Sect. A 86 , 61-64. (1980)\nInstitute for Advanced Study, Princeton, NJ 08540, USA\nE-mail address :bourgain@ias.edu\nUniversity of California, Irvine, Department of Mathematics, Irvine, CA 92697-3875,\nUSA\nE-mail address :aklein@uci.edu"    },    {        "title": "1112.2156v1.Stabilizer_States_as_a_Basis_for_Density_Matrices.pdf",        "content": "arXiv:1112.2156v1  [quant-ph]  9 Dec 2011Stabilizer States as a Basis for Density Matrices\nSimon J. Gay\nSchool of Computing Science, University of Glasgow, UK\nAugust 28, 2018\nAbstract\nWe show that the space of density matrices for n-qubit\nstates, considered as a (2n)2-dimensional real vector\nspace, has a basis consisting of density matrices of sta-\nbilizer states. We describe an application of this result\nto automated verification of quantum protocols.\n1 Definitions and Results\nWe are working with the stabilizer formalism [5], in\nwhich certain quantum states on sets of qubits are repre-\nsented by the intersection of their stabilizer groups with\nthe group generated by the Pauli operators. The sta-\nbilizer formalism is defined, explained and illustrated in\na substantial literature; good introductions are given by\nAaronson and Gottesman [1] and Nielsen and Chuang\n[7, Sec. 10.5].\nIn this paper we only need to use the following facts\nabout stabilizer states.\n1. The standard basis states are stabilizer states.\n2. The set of stabilizer states is closed under applica-\ntion of Hadamard ( H), Pauli ( X,Y,Z), controlled\nnot (CNot), and phase ( P=/parenleftbigg1 0\n0i/parenrightbigg\n) gates.\n3. The set of stabilizer states is closed under tensor\nproduct.\nNotation 1 Write the standard basis for n-qubit states\nas{|x/an}bracketri}ht |0/lessorequalslantx <2n}, considering numbers to stand\nfor their binary representations. We omit normalization\nfactors when writing quantum states.\nDefinition 1 LetGHZn=|0/an}bracketri}ht+|2n−1/an}bracketri}htandiGHZn=\n|0/an}bracketri}ht+i|2n−1/an}bracketri}ht, asn-qubit states.\nLemma 1 For alln,GHZnandiGHZnare stabilizer\nstates.\nProof:By induction on n. For the base case ( n= 1), we\nhave that |0/an}bracketri}ht+|1/an}bracketri}htand|0/an}bracketri}ht+i|1/an}bracketri}htare stabilizer states, by\napplying Hand then Pto|0/an}bracketri}ht.\nFor the inductive case, GHZnandiGHZnare obtained\nfromGHZn−1⊗|0/an}bracketri}htandiGHZn−1⊗|0/an}bracketri}ht, respectively, by\napplying CNotto the two rightmost qubits. /squareLemma 2 If0/lessorequalslantx,y <2nandx/ne}ationslash=ythen|x/an}bracketri}ht+|y/an}bracketri}htand\n|x/an}bracketri}ht+i|y/an}bracketri}htare stabilizer states.\nProof:By induction on n. For the base case ( n= 1),\nthe closure properties imply that |0/an}bracketri}ht+|1/an}bracketri}ht,|0/an}bracketri}ht+i|1/an}bracketri}htand\n|1/an}bracketri}ht+i|0/an}bracketri}ht=|0/an}bracketri}ht−i|1/an}bracketri}htare stabilizer states.\nFor the inductive case, consider the binary represen-\ntations of xandy. If there is a bit position in which x\nandyhave the same value b, then|x/an}bracketri}ht+|y/an}bracketri}htis the tensor\nproduct of |b/an}bracketri}htwith an ( n−1)-qubit state of the form\n|x′/an}bracketri}ht+|y′/an}bracketri}ht, wherex′/ne}ationslash=y′. By the induction hypothesis,\n|x′/an}bracketri}ht+|y′/an}bracketri}htis a stabilizer state, and the conclusion follows\nfrom the closure properties. Similarly for |x/an}bracketri}ht+i|y/an}bracketri}ht.\nOtherwise, the binary representations of xandyare\ncomplementary bit patterns. In this case, |x/an}bracketri}ht+|y/an}bracketri}htcan\nbe obtained from GHZnby applying Xto certain qubits.\nThe conclusion follows from Lemma 1 and the closure\nproperties. The same argument applies to |x/an}bracketri}ht+i|y/an}bracketri}ht,\nusingiGHZn. /square\nTheorem 1 The space of density matrices for n-qubit\nstates, considered as a (2n)2-dimensional real vector\nspace, has a basis consisting of density matrices of n-\nqubit stabilizer states.\nProof:This is the space of Hermitian matrices and its\nobvious basis is the union of\n{|x/an}bracketri}ht/an}bracketle{tx| |0/lessorequalslantx <2n} (1)\n{|x/an}bracketri}ht/an}bracketle{ty|+|y/an}bracketri}ht/an}bracketle{tx| |0/lessorequalslantx < y < 2n} (2)\n{−i|x/an}bracketri}ht/an}bracketle{ty|+i|y/an}bracketri}ht/an}bracketle{tx| |0/lessorequalslantx < y < 2n}.(3)\nNow consider the union of\n{|x/an}bracketri}ht/an}bracketle{tx| |0/lessorequalslantx <2n} (4)\n{(|x/an}bracketri}ht+|y/an}bracketri}ht)(/an}bracketle{tx|+/an}bracketle{ty|)|0/lessorequalslantx < y < 2n}(5)\n{(|x/an}bracketri}ht+i|y/an}bracketri}ht)(/an}bracketle{tx|−i/an}bracketle{ty|)|0/lessorequalslantx < y < 2n}.(6)\nThis is also a set of (2n)2states, and it spans the space\nbecause we can obtain states of forms (2) and (3) by\nsubtracting states of form (4) from those of forms (5)\nand (6). Therefore it is a basis, and by Lemma 2 it\nconsists of stabilizer states. /square\n12 Applications\nThe stabilizer formalism can be used to implement an\nefficient classical simulation of quantum computation, if\nquantum operations are restricted to those under which\nthe set of stabilizer states is closed — i.e. the Clifford\ngroup operations. This result is the Gottesman-Knill\nTheorem [6]. Gay, Nagarajan and Papanikolaou [3, 4, 8]\nhave applied it to formal verification of quantum sys-\ntems, adapting model-checking techniques [2] from clas-\nsical computer science. A simple example of model-\nchecking is the following.\nConsider a quantum teleportation protocol as a sys-\ntem with one qubit as input and one (different) qubit\nas output; call this system Teleport. Also consider the\none-qubit identity operator I. Then the specification\nthat teleportation should satisfy is that Teleport =I,\nwhere equality means the same transformation of a one-\nqubit state. The aim of model-checking in this context\nis to automatically verify that this specification is satis-\nfied, by simulating the operation of the two systems on\nall possible inputs. For this to be possible, the telepor-\ntation protocol is first expressed in a formal modelling\nlanguage analogous to a programming language.\nTheapproachofGay, NagarajanandPapanikolaoure-\nduces the problem to that of simulating teleportation on\nstabilizer states as inputs, which can be done efficiently\nbecause the teleportation protocol itself only uses Clif-\nford group operations. Correctness of teleportation on\nstabilizer states is interpreted as evidence for, although\nnot proof of, correctness on arbitrary states. Now, how-\never, we can draw a stronger conclusion.\nThe teleportation protocol defines a superoperator;\nthis can be proved, for example, by the techniques of\nSelinger [9], who uses superoperators to define the se-\nmantics of a programming language that can certainly\nexpress teleportation. Superoperators, among other\nproperties, are linear operators on the space of density\nmatrices. To check equivalence of two superoperators, it\nis therefore sufficient to check that they have the same\neffect on all elements of a particular basis. By taking a\nbasis that consists of stabilizer states, this can be done\nefficiently.\nMoreover, the number of stabilizer states on nqubits\nis approximately 2(n2)/2, whereas the dimensionality of\nthe spaceof n-qubit densitymatricesis only(2n)2= 22n.\nIt is therefore more efficient to model-check on a basis\nthan on all stabilizer states.\n3 Conclusion\nWe have proved that the space of n-qubit density ma-\ntrices has a basis consisting of stabilizer states, and ex-\nplained how this result can be used to improve the effi-\nciency and strengthen the results of model-checking for\nquantum systems.References\n[1] S.AaronsonandD.Gottesman. Improvedsimulation\nof stabilizer circuits. Physical Review A , 70:52328,\n2004.\n[2] E. M. Clarke, Jr., O. Grumberg, and D. A. Peled.\nModel Checking . MIT Press, 1999.\n[3] S.J.Gay,N.Papanikolaou,andR.Nagarajan.QMC:\na model checker for quantum systems. In CAV\n2008: Proceedings of the 20th International Confer-\nence on Computer Aided Verification , number 5123\nin Lecture Notes in Computer Science, pages 543–\n547. Springer, 2008.\n[4] S.J.Gay,N.Papanikolaou,andR.Nagarajan. Speci-\nfication and verification of quantum protocols. In Se-\nmantic Techniques in Quantum Computation , pages\n414–472. Cambridge University Press, 2010.\n[5] D. Gottesman. Class of quantum error-correcting\ncodes saturating the quantum Hamming bound.\nPhysical Review A , 54:1862, 1996.\n[6] D. Gottesman. Stabilizer Codes and Quantum Er-\nror Correction . PhD thesis, California Institute of\nTechnology, 1997.\n[7] M. A. Nielsen and I. L. Chuang. Quantum Computa-\ntion and Quantum Information . Cambridge Univer-\nsity Press, 2000.\n[8] N. Papanikolaou. Model Checking Quantum Proto-\ncols. PhD thesis, University of Warwick, 2009.\n[9] P. Selinger. Towards a quantum programming lan-\nguage.Mathematical Structures in Computer Sci-\nence, 14(4):527–586, 2004.\n2"    },    {        "title": "1112.3045v2.Spectral_properties_of_one_dimensional_spiral_spin_density_wave_states.pdf",        "content": "arXiv:1112.3045v2  [cond-mat.str-el]  6 Mar 2012Spectral properties of one-dimensional spiral spin densit y wave states\nDirk Schuricht\nInstitute for Theory of Statistical Physics, RWTH Aachen, 5 2056 Aachen, Germany and\nJARA-Fundamentals of Future Information Technology\n(Dated: November 10, 2018)\nWe provide a full characterization of the spectral properti es of spiral spin density wave (SSDW)\nstates which emerge in one-dimensional electron systems co upled to localized magnetic moments or\nquantumwires with spin-orbit interactions. Wederive anal ytic results for thespectral function, local\ndensity of states and optical conductivity in the low-energ y limit by using field theory techniques.\nWeidentify various collective modes and showthat the spect rum strongly depends on the interaction\nstrength between the electrons. The results provide charac teristic signatures for an experimental\ndetection of SSDW states.\nPACS numbers: 71.10.Pm, 71.35.Cc, 72.15.Nj\nIt is by now well established that interactions have\ndrasticeffects onone-dimensionalsystems.1Forexample,\nthe spectral properties of Luttinger liquids, which one\nmay regards as the standard model of one-dimensional\nelectrons, show spin-charge separation as well as non-\ntrivial power laws which depend on the interaction\nstrength.2The system decomposes into two gapless sec-\ntors describing collective charge and spin excitations\nwhich possess linear dispersion relations with velocities\nvcandvs, respectively. More general situations can be\ndescribed by adding appropriate terms to the Luttinger\nliquid Hamiltonian. For example, spin-orbit interactions\n(SOI) lead to a mixing of charge and spin degrees of\nfreedom, which strongly affects the spectral function but\nleaves the system gapless.3\nA more profound change in the low-energy properties\nhappens when the system develops a gap. The proto-\ntype scenario1for this situation is the Mott transition at\nwhich a gap in the charge sector opens provided the sys-\ntem is at commensurate filling. The spectral function of\nthe resultingMott insulator(MI)4still showsspin-charge\nseparation but the charge excitations possess a mass ∆.\nA similar situation is encountered in systems with SOI\nand an additional magnetic field, where a spin gap opens\nwhile the charge sector remains gapless.5On the other\nhand, if the SOI are spatially modulated with a wave\nlength commensurate with the Fermi wave length, the\nsystem develops gaps in both sectors.6In all three situa-\ntions the system separates into charge and spin sectors,\ni.e., there is no non-trivial mixing of gapless and gapped\ndegrees of freedom.\nAs it was shown by Braunecker et al.,7this is no longer\nthe case if one considers the interplay of one-dimensional\nelectronic degrees of freedom and localized magnetic mo-\nments such as nuclear spins. In such systems feedback ef-\nfects between the conduction electrons and nuclear mag-\nnetic moments can, at sufficiently low temperatures, sta-\nbilize the coexistence of electron density order and he-\nlical magnetic order. Specifically they showed that the\nelectronic subsystem develops a gap in a superposition\nof charge and spin degrees of freedom accompanied by\na spiral spin density wave (SSDW)8following the helicalmagneticorder. Asthe originalelectronspossessahighly\nnon-trivial decomposition into the gapped and gapless\nsectors one expects the system to possess rather unusual\nspectral properties. We note that similar situations are\nencountered in Luttinger liquids coupled to extended fer-\nromagnetic domain walls9as well as in quantum wires\nwith SOI in a magnetic field along the wire provided the\nband shift induced by the SOI is commensurate with the\nFermi momentum.10\nIn this Rapid Communication we provide a detailed\nanalysis of the spectral properties of such SSDW states.\nWe show that the spectral function is strongly asymmet-\nricinthe spincomponentsandidentifyvariousdispersing\ncollective modes. The calculated local density of states\n(LDOS) exhibits typical features of both the Luttinger\nliquid and the MI. In particular we are able to derive\nexact results for the appearing power-law exponents. Fi-\nnally we show that the optical conductivity possesses\ncharacteristic singularities intimately related to the in-\nteractions between the electrons. Taken together our re-\nsults provide a set of specific signatures necessary for the\nexperimental identification of SSDW states.\nModel.Experimentalsystemsinwhichsuchstatesmay\nexist require the interaction between electrons and nu-\nclearspins in one dimension as it is, for example, the case\nin13C single-wall carbon nanotubes11or GaAs quantum\nwires.12Braunecker et al.7showed that in these systems\nthe conduction electrons mediate an RKKY interaction\nbetween the nuclear spins. The feedback of the resulting\nhelical magnetic order causes the formation of a SSDW\nstate and generates a gap in the electronic subsystem,\nwhichcanbedescribedbytheeffectivelow-energyHamil-\ntonian (which is also relevant10to quantum wires with\nSOI showing a spin-selective Peierls transition)\nH=H++H−, (1)\nH+=v+\n16π/integraldisplay\ndx/bracketleftBig/parenleftbig\n∂xΦ+/parenrightbig2+/parenleftbig\n∂xΘ+/parenrightbig2/bracketrightBig\n+B\n2π/integraldisplay\ndxcos/parenleftbig/radicalbig\nK/4Φ+/parenrightbig\n, (2)\nH−=v−\n16π/integraldisplay\ndx/bracketleftBig/parenleftbig\n∂xΦ−/parenrightbig2+/parenleftbig\n∂xΘ−/parenrightbig2/bracketrightBig\n.(3)Here Φ ±are canonical Bose fields and Θ ±are their dual\nfields. The velocities v±and the Luttinger parameter K\nare related to the original parameters in the charge and\nspin sectors via\nv+=1\nK/bracketleftbigg\nvcKc+vs\nKs/bracketrightbigg\n, v−=1\nK/bracketleftbiggvc\nKs+vsKc/bracketrightbigg\n,(4)\nandK=Kc+1/Ks. ForK <4 the cosine term in (2)\nis relevant and generates a gap ∆, which is a function13\nof the bare Overhauser field BandK. We will view it\nhere as a phenomenological parameter replacing Bsuch\nthat we are left with the four free parameters Kc,s,vc/vs,\nand ∆. For repulsive electron-electron interactions one\nhas 0<Kc<1. Furthermore, in the derivation of (1) it\nwas assumed that7Ks≥1, whereKs= 1 corresponds to\nthe spin rotational invariant situation.\nThe bosonic fields appearing in (1) are related to the\nslowly varying right- and left-moving Fermi fields defined\nvia\nΨσ(x) =eikFxψRσ(x)+e−ikFxψLσ(x),(5)\nby the relations7(l= R/L≡ ±,σ=↑,↓≡ ±)\nψ±±=η±±√\n2πexp/parenleftbig\n∓iaΦ+/parenrightbig\nexp/parenleftbig\n±ibΦ−−icΘ−/parenrightbig\n,(6)\nψ±∓=η±∓√\n2πexp/parenleftbigi\n2√\nKΘ+∓i√\nK\n4Φ+/parenrightbig\nexp/parenleftbig\nidΘ−/parenrightbig\n,(7)\nwhere the Klein factors ηlσsatisfy the anticommutation\nrelations {ηlσ,ηl′σ′}= 2δllδσσ′and the constants are de-\nfined bya= (KcKs−1)/(4Ks√\nK),b=/radicalbig\nKc/(KKs)/2,\nandc,d= (KcKs±1)/(4√KKcKs).\nThe excitations in the sine-Gordon model (2) are\nsolitons and antisolitons with mass ∆, and, if K <\n4/(N+1),breather(soliton-antisoliton)boundstates Bn,\nn= 1,...,N, with masses ∆ n= 2∆sin(nπK/(8−2K)).\nWe note that there exists at least one breather for repul-\nsive interactions. The solitons and antisolitons possess\na relativistic dispersion relation E2= ∆2+v2\n+P2; the\ndispersion relation for the nth breather is obtained by\nreplacing ∆ by ∆ n. The sector (3) describes gapless col-\nlective excitations propagating with velocity v−.\nGreen’s function. The spin-resolved spectral function\nand LDOS discussed below follow from the spin-diagonal\nGreen’s function in Euclidean space, which is defined by\nGσ(τ,x) =/summationdisplay\nll′eilkFxGll′\nσ(τ,x), (8)\nwhereGll′\nσ(τ,x) =−∝an}bracketle{tTτψlσ(τ,x)ψ†\nl′σ(0,0)∝an}bracketri}ht,τ= itde-\nnotes Euclidean time, and Tτis theτ-ordering operator.\nDue to the decompositionofthe Hamiltonian the Green’s\nfunction factorizesinto a product of correlationfunctions\nin the gapped and gapless sectors. The correlation func-\ntions in the gapless sector can be straightforwardly cal-\nculated by using standard methods.1The derivation in\nthe gapped sector employs the integrability of the sine-\nGordon model via a form-factor expansion14,15with theresults (τ >0)\nGRR\n↑(τ,x) =−1\n2π1\n(v−τ−ix)2(b+c)21\n(v−τ+ix)2(b−c)2\n×G2\na/bracketleftBigg\n1+λ2sin22π√\nKa\n4−K\nπsin2πK\n4−KK0/parenleftBig\n∆1/radicalBig\nτ2+x2/v2\n+/parenrightBig\n+.../bracketrightBigg\n(9)\nand\nGRR\n↓(τ,x) =−1\n2π1\n(v−τ−ix)2d21\n(v−τ+ix)2d2\n×\nZ1\nπ/parenleftbiggv+τ+ix\nv+τ−ix/parenrightbigg1\n4\nK1/parenleftBig\n∆/radicalBig\nτ2+x2/v2\n+/parenrightBig\n+...\n.\n(10)\nHereK0/1denote modified Bessel functions16and the\nconstantsGa,λ, andZ1(which depend on KcandKs)\nwere obtained in Refs. 17 and 18. The subleading terms\nin (9) and (10) involve heavier breathers Bn,n≥2 (for\nK <4/3), or multi-particle states. Furthermore we have\nGLL\nσ(τ,x) =GRR\n¯σ(τ,−x),Gll\nσ(−τ,−x) =Gll\nσ(τ,x), and\nGRL\nσ=GLR\nσ= 0 with ¯σ=∓forσ=±.\nSpectral function. The spectral function can be ex-\nperimentally probed using angle-resolved photoemission\nmeasurements. We can obtain the spin-resolved spectral\nfunctionAσ(ω,p) =A>\nσ(ω,p) +A<\nσ(ω,p) directly from\nthe Green’s function (8) via\nA≷\nσ(ω,p) =∓/integraldisplaydt\n2πdx\n2πei(ωt−px)Gσ(τ≷0,x)/vextendsingle/vextendsingle/vextendsingle\nτ→it±δ,\n(11)\nwhereδis small but finite and mimics broadening by in-\nstrumental resolution and temperature in experiments.\nThe form-factor expansions (9) and (10) result in a simi-\nlar expansion for the spectral function, where each term\ncan be expressed as an integral over a hypergeomet-\nric function depending on the parameters b,c, andd.\nWe note that the multi-particle corrections to (9) and\n(10) yield negligible19contributions at low energies and\nvanish for |ω|<2∆. The spectral function satisfies\nAσ(−ω,kF−q) =Aσ(ω,kF+q) andAσ(ω,−kF−q) =\nA¯σ(ω,kF+q).\nThe spectral function in the vicinity of p=kFis shown\nin Fig. 1. We observe a strong spin dependence. The up-\nspincomponent isdominated byalinearlydispersingfea-\nture following ω=v−qwhich originates from the gapless\nexcitations of (3). In addition the existence of breathers\nresults in dispersing features at ω=±(∆2\nn+v2\n+q2)1/2\nwhich are, however, very weak (see the arrows in the in-\nset in Fig. 1). From our analytical results we can extract\nthe power laws at the dispersing features, e.g., for fixed\nq>0 andδ→0+, we obtain\nA↑(ω,kF±q)∝(ω−v−q)2(b∓c)2−1, ω→v−q+,(12)\nwhile atq= 0 the exponent is 4( b2+c2)−2. The mea-\nsurement of these power laws can be used to extract the\n2-6 -4 -2 0 2 4 6\nω/∆-6-4-2024vsq/∆=6Aσ(ω,kF+q)A (ω,kF+q)\nA (ω,kF+q)\n-2 0 2vsq/∆=1\n-202\nFIG. 1. (Color online) Spectral function Aσ(ω,kF+q) for\nKc= 0.5,Ks= 1,vc= 1.2vs,σ=↑(solid black lines),\nandσ=↓(dashed red lines). The curves are constant q-\nscans which have been offset along the y-axis with respect to\none another. We stress that the up-spin component possesses\nspectral weight within the gap ∆ and that the dispersions\ncross at vsq0/∆≈ ±2.6. The arrows in the inset indicate the\nenergies where processes involving B1set in.\ninteraction strength encoded in the Luttinger parame-\ntersKcandKs. We note that the derivation7of the\neffective low-energy model (1) relied on a linear single-\nparticle dispersion, whereas in any realistic microscopic\nmodel some curvaturewill be present. Although this cur-\nvature is irrelevant in the renormalization-groupsense, it\nhas been shown20to affect the power laws in the spec-\ntral function. However, as was argued in Ref. 7 for both\nsingle-wall carbon nanotubes and GaAs quantum wires\nthe curvature-induced deviations from (1) are negligible.\nWefurthernotethatapotentiallypresentmomentumde-\npendence of the two-particle interaction21was neglected\nin the derivation7of (1).\nThe down-spin component of the spectral function is\ndominated for q>0 by a dispersing mode following ω=\nωq≡(∆2+v2\n+q2)1/2for which we find\nA↓(ω,kF+q)∝(ω−ωq)4d2−1, ω→ω+\nq.(13)\nInaddition, thereisaveryweakfeatureat ω=−ωq. The\ndispersing modes for q <0 are obtained using the sym-\nmetry of the spectral function. These dispersing features\noriginate from the massive soliton excitations of (2), in\nparticular the spectral function vanishes for |ω|<∆.\nFor generic system parameters one has vc> vsand\nKcKs<1, which results in a crossing of the dispersing\nup- and down-spin features at q0=±∆/(v2\n−−v2\n+)1/2.\nIn the non-interacting case, Kc=Ks= 1,vc=vs≡v,\nthe contributions from the breathers vanish and the\nremaining dispersing features become δ-functions, e.g.,\nA↑(ω,kF+q)∝δ(ω−vq).\nThe vanishing of A↓(ω,kF+q) andA↑(ω,−kF−q) for\n|ω|<∆ shows that the system effectively acts as a spin\nfilter at sufficiently low energies. As was pointed out in00.511.522.5\nω/∆00.51ρ(ω)\n00.511.522.5\nω/∆Kc=0.5 Kc=0.2\nB1 B1B2\nFIG. 2. LDOS for Kc= 0.5 (left panel) and Kc= 0.2 (right\npanel),Ks= 1, and vc= 1.2vs. We observe a power-law\nbehavior for ω→0 [see Eq. (14)]. The singularity at ω= ∆\nis only present for Kc>√\n5−2 [see Eq. (15)]. The arrows\nindicate the energies where single-breather processes set in.\nRef.7, theexistenceofagapin onehalfoftheconduction\nchannelsyieldsareductionoftheconductancebyafactor\nof2. Wenote, however,thatanexperimentaldetectionin\ntransportexperiments may be hampered by the influence\nof the attached contacts.22\nFinally let us compare our results to the one-\ndimensional MI, whose spectral function4does not de-\npend on the spin, possesses a gap, and clearly displays\npropagating charge and spin excitations. Thus although\nthe low-energy theories of the MI and the SSDW state\nare both given by a sum of a gapped and a gaplesssector,\ntheirspectralfunctionsdiffersignificantlyduetothenon-\ntrivial decomposition of the electronic degrees of freedom\ninto elementary excitations in the latter. In particular,\nthe dispersing features in the SSDW state do not permit\na simple interpretationin terms ofindividually propagat-\ning charge and spin excitations, i.e., a direct observation\nofspin-chargeseparationin aSSDWstateisnotpossible.\nFurthermore the operator e∓iaΦ+appearing in (6) pos-\nsesses a non-vanishing vacuum expectation value17and\nthus gives rise to spectral weight at energies |ω|<∆.\nLDOS.Scanning tunneling microscopy (STM) and\nspectroscopy experiments measure the tunneling current\nbetween the sample and the STM tip as a function of the\nposition of the tip and the applied voltage. The tunnel-\ning current is directly related to the LDOS of the sample,\nwhich is thus accessible in STM experiments. The LDOS\nρ(ω)isobtainedfromthe spectralfunction byintegrating\nover the momentum. The result is spin independent and\nplotted in Fig. 2. We note that the LDOS has recently\nbeen calculated23by using a self-consistent harmonic ap-\nproximation for the gapped sector, i.e., by expanding the\ncosine in (2) up to quadratic order.\nThe low-energy behavior of the LDOS can be deter-\nmined from the first term in (9). We find for |ω|<∆\nρ(ω)∝ |ω|α, α= 4(b2+c2)−1 =/bracketleftbig\n1+Kc(Ks−2)/bracketrightbig2\n4Kc(1+KcKs),\n(14)\nin agreement with the harmonic approximation.23This\nagain highlights the existence of spectral weight within\n30 1 2 3\nω/∆00.005\nσ(ω)\n0 1 2 3\nω/∆Kc=0.5\nB1B1B3\nB1B2\nssKs=1Kc=0.2\nKs=1.5\nD D\nss\nFIG. 3. Conductivity for vc= 1.2vsandK= 1.5 (left panel)\nas well as K= 0.87 (right panel). We have convoluted the\nδ-peaks by a Lorentzian. In both systems we observe the\nDrude peak ( D) atω= 0, the δ-peak from the first breather\nB1, and the soliton-antisoliton continuum ( s¯s). In the right-\nhand panel we further observe the δ-peak from B3and the\nB1B2continuum, which possesses a singularity at the lower\nthreshold.\nthe gap in contrast to the MI.4In addition the LDOS\npossesses a pronounced feature at ω= ∆ at which the\ngapped sector (2) starts to contribute. For ω→∆+the\nleading term in (10) results in\nρ(ω)∝(ω−∆)β+...,\nβ= 4d2−1\n2=1+K2\ncKs(Ks−2)−2Kc(1+Ks)\n4Kc(1+KcKs),\n(15)\nwhere the dots represent the regular background (14).\nWe note that except for the non-interacting case, where\nβ=−1/2, the exponent differs from the corresponding\nresult of Ref. 23, as the harmonic approximation applied\nthere does not correctly capture the Lorentz spin of soli-\ntons and antisolitons.\nThe most pronounced feature of the LDOS is found\natω= ∆, which is identical to the gap observed in\nA↓(ω,kF+q). Furthermore, the exponents αandβare\nfixed by the Luttinger parameters determined from the\nspectral function thus providing a consistency check be-\ntweenthemeasurements. Alternatively,ifonlytheLDOS\nis accessible but the system is spin-rotational invariant\n(Ks= 1), there is a fixed relation between αandβ, e.g.,\nβvanishes for Kc<√\n5−2. In any case the breather\ncontributions to the LDOS are negligible (see Fig. 2).\nDue to the translational invariance of the system the\nLDOS shows no signatures of propagating quasiparticles.\nIn order to identify such features in STM experiments it\nis thus neccessary to study the spatial Fourier transform\nof the LDOS in systems with boundaries or impurities.24\nConductivity. Finally let us discuss the optical\nconductivity1σ(ω) which is obtained from the current-\ncurrent correlation function. With the current density1,7\nje=e\nπ∂tΦc=e\nπ√\nK/bracketleftBigg\nKc∂tΦ+−/radicalbigg\nKc\nKs∂tΦ−/bracketrightBigg\n(16)it follows that the conductivity is the sum of the con-\nductivities in the gapped and gapless sectors, σ(ω) =\nσ+(ω) +σ−(ω). For the clean system we consider here\nthe latter is given by1a Drude peak ∝δ(ω) while the\nformer can be calculated analogously to a MI.25Thus\nσ+(ω) is given by a sum of δ-functions originating in\nsingle-breather processes and, at higher frequencies, a\nmany-particle continuum σcont\n+(ω). As the current op-\nerator couples only to single-breather states with nodd\nwe find\nσ+(ω) =(N+1)/2/summationdisplay\nj=1ajδ(ω−∆2j−1)+σcont\n+(ω),(17)\nwhere the coefficients ajare fixed by the single-breather\nform factors14,15,25of the current operator.\nFor 4/3≤K <2 only the breather B1exists. As\nshown in the left-hand panel in Fig. 3 the conductivity\nis dominated by two δ-peaks at low energies, while at\nω>2∆ the soliton-antisoliton ( s¯s) continuum shows up.\nIn particular the optical gap is twice the gap observed in\nthe spectral function or the LDOS. The leading correc-\ntions are given by the three-particle process s¯sB1which\nsets in atω= 2∆+∆ 1. ForK <4/3the second breather\nstarts to contribute in the two-breather continuum, but\nnot as a single-particle peak as the current operator does\nnot couple to B2. As shown in the right-hand panel in\nFig. 3, for 2 /3≤K <1 a thirdδ-peak originating in B3\nexists. The two-particle continuum is dominated by the\ntwo-breather process B1B2while the soliton-antisoliton\ncontribution is strongly suppressed as its spectral weight\nis transferred to the coherent B1-peak. (B1is as¯sbound\nstate.) We stress that the B1B2-continuum possesses a\nsingularity at the threshold ω→(∆1+∆2)+. The lead-\ning corrections are given by the three-particle processes\nB1B1B1. We note that the condition K <1 requires the\nbreaking of the spin rotation invariance, i.e., Ks>1. If\nKis further decreased, the conductivity develops more\nfeatures due to the existence of additional breathers.\nIncontrasttothespectralfunctionandLDOS,thecon-\nductivity allows an efficient spectroscopy of the individ-\nual breather masses. In particular, the knowledge of the\nLuttinger parameters and thus Kgives a precise predic-\ntion for the number and masses of detectable individual\nbreather states as well as features in the many-particle\ncontinuum. This provides a non-trivial consistency check\nbetween measurements of the optical conductivity and\nthe spectral function or LDOS which has to be satisfied\nfor a definite detection of a SSDW state.\nConclusions. In this Rapid Communication we have\npresented a complete characterization of the spectral\nproperties of SSDW states, which appear7in one-\ndimensional electron systems coupled to nuclear spins or\nquantum wires in a magnetic field and with SOI that are\ncommensurate with the Fermi momentum.10We calcu-\nlated the spin-resolved spectral function, LDOS, and op-\ntical conductivity. We identified various collective modes\nand showed that the spectrum strongly depends on the\n4interaction strength between the electrons. We discussed\nvarious consistency checks between the observables and\nthus between measurements of the spectral function, the\nLDOS, and the optical conductivity which should allow\nan unambiguous detection of a SSDW state in future ex-\nperimentson13Csingle-wallcarbonnanotubesandGaAsquantum wires.\nIwouldliketothankBerndBraunecker,VolkerMeden,\nPascal Simon, and particularly Fabian Essler, for useful\ndiscussions. 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B 84, 045122 (2011).\n25D. Controzzi, F. H. L. Essler, and A. M. Tsvelik, Phys.\nRev. Lett. 86, 680 (2001); F. H. L. Essler, F. Gebhard,\nand E. Jeckelmann, Phys. Rev. B 64, 125119 (2001).\n5"    },    {        "title": "1203.6856v1.Propagation_of_Initially_Excited_States_in_Time_Dependent_Density_Functional_Theory.pdf",        "content": "arXiv:1203.6856v1  [physics.chem-ph]  30 Mar 2012Propagation of Initially Excited States in Time-Dependent Density Functional Theory\nPeter Elliott and Neepa T. Maitra\nDepartment of Physics and Astronomy, Hunter College and the City\nUniversity of New York, 695 Park Avenue, New York, New York 10 065, USA\nMany recent applications of time-dependent density functi onal theory begin in an initially excited\nstate, and propagate it using an adiabatic approximation fo r the exchange-correlation potential. This\nhowever inserts the excited-state density into a ground-st ate approximation. By studying a series\nof model calculations, we highlight the relevance of initia l-state dependence of the exact functional\nwhen starting in an excited state, and explore the errors inh erent in the adiabatic approximation that\nneglect this dependence.\nI. INTRODUCTION\nTime-dependent density functional theory (TDDFT)\nis an exact reformulation of the time-dependent quan-\ntum mechanics of many-electron systems [1, 2] that\noperates by mapping a system of interacting elec-\ntrons into one of fictitious non-interacting fermions, the\nKohn-Sham (KS) system, reproducing the same time-\ndependent density of the true system. By the theorems\nof TDDFT, all properties of the true interacting system\nmay be obtained from the KS system, in principle. In\npractice, approximations are needed for the exchange-\ncorrelation effects: in particular, for the exchange-\ncorrelation (xc) potential, vXC[n;Ψ(0),Φ(0)](r,t), a cru-\ncial term in the single-particle KS equations determining\nthe evolution of the ficitious non-interacting fermions.\nThis functionally depends on the one-body density,\nn(r,t), the initial many-body state of the interacting sys-\ntemΨ(0) , and the initial state of the KS system Φ(0) .\nFrom the inception of the theory, subtleties were\nnoted in the functional-dependences, which make good\napproximations for the xc potential more challenging\nto derive than for its counterpart in the (much older)\nground-state density functional theory [3–5]. One of\nthese is memory: the dependence of the potential at time\nton the density at earlier times, and on the initial-states\nΨ(0) andΦ(0) . Efforts to include memory-dependence\nin functionals [6–14], are not widely used, and have\nall focussed on the dependence on the history of the\ndensity, neglecting the dependence on the initial states.\nThe vast majority of applications have been in linear re-\nsponse from a non-degenerate ground-state, and there\nthis initial-state dependence (ISD) is redundant: by the\nHohenberg-Kohn theorem, a non-degenerate ground-\nstate is a functional of its own density. In fact, the simple\nadiabatic approximation, where the instantaneous den-\nsity is input into a ground-state approximation, neglect-\ning any memory-dependence, has shown remarkable\nsuccess for a great range of excitations and spectra. All\ncommonly available codes with TDDFT capabilities are\nwritten assuming the adiabatic approximation. Over the\nyears it has been understood that certain types of exci-\ntations cannot be captured by the adiabatic approxima-\ntion, e.g. double excitations [15], excitonic series in op-\ntical response of semiconductors [16], and there is bothon-going research in developing frequency-dependent\nkernels to deal with this, as well as understanding when\nnot to trust the calculated adiabatic TDDFT spectra.\nWith the results and understanding of recent years\nlending a level of comfort with calculations of spec-\ntra and response, TDDFT has now entered a more ma-\nture stage, and with that, comes more adventurous ap-\nplications. In particular, for electron dynamics in real\ntime, evolving under strong laser fields, or coupled\nelectron-nuclear dynamics following photo-excitation\n(e.g. Refs. [17–19]). The role of memory is less well-\nunderstood in these applications; studies on model sys-\ntems have shown that sometimes memory-dependence\nis essential [20–24], other times it is not important at\nall [25, 26]. Moreover a new element enters: the initial-\nstate dependence that was conveniently and correctly\nbrushed aside in the linear response regime, now raises\nits head. In applications such as modeling solar cell pro-\ncesses, the initial photo-excitation of the electronic sys -\ntem is not dynamically modeled; instead the dynamics\nbegins with the electronic system assumed to be in an\nexcited state. The initial state is not the ground-state, ye t\nno truly initial-state dependent functionals are availabl e\ntoday, hence we ask how large are the errors in such cal-\nculations? Knowing that the true interacting system be-\ngins in a certain excited state, is there an optimal choice\nfor the initial KS state when propagating with an adia-\nbatic approximation? How large are the errors due to\nISD compared to those due to history-dependence?\nIn this paper, we begin to answer these questions\nby considering a series of model calculations of two-\nelectron systems. We start by reviewing the underlying\ntheorems of TDDFT and the subtleties of ISD, even in sit-\nuations where we may not expect it. We show how ISD\nleads to a non-zero xc potential even for non-interacting\nelectrons.\nThen, we move to electron dynamics in the\nmodel soft-Coulomb helium atom, performing adia-\nbatic TDDFT calculations with different initial-states\nand comparing with exact dynamics. Here we must\ndissect other sources of error in usual TDDFT calcula-\ntions, such as using an initial KS excited state whose\ndensity does not equal the true excited state density, in\norder to properly ascribe the influence of ISD. Finally\nwe make the first step towards investigating how ISDcan affect an area currently of much interest, namely\ncoupled electron-ion dynamics, by performing Ehren-\nfest calculations for a model LiH system. We work in\natomic units throughout ( e2=me=/planckover2pi1= 1).\nII. INITIAL-STATE DEPENDENCE IN TDDFT\nIn TDDFT, one evolves a set of single-particle orbitals\n{φj(r,t)}with a one-body KS potential:\ni∂\n∂tφj(r,t) =/parenleftbigg\n−1\n2∇2+vS(r,t)/parenrightbigg\nφj(r,t) (1)\nvS(r,t) =vext(r,t)+vXC[n](r,t)+vXC[n;Ψ0,Φ0](r,t)(2)\nwherevH(r,t) =/integraltext\nn(r′,t)/|r−r′|d3r′is the usual Hartree\npotential and vXC(r,t)is the xc potential which depends\non the entire history of the density, the interacting initia l\nstateΨ0, and the initial KS wavefunction Φ0.\nThe origin of the initial-state dependence in the func-\ntionals is that the one-to-one Runge-Gross mapping [1]\nbetween time-dependent densities and potentials holds\nfor a fixed initial-state. Thus, the external potential vext\nfunctionally depends on the density and the true ini-\ntial state Ψ(0) , and the KS potential vSdepends on the\ndensity and the KS initial state Φ(0) . These functional\ndependences are not directly relevant themselves in a\npractical calculation, because there only the xc potential\nvXCneeds to be approximated. However, they lend their\ndependences to vXCvia Eq. (2), which must therefore de-\npend on both initial states and the density.\nThere are two known situations in which there is\nno initial-state dependence. When the initial state is a\nground-state, then, by the Hohenberg-Kohn theorem of\nground-state DFT, it is itself a functional of its own den-\nsity. ISD is redundant, as the information about the ini-\ntial state is contained in the initial density. (See also\nSec. II A). The other situation is for one-electron sys-\ntems: starting in anyinitial state, there is only one poten-\ntial that can yield a given density-evolution [27]. In all\nother situations, it is assumed that ISD cannot be sub-\nsumed into a density-dependence; explicit demonstra-\ntions for two electrons can be found in Refs. [22, 27].\nRef. [28] derived an exact condition relating the depen-\ndence on the history of the density to ISD. This condition\nis likely violated by any history-dependent functional\napproximation that has no ISD.\nTechnically, one may choose any initial KS wave-\nfunction that has the same nand˙nas the true initial\nstate [29]. Usually a single Slater-determinant of Nspin-orbitalsφiis selected, with the required property that\nN/summationdisplay\ni=1|φi(r,0)|2=n(r,0) and (3)\n−∇·ImN/summationdisplay\ni=1φ∗\ni(r,0)∇φi(r,0) = ˙n(r,0) (4)\nwhere\nn(r,0) =N/summationdisplay\nσ/integraldisplay\ndx2···/integraldisplay\ndxN|Ψ(x,x2···xN,0)|2(5)\n˙n(r,0) =−N∇·Im/braceleftBigg/summationdisplay\nσ/integraldisplay\ndx2···/integraldisplay\ndxNΨ∗(x,x2xN,0)\n∇Ψ(x,x2,,xN,0)/bracerightBigg\n(6)\nwherex= (r,σ)and/integraltext\ndx=/summationtext\nσ/integraltext\nd3x.\nIn this paper we will consider different choices of Φ(0)\nfor two-electron systems that begin in the first excited\nsinglet state of the true system, denoted Ψ∗. Specifi-\ncally we will, at various points, investigate the follow-\ning three forms:\n(a) An excited KS singlet state with two occupied or-\nbitalsφ0andφ1, whose spatial part has the form\nΦ∗(r1,r2) =1√\n2(φ0(r1)φ1(r2)+φ0(r2)φ1(r1)).(7)\nNote that this state is a sum of two Slater determinants.\n(b) A ground KS singlet state, Φgs, which, for two elec-\ntrons corresponds to a single doubly-occupied orbital\nΦgs[n∗](r1,r2) =φ(r1)φ(r2), (8)\nwith\nφ(r) =/radicalbig\nn∗(r)/2 (9)\nwheren∗(r) = 2/integraltext\n|Ψ∗(r,r′)|2d3r′is the density of the\ninitial excited state. This is a single Slater-determinant ,\nand is the usual choice when the true initial state is a\nground state.\n(c) A spin-symmetry-broken excited KS state of the form\nΦ∗\nSB(r′σ′,rσ) =1√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ0(r)δσ↑φ0(r′)δσ′↑\nφ1(r)δσ↓φ1(r′)δσ′↓/vextendsingle/vextendsingle/vextendsingle/vextendsingle(10)\nThis state is not a spin-eigenstate (although it does have\n/an}b∇acketle{tS/an}b∇acket∇i}ht= 0), but is a valid choice for an initial KS state, hav-\ning the same total density as the true system. The up and\ndown spin-densities are not equal, and we shall evolve\nthem in different spin-up and spin-down KS potentials,\nbut, unlike in spin-DFT, we do not consider the spin-\n2densities separately meaningful; only their sum is con-\nsidered as an observable [30]. Instead, (c) will be used\nto illustrate the effects of orbital-specific functionals, as\nwill be discussed more later.\nNow the exact KS potential differs depending on\nwhich choice (a), (b), or (c), is made, but any adiabatic\napproximation is identical for them all. Almost all the\ncalculations being run today, certainly all that are coded\nin the commonly available codes, utilize such an ap-\nproximation, and insert the instantaneous density into\na ground-state approximation:\nvadia\nXC[n;Ψ(0),Φ(0)](r,t) =vgs\nXC[n(t)](r). (11)\nIt is not surprising that there are errors inherent in such\nan approximation but one question we hope to shed\nlight on by our investigations here, is how much of this\nerror is due to the lack of ISD, rather than the lack of\nhistory-dependence (dependence on n(r,t′< t)). Only\nthe latter occurs in usual calculations that begin in an\ninitial ground-state, where, as mentioned earlier, ISD\nmay be subsumed into density-dependence.\nA. Beginning in the ground-state: ISD or not?\nMany practical situations start with the system in its\nground-state; in fact most calculations, except in the\nmost recent years, have assumed an initial ground-state.\nThis is fortunate for TDDFT, since, at least at short times,\nthe adiabatic approximation with its ground-state func-\ntionals, which have become increasingly sophisticated\nover the years, could then be expected to work reason-\nably well. However, we will show that the the subtleties\nof initial-state dependence can appear even in this case.\nA completely legitimate choice for the KS initial state\nwould be the true ground-state wavefunction as it triv-\nially satisfies the initial conditions Eq. (3) and Eq. (4).\nThe exact KS potential, for this choice, reproduces the\ndensity evolution of the exact wavefunction propagat-\ning in the interacting system by propagating the exact\nwavefunction in a non-interacting system.\nGiven that at the initial time we are dealing with a\nground-state, one might expect that the adiabatic ap-\nproximation would be exact, at least initially, however\nthis is not the case. The adiabatic approximation re-\nturns the xc potential for a ground-state DFT calcula-\ntion, where the KS wavefunction is the ground-state KS\nwavefunction. If we were to start the KS calculation\nwith this ground-state KS wavefunction, the adiabatic\napproximation is then exact at the initial time.\nHowever the initial interacting wavefunction is not a\nground state of the non-interacting KS system, and so,\nchoosing this as the initial KS state means the adiabatic\napproximation will be in error from the start.\nThus care must be taken when using the short-hand\nthat there is no initial-state dependence when starting in\nthe ground-state. More precisely, what is meant is that-4 -2 0 2 400.20.40.60.8\nn*(x)\n-4 -2 0 2 4x0123\nvC[n*,Ψ∗,ΦGS](x)\n-10 0 10-20\n-4 0 408-10-5051000.20.40.6\nn*(x)\n-10-50510x00.511.5\nvC[n*,Ψ∗,ΦGS](x)\nFIG. 1: Top Panels: The first excited-state density, n∗(x),\nfor two non-interacting electrons in a one-dimensional har -\nmonic potential (left) and soft-Coulomb potential (right) .\nLower Panels: The corresponding correlation potentials,\nvC[n∗,Ψ∗,Φgs[n∗]](x), for an initial KS wavefunction chosen\nto have ground-state form but yielding the initial excited- state\ndensityn∗(case (b) of text). The insert shows the KS potential\n(solid line) compared with the external potential (dashed l ine),\nchosen as harmonic and soft-Coulomb, respectively. When th e\ninitial KS state is chosen to be Φ∗(case (a) in text), on the other\nhand, the correlation potential is zero.\n1)the initial KS orbitals are chosen to be those of the\nKS ground-state wavefunction, and 2)the dependence\non initial state for this case can be subsumed into the\ndensity via the theorems of ground-state DFT.\nIII. NON-INTERACTING ELECTRONS\nThe importance of initial-state dependence is strik-\ningly evident even for the hypothetical case of non-\ninteracting electrons. Due to ISD, the xc potential is not\nalways equal to zero, which may be surprising at first\nsight, given that there is no interaction. Although no-\none in their right mind would perform KS calculations\nfor non-interacting electrons for practical purposes, it i s\ninstructive to consider how the KS system behaves in\nthis case. In particular, the studies suggest what is the\nbest choice of KS initial state for a given true initial state\nwhen an adiabatic approximation is used. Since the adi-\nabatic approximation is designed for ground-states, is\nthe error least if we always choose a KS state that is a\nground-state?\nIn the following we consider the “true” system to be\nnon-interacting, i.e. we scale the electron-electron inte r-\naction by λin the limit that λ→0and consider terms\nzero-th order in λonly, e.g. the Hartree potential van-\nishes. We then consider the xc potential when the KS\nsystem is started in different allowed initial-states.\nConsider two non-interacting electrons prepared in\nan excited state Ψ∗and evolving in some potential\nvext(t). To start the KS evolution, any initial state of the\n3same density n, and first time-derivative, ˙n, zero in this\ncase, may be chosen. We consider the initial state choices\n(a) and (b) introduced in Sec. II, which become here: (a)\nΦ(0) = Ψ∗, and (b) Φ(0) = φ0(r)φ0(r′) = Φgs,φ0=/radicalbig\nn∗/2, wheren∗is the density of Ψ∗.\nFor choice (a), the exact xc potential vanishes,\nvXC[n;Ψ∗,Φ∗](r,t) = 0 . This can be most easily seen by\ninvoking the uniqueness property of the Runge-Gross\nmapping: for a fixed particle-interaction, only one po-\ntential (vext(t)) can yield a given density-evolution from\na given initial state. Since Φ∗= Ψ∗for choice (a), we\nconclude vXC(t) = 0 . This result can also be seen from\nthe general formula [29]:\n∇·[n(r,t)∇(vH(r,t)+vXC(r,t))] =q(r,t)−qS(r,t)(12)\nwhere\nq(r,t) =1\ni/planckover2pi1∇·/an}b∇acketle{tΨ(t)|[ˆj(r),ˆT+ˆVee]|Ψ(t)/an}b∇acket∇i}ht(13)\nqS(r,t) =1\ni/planckover2pi1∇·/an}b∇acketle{tΦ(t)|[ˆj(r),ˆT]|Φ(t)/an}b∇acket∇i}ht (14)\nandˆj(r)is the current-density operator. When the two\ninitial wavefunctions are the same, the RHS vanishes at\nt= 0 in the non-interacting limit, and we are left with\nvH(t= 0) = 0 , andvXC(t= 0) = 0 . Stepping forward in\ntime, it follows that vXC(t) = 0 for all times.\nNow consider choice (b). In this case,\nvXC[n;Ψ∗,Φgs](r,0)is non-zero: the RHS of Eq. (12) at\nt= 0 is no longer zero in the non-interacting limit, due\nto the difference in the initial wavefunction in qandqS.\nThat this is non-zero is also expected by the fact that vXC\nmust be such that vS=vext+vHXCevolves the initial\nΦgswith the same density for all time as the different\ninitial state Ψ∗has when evolved in vext. So, ISD leads\nto a non-zero xc potential, even when the electrons do\nnot interact.\nFor an explicit demonstration, consider the initial\ntime and realize that vXC(t= 0) depends entirely on the\ninitial states, as can be seen from Eq. (12), and not on\nthe choice of the external potential. Figure 1 plots this\npotential vXC(t= 0) for the first excited state of an ex-\nternal harmonic potential (1\n2x2, on the left), and a soft-\nCoulomb potential ( −2/√\nx2+1, on the right) in one-\ndimension. The top panels show the density, and the in-\nsets show the external potential in which Ψ∗is an eigen-\nstate, as well as the the KS potential vext+vCin which\nΦgsis an eigenstate (the ground-state, since there are no\nnodes). Because for two electrons in a spin-singlet with\none doubly-occupied orbital, vX=−vH/2 = 0 in the\nnon-interacting limit, this effect appears entirely in the\ncorrelation potential, and can be interpreted as static cor-\nrelation : due completely to the non-single-determinantal\nstructure of the true initial state. In case (a), this effect\nvanishes, because the KS initial state is chosen also not\nto be a SSD.\nWe now ask what an adiabatic potential would give:approximating vXC[n;Ψ(0),Φ(0)] by a ground-state po-\ntentialvgs\nXC[n]. To distinguish errors arising from the\nchoice of approximate ground-state functional itself,\nwe consider an “adiabatically exact” potential [21, 25],\nvadia−ex\nXC[n]. We shall define this generally in the next sec-\ntion, but for now it suffices to define it such that if both\nthe true and KS wavefunctions at all times were in fact\nground states of some potential, then the exact xc po-\ntential is the exact ground-state one, vXC[n;Ψgs,Φgs] =\nvadia−ex\nXC[n]. Consider again just the initial time, t=\n0. In the non-interacting limit, for both the true and\nKS wavefunctions to be ground-states that have the\ndensity of the excited state Ψ∗, thenΨ(0) = Φ(0) =/radicalbig\nn0(r)n0(r′)/2, and we rapidly conclude\nthat\nvXC[n,Ψgs,Φgs](r,t) =vadia−ex\nXC[n](r,t) = 0 (15)\nin the non-interacting limit. Applying now this adia-\nbatic approximation to the case of the initial true excited\nstateΨ∗considered above, we conclude that the adiabatic\napproximation is exact for choice (a) , when the initial KS\nstate is also chosen excited, as we had argued there the\nexactvXCalso vanishes. It is inaccurate for choice (b),\nwhen the initial KS state is chosen to be a ground-state\none, where the exact correlation potential is non-zero.\nThis study suggests that in the general interacting\ncase, errors in adiabatic TDDFT will be least when an\ninitial KS state that most closely resembles the configu-\nration of the true excited state is chosen. This expecta-\ntion is indeed borne out in the following studies.\nIV . MODEL TWO-ELECTRON SOFT-COULOMB\nINTERACTING SYSTEMS\nThe case of non-interacting electrons illustrated the\nimportance of ISD, while offering a hopeful diagnosis\nfor the adiabatic approximation; the latter becomes ex-\nact if the initial state is chosen appropriately. Of course,\nelectrons do interact, and now we turn to studying the\neffect of ISD on the dynamics of interacting systems.\nWhen running an approximate TDKS calculation\nstarting in an excited state of the interacting problem,\nthree separate sources of error come into play. First,\nexcited states of the exact ground-state KS potential\nvS[n0](r)do not have the same densities as interacting\nexcited states of the potential vext(r)whose ground-state\ndensity is n0(r). Yet these are usually the ones chosen in\npractice. We discuss this problem in Section IV A. The\nsecond source of error is the central one for this paper:\nthe use of the adiabatic approximation, when, from the\nstart, we have an excited state. We consider first the\n“adiabatically exact” approximation, mentioned also in\nSec. III, but considered now for interacting electrons, in\nSection IV B. This will allow us to separate the errors\nfrom the choice of ground-state functional approxima-\ntion used in the adiabatic approximation, which is the\n4-10 -5 0 5 10x00.20.40.6n*(x)\nFIG. 2: Exact excited-state density (solid line) compared t o the\nexact excited-state density of the exact ground-state KS po ten-\ntial (dashed line) for the soft-Coulomb Helium atom.\nthird issue. In Sections IV D, and V, we will study the\neffect that missing initial-state dependence has in prac-\ntice, on electron dynamics in model two-electron sys-\ntems when begun in the first singlet excited state of the\nsystem.\nIn our model one-dimensional Helium atom, the two\nelectrons interact via a soft-Coulomb electron-electron\ninteraction, vee(x1,x2) = 1//radicalbig\n(x1−x2)2+1and live in\nan external soft-Coulomb potential,\nvext(x) =−2//radicalbig\nx2+1. (16)\nSuch a model, straightforward to solve numerically, is\npopular in understanding and analyzing electron inter-\nactions, in both the ground-state and in strong-field dy-\nnamics [31], inside and outside the density-functional\ncommunity. In our model of the diatomic LiH molecule,\nthe soft-Coulomb interaction is also used, while the ex-\nternal potential has an asymmetric soft-Coulomb well\nstructure (Sec. V).\nBefore proceding, we make a computational note. The\ntrue dynamics are propagated on a real-space grid using\nthe exponent midpoint rule Taylor-expanded to fourth\norder. Imaginary time propagation along with Gram-\nSchmidt orthogonalisation is first performed to find\nthe lowest eigenstates. The adiabatic exact-exchange\n(AEXX) dynamics uses the Crank-Nicolson method\nwith an explicit predictor step for the Hartree-exchange\npotential. In both cases, a timestep of 0.001is used with\na grid spacing of 0.1. When possible these calculations\nwere tested for accuracy against the parallelized code\nOCTOPUS[32], which was also used for LDA and LDA-\nSIC runs.\nA. The Excited-State Density\nThe theorems of TDDFT require that the KS initial\nstate has the same n(r,0)and˙n(r,0)as the true inter-\nacting system (Sec. II). In this section we discuss the\ndifficulty of fulfilling this requirement when the initial\nstate is not a ground-state.-10 -5 0 5 10-0.8-0.400.4vXC[n*,Ψ∗,Φ∗](x)\nvXC[n*,Ψ∗,Φgs](x)\n-10 -5 0 5 10x-2-1.5-1-0.50\nvS[n*,Φ∗](x)\nvS[n](x)\nvS[n*](x)\nvext(x)\nFIG. 3: Top Panel: The exact xc potentials for the two choices\nof initial state, Φ∗(solid line) and Φgs(dot-dashed line). Lower\nPanel: The KS potentials for which the two initial state choi ces,\nΦ∗(solid line) and Φgs(dot-dashed line) are eigenstates. Also\nshown are the exact ground-state KS potential (dashed line)\nfor the soft-Coulomb Helium potential (dotted line).\nWhen the true initial state is a ground-state, the nat-\nural and usual choice for the KS initial state is the non-\ninteracting ground-state, and this can be found by solv-\ning the ground-state KS equations. However, if we want\nto compute the dynamics of an excited state, we en-\ncounter the problem that there is no DFT scheme to find\nexcited state densities[34]. Furthermore, even if we use\na more computationally expensive higher-level wave-\nfunction method to calculate the state Ψ(0) , we still must\nthen choose an initial Φ(0) in which to start the KS cal-\nculation. If we start the interacting system in a first ex-\ncited state of some vext(r), the results of the earlier sec-\ntions suggest a good choice for adiabatic TDDFT is to\nstart the KS system in a corresponding non-interacting\nexcited state where one electron is excited from the high-\nest occupied to lowest unoccupied orbital of some ˜vS\n(as in form (a) of Sec. II). This ˜vS(r)however cannot\nbe the ground-state KS potential vS(r)corresponding\nto the interacting vext, because this would not satisfy\nEq. (3):vS(r)yields the same ground-state density in a\nnon-interacting system that vext(r)yields in the interact-\ning one, but the densities of their excited-states are dif-\nferent. Yet, the usual practice isto use the correspond-\ning excited-state of the ground-state KS potential [33].\nThis is partly because the density of the interacting ex-\ncited state is not often known anyway. In Fig. 2 we\ncompare the exact density of the first excited interact-\ning singlet state in the soft-Coloumb He atom (Eq. 16)\nwith the density of the excited state of the correspond-\ning ground-state KS potential. Although matching the\ngeneral shape, the latter is too narrow and misses some\nstructure.\n5To be able to separate the error from not quite hav-\ning the exact initial density from the error from using\nan adiabatic approximation, we now search for a non-\ninteracting system where the first singlet excited-state\ndensity isexactly equal to the true excited-state den-\nsityn∗, for our model He atom. Several such potentials\nmay exist [34], and in the lower panel of Fig. 3, we plot\none such KS potential, denoted vS[n∗,Φ∗](x)and com-\npare to the KS potential where n(x)is the exact ground-\nstate density vS[n](x,0)of the model He atom. (The first\nexcited state densities of these potentials are precisely\nthose shown in Fig. 2). We also show vS[n∗](x,0), i.e.\nthe KS potential for which n∗(x)is the density of the\nground-state. This corresponds to choice (b) in Sec. II\nfor the initial KS state.\nOnce a valid KS state is found satisfying Eqs. (3)\nand (4), the initial xc potential is completely determined:\nEqs. (12) and (13) show that at the initial time, vXCis a\nfunctional of just the two initial states Ψ(0) andΦ(0) .\nThis can be added to any external potential, prescribed\nby the physical problem at hand, and Hartree-potential,\ndetermined by the initial density, to start the evolution.\nIn other words, the choice of the initial KS state funda-\nmentally points to an xc potential, rather than to a KS\npotential, and so in the top panel of Fig. 3 we plot the\nxc potentials corresponding to choices (a) and (b) of the\ninitial KS state. In the following examples, we take the\ninitialvextto be that of the soft-Coulomb Eq. (16); some-\ntimes subsequently an external field is added.\nWe have therefore, now found initial states of the form\n(a) and (b) of Sec II, both yielding the same density\nas that of the true excited-state, meeting the conditions\nEq. (3) and (4). We can now move forward to investigate\nthe time-dependent properties.\nB. The Exact Adiabatic Approximation\nWe next move to the error introduced by using the\nadiabatic approximation.\nTo isolate this error we will calculate the adiabatically\nexact potential at the initial time, and start in a KS state\nof the exact same density as the true state, as found in\nSection IV A.\nThe adiabatically exact potential is defined by [25]\nvadia−ex\nXC[n] =vadia−all\nS[n]−vadia−all\next[n]−vH[n](17)\nwherevadia−all\nS[n](r,t)is the potential for which\nn(r,t)is the non-interacting ground state density and\nvadia−all\next[n](x)is the potential for which interacting elec-\ntrons have n(r,t)as their ground-state density. If both\nthe true and KS wavefunctions are actually always in\nsome ground-state, then Eq. (17) becomes the exact xc\npotential. We shall consider this potential for initially\nexcited states, only at the initial time.\nThe non-interacting potential vadia−all\nS[n∗](r,0)may\nbe easily found by inverting the KS equation for a-10 -5 0 5 10-0.8-0.400.4vXC[n*,Ψ∗,Φ∗](x,0)\nvXC[n*,Ψ∗,Φgs ](x,0)\nvXCadia-ex [n*](x,0)\n-10 -5 0 5 10x-1.5-1-0.50\nvS[n*,Φ∗](x,0)\nvS[n*,Φgs](x,0)\nvSadia (x,0)\nFIG. 4: Top Panel:The exact xc potentials and the exact adia-\nbatic xc potential. Lower Panel: The exact and adiabaticall y-\nexact KS potentials.\n-20 -10 0 10 20x00.40.8n(x)\n-20 -10 0 10 20x00.40.8n(x)\n-20 -10 0 10 20x00.40.8n(x)\n-20 -10 0 10 20x00.40.8n(x)T=10au\nT=20au T=15auT=5au\nFIG. 5: The density at times 5au,10au,15au, and20au for ini-\ntial states Φ∗(solid line) and Φgs(dashed line) propagating in\nthe static exact-adiabatic initial potential. Also shown i s the\nexact excited-state density (dotted line) which is the exac t so-\nlution at each time.\ndoubly-occupied orbital, φ(r) =/radicalbig\nn∗(r)/2wheren∗(r)\nis the density of the initial state Ψ∗.\nThe interacting potential, vadia−all\next[n](r,0)is solved for\nusing an iterative technique whereby the external po-\ntential is updated based on the difference between the\nground-state density of the current iteration and the\ndensity we are targeting. This is based on the inversion\nalgorithm of Ref. 35, but generalized to the interacting\ncase[25]. We also increase the update to the potential in\nregions of low density by simply using the inverse of the\ndensity (up to a maximum value) as a weighting factor,\nsimilar to Ref. 30. When the density converges to the\ntarget density, we have found vadia−all\next[n](x).\n6The dotted line in the lower panel of Fig. 4, shows\nat the initial time, the full KS potential with the adia-\nbatically exact xc potential, namely vadia\nS(x) =vext(x) +\nvH[n](x)+vadia−ex\nXC[n](x), choosing vext(t= 0) as the soft-\nCoulomb potential of Eq. 16, for the first excited singlet\nstateΨ∗. Alongside, we compare this to the exact KS po-\ntentials found in Sec. IV A (see Fig. 3), for the initial state\nchoices of case (a) and (b) respectively. The adiabatically\nexact KS potential vadia\nS(x)tracks the shape of the ex-\nact KS potential vS[n∗,Ψ∗,Φ∗]although misses structure.\nThe difference between them is the difference in their xc\npotentials shown in the upper panel of Fig. 4. The dif-\nference is much larger for the comparison with the exact\nKS potential when the initial KS state is chosen as the\nground-state, (i.e. vadia−ex\nXC[n∗]is closer to vXC[n∗,Ψ∗,Φ∗]\nthan it is to vXC[n∗,Ψ∗,Φgs[n∗]]); in this case, there is a\nlarge static-correlation contribution to the exact xc po-\ntential. The results here are consistent with the non-\ninteracting case discussed in Sec. III: with interaction,\nthe adiabatic approximation is no longer exact for the\nchoice of Φ∗, however it is still the better choice over\nΦgs.\nThe fact that the adiabatically exact potential does not\nequal the true potential, even at the initial time, leads to\nerroneous dynamics. To illustrate this effect we propa-\ngate our exact initial wavefunctions in the adiabatically\nexact potential and compare with exact dynamics. Since\ncalculating at each time is computationally demanding\nand delicate, we will simply evolve in time without any\nadditional perturbing potentials, and hold the adiabatic\npotential fixed to its initial value. The idea is that since\nthe exact evolution is static, the adiabatically exact po-\ntential is static and equals its initial value at all times; i f\nthis was a good approximation, it would yield density-\ndynamics that were close to static. The deviation of the\nadiabatic propagation from the exact is a measure of its\nerror. Note such a calculation does not treat the adia-\nbatic potential self-consistently; later calculations wi th\napproximate adiabatic functionals suggest having self-\nconsistency decreases the error somewhat.\nIn Fig. 5, we plot snapshots of the density at 5au in-\ntervals until T= 20 au for the two choices of exact initial\nstates discussed above, evolving in the fixed adiabati-\ncally exact potential plotted in Fig. 4. As each time we\nshow the exact static excited-state density. While both\nchoices show density ’leaking’ out, the Φ∗choice is more\nsuccessful in preventing this, as might be expected from\nthe above discussions, but cannot stop the density melt-\ning away in its outer regions. The Φgs[n∗]choice is poor\nfrom the start, jettisoning density outwards and becom-\ning too narrow.\nThis error in the adiabatically exact evolution is\ncaused entirely by initial-state dependence: for the anal-\nogous calculation beginning in the interacting ground-\nstate and starting with the ground-state KS wavefunc-\ntion, the adiabatic approximation would be exact at all\ntimes when there are no perturbing fields.-15 -10 -5 0 5 10 15x-0.6-0.5-0.4-0.3-0.2-0.10vXC[n](x)\nADIA-EX\nAEXX\nFIG. 6: The exact adiabatic xc (solid) and AEXX (dashed) po-\ntentials evaluated on the true excited-state density n∗. (Note\nthat the solid line here is the same as the dotted line in the to p\npanel of Fig. 4\n-15 -10 -5 0 5 10 15x-0.4-0.200.20.40.60.8φ(x)EXACT\nKS\nEXX\nLDA\nFIG. 7: The ground and first excited orbitals from ground-sta te\nEXX and LDA, compared to those from the exact GS KS poten-\ntial. Also shown are those orbitals found in Sec. IV A of this\nconfiguration that yield the exact excited-state density.\nC. Approximate GS functionals\nFinally we look at the third source of error, namely us-\ning an approximate ground-state xc functional instead\nof the exact one. This error contributes at both the\nground-state level in obtaining the initial orbitals, and\nin the adiabatic functional when computing dynamics.\nWe first turn our attention to the latter and investigate\nthe adiabatic exact-exchange functional, AEXX, which,\nfor a two-electron system, is simply vAEXX\nXC[n](r,t) =\n−vH[n](r,t)/2. In Fig. 6 we compare the adiabatically ex-\nact xc potential found in the previous section to AEXX,\nat the initial time. The undulations of the exact adiabatic\nxc potential that are missing in the AEXX represent cor-\nrelation; they are relatively small in this particular case\nonce added to the external potential, so we expect that\nAEXX dynamics fairly approximates the adiabatically\nexact dynamics in this situation at least for short times.\nWe now return to the error of not having the correct\ninitial density (Sec. IV A).\nIn a usual TDDFT calculation, a ground-state DFT cal-\nculation is first performed and the orbitals from this are\nused to create the initial wavefunction. So we would\nstart with the ground-state KS density of Fig. 2 except\n7here there is a further error that will be made, as an ap-\nproximate ground-state xc functional is used instead of\nthe exact one.\nIn Fig. 7 we plot the ground- and first-excited- KS\norbitals from ground-state DFT calculations using EXX\nand LDA for the soft-Coulomb helium atom, and com-\npare to the exact KS orbitals. We also plot, for com-\npleteness, the pair of orbitals found for this configu-\nration that yield the exact interacting excited-state den-\nsity, i.e. these are the exact orbitals which, when singly-\noccupied, make up the densities shown in Fig. 2.\nIf we first compare the approximate orbitals to the ex-\nact KS orbitals of the soft-Coulomb potential, we find\nboth LDA and EXX perform very well, particularly for\nthe ground-state orbital. Their ground-state KS poten-\ntials are however quite different: it is well known that\nthe LDA potential is much too shallow, with a much too\nrapid exponential decay away from the nucleus, while\nthe EXX is much closer to the exact, with the correct\nasymptotic decay. This is reflected in the first-excited\norbital in LDA, which is more diffuse than EXX and the\nfirst-excited orbital of the exact soft-Coulomb KS po-\ntential. The asymptotic behavior will be important es-\npecially when we turn on an electric field, so we will\n(mostly) use the EXX orbitals to build our initial KS\nwavefunctions.\nIn the next section we will add the spin-symmetry-\nbroken initial state (c) introduced in Section II to our\ninvestigations; while no longer a spin eigenstate like\nthe exact case or Φ∗andΦgs, this still has the same to-\ntal density as the exact system. We will simply con-\nstruct this from the same EXX orbitals composing Φ∗,\nbut will evolve the two orbitals using different xc po-\ntentials, v↑\nXCandv↓\nXC. For exact-exchange, we have\nv↑↓\nHXC[n](x) =vH[n↓↑](x)for two electrons. We are\nhowever notdoing spin-DFT since the individual spin-\ndensities are not physical ones; we only ever consider\ntheir sum as observable. (Indeed, even the initial spin-\ndensities are wrong). The point of considering such an\ninitial state and dynamics is that, for two electrons, it\nis an example of dynamics in orbital-specific potentials,\nas would occur in generalized KS approaches [36–38].\nOrbital-dependent functionals require an optimized ef-\nfective potential approach to find a single potential for\nall the KS orbitals to live in. This procedure is nu-\nmerically very intensive, so often a generalized KS ap-\nproach is used, that relaxes the condition that all orbitals\nevolve under the same potential; moreover, the latter\nhas shown to have certain advantages over the OEP\nin certain cases, e.g. better band-gaps. The interest in\norbital-dependent functionals is that that they can work\nless hard to capture both spatially-non-local and time\nnon-local- density-dependence since the orbitals them-\nselves are non-local functionals of the density.\nWe note nevertheless that it can be shown there is no\nISD for two-electron dynamics with spin-TDDFT; this\nfollows straightforwardly generalizing the arguments of\nRef. 27 to each spin-density. This is not the case for our0 50 100 150t-0.08-0.0400.040.08d(t)EXACT\nEXX\nLDASIC\nLDA\nFIG. 8: The exact dipole moment and those from TDDFT cal-\nculations starting in the self-consistent ground-state KS wave-\nfunction and propagated with the corresponding adiabatic a p-\nproximation, all under the influence of the electric field de-\nscribed in the text.\n0 50 100 150t-4-3-2-10123d(t)\nΦ∗ EXPHI\nΦ∗ EXX\nFIG. 9: The dipole moments when beginning in Φ∗composed\nof EXX orbitals (dashed line) and the exact orbitals of Sec. I V A\n(solid line) in the electric field (see text).\nspin-broken case, however, as here we are notattempt-\ning to reproduce the exact spin-densities but rather the\ntotal density evolution; only n(r) =n↑+n↓will be con-\nsidered meaningful.\nHaving delineated and explored the three possible\nsources of error in an adiabatic TDDFT calculation of\nexcited state dynamics, we are now ready to run a typi-\ncal adiabatic TDDFT calculation on our model systems,\nstarting with our different choices of initial KS states,\nand compare with the exact result.\nD. Propagation in an electric field\nWe simulate electric-field driven dynamics in the soft-\nCoulomb helium model, using adiabatic TDDFT. We ap-\nply a relatively weak oscillating electric field field of am-\nplitude0.01au with an off-resonant frequency of 0.2a.u.\nWe run for 6-cycles including a trapezoidal envelope\nconsisting of a 2-cycle linear switch on and a 2-cycle\nswitch off.\nAs a point of reference, we first look at the perfor-\nmance of TDDFT when starting in the ground state . We\n80 50 100 150t-2-1012d(t)\nEXACT\nΦ∗\nΦ∗\nSB\nΦgs\nFIG. 10: The dipole moments for dynamics in the given electri c\nfield starting in the Φ∗,Φ∗\nSB,Φgsinitial KS wavefunctions and\nusing AEXX, compared to exact.\n0 50 100 150t-1012d(t)EXACT\nLSDA\nEXX\nLSDASIC\nFIG. 11: Dynamics for the spin-broken case, Φ∗\nSB, for differ-\nent orbital-specific adiabatic functionals, EXX, LSDASIC, and\nLSDA, compared to the exact.\npropagate the doubly-occupied EXX ground-state or-\nbital in the electric field, using AEXX, and compare to\nthe exact dynamics of the initial interacting ground-\nstate. The resulting dipole moment is very accurate as\ncan be seen in Fig. 8. In fact, almost exact dynam-\nics can be achieved in this case with an adiabatic ap-\nproximation: the same figure shows the result of using\nthe adiabatic self-interaction corrected LDA approxima-\ntion (LDASIC) to propagate the LDASIC ground-state\norbital. The dipole moment lies practically on top of the\nexact curve. Including correlation in a self-interaction\nfree functional therefore improves over exact exchange,\nbut an adiabatic approximation is certainly adequate in\nthis case. We also show for comparison the LDA result.\nThe adiabatic approximation is however not so rosy,\nas we shall see, in the case of an initially excited state.\nThe initial interacting state is the first singlet excited\nstate (E=−1.705au).\nThe typical TDDFT calculation will begin in an ex-\ncited state determined by the orbitals of the corre-\nsponding ground-state KS system, but as discussed in\nSec. IV A, this does not have quite the right density to\nstart with. Since in our model system, we found an ini-\ntial KS state of the correct density, we first check theerror that using the AEXX orbitals make. We plot in\nFig. 9 the dipole moments for AEXX calculations start-\ning in the Φ∗initial state with exact orbitals and with\nthe ground-state EXX orbitals (i.e. using the solid and\ndotted orbitals of Figure 7 respectively). As\npreviously anticipated, the two calculations perform\nsimilarly, especially at shorter times.\nAt last we are ready to study the excited-state elec-\ntron dynamics, using, as in practical calculations, an ap-\nproximate ground-state KS potential (in this case EXX)\nto generate the orbitals composing the various initial\nstate choices. In Fig. 10, we plot the dipole mo-\nment for each choice of initial state. The spin-broken-\nsymmetry case gives the best dynamics; as mentioned\nearlier, functionals expressed directly in terms of instan -\ntaneous orbitals automatically contain time non-local\nand spatially-non-local density-information. In this par -\nticular case the orbital-specific EXX potentials, obtained\nfrom spin-scaling the (spin-restricted) AEXX potential,\nare self-interaction-free, while the latter is not for the\ncase of the excited two-orbital singlet state.\nUnlike in the propagation of the ground-state, adding\ncorrelation to the adiabatic potential does not signifi-\ncantly improve the results, as can be seen in Fig. 11;\nthe symmetry-broken LSDASIC and EXX calculations\nare both off by roughly the same amount. Fig. 11 also\nshows the LSDA result, clearly worse than the others by\ncomparison.\nTheΦ∗initial state has the correct spin symmetry and\nis closest in character to the true excited state, being a\ndouble Slater determinant. It gives better dynamics than\ntheΦgs[n∗]initial state, whose dynamics are completely\nincorrect. This is consistent with the results in the previ-\nous sections. We can say with confidence that it is ISD,\nand its underlying static correlation that it causing poor\ndynamics for this doubly-occupied-singlet case.\nWe can understand the poor result for the initial state\nof the Eq. (8) form also by looking at the exact adiabatic\nxc potential and vXC[n∗,Ψ∗,Φgs[n∗]](x), both shown in\nFig. 4. This form treats the density as if it were a ground-\nstate and uses one doubly occupied orbital, inversion of\nthis orbital yields vadia−all\nS[n](x), i.e. this is the potential\nin which the wavefunction is static at the initial time.\nHowever in a calculation using the adiabatic approxi-\nmation , it will fall into a much deeper potential, dras-\ntically effecting its dynamics. Without the initial-state\ndependence of the xc potential giving a bump similar to\nFig. 1, it will perform poorly.\nV . MODEL LIH DYNAMICS\nOur final example is perhaps the most topical one\nwhere the initial state is an excited one: coupled\nelectron-ion dynamics after photo-excitation. The ini-\ntial excitation is assumed to place the initial nuclear\nwavepacket on an excited potential energy surface, ver-\ntically up from its ground-state equilibrium. Field-free\n90 2 4 6 8 10 12\nR-1.4-1.3-1.2-1.1-1-0.9ε(R)\nFIG. 12: The exact Born-Oppenheimer potential energy sur-\nfaces (solid lines) and the EXX KS Born-Oppenheimer surface s\n(dashed lines). Note that due to degeneracy between the low-\nest two surfaces, these calculations cannot be converged fo r\nlarge values of R, the interatomic separation.\ndynamics on the excited surface ensues; usually the nu-\nclei are treated classically, coupled in either an Ehren-\nfest or surface-hopping scheme to the quantum elec-\ntron dynamics. For Ehrenfest dynamics in the TDDFT\nframework [33], and the simplest type of surface-\nhopping [39], the electrons start in the KS excited-state\nobtained simply from promoting an electron from the\nhighest occupied orbital in the ground-state KS config-\nuration to a virtual orbital. This initial state is then\nevolved in the time-dependent external potential caused\nby the moving ions. The dynamics of the ions is given by\nsimple Newtonian mechanics where the electronic sys-\ntem provides a force/integraltextn(r,t)∇Rvext(r;{R}), where{R}\nrepresent the nuclear coordinates. All the three errors\nmentioned earlier therefore raise their heads: the initial\ndensity is not that of the true excited state (even if the ex-\nact xc potential was used), an adiabatic approximation\nis used to propagate it, and this involves an approxi-\nmate ground-state functional. It should be noted that in\nthe more accurate surface-hopping method of Ref. [40],\nwhere correct TDDFT surfaces are used to provide the\nnuclear forces [41], the first problem does not arise; the\nmethod in fact circumvents having to define the elec-\ntronic state.\nWe will use the parameters found in Ref. 42 for a\none-dimensional two-electron lithium hydride model,\nwhere\nvext(x) =−1//radicalbig\nx2+0.7−1//radicalbig\nx2+2.25 (18)\nandvnn(R) =−1.0/√\nR2+1.95. We integrate Newton’s\nequations of motion using the leapfrog algorithm, and\nfor simplicity, we use the same time step for both elec-\ntrons and ions.\nThe Born-Oppenheimer potential energy surfaces are\nshown in Fig. 12 for both the exact system and those\nof the bare KS system calculated within spin restricted\nEXX.\nThe ground state surface has a minimum at R= 1.55,\nwhereas the EXX has the gs equilibrium at R= 1.45.0 1000 2000 3000 4000 5000t0510152025R(t)0 300 6000510\nΦ*\nSB\nΦ*\nΦgsEXACT\nFIG. 13: The interatomic distance, R(t), for Ehrenfest dynam-\nics starting in the first excited-state and propagated freel y.\nThe various KS initial wavefunctions ( Φ∗,Φ∗\nSB,Φgs) are prop-\nagated with the adiabatic exact exchange approximation. In -\nsert: short-time dynamics for the various initial states.\nNear equilibrium the surfaces are somewhat similar.\nThe spin-restricted EXX surfaces however encounter the\nwell-known fractional charge problem as Rincreases,\nwith the calculations becoming extremely difficult to\nconverge and the lowest two surfaces collapsing onto\neach other. In the exact surface, an avoided crossing\ncan be seen between the ground and first excited state\naroundR= 4, where the latter establishes ionic charac-\nter, eventually decaying as −0.896−1/x, while the neu-\ntral dissociation curve flattens out. In fact there are mul-\ntiple avoided crossings at larger distances, the next be-\ning around R= 10 . Unrestricted EXX calculations break\nthe spin-symmetry of the true ground-state at a critical\nseparation, but describe dissociation and the shape of\nthe potential energy surfaces at larger distances qualita-\ntively [43, 44].\nWe vertically excite the system at the ground-state\nequilibrium geometry into the first excited singlet state\nand then begin the propagation with the initial nuclear\nmomentum chosen as zero. In Fig. 13, we show the in-\nternuclear separation as a function of time for our three\nchoices of initial states, calculated within EXX. For Φ∗\nwe make the usual practise of taking φ0in Eqs. 7 and\n10 as the HOMO and φ1as the LUMO; likewise for the\nspin-symmetry-broken version Φ∗\nSB. The density of this\nstaten∗is then used to define the doubly-occupied or-\nbital in the SSD state Φgs[n∗](Eq. 8).\nFor this particular system, the exact system remains\non the first excited surface; the nuclear coordinate os-\ncillates between the turning points at R= 1.5and\nR= 11.165. The ground state Φgscompletely fails to\ncapture this behavior, making only very small oscilla-\ntions around its initial position. The excited states Φ∗\nandΦ∗\nSBshow reasonable agreement for short times,\nwith the spin-broken state once again edging out the\nother. For longer times however, the spin-broken state is\ndramatically wrong, yielding nuclei moving away from\neach other with constant speed. The excited state Φ∗\nshows behavior more qualitatively exact but, because of\nthe deviation of its potential energy curve compared to\n10the exact one, the oscillation period is quite wrong.\nThe conclusions regarding ISD in this example, at\nleast at short times, are consistent with the results\nthroughout this paper and support the idea that when\nusing an adiabatic approximation choosing a KS initial-\nstate with a configuration close to that of the exact in-\nteracting initial state yields the better result. However,\nthe behavior at longer times in this problem highlights,\nabove all, the need for accurate ground-state function-\nals: EXX produces poor nuclear dynamics when Φ∗is\nchosen as the KS state primarily because the shape of\nits potential energy surfaces is wrong – the ground state\nas well as the excited state ones. Any adiabatic approx-\nimation that does not have strong correlation will not\nperform well. Despite giving improved potential energy\nsurfaces, the spin-broken EXX approach fails for dynam-\nics for longer times. We believe this is because as the\nmolecule dissociates, symmetry-breaking localizes each\nelectron on one nucleus or the other, with each evolv-\ning according to a different xc potential; each atom sees\njust a neutral atom, leading to a net zero potential, and\na constant velocity (somewhat as if the molecule was\ndissociating on the ground-state surface). It should be\nnoted that the spin-broken solution is not evolving on\nthe spin-broken PES. e.g. even at the initial time, at the\nequilibrium geometry there is no symmetry-breaking in\na spin-unrestricted approach, so the spin-broken surface\nis on top of the spin-restricted surface showed in the\nfigure. However the spin-broken wavefunction (c) is\nevolved using different potentials for each orbital, un-\nlike the evolution dictated by either the spin-restricted\nEXX surface or the unrestricted EXX (the symmetry-\nbreaking point occurs around R = 2.5).\nVI. CONCLUSIONS AND OUTLOOK\nUntil relatively recently, ISD fell in the realm of a the-\noretical curiosity, but due to an increasing number of\ntopical applications beginning with the system not in its\nground-state, ISD is now also of much practical concern.\nAlmost all calculations today use adiabatic functionals\nthat neglect the ISD that the exact functionals are known\nto have.\nBy considering several choices of initial KS states\nwhen the initial interacting state is excited, we explored\nthe effect of ISD on dynamics in several exactly-solvable\nmodel systems and the performance of the adiabatic ap-\nproximation. We noted there are three sources of error:\nexcited KS initial states do not have the same density\nas the corresponding true excited states even when the\nexact functional is used, the use of the adiabatic approx-imation to propagate the initial state, and the ground-\nstate functional generating the adiabatic approximation\nitself being approximate. By separating these errors\nfrom each other, we were able to properly assign how\nbadly they impacted the dynamics in model systems, al-\nlowing us to see the effect of ISD more clearly.\nWhen the initial KS state has a configuration that is\nsignificantly different than the true state, the adiabatic\napproximation fails severely – even for non-interacting\nelectrons, where there is a significant error that can be\ninterpreted as a static correlation effect. The optimal\nchoice for the KS initial state is one whose configuration\nis most similar to the true excited state; this is in fact\nthe usual practise in recent calculations (e.g. Ref. [33]).\nThe error in using an adiabatic xc functional to propa-\ngate such a state, however, is significantly larger than\nthe error the same functional produces when describ-\ning the propagation of an initial true ground-state, as\ndemonstrated by the model soft-Coulomb helium atom\nexample. For two electrons, the spin-broken initial\nKS state, with the two orbitals evolved under different\nspin-decomposed potentials (depending on the instan-\ntaneous spin-densities), often appeared to be the best\nchoice, suggesting that orbital-specific potentials (al-\nlowed by the generalized KS scheme) could be a useful\nfuture avenue of research for TDDFT. However, how to\nobtain such potentials in a general N-electron case is not\nobvious (at least without the need for empirical param-\neters). Generally, orbital-dependent functionals treate d\nwithin the OEP may be promising for these problems\nwhere memory-(of the density)-dependence, including\nISD at the KS level only, is naturally captured by the\nKS orbitals; again, devising suitable orbital-dependent\nfunctionals with the appropriate ISD is a direction for\nfuture research.\nThe example of coupled electron-nuclear dynamics\nin the model LiH molecule, although supporting the\nearlier results of the optimal choice of a KS initial\nstate, above all, highlighted the need for more accu-\nrate ground-state functional approximations for disso-\nciation.\nWe do not provide in this work an approximation that\nincludes ISD, both of the true interacting state and the\nKS state; this is likely a difficult task. But what is clear\nfrom the results in this paper, and the stage of the field,\nis that now is the right time to address this issue.\nAcknowledgments Financial support from the National\nScience Foundation (CHE-1152784), and a grant of com-\nputer time from the CUNY High Performance Comput-\ning Center under NSF Grants CNS-0855217 and CNS-\n0958379, are gratefully acknowledged.\n[1] E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984).\n[2]Fundamentals of Time-Dependent Density Functional The-\nory, (Lecture Notes in Physics 837) , eds. M.A.L. Marques,N.T. Maitra, F. Nogueira, E.K.U. Gross, and A. Rubio,\n(Springer-Verlag, Berlin, Heidelberg, 2012).\n[3] P . Hohenberg and W. Kohn, Phys. Rev. 136, B 864 (1964).\n11[4] W. Kohn and L.J. Sham, Phys. Rev. 140, A 1133 (1965).\n[5] W. Kohn, Rev. Mod. Phys. 71, 1253 (1999).\n[6] E.K.U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850\n(1985); 57, 923 (1986) (E).\n[7] J.F. Dobson, Phys. Rev. Lett. 73, 2244 (1994).\n[8] G. Vignale and W. 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Chem. Phys. 129, 124108 (2008).\n[19] J. Gavnholt, A. Rubio, T. Olsen, K. S. Thygesen, and J.\nSchiøtz, Phys. Rev. B 79, 195405 (2009).\n[20] I. DAmico and G. Vignale, Phys. Rev. B 59, 7876 (1999).\n[21] P . Hessler, N.T. Maitra, and K. Burke, J. Chem. Phys. 117,\n72 (2002).\n[22] N. T. Maitra and K. Burke, Chem. Phys. Lett. 359, 237\n(2002).\n[23] C.A. Ullrich, J. Chem. Phys. 125, 234108, (2006).\n[24] J. I. Fuks, N. Helbig, I. V . Tokatly, and A. Rubio, Phys. R ev.\nB.84, 075107 (2011).\n[25] M. Thiele, E.K.U. Gross, S. Kummel, Phys. Rev. Lett. 100,\n153004 (2008).\n[26] R. Baer, J. Mol. Struct.: THEOCHEM 914, 19 (2009).\n[27] N.T. Maitra and K. Burke, Phys. Rev. A 63, 042501 (2001);64039901(E).\n[28] N.T. Maitra, K. Burke, and C. Woodward, Phys. Rev. Lett.\n89, 023002 (2002).\n[29] R. van Leeuwen, Phys. Rev. Lett. 82, 3863 (1999).\n[30] I. Dreissigacker and M. Lein, Chem. Phys. 391, 143 (2011).\n[31] J. Javanainen, J.H. Eberly, and Q. Su, Phys. Rev. A 38, 3430\n(1988); W.-C. 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Rubio, (Springer, Berlin, Heidelberg, 2012).\n[38] A. G¨ orling and M. Levy, J. Chem. Phys. 106, 2675 (1997).\n[39] C.F. Craig, W.R. Duncan, and O.V . Prezhdo, Phys. Rev.\nLett. 95, 163001 (2005).\n[40] E. Tapavicza, I. Tavernelli, and U. Rothlisberger, Phy s.\nRev. Lett. 98, 023001 (2007).\n[41] N. T. Maitra, J. Chem. Phys. 125, 014110 (2006).\n[42] D.G. Tempel, T.J. Mart´ ınez, and N.T. Maitra, J. Chem. T he-\nory and Comput., 5, 770 (2009).\n[43] M.E. Casida, F. Gutierrez, J. Guan, F-X. Gadea, D.\nSalahub, and J-P . Daudey, J. Chem. Phys. 113, 7062 (2000).\n[44] J.I. Fuks, A. Rubio, and N.T. Maitra, Phys. Rev. A 83,\n042501 (2011).\n12"    },    {        "title": "1203.6873v1.Density_functional_theory_with_adaptive_pair_density.pdf",        "content": "arXiv:1203.6873v1  [cond-mat.str-el]  30 Mar 2012Density functional theory with adaptive pair density\nJ. Lorenzana,1Z.-J. Ying,1and V. Brosco1\n1ISC-CNR and Department of Physics, Sapienza University of R ome, Piazzale Aldo Moro 2, I-00185, Rome, Italy\n(Dated: September 7, 2017)\nWeproposeadensityfunctionaltofindthegroundstateenerg yanddensityofinteractingparticles,\nwhereboththedensityandthepairdensitycanadjustinthep resenceofaninhomogeneouspotential.\nAs a proof of principle we formulate an a priori exact functional for the inhomogeneous Hubbard\nmodel. The functional has the same form as the Gutzwiller app roximation but with an unknown\nkinetic energy reduction factor. An approximation to the fu nctional based on the exact solution of\nthe uniform problem leads to a substantial improvement over the local density approximation.\nPACS numbers: 71.15.Mb, 71.27.+a, 71.10.Fd\nMost of our theoretical understanding of condensed\nmatter and complex molecules stems from density func-\ntional theory (DFT) computations [1]. Practical im-\nplementations rely on the local density approximation\n(LDA) or its gradient generalizations which often pro-\nvide accurate results at the modest cost of a Hartree\nlike computation [2]. These methods however fail in sys-\ntems at or close to the Mott insulating regime [3], when\nthe tunneling matrix element of electrons becomes small\ncompared with the typical electron-electronrepulsion en-\nergies. This can be seen already at the level of an H 2\nmolecule which is stretched to produce two separated H\natoms. LDA performs reasonably well at the equilib-\nrium distance, when electrons have substantial tunneling\namong the atoms, but fails in the molecular analog of\nthe Mott regime, when each electron is localized on one\nH atom [4, 5].\nGeneralizing to the local spin density approximation\nimprovesthe energyatthe costofanartificialbreakingof\nthe symmetry.[6] While this can be formally justified,[7]\nin practice it leads to unwanted features. For example\nin the case of a strongly correlated metal the artificial\nbreaking of symmetry will give rise to a Fermi surface\nwith the wrong Luttinger volume.[8]\nThe breakdown of unpolarized LDA in the H molecule\ncan be traced back to the poor treatment of each H atom\nseparately.[4, 5] The density close to the center of the\nH atom is 0.32 a.u. LDA essentially assumes that this\nportion of the system behaves as a uniform electron gas\nwith the same density (Wigner-Seitz radius parameter\nrs∼0.91). However the two systems have radically dif-\nferent pair distribution functions which leads to differ-\nent electron-electron interaction energies (zero for the H\natom). LDAthus introduces a spuriousinteractionofthe\nelectron with itself, the so-calledself-interactionerror[9].\nA way to mitigate this problem would be to have a func-\ntional theory which depends both on the density and the\npair density (DPDFT) so that it can discriminate be-\ntween situations with the same density but different pair\ndensities.\nThe idea to involve the pair density in electronic struc-\nture computations is olderthan DFT itself [10, 11]. Morerecently various works explored the possibility to define\nan energy functional based on the pair-density alone [12–\n15] or explore the possibility to adjust the spatially aver-\naged pair density[16] but face serious problems on find-\ning physically acceptable pair densities.[15] Our proposal\ndiffers in that we still keep the density as the basic vari-\nable but we use partial information on the pair density\nas an auxiliary variable which gives the functional more\nsensitivity to correlation. In this respect our functional\nis similar to the proposal of Ref. [7] with the difference\nthat it does not need an artificial breaking of symmetry.\nToshowthefeasibilityandtheusefulnessoftheformal-\nism in strongly correlated systems we present a DPDFT\nfor the one-dimensional Hubbard model i.e.electrons on\na lattice with strong local interaction. The functional is\ninspiredontheGutzwillerapproximation(GA)[17–19]in\nthe same way as Kohn-Sham approach is inspired on the\nHartreeapproximation.[20] It is thus in principle exactas\nKohn-Sham theory represents the “exactification” of the\nHartree approximation[1]. We develop an approximation\nto the unknown functional which becomes exact for uni-\nform systems and involves local or semilocal quantities\nlike LDA and its gradient generalizations, but which is\nmore accurate and goes beyond LDA in the sense that it\nbecomes highly non-local when expressed as a standard\nfunctional of the density alone.\nWe consider a one-dimensional inhomogeneous Hub-\nbard model H=Ht+HU+Hvwith\nH=−t/summationdisplay\nxσ(c†\nxσcx+1σ+H.c.)+/summationdisplay\nxUxnx↑nx↓+/summationdisplay\nxσvxnxσ,\nherecxσandc†\nxσindicate the annihilation and creation\noperator of electrons with spin σon sitexandnxσ≡\nc†\nxσcxσ.tis the nearest-neighbor hopping amplitude\nwhileUxandvxdenote respectively the Hubbard inter-\naction energy and the external potential on site x. For\nreasons to become clear below we allow the interaction\nenergy to be site dependent.\nThe Hohenberg and Kohn theorem [21, 22] guar-\nantees that there exists a functional, Ev[{ρx}] =\nF[{Ux,ρx}]+/summationtext\nxvxρx, which, when minimized with re-\nspect to the density provides the exact ground state en-2\nergy.F[{Ux,ρx}] is a “universal”functional independent\nofvx, but depending on the Ux’s, which represents the\ncontribution of Ht+HUto the energy of a system with\ndensityρx. A formal DPDFT for this model can be ob-\ntained by performing the Legendre transform,[23]\nT[{dx,ρx}] = max\n{Ux}/parenleftBigg\nF[{Ux,ρx}]−/summationdisplay\nxUxdx/parenrightBigg\n(1)\nwheredx=/an}bracketle{tnx↑nx↓/an}bracketri}htis the on-site pair density or double\noccupancy. Identifying the last term in the brackets with\nthe interaction energy we arrive at the conclusion that\nT[{dx,ρx}] is the interacting kinetic energy of the model\nwith the specified pair density and density distributions.\nThis is a universal functional which does not depend on\nthe specific form of Uxnorvx. Below we develop an\napproximation for this functional.\nThe ground state energy of the system is obtained by\nminimizing the functional,\nEUv[{dx,ρx}] =T[{dx,ρx}]+/summationdisplay\nxUxdx+/summationdisplay\nxvxρx(2)\nwith respect to dxandρx. In the new scheme Hthas\nbecome the fixed part of the Hamiltonian while both HU\nandHvare considered as problem dependent. We will\nshow below that even in the conventional case in which\nUxis homogeneous the new functional is quite conve-\nnient.\nEq. (1) yields,\n∂F[{Ux,ρx}]\n∂Ux′=dx′.\nInversion of this expression determines the set of Uxa\nsystem must have to have the given dxandρx. SinceHv\nandHUare determined by ρxanddx, the wave-function\nand all physical quantities are functionals of ρxanddx.\nThe interacting kinetic energy can be written as\nT=−t/summationdisplay\nx(ρx,x+1+ρx+1,x)\nwhereρx,x′denotes the one body density matrix, ρx,x′≡/summationtext\nσ/an}bracketle{tc†\nxσcx′σ/an}bracketri}htand is also a functional of ρxanddx.\nTo proceed we search for a non-interacting system sat-\nisfying the, to be determined, Kohn-Sham like equations,\n−/summationdisplay\nδ=±1teff\nx,x+δϕ(ν)\nx+δσ+veff\nxϕ(ν)\nxσ=ενϕ(ν)\nxσ,(3)\nandwhichhasthe samedensityofthe interactingsystem.\nDefining the non-interacting one-body density matrix as\nthe sum restricted to the occupied Kohn-Sham orbitals,\nρ0\nx,x′=/summationdisplay\nσν∈occ.ϕ∗(ν)\nσ(x)ϕ(ν)\nσ(x′), (4)\nand the density as ρ0\nx≡ρ0\nx,x, we require thus that, ρx=\nρ0\nx.In addition we introduce the kinetic energy reduction\nfactors which are also functionals of dandρand satisfy\nρx,x±1=qx,x±1[{dx′,ρx′}]ρ0\nx,x±1. (5)\nWiththesedefinitionsthekineticenergyfunctionalcan\nbe written in terms of the non interacting density matrix\nas follows\nT[{dx,ρ0\nx}] =−t/summationdisplay\nx′,δ=±1qx′,x′+δ[{dx,ρ0\nx}]ρ0\nx′,x′+δ.(6)\nSimilarly to the exchange correlation potential in the\nstandard Kohn-Sham approach, qencodes all the com-\nplication of the many-body problem in an exact way.\nMinimizing Eqs. (2) with respect to the orbitals and\nusing Eq. (6) we arrive at\nteff\nx,x±1=tqx,x±1[{dx′,ρ0\nx′}], (7)\nveff\nx=vx−t/summationdisplay\nx′,δ∂qx′,x′+δ\n∂ρxρ0\nx′,x′+δ.(8)\nEqs. (3)-(8) define the kinetic energy reduction func-\ntionalqx,x+δ[{dx′,ρx′}] and the associated Kohn-Sham\nsystem.\nFor practical computations one needs to introduce\napproximations. In analogy with the Gutzwiller\napproximation[17, 19] we assume the kinetic energy re-\nduction functional factorizes in terms of functions of the\nlocal densities, qx,y=z(ρx,dx)z(ρy,dy). In the following\nwe refer to this approximation as “factorized approxi-\nmation” (FA). In the spirit of LDA,[20, 24] we approxi-\nmate the functional dependence of z(ρx,dx) by request-\ning that the functional yields the exact energy in the case\nof a uniform system which we obtain in one-dimension\nby solving numerically the exact Bethe-ansatz integral\nequations[25] (in higher dimension a numerical solution\ncan be used). Thus z(d,ρ) is calculated from the con-\nditionz2(d,ρ)T0(ρ) =TBA(d,ρ),whereT0(ρ) denotes\nthe kinetic energy of the non-interacting uniform system.\nTBA(d,ρ) is obtained as a function of the double occu-\npancyd, from the exact Bethe-ansatz energy E(U,ρ) by\nthe Legendre transform Eq. (1) for a system with uni-\nformUand density ρ. It is also useful to compute the\nfunctionUp(ρ,d) which yields the Hubbard Ua uniform\nsystem with density ρ=ρxmust have, to yield the dou-\nble occupancy dxof the non-uniform system at the given\nsite. We indicate as d(ρ,Up) the inverse of Up(ρ,d).\nThe functional Eq. (2) has to be minimized with re-\nspect to the local double occupancies dx, and the Kohn-\nShamorbitalsleadingtoEq.(3). Wecanusethe function\nd(ρ,Up) to eliminate dxin favor of a site dependent ef-\nfective interaction, termed the “pseudointeraction”, Up\nx\nwith respect to which the minimization is actually done\n[dx→d(ρx,Up\nx)]. This change of variables allows us to\navoid the problem of minimizing with respect to a con-\nstrained variable. In the case of a homogeneous exter-\nnal potential vx=V, the minimum is attained when\nUp\nx=U, and one recovers the exact Bethe ansatz result.3\n/s48 /s49/s48 /s50/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s48 /s49/s48 /s50/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s48 /s49/s48 /s50/s48/s48/s50/s48/s52/s48/s54/s48\n/s48 /s49/s48 /s50/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s32/s76/s68/s65\n/s32/s71/s65\n/s32/s68/s80/s68/s70/s84/s45/s70/s65/s32/s97\n/s32/s32/s86/s32/s47/s32/s116/s32/s85/s32/s47/s32/s116/s32/s61/s32/s52/s32\n/s32/s32/s40/s32/s101/s32/s45/s32/s101\n/s69/s120/s97/s99/s116/s32/s41/s32/s47/s32/s116\n/s32/s85/s32/s47/s32/s116/s32/s32/s76/s68/s65\n/s32/s71/s65\n/s32/s68/s80/s68/s70/s84/s45/s70/s65/s86/s32/s47/s32/s116/s32/s61/s32/s53/s98\n/s40/s32/s101/s32/s45/s32/s101\n/s69/s120/s97/s99/s116/s32/s41/s32/s47/s32/s116/s32/s32\n/s32/s111/s100/s100/s32/s120\n/s32/s101/s118/s101/s110/s32/s120/s32/s85/s32/s47/s32/s116/s32/s61/s32/s52/s32/s100\n/s85\n/s120/s112\n/s32/s47/s32/s116/s32\n/s32/s32/s86/s32/s47/s32/s116/s32\n/s32/s76/s68/s65\n/s32/s71/s65\n/s32/s68/s80/s68/s70/s84/s45/s70/s65/s32/s32/s85/s32/s47/s32/s116/s32/s61/s32/s52/s32/s99\n/s32/s86/s32/s47/s32/s116/s32/s40/s32/s117/s32/s45/s32/s117\n/s69/s120/s97/s99/s116/s32/s41/s32/s47/s32/s116\nFIG. 1: (a) Error on the energy per site, e, for a periodic\nHubbard chain of length L= 16 with a binary potential and\nN= 6 electrons as a function of the potential strength V.\nNotice that as for the LDA, DPDFT-FA becomes asymptot-\nically exact for a negligible external potential but as the G A\napproximation gives a small error in the case of a strong inho -\nmogeneous potential where LDA fails. (b) Error as a function\nof the interaction strength. (c) Error in the interaction en -\nergy per site u=U/summationtext\nxdx/Las a function of V. (d) The\npseudointeraction (defined in the text) at odd and even sites\nas a function of the potential strength.\nDPDFT is a variant of DFT. Indeed once a functional\nforTis known we can define a functional of the den-\nsity alone by minimizing over dx, i.e.F[{U,ρx}] =\nmin{dx}(T[{dx,ρx}]+U/summationtext\nxdx). Interestingly, as we\nknow from previous works[26], this procedure leads to\na highly non-local functional of the density starting from\na nearly local functional of variables ρx,dx.\nIn order to test the functional we solved the DPDFT-\nFAequationsforaHubbardchainwithabinarypotential\nvx= (−1)xV(known as ionic Hubbard model[27, 28])\nand compare it with standard LDA derived from the\nexact Bethe ansatz solution[24] and with exact results\nobtained by Lanczos diagonalization with the ALPS\npackage[29].\nFig. 1 shows that DPDFT-FA performs much better\nthan LDA when the system becomes highly inhomo-\ngeneous (a) and strongly interacting (b). The double\noccupancy in LDA is independent of the environment\nwhich leads to a poor approximate interaction energy for\nstrongly inhomogeneous systems (c). In DPDFT-FA the\ndouble occupancy is allowed to adapt, so that localized\nelectrons, for large values of the potential, tend to have\na small double occupancy and a small interaction energy\n(c), explaining the better performance of DPDFT-FA re-\nspect to LDA. The enhanced pseudointeraction on the\nmore charged sites [odd xin (d)] leads to a reduced dou-\nble occupancy and explains the small interaction energy.\nThe behavior of DPDFT-FA is qualitatively similar tothe GA except at small Vwhere the GA is obviously\nnot exact. The errors in other quantities like kinetic en-\nergy, potential energy and density (not shown) are also\ngenerically smaller in DPDFT-FA than in LDA.\nThese results suggest that the self-interaction (SI)\nshould be strongly reduced in DPDFT-FA with respect\nto LDA. In order to verify that this is the case we have\nsolved the problem of one electron with one attractive\nimpurity site in a chain. The potential is given by\nvx=−Vδx,0. This is the lattice analogue of the hy-\ndrogen atom in the continuum. Being a single electron\nproblem, the exact solution has dx= 0 for all Uwhile\nboth LDA and DPDFT-FA yield a finite dx. We define\nthe self-interaction error as the spurious interaction en-\nergy of the single electron problem, ESI=U/summationtext\nxdx.\nIn Fig. 2(a) we plot the self-interaction error as a func-\ntion of the potential strength at the impurity site, with\ninteractionparameter U= 4t. For largeVthe chargebe-\ncomes localized at the impurity site with ρ0∼1. In LDA\nthe interaction energy corresponds to that of a nearly\nhalf-filled uniform Hubbard model which is clearly a very\nbad approximation thus ESI(red dashed line) increases\nand tends to saturate at a large value. In DPDFT-\nFA (blue solid line) ESIstarts with a slower increase\nand then decays to very small values. In this case the\ntotal double occupancy ESI/Ubecomes small showing\nthe adaptability of the pair density to the local envi-\nronment. As shown in Fig. 2(b), the reduction of the\nself-interaction error is large for a wide range of the in-\nteraction.\nIt is interesting to compare DPDFT with traditional\nKohn-Sham DFT for the same model[24]. In the lat-\nter the difference between the interacting and the non-\ninteracting kinetic energy is absorbed additively in the\nexchange-correlation potential. Here the correction is\nmultiplicative and included in the kinetic energy reduc-\ntion factor functional q.\n/s48 /s49/s48 /s50/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s48 /s49/s48 /s50/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s32/s86/s32/s47/s32/s116/s97/s85/s32/s47/s32/s116/s32/s61/s32/s52\n/s32/s76/s68/s65\n/s32/s68/s80/s68/s70/s84/s45/s70/s65\n/s32/s32/s69\n/s83/s73/s32/s47/s32/s116\n/s32/s85/s32/s47/s32/s116/s32/s32/s76/s68/s65\n/s32/s68/s80/s68/s70/s84/s45/s70/s65/s98/s86/s32/s47/s32/s116/s32/s61/s32/s53\n/s32/s32\nFIG. 2: Self-interaction error for a chain of L= 12 sites, a sin-\ngle electron and a potential in which one site has strength −V\nas a function of V(a) andU(b). For large potential strength\nthe density at the impurity becomes close to one. In LDA\nthis fixes the double occupancy to an unphysical value while\nin DPDFT-FA the double occupancy adjusts to small values\nwhen the electron localizes leading to a small self-interac tion.4\nFormally the minimization in DFT has to be restricted\nto densities, and here also double occupancies, that cor-\nrespond to a physical wave function (N-representability).\nWhile this may appear as a severe difficulty[15] it is of-\nten not a problem in practical implementations. It has\nindeed not hampered the development of DFT methods.\nIn our case the problem is not more severe than in LDA\nbecause each portion of the system is approximated by a\nuniform system but with a modified interaction.\nWe have shown DPDFT at work in the lattice but a\nsimilar functional can be defined in the continuum. Ide-\nally one would like to use the pair density, γσσ′(r,r′) =\n/an}bracketle{tψ†\nσ(r)ψ†\nσ′(r′)ψσ′(r′)ψσ(r)/an}bracketri}ht, as a variable, with ψσ(r) the\nfield operator at point rwith spinσ. This, however,\nwould be quite cumbersome as in each point a full func-\ntionmustbedetermined. Amorepracticalapproachisto\nimpose a spatial dependent constraint on the pair den-\nsity and use such a constraint to parameterize families\nof physical pair density functions. For example we can\ndefine,\nD(r) =/summationdisplay\nσσ′/integraldisplay\nd3r′γσσ′(r,r′)θ(a−|r−r′|),\nwithθthe Heaviside function and ais an appropri-\nately chosen cutoff radius. Dmeasures the double oc-\ncupancy probability within a sphere of radius a. These\nor other constraints can be implemented[23] by replacing\nthe physical interaction w(r,r′) with a fictitious inter-\naction. In the present example the fictitious interaction\nwould read w(r,r′)+U(r)θ(a−|r−r′|). As in the lat-\ntice, the corresponding Hohenberg and Kohn functional\nhasU(r) as a variable, F[n(r),U(r)] which allows to de-\nfine the functional\nG[n(r),D(r)] = max\nU(r)/parenleftbigg\nF[n(r),U(r)]−/integraldisplay\nd3rU(r)D(r)/parenrightbigg\n.\nleading to a theory where both n(r) andD(r) are fun-\ndamental variables similar to the lattice but with the\ndifference that Gis not the kinetic energy. Solution of\nthe uniform problem in the presence of the fictitious in-\nteraction with a constant U(r) =Upcould serve as a\nbasis for approximate functionals which converge to the\nLDA in the uniform case but have an adaptive exchange\ncorrelation hole in non-uniform situations.\nWe have conceptually shown how a DFT which uses\nthe pair density as an auxiliary variable can be intro-\nduced and we have developed an approximation for the\nHubbard model inspired on the GA combined with LDA\nideas. Formally DPDFT can also be defined in the con-\ntinuum with a local variable D(r) playing the role of the\ndouble occupancy in the lattice. Approximate function-\nals based on this approach should allow more control on\nthe correlationsbuilt on the underlyingwave-functionre-\nspect to what LDA does, and could help to extend thesuccess of DFT methods to strongly correlated systems\nwhere correlations are substantially different from those\nof the homogeneous electron gas.\nWe thank A. Filippetti and V. Fiorentini for useful\ndiscussions. We are in debt with P. Gori-Giorgi for a\ncritical riding of the manuscript and many valuable sug-\ngestions. This work was supported by IIT-Seed project\nNEWDFESCM.\n[1] W. Kohn, Rev. Mod. Phys. 71, 1253 (1999).\n[2]A Primer in Density Functional Theory (Lecture Notes\nin Physics) (v. 620) , 1 ed., edited by C. Fiolhais, F.\nNogueira, and M. Marques (Springer, Berlin, 2003).\n[3] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys.\n70, 1039 (1998).\n[4] E. J. Baerends and O. V. Gritsenko, J. Phys. Chem. A\n101, 5383 (1997).\n[5] A.J.Cohen, P.Mori-S´ anchez, andW.Yang, Science 321,\n792 (2008).\n[6] O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13,\n4274 (1976).\n[7] J. P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51,\n4531 (1995).\n[8] J. M. Luttinger, Phys. Rev. 119, 1153 (1960).\n[9] J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048\n(1981).\n[10] C. A. Coulson, Rev. Mod. Phys. 32, 170 (1960).\n[11] A. J. Coleman, Rev. Mod. Phys. 35, 668 (1963).\n[12] P. Ziesche, Phys. Lett. A 195, 213 (1994).\n[13] F. Furche, Phys. Rev. A 70, 022514 (2004).\n[14] M.LevyandP.Ziesche, J.Chem.Phys. 115, 9110(2001).\n[15] N. Schuch and F. Verstraete, Nature Phys. 5, 732 (2009).\n[16] P. Gori-Giorgi and A. Savin, Phil. Mag. 86, 2643 (2006).\n[17] M. C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963).\n[18] D. Vollhardt, Rev. Mod. Phys. 56, 99 (1984).\n[19] F. Gebhard, Phys. Rev. B 41, 9452 (1990).\n[20] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).\n[21] K. Sch¨ onhammer, O. Gunnarsson, and R. M. Noack,\nPhys. Rev. B 52, 2504 (1995).\n[22] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864\n(1964).\n[23] P. W. Ayers and P. Fuentealba, Phys. Rev. A 80, 032510\n(2009).\n[24] N. A. Lima, M. F. Silva, L. N. Oliveira, and K. Capelle,\nPhys. Rev. Lett. 90, 146402 (2003). Differently from this\nreference to define the LDA we did not used an approx-\nimate formula for the energy but worked with the exact\nBethe ansatz expression.\n[25] E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445\n(1968).\n[26] G. Seibold and J. Lorenzana, Phys. Rev. Lett. 86, 2605\n(2001).\n[27] N. Nagaosa and J.-I. Takimoto, J. Phys. Soc. Japan 55,\n2735 (1986).\n[28] T. Egami, S. Ishihara, and M. Tachiki, Science 261, 1307\n(1993).\n[29] B. Bauer et al.J. Stat. Mech. P05001 (2011); F. Albu-\nquerque et al., J. Mag. and Mag. Mat. 310, 1187 (2007)."    },    {        "title": "1204.0399v3.Constraining_the_nuclear_equation_of_state_at_subsaturation_densities.pdf",        "content": "arXiv:1204.0399v3  [nucl-th]  28 May 2015Constraining the nuclear equation of state at subsaturatio n densities\nE. Khan,1J. Margueron,1and I. Vida˜ na2\n1Institut de Physique Nucl´ eaire, Universit´ e Paris-Sud, I N2P3-CNRS, F-91406 Orsay Cedex, France\n2Centro de Fsica Computacional, Department of Physics,\nUniversity of Coimbra, PT-3004-516 Coimbra, Portugal\n(Dated: November 8, 2018)\nOnly one third of the nucleons in208Pb occupy the saturation density area. Consequently nuclea r\nobservables related to average properties of nuclei, such a s masses or radii, constrain the equation\nof state (EOS) not at saturation density but rather around th e so-called crossing density, localised\nclose to the mean value of the density of nuclei: ρ≃0.11 fm−3. This provides an explanation for\nthe empirical fact that several EOS quantities calculated w ith various functionals cross at a density\nsignificantly lower than the saturation one. The third deriv ative M of the energy at the crossing\ndensity is constrained by the giant monopole resonance (GMR ) measurements in an isotopic chain\nrather than the incompressibility at saturation density. T he GMR measurements provide M=1110\n±70 MeV (6% uncertainty), whose extrapolation gives K ∞=230±40 MeV (17% uncertainty).\nPACS numbers: 21.10.Re, 21.65.-f, 21.60.Jz\nConstraining the nuclear equation of state (EOS) and\nreducing the uncertainties on nuclear matter properties\nin dense stellar objects such as neutron stars and super-\nnovae is one of the major goals of nuclear physics. To\ndo so observables such as masses, radii or energy cen-\ntroid of the isoscalar Giant Monopole Resonance (GMR)\nare measured in finite nuclei. Relating the EOS to such\nobservables is usually undertaken in several ways. The\nliquiddropexpansion[1]isoneofthemostusedmethods.\nIn its generalised version, one performs a development of\na quantity (the mass for instance) considered at satura-\ntion density (volume term), adding several terms such as\nthe surface one. It should be noted that taking the vol-\nume term at saturation density is based on the fact that\nthe inner part of the nuclei density is close to the sat-\nuration. Another method is the so-called Local Density\nApproximation (LDA) [2]. In this approach the global\nproperties in nuclei are obtained from considering nuclei\naslocalpiecesofnuclearmatter. Finally, anothermethod\nis based on the microscopic approach, relying on energy\ndensity functionals (EDF): using various EDF’s, a corre-\nlationdiagramisdrawnbetweenthepredictedobservable\nand a related property of the EOS. The measurement of\nthe observable allows to validate an EDF and the corre-\nsponding property of the EOS [3–6]. For instance, the\nneutrons skin measurement is correlated with the slope\nof the symmetry energy. It should be noted that usually\nEDFaredesignedusingmasses, radiibut alsothe nuclear\nincompressibility [7, 8].\nEach of these methods comprise however limitations.\nThe liquid drop expansion is known not to be accurate\nenough in the case of the incompressibility at the sat-\nuration density [9, 10], although the inclusion of higher\nother terms is relevant [11]. The liquid drop expansion\nof the mass has been successful, providing an accurate\ndetermination of the saturation energy. This quantity is\nnow used in EDF determination, but this is an excep-\ntional case, related to the profusion of data available on\nmasses. In the case of the LDA, its validity is generallyquestionablefor finite size systems. Finally, in the caseof\nthe microscopicapproach, the EDF employedhas usually\nbeen adjusted to data on magic nuclei, whereas most ap-\nplications are aimed to be used for deformed open-shell\nnuclei. However there may be a general consensus that\nthe microscopic approach should be used in fine, because\nof its accuracy and reliability.\nIt seems for now difficult to straightforwardly deter-\nmine the nuclear incompressibility even with the micro-\nscopic method. The earliest microscopic analysis came\nto a value of K ∞=210 MeV [9], but with the advent of\nmicroscopic relativistic approaches, a value of K ∞=260\nMeVwasobtained [12]. It has been shownthat this value\nis not determined accurately and that the density depen-\ndence of the EDF as well as pairing effects (and therefore\nthe shell structure) have an impact on the determination\nof K∞[13–15]. Using K ∞in the design of EDF may\nnot be a sound approach since it cannot be safely deter-\nmined by the microscopicmethod. Thereforethe method\nof determination of the nuclear incompressibility has to\nbe rethought and more generally, it is necessaryto clarify\nthe link between nuclear matter EOS determination and\nnuclear observables.\nFirst it should be noted that the liquid drop expan-\nsion is not a perturbative one since the surface properties\nof the nucleus are almost as important as the bulk one.\nHence it may be a misleading idea to consider the nu-\ncleus as mainly composed of nucleons at saturation den-\nsity with a few at the surface. Let us consider the case of\n208Pb which is usually considered as a benchmark for ex-\ntracting bulk properties. Fig. 1 shows the total density\ncalculated in the Skyrme Hartree-Fock (HF) approach.\nThe lowerpartshowsthe usualrepresentationofthe den-\nsity whereas the upper part displays an equivalent repre-\nsentation with an X axis scaled as r3. This allows to take\ninto account the increase of nucleons per volume unit;\nthe total number of nucleons corresponds to the integral\nof constant steps of the radial density represented on the\nupper part of Fig. 1. It is now perceptible that about2\n0 2.25.87.4 8.4 9.3 10r (fm)00.050.10.150.2 ρ  (r) (fm-3)\n0 5 10 15\nr (fm)00.050.10.150.2 ρ  (r) (fm-3)208Pb\n208Pb\nI=4π   ρ(r)r2dr4π   ρ(r)d(r3)\n3I=\nFIG. 1. Matter density of208Pb calculated with the HF ap-\nproach using the SLy4 functional with different X axis scales .\nTop: representation taking into account the nucleons’ dist ri-\nbution in the nucleus. Bottom: usual representation\none third of the nucleons of the208Pb nucleus lie in the\nsaturation density area, whereas two thirds are localised\nin the surface at a density lower than the saturation one.\nTherefore even in heavy nuclei, the contribution of the\nsurface is larger with respect to the volume one, raising\nthe question of the legitimacy of constraining EOS quan-\ntities at saturation density by measurements of nuclear\nobservables.\nAnother illustration is given by calculating the mean\ndensity of the208Pb nucleus. Using a Skyrme-HF cal-\nculation, one obtains <ρ>= 0.12 fm−3in208Pb with\na variance/radicalbig\n<ρ2>−<ρ>2=0.04 fm−3. Therefore the\ndensity value characterising a nucleus is not the satura-\ntion one ( ρ0=0.16 fm−3) but a significantly lower one,\nwith a range spanning from ρ0/2 toρ0. It should be\nnoted that the dependence of the mean density on the\nnucleus is rather weak: 0.11 fm−3for120Sn. In light\nnuclei, the mean density drops down as expected: 0.07\nfm−3for40Ca.\nConsequently the measurement of an averaged observ-\nable in a nucleus is more properly related to a correlated\nEOS quantity defined around the mean density than at\nthe saturation density. This fact is illustrated on Fig.\n2, wherethedensity-dependent incompressibility, defined\nby [16, 17]\nK(ρ) =9\nρχ(ρ)= 9ρ2∂2E(ρ)/A\n∂ρ2+18\nρP(ρ) (1)\nand obtained from various EDF’s is plotted with sev-\neral Skyrme, Gogny and relativistic interactions, all de-\nsigned to describe observables in nuclei such as masses\nand radii. They intersect around the crossing density\nρc≃0.7ρ0≃0.11 fm−3. The existence of a crossing den-\nsityhasbeenempiricallynoticedinpreviousworksonthe0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1\nρ/ρ0-100-50050100150200250300350400K(ρ)   (MeV)SLy230a\nSLy230b\nSGII\nSKM*\nSVII\nRATP\nSKX\nMSk5*\nSKP\nT6\nSkI2\nLNS\nBSK16\nT44\nSkz2\nSLy5\nSk272\nSk255\nDDME2\nFSUGold\nNL3\nD1S\nD1P\nFIG. 2. (Color online) EOS incompressibility calculated wi th\nvarious functionals, showing the crossing density around 0 .7\nρ0≃0.11 fm−3\nsymmetry energy (Fig. 2 of Ref. [6]), pairing gap (Fig.\n2 of Ref. [18]) or the neutron EOS (Fig. 2 of Ref. [3]\nand Fig. 1 of Ref. [4]), and we provide here an explana-\ntion, related to the mean density: when designing EDF\nwith nuclear observables, the corresponding EOS is con-\nstrained not at the saturation density but rather around\nthe mean density (the crossing density). In the case of\nthe symmetry energy, the value at a density ρ≃0.11\nfm−3is taken to be around 25 MeV, a value close to the\nsymmetry energy coefficient of the liquid drop expansion\n[5, 6, 19], as an empirical prescription. This last value\ncontains both a volume and surface terms, and thus rep-\nresents the symmetry energy extracted from nuclei ob-\nservables. For the incompressibility, the GMR is known\nto be related to the mean square radius of the nucleus by\nthe energy weighted sumrule [2]. In the design of EDF’s,\nthe considered constraint on nuclear radii induces a con-\nstraint on the compression mode, likely explaining the\ncrossing around 0.7 ρ0observed on Fig. 2. This shows\nthe universality of the crossing effect, arising from the\nconstraints encoded in the EDF from nuclei observables.\nDue to this crossing area, a larger K ∞=K(ρ0) value for a\ngiven EDF can be compensated by lower values of K( ρ)\nat sub-crossing densities, so to predict a similar energy\nof the GMR in nuclei: the GMR centroid is related to\nthe integral of K( ρ) over a large density range [17]. This\nallows to understand how EDF with different K ∞can\npredict a similar energy of the GMR, as noted in [15].\nVarious EDF’s shall exhibit various density dependen-\ncies around the crossing point. At first order the deriva-\ntive of the pairing gap (or incompressibility or energy)\nat this point will differ between various EDF’s. Comple-\nmentary measurements in nuclei are needed to charac-\nterise these derivatives. For instance in Ref. [3, 4] the\nderivative of the neutron EOS around ρc≃0.11 fm−3is3\nfound to be constrainedbyneutron skinmeasurementsin\n208Pb. The associated interpretation was that the 0.11\nfm−3density was considered as the neutron saturation\ndensity in nuclei. We provide an alternative and general\nview: 0.11 fm−3corresponds to the crossing density, as\nseen on Fig. 2.\nWe apply this method to the determination of the nu-\nclear incompressibility. When the Giant Monopole Reso-\nnanceismeasuredandwellreproducedbyagivenEDF,it\nshall therefore not be correlated with the incompressibil-\nity of EOS at saturation density but rather with its first\nderivative M around the crossing density. It should be\nrecalled that the crossingdensity exists because the EDF\nare determined by including nuclear observables such as\nmasses and radii in their fit. On Fig. 2 the crossing\npoint at ρc≃0.7ρ0yields K( ρc)≃40 MeV for the in-\ncompressibility (this is analogous to the fixed symmetry\nenergy taken to be 25 MeV, as discussed above). To be\nconsistent with a generalised liquid drop expansion [20],\nthe derivative M of the incompressibility at this point is\ndefined by:\nM= 3ρK′(ρ)|ρ=ρc (2)\nwhere K′(ρ) is the derivative of the incompressibility\ndensity dependent term defined in Eq. (1).\nMost of the GMR measurements have been performed\non208Pb and data on other nuclei like the Sn isotopic\nchain is relevant [21]. In Fig. 3 the GMR prediction\nusing the constrained Hartree-Fock (CHF) method as a\nfunctionofMisshownforvariousSkyrmeEDF’sin208Pb\nand120Sn. The CHF method is a sum rule approach to\ncalculate the centroid energy of the isoscalar GMR:\nEISGMR=/radicalbiggm1\nm−1. (3)\nThem1moment is evaluated by the double commuta-\ntor using the Thouless theorem [2] and for the m−1mo-\nment, the CHF approach is used [14, 22, 23]: the CHF\nHamiltonian is built by adding the constraint associated\nwith the IS monopole operator. The CHF method has\nthe advantage to very precisely predict the centroid of\nthe GMR using the m−1sumrule. To be comprehensive,\nFig. 3 also displays values for several relativistic and the\nGogny D1S EDF’s in the case of208Pb.\nThe value of M deduced from both nuclei is compatible\nwithin few percents. In order to derive a sound value of\nM, we have further performed similar calculations in the\n112−124Sn isotopes, as well as in90Zr and144Sm nuclei,\nleading to M ≃1100 MeV ±70 MeV. The contributions\nto the uncertainty shall come from the small variance\nbetween the mean density in the respective nuclei com-\npared to the crossing density, the isospin dependence of\nthe incompressibility and the pairing effects. The linear\ncorrelation observed on Fig. 3 is striking, since the inter-\nactions employed can have a symmetry energy spanning1000 1100 1200 1300 1400 1500\nM (MeV)131415161718EGMR (MeV)120Sn\n208Pb\nSkM*SLy5Sk255Sk272\n120Sn\nExp.\nSkM*SLy5Sk255Sk272\n208Pb\nExp. DDME2\nD1SFSUGoldNL3\nFIG. 3. Centroid of the GMR in208Pb and120Sn calculated\nwith the CHF method vs. the value of M for various function-\nals. The experimental values for208Pb and120Sn are taken\nfrom Ref. [24] and [21], respectively with respective error bars\nof±200 keV and ±100 keV\nfrom 30 to 37 MeV. This shows that the present results\nmoderately depend on the isospin asymmetry of the sys-\ntem studied. The neutron vs. proton asymmetry pa-\nrameter δ=(N-Z)/A also remains rather constant in the\nconsidered nuclei for the GMR such as208Pb and sta-\nble Tin isotopes: δ≃0.2, validating the present isospin\nindependent approach.\nIt should be noted that the present method clarifies\nthe issue of determining different incompressibility val-\nues at saturation when using either the Pb or the Sn\ndata [14, 25]: a close M value is found with these two\ndata sets. However the remaining discrepancy of the M\nvalues deduced from the Sn and the Pb measurements\nshows that the proper density dependence of the EDF’s\nhas not been revealed yet. Another striking feature is\nthat the Gogny and relativistic EDF’s are found on the\nsamelinearcorrelationthanthe Skyrmeone, showingthe\nuniversality of the E GMRvs. M correlation, contrary to\nthe EGMRvs. K∞one: it is well-known that relativistic\nEDF’s can predict a similar E GMRin208Pb but with a\nsignificantly larger value of K ∞[12].\nLet us now investigate why a clear correlation exists\nbetween the centroid of the GMR and K ∞in the specific\ncase of the Skyrme functionals. [15]. Fig. 4 displays\nthe relative contribution of the kinetic, central (t 0), fi-\nnite range (t 1,t2) and density (t 3) terms of the Skyrme\nfunctionals to the EOS and its derivative values at the\ncrossing density. A striking regularity is observed among\nthe functionals used. The dominant terms are the central\nand the density ones, but the central term vanishes from\nthe second derivative of the EOS and beyond, allowing\nthe density term (in ρα) to dominate alone. Therefore\nthe second derivative terms and beyond are correlated\nto each other, implying that the correlation between M4\n-0.6-0.4-0.200.20.4 E/A \n-0.6-0.4-0.200.20.4 d(E/A)/d ρ\n0 10 20 30\n# Skyrme-0.200.20.40.60.81d2(E/A)/dρ2\n0 10 20 30\n# Skyrme-0.8-0.6-0.4-0.200.2 d3(E/A)/dρ3a) b)\nc)d)\nFIG. 4. (Color online) Positive or negative relative contri -\nbution of the kinetic (circles), central (squares), finite r ange\n(diamonds) and density (triangles) terms to the equation of\nstate (a), its first (b), second (c) and third (d) derivatives at\nthe crossing density, calculated for 36 Skyrme functionals\nand E GMRis propagated to the one between K ∞and\nEGMR: a linear correlation is preserved between M and\nK∞due to the vanishing ofthe centralterm from the sec-\nond derivative and beyond of the equation of state. In\nother words the density dependence of the incompress-\nibility, driven by these derivatives, is correlated to M.\nTherefore K ∞is correlated to M, and this outcome is\nsimilar for all Skyrme functionals as seen on Fig. 4: the\n(EGMR,K∞) linear correlation may be due to an arte-\nfact from the specific density dependence of the Skyrme\nEDF’s.But this linear correlationvanishes when including rel-\nativistic EDF predictions, as mentioned above. M shall\nbe the exclusive quantity which can be deduced from\nGMR measurements, whereas the K ∞determination re-\nlies on additional density dependence assumptions. Us-\ning the M value at the crossing point, a linear expansion\nprovidesK ∞=230MeV. An uncertaintyof ±40MeVcan\nbe inferred from the spreading of K ∞values on Fig. 2\nobtained with the various functionals, which is of the or-\nder of±17%. It is therefore argued that measurements\nin nuclei constrain the EOS around the crossing density\n(namely the derivative of the EOS quantity at the cross-\ning density) and deducing values at saturation density\nremains a mainly model-dependent extrapolation. In the\ncase of nuclear incompressibility, it is proposed to change\nthe usual E GMRvs. K ∞correlation plot (only valid in\nthe specific density dependence of Skyrme EDF’s) and\nto replace it by a more reliable and universal E GMRvs.\nM plot (Fig. 3), where M is the derivative of the incom-\npressibility at the crossing density. The measurement of\nthe GMR in more neutron-rich nuclei [26, 27] will cer-\ntainly open the possibility to study a part of the isospin\ndependence of the incompressibility.\nThis work has been partly supported by the ANR\nNExEN and SN2NS contracts, the Institut Universi-\ntaire de France, by CompStar, a Research Networking\nProgramme of the European Science Foundation and\nby the initiative QREN financed by the UE/FEDER\nthrough the Programme COMPETE under the projects,\nPTDC/FIS/113292/2009, CERN/FP/109316/2009 and\nCERN/FP/116366/2010. J.M. would like to thank K.\nBennaceur and J. Meyer for useful discussions during the\ncompletion of this work.\n[1] C.F. Von Weizs¨ acker, Zeitschrift fr Physik 96 (1935) 43 1;\nH.A. Bethe and R.F. Bacher, Rev. Mod. Phys. 8 (1936)\n82.\n[2] P. Ring, P. Schuck, The Nuclear Many-Body Problem,\nSpringer-Verlag, Heidelberg, (1980).\n[3] B.A. Brown, Phys. Rev. Lett. 85, 5296 (2000).\n[4] S. Typel and B.A. Brown, Phys. Rev. C64, 027302\n(2001).\n[5] M. Centelles, X. Roca-Maza, X. Vi˜ nas and M. Warda,\nPhys. Rev. Lett. 102, 122502 (2009).\n[6] J. Piekarewicz, Phys. Rev. C83, 034319 (2011).\n[7] M. Bender, P.-H. Heenen, and P.-G. Reinhard Rev. Mod.\nPhys. 75, 121 (2003).\n[8] D. Lunney, J.M. Pearson, and C. Thibault, Rev. Mod.\nPhys. 75, 1021 (2003).\n[9] J.-P. Blaizot, Phys. Rep. 64, 171 (1980).\n[10] J.M. Pearson, Phys. Lett. B271, 12 (1991).\n[11] S.K. Patra, M. Centelles, X. Vi˜ nas and M. Del Estal,\nPhys. Rev. C65 (2002) 044304.\n[12] D. Vretenar, T. Niksic and P. Ring, Phys. Rev. C68\n(2003) 024310.\n[13] J. Li, G. Col` o and J. Meng, Phys. Rev. C78, 064304\n(2008).[14] E. Khan, Phys. Rev. C80, 011307(R) (2009).\n[15] G. Col` o, N.V. Giai, J. Meyer, K. Bennaceur and P.\nBonche, Phys. Rev. C70, 024307 (2004).\n[16] A.L. Fetterand J.D. Walecka, Quantum Theory of Many-\nParticle Systems (McGraw-Hill, New York, 1971).\n[17] E. Khan, J. Margueron, G. Col` o, K. Hagino and H.\nSagawa, Phys. Rev. C82, 024322 (2010).\n[18] E. Khan, M. Grasso and J. Margueron, Phys. Rev. C80,\n044328 (2009).\n[19] C.J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. 86,\n5647 (2001).\n[20] C. Ducoin, J. Margueron, C. Provid˜ encia and I. Vida˜ na ,\nPhys. Rev. C83, 045810 (2011).\n[21] T. Li et al., Phys. Rev. Lett. 99, 162503 (2007).\n[22] O. Bohigas, A.M. Lane and J. Martorell, Phys. Rep. 51,\n267 (1979).\n[23] L. Capelli, G. Col` o and J. Li, Phys. Rev. C79, 054329\n(2009).\n[24] D.H. Youngblood et al., Phys. Rev. C69 034315 (2004).\n[25] J. Piekarewicz, Phys. Rev. C76, 031301 (2007).\n[26] C. Monrozeau et al., Phys. Rev. Lett. 100, 042501 (2008) .\n[27] M. Vandebrouck et al., Exp. E456a performed at GANIL\n(2010)."    },    {        "title": "1204.2207v1.Superfluid_Local_Density_Approximation__A_Density_Functional_Theory_Approach_to_the_Nuclear_Pairing_Problem.pdf",        "content": "arXiv:1204.2207v1  [nucl-th]  10 Apr 2012August 16, 2018 17:9 World Scientific Review Volume - 9in x 6in ws-rv9x6˙bulgac\nChapter 1\nSuperfluid Local Density Approximation: A Density\nFunctional Theory Approach to the Nuclear Pairing\nProblem\nAurel Bulgac\nDepartment of Physics\nUniversity of Washington, Seattle, WA 98195-1560, USA\nI describe the foundation of a Density Functional Theory app roach to in-\ncludepairingcorrelations, whichwasappliedtoavarietyo fsystemsrang-\ning from dilute fermions, toneutron stars and finite nuclei. Ground state\nproperties as well as properties of excited states and time- dependent\nphenomena can be achieved in this manner within a formalism b ased on\nmicroscopic input.\n1. Why use a Density Functional Theory approach?\nThe calculations of the ground and excited state properties of com plex nu-\nclei represent an ongoing challenge for several decades. Qualitat ively we\nknow a lot about these nuclear properties, however, extracting t hese prop-\nerties from a microscopic approachis still one of the most difficult pro blems\nin theoretical physics. The atomic nucleus presents a number of diffi culties,\nnot all of them shared by other quantum many-body systems. The exis-\ntence of magic numbers is undoubtedly related to the existence of a well\ndefined meanfield, in which nucleons move inside the nuclear medium. Th e\nexperimental confirmation of the existence of superdeformed nu clei and the\nsemi-quantitative explanation of the equilibrium shape deformations based\non semiclassical arguments represent another strong argument in favor of\nthe existence of a well defined meanfield even in strongly deformed n uclei.\nPairing correlations, which energetically represent a rather small d ecora-\ntion of the bulk nuclear binding energy, can equally well be described in\nterms of a generalized meanfield. At the same time the interaction am ong\nnucleons is strong and the description of the nuclear binding and str ucture\nin terms of the bare interaction among nucleons is still a formidable ch al-\nlenge for the many-body theory, apart form brute force method s (quantum\n1August 16, 2018 17:9 World Scientific Review Volume - 9in x 6in ws-rv9x6˙bulgac\n2 A. Bulgac\nMonte Carlo (QMC) and no-core shell model) capable of handling so fa r\nonly very light nuclei. In condensed matter theory a somewhat para llel\napproach was developed, in which the existence of well defined ferm ionic\nquasiparticle excitations played a fundamental role, the so called La ndau\nFermi liquid theory,1,2subsequently extended to nuclei by Migdal and col-\nlaborators.3Landau took it as a given that the ground state properties\nof strongly interacting systems cannot be calculated and have to b e con-\nstructed in a phenomenological manner. In order to describe excit ed states\none had to introduce additionally residual interactions and the emer ging\nformalism was equivalent to the linear response theory.1\nInthe1970’saseriouseffortwasmountedtocalculatenuclearprop erties\non an almost theoretically self-consistent approach, using effectiv e interac-\ntions based on a G-matrix approach to nuclear matter.4The approach\nwas cumbersome, the theoretical errors were hard to evaluate, especially\nsince a number of shortcuts (based on the current dominating pre vailing\ntheoretical “intuition”) were adopted. The use of the term “effec tive” be-\ncame so widespread that one can hardly find a universal meaning, of ten\nbeing used just as a misnomer for a semi-phenomenological approac h of\none kind or another. Since the late 1970’s and early 1980’s we witness the\npowerful rise of the pure phenomenological approach to calculatin g ground\nand excited state nuclear properties using either Skyrme,5Gogny6or rela-\ntivistically inspired7type of “effective” in-medium nuclear forces, with an\nimmense proliferation of various parameterizations. There were a n umber\nof little nagging problems with all these models. The Skyrme interactio n\nwas formally interpreted as “effective” in-medium nucleon-nucleon in terac-\ntion, which can be used in a Hartree-Fock like calculation.8Certain con-\ntributions arising from this kind of interaction lead to undesirable effe cts\nespecially in deformed nuclei and were unceremoniously dropped. Sk yrme\nparametrizationoftheinteractionwasusedonlyintheparticle-hole channel\n(similarly in relativistic models), and a totally separate phenomenologic al\napproach was used in the particle-particle channel, in order to trea t the\npairing correlations. Theoretically one can bring hand-waving argum ents\nthat such an approach was meaningful, as a similar conclusion was rea ched\nin the more formal Landau Fermi liquid approach,2,3where one can clearly\nshow that one obtains different contributions to the irreducible diag rams\nin particle-hole and particle-particle channels respectively, even th ough a\nnumber of sum rules leads to certain correlationsamong them. At th e same\ntime Gogny and his followers insist in using the same two-body “effectiv e”\ninteraction in both the particle-hole and particle-particle channels, evenAugust 16, 2018 17:9 World Scientific Review Volume - 9in x 6in ws-rv9x6˙bulgac\nSuperfluid Local Density Approximation: A Density Function al Theory Approach to the Nuclear Pairing Problem 3\nthough it is hard to make a many-body theory argument in favor of s uch\nan approach.\nIn the 21-st century the prevailing theoretical attitude changed . It was\nrealized and also became widely (though not unanimously) accepted t hat\n“effective” in-medium nuclear forces of one kind or another lack a cle ar\ntheoretical underpinning (in particular these “effective” in-medium forces\nare not observables and moreover cannot be defined uniquely), th at their\nuse was merely a means to obtain an Energy Density Functional (EDF )\n(which, however, can be phenomenologically parameterized directly ), and\nthat an entirely new theoretical concept makes more sense, name ly that of\nan Density Functional Theory (DFT), formally introduced by Kohn, Ho-\nhenberg and Sham in the 1960’s for electron systems.9DFT is in principle\nan exact approach, in which the role of the many-body wave functio n is\nreplaced with the one-body density distribution, with the caveat th at one\nneeds to find an accurate energy density functional. The solution o f the\nmany-body Schr¨ odinger equation is thus replaced with a significant ly much\nsimpler meanfield-like approach. DFT, particularly in its so called Local\nDensity Approximations (LDA) of Kohn and Sham, proved to be widely\nsuccessful in chemistry and condensed matter calculations of nor mal sys-\ntems, where pairing correlation are absent. Fayans10was perhaps the first\nto introduce DFT into nuclear structure calculations in a spirit very s im-\nilar to condensed matter physics, namely by fitting an LDA functiona l to\nQMC results for infinite homogeneous matter, and adding a small num ber\nof phenomenological gradient correction terms.\n2. DFT for a system with pairing correlations\nThe DFT extension to superfluid systems has been performed in sev eral\nwaysso far, but by followingasimilaridea: one needs to addthe anoma lous\ndensity in order to describe the presence of a new order paramete r. The\nfirst extension due to Gross and collaborators11added a dependence of\nthe EDF on the nonlocal anomalous density matrix. In this way the gr eat\nadvantage of the LDA was nullified, as in this case one has to solve non local\nor integro-differential equations. Especially in the case of nuclear s ystems,\nwhere pairing correlations are relatively weak and the size of the Coo per\npair is larger than the radius of the nucleon-nucleon interaction, th is kind\nof approach looks like an overkill. One expects that a large size and a\nweakly bound two-fermion Cooper pair could be accurately describe d using\na zero range interaction. One can easily show however that in such a caseAugust 16, 2018 17:9 World Scientific Review Volume - 9in x 6in ws-rv9x6˙bulgac\n4 A. Bulgac\nthe emerging equations lead to an ultraviolet (UV) divergent anomalo us\ndensity.12,13That was the main reason why Gross and collaborators11\nresorted to a non-local extension of DFT to superfluid fermion sys tems. A\nnatural UV-cutoff in electronic systems is provided by the Debye fr equency.\nIn nuclear systems the situation is quite different, as there are no p honon\ninduced pairing correlations.\nA similar reasoning for dealing with UV-divergence was implicit as well\nin the Gogny parametrization,6which uses a finite range “effective” in-\nmedium interaction, where the finite radius of the interaction provid es the\nneeded UV-cutoff in the particle-particlechannel. In the particle-h olechan-\nnel a finite range can be converted quite accurately (particularly in the case\nofexchangeterms) intoa number contactterms with spatial deriv atives,8,14\nusing the density matrix expansion approach. In Skyrme-like EDFs u sed in\nliterature most of the time practitioners introduce an arbitrary cu toff pa-\nrameter, which is often used as an additional phenomenological par ameter.\nThe nuclear pairing gap is small and the size of the nuclear Cooper pair\nis rather large in comparison with the nucleon-nucleon interaction ra dius,\nand it is hard to make the case that a finite range of the interaction is\nneeded and that is responsible for the stabilization of the calculation s. The\ntheoretical situation here is totally similar to the nucleon-nucleon int erac-\ntion for energies below the π-threshold15where a rigorous treatment using\ncontact interaction terms is easy and accurate as well to implement .\nThe nature of the UV-divergence in the pairing channel was clarified a\nlong time ago12and a simple regularization scheme was later introduced.13\nIn the case of two particles interacting with a finite range interactio n the\ns-wave function for either bound states or scattering states beh aves as\n∝1/|r1−r2|outside the interaction range. In the case of scattering states\nthewavefunctionatlowenergiesisexp( ik·r)+f(k)exp(ik|r1−r2|)/|r1−r2|.\nOne can showthat the anomalousdensity matrix ν(r1,r2) satisfies an equa-\ntion almost identical to the Schr¨ odinger equation for two interact ing par-\nticles12and that ν(r1,r2)∝∆((r)/|r1−r2|when|r1−r2| →0 (for a\nzero-range interaction, where r= (r1+r2)/2) and exactly this is the rea-\nson why the diagonal anomalous density diverges. This divergence a nd\nits amplitude has physical meaning and one cannot simply hide it under\nthe rug by introducing a ill-defined cutoff. Fermi devised a very simple\napproach to deal with this kind of situation, without making recours e to\neither arbitrary UV-cutoffs or to fictitious finite range effects, by introduc-\ning a pseudo-potential.16Fermi’s pseudo-potential approach can be imple-\nmented in a straightforward manner in treating pairing correlations .13InAugust 16, 2018 17:9 World Scientific Review Volume - 9in x 6in ws-rv9x6˙bulgac\nSuperfluid Local Density Approximation: A Density Function al Theory Approach to the Nuclear Pairing Problem 5\nparticular, in order to describe the value of the s-wave pairing gap only\none coupling constant is needed for both protons and neutrons, a s expected\nfrom isospin invariance.17\nIn a parallel approach a many-body perturbative approach to EDF , us-\ning renormalized bare NN (and NNN) interactions was suggested by F urn-\nstahl and collaborators.18Only a few study cases have been considered so\nfar and only in the absence of pairing correlations.\nA quite successful DFT approach for fermion systems with pairing c or-\nrelations has been developed and applied to a diverse sample of physic al\nsystems13,17,19–26and was dubbed the Superfluid Local Density Approxi-\nmations (SLDA) as a direct generalization of the LDA approach of Ko hn\nand Sham. In LDA/SLDA single-particle wave functions appear explic -\nitly and often this type of approach is referred to as orbital based DFT.\nIn SLDA one constructs a local EDF in terms of various densities (sp in\ndegrees of freedom are not shown for the sake of simplicity)\nρ(r) =/summationdisplay\nn|vn(r)|2, τ(r) =/summationdisplay\nn|∇vn(r)|2, ν(r) =/summationdisplay\nnun(r)v∗\nn(r).(1)\nIf spin-orbit interaction is present an additional density should be a dded.\nThe sums in the definition of these densities are all performed up to a n\nUV-cutoff, in order to avoid the UV-divergence of the kinetic energ y and\nanomalous densities. Even though an explicit UV-cutoff appears in th e\nSLDA formulation, no dependence of the observables on this UV-cu toff\nexists once this cutoff is chosen appropriately. Both kinetic energy density\nτ(r) and the anomalous density ν(r) diverge in a similar fashion and their\ncontribution to EDF is handled by introducing a well chosen counter t erm,\nandsubsequentlytheentireformalismbecomesdivergenceandcou nterterm\nfree.13One can show that a unique combination\n/planckover2pi12\n2meff(r)τ(r)−∆(r)ν∗(r) (2)\nis divergence free, where meff(r) is the effective mass, ∆( r) =−geff(r)ν(r)\nis the pairing gap and geff(r) is the renormalized position dependent cou-\npling constant defining the strength of the pairing correlations.13This\nimplies that both kinetic energy and anomalous densities appear in EDF\nin this combination alone.\nSince DFT does not provide any constructive recipes for the EDF, a ny\nsuggested approach needs validation. Unlike in the case of a phenom eno-\nlogical approach in this case agreement with experiment is not the pr oof\nthat the championed approach is correct. One needs to explicitly sh ow thatAugust 16, 2018 17:9 World Scientific Review Volume - 9in x 6in ws-rv9x6˙bulgac\n6 A. Bulgac\nthe solution of the Schr¨ odinger equation for a many-body fermion system,\nin which pairing correlations are present and the corresponding DFT in-\ncarnation produce the same results for the ground state energy and ground\nstate one-body density distribution. Fortunately, an extremely in teresting\nsystem, which has mesmerized theorists across most physics subfi elds as\nwell as experimentalists in cold atom physics, exists. This is the unitar y\nFermi gas (UFG), a system of fermions with spin-up and spin-down, inter-\nacting with a zero-range interaction and an infinite scattering lengt h.26,27\nThe UFG has properties very similar to the properties of dilute neutr on\nmatter26–29as envisioned by Bertsch in 1999.30In this case it is possible to\ncalculate with relatively high controlled accuracy the energy and a nu mber\nof properties of the ground state of a large series of such system s, with\nvarious particle numbers with spin-up and down, both in the case of h omo-\ngeneous systems and systems in external confining potentials. Th e results\nfor the ground state properties of the homogeneous state - spe cifically the\nground state energy, the pairing gap and the quasiparticle excitat ion spec-\ntrum (effective mass) - are used to build the SLDA EDF. In the case o f a\nUFG the only dimensional scale in the system is the inter-particle dista nce\nand this fact constrains the possible EDF structure. Simple dimensio nal\narguments show that apart from three dimensionless constants α, βand\nγthe (un-renormalized) EDF of an unpolarized ( N↑=N↓) UFG has the\nsimple structure:\nESLDA[n,τ,ν] =/planckover2pi12\nm/bracketleftbiggα\n2τ(r)+β3(3π2)2/3\n10n5/3(r)+γ|ν(r)|2\nn1/3(r)/bracketrightbigg\n,(3)\nand an additional Vext(r)n(r) term, for an arbitrary external potential in\nwhich the systemmight ormight not reside. Asdiscussed above, the kinetic\nenergy and the anomalous densities diverge, and a well defined reno rmal-\nization procedure was devised,12,13,26which amounts to using these two\ndensities in a unique combination in SLDA functional, see Eq. (2). The\ninfinite matter QMC calculations28(whereVext(r)≡0 andn(r) =const.)\nprovide enough information to determine the dimensionless constan tsα, β\nandγ.\nAn independent seriesofQMCcalculationsofalargenumber ofsyste ms\nwith variousnumbersoffermions N↑andN↓in anexternalharmonictrap31\nprovide results for the ground state properties of inhomogeneou s systems.\nAt this point one can use the SLDA functional to predict the proper ties of\nthese finite systems, see following table for a sample of results:19,26\nThe degree of agreement between the QMC results for the finite inh o-August 16, 2018 17:9 World Scientific Review Volume - 9in x 6in ws-rv9x6˙bulgac\nSuperfluid Local Density Approximation: A Density Function al Theory Approach to the Nuclear Pairing Problem 7\nA sample of ground state energies for unitary fermions in a harmonic trap\n(N↑,N↓)EQMC ESLDA Error\n(2,1) 4.281±0.004 4.417 3 .2%\n(2,2) 5.051±0.009 5.405 7%\n(3,2) 7.61±0.01 7.602 0 .1%\n(3,3) 8.639±0.03 8.939 3 .5%\n(4,3) 11.362±0.02 11.31 0 .49%\n(4,4) 12.573±0.03 12.63 0 .48%\n(5,5) 16.806±0.04 16.19 3 .7%\n(6,6) 21.278±0.05 21.13 0 .69%\n(7,6) 24.787±0.09 24.04 3%\n(7,7) 25.923±0.05 25.31 2 .4%\n(8,8) 30.876±0.06 30.49 1 .2%\n(9,9) 35.971±0.07 34.87 3 .1%\n(10,10) 41.302±0.08 40.54 1 .8%\n(11,10) 45.474±0.15 43.98 3 .3%\n(11,11) 46.889±0.09 45.00 4%\nmogeneous systems and the correspondingSLDA is a measure of th e ability\nof the DFT to describe strongly interacting superfluid fermionic sys tems.\nOne should keep in mind that the accuracy of the QMC results is cur-\nrently at the level of 5% (the least accurate quantity being the pair ing gap\n∆UFG(QMC) = 0.504(24)εFand ∆ UFG(exp.)≈0.45(5)εF), and that the\nQMC results for the infinite matter and finite systems were obtained by dif-\nferent groups, using somewhat different numerical approaches. Apart form\nthis reduced sample of results presented in this table many other th eoret-\nical results (thermodynamic properties, collective states, therm odynamic\nand quantum phase transitions)19,26and also extensive comparisons with\nresults of many experiments are available in literature. The good qua lity\nof the agreement shown here, along with the theoretical argumen ts pre-\nsented in favor of the SLDA functional above lend strong support to the\nassertionthatsuperfluid correlationsin fermionicsystemscanbea ccurately\ndescribed within the DFT approach. The quality of the agreement be tween\nthe QMC results for finite systems in harmonictraps and the corres ponding\nSLDAresults(in particulartoeven-oddstaggering)iseven moresu rprising,\nas one might have expected that derivative corrections might exist , specifi-\ncallyaterm /planckover2pi12|∇n(r)|2/mn(r), proportionaltoanundefined dimensionlessAugust 16, 2018 17:9 World Scientific Review Volume - 9in x 6in ws-rv9x6˙bulgac\n8 A. Bulgac\nconstant. One can make the argument that such a term can be exp ected if\nthe interaction has a finite range, and also that the introduction of such a\nterm would completely destroy the agreement between the SLDA an d the\ncorresponding QMC results for the finite UFG systems.19,26\nVery natural theoretical arguments allow us to extend the validity of\nthe SLDA functional even further. By invoking local Galilean covaria nce\none can show that in the un-renormalized SLDA functional (3) one h as to\nadd to perform the replacement\nα/planckover2pi12\n2mτ(r)→/planckover2pi12\n2mτ(r)+(α−1)/planckover2pi12\n2m/bracketleftbigg\nτ(r)−j2(r)\nn(r)/bracketrightbigg\n, (4)\nwherej((r) is the one-body current density.19,26A straightforward exten-\nsion exists as well to the case of a polarized UFG, when N↑∝negationslash=N↓.19,26By\nchanging the ratio N↓/N↑a UFG undergoes a number of quantum phase\ntransitions, from a uniform superfluid to a Larkin-Ovchinnikov phas e, in\nwhich the order parameter oscillates in space, and further to a sup erfluid\nwith relatively weak p-wave pairing and a normal state at very small tem-\nperatures.26The SLDA extension to include currentsallowsthe description\nof excited states, in particular vortices, and of time-dependent p henomena.\nThe existence of a DFT extension to time-dependent processes ha s been\nproven for quite some time.32The TDSLDA extension allowed so far the\nstudy of a large number of phenomena in a UFG system: the excitatio n of\nthe Anderson-Higgs modes,21in which the magnitude of the pairing field is\nexcited with a large amplitude; the Anderson-Bogoliubovsound mode s; the\nvortex structure20and the dynamical generation of vortices and their non-\ntrivial dynamics,22in particular the first microscopic description of vortex\ncrossing and reconnection in a fermionic superfluid leading to quantu m tur-\nbulence predicted by Feynman in 1956; the generation of quantum s hock\nwaves and domain walls in the collision of two UFG clouds.23\n3. Nuclear DFT\nThe nuclear EDF is slightly more complicated, as one has to embed the d e-\npendence on proton and neutron densities.17,25The general principle how-\never is very similar, the nuclear EDF has to satisfy all required symme tries:\nrotational and translational invariance, parity and isospin symmet ry, gauge\nsymmetry and Galilean invariance. We will not discuss here questions r e-\nlated to symmetry restoration, quantization of large amplitude mot ion, and\nthe DFT stochastic extension, which were addressed recently else where.33August 16, 2018 17:9 World Scientific Review Volume - 9in x 6in ws-rv9x6˙bulgac\nSuperfluid Local Density Approximation: A Density Function al Theory Approach to the Nuclear Pairing Problem 9\nAn un-renormalized local nuclear EDF should have the following struc ture:\nESLDA=/planckover2pi12\n2mpτp(r)+/planckover2pi12\n2mnτn(r)+εN[ρp(r),τp(r),ρn(r),τn(r),...]\n+εS[ρp(r)+ρn(r)](|νp(r)|2+|νn(r)|2)\n+εS′[ρp(r)+ρn(r)](ρp(r)−ρn(r))(|νp(r)|2−|νn(r)|2)\n+e2/integraldisplay\nd3r′ρch(r)ρch(r′)\n|r−r′|+εxc[ρp(r),ρn(r)]. (5)\nwhere the isospin symmetry is broken by the small difference betwee n the\nprotonmpand neutron mnmasses and the Coulomb interaction. The\nCoulomb interaction has a Hartree contribution in terms of the char ge den-\nsityρch(r)(whichisdifferentfrom ρp(r)mostlyduetothefiniteprotonsize)\nand an exchange-correlation contribution εxc[ρp(r),ρn(r)], which is neither\na simple Fock term nor its Slater approximation. Phenomenologicalst udies\nof nuclear mass formulas show that a better fit can be typically obta ined\nby neglecting the Coulomb exchange,34and a many-body analysis of the\nCoulomb exchange in a nuclear medium show that its magnitude is signifi-\ncantlyreduced35andthiseffectisencodedintheterm εxc[ρp(r),ρn(r)]. The\ntermεN[ρp(r),τp(r),ρn(r),τn(r),...], where the ellipses stand for other den-\nsitiessuchasspindensitiesandderivativesof ρp,n(r), shouldbeasymmetric\nfunction of the proton and neutron densities in order to satisfy th e isospin\ninvariance. The first five terms should be chosen so as to describe c or-\nrectly the QMC results for infinite neutron and symmetric nuclear ma tter.\nThis is how Fayans has defined the first implementation of the Kohn-S ham\nDFT approach to nuclei.10In the case of infinite matter various derivative\nterms give a vanishing contribution and their contribution to the nuc lear\nEDF has to be determined from the properties of finite nuclei. The te rm\nεS[ρp(r)+ρn(r)](|νp(r)|2+|νn(r)|2) is the most important term describing\nthe nuclear pairing correlations, which also satisfies isospin invarianc e and\nit has been used to describe odd-even effects in more than 200 nucle i with a\nsimple constantvolumepairing( εS≡const.).17,25The sameformalismwas\nused to demonstrate that a vortex in neutron matter develops a la rge den-\nsity depletion at its core, a feature which controlsthe pinning mecha nism.24\nThe additional term εS′[ρp(r)+ρn(r)](ρp(r)−ρn(r))(|νp(r)|2−|νn(r)|2) is\nalso isospin symmetric, but at this time it is not entirely clear whether\nsuch a term is present and whether its presence is required. In man y\nphenomenological nuclear mass studies36various authors introduce inde-\npendent coupling constants in the particle-particle channel for ne utron and\nprotons, which is a clear violation of the isospin symmetry. What is eve nAugust 16, 2018 17:9 World Scientific Review Volume - 9in x 6in ws-rv9x6˙bulgac\n10 A. Bulgac\nmore puzzling is the fact that these phenomenological studies claim t hat\nthe strength of the coupling is stronger in the proton channel tha n in the\nneutron channel, which is hard to substantiate microscopically. Ano ther\ninconsistency of these phenomenological approaches is the introd uction of\ndifferentcouplingstrengthsforevenandoddsystems,36whichisnotneeded\nand is theoretically unjustified.17It is natural to expect that the pairing\ninteractionis weakerin theprotonchannel, due tothe repulsivecha racterof\nthe Coulomb interaction.37Another problem with approaches which try to\nbe more microscopic in this respect37,38is the adoption of the pairing gaps\ncalculated in the weak coupling BCS approximation, which are significan tly\nhigher than the actual values of the pairing gaps in infinite matter.29It is\nknown since 196139that pairing gaps are reduced by a factor (4 e)1/3≈2.2,\nwhen one accounts for the contribution of the induced interaction s, which\nis also confirmed by full QMC calculations of the UFG and of the neutro n\nmatter.28,29UnfortunatelytherearenoQMCcalculationsoftheprotonand\nneutron pairing gaps in symmetric nuclear matter, and thus the pos sible\ndependence of these gaps on the isospin composition of the nuclear matter\nis still unknown. The divergences of the kinetic energy and anomalou s den-\nsities in (5) should be dealt with as described above,12,13,19,26see Eq. (2).\nThe Galilean invariance is retrieved by using the recipe described abov e in\nthe case of kinetic energy density19,26by including proton and neutron cur-\nrentdensities, seeEq. (4) andsimilarextensionsforotherdensitie s.40With\nthe inclusion ofcurrentsthe DFT canbe used to describeexcited st atesand\ntime-dependent phenomena and collisions.21–23,25,33The first application\nof the TDSLDA extension to a nuclear process was described recen tly25by\ncalculating for the first time the excitation of the giant dipole resona nces in\ndeformed open-shell heavy nuclei without any unjustified approx imations\nand the agreement with experimental data is very good, without th e need\nof any fitting parameters. Formally the solution of the TDSLDA equa tions\nappears as a time-dependent Hartree-Bogoliubov formalism in 3D an d the\ncomplexityoftheseequationsrequirestheuseofleadershipclassc omputers,\nas it amounts to solving tens to hundreds of thousands of coupled n onlin-\near time-dependent 3D partial differential equations for tens to h undreds\nof thousands of time steps. With TDSLDA a microscopic treatment o f low\nenergy nuclear collisions, analogous to the collisions of superfluid ato mic\nclouds performed recently,23and induced nuclear fission in particular is\nwithin reach for the first time.\nThe authorthanks his collaboratorsM.M. Forbes, Y.-L. Luo, P. Mag ier-\nski, K.J. Roche, V.R. Shaginyan, I. Stetcu, S. Yoon, and Y. Yu, and theAugust 16, 2018 17:9 World Scientific Review Volume - 9in x 6in ws-rv9x6˙bulgac\nSuperfluid Local Density Approximation: A Density Function al Theory Approach to the Nuclear Pairing Problem 11\nsupport received from the US Department of Energy through the grants\nDE-FG02-97ER41014 and DE-FC02-07ER41457.\nReferences\n1. D. Pines and P. Nozi` eres, The theory of Quantum Liquids , (Cambridge, MA,\nPerseus, 1999).\n2. A.A. Abrikosov, L.P. Gorkov, L.E. Dzyaloshinski, Methods of Quantum Field\nTheory in Statistical Physics , (Dover, 1975).\n3. A.B. Migdal, Theory fo Finite Fermi Systems, and Applications to Atomic\nNuclei, (Interscience Publishers, N.Y. 1767).\n4. H. Bethe, Annu. Rev. Nucl. Sci. 21, 93 (1971).\n5. D. Vautherin and D.M. Brink, Phys. Rev C 5, 626 (1972).\n6. D. Gogny, in Proc. Int. Conf. on nuclear physics , J. de Boer, H.J. Mang eds.,\n(Munich, North-Holland, Amsterdam, 1973); in Nuclear self-consistent fields ,\nG. Ripka, M. 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C ontinenza,\nPhys. Rev. Lett. 96, 047003 (2006)\n12. A. Bulgac, IPNE FT-194-1980, Bucharest, arXiv:nucl-th /9907088.\n13. A. Bulgac and Y. Yu, Phys. Rev. Lett. 88, 042504, (2002); A. Bulgac, Phys.\nRev. C65, 051305(R) (2002).\n14. B. Gebremariam, T. Duguet, and S.K. Bogner, Phys. Rev. C 82, 014305\n(2010).\n15. D.B. Kaplan, M.J. Savage, and M.B. Wise, Phys. Lett. B424, 390 (1998).\n16. J.M. Blatt and V.F. Weiskopf, Theoretical Nuclear Physics , (Wiley, New\nYork, 1952), pp. 7476; K. Huang, Statistical Mechanics , (John Wiley & Sons,\nNew York, 1987), pp 230238.\n17. Y. Yu and A. Bulgac, Phys. Rev. Lett. 90, 222501 (2003).\n18. J.E. Drut, R.J. Furnstahl, and L. Platter, Prog. Part. Nu cl. Phys. 64, 120\n(2010).\n19. A. Bulgac, Phys. Rev. A 76, 040502(R) (2007).August 16, 2018 17:9 World Scientific Review Volume - 9in x 6in ws-rv9x6˙bulgac\n12 A. Bulgac\n20. A. Bulgac and Y. Yu, Phys. Rev. Lett. 91, 190404 (2003).\n21. A. Bulgac and S. Yoon. Phys. Rev. Lett. 102, 085302 (2009).\n22. A. Bulgac, Y.-L. Luo, P. Magierski, K.J. Roche, and Y. Yu, Science, 332,\n1288 (2011).\n23. A. Bulgac, Y.-L. Luo, and K.J. Roche, Phys. Rev. Lett. 108, 150401 (2012).\n24. Y. Yu, and A. Bulgac, Phys. Rev. Lett. 90, 161101 (2003).\n25. I. Stetcu, A. Bulgac, P. Magierski, and K.J. Roche, Phys. Rev. C 84,\n051309(R) (2011).\n26. A. Bulgac, M.M. Forbes, and P. Magierski, The Unitary Fermi Gas: From\nMonte Carlo to Density Functionals , arXiv:1008.3933, chapter 9, pp 305-373,\nin Ref.27\n27.BCS-BEC Crossover and the Unitary Fermi Gas (Lecture Notes in Physics,\nVol. 836), ed. W. Zwerger (Springer Heidelberg, 2012).\n28. J. Carlson and S. Reddy, Phys. Rev. Lett. 95, 060401 (2005); Phys. Rev.\nLett.100, 150403 (2008).\n29. A. Gezerlis and J. Carlson, Phys. Rev. C 81, 025803(R) (2010).\n30. The Many-Body Challenge Problem (MBX) formulated by G. F . Bertsch in\n1999; G.A. Baker, Phys. Rev. C 60(5), 054311 (1999); Int. J. M od. Phys. B\n15(10-11), 1314 (2001).\n31. D. Blume, J. von Stecher, C.H. Greene, Phys. Rev. Lett. 99, 233201 (2007);\nD. Blume, Phys. Rev. A 78, 013635 (2008).\n32. E. Runge, E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984); see URL\nwww.tddft.org forreports onmanycurrentdevelopmentsincondensedmatte r\ntheory and chemistry.\n33. A. Bulgac, J. Phys. G: Nucl. Part. Phys. 37, 064006 (2010).\n34. B.A. Brown, Phys. Rev. C 58, 220 (1998); B. A. Brown, W. A. Richter, and\nR. Lindsay, Phys. Lett. B 483, 49 (2000).\n35. A. Bulgac and V.R. Shaginyan, Nucl. Phys. A 601, 103 (1996); Phys. Lett.\nB469, 1 (1999).\n36. D. Luney, J.M. Pearson, and C. Thibault, Rev. Mod. Phys. 75, 1021 (2003);\nM. Kortelainen, J. McDonnell, W. Nazarewicz, P.-G. Reinhar d, J. Sarich,\nN. Schunck, M.V. Stoitsov, and S.M. Wild, Phys. Rev. C 85, 024304 (2012);\nG.F. Bertsch, C.A. Bertulani, W. Nazarewicz, N. Schunck, an d M.V. Stoitsov,\nPhys. Rev. C 79, 034306 (2009).\n37. T. Lesinski, T. Duguet, K. Benaceur, and J. Meyer, Eur. Ph ys. J.A 40, 121\n(2009).\n38. S. Goriely, N. Chamel, and J.M. Pearson, Phys. Rev. Lett. 102, 152503\n(2009).\n39. L.P. Gorkov and T.K. Melik-Barkhudarov, Sov. Phys. JETP 13, 1018 (1961);\nH. Heiselberg, C.J. Pethick, H. Smith, and L. Viverit, Phys. Rev. Lett. 85,\n2418 (2000).\n40. Y.M. Engel, D.M. Brink, K. Goeke, S.J. Krieger, D. Vauthe rin, Nucl. Phys.\nA 249, 215 (1975); M. Bender, P.H. Heenen, P.-G. Reinhard, Rev. Mo d. Phys.\n75, 121 (2003)."    },    {        "title": "1206.4872v1.Theoretical_method_for_the_study_of_the_excited_states_of_a_system.pdf",        "content": "arXiv:1206.4872v1  [quant-ph]  21 Jun 2012Theoretical method for the study of the excited states of a sy stem\nRam´ on Alain Miranda Quintana∗\nDepartment of Radiochemistry, Higher Institute of Technol ogies and Applied Sciences,\nAve. Salvador Allende y Luaces, Quinta de los Molinos,\nPlaza de la Revoluci´ on, POB 6163, 10600, Havana, Cuba.\nA novel, exact, theoretical method for the study of the excit ed states of a system is presented. It is\ndemonstrated how to transform the excited state problem of o ne Hamiltonian into the ground state\nproblem of an auxiliary one. From this, a new exact density fu nctional suitable for excited states is\nconstructed. These results make the excited states of a syst em accessible through all ground state\ntheoretical techniques.\nPACS numbers: 03.65.-w, 71.15.Qe, 31.15.ec, 71.15.Mb\nKeywords: Excited states, Variational principle, Density functional theory\nIn spite of the development in the course of the years of many soph isticated and powerful theoretical tools, the\nvariational principle of Rayleigh and Ritz (RR) keeps its central role in quantum theory. This principle, besides its\nmany practical applications, has also served as starting point for t he development of new theoretical methods, the\ndensity functional theory (DFT) being an example of this1–3. But the classical formulation of the variational principle\nis restricted to the study of ground states or to the first excited states having a symmetry which is different from\nthe symmetry of the ground state. Given the importance of the an alysis of the excited states of a system, a great\neffort has been put in the search for generalizations of the RR princ iple in ways that can be used to study them4–8.\nHere we demonstrate a set of theorems that allow us to apply the or iginal RR principle to excited states. In doing\nso, from a Hamiltonian H0and with some knowledge of its ground state, a new Hamiltonian H1is built, which has\nthe characteristic that its ground state exactly coincides with the first excited state of H0. A DFT type variational\nprinciple for the new Hamiltonian is obtained, which enable us to constr uct a new exact density functional suitable\nfor the study of excited states. These results make the excited s tates of a system accessible through all ground state\ntechniques.\nLet us start giving some definitions. Let H0be a Hamiltonian and E0the set of its eigenvalues. We will denote by\nV0the set of the eigenvectors of H0, ifEk∈E0,VEk\n0will represent the subset of V0associated with the eigenvalue Ek.\nThe set{|Ekβ/a\\}bracketri}ht}corresponds to an orthogonal basis of VEk\n0whereβcan be interpreted as the eigenvalues of a set of\nobservables which, along with the energy, constitute a complete se t of observables for the system under study9. Let\nus suppose that E0=minE0and that we know a {|E0β/a\\}bracketri}ht}. In this case we can build a Hamiltonian H1according to:\nH1=H0+K/summationdisplay\nβ|E0β/a\\}bracketri}ht/a\\}bracketle{tE0β| (1)\nIn this expression Kis a real constant that will be analyzed later. In full analogy as it has been done before, the sets\nE1,V1andVEk\n1corresponding to H1can be defined.\nNow, startingfrom the definition of H1, some ofits propertieswill be analyzed. In the following, Halmos’ /squaresolidindicates\nthe conclusion of a proof.\nTheorem 1: IfE1=min(E0\\{E0}), then:∀ψ,/a\\}bracketle{tψ|H1|ψ/a\\}bracketri}ht ≥min{E1,E0+K}\nProof:BecauseH0is an observable we can write9:\n∀|ψ/a\\}bracketri}ht,|ψ/a\\}bracketri}ht=/summationdisplay\nEkβc(Ekβ)|Ekβ/a\\}bracketri}ht;Ek∈E0,c(Ekβ) =/a\\}bracketle{tEkβ|ψ/a\\}bracketri}ht (2)\nWhere/summationtext\nEkβ|c(Ekβ)|2= 1.\nThen:\n/a\\}bracketle{tψ|H1|ψ/a\\}bracketri}ht=/a\\}bracketle{tψ|H0|ψ/a\\}bracketri}ht+K/summationdisplay\nβ|c(E0β)|2(3)\nDoing/summationtext\nβ|c(E0β)|2≡twe will have that:\n∀|ψ/a\\}bracketri}ht,/a\\}bracketle{tψ|H1|ψ/a\\}bracketri}ht ≥minTψ/a\\}bracketle{tϕ|H0|ϕ/a\\}bracketri}ht+Kt (4)\nThe minimum is searched over the set Tψ={|ϕ/a\\}bracketri}ht:/summationtext\nβ|/a\\}bracketle{tE0β|ϕ/a\\}bracketri}ht|2=t}.2\nIt is then evident that, provided /a\\}bracketle{tϕ|ϕ/a\\}bracketri}ht= 1:minTψ/a\\}bracketle{tϕ|H0|ϕ/a\\}bracketri}ht=E0t+E1(1−t) so:\n∀|ψ/a\\}bracketri}ht,/a\\}bracketle{tψ|H1|ψ/a\\}bracketri}ht ≥ǫ(t) =E1+(E0+K−E1)t (5)\nEvery linear functional ǫ(t) will reach its lower value in the frontier of its domain of definition, in th is case, int= 0\nor int= 1. Checking that, ǫ(0) =E1andǫ(1) =E0+Kends up proving the Theorem. /squaresolid\nTheorem 2: V0⊆V1\nProof:|ψ/a\\}bracketri}ht ∈V0⇒ |ψ/a\\}bracketri}ht=/summationtext\nβc(Ekβ)|Ekβ/a\\}bracketri}ht;Ek∈E0,c(Ekβ) =/a\\}bracketle{tEkβ|ψ/a\\}bracketri}htSo:\nH1|ψ/a\\}bracketri}ht=H0|ψ/a\\}bracketri}ht+K/summationdisplay\nβ′|E0β′/a\\}bracketri}ht/a\\}bracketle{tE0β′|ψ/a\\}bracketri}ht (6a)\nH1|ψ/a\\}bracketri}ht=H0|ψ/a\\}bracketri}ht+K/summationdisplay\nβ′|E0β′/a\\}bracketri}ht/summationdisplay\nβc(Ekβ)/a\\}bracketle{tE0β′|Ekβ/a\\}bracketri}ht (6b)\nWhere/a\\}bracketle{tE0β′|Ekβ/a\\}bracketri}ht=δE0Ekδβ′β, beingδijthe Kronecker’s delta.\nLet us consider the cases: a) Ek/\\e}atio\\slash=E0and b)Ek=E0.\na)Ek/\\e}atio\\slash=E0⇒ /a\\}bracketle{tE0β′|Ekβ/a\\}bracketri}ht= 0∀β,β′, then:\nH1|ψ/a\\}bracketri}ht=H0|ψ/a\\}bracketri}ht=Ek|ψ/a\\}bracketri}ht ⇒ |ψ/a\\}bracketri}ht ∈V1 (7)\nThus, an eigenket of H0corresponding to an eigenvalue Ek/\\e}atio\\slash=E0will be as well an eigenket of H1corresponding\nto the same eigenvalue.\nb)Ek=E0⇒ /a\\}bracketle{tE0β′|Ekβ/a\\}bracketri}ht=/a\\}bracketle{tE0β′|E0β/a\\}bracketri}ht=δβ′β, then:\nH1|ψ/a\\}bracketri}ht=H0|ψ/a\\}bracketri}ht+K/summationdisplay\nβ′c(E0β′)|E0β′/a\\}bracketri}ht=E0|ψ/a\\}bracketri}ht+K|ψ/a\\}bracketri}ht= (E0+K)|ψ/a\\}bracketri}ht ⇒ |ψ/a\\}bracketri}ht ∈V1 (8)\nThus, an eigenket of H0corresponding to the eigenvalue E0will be as well an eigenket of H1corresponding to the\neigenvalueE0+K./squaresolid\nFrom this proof it follows that ( E0\\{E0})∪{E0+K} ⊆E1.\nTheorem 3: K >0⇒(E0\\{E0})∪{E0+K}=E1\nProof:(byreductio ad absurdum ) Let us suppose that:\n���˜E /∈(E0\\{E0})∪{E0+K}:∃|˜E/a\\}bracketri}ht /\\e}atio\\slash=|0/a\\}bracketri}ht;H1|˜E/a\\}bracketri}ht=˜E|˜E/a\\}bracketri}ht (9)\nFrom Theorem 1 and the RR principle: K >0⇒˜E/\\e}atio\\slash=E0.\nFor beingH0an observable we can write9:\n|˜E/a\\}bracketri}ht=/summationdisplay\nEkβc(Ekβ)|Ekβ/a\\}bracketri}ht;Ek∈E0,c(Ekβ) =/a\\}bracketle{tEkβ|˜E/a\\}bracketri}ht (10)\nBut, according to Theorem 2: ∀Ekβ,Ek∈E0,|Ekβ/a\\}bracketri}ht ∈V1then we will have, by hypothesis:\n∀Ekβ,Ek∈E0,c(Ekβ) =/a\\}bracketle{tEkβ|˜E/a\\}bracketri}ht= 0 (11)\nThe contradiction ends up proving the Theorem. /squaresolid\nThe previous results allow us to conclude with the following:\nCorollary: K >0⇒V1=V0, even more, V˜Ek\n1=VEk\n0,˜Ek=Ek+δ0kK\nProof:From the proof of Theorem 2 it follows that VEk\n0⊆V˜Ek\n1, so it suffices to proof that V˜Ek\n1⊆VEk\n0.\nBe|ψ/a\\}bracketri}ht ∈V˜Ek\n1, according to Theorem 3 this implies that ˜Ek∈(E0\\{E0})∪{E0+K}.\nFor beingH0an observable we can write9:\n|ψ/a\\}bracketri}ht=/summationdisplay\nEk′βc(Ek′β)|Ek′β/a\\}bracketri}ht=/summationdisplay\nβc(Ekβ)|Ekβ/a\\}bracketri}ht+/summationdisplay\n(Ek′/negationslash=Ek)βc(Ek′β)|Ek′β/a\\}bracketri}ht;Ek′∈E0,c(Ek′β) =/a\\}bracketle{tEk′β|ψ/a\\}bracketri}ht(12)\nIn this expression when |ψ/a\\}bracketri}ht ∈V˜E0\n1we will take k= 03\nFrom Theorem 2:/summationtext\nβc(Ekβ)|Ekβ/a\\}bracketri}ht ∈V˜Ek\n1, then:\n/summationdisplay\n(Ek′/negationslash=Ek)βc(Ek′β)|Ek′β/a\\}bracketri}ht=|ψ/a\\}bracketri}ht−/summationdisplay\nβc(Ekβ)|Ekβ/a\\}bracketri}ht ∈V˜Ek\n1 (13)\nfor being V˜Ek\n1a linear subspace.\nDenoting the set of all pure states of the system by Wwe will have, for being H1an observable9,10:\nW=⊕kV˜Ek\n1 (14)\nwhere⊕kstands for the direct sum of the different subspaces. But from (1 4)10:\nV˜Ek\n1∩/summationdisplay\nk′/negationslash=kV˜Ek′\n1={|0/a\\}bracketri}ht} (15)\nThen:\n|ψ/a\\}bracketri}ht=/summationdisplay\nβc(Ekβ)|Ekβ/a\\}bracketri}ht ∈VEk\n0/squaresolid (16)\nLet us analyze the case:\nK >E 1−E0 (17)\nfrom which we can directly infer that K >0.\nIn virtue of the properties proven before results that, with this s election ofK, we have built a Hamiltonian whose\nground state satisfies E1=minE1at the same time that VE1\n1=VE1\n0. That is, the study of the first excited state of\nH0reduces to the study of the ground state of H1, for which we can make use of the techniques classically prescribed\nfor the study of minimum energy states.\nAmong the requirements needed for the construction of H1, the necessity of having solutions for the ground state\nofH0that forms an orthogonal basis for this subspace stands out. Th is, besides been computationally expensive,\ncan reminds us the known fact that the RR principle can be applied to t he study of excited states under certain\nconditions11. Ineffect, ifweworkwith theexpression minP0/a\\}bracketle{tψ|H0|ψ/a\\}bracketri}ht, whereithasbeenpointedoutthatthe minimum\nmust be searched over the set P0={|ψ/a\\}bracketri}ht:/a\\}bracketle{tψ|ϕ/a\\}bracketri}ht= 0∀|ϕ/a\\}bracketri}ht ∈VE0\n0}, we will obtain the energy and eingenfunctions\ncorresponding to the first excited state of H0. This procedure is different from the one presented here: in the\nexpression of the RR principle corresponding to H1allNparticle functions fulfilling the Pauli principle are included.\nThat is, the expression min/a\\}bracketle{tψ|H1|ψ/a\\}bracketri}htcan be used directly, exactly as the original RR principle min/a\\}bracketle{tψ|H0|ψ/a\\}bracketri}ht, because\nwe are dealing with a ground state. In this new method there is no nee d to know the set P0.\nThe other condition for H1having the desired properties, namely K > E 1−E0, is of the outermost importance.\nEven when K >0, but with Knot large enough to fulfill the above condition, the application of the RR principle to\nH1will end in a ground state with energy E0+Kwhose eigenfunctions will be those of the ground state of H0. In\norder to ensure that (17) holds true we must work with Klarge enough. As a way to check if we have made a right\nchoice is enough to determine the min/a\\}bracketle{tψ|H1|ψ/a\\}bracketri}ht, if this value is different from E0+Kthe chosen Kvalue is correct.\nAnother advantage of being able to apply the RR principle to H1lies in the possibility of formulate a DFT type\nvariational principle for the first excited state of the Hamiltonian if w e take, in the spirit of Levy’s constrained\nsearch2,3:\nE1=minρE[ρ] (18)\nWhere the minimum is searched over all the N-representable densities, being:\nE[ρ] =FH0[ρ]+/integraldisplay\nρ(r)υ(r)dr (19a)\nFH0[ρ] =infψ→ρ/a\\}bracketle{tψ|\nT+W+K/summationdisplay\nβ|E0β/a\\}bracketri}ht/a\\}bracketle{tE0β|\n|ψ/a\\}bracketri}ht (19b)\nIn the above expressions υ(r) stands for the external potential present in H0,TandWcorrespond to the kinetic\nenergy and particle-particle interaction operators respectively. The subindex in FH0emphasizes the non-universality4\nofthis functional, anunfavorablefeature that this method share swith anotherDFT extensionstoexcited states3,12–14.\nHowever, it must be noticed that the non-universal character is r espect to the excited state density. The functional\nis in fact an universal bifunctional, depending on the electronic dens ity of the excited and ground states15,16.\nApplications of this scheme within DFT can be expected in two different ways. First, the derivation of formal\nconstraints (of the type of coordinate scaling, asymptotic behav ior, etc.) for density functionals designed for excited\nstates. For this, besides using traditional tools as the virial and He llmann-Feynman theorems, we can use a novel\nproperty of this functional. The fact that ∀K:K >E 1−E0E[ρ] always reach the same minimum value in the same\ndensity functions will contribute to shed some light in the formal str ucture of these functionals. Second, practical\napplications in DFT can be derived by means of the optimized potential method16. Here it is not a problem that\nthe energy partition for the excited state functionals not needs t o be the same that for the ground state functionals,\nsince the adiabatic connection can help us to develop new exchange a nd correlation functionals suitable for excited\nstates15,17.\nTo conclude let us point out that the method presented here can be extended to the study of an arbitrary excited\nstate if we have enough information about the stationary states b elow it. That is, the study of VEk\n0,Ek∈E0is\npossible using techniques designed to ground states if {|Ek′β/a\\}bracketri}ht}∀Ek′∈E0,Ek′<Ekare known.\nAcknowledgments\nI am indebted to Alberto Beswick, Ernesto Altshuler, Marco Mart´ ın ez and Alfo J. Batista for their critical reading\nof the manuscript.\n∗Electronic address: alain@instec.cu\n1P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864, (1964).\n2M. Levy, Proc. Natl. Acad. Sci. U.S.A. 76, 6062, (1979).\n3R. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989). pp.\n56-60, 204-208.\n4J. K. L. McDonald, Phys. Rev. 46, 828 (1934).\n5A. K. Theophilou, J. Phys. C 12, 5419 (1979).\n6E. K. U. Gross, L. N. Oliveira and W. Kohn, Phys. Rev. A 37, 2805 (1988).\n7E. K. U. Gross, L. N. Oliveira and W. Kohn, Phys. Rev. A 37, 2809 (1988).\n8L. N. Oliveira, E. K. U. Gross and W. Kohn, Phys. Rev. A 37, 2821 (1988).\n9P. A. M. Dirac, The Principles of Quantum Mechanics (3rd edn. Oxford University Press, New York, 1947).\n10E. S. Suhubi, Functional Analysis (Kluwer Academic Publishers, The Netherlands, 2003).\n11I. N. Levine, Qu´ ımica Cu´ antica , Spanish translation (5th edn. Pearson Educaci´ on S. A., Ma drid, 2001).\n12S. M. Valone and J. F. Capitani, Phys. Rev. A 23, 2127 (1981).\n13R. M. Dreizler and E. K. U. Gross, Density Funtional Theory. An Approach to the Quantum Many-B ody Problem (Springer-\nVerlag, Berlin Heidelberg, 1990). pp. 32-36.\n14E. H. Lieb, in Density Functional Methods in Physics , edited by R. M. Dreizler and J. da Providencia (Plenum Press , New\nYork, 1985) p. 31.\n15M. Levy and ´A. Nagy, Phys. Rev. Lett. 83, 4361 (1999).\n16´A. Nagy and M. Levy, Phys. Rev. A 63, 052502 (2001).\n17M. Levy and A. G¨ orling, Phys. Rev. A 53, 3140 (1996)."    },    {        "title": "1206.5630v1.Separable_states_and_the_SPA_of_a_positive_map.pdf",        "content": "arXiv:1206.5630v1  [quant-ph]  25 Jun 2012Separable states and the SPA of a positive map\nErling Størmer\n06-21-2012\nWe introduce a necessary condition for a state to be separable and apply\nthis condition to the SPA of an optimal positive map and give a proof of the\nfact that the SPA need not be separable.\nIntroduction\nIt is often very difficult to verify whether a givenstate on a tensorp roductMn⊗\nMm,Mnbeing the nxn matrices, is separable. There are necessary conditio ns\nlikethePPTconditionwhicharereasonablyeasyto verify, but nosuc hsufficient\nconditions. In the present paper we introduce two necessary con ditions which\nare very easy to verify, and then apply them to show that the SPA o f some\noptimal maps introduced by Ha and Kye [3] do not correspond to sep arable\nstates, yielding a counter example to the conjecture in [5] stating that the SPA\nof an optimal unital map is separable. It was announced in [3] that so me of\ntheir maps yield counter examples of the conjecture and that proo fs will appear\nin [4].\nAnother consequence of our analysis is that if φ:Mn→Mnis positive, then\nthe mapφ(1)Tr+φis super-positive, i.e. entanglement breaking, a result also\nshown in [8].\nLet us recall some terminology. A linear map φ:Mn→Mmis positive,\nwrittenφ≥0, ifφ(a)≥0 whenever a≥0.φis completely positive if it is the\nsum of maps of the form AdVdefined byAdV(a) =V∗aV, where V is a linear\noperatorfrom CmtoCn, see [1].φis called optimal if there exists no completely\npositivemap ψ:Mn→Mmsuchthatφ−ψ≥0. Ifφisunital, i.e. φ(1) = 1, then\nthe structural physical approximation, the SPA of φ, is the Choi matrix (also\ncalled entanglement witness) of a completely positive map closely asso ciated\nwithφ, see Section 2. We recall that if ( eij) is a complete set of matrix units\nforMnthen the Choi matrix Cφforφis the matrix/summationtexteij⊗φ(eij)∈Mn⊗Mm.\nThenφis completely positive if and only if Cφis positive, see [1]. φis called\nsuper-positive, or entanglement breaking, if Cφis separable, i.e. a sum of the\nformCφ=/summationtextai⊗biwithai,bi≥0.\n11 Separable states\nLet (eij) be a complete set of matrix units for Mn. Then the set ( eij⊗ekl)ijkl\nis a complete set of matrix units for Mn⊗Mn. Hence, if a∈Mn⊗Mnthena\nis of the form\na=/summationdisplay\nijkla(ij)(kl)eij⊗ekl.\nWe introduce the following notation.\nS(a) =/summationdisplay\na(ij)(ij),T(a) =/summationdisplay\na(ij)(ji).\nS(a) and T(a) give necessary conditions for a state with density op eratorato\nbe separable. Indeed we have:\nProposition 1 Ifais a separable density operator in Mn⊗Mnthen0≤S(a)≤\n1,0≤T(a)≤1\nProof.We first consider the case when a=b⊗cwithb,c≥0. SinceTr(a)≤1\nwe may scale bandcsoTr(b),Tr(c)≤1. Letb= (bij),c= (ckl). Then\na(ij)(kl)=bijckl.\nHence\nT(a) =/summationdisplay\nbijcji.\nSincebc=/summationtextbijcjkeik, we have\n0≤Tr(bc) =/summationdisplay\nbijcji=T(a).\nHence, if we write TrforTr⊗Tr\nT(a)≤/bardblb/bardblTr(c)≤Tr(b)Tr(c) =Tr(b⊗c)≤1.\nIn the general case a=/summationtextbk⊗ckwithbk,ck≥0. Thus\n0≤T(a) =/summationdisplay\nT(bk⊗ck)≤/summationdisplay\nTr(bk⊗ck) =Tr(a) = 1.\nThe proof for S(a) is similar replacing ckby its transpose ( ck)t. The proof is\ncomplete.\nIn order to study separability Werner [9] considered automorphis ms of the\nformAdU⊗UonMn⊗Mn. He showed the following result; for a detailed proof\nsee Section 7.4 in [8].\nLemma 2 Leta∈Mn⊗Mnbe invariant under all automorphisms AdU⊗U\nwithUunitary inMn. Thenais of the form a=α1⊗1+βVwithα,β∈C\nandVthe flipb⊗c→c⊗bonMn⊗Mn.\n2Let G denote the compact group G={AdU⊗U:U∈Mnunitary}. Let\ndUbe the normalized Haar measure on G. Then\nP(a) =/integraldisplay\nGAdU⊗U(a)dU\nis the unital trace preserving projection of Mn⊗Mnonto the fixed point algebra\nof G, which by Lemma 2.2 is the linear span of 1 ⊗1 andV.\nProposition 3 Leta∈Mn⊗Mn. Then\nT(a) =Tr(aV) =Tr(P(a)V).\nProof.We have\nTr(P(a)V) =/integraldisplay\nGTr(AdU⊗U(a)V)dU\n=/integraldisplay\nGTr(aAdU∗⊗U∗(V))dU\n=/integraldisplay\nGTr(aV)dU\n=Tr(aV).\nTo show the other equality we first note that V=/summationtexteij⊗eji. Indeed, if ξiis\nan orthonormal basis for Cnsuch thateijξk=δjkξi, then\neij⊗eji(ξk⊗ξl) =/summationdisplay\nδjkξi⊗δilξj=ξl⊗ξk,\nproving the assertion. If a=/summationtexta(ij)(kl)eij⊗ekl, we get\nTr(aV) =Tr(/summationdisplay\nijkla(ij)(kl)eij⊗ekl/summationdisplay\nrsers⊗esr)\n=Tr(/summationdisplay\nj=r,l=sa(ij)(kl)eis⊗ekr)\n=Tr(/summationdisplay\nijkla(ij)(kl)eil⊗ekj)\n=/summationdisplay\nija(ij)(ji)\n=T(a),\nproving the proposition.\nTheorem 4 Letabe a density matrix in Mn⊗Mn, and letψdenote the positive\nmap ofMnintoMngiven by\nψ(b) =n(n2−1)\nnT(a)−1)Tr(a)+at,\n3t being the transpose map. Then we have:\n(i)Cψ=n(n2−1)\nnT(a)−1P(a).\n(ii)cTr+tis super-positive if and only if c≥1.\n(iii)P(a)is separable if and only if T(a)≤1.\nProof.By Lemma 2 P(a) =α1⊗1+βV, hence\nTr(a) =Tr(P(a)) =αn2+βn.\nBy Lemma 3, since Tr(V) =n,\nT(a) =Tr(P(a)V) =Tr((α1⊗1+βV)V) =Tr(αV+β1⊗1) =αn+βn2.\nSolving the last two equations for αandβwe get\nα=n−T(a)\nn(n2−1), β=nT(a)−1\nn(n2−1)\nHence\nP(a) =n−T(a)\nn(n2−1)1⊗1+nT(a)−1\nn(n2−1)V.\nSinceCTr= 1⊗1, andCt=Vfor the transpose map t, we find\nn(n2−1)\nnT(a)−1P(a) =n−T(a)\nnT(a)−11⊗1+V=Cψ,\nproving (i).\nIn the special case when a=e⊗ewithea 1-dimensional projection we have\nT(a) = 1, so by (i)\nn(n+1)P(a) =Cψ=n−1\nn−11⊗1+V= 1⊗1+V,\nsoψ=Tr+t. Thus, ifφis a positive map, then\nTr(CψCφ) =n(n+1)/integraldisplay\nGTr(AdU(e)⊗AdU(e)Cφ)dU≥0,\nsinceTr(b⊗cCφ)≥0 for allb≥0,c≥0 whenφis a positive map. Thus Tr+t\nis in the dual cone of all positive maps, so that Tr+tis super-positive, see [7].\nIfc≥1 then\ncTr+t= (c−1)Tr+(Tr+t)\nis the sum of two super-positive maps, so is super-positive. If ιdenotes the\nidentity map on Mn, thenTr+ι= (Tr+t)◦t.Since the composition of a\nsuper-positivemap and a positive map is super-positive, Tr+ιis super-positive,\nand as above cTr+ιis super-positive for c≥0.\n4To complete the proof of (ii) let 0 < c <1. We have CTr= 1⊗1 and\nCι=/summationtext\nijeij⊗eij=newith e a 1-dimensional projection. We get\nTr(CcTr+ιCTr−ι) =Tr((c1⊗1+ne)(1⊗1−ne))\n=Tr(c1⊗1−cne+ne−n2e)\n=cn2−cn+n−n2\n= (c−1)(n2−n)\n<0.\nSinceTr−ιis a positive map, it follows that cTr+ιis not in the dual cone of\nthe positive maps, hence is not super-positive by [7], and since ( cTr+t)◦t=\ncTr+ι,cTr+ιis not super-positve either, completing the proof of (ii). By (i)\nand (ii) we have P(a) is separable if and only if\nn−T(a)\nnT(a)−1≥1,\nhence if and only if T(a)≤1, proving (iii). The proof is complete.\nA consequence of the above proof is the following corollary, which is a lso\nshown in [8].\nCorollary 5 Letφ:Mn→Mnbe a positive map. Then the map a→φ(1)Tr(a)+\nφ(a)is super-positive.\nProof.φ(1)Tr+φ=φ◦(Tr+ι) is super-positive since the composition of a\nsuper-positive map and a positive map is super-positive.\n2 The SPA of a map\nLetφbe a unital map of Mninto itself. Let W=1\nnCφ. ThenTr(W) =n\nFollowing [2], for 0 ≤t≤1 we put\n˜W(t) =1−t\nn21⊗1+tW.\nThen˜W(t) hastrace1. Write W=W+−W−withW+,W−≥0andW+W−=\n0, and similarly for Cφ. By [2], equation 14,\nt∗= (1+n2/bardblW−/bardbl)−1\nis the maximal t such that ˜W(t)≥0. The SPA of φ,SPA(φ),is defined as\nSPA(φ) =˜W(t∗)\n=1\nn2(1−1\n1+n2/bardblW−/bardbl)1⊗1+1\n1+n2/bardblW−/bardblW\n=1\nn2(1+n2/bardblW−/bardbl)(n2/bardblW−/bardbl1⊗1+n2W)\n5=1\n1+n/bardblC−\nφ/bardbl(1\nn/bardblC−\nφ/bardbl1⊗1+1\nnCφ)\n=1\nn+n2/bardblC−\nφ/bardbl((/bardblC−\nφ/bardbl1⊗1+Cφ)\nWe have just proved the first part of\nProposition 6 Ifφis a unital positive map of Mninto itself, then\nSPA(φ) =1\nn+n2/bardblC−\nφ/bardbl(/bardblC−\nφ/bardbl1⊗1+Cφ).\nFurthermore, SPA(φ)is the density for a state, which is separable if /bardblC−\nφ/bardbl= 1.\nProof.ClearlyTr(SPA(φ) = 1, and by choice of t∗,SPA(φ)≥0 This follows\nalso from the fact that /bardblC−\nφ/bardbl1⊗1+Cφ≥0. The last statement follows from\nCorollary 5.\nProposition 7 Ifφis unital and SPA(φ)is separable, then both\nT(Cφ), S(Cφ)≤n+n(n−1)/bardblC−\nφ/bardbl.\nProof.By Propositions 1 and 6 0 ≤T(SPA(φ))≤1. SinceT(1⊗1) =nit\nfollows from Proposition 6 that\n/bardblC−\nφ/bardbln+T(Cφ)≤n+n2/bardblC−\nφ/bardbl,\nso that\nT(Cφ)≤n+(n2−n)/bardblC−\nφ/bardbl.\nThe proof for S(Cφ) is similar.\nIt follows that if either S(Cφ) orT(Cφ) is greater than n+n(n−1)/bardblC−\nφ/bardbl,\nthenSPA(φ) is entangled. Thus to give an example of this one should look for\nφsuch that /bardblC−\nφ/bardblis small. This will be done in the next section.\n3 An example\nRecall that a positive map φis optimal if φ−ψis never a non-zero positive map\nforψcompletely positive, or equivalently Cφ−ais never the Choi matrix for a\npositive map when a≥0. There is a conjecture [5] that the SPA of an optimal\npositive map is separable. Following the analysis of some generalization s of the\nChoi map of M3into itself by Ha and Kye [3] we shall apply our previous results\nto give a counter example to the conjecture. In [3] it is stated that the same\nclass of maps yield counter examples to the conjecture. Following th eir notation\nwe let fora,b,c≥0, andπ≤θ≤π, φ(a,b,c,θ) be the map of M3into itself\ndefined by\n6φ(a,b,c,θ) =\nax11+bx22+cx33 −eiθx12 −e−iθx13\n−e−iθx21cx11+ax22+bx33 −eiθx23\n−eiθx31 −e−iθx32bx11+cx22+ax33\n\nwherex= (xij)∈M3For simplicity write φforφ(a,b,c,θ). Then the Choi\nmatrix forφis given by\nCφ=\na0 0 0 −eiθ0 0 0 −e−iθ\n0c0 0 0 0 0 0 0\n0 0b0 0 0 0 0 0\n0 0 0 b0 0 0 0 0\n−e−iθ0 0 0 a0 0 0 −eiθ\n0 0 0 0 0 c0 0 0\n0 0 0 0 0 0 c0 0\n0 0 0 0 0 0 0 b0\n−eiθ0 0 0 −e−iθ0 0 0 a\n\nThe interesting information on Cφis obtained from the 3x3 submatrix\nP(a,θ) =\na−eiθ−e−iθ\n−e−iθa−eiθ\n−eiθ−e−iθa\n\nLetpθ=max{2Reei(θ−π/3),2Reeiθ,2Reei(θ+π/3)}.Then 1 ≤pθ≤2, and\nP(a,θ)≥0 if and only if a≥pθ. By [3], Theorem 2.2 we have:\nLemma 8 The mapφ(a,b,c,θ)is positive if and only if\n(i)a+b+c≥pθ, and\n(ii)a≤1⇒bc≥(1−a)2.\nBy [3], Theorem 4.1 it follows that φhas the spanning property, hence is\noptimal by [6], under the following conditions:\nLemma 9 Supposeφ(a,b,c,θ)is positive and 1< pθ<2thenφ(a,b,c,θ)is\noptimal if 0≤a<1, andbc= (1−a)2.\nWe can now give examples of optimal maps for which the SPA is not sepa -\nrable. Let\nP=\n1 1 1\n1 1 1\n1 1 1\n\nTheorem 10 There exist a,b,c≥0and0≤θ≤πsuch thatφ(a,b,c,θ)is\noptimal while its SPA is entangled.\n7Proof.Let 0<ǫ≤1/4. Chooseδ>0 such that the following conditions hold:\n(i) Ifθ=π−δ, then 1<pθ<1+ǫ.\n(ii)Re(−eiθ)>1−ǫ.\n(iii)/bardblP(a,θ)−P/bardbl<ǫ, wherea=pθ−ǫ.\nLetb=ǫand\nc=(1−a)2\nb=(1−pθ+ǫ)2\nǫ.\nSince 1<pθ<1+ǫ,0<1−pθ+ǫ<ǫ.Thus 0<c<ǫ. We have\na+b+c=pθ−ǫ+ǫ+c>pθ>1.\nThus by Lemmas 7 and 8 φ=φ(a,b,c,θ) is an optimal map. Furthermore\n1<φ(1) = (a+b+c)1<pθ−ǫ+ǫ+ǫ=pθ+ǫ<1+2ǫ,\nand soψ=/bardblφ/bardbl−1φis unital.\nSincea=pθ−ǫ<pθwehave,asremarkedabove,that P(a,θ)isnotpositive,\nand by (iii) its minimal eigenvalue lies in the interval ( −ǫ,0). It follows that the\nminimal eigenvalue for Cφ,− /bardblC−\nφ/bardbl, satisfies\n/bardblC−\nφ/bardbl<ǫ.\nWe now show that the conclusion of Proposition 7 is false for ψand hence that\nφis not separable. From the form of Cφwe compute, denoting for simplicity of\nnotationk=/bardblφ/bardbl−1, soψ=kφwith (1+2 ǫ)−1<k<1.\nS(Cψ) =kS(Cφ)\n=k(3a+3/squaresmallsolid2Re(−ei(π−δ)))\n>3k(a+2(1−ǫ)), by(ii),\n= 3k(pθ−ǫ+2−2ǫ)\n>3k(1+2−3ǫ)\n= 9k(1−ǫ)\n>91−ǫ\n1+2ǫ\n≥9/2\n>3+3/squaresmallsolid2/bardblC−\nφ/bardbl\n>3+3/squaresmallsolid2/bardblC−\nψ/bardbl\n,\nwhere we used that /bardblC−\nφ/bardbl< ǫ≤1/4. Thus by Proposition 7 SPA(ψ) is not\nseparable, hence the same holds for SPA(φ). The proof is complete.\n8References\n[1] M-D. Choi, Completely positive linear maps on complex matrices , Linear\nAlg. and Applic. 10(1975), 285-290.\n[2] D. Chruscinski and J. Pytel, Optimal entanglement witnesses from gener-\nalized reduction and Robertson maps , J.Phys.A, 44 (2011), 165304.\n[3] K-C. Ha and S-H. Kye, Entanglement witnesses arising from Choi type\npositive linear maps , arXiv 1205.2921(quant-ph), May 2012.\n[4] K-C. Ha and S-H. Kye, The structural physical approximations and opti-\nmal entanglement witnesses , preprint.\n[5] J.K. Korbicz, M.L. Almeida, J. Bae, M. Lewenstein, and A. Acin,\nPhys.Rev.A, 78 (2008), 062105.\n[6] M. Lewenstein, B. Kraus, J.I. Cirac, and P. Horodecki, Phys.Rev .A, 62\n(2000), 052310.\n[7] E. Størmer, Duality of cones of positive maps , Munster J. Math. 2 (2009),\n299-309.\n[8] E. Størmer, Positive linear maps of operator algebras , Springer-Verlag, to\nappear.\n[9] R.F. Werner, Quantum states with Einstein-Podolsky-Rosen correlation\nadmitting a hidden-variable model , Phys.Rev.A. 40 (1989), 4277-4281.\nDepartment of Mathematics, University of Oslo, 0316 Oslo, Norway .\ne-mail erlings@math.uio.no\n9"    },    {        "title": "1208.1910v1.Comparison_of_Density_Functional_Approximations_and_the_Finite_temperature_Hartree_Fock_Approximation_in_Warm_Dense_Lithium.pdf",        "content": "arXiv:1208.1910v1  [cond-mat.mtrl-sci]  9 Aug 2012Comparison of Density Functional Approximations\nand the Finite-temperature Hartree-Fock Approximation in Warm Dense Lithium\nValentin V. Karasiev,∗Travis Sjostrom, and S.B. Trickey\nQuantum Theory Project, Departments of Physics and of Chemi stry,\nP.O. Box 118435, University of Florida, Gainesville FL 3261 1-8435\n(Dated: 19 July 2012)\nWe compare the behavior of the finite-temperature Hartree-F ock model with that of thermal den-\nsity functional theory using both ground-state and tempera ture-dependent approximate exchange\nfunctionals. The test system is bcc Li in the temperature-de nsity regime of warm dense matter\n(WDM). In this exchange-only case, there are significant qua litative differences in results from the\nthree approaches. Those differences may be important for Bor n-Oppenheimer molecular dynamics\nstudies of WDM with ground-state approximate density funct ionals and thermal occupancies. Such\ncalculations require reliable regularized potentials ove r a demanding range of temperatures and den-\nsities. By comparison of pseudopotential and all-electron results at T = 0K for small Li clusters of\nlocal bcc symmetry and bond-lengths equivalent to high dens ity bulk Li, we determine the density\nranges for which standard projector augmented wave (PAW) an d norm-conserving pseudopotentials\nare reliable. Then we construct and use all-electron PAW dat a sets with a small cutoff radius which\nare valid for lithium densities up to at least 80 g/cm3.\nI. INTRODUCTION\nWarm dense matter (WDM) encompasses the region\nbetween conventional condensed matter and plasmas.\nWDM occurs on the pathway to inertial confinement fu-\nsion and is thought to play a significant role in the struc-\nture of the interior of giant planets. The theoretical and\ncomputational description of WDM is important for un-\nderstanding and performing experiments in which WDM\nis created [1]. Two parameter ranges which are very dif-\nferent from those in standard condensed matter physics\ncharacterize WDM: elevated temperature (from one to a\nfew tens of eV) and high pressure (up to thousands of\nGPa). These ranges are challenging computationally be-\ncause the standard solid state physics methods become\nveryexpensive(due tohightemperature)orstandardap-\nproximations used in those methods cease to work (due\nto high material density). From the plasma side, the\ntemperature and pressure are not high enough to employ\nclassical approaches.\nA combination of a quantum statistical mechanical de-\nscription of the electrons, and classicalmolecular dynam-\nics for ions is a standard theoretical and computational\napproach to WDM at present. Usually the quantum sta-\ntistical mechanics is handled via finite-temperature den-\nsity functional theory (ftDFT) [2–4]. There is a sub-\nstantial literature, too large to review here, about such\ncalculations at zero temperature via Born-Oppenheimer\nmolecular dynamics (BOMD) or Car-Parrinello MD,\nwith DFT implemented via the Kohn-Sham (KS) pro-\ncedure for the electronic degrees of freedom. The per-\ntinent point is that the same techniques can be applied\n∗Electronic address: vkarasev@qtp.ufl.eduto the finite-temperature case [5–19]. The combination,\ncalledab initio molecular dynamics (MD), is computa-\ntionally costly at high temperature (for a given density)\nbecause of the large number of partially occupied KS or-\nbitals which must be taken into account.\nThe great majority of the reported finite-temperature\nab initio MD calculations use zero-temperature\nexchange-correlation (XC) functionals, Exc, with\nFermi thermal occupations to construct the electron\ndensity. In such calculations, the only T-dependence in\nthe XC contribution to the free energy Fxcis through\nthe T-dependence of the electron density:\nFxc[n(r,T),T]≈Exc[n(r,T)], (1)\nwithn(r,T) the electron number density at temperature\nT.\nMost ftDFT calculations with ground state XC func-\ntionals seem to have been done with the Vasp[20] or\nAbinit[21] codes using either the local density approx-\nimation (LDA) for Exc[22–24] or the Perdew-Burke-\nErnzerhof generalized gradient approximation (GGA)\nfunctional [25].\nThe orbital-free density functional theory (OF-DFT)\ntreatment of electronic degrees of freedom is a less ex-\npensive alternativeto orbital-dependent methods such as\nKS. OF-DFT in principle provides the same quantum-\nmechanical treatment of electrons as KS DFT, but the\nlack of accurate orbital-free approximations for the ki-\nnetic energy functionals has limited the use OF-DFT,\nevenatstandardconditions. Incontrast,thehighdensity\nof the WDM regime is favorable for use of the OF-DFT\napproach, which is a motive for developing functionals.\nThe standard KS approach clearly must be used to test\nand calibrate such OF-DFT functionals. The limitations\nand consequences of various choices in those thermal KS\ncalculations have not seen much detailed attention how-2\never. Two closely related sets of potentially significant\nissues occur.\nFirst, the use of ground-state functionals in a ftDFT\ncalculation inevitably raises a topic for fundamental\nDFT, namely, the adequacy, accuracy, and scope of Eq.\n(1). Relative to the number of calculations, there are\ncomparativelyfew studies to assessthis approachagainst\nothers [7, 8, 14, 16–19, 26]. Ref. 7 shows that the maxi-\nmum density of the Al shock Hugoniot is increased about\n5% or less by use of a temperature-dependent functional\nofthe Singwi-Tosi-Land-Sj¨ olander(STLS) type[27]. Ref.\n26 made essentially the same comparison but with re-\nspect to simple Slater exchange (in Hartree atomic units)\nEx[n] =/integraldisplay\ndrn(r)ǫx,S[n(r)]\nǫx,S[n(r)] :=Cx,Sn1/3(r)\nCx,S:=−3\n4/parenleftbigg3\nπ/parenrightbigg1/3\n. (2)\nand with the added complication [for the purpose of as-\nsessing Eq. ( 1)] of use of an OF-DFT approximation.\nRef. 8 compared calculations for ground-state LDA and\nPW91 GGA [28] functionals. Faussurier et al.[14] com-\npared the electrical conductivity of Al computed with\nthe T-dependence from classical-map hypernetted chain\nscheme [29] versus ground-state LDA. They concluded\nthat the effects on conductivity are small in the WDM\nregime but become increasingly important as the energy\ndensityincreases. W¨ unsch et al.[18]reversedtheperspec-\ntive and used ftDFT calculations with a ground-state\nXC functional to calibrate hypernetted chain approxi-\nmations, hence assumed the validity of Eq. ( 1). Vinko\net al.compared ground-state GGA calculations of free-\nfree opacity for Al with an RPA model and found semi-\nquantitative agreement at lower photon energies with in-\ncreasing disagreement at higher ones, all over the range\n0≤T≤10 eV [17]. As an aside, we note that the same\nissues of use of ground-state approximate XC function-\nals in a T-dependent context can arise in average-atom\nmodels [30–34].\nThe second set of issues involves computational tech-\nnique. The primary focus is control of the effects of pseu-\ndopotentials (or regularization of the nuclear-electron in-\nteraction). These are ubiquitous in the highly refined\ncodes in use for both WDM and ground state calcula-\ntions. Clear insight into the behavior and limitations of\nfunctionals requires that the regularized potentials not\nintroduce artifacts of their own. The challenge is to test\nthosepotentialsagainsthigh-qualityall-electron(AE)re-\nsults over the appropriate density range.\nAn obvious issue associated with pseuodpotentials is\nthe effect of a finite core radius upon compressibility\n(hence, equation of state). Ref. [35] shows that a norm-\nconserving pseudopotential for boron with the standardcutoff radius ( rc= 1.7 Bohr) is not transferable to the\nhigh material density regime. In that work, the au-\nthors built an “all-electron” pseudopotential with small\nrc= 0.5 Bohr and tested its transferability to very high\nmaterial density by comparison with the Thomas-Fermi\n(TF) limit calculated using an average-atom model [30].\nAnother issue is the extent to which removal of core\nelectrons has an unphysical effect on the distribution\nof ionization. A related issue is the effect that remov-\ning core levels has on Fermi-Dirac occupation numbers.\nAt fixed density, such core levels should be progressively\ndepopulated with increasing temperature. Does the de-\npopulation of pseudo-density levels behave correctly? A\nsignificant computational practice issue is the minimum\nmagnitude threshold for retention of occupation num-\nbers. That threshold is directly related to basis set size\nor, equivalently, the plane-wave cutoff. We know of only\nonestudy ofany ofthese questions [36]. In it, all-electron\ncalculations with the full-potential linearized muffin-tin\norbital methodology were used to benchmark projector\naugmented wave (PAW) calculations with a plane wave\nbasis. Twometals,AlandW,weretreatedat T/ne}ationslash= 0K.At\nleast for W, it appears that different XC functionals were\nused for the comparison. Additionally, Ref. 36 used the\nfree-electron expression for the non-interacting electronic\nentropy, rather than the proper explicit dependence on\noccupation numbers fi:\nSs=−kB/summationdisplay\ni{filnfi+(1−fi)ln(1−fi)}.(3)\nDespite these differences, to the extent that their topics\nand ours overlap, the findings are consistent.\nTo establish a basis for comparison, first we consider\nthe issues of regularized potentials. We consider both or-\ndinary pseudopotentials (PPs) and the pseudopotential-\nlike PAW technique. Those tests are against all-electron\n(bare Coulomb nuclei potential) calculations for small Li\nclusters of bcc symmetry. We establish a PAW which\ndemonstrably is reliable for the density range of inter-\nest. Then we study the behavior and limits of the\nuse of ground-state X functionals in ftDFT by compari-\nson of finite-temperature Hartree-Fock (ftHF) and DFT\nexchange- onlyresults. For clarity of interpretation, all\nthe bulk solid calculations reported here were performed\nat fixed ionic positions corresponding to an ideal bcc\nstructure for Li.\nII. CODES\nWe used the atompaw code [37] to form the PAWs.\nFor periodic systems, we used three codes, Abinitvers.\n6.6 [21], Vaspvers. 5.2 [20], and Quantum-Espresso\nver. 4.3 [38]. All three are plane-wave, PP codes. All\nthree also implement PAWs. AbinitandQuantum-\nEspresso are open source. Technical details of the ftHF3\ncalculations are discussed below. For the all-electron cal-\nculations on finite clusters, we used conventional molec-\nular gaussian basis techniques as embodied in the Gaus-\nsian 03 program [39].\nIII. REGULARIZED POTENTIALS\nDiverse PP techniques commonly are used in KS cal-\nculations to reduce computational cost by excluding the\ncore electrons from the self-consistent field (SCF) pro-\ncedure and to regularize the singular external potential\nin order to use an efficient, compact plane wave basis\nset. Excludingcoreelectronsimplicitlyinvokesthefrozen\ncore approximation ( i.e., the omission of core electrons\nfrom the SCF procedure). That approximation generally\nis well-justified in standard conditions. There, the core\nelectrons are uninvolved in chemical bonding and their\nstate is essentially independent of the chemical environ-\nment. The validity of this justification is not obvious for\nthe WDM regime. In it, all electrons become important\nfor correct evaluation of the Fermi occupancy at high\ntemperature and correct description of the electron den-\nsity at high external pressure. As a consequence, it is\nmandatory to include at least some core electrons in the\nsolution of the relevant Euler equation (DFT or finite-\ntemperature HF) in the WDM regime. For light atoms\nthis may mean an all-electron PP. Those are, of course,\na particular form of regularized potential.\nGenerationofPPsusuallyischaracterizedbycutoff(or\npseudization) radii, rc. Values of rcarea compromisebe-\ntween softness of the PP (for compactness of plane wave\nbasis sets) and correct description of the one-electron\norbitals close to the nucleus. Standard PPs are devel-\noped for use under near-equilibrium condensed matter\nand molecular conditions, hence their transferability to\nthe WDM regime needs to be explored. For example,\ncommonly rcis assumed to be somewhat smaller than\nhalf the nearest-neighbordistance between atoms so that\nthere is no core overlap. There is no guarantee that such\nequilibrium prescriptionsare satisfactoryfor WDM stud-\nies.\nA. Basic PAW formalism\nPAW concepts are summarized in Ref. 40. We out-\nline the relevant points here. The PAW valence electron\nenergy is comprised of a pseudo-energy evaluated using\na smooth pseudo-density and pseudo-orbitals plus atom-\ncentered corrections. An energy correction centered on\natomais evaluated using an augmentation sphere of\nradiusra\nc. Within each sphere, the correction replaces\nthe valence pseudo-energy of atom a,˜Ea\nv, by the valence\nenergyEa\nvgenerated from the valence part of the all-electron atomic density\nEv=˜Ev+/summationdisplay\na/parenleftBig\nEa\nv−˜Ea\nv/parenrightBig\n. (4)\nDetailed descriptions of each term in Eq. ( 4) are given,\nfor example, in Ref. [41]. Here the issue is treatment of\ncore density contributions to the XC energy, as discussed\nin that reference. In the scheme due to Bl¨ ochl [42], the\nXC energy is expressed as\nExc=Exc[˜n+ ˜nc]+/summationdisplay\na/parenleftBig\nExc[na+na\nc]−Exc[˜na+ ˜na\nc]/parenrightBig\n,\n(5)\nwherenaandna\ncareatom-centeredvalence andcoreelec-\ntronchargedensitiescorrespondingtoall-electronatomic\norbitals, ˜ naand ˜na\ncare atom-centered valence and core\nelectron pseudo-densities, and ˜ n, ˜ncare total valence and\ncoreelectronpseudo-densities. Theideabehind Eq.( 5)is\nthat the third term, which corresponds to atom-centered\ncontributions of pseudo-densities (evaluated within aug-\nmentation spheres, radii ra\nc), cancels the corresponding\natom-centered pseudo-density contributions (evaluated\nover all space) in the first term, and the canceled contri-\nbution is replaced by the second term, which is evaluated\nwith atom-centered all-electron densities (again within\nthe augmentation spheres only).\nThe Kresse scheme [43] introduces a valence compen-\nsation charge density, ˆ n, as well. Its purpose is to repro-\nduce the multipole moments of the all-electron charge\ndensity outside the augmentation spheres [41]. For the\nXC contribution, ˆ nis added to the pseudo-densities in\nthe functionals in Eq. ( 5) to give\nExc=Exc[˜n+˜nc+ˆn]+/summationdisplay\na/parenleftBig\nExc[na+na\nc]−Exc[˜na+˜na\nc+ˆna]/parenrightBig\n.\n(6)\nThis procedure can cause problems with GGA XC func-\ntionals; see Ref. 40.\nThere are what are called all-electron PAWs, which\nin essence are regularized potentials for all-electron cal-\nculations. In customary notation, an “N-electron”\nPAW retains N electrons in the valence. Thus a 3-\nelectron(“3 e−”)PAWcalculationforLiisanall-electron,\nregularized-potential calculation.\nB. PAW and high density lithium\nWe tested the PAW approach by calculating the pres-\nsure of bcc Li over a large range of material densi-\nties, from approximate equilibrium, ρLi= 0.5 g/cm3, to\nρLi= 25.0 g/cm3, all at T = 100K. (The equilibrium\ndensity from simple Slater LDA all-electron calculations\nis 0.54 g/cm3, or lattice constant 6.59 Bohr, close to the\nexperimental value; see Ref. 44. Newer LDAs give some-\nwhatcontractedresults; seebelow.) ThreedifferentPAW4\ndata sets were used for each LDA and GGA exchange-\ncorrelation functional: (i) the standard set with compen-\nsation charge density included from Ref. 45, (ii) a set\nwith the same cutoff radius ( rc= 1.61 Bohr) but with-\nout compensation charge density, and (iii) a set we gen-\nerated with rc= 0.80 Bohr and no compensation charge\ndensity. The Perdew-Wang (PW) and Perdew-Zunger\n(PZ) LDAs [23, 24] and Perdew-Burke-Ernzerhof GGA\n[46] (PBE) XC functionals were used.\nThe upper segment of Table Icompares the calculated\nbcc Li equilibrium lattice constants and bulk moduli for\nthe various combinations. These were done with Abinit.\nThe lattice constant and bulk modulus were obtained\nby fitting the calculated total energies per cell to the\nstabilized jellium model equation of state (SJEOS) form\n[47]. One sees that the exclusion of the compensation\ndensity slightly decreases the lattice constant for both\nPW and PBE functionals. The results are essentially\nunchanged when the rcvalue is decreased to 0.80 Bohr.\nTableIalso summarizes results obtained using both\nQuantum-Espresso andVasp. The lattice constant\nand bulk modulus again were obtained via fitting to the\nSJEOS form in all cases. The results for Vaspcome from\nusing the PAW pseudopotentials supplied with the code\nitself. There is excellent agreement between Quantum-\nEspresso andAbinitresults when the same 3 e−PAW\ndata set is used. The VaspPBE 3e−results do not\nagree as well, consistent with the findings of Ref. 40 re-\ngarding the effects of the valence compensation charge\ndensity contribution. Two LDA PPs also were used with\nQuantum-Espresso , namely the 1 e−Von Barth-Car\nand 3e−norm-conserving pseudopotentials (both taken\nfrom the Quantum-Espresso web page). The lattice\nconstant corresponding to the first of these PPs is un-\nderestimated as compared to other PZ LDA calcula-\ntions, independent confirmation of the importance of the\n3e−treatment. For the 1 e−(Vanderbilt ultrasoft) and\n3e−(norm-conserving) PBE PPs (again taken from the\nQuantum-Espresso web page), the lattice constant is\nslightly overestimated and the bulk modulus is underes-\ntimated by the 1 e−pseudopotential. The 3 e−results are\nin nearly perfect agreement with the PAW data.\nTo assess the PAW method for high material density,\nwe compared PAW and true all-electron (bare Coulomb\npotential) results for two small lithium clusters with lo-\ncal bcc symmetry; see Fig. 1. The interatomic dis-\ntances in both clusters were set equal to the nearest-\nneighbordistancein bulk bcc-Li fordensities in the range\n0.5≤ρLi≤150 g/cm3. The all-electron calculations\nweredonewith the Gaussian 03 codeandtwobasissets,\n6-311++G(3df,3pd) and cc-pVTZ. For the LDA calcula-\ntions, we used the Vosko, Wilk, and Nusair parameter-\nization (VWN) [22]; it is very close to the PZ parame-\nterization and based on the same data. For the GGA\nfunctional we used PBE. For high densities, ρLi≥50g/cm3, we did additional calculations with cc-pV5Z (8-\natom cluster) and cc-pVQZ (16-atom cluster) basis sets.\nThe PAW calculations were done with the Abinitcode.\nIn it, the clusterswerecentered in a largecubic super-cell\nof sizeL. For the standard PAW, we used L= 15˚Awith\nan energy cutoff 1000 eV, while L= 12˚Awith energy\ncutoff 3000 eV was used for the small rcPAW. The PZ\nLDA and PBE GGA functionals were used in these cal-\nculations. Note that the difference in behavior between\nPZ and PW is essentially negligible for the purposes of\nthis study.\nFig.2shows all-electron and PAW LDA total ener-\ngies for the two clusters as a function of distance corre-\nsponding to the stated bulk density. Fig. 3shows the\ncorresponding GGA results. The behavior of the two\nclusters is quite similar. For the standard PAW data set\n(labeled “(i)” previously), the total energy starts to de-\nviate from the all-electron (AE) values at approximately\nρclust−crit1\nLi = 8.0g/cm3. ForthestandardPAWsetwith-\nout compensation density [set (ii)], the critical density\nρclust−crit2\nLiis approximately 25 g/cm3. In contrast, the\nPAWwithsmall rcandnocompensationdensity[set(iii)]\ngives essentially perfect agreement with the AE results\nforthewholedensityrange. Fordensitiesupto30g/cm3,\ntwo basis sets, 6-311++G(3df,3pd) and cc-pVTZ, give\nessentially the same quality results. At high density (50\ng/cm3and up), the cc-pVTZ basis set energies lie above\nthe values corresponding to the 6-311++G(3df,3pd) ba-\nsis. For those high densities, AE calculations done with\nthe larger cc-pV5Z (8-atom cluster) and cc-pVQZ (16-\natom cluster) basis sets lower the total energy to the\n6-311++G(3df,3pd) level (16-atom cluster) or slightly\nlower (8-atom cluster). Once again there is essential per-\nfect agreement with the set (iii) PAW plane wave results.\nThe corresponding PBE GGA comparison of PAW\nand AE results, Fig. 3, shows that the critical densities\nfor each PAW data set are almost identical for the 8-\natom and 16-atom clusters. For the PAW data set (i),\nρclust−crit1\nLi ≈6.0 g/cm3is slightly lower than for the\nLDA case. For PAW data set (ii), the critical density\nis essentially the same as for LDA ( ≈25 g/cm3). Once\nagain, the small rcPAW data set (iii) gives good agree-\nment with the AE results up to the maximum density\nconsidered (150 g/cm3). We conclude that PAW data\nset (iii), namely rc= 0.80 Bohr and no compensation\ncharge, may serve for making reference KS calculations\nin the high density regime.\nThe remaining validation issue is the effect of PAW\nor PP on the calculated pressure. A study [15] of the\nEOS for warm, dense LiH found that the 3 e−PAW for\nLi in VASP calculations was necessary for T = 2, 4, and\n6 eV and densities twice that of ambient and greater.\nFig.4shows the bulk bcc Li pressure as a function of\nmaterial density at T=100 K calculated using Abinit\nwith the same three PAW data sets as before for both5\nTABLE I: Equilibrium lattice constant for bcc-Li, a(Bohr) and bulk modulus, B(GPa).\nLDA GGA\nMethod rc PW PZ PBE\na B a B a B\nAbinit(3e−, PAW, c.ch.a) 1.61 6.363 15.1 – – 6.513 14.0\nAbinit(3e−, PAW) 1.61 6.363 15.0 6.360 15.1 6.497 13.8\nAbinit(3e−, PAW) 0.80 6.362 15.1 6.359 15.1 6.497 13.9\nQ-Espreso (3e−, PAW, c.ch.a) 1.61 6.364 15.0 – – 6.516 14.1\nQ-Espreso (3e−, PAW, c.ch.a) 0.80 6.362 15.0 – – 6.497 13.8\nQ-Espreso (1e−)b– – – 6.320 15.1 6.724 11.9\nQ-Espreso (3e−)c– – – 6.362 15.1 6.500 13.8\nVasp(1e−, PAW, c.ch.a) 2.05 – – 6.373 15.0 6.514 13.7\nVasp(3e−, PAW, c.ch.a) 1.55-2.00 – – 6.362 14.9 6.505 13.8\naCompensation charge density is included.\nbLDA: PZ exchange-correlation, nonlinear core-correction Von\nBarth-Car; GGA: PBE exchange-correlation, nonlinear core -\ncorrection, Vanderbilt ultrasoft pseudopotentials.\ncPZ and PBE semicore state sin valence Troullier-Martins pseu-\ndopotentials.\nFIG. 1: Bcc Li 8(left panel) and Li 16(right panel) clusters\nused to test PAW calculations.\n0 10 20 30 40 50 60 70 80 90 100110120130140150\nρLi (g/cm3)−60−50−40−30Etot(Hartree)AE, 6−311++g(3df,3pd)\nAE, cc−pVTZ\nPAW, rc=1.61 Bohr, c.ch.\nPAW, rc=1.61 Bohr\nPAW, rc=0.80 Bohr\nAE, cc−pV5Z\nLi8, LDA\n0 10 20 30 40 50 60 70 80 90 100110120130140150\nρLi (g/cm3)−120−110−100−90−80−70−60−50−40Etot(Hartree)AE, 6−311++g(3df,3pd)\nAE, cc−pVTZ\nPAW, rc=1.61 Bohr, c.ch.\nPAW, rc=1.61 Bohr\nPAW, rc=0.80 Bohr\nAE, cc−pVQZ\nLi16, LDA\nFIG. 2: All-electron (VWN correlation) and PAW (PZ cor-\nrelation) LDA total energies for the Li 8and Li 16clusters.0 10 20 30 40 50 60 70 80 90 100110120130140150\nρLi (g/cm3)−60−50−40−30Etot(Hartree)AE, 6−311++g(3df,3pd)\nAE, cc−pVTZ\nPAW, rc=1.61 Bohr, c.ch.\nPAW, rc=1.61 Bohr\nPAW, rc=0.80 Bohr\nAE, cc−pV5Z\nLi8, GGA\n0 10 20 30 40 50 60 70 80 90 100110120130140150\nρLi (g/cm3)−120−110−100−90−80−70−60−50−40Etot(Hartree)AE, 6−311++g(3df,3pd)\nAE, cc−pVTZ\nPAW, rc=1.61 Bohr, c.ch.\nPAW, rc=1.61 Bohr\nPAW, rc=0.80 Bohr\nAE, cc−pVQZ\nLi16, GGA\nFIG. 3: All-electron and PAW GGA total energies for the\nLi8and Li 16clusters.\nthe PW LDA and PBE XC functionals. (Use of the PZ\nLDAfunctional givesresults indistinguishablefrom those\nfrom PW LDA on the scale of the figure.) One sees\nthat the standard PAW data set (i) starts to overesti-\nmate the pressure at ρbulk−crit1\nLi= 6.0 g/cm3for LDA\nand at a slightly lower value for PBE. PAW data set (ii)\nproduces results which agree with the reference calcula-\ntions (i.e., those from PAW set (iii)) for densities up to\nρbulk−crit2\nLi= 15.0 g/cm3. Comparison of critical density\nvalues in the clusters and in bulk shows that ρbulk−crit1\nLi6\n0 5 10 15 20 25\nρLi (g/cm3)0500010000150002000025000P (GPa)PAW LDA, rc=1.61 Bohr, c.ch.\nPAW LDA, rc=1.61 Bohr\nPAW LDA, rc=0.80 Bohr\n0 5 10 15 20 25\nρLi (g/cm3)0500010000150002000025000P (GPa)PAW GGA, rc=1.61 Bohr, c.ch.\nPAW GGA, rc=1.61 Bohr\nPAW GGA, rc=0.80 Bohr\nFIG. 4: Pressure vs. material density from PAW LDA (PW\ncorrelation) (leftpanel)andfromPAWGGA(PBEXC)(right\npanel) calculations for bulk bcc-Li.\n0.6 2 5 10\nρLi (g/cm3)101001000P (GPa)\nLDA PAW 3e-, rc=0.80 Bohr (Abinit)\nLDA PAW 1e- (Vasp)\nLDA PAW 3e- (Vasp)\nBoetger Albers\n1e- (Quantum-Espresso)\n3e- (Quantum-Espresso)\n0.6 2 5 10\nρLi (g/cm3)101001000P (GPa)\nGGA PAW 3e-, rc=0.80 Bohr (Abinit)\nGGA PAW 1e- (Vasp)\nGGA PAW 3e- (Vasp)\n1e- (Quantum-Espresso)\n3e- (Quantum-Espresso)\nFIG. 5: Validation of Vasp1e−, 3e−PAW,Quantum-\nEspresso 1e−, 3e−PZ LDA (left panel) and Vasp1e−, 3e−\nPAW,Quantum-Espresso 1e−, 3e−PBE GGA(rightpanel)\npseudopotential calculations: pressureas afunction ofde nsity\nfor bcc-Li calculated at T=100 K.\nis slightly lowerthan ρclust−crit1\nLi. A crude linear extrapo-\nlation of the results from PAW data sets (i) and (ii) gives\nan estimated lower bound for the critical bulk density for\nthe reference PAW data set ρbulk−crit3\nLi to be 80 g/cm3.\nAdditional tests would be needed to get the actual value\nofρbulk−crit3\nLi. Such a determination is not required for\nthe present purposes.\nWe observe that for fcc Al at T=0 K, Levashov et\nal.[36] found that the standard VASP PAW pressures\nbegan to deviate materially from all-electron values at\nabout a compression of seven. Since it was standard\nVASP, presumably that PAW included charge compen-\nsation, hence their result should correspond to our set (i)\nLi results, those labeled “PAW, rc= 1.61 bohr, c.ch.” in\nFig.4. It is clear that the deviation they found in fcc\nAl is at similar but modestly lower compression than we\nfind for bcc Li.\nFigure5compares pressure versus material density\n(T=100 K) for LDA (left panel) and PBE GGA (rightpanel) for material densities in the range 0 .6−10.0\ng/cm3obtained from VaspandQuantum-Espresso\nusing standard PPs (the PAW provided with the Vasp\npackage and the norm-conserving PP taken from the\nQuantum-Espresso web page), and reference results\nobtained with our PAW data set (iii) (for both the PZ\nLDA and PBE GGA XC functionals.) For LDA, we also\nshow the earlier all-electron results by Boettger and Al-\nbers [48]. Observe first that our designation of the PAW\n(iii) as areference is substantiated by the agreementwith\nthe all-electron LDA calculation. The Vasp1e−PAW\nLDA results start to deviate from the reference values\nat about 3.0 g/cm3. However the 3 e−PAW LDA pres-\nsure from Vaspis in good agreement for densities up\nto 6.0 g/cm3.Quantum-Espresso results calculated\nwith the 1 e−PZ LDA pseudopotential start to deviate\nfrom the reference results for density between 2 and 3\ng/cm3, whereas the 3 e−potential produces results which\nagree virtually perfectly for the full density range. For\nthe GGA case, the right-hand panel of Fig. 5shows that\nthe situation is verysimilarexcept that both the 1 e−and\n3e−PAWVaspcalculationsstart todeviate fromthe ref-\nerence results at almost the same density ( ≈3 g/cm3).\nQuantum-Espresso 1e−calculations overestimate the\npressure for ρLi>0.8 g/cm3. However, the Quantum-\nEspresso 3e−results are in virtually perfect agreement\nwith the reference PAW results for the whole range of\ndensities.\nIV. FINITE TEMPERATURES\nA. Pseudopotentials and level populations\nIn finite-temperature calculations (either KS or HF),\nthere is non-zero occupation of one-electron levels which\ncorrespond to empty levels at T = 0 K (virtual states or\nsimply “virtuals”). Satisfaction of some computational\nthresholdforthesmallestnon-negligibleoccupationnum-\nber requires an increasingly large set of those virtuals to\nbe considered with increasing T. Concurrently there is\ndepopulation of levels fully occupied at T = 0 K. One\nwould hope that PP methods which treat all electrons\nself-consistentlywouldbe applicableforsuch finite-T cal-\nculations. A related issue is the validity of using PPs\nwhich remove some of the core. A rough estimate of the\nrelevant scale comes from taking the 1 sionization poten-\ntialforthe Liatomto beapproximatelythe magnitudeof\nthe LDA Kohn-Sham 1 seigenvalue, about 51 eV. Then\nPPtreatmentofLi1 selectronsascoremightbeexpected\nto be applicable for temperatures much smaller than 51\neV. The question is the validity of any estimate of this\nsort, in particular, how much smaller? We remark that\nLevashov et al.[36] found that for ambient density Al,\nthe PAW pressure deviated from the all-electron value at7\n0 20 40 60 80 100\nT (kK)−5050150250350450P (kBar)LDA (1e− )\nLDA (3e− )\nρLi=0.5 g/cm3\n0 20 40 60 80 100\nT (kK)100300500700900P (kBar)LDA (1e− )\nLDA (3e− )\nρLi=1.0 g/cm3\nFIG. 6: Comparison of pressure vs. temperature for bcc-Li\nobtained with 3 e−and 1e−pseudopotentials for the Perdew-\nZunger LDA exchange-correlation functional as implemente d\ninQuantum-Espresso (2-atom unit cell, 9x9x9 k-mesh, 128\nbands). Left panel: ρLi= 0.5 g/cm3. Right panel: ρLi= 1.0\ng/cm3.\nabout T= 5-6eV. This islessthan 10%ofthe magnitude\nof the LSDA 2p atomic KS eigenvalue (about 70 eV).\nFirst consider the comparative performance of the\nPPs. Figure 6shows the hydrostatic pressure as a\nfunction of temperature calculated using 1 e−and 3e−\nnorm-conserving pseudopotentials for the bcc-Li struc-\nture (fixed nuclear positions, ρLi= 0.5 and 1.0 g/cm3).\nNotice that this calculation uses a ground-state XC func-\ntional: there is no explicit temperature dependence in\nthe PZ LDA XC functional. If the number of bands\ntaken into account for a two-atom unit cell for ρLi= 0.5\ng/cm−3is 128, the occupation number of the highest\nenergy bands is of the order of 10−6−10−7. Observe\nthat the results from the 1 e−PP are in almost perfect\nagreement with those from the 3 e−calculations for T\nup to 75,000 K, with small disagreement appearing at\nhigher temperatures. For low to moderate compression,\nit appears that the range of applicability of standard 1 e−\nnorm-conserving pseudopotentials is at least up to T =\n100 kK or about 8 – 9 eV. This fits the rough argument\nbased on the Li 1 sKS eigenvalue, with the criterion for\n“much smaller” being of order 20 % at most.\nNext comesthe matter ofsignificant fractionaloccupa-\ntion of ever-higher energy orbitals with increasing tem-\nperature. A related issue is energy level shifting and\nreordering with increased density, an effect known for\nT = 0K Li [44, 49].\nAgain, we did calculations with ground-state XC on\nbcc Li, with material density from 0.6 to 4.0 g/cm3. For\nall temperatures and densities, we used a 7 ×7×7\nMonkhorst-Pack k-grid [50], a two-atom unit cell, and\nincluded 128 bands with a plane wave energy cutoff of\n150 Ry. The calculations were done with Quantum-\nEspresso and the 1 e−PP just mentioned. The upper\npanel of Figure 7shows a sample of the orbital eigen-\nvalues for a single k-point, Γ, as a function of material\ndensity for T = 100 kK. The T = 100 K plot is ab-\nsolutely indistinguishable, due to relatively small differ-\nences in the eigenvalues. The eigenvalues are labeled in-1 0 1 2 3 4 5 6 7 8 9 10\n 0.5 1 1.5 2 2.5 3 3.5 4Orbital energy (Hartree)\nρLi (g/cm3)\n-0.5 0 0.5 1 1.5\n 0.5 1 1.5 2 2.5 3 3.5 4Occupation number\nρLi (g/cm3)1\n5\n10\n15\n20\n40\n60\n80\n 0 0.1 0.2 0.3 0.4\n 0.5 1 1.5 2 2.5 3 3.5 4Occupation number\nρLi (g/cm3)\nFIG. 7: Top: orbital energies for the Γ point, at 100 kK.\nBottom left: single-spin occupation number fiof the levels\nplotted for T = 100 K. Bottom right: same but for T = 100\nkK. The legend for the level number, given in the upper right,\nis for all plots.\norder of increasing energy, ε1lowest,ε128highest. The\nmain point to be noticed is that as the density increases,\nthe spread in the lower half (roughly) of the eigenval-\nues increases. Those are the eigenvalues most pertinent\nto the calculation, in the sense that at a given tempera-\nture, excited levels will be depopulated at higher densi-\nties compared to the corresponding levels at lower densi-\nties. (An exception would be a pressure-induced switch\nin level-ordering.) The lower panels of Fig. 7show the\noccupation numbers for those same eigenvalues. At low\ntemperature and low density, the results are as expected,\nan almost square-wave Fermi distribution with the low-\nest band fully occupied (since there are two electrons\nin the unit cell) and the higher bands unoccupied. At\nhigher densities, some k-points, including the Γ point, as\nshowninthe lowerleft panel, fordensitiesabove3g/cm3,\nhave no occupation while others have two occupied lev-\nels. This repopulation is a consequence of changes in\nthe KS orbitals caused by changes in the external poten-\ntial, hence also in the effective KS potential. The lower\nrightpanelshowsthatathighertemperaturesthereisnot\nonly a temperature dependence of the occupation num-\nbers, but a significant density dependence because of the\nspreading of the orbital energy levels.\nNext we consider the number of bands required for\na stipulated precision, given by a minimum occupation\nnumber threshold, as a function of temperature and den-\nsity. For the calculations just discussed, we calculated a\nzone-averagedband occupation. For a band of composite8\n 1e-10 1e-08 1e-06 0.0001 0.01 1\n 0 20 40 60 80 100 120 140Average occupation\nBand numberρ=0.6\nρ=1.0\nρ=2.0\nρ=3.0\nρ=4.0\n10-6\nFIG. 8: Zone-averaged band occupation numbers for all\nbands at T = 100 kK, for the various material densities listed .\nindexi, we sum the occupations of the εilevel multiplied\nby the k-point integration weight for all k-points. This\nzone-averaged occupation is plotted for T = 100kK in\nFigure8. One sees clearly that, for a given threshold in\noccupation number, for example 10−6, the number of re-\nquired bands decreases significantly with increasing den-\nsity. This decrease again is due to changes in the KS or-\nbitals. AtleastatT = 0K,itlonghasbeenknown[44,49]\nthat as Li is compressed, its band structure initially be-\ncomeslesslikethe homogeneouselectrongas(HEG) than\nthe bcc zero-pressure bands. However, eventually, the\nsystem passes over to a Thomas-Fermi-Dirac equation of\nstate, signifying near-perfect but spread parabolic bands\n(see upper panel of Fig. 7) and corresponding HEG oc-\ncupations.\nB. Exchange free energy\nThough it originated in the Greens function for-\nmalism of many-fermion theory, the finite-temperature\nHartree-Fock approximation is the thermodynamical\ngeneralization of the variational optimization of a single-\ndeterminant trial wave function which is ubiquitous in\nquantum-chemistry and molecular physics as the HF ap-\nproximation [51].\nTo summarize, the thermal generalization of the fa-\nmiliar HF single-determinantal exchange energy may be\nexpressed in terms of the one-electron reduced density\nmatrix (1-RDM)\nFx[n] :=−/integraldisplay\ndx1dx2{g12¯Γ(1)(x1|x′\n2)\nׯΓ(1)(x2|x′\n1)}x′\n1=x1,x′\n2=x2,(7)\nwherex:=r,sis a composite space-spin variable, g12=\n1/2|r1−r2|, and the 1-RDM is defined in terms of therelevant orbitals {ϕi}and occupation numbers {fi}\n¯Γ(1)(x1|x′\n1) :=∞/summationdisplay\nj=1fjϕj(x1)ϕ∗\nj(x′\n1),(8)\nsubject to\nfj≡f(εj−µ) = [1+exp( β(εj−µ))]−1(9)\nand\n/integraldisplay\ndxϕi(x)ϕ∗\nj(x) =δij\n∞/summationdisplay\nj=1fj=N , (10)\nwithβ:= 1/kBT as usual. Here µis the chemical poten-\ntial (determined by Eq. ( 10)) and the εjare the eigen-\nvalues of the associated one-particle ftHF equation.\nThe analogue to ftHF in DFT is called finite-\ntemperatureexactexchange(ftEXX hereafter)DFT [52].\nIn its pure Kohn-Shamform, ftEXXdefines the exchange\nfree energy formally identically with ftHF, but evaluates\nthe density n(r,T) from orbitals which follow from a true\nKSprocedure,that is, fromaone-bodyHamiltonianwith\na local (multiplicative) exchange potential. That poten-\ntial follows from the system response function, δn/δvKS.\nA full ftDFT calculation (not exchange only) would have\na correlation free energy functional and associated KS\npotential as well.\nGround state DFT with so-called hybrid approximate\nexchange functionals has a similar structure for the total\nenergy, in the sense that hybrids have contributions both\nfrom single-determinant exchange and from exchange-\ncorrelation functionals which are explicitly density de-\npendent. InsteadofaKSprocedure,onecangofromsuch\nahybridexpressiondirectlytocoupledone-electronequa-\ntions by explicit variation with respect to the orbitals. In\nground-state theory with a hybrid functional, this proce-\ndure sometimes is called generalized KS. The relevant\npoint is that the same approach applies directly to ftHF.\nSimply switch off the explicit density functionals for ex-\nchange and correlation and leave the exchange functional\nwhich comes from the trace over single-determinants.\nSince the capacity to do hybrid DFT as a generalized KS\napproachexists in both VaspandQuantum-Espresso ,\none sees that such coding is immediately exploitable for\ndoing ftHF.\nIn parallel with ground state DFT, an LDA may be\nobtainedfromconsideringthefinite-THEG.Itsexchange\nfree energy is given in first-order perturbation theory by\nFHEG\nx=−V\n(2π)6/integraldisplay /integraldisplay\ndkdk′4π\n|k−k′|f(k)f(k′) (11)\nwheref(k) = [1+exp( β(k2/2−µHEG))]−1, andVis the\nsystem volume. With the chemical potential expanded9\nto the same order as well, µHEG=µHEG\n0+µHEG\nx, the\nexchange portion is\nµHEG\nx(n,T) =δFHEG\nx\nδn(12)\nIf expressed in closed form, this result may be used\nas the finite-T LDA, with exchange free energy per\nelectron fLDA\nx(n(r,T),T) = (FHEG\nx/nV)|n=n(r,T), and\nvLDA\nx(n(r,T),T) = µHEG\nx(n,T)|n=n(r,T). Here we\nused the parametrization given by Perrot and Dharma-\nwardana [53]. The LDA exchange free energy is then\nFLDA\nx[n(r),T] =/integraldisplay\nfLDA\nx(n(r,T),T)n(r,T)dr.(13)\nThe one-particle density follows by obvious analogy with\nEqs.8-10.\nC. Finite-temperature Hartree-Fock and DFT\nX-only calculations\nTo study the importance of using an explicitly T-\ndependentexpressionfortheexchangefreeenergy(rather\nthan a calculation with a ground-state X functional) and\nto estimatethe qualityofthe T-dependent exchangefree-\nenergy functional defined by Eq. ( 13), we compare ftHF\ncalculationswhich usethe exactexchangefree energyEq.\n(7), Kohn-Sham calculations with T-independent LDA\nexchange for the exchange free energy, Fx≈ELDA\nx, and\nKScalculationsdonewiththeT-dependentexchangefree\nenergy functional FLDA\nx. In the following discussion, the\nground-statefunctional calculationsarelabeled “LDAx”,\nwhile those which used the explicitly T-dependent LDA\nare labeled “LDAx(T)”. All the calculations were done\nwithQuantum-Espresso using the 1 e−PZ LDA pseu-\ndopotential taken from the Quantum-Espresso web\npage. WetreatedbccLiwithfixednuclearpositions, here\nwith densities between ρLi= 0.6 and 1.8 g/cm3and tem-\nperatures between 100K and 100kK. At these densities\nand temperatures, the Quantum-Espresso 1e−pseu-\ndopotential is adequate; recall Sections IIIBandIVAas\nwell as Figs. 5and6.\nConvergence of the ftHF and the LDAx calculations\nwith respect to the k-mesh for the bcc-Li 2-atom unit\ncell requires attention. It is known [54] that T=0 K LDA\ncalculations on bcc Li exhibit misleading convergencebe-\nhavior at a relatively coarse k-mesh density. We tested\nfor the smallest real spacecell size used, correspondingto\nbulk density ρLi= 1.8 g/cm3. The ftHF total free energy\ncalculations converge much more slowly than the ftDFT\ncalculations. Forthe moderate7 ×7×7k-mesh, the DFT\ncalculationsareconvergedtoaniteration-to-iterationdif-\nference of 0.02 eV per atom, while the HF calculations\nconverge to the same precision only upon reaching the\nmuch denser 17 ×17×17 mesh. Moreover, the HF calcu-\nlation exhibits a potentially misleading energy minimum0 10 20 30 40 50 60 70 80 90 100\nT (kK)−10123456Fx(T)−Fx(100K) per atom (eV)HF\nLDAx\nLDAx(T)\nρLi=0.6 g/cm3\n0 10 20 30 40 50 60 70 80 90 100\nT (kK)−10123456Fx(T)−Fx(100K) per atom (eV)HF\nLDAx\nLDAx(T)\nρLi=1.2 g/cm3\nFIG. 9: Comparison of finite-temperature HF, ground\nstate LDA X- only(LDAx) and T-dependent LDA X- only\n(LDAx(T)) exchange free energy differences ∆ Fx(T) =\nFx(T)−Fx(100K)peratomasafunctionofelectronic temper-\nature T. Left panel: ρLi= 0.6 g/cm3; right panel: ρLi= 1.2\ng/cm3.\nat 15×15×15. Thek-mesh convergence becomes faster\nwith increasing T. For example, at T = 100 kK, both\nHF and LDAx calculations already are converged at the\n3×3×3k-mesh. In all calculations, both HF and DFT,\npresented in this section, the 25 ×25×25k-mesh was\nused.\nFigure9compares changes in the exchange free energy\ncontribution with increasing T relative to 100K values,\nFx(T)− Fx(100K). The T-independent LDA exchange\nfree energy practically does not change over that range,\ni.e.,FLDA\nx[n(r,T)]≈ FLDA\nx[n(r,100K)]. In contrast, the\nHF exchangefree energyincreasessignificantly(byabout\n4-5 eV per atom) with increasing T. The T-dependent\nLDA exchange free energy reproduces the HF behavior\nat least qualitatively.\nThe exchange free energy is, of course, a small portion\nof the total free energy. Figure 10shows total free energy\ndifferences ∆ Ftot(T) =Ftot(T)−Ftot(100K) as a func-\ntion of electronic temperature. The free energy is mono-\ntonicallydecreasingwith increasingT, in agreementwith\nnon-negativityof the entropyevaluated fromthe thermo-\ndynamic relation S=−∂F\n∂T/vextendsingle/vextendsingle/vextendsingle\nN,V. The DFT X- onlytotal\nfree energies from T-independent LDA X lie below the\ncorresponding ftHF values for all T and both densities.\nAt T≈20 kK, the interval is about 2 eV/atom, growing\nto about 4-5 eV/atom by 40 kK. The T-dependent LDA\nX gives total free energy behavior much closer to that of\nftHF, with discrepancies not exceeding 1 −2 eV/atom.\nThe effect of explicit T-dependence in exchange upon\nthe pressure may be estimated from the difference be-\ntween the ftHF or LDAx(T) and the LDAx values. Fig-\nure11provides this comparison. As a function of T, the\npressure from ftHF starts below the LDAx curve, then\ncrosses and goes above it at about 55-75 kK, depend-\ning upon the material density. The temperature of this\ncrossingpointincreasesslightlywithmaterialdensity. To\nisolate the effect of exact T-dependent X upon the pres-\nsure, we consider the difference between ftHF and LDAx10\n0 20 40 60 80 100\nT (kK)−25−20−15−10−50Ftot(T)−Ftot(100K) per atom (eV)HF\nLDAx\nLDAx(T)\nρLi=0.6 g/cm3\n0 20 40 60 80 100\nT (kK)−25−20−15−10−50Ftot(T)−Ftot(100K) per atom (eV)HF\nLDAx\nLDAx(T)\nρLi=1.2 g/cm3\nFIG. 10: Comparison of finite-temperature HF, ground\nstate LDA X- only(LDAx) and T-dependent LDA X-\nonly(LDAx(T)) total free energy differences ∆ Ftot(T) =\nFtot(T)−Ftot(100K) per atom as afunction ofelectronic tem-\nperature. Left panel: ρLi= 0.6 g/cm3; right panel: ρLi= 1.2\ng/cm3.\nvalues, offset by the near-zero-temperature difference\n∆PHF−LDAx(T) =PHF(T)−PLDAx(T) (14)\nat T = 100K, i.e.,\n∆∆PHF−LDAx(T) = ∆ PHF−LDAx(T)\n−∆PHF−LDAx(100K).(15)\nOne can see from the right-hand panels of Fig. 11that\nthe maximum magnitude of this difference at T= 100\nkK is about 10 % for all material densities considered.\nFor low temperatures, the effect of exact T-dependent\nX on pressure is stronger. For example, at 30 kK\n∆∆PHF−LDAx(30kK)≈5 GPa for material density 1.0\ng/cm3, the shift is ≈30% of the HF pressure (about\n15 GPa) at that T. Again, the LDAx(T) and ftHF\ntemperature-dependence resemble one another qualita-\ntively whereas the LDAx result does not. Note that the\nLDAx(T) crossingtemperaturewith respecttotheLDAx\ncurve increases much more rapidly with increasing mate-\nrial density than for ftHF. For material density ρLi= 0.6\ng/cm3, both curves cross at T ≈60 kK. At ρLi= 1.2\ng/cm3the LDAx(T) pressures crosses the LDAx curve\nat T≈100 kK, higher than the temperature of the HF-\nLDAx crossing point T ≈70 kK. With increasing ma-\nterial density the shift between LDAx(T) and ftHF in-\ncreases especially for T ≥50 kK.\nV. CONCLUSIONS\nDetailed computational examination of the applica-\nbility of standard PP and PAW methods to the WDM\nregime, with bulk Li as the test system, yields several\ninsights. By unambiguous comparison with all-electron\nresults from small Li clusters of bcc-derived symmetry,\nwe find that the PAW scheme requires a small augmen-\ntation sphere radius, that the compensation-charge term\nis not helpful, and that all electrons must be treated in0 20 40 60 80 100\nT (kK)−55152535455565P (GPa)P(HF)\nP(LDAx)\nP(LDAx(T))\nρLi=0.6 g/cm3\n0 20 40 60 80 100\nT (kK)−10−50510∆P (GPa)∆P=P(HF)−P(LDAx)\n∆P=P(LDAx(T))−P(LDAx)\nρLi=0.6 g/cm3\n0 20 40 60 80 100\nT (kK)0102030405060708090P (GPa)P(HF)\nP(LDAx)\nP(LDAx(T))\nρLi=0.8 g/cm3\n0 20 40 60 80 100\nT (kK)−10−50510∆P (GPa)∆P=P(HF)−P(LDAx)\n∆P=P(LDAx(T))−P(LDAx)\nρLi=0.8 g/cm3\n0 20 40 60 80 100\nT (kK)102030405060708090100110P (GPa)P(HF)\nP(LDAx)\nP(LDAx(T))\nρLi=1.0 g/cm3\n0 20 40 60 80 100\nT (kK)−10−50510∆P (GPa)∆P=P(HF)−P(LDAx)\n∆P=P(LDAx(T))−P(LDAx)\nρLi=1.0 g/cm3\n0 20 40 60 80 100\nT (kK)0102030405060708090100110120130140P (GPa)P(HF)\nP(LDAx)\nP(LDAx(T))\nρLi=1.2 g/cm3\n0 20 40 60 80 100\nT (kK)−15−10−50510∆P (GPa)∆P=P(HF)−P(LDAx)\n∆P=P(LDAx(T))−P(LDAx)\nρLi=1.2 g/cm3\nFIG. 11: Effect of temperature-dependent exchange on pres-\nsure. Left panels: pressure as a function of electronic tem-\nperature as predicted by HF, LDAx, and LDAx(T) calcu-\nlations for ρLi= 0.6,0.8,1.0 and 1 .2 g/cm3. Right pan-\nels: differences in pressure between calculations with T-\ndependent and T-independent X, P(HF) −P(LDAx) and\nP(LDAx(T)) −P(LDAx).\nthe SCF calculation. We have constructed such PAW\ndata sets for LDA and GGA functionals and used them\nto generate reference data.\nWe have located the maximal material density of bulk\nbcc-Li usable for standard PPs in Vasp,Abinit, and\nQuantum-Espresso codes. And we havedelineated the\nvalidityofusingsuchPPsathighTbycomparisonof1 e−\nand 3e−PP results. The transferabilityof PPs and PAW\ndatasetsdevelopedfornear-equilibriumconditionstothe\nWDM regime is conditional. At near-equilibrium densi-\nties it appearsto be acceptable, but not at high densities.\nClearly, such transferability should not be assumed.\nWith these issues settled, we have found that there is\nnon-trivial effect of explicit T-dependence in the X func-11\ntional in the specific sense of comparison with ftHF. In\nparticular, the LDA T-dependent exchange contribution\nto the total free energy is much closer to the exact HF\nexchange value than is the contribution from exchange\napproximatedbytheLDAground-stateXfunctional. Al-\nthough the exchange free energy is a small portion of the\ntotal free energy, this difference carries over into clearly\nsignificant differences in the equation of state. Thus, the\neffect of explicit T-dependence in X is relevant for an\naccurate characterization of the Li equation of state in\nthe WDM regime. We suspect that this may be gen-\nerally true of WDM systems. 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Dacorogna and M.L. Cohen, Phys. Rev. B 34, 4996\n(1986)."    },    {        "title": "1208.3275v1.alpha_cluster_correlations_and_symmetry_breaking_in_light_nuclei.pdf",        "content": "arXiv:1208.3275v1  [nucl-th]  16 Aug 2012KUNS-2411,RIKEN-QHP-30\nα-cluster correlations and symmetry breaking in light nucle i\nYoshiko Kanada-En’yo\nDepartment of Physics, Kyoto University, Kyoto 606-8502, J apan\nYoshimasa Hidaka\nTheoretical Research Division, Nishina Center, RIKEN, Wak o 351-0198, Japan\nα-cluster correlations in the ground states of12C and16O are studied. Because of the αcor-\nrelations, the intrinsic states of12C and16O have triangle and tetrahedral shapes, respectively.\nThe deformations are regarded as spontaneous symmetry brea king of rotational invariance, and the\nresultant oscillating surface density is associated with a density wave (DW) state caused by the\ninstability of Fermi surface with respect to a kind of 1 p-1hcorrelations. To discuss the symmetry\nbreaking between uniform density states and the oscillatin g density state, a schematic model of a\nfew clusters on a Fermi gas core in a one-dimensional finite bo x was introduced. The model analysis\nsuggests structure transitions from a Fermi gas state to a DW -like state via a BCS-like state, and to\na Bose Einstein condensation (BEC)-like state depending on the cluster size relative to the box size.\nIt was found that the oscillating density in the DW-like stat e originates in Pauli blocking effects.\nI. INTRODUCTION\nNuclear deformation is one of the typical collective motions in nuclear systems. It is known that ground states of\nnuclei often have static deformations in the intrinsic states, which are regarded as spontaneous symmetry breaking of\nthe rotational invariance due to collective correlations. Needless t o say, the broken symmetry in the intrinsic states is\nrestored in nuclear energy levels, because total angular momenta are good quanta in energy eigenstates of the finite\nsystem. Not only normal deformations of axial symmetric quadrup ole deformations but also triaxial, octupole, and\nsuper deformations have been attracting interests in these deca des. To investigate those deformation phenomena\nmean-field approaches have been applied, in particular, for heavy m ass nuclei.\nIn light nuclearsystems, further exotic shapes due to cluster str uctureshavebeen suggested. Forinstance, a triangle\nshape in12C and a tetrahedral one in16O have been discussed based on the cluster picture that12C and16O are\nconsidered to be 3 αand 4αsystems. In old days, to understand spectra of12C and16O non-microscopic α-cluster\nmodels havebeen applied [1, 2]. From vibrations of the triangle struct ure of theeαparticles and the tetrahedral one of\nfourαs, Wheeler has suggested the low-lying J= 3 states in12C and16O [1], which are now considered to correspond\nto the lowest negative-parity states12C(3−\n1, 9.64 MeV) and16O(3−\n1, 6.13 MeV) established experimentally. In 1970’s,\ncluster structures of the ground and excited states in12C and16O have been investigated by using microscopic and\nsemi-microscopic cluster models [3–11], a molecular orbital model [12], and also a hybrid model of shell model and\ncluster model [13].\nFor12C, the ground state is considered to have the triangle deformation because of the 3 α-cluster structure. In\naddition, a further prominent triangle 3 αstructure has been suggested in12C(3−\n1, 9.64 MeV). Although the cluster\nstructure of the ground state,12C(0+\n1), may not be so prominent as that of the12C(3−\n1), theJπ= 0+\n1and 3−\n1states\nare often described by the rotation of the equilateral triangle 3 αconfiguration having the D3hsymmetry. In contrast\nto the geometric configuration suggested in12C(0+\n1) and12C(3−\n1), a developed 3 α-cluster structure with no geometric\nconfiguration has been suggested in the 0+\n2state assigned to12C(0+\n2, 7.66 MeV) by (semi-)microscopic three-body\ncalculations of αclusters [5, 7, 8, 10]. In the 0+\n2state, three αparticles are weakly interacting like a gas, for which\nthe normal concept of nuclear deformation may be no longer valid. F or the 0+\n2state, Tohsaki et al.has proposed a\nnew interpretation of a dilute cluster gas state, where αparticles behave as bosonic particles and condensate in the\nSorbit [14]. This state is now attracting a great interest in relation with the Bose Einstein condensation (BEC) in\nnuclear matter [15].\nLet us consider the cluster phenomena in12C from the viewpoint of symmetry breaking. If the symmetry of the\nrotational invariance is not broken, a nucleus has a spherical shap e. In the intrinsic state of12C(0+\n1), the spherical\nshape changes to the triangle shape via the oblate shape because o f theα-cluster correlation. It is the symmetry\nbreaking from the rotational symmetry to the axial symmetry, an d to theD3hsymmetry. In the group theory, it\ncorresponds to O(3) →O(2)×Z2→D3h, where the symmetry breaking from the continuous (rotational) g roup to\nthe discrete (point) group occurs. In the excited state,12C(0+\n2), the system again may have the continuous group\nO(3) symmetry. It indicates that the cluster correlations in12C(0+\n1) and12C(0+\n2) may have different features in terms\nof symmetry breaking. The triangle shape with the D3hsymmetry in12C(0+\n1) is characterized by the geometric\nconfiguration, while the12C(0+\n2) has no geometric configuration. Now a question arises: what is the mechanism of the\nsymmetry breaking of the continuous group in12C(0+\n1), which is restored again in12C(0+\n2). One of the key problems2\nis the geometric configuration because of αcorrelations in the ground state. The triangle state has oscillating s urface\ndensity along the oblate edge. It can be understood by the density wave (DW)-like correlation caused by the 1 p-1h\ncorrelationcarryingafinite momentum inanalogytothe DWin infinite ma tterwith inhomogeneousperiodicdensities,\nwhich has been an attractive subject in various field such as nuclear and hadron physics [16–34] as well as condensed\nmatter physics [35, 36]. Indeed, in our previous work, we interpret ed the triangle shape as the edge density wave on\nthe oblate state by extending the DW concept to surface density o scillation of finite systems [37]. Then the structures\nof12C(0+\n1) and12C(0+\n2) may be associated with the DW and the BEC phases in infinite matter, respectively. The\nmechanism of the geometric triangle shape in the finite system may giv e a clue to understand an origin of DW in\ninfinite matter.\nSimilar to12C, a geometric configuration with a tetrahedral shape in16O has been discussed in theoretical studies\nwith cluster model calculations [1, 3, 11] and also with Hartree-Fock calculations [38–40]. The excited state,16O(3−\n1,\n6.13 MeV), is understood by the tetrahedron vibration or the rota tion of the tetrahedral deformation with the Td\nsymmetry, while the static tetrahedral shape in the ground state has not been confirmed yet. The tetrahedron\nshape in16O(0+\n1) and16O(3−\n1) is supported in analysis of experimental data such as E3 transition strengths for\n3−\n1→0+\n1[41] andα-transfer cross sections on12C [42]. The tetrahedral shape tends to be favored in cluster-mode l\ncalculations [3, 11]. However, Hartree-Fock calculations with tetra hedral deformed mean-field potentials usually\nsuggest the spherical p-shell closed state as the lowest solution for16O except for the calculations using effective\ninteractions with specially strong exchange forces [38–40]. If the g round state of16O has the tetrahedral shape, it\nmay suggest the breaking of the O(3) symmetry into the Tdsymmetry. In the excited 0+states of16O,12C+αcluster\nstructure was suggested in the 0+\n2state at 6.05MeV [4, 9, 10, 43, 44]. Moreover, in analogyto the 3 α-cluster gas state\nof12C(0+\n2), a 4α-cluster gas state with the αcondensation feature has been suggested recently in a 0+state above\nthe 4αthreshold [45, 46]. Similarly to12C, the possible tetrahedral shape in16O may lead to symmetry breaking of\nthe continuous group in16O(0+\n1), which is restored in higher 0+states. Again, the geometric configuration due to α\ncorrelations in the ground state should be one of the key problems.\nOur aim is to clarify the α-cluster correlations with geometric configurations in the ground s tates of12C and\n16O, and understand the mechanism of the symmetry breaking from c ontinuous (rotational) groups into discrete\n(point) groups. We first confirm the problem whether the triangle a nd tetrahedron shapes are favored in the intrinsic\nstates of12C and16O. For this aim, we use a method of antisymmetrized molecular dynamic s (AMD) [47–49] and\nperform microscopic many-body calculations without assuming exist ence of any clusters nor geometric configurations.\nVariation after the spin-parity projections (VAP) is performed in t he AMD framework [50]. The AMD+VAP method\nhas been proved to be useful to describe structures of light nucle i. With this method, one of the authors, Y. K-E., has\nsucceeded to reproduce various properties of ground and excite d states of12C [50, 51], and confirmed the formation\nand development of three αclusters in12C in the microscopic calculations with no cluster assumptions for the fi rst\ntime. The result was supported also by the work using the method of Fermionic molecular dynamics [52], which shows\na similar method to the AMD.\nIn this paper, we apply the AMD+VAP method to16O as well as12C and analyze the intrinsic shapes of the\nground states. We show that the geometric configurations having the approximate D3handTdsymmetry arise in\nthe ground states of12C and16O, respectively. To discuss appearance of the geometric configur ations, we perform\nanalysis using a simple cluster wave functions of Brink-Bloch (BB) α-cluster model [53], while focusing on Pauli\nblocking effect on rotational motion of an αcluster. Important roles of the Pauli blocking effect in appearance of\ngeometric configurations are described. We also introduce a schem atic model by considering clusters on a Fermi gas\ncore in a one-dimensional (1D) finite box, which can be linked with clust ers at surface in a 3 αsystem. By analyzing\nthe 1D-cluster wave function, in particular, looking at Pauli blocking effects from the core and those between clusters,\nwe try to conjecture what conditions favor BCS-like, DW-like, and B EC-like correlations.\nThis paper is organized as follows. In the next section, intrinsic shap es and cluster formation in the ground states\nof12C and16O are investigated based on the AMD+VAP calculation. In Sec. III a P auli blocking effect in α-cluster\nsystems and its role in α-cluster correlations is described by analysis of BB α-cluster wave functions. In Sec. IV, the\nschematic model of clusters on the Fermi gas core in the 1D finite bo x is introduced and the roles of Pauli blocking\neffects inα-cluster correlations are discussed. Summary and outlook are give n in Sec. V. The relations between\n3α- and 4α-cluster wave functions and triangle and tetrahedral deformed m ean-field wave functions are explained in\nappendix A. In appendix B and C, features of weak-coupling wave fu nctions in the 1D-cluster model are described.\nII. SHAPES AND CORRELATIONS IN THE GROUND STATES OF12C AND16O\nWe discuss here intrinsic deformations of the ground states of12C and16O based on the AMD+VAP calculation.\nTheAMDmethodhasbeenappliedforvariouslightmassnucleiandhas beensuccessfulindescribingclusterstructures\nas well as shell-model-like structures in light-mass nuclei. In the pres ent work, the AMD+VAP method, i.e., variation3\nafter spin and parity projections in the AMD framework, is applied to12C and16O. For the details of the framework,\nthe reader is refereed to, for instance, Refs. [49, 50].\nA. Variation after projection with AMD wave function\nIn the AMD framework, we set a model space of wave functions and perform the energy variation to obtain the\noptimum solution in the model space.\nAn AMD wave function is given by a Slater determinant of Gaussian wav e packets,\nΦAMD(Z) =1√\nA!A{ϕ1,ϕ2,...,ϕA}, (1)\nwhere theith single-particle wave function is written by a product of spatial ( φ), intrinsic spin ( χσ), and isospin wave\nfunctions (χτ) as\nϕi=φXiχσ\niχτ\ni, (2)\nφXi(rj) =/parenleftbigg2ν\nπ/parenrightbigg4/3\nexp/braceleftbig\n−ν(rj−Xi√ν)2/bracerightbig\n, (3)\nχσ\ni= (1\n2+ξi)χ↑+(1\n2−ξi)χ↓. (4)\nφXiandχσ\niarespatialandspinfunctions, and χτ\niistheisospinfunction fixedtobeup(proton)ordown(neutron). A c-\ncordingly,anAMDwavefunctionisexpressedbyasetofvariational parameters, Z≡ {X1,X2,···,XA,ξ1,ξ2,···,ξA}.\nThe width parameter νrelatesto the sizeparameter basν= 1/2b2and it is chosento be ν= 0.19fm−2that minimizes\nenergies of12C and16O.\nThe center positions X1,X2,···,XAofsingle-nucleonwavepacketsareindependently treatedas varia tionalparam-\neters. Thus existence of any clusters are not a prioriassumed in the AMD framework. Despite of it, the model wave\nfunction can describe shell-model structures and cluster struct ures because of the antisymmetrizer and the flexibility\nof spatial configurations of Gaussian centers. If a cluster struc ture is favored in a system, the corresponding cluster\nstructure is automatically obtained in the energy variation.\nFor even-even nuclei, the ground states are known to be Jπ= 0+states, i.e., they are symmetric for rotation.\nIntrinsic deformation is understood as spontaneous symmetry br eaking with respect to the rotational invariance,\nwhich is restored in the Jπ= 0+ground states. It means that when an intrinsic state has a deform ation the ground\nstate is constructed by the spin and parity projections from the in trinsic state. In more general, spin and parity\nare good quanta in energy eigenstates of nuclei because of the inv ariance of the Hamiltonian for rotation and parity\ntransformation. Therefore, to express a Jπstate, an AMD wave function is projected onto the spin-parity eige nstate,\nΦ(Z) =PJπ\nMKΦAMD(Z), (5)\nwherePJπ\nMKis the spin-parity projection operator.\nToobtainthewavefunctionfora Jπstate, theenergyvariationisperformedforthespin-parityproj ectedAMDwave\nfunction Φ( Z) with respect to variationalparameters {Z}. This method is called variationafter projection(VAP). The\nAMD+VAP method has been applied to various light nuclei for structu re study of ground and excited states. For the\nground states of12C and16O, we perform the variation of the energy expectation value ∝an}bracketle{tΦ(Z)|H|Φ(Z)∝an}bracketri}ht/∝an}bracketle{tΦ(Z)|Φ(Z)∝an}bracketri}ht\nfor theJπ= 0+projected wave function and get the optimum parameter set {Z}that minimizes the energy. Then,\nthe AMD wavefunction Φ AMD(Z) given by the optimized {Z}is regardedas the intrinsic wavefunctions of the ground\nstate.\nAnAMDwavefunctionisexpressedbyasingleSlaterdeterminant; ho wever,thespin-parityprojectedwavefunction\nisno longeraSlaterdeterminantbut it isalinearcombinationofSlaterd eterminantsexceptforthe casethatthe AMD\nwave function before the projection is already a spin-parity Jπeigenstate. If the intrinsic state has a deformation the\nprojected wave function contains some kinds of correlations beyo nd Hartree-Fock approximation.\nB. Intrinsic structures of12C and16O\nWe apply the AMD+VAP method to the ground states of12C and16O, and discuss their intrinsic structures.4\n1. Density distribution\nThe ground states of12C and16O have the intrinsic deformations. The density distribution of the int rinsic wave\nfunctions Φ AMDare shown in Fig. 1. The result for12C shows a triaxial deformation with a triangle feature, while\n16O has a deformation with a tetrahedral feature. The quadrupole d eformation parameters ( β,γ) evaluated by the\nquadrupole moments are ( β,γ) = (0.31,0.13) for12C and (β,γ) = (0.25,0.09) for16O. The triangle and tetrahedral\nshapes are caused by α-cluster correlations. Strictly speaking αclusters are not ideal (0 s)4clusters but somewhat\ndissociated ones. Moreover, the triangle and tetrahedron are no t regular but distorted as an αcluster is situated\nslightly far from the other αs.\n16(a)   C\n 0 0.1 0.2 0.3 0.4 0.5 0.6\nz  (fm)x  (fm)\n-5  5-5 5\n 0 0.1 0.2 0.3 0.4 0.5 0.6\nx  (fm)y  (fm)\n-5  5-5 5\n 0 0.1 0.2 0.3 0.4 0.5 0.6\ny  (fm)z  (fm)\n-5  5-5 5\n 0 0.1 0.2 0.3 0.4 0.5 0.6\nx  (fm)y  (fm)\n-5  5-5 5\n 0 0.1 0.2 0.3 0.4 0.5 0.6\ny  (fm)z  (fm)\n-5  5-5 5\n 0 0.1 0.2 0.3 0.4 0.5 0.6\nz  (fm)x  (fm)\n-5  5-5 5(b)   O12\nFIG. 1: (color on-line) Density distributions in the intrin sic states for the ground states of (top)12C and (bottom)16O obtained\nby the AMD+VAP calculation. The density integrated on the z,x, andyaxes is plotted on the x-y,y-z, andz-xplanes.\n2. Oscillation of the surface density\nLet us consider the deformations of12C and16O from the viewpoint of symmetry breaking. The highest symmetry\nis the sphere, which is realized in the closed p3/2-shell andp-shell states for12C and16O, respectively. As a higher\nsymmetry can break into lower symmetry due to multi-nucleon corre lations, the deformation mechanism of12C is\ninterpreted as follows. Because of the α-cluster correlations, the rotational symmetry in the sphere bre aks to the\naxial symmetry in an oblate deformation and changes into the D3hsymmetry in the regular triangle 3 αconfiguration,\nand then it breaks to the distorted triangle. The symmetry change , spherical →oblate→triangle, corresponds to\nO(3)→O(2)×Z2→D3h. Similarly, the deformation of16O is understood as the rotational symmetry in the sphere\nbreaks into the Tdsymmetry in the regular tetrahedral 4 αconfiguration, and it breaks to the distorted tetrahedron.\nThe structure change from the spherical to the tetrahedron is t he breaking of the O(3) symmetry to the Tdsymmetry.\nNote that the continuous (rotational) groups break to the discre te (point) groups in the triangle and tetrahedron\ndeformations.\nWe discuss the connection of cluster correlations with the density w ave, which is characterized by the static density\noscillation at the surface. The surface density oscillation occurs at the symmetry breaking from the axial symmetry\n(oblate shape) to the D3hsymmetry (triangle) in12C and that from the rotational symmetry (sphere) to the Td\nsymmetry (tetrahedron) in16O. The triangle deformation contains the ( Y−3\n3−Y+3\n3)/√\n2 component with the Y0\n2\ndeformation in the density, while the tetrahedral one has the (√\n5Y0\n3+√\n2Y−3\n3−√\n2Y+3\n3)/3 component, which can be\ntransformed to ( Y−2\n3+Y+2\n3)/√\n2 by the rotation. Indeed, as described in appendix A, an ideal 3 α(4α)-cluster wave\nfunction with the triangle(tetrahedral) configurationhas the den sity havingthe finite components of ( Y−3\n3−Y+3\n3)/√\n2\n((√\n5Y0\n3+√\n2Y−3\n3−√\n2Y+3\n3)/3) and it can be described by the DW-type particle-hole correlations in case of weak\ndeformations. Note that the DW in the triangle shape is characteriz ed by the particle-hole correlations on the Fermi\nsurface carrying the finite angular momentum ( l,m) = (3,±3) and that in the tetrahedral one is given by the particle-\nhole correlations with ( l,m) = (3,±2) as described in the appendix. Namely, the symmetry breaking is ch aracterized\nby the (Y−3\n3−Y+3\n3)/√\n2 component whose amplitudes linearly relate to the order paramete r of the DW correlations\nas shown in Eqs. (A3), (A6), and (A7). In other words, the symme try broken states have finite ( Y−3\n3−Y+3\n3)/√\n2\ncomponents in surface density and show the oscillation density with t he wave number three.5\nTo analyze the surface density oscillation in the ground states of12C and16O, we perform the multipole decompo-\nsition of the intrinsic density obtained with the AMD+VAP calculation\nρ(r=R0,θ,φ) = ¯ρ(R0)/summationdisplay\nlmαlmYm\nl(θ,φ), (6)\natacertainradius r=R0, anddiscussthe( Y−3\n3−Y+3\n3)/√\n2components. Wetake R0tobetherootmeansquareradius\nof the intrinsic state. ¯ ρ(R0) is determined by the normalization α00= 1, andαlmhas the relation αlm= (−1)mα∗\nl−m\nbecause the density ρ(r=R0,θ,φ) is real.\nThe density plot at r=R0on theθ-φplane for12C and that at r=R0andθ=π/2 as a function of φare shown in\nFig. 2. As seen clearly, the density on the oblate edge shows the osc illation with the approximate wave number three\nperiodicity, which comes from the α-cluster correlation. In the right panel of Fig. 2, we also plot the de nsity for the\nidealD3hsymmetry given only by the ( l,m) = (0,0) ,(2,0),(3,3) and (3, −3) components to show somewhat distortion\nof the AMD+VAP result from the ideal D3hsymmetry. In the coefficients |αlm|shown in Fig. 4, it is found that the\nY±3\nlcomponents are actually finite indicating that the axial symmetry is b roken to the triangle shape in12C.\nFor the density in16O, theθ-φplot is shown in Fig. 3, and the coefficients of the multipole decompositio n are\nshown in Fig. 4. The tetrahedral component√\n5Y0\n3/3+√\n2Y+3\n3/3−√\n2Y−3\n3/3 is shown by the hatched boxes at α30\nandα33in Fig. 4. The open boxes indicate the distortion components from th e tetrahedron. The distortion exists\nin the axial symmetry components, α30,α20, andα10, coming from the spatial development of an αcluster from the\nothers as explained before.\nThus, the intrinsic states of12C and16O show the surface density oscillation with the ( Y−3\n3−Y+3\n3)/√\n2 component.\nThe wave number three oscillation characterized by the ( Y−3\n3−Y+3\n3)/√\n2 component is understood by the α-cluster\ncorrelations with triangle and tetrahedral deformations, which ar e interpreted as the DWs on the oblate and spherical\nshapes, i.e., the spontaneous symmetry breaking of axial symmetr y→D3hand rotational symmetry →Tdsymmetry.\n 0 0.5 1\nθ  (π rad) 0\n 0.5\n 1\n 1.5\n 2φ  (π rad)\n 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16\n 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16\n 0 0.5 1 1.5 2Density  (fm-3)\nφ  (π rad)\nFIG. 2: (color on-line) Left: density at r=R0for12C calculated by the AMD+VAP is plotted on the θ-φplane. Right: that at\nr=R0andθ=π/2 line (the solid line). The density for the ideal D3hsymmetry given only by the ( l,m) = (0,0) ,(2,0),(3,3)\nand (3,−3) components is also plotted (the dashed line). R0is taken to be 2.53 fm.\n 0 0.5 1\nθ  (π rad) 0\n 0.5\n 1\n 1.5\n 2φ  (π rad)\n 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16\nFIG. 3: (color on-line) density at r=R0for16C calculated by the AMD+VAP is plotted on the θ-φplane.R0is chosen to\nbe 2.81 fm.6\n-0.2 0 0.2 0.4\nα10α11α20α21α22α30α31α32α33-0.2 0 0.2 0.4\nα10α11α20α21α22α30α31α32α3312C\n16O\nFIG. 4: The coefficients αlmof the multipole decomposition of the intrinsic density of ( top)12C and (bottom)16O calculated by\nthe AMD+VAP. The hatched areas in the bottom panel are the tet rahedron component√\n5Y0\n3/3+√\n2Y+3\n3/3−√\n2Y−3\n3/3 defined\nbyαhatch\n30≡/radicalbig\n5/2α33andαhatch\n33≡α33. The open area for the Y0\n3component is defined by the relation α30=αopen\n30+αhatch\n30.\nIII. PAULI BLOCKING EFFECT IN αCORRELATIONS\nAs shown in the previous section, the intrinsic states of12C and16O obtained with the AMD+VAP calculation\ncontain the α-cluster correlations with the triangle and tetrahedral deformat ions. For12C, the triangle deformation\nhasbeen suggestedin 3 α-clustermodels, in which 3 αstate with aregulartriangleconfigurationisthe energyminimum\nstate [3, 7, 10]. In contrast to such the geometric configuration a s the triangle, the second 0+state of12C is considered\nto be a cluster gas state where 3 αclusters are freely moving in dilute density like a gas without any geome tric\ncorrelations between clusters [7, 8, 10, 14]. The 0+\n2state is associated with the αcondensation in analogy to BEC\nbecause it is regarded as a system of three αs occupying the same Sorbit [14].\nFrom the viewpoint of symmetry breaking, the symmetry is broken in the 0+\n1state with the triangle deformation\nand it is restored in the 0+\n2state. In the 0+\n1state, the axial symmetry breaks down to the D3hsymmetry due to the\n3α-cluster structure. One of the characteristics of the 0+\n1state is the oscillating surface density caused by the angular\ncorrelation of αclusters. Then the structure change from the 0+\n1to the 0+\n2is expected to connect with the transition\nfrom the symmetry broken state with the angular correlation to th e symmetric state with no (or less) correlation\nbetweenαclusters.\nThe origin of the angular correlation in the 0+\n1state and the transition into the uncorrelated 0+\n2state can be\nunderstood by the Pauli blocking effect between clusters as follows . Let us here consider motion of an αcluster\naround a 2 αcore in the BB 3 α-cluster model wave function. In the BB 3 α-cluster model, αclusters are located\naround certain positions S1,S2, andS3, and the wave function is written as\nΦBB(S1,S2,S3) =1√\nA!A{ΠτσφS1XτσφS2XτσφS3Xτσ}, (7)\nwhereXτσis the spin-isospin wave function with τ={p,n}andσ={↑,↓}. We assume that 2 αclusters placed at\nS1= (0,0,d/2) andS2= (0,0,−d/2) form the core. The third αis placed at S3= (0,y,z) fory=rcosθy,z=rsinθy\non the (y,z)-plane (see Fig. 5). Because of Pauli blocking effect between clust ers, the motion of the third αaround\nthe core is restricted in the Pauli allowed region. Particularly when th eαexists near the core, rotational motion is\nstrongly blocked because of the existence of other αclusters. The Pauli allowed and forbidden areas for the rotation\nof the angle θyfor the third αcenter in the ( y,z)-plane are presented in Fig. 5. In the figure, the norm N3α(y,z) =\n∝an}bracketle{tΦBB(S1,S2,S3)|ΦBB(S1,S2,S3)∝an}bracketri}htof the BB wave function Φ BB(S1,S2,S3) with the parameters, S1= (0,0,d/2),\nS2= (0,0,−d/2), andS3= (0,y,z) is shown in the ( y,z)-plane.dis fixed and taken to be a small value. The norm\nis normalized by the value for the αon the y-axis,\n˜N3α(y,z)≡N3α(y=rcosθy,z=rsinθy)\nN3α(y=r,z= 0). (8)\nThe area with ˜N3α∼1 is the allowed region where the αfeels no Pauli blocking with respect to the rotational\nmotion, while the ˜N3α∼0 region is the blocked region where it feels the strong Pauli blocking e ffect fromαclusters\nof the core. It means that, when the third αexists near the core, its angular motion is blocked by the 2 αcore. As\na result, the third αis confined in the Pauli allowed region around the yaxis, and it has the angular correlation7\nagainst the 2 αdirection. Consequently, a compact 3 αstate has a geometric structure of the triangle deformation\nand it has the surface density oscillation. On the other hand, as the cluster develops specially the Pauli blocking\neffect becomes weak. When the αis far enough from the core, it can freely move in the rotational mod e. Then the\nangular correlation vanishes and the system transits to angular-u ncorrelated state. We note that in cluster physics\nthis transition is known to be the change between strong cluster co upling and weak cluster coupling states, where the\nangular momentum of the inter-cluster motion couples with inner spin s of clusters strongly and weakly, respectively\n(see Fig. 5). What we call the cluster coupling is coupling between clus ters and it is different from the terminology of\nstrong and weak couplings in the BEC-BCS crossover phenomena, w hich relate to the coupling between nucleons in\na pair (or in a cluster).\nThe 0+\n1state of12C is considered to be the compact 3 αstate in the strong Pauli blocking regime and corresponds\nto the angular correlated state with the surface density oscillation due to the triangle deformation. In the12C(0+\n2),\nclusters spatially develop well and the system goes to the non-angu lar-correlation state. Strictly speaking, in the\n12C(0+\n2), all clusters develop to form a uncorrelating three αstate associated with the αcondensation, where the\nradial motion is also important as well as the angular motion. Neverth eless, we can say that, for the restoration of the\nbroken symmetry with the surface density oscillation in the 0+\n1to the rotational symmetry in the 0+\n2, the transition\nfrom the angular correlated state to the uncorrelated state in th e cluster development is essential.\nIn the above discussion, we consider the angular motion in the intrins ic (body-fixed) frame, i.e., the y-zplane.\nSince the system is symmetric for the rotation around the z-axis, the motion of the third αis free for the z-rotation.\nThis rotational mode around the zaxis is nothing but the projection onto the Jz= 0, which is usually performed in\ntheJπ= 0+projection of the intrinsic state.\nAlso in16O, the angular motion of an αcluster at the surface is blocked by other three αs. The Pauli blocking\neffect between αclusters causes the angular correlation of the tetrahedral confi gurations in a compact 4 αstate. For\ntransition from the angular correlated cluster state of the groun d state to the uncorrelated cluster state like a cluster\ngas, at least two αclusters need to spatially develop to move freely in the allowed region w ithout feeling the Pauli\nblocking effect. The 4 α-cluster gas state in16O has been suggested near the 4 αthreshold energy. The suggested\nexcitation energy Ex∼15 MeV is almost twice of Ex= 7.66 MeV for the 3 α-cluster gas state in12C. This might\ncorrespond to the energy cost of the spatial development for tw oαs in16O compared with that for an αin12C.\n-2-1 0 1 2\ny ν1/2-2-1 0 1 2 z ν1/2\n 0 0.2 0.4 0.6 0.8 1(y,z)\nfree motionz\ny\nPauli blocking(d) Weak(a) (b)\n(c) Strong\ncluster coupling cluster coupling\nFIG. 5: (a) The Pauli allowed and forbidden regions for the ro tation of the angle θyfor the third αin the (y,z)-plane around\nthe 2αcore. The norm ˜N3α(y=rcosθy,z=rsinθy) normalized at ( y= 0,z=r) is presented. The ˜N3α∼1 and˜N3α∼0\nregions are the Pauli allowed and forbidden regions, respec tively. (b) A schematic figure for the position of the third αaround\nthe 2αcore. (c) A schematic figure of the strong cluster coupling st ate for a compact 3 αstate with the strong the Pauli blocking\neffect, and (d) that of the weak cluster coupling state. See th e text.8\nIV. CLUSTERS ON A FERMI GAS CORE IN ONE DIMENSIONAL BOX\nA. Concept of one-dimensional cluster model\nAs mentioned, the triangle deformation of the12C ground state can be interpreted as the density wave on the\naxial symmetric oblate state and it is characterized by the static su rface density oscillation with the wave number\nthree on the edge of the oblate intrinsic state. The symmetry brea king in the ground state originates in the 4-body\ncorrelation in αclusters. The axial symmetry is broken by the angular correlation b etweenαclusters due to the Pauli\nblocking effect. As αclusters develop, the Pauli blocking weakens and the angular corre lation vanishes. Then the\nsystem transits from the symmetry broken state of the12C(0+\n1) to the symmetry restored state of the12C(0+\n2), where\nαclusters are freely moving in a dilute density like a gas.\nIn this section, we consider Pauli blocking effects in a schematic mode l of two clusters on a Fermi gas core in a\none-dimensional (1D) finite box, and discuss how the translational invariance is broken to form an oscillating density\n(inhomogeneous) state and how the broken symmetry is restored to a uniform density (homogeneous) state.\nLet us consider a schematic model for a simplified 3 αsystem consisting of two αclusters around the αcore. To\nconcentrate only on angular correlations between two αs, we ignore the radial coordinate degree of freedom and\nassume that two αclusters are moving at a certain distance r0from the core. Taking the body-fixed frame so as one\nof twoαclusters on the z-axis and the other αin they-zplane, then, the angular motion between two αs can be\nreduced to the 1D problem, where the first cluster sits around the origin and the second cluster exists in a finite size\nL= 2πr0box with the periodic boundary condition. For the core effect, we ta ke into account only the Pauli blocking\nfrom the core particles, which is treated here by a Fermi gas core f or simplicity.\nWe extend the 1D-cluster model and treat the cluster size b, the box size L, and the core Fermi momentum kc\nas parameters. In this model, we do not perform energy variation n or calculate energy eigenstates. Moreover, the\nmechanism of cluster formation and the dynamical change of the clu ster structure or the cluster size are beyond our\nscope. We give a model wavefunction with fixed parameters b,L,kcbased on simple ansatz, and analyze Pauliblocking\neffect between a cluster and the core and that between two cluste rs in the given wave function. From behavior of\nthe wave function and features of the Pauli blocking effects, we co njecture how the transition between the Fermi gas,\nBCS-like, DW-like, and BEC-like states occurs.\nB. Formulation of one-dimensional cluster model\nWe explain details of the 1D-cluster model wave function. Two n-body clusters are formed on a Fermi gas core in\na 1D box with the box size L= 2πr0. For anαclustern= 4 and a cluster consists of p↑,p↓,n↑, andn↓. Basis\nsingle-particle wave functions in the periodic boundary condition are given ase+ikx/√\n2Lande−ikx/√\n2L, where the\nmomentum k= 2π˜k/L. Here˜kis the dimensionless momentum ˜k=kr0and takes ˜k= 0,1,2,3,···. Core nucleons\nare assumed to form a Fermi surface at kc. It means that k≤kcorbits are occupied by core nucleons and they are\nforbidden single-particle states for constituent nucleons of clust ers. Using the dimensionless parameter ˜kc=kcr0, the\nlowest allowed momentum is kF=˜kF/r0for˜kF≡˜kc+1 andk≥kForbits are allowed.\nWe assume the first cluster ( α1) wave function that localized around x= 0 in a Gaussian form,\nψα1= Πχφ0(x1χ)Xχ, (9)\nχ=τσ={p↑,p↓,n↑,n↓}, (10)\nφ0(x) =1√\n2L/summationdisplay\nk≥kFf(k)/parenleftbig\ne+ikx+e−ikx/parenrightbig\n. (11)\nf(k) is the Fourier transformation of the Gaussian with the cluster size band given as\nf(k) =n0exp(−b2k2\n2), (12)\nwheren0is determined by the condition,\n/summationdisplay\nk≥kF|f(k)|2= 1, (13)\nto normalize φ0(x) in the box. The wave function indicates that four species of nucleo ns,p↑,p↓,n↑,n↓, occupy the\nsame spatial orbit with the Gaussian form forbidden partially by the F ermi gas core. If b/LandkFare small enough,9\nthe wave function is localized well around x= 0. Since there is no identical Fermion in a cluster, the antisymmetriz er\ncan be omitted in the expression.\nNext we consider the second cluster ( α2). Assuming that α2is localized around a position x=s, the wave function\nis given by the shifted function,\nψs\nα2=/productdisplay\nσφs(x2χ)Xχ, (14)\nφs(x) =1√\n2L/summationdisplay\nk≥kFf(k)/parenleftBig\ne+ik(x−s)+e−ik(x−s)/parenrightBig\n. (15)\nThen, the normalized wave function of the 2 αsystem with the parameter sis written as\nΨs=N−2\nPB(s)1\n8!A/braceleftbig\nψα1ψs\nα2/bracerightbig\n, (16)\nNPB(s)≡ |1√\n2A{φ0φs}|2(17)\n=1\n2|φ0(x1)φs(x2)−φ0(x2)φs(x1)|2(18)\n= 1−∝an}bracketle{tφ0φs|φsφ0∝an}bracketri}ht. (19)\nNPB(s) is the overlap norm for two identical nucleons. It is a function of th e parameter sfor the localization center\nof the second cluster. N4\nPBmeans the overlap norm for the 2 αsystem and it is an indicator to evaluate the Pauli\nblocking effect between two clusters. In the case that two cluster s feel no Pauli blocking, N4\nPB(s) = 1, while in the\ncase that they feel complete Pauli blocking, N4\nPB= 0. It means that N4\nPB(s) stands for the Pauli ”allowedness” for\nα2arounds.\nThe density distribution for χ=τσparticles in the 2 αstate is\nρχ\ns(x) =∝an}bracketle{tΨs|/summationdisplay\ni∈χδ(x−xi)|Ψs∝an}bracketri}ht. (20)\nNote that the density distributions for all kinds of nucleons are the same in the present cluster model and the total\nnuclear density is/summationtext\nχρχ\ns(x) = 4ρχ\ns(x). In a similar way, the χ=τσdensity for the one- αstateψα1is\nρχ\nα1(x) =∝an}bracketle{tψα1|/summationdisplay\ni∈χδ(x−xi)|ψα1∝an}bracketri}ht. (21)\nIn the present model, we can express all the formulation with the dim ensionless parameters ˜L=L/r0= 2π,\n˜b=b/r0, ˜s=s/r0, ˜x=x/r0,˜k=kr0,˜kc=kcr0,˜kF=kFr0, and so on. The dimensionless densities are also defined\nas ˜ρ=ρr0. Then, the state |Ψs∝an}bracketri}htis specified with the dimensionless parameters ( ˜kF,˜b,˜s) and does not depend on\nthe scaling factor r0. Since the number of clusters is fixed to be two, a larger volume size Lcorresponds to a lower\ncluster density. Because of the scaling, a larger volume size L= 2πr0with a fixed cluster size bis equivalent to a\nsmaller cluster size bwith a fixed L. That means, the parameter ˜bindicates the cluster size relative to the box size\nand also corresponds to the cluster density. The ˜kF=˜kc+ 1 is the lowest allowed momentum just above the core\nFermi momentum ˜kc, and it relates to the density of the core particles.\nC. Pauli blocking effect from the core to one cluster\nLet us describe the structure change of one cluster on the Fermi gas core in the 1D-cluster model. The coefficients\nf(˜k) of the Fourier transformation and the density distribution ρχ\nα1(˜x) for theα1cluster are shown in Figs. 6 and 7 for\n˜kF= 1 and ˜kF= 2, respectively. Because of the Pauli blocking effect from the cor e as well as the finite volume effect,\nthe structure of the cluster changesfrom the originalGaussian f orm depending of the parameters ˜band˜kF. In the case\nof small˜b, which corresponds to the small cluster size bor the large volume size L, the coefficient f(˜k) is distributed\nwidely and has a long tail toward the high momentum region, and the de nsity is localized well around ˜ x= 0 (and\nalso ˜x= 2πfor the periodic boundary). With increase of ˜kF, the low momentum components are truncated and the\ndensity shows a oscillating tail. Nevertheless, if the cluster size ˜bis small enough, the density is still localized. With\nincrease of the cluster size ˜b, the density localization weakens and the density approaches the p eriodic one. Then,10\nthe component of the lowest allowed orbit ˜kF=˜kc+ 1 is dominant and the components of higher orbits decrease.\nThe localization declines more rapidly in the case of the larger ˜kF. It means that, as the ˜bincreases or as the ˜kF\nincreases, the spatial correlation between nucleons (inner corre lation) in a cluster becomes weak. We call the case of\nthe weak localization that the wave function has the dominant ˜kFcomponent and minor ˜k >˜kFcomponents “the\nweak coupling regime”, and the opposite case of the strong localizat ion with significant ˜k >˜kFcomponents “the\nstrong coupling regime”. In the weak coupling limit, the one-cluster d ensity goes to cos2(˜kF˜x)/π.\nD. Pauli blocking between two clusters\nFor twoαclusters expressed by the cluster wave function Ψ s, the clusters α1andα2are assumed to be localized\naround ˜x= 0 and ˜x= ˜s, respectively. The Pauli blocking effect between α1andα2is evaluated by the overlap norm\nN4\nPB(˜s), which is an indicator for Pauli allowedness. The Pauli allowedness N4\nPB(˜s)∼0 means that the ˜ sregion is\nblocked by the α1cluster and is a forbidden area for α2. The middle panels of Figs. 6 and 7 show the allowedness\nN4\nPB(˜s) (thin solid lines) as well as NPB(˜s) (dashed lines) plotted as a function of ˜ s. In principle, the Pauli blocking\neffect reflects the probability of the α1cluster, and therefore, the forbidden region corresponds to th e relatively high\ndensity region of the α1cluster. In case of a small cluster size ˜b, the allowed region exists widely and the forbidden\nregion exists only in the small area close to ˜ x= 0 and ˜x= 2π. That is to say, the second cluster feels almost no\nPauli blocking effect except for the region near the first cluster ( α1). With increase of ˜b, the forbidden area spreads\nin a wide region around ˜ x= 0 and ˜x= 2π, and the allowed region for α2becomes narrow. In the weak coupling\nregime, where the α1density is periodic, the allowed region with N4\nPB(˜s)∼1 also shows the periodicity. Reflecting\nthe cos2(˜kFx) periodicity of ρα1(x), the areas ˜ s≈π2m/2˜kF(m= 0,···,2˜kF) are the forbidden regions, while the\nareas ˜s= ˜sj=π(2j−1)/2˜kF(j= 1,···,2˜kF) are the allowed regions.\nAs mentioned, the parameter ˜bcorresponds to the cluster size relative to the box size. When the c luster density is\nlow enough, the Pauli blocking effect is weak and almost all region is allow ed for theα2cluster except for the position\nclose to the α1. On the other hand, in the case of the high cluster density the Pauli blocking effect is strong and the\nallowedsregion is restricted in the periodic regions.\nE. Transitions from strong coupling to weak coupling regime s\nWe first discuss the features of the two cluster wave function Ψ swith a fixed parameter ˜ s. It means that the center\nof the second cluster is located around the fixed position. Later, w e will discuss how the spatial correlations between\nclusters (inter-cluster correlation) can be affected by the Pauli b locking.\nWe choose ˜ sj=π(2j−1)/2˜kFwithj= 1 andj=˜kF, which are the allowed positions ˜ sjnearest to ˜ x= 0\nand ˜x=πand corresponds to the smallest and largest inter-cluster ( α1-α2) distances, respectively. The density\ndistribution ρχ\ns(˜x) in the 2α-cluster wave function Ψ sare shown in the right panels of Figs. 6 and 7 for ˜kF= 1 and\n˜kF= 2. In the strongcouplingregime, forinstance, the ( ˜kF,˜b) = (1.0,0.25)state, the densityshowsthe cleartwopeak\nstructure and indicates that two clusters are well isolated without almost no overlap. As the ˜bincreases, the overlap\nregion between clusters gradually increases and the density chang es to the oscillation structure, in particular, in the\ncase ofj=˜kF. The density oscillation is remarkable, for instance, in the ( ˜kF,˜b) = (1.0,1.0) and (˜kF,˜b) = (2.0,0.75)\nstates, which shows the 2 ˜kF+ 1 periodicity. With further increase of ˜b, the density oscillation weakens and finally\ndisappears to the uniform density, and the system goes to the Fer mi gas limit with the Fermi momentum ˜kF.\nIn the present model, we put the α2cluster around the fixed position ˜ s. For more realistic wave functions of two α\nclusters, one should extend the model by taking into account the m otion of the α2cluster relative to the α1cluster.\nMicroscopically, it corresponds to superposing ψs\nα2with various values of the parameter sas\nψα2=/integraldisplay\nd˜sF(˜s)ψs\nα2. (22)\nThe spatial correlation between clusters (inter-cluster correlat ion) is expressed by the weight function F(˜s). On the\nother hand, the correlation between nucleons inside a cluster (inne r correlation) is described by the intrinsic structure\nof a single-cluster wave function of ψα1orψs\nα2determined by the parameters ˜band˜kF, and it is given by hand in the\npresent model, where the cluster formation and its intrinsic struct ure area prioriassumed.\nMoreover, the wave function should be projected to the total mo mentumKG= 0 for the center of mass motion\n(c.m.m.) of two clusters so that the translational invariance is resto red in the finite system.11\nIn the following discussions, we do not treat the α2-cluster motion explicitly, but conjecture how the Pauli blocking\nmay restrict the motion of the α2cluster and affect to the spatial correlations between clusters (in ter-cluster corre-\nlations). Since the area of high α1density is blocked as mentioned, the α2cluster may move in the allowed regions\nwith large N4\nPB(˜s). We assume that the interaction between clusters is weak and the Pauli blocking effect, which acts\nlike an effective repulsion, gives the most important contribution in th e relative motion between clusters.\n1. BEC-like state\nLet us consider the strong coupling regime, where the cluster size ˜bis small enough. The N4\nPB(˜s) curve shows\na wide open window of the allowed region as in the case of ( ˜kF,˜b) = (1.0,0.25). Since the Pauli blocking effect is\nweak, theα2can move in the wide allowed region almost freely. It means the strong inner correlation but almost no\ninter-cluster correlation. In such the uncorrelated cluster stat e, two clusters may condensate approximately the zero\nmomentum state in the ground state similarly to the BEC phase.\n2. DW-like state\nAs the˜bincreases, the open window for the allowed region closes and α2no longer can move freely. Instead,\nthe allowed region becomes discrete, and the α2-cluster center may be confined around the allowed ˜ svalues, ˜sj=\nπ(2j−1)/2˜kF(j= 1,···,2˜kF).\nFor a possible wave function for α2, one may consider a superposition of wave functions with ˜ sjas\nψα2=/summationdisplay\njF(˜sj)ψsjα2. (23)\nAs seen in the density distribution ρχ\ns(˜x) of two cluster states shown in Figs. 6 and 7, the density oscillation s hows\nthe 2˜kF+1 periodicity for any ˜ sjvalues.\nThe density oscillation can be remarkable provided that the amplitude F(˜sj) forj=˜kF(nearest ˜sjto the middle\npoint ˜x=π) is relatively larger than those for ˜ sjaround ˜s= 0 and 2πbecause of the effective repulsion between\nclusters due to the Pauli blocking effect. In other words, possible lo calization of the amplitude F(˜sj) because of the\nPauli blocking effect causes the spatial correlation between cluste rs (inter-cluster correlation). In such the case, the\ndensity oscillation shows the clear 2 ˜kF+ 1 periodicity whose origin is the DW-type correlations. Indeed, the state\ncontains dominantly the coherent 1 p-1hcomponents of a ±˜kF+1 particle and a ±˜kFhole on the Fermi surface at\n˜kF. The 1p-1hcorrelation carries the 2 ˜kF+1 momentum and causes the 2 ˜kF+1 periodicity. This correlation is the\nsimilar to that of the DW phase. In the weak coupling approximation, t he spatial wave function Eq. (11) for an α\ncluster is approximated by the dominant ˜kFcomponent with a minor mixing of the ˜kF+1 component as\nφ0(x) =1√\n2L/parenleftbig\ncos˜kF˜x+ǫcos(˜kF+1)˜x/parenrightbig\n. (24)\nIn this approximation, it can be proved that the 2 αwave function Ψ sforj=˜kFactually contains the dominant\nDW-type 1p-1hcorrelation in the particle-hole representation (See appendix B).\nIn the opposite case that there exists attractive force between clusters, the amplitudes F(˜sj) may gather to smaller\n˜sj. It corresponds to the exciton(Exc)-type correlations describ ed by the coherent 1 p-1hcomponents of a ±˜kF+ 1\nparticle and a ∓˜kFhole on the Fermi surface as described in appendix B. In this case, t he 1p-1hcarriesthe momentum\n|˜kF+ 1∓˜kF|= 1, which suggests that spatial density oscillation is not so remarka ble. Indeed, such the feature is\nseen in the weaker density oscillation in ρχ\ns(˜x) for ˜s= ˜sj(j= 1) shown in Fig. 7 (dashed lines in the right panels) for\n˜kF= 2. We should comment that the system for kF= 1 is a special case that the small ˜ sregion, for instance, the\nregion ˜s<π/4 is forbidden and the Exc-type correlations are suppressed.\n3. BCS-like state\nWith further increase of ˜b, the˜kFcomponent becomes more dominant. In the case f(˜k)≪f(˜kF) for˜k >˜kF, the\nstructure of the Pauli allowed area for α2approaches the pure periodic one following the periodicity of the α1density\nρα1(˜x)≈cos2(˜kF˜x)/√π. Then, a linear combination of ψsj\nα2with an equal weight may be the lowest state to restore12\nthe translationalinvariance because Ψ sfor different ˜ sjmay degenerate energetically. It means that the state no longer\nhas the spatial correlation between clusters (inter-cluster corr elation) and it corresponds to a BCS-like state. Namely,\nthe BCS-like state has the weak inner correlation and no inter-clust er correlation.\nAs described in appendix B, in the weak coupling approximation, the to tal c.m.m. momentum KG= 0 state\nprojected from the equal weight linear combination of two cluster w ave functions Ψ sis equivalent to the BCS-like\nstate containing 2 p-2hconfigurations of a ( ˜kF+1,−˜kF−1)χαχβparticle pair and a ( ˜kF,−˜kF)χαχβhole pair [see\nEq. (C5)]. In the 2 p-2hstate, all kinds of pairing is coherently mixed so as to keep the spin-is ospin symmetry of α\nclusters.\n4. Fermi-gas state and correlations\nIn the large ˜blimit, excited components of ˜k >˜kFvanish and the system goes to the Fermi gas (FG) state with\nthe Fermi surface at ˜kF. Needless to say, the FG state has no inner correlation nor inter-c luster correlation. This is\nnothing but the uncorrelated state. On the other hand, the corr elated states are characterized by configurations of\nexcited˜k >˜kFcomponents. In the weak coupling regime, we can recognize DW-like, Exc-like, BCS-like states, or\nmixing of them by correlations in 1 p-1hand 2p-2hconfigurations on the Fermi surface.\n5. Diagram of structure transitions\nAs discussed above, in the present schematic model for two cluste rs on the Fermi gas core in the 1D box, the cluster\nstate is expected to show the BEC-like, the DW/Exc-like, BCS-like, o r FG behaviors depending on the cluster size ˜b\nand the lowest allowed momentum ˜kF. We here assume the criterion to judge a correlation type for a wav e function\nwith given ˜band˜kFvalues, and show a diagram of structure transitions on the ˜b-˜kFplane.\nFor the criterion, we define ∆ ˜kby the deviation of ˜kfrom the lowest allowed momentum ˜kF.\n(∆˜k)2=/summationdisplay\n˜k≥˜kFf2(˜k)(˜k−˜kF)2. (25)\nIt indicates the deviation from the FG state |HF∝an}bracketri}ht. In the case ∆ ˜k≪1,˜k≥˜kF+2 components are negligible and\nthe coefficient ǫ≡f(˜kF+1) of the ˜kF+1 component approximates ǫ≈∆˜k. In the ∆ ˜k= 0 limit, the system goes\nto the FG state. For the criterion to classify a wave function with giv en˜band˜kFto a correlation type, we use the\ndeviation ∆ ˜kand the Pauli allowedness N4\nPB(˜s) at the middle point of the box ˜ s=˜L/2 =π.\nIn Table I, we list the criterion for various correlation types. In the table, we also show the typical examples of the ˜b\nvalues for ˜kF= 1 and ˜kF= 2 that satisfy the criterion. The densities and the Pauli allowednes s for the corresponding\nstates are already shown in Figs. 6 and 7. The BEC-like state is expec ted to appear when the Pauli allowed region is\nwidely open. Then, we adopt the condition N4\nPB(˜s=π)>0.8 for the BEC-like state. The DW-like and/or Exc-like\nstates may appear when the Pauli allowed region is restricted. For t his condition, we use N4\nPB(˜s=π)<0.1. The\nDW/Exc-like states may change to the BCS-like state when the α1-cluster density ρα1(x) approaches cos2(˜kFx)/√π\nand the Pauli allowed area for α2becomes almost periodic. For the criterion that ρα1(˜x)≈cos2(˜kF˜x)/√π, we adopt\nthe ratio of the α1density at ˜x= 0 to that at ˜ x=˜L/2 =π. In the weak coupling regime of ∆ ˜k≪1, the ratio\nρχ\nα1(˜x= 0)/ρχ\nα1(˜x=π) is approximately given by 1 −4ǫ≈1−4∆˜k, therefore, we use ∆ ˜kfor the measure. We apply\n∆˜k >0.05 for the DW/Exc-like states. This approximately corresponds to the ratioρχ\nα1(˜x= 0)/ρχ\nα1(˜x=π)<0.8.\nWith the decrease of ∆ ˜k, the 2αwave function may gradually change to the FG state via the BCS-like s tate. Since\nthe higher momentum components ˜k >˜kFnearly equals to (∆ ˜k)2in the weak coupling regime, we use ∆ ˜kas the\nmeasure for structure transitions from the DW/Exc-like to the FG state as listed in the table. For instance, the\ncondition ∆ ˜k<0.001 for the FG state indicates the contamination of ˜k>˜kFcomponents is less than 10−6.\nThe diagram of the structure transitions between the FG state, B CS-like state, DW/Exc-like state, and BEC-like\nstate on the ˜kF-˜bplane is shown in Fig. 8.\nFor a system with higher ˜kF, nucleons in a cluster couple more weakly to each other because of P auli blocking\nfrom core nucleons, and therefore the FG region is wider in the diagr am. However, one should care about that the\nassumption of the sharp surface for the core Fermi momentum mig ht be inadequate, in particular, in case of high\n˜kF. The core Fermi surface may diffuses in correlated states. If the surface diffuses, the lower orbitals below ˜kFare\npartially allowed for nucleons in clusters. Then, the weakening of the cluster localization by the Pauli blocking from\ncore nucleons can be quenched. The present model should be exte nded by incorporating the surface diffuseness of the\ncore Fermi surface.13\nTABLE I: For the criterion to classify the 1D-cluster wave fu nction with given ˜band˜kFinto various types of correlation. The\ndeviation ∆ kand the Pauli allowedness N4\nPB(˜s=˜L/2 =π) are used for the criterion. The typical examples of the ˜bvalues for\n˜kF= 1 and ˜kF= 2 that satisfy the criterion are also shown in the table.\ncorrelation criterion example\n˜kF= 1˜kF= 2\nFG ∆ ˜k <0.001 ˜b= 2.0\nFG-BCS crossover 0 .001≤∆˜k <0.005 ˜b= 2.0˜b= 1.5\nBCS-like 0 .005≤∆˜k <0.01\nBCS-DW/Exc crossover 0 .01≤∆˜k <0.05 ˜b= 1.5˜b= 1.25\nDW/Exc-like 0 .05≤∆˜kandN4\nPB(˜L/2)<0.1˜b= 1.0˜b= 0.75\nDW/Exc-BEC crossover 0 .1≤ N4\nPB(˜L/2)<0.8˜b= 0.5˜b= 0.5\nBEC-like 0 .8<N ∗4\nPB(˜L/2) ˜b= 0.25˜b= 0.25\nWe alsoshould comment that, in a real system, twoparameters ˜band˜kFareuncontrollable in general. As explained\nbefore,˜bindicates the cluster size relative to the box size (the cluster densit y), while ˜kFrelates to the core density.\nThe size of clusters on the Fermi gas core should be determined dyn amically as a consequence of nuclear interactions\nbetween constituent nucleons in clusters. It should be also affecte d by the neighboring cluster, i.e., cluster density\nas well as ˜kF. Moreover, the ratio of cluster nucleons to the core nucleons sho uld be determined dynamically. The\nextension of the present model by taking into account dynamical c hange of cluster structure with the use of effective\nnuclear interactions is an important issue to be solved in future stud y.\nF. Correspondence to finite nuclei\nAs shown in the AMD+VAP calculation, three αclusters are formed in the ground state of12C even though\nthe existence of any clusters is not a priori assumed in the framework. Once three αclusters are formed, it can\nbe associated with the schematic 1D-cluster model. Angular correla tion between two αclusters around the αcore\ncorresponds to the spatial correlation between two αclusters in the 1D-cluster model with ˜kF= 1. The cluster size\nbis 1.62 fm for ν= 0.19 fm−2used in the AMD calculations. If we adopt the r.m.s. matter radius rα= 1.72 fm\nof the core αas the radial size r0, the dimensionless ˜b=b/r0is estimated to be ˜b= 0.94. Or if we use the r.m.s.\nradius of cluster positions of 3 αs evaluated from r2\n0+r2\nα=R2\n0, we get ˜b= 0.87. In both cases, ˜b∼1. As already\ndescribed, the state with ( ˜kF,˜b) = (1.0,1.0) in the 1D-cluster model shows the remarkable density oscillation w ith\nthe wave number three in the DW-like regime. It is consistent with the intrinsic structure with the12C(0+\n1) obtained\nwith the AMD+VAP calculation. Indeed, the12C(0+\n1) has the (Y−3\n3−Y+3\n3)/√\n2 component and the surface density\nshown in Fig. 2 is similar to the oscillating density in the 1D-cluster state for (˜kF,˜b) = (1.0,1.0) in Fig. 6.\nAs discussed in Ref. [37], the αcorrelation in the pentagon ground state of28Si can be interpreted as the density\nwave on the sd-shell oblate state. The28Si ground state is associated with the 1D-cluster model with ˜kF= 2\nconsidering a core consisting of the spherical16O and four nucleons in oblate orbits. Then, the pentagon shape can\nbe understood again by the DW-like state in the 1D-cluster model wa ve function with the 2 ˜kF+1 = 5 periodicity.\nIn case of the αcorrelation in the16O ground state, the tetrahedron shape can not be connected dir ectly to the 1D\nproblem. However, when we focus only on the ( Y−3\n3−Y+3\n3)/√\n2 component of the intrinsic density in the16O ground\nstate, the density oscillation is characterized by the wave number t hree periodicity similar to that of the triangle\nshape in12C(0+\n1) and the deformation feature is associated with the DW-type corr elation in the 1D cluster model as\nin12C.\nV. SUMMARY AND OUTLOOK\nWe investigated α-cluster correlations in the ground states of12C and16O while focusing on the surface density\noscillation in the intrinsic states. The intrinsic states of12C and16O obtained by the AMD+VAP method show\ntriangle and tetrahedral shapes, respectively, because of the αcorrelations. The formation of αclusters in these\nstates was confirmed in the AMD framework, in which existence of an y clusters are not a priori assumed. The\nintrinsic deformations are regarded as spontaneous symmetry br eaking of rotational invariance. It was shown that14\nthe oscillating surface density in the triangle and tetrahedral shap es is associated with that in DW states caused by\nthe instability of Fermi surface with respect to a kind of 1 p-1hcorrelations.\nTo discuss the symmetry breaking between uniform density states and the oscillating density states, a schematic\nmodel of a few clusters on a Fermi gas core in a one-dimensional finit e box was introduced. In the model analysis,\nwe conjecture structure transitions from a Fermi gas state to a DW-like state via a BCS-like state, and to a BEC-like\nstate depending on the cluster size relative to the box size.\nIn both analyses of the BB-cluster model and the schematic 1D-clu ster model, Pauli blocking effects are found\nto play an important role in the DW-like state. The breaking of the tra nslational invariance in the DW-like state\noriginates in the Pauli blocking effect between clusters, which acts a s an effective inter-cluster repulsion and restricts\ncluster motion.\nIn the present analysis with the schematic 1D-cluster model, we do n ot perform energy variation nor calculate\nenergy eigenstates. Moreover, the mechanism of cluster format ion and the dynamical change of the cluster structure\nor the cluster size are beyond our scope in the present paper. The extension of the present model by taking into\naccount dynamical change of cluster structure with the use of eff ective nuclear interactions is an important issue to\nbe solved in future study. Also the cluster and core formations as w ell as diffuseness of the core Fermi surface should\nbe studied in more realistic models. Furthermore, the assumption th at the interaction between clusters is weak and\nthe Pauli blocking effect gives the major contribution to the inter-c luster motion may be too simple. To clarify which\nstate of BCS-like, DW-like, or EXc-like ones realizes it is essential to e xplicitly solve the problem of inter-cluster\nmotion by taking into account nuclear forces or inter-cluster inter actions.\nIt would be interesting to associate the present picture for cluste rs in the 1D finite box with phase transitions in\ninfinite matter. In the extensionofthe presentmodel toinfinite ma tter problem, oneshouldtakecareofthe differences\nbetween finite systems and infinite systems as follows. Firstly, mome ntumkvalues in a finite box is discrete because of\nboundary condition, while they are continuum values in infinite matter . In the description with discrete momentum,\nlong range correlations beyond the box size Lis not taken into account. What we call the BCS-type correlation in\nthe present 1D-cluster model is the correlation in the range of the box size at most. In second, the total momentum\nof c.m.m. should be projected to zero in finite system, while it is not nec essarily zero in infinite systems. In spite of\nthose differences, the 1D-cluster model may give a hint to underst and an origin of DW in infinite matter.\nAcknowledgments\nThe authors thank to nuclear theory group members of departme nt of physics of Kyoto University for valuable\ndiscussions. Discussions during the YIPQS long-term workshop ”DC EN2011” held at YITP are helpful to advance\nthis work. The computational calculations of this work were perfor med by using the supercomputers at YITP. This\nwork was supported by Grant-in-Aid for Scientific Research from J apan Society for the Promotion of Science (JSPS)\nGrant Number [No.23340067, 24740184]. It was also supported by t he Grant-in-Aid for the Global COE Program\n“The Next Generation of Physics, Spun from Universality and Emerg ence” from the Ministry of Education, Culture,\nSports, Science and Technology (MEXT) of Japan.\nAppendix A: Triangle and tetrahedral deformations in the 3α- and4α-cluster structures\nWe here describe triangle and tetrahedral deformations and part icle-hole representations of 3 α- and 4α-cluster\nwave functions. As explained in Ref. [37], the Brink-Bloch (BB) 3 α-cluster wave functions [53] with a regular triangle\nconfiguration in the small inter-cluster distance case can be rewrit ten as,\nΦBB\n3α(ǫ)≈/productdisplay\nτσ{φ00Xτσ(φ1−1+ǫφ2+2)Xτσ(φ1+1−ǫφ2−2)Xτσ}. (A1)\nHereφlmis the single-particle orbit in the harmonic oscillator potential, Xτσis the spin-isospin wave function with\nτ={p,n}andσ={↑,↓}.ǫis the order parameter for the axial symmetry breaking and it corr esponds to u/v\nin the BCS theory. Here ǫis considered to be small to omit ǫ2and higher terms and taken to be a real value. In\nprinciple,ǫcan be a complex value but the phase is arbitrary and can be changed by the rotation around the zaxis\n(transformation of the polar coordinate ϕ). The density of the ΦBB\n3α(ǫ) state is\nρ(r) =4\nπ3/2b3e−r2\nb2/braceleftbigg\n1+2r2\nb2sin2(θ)+ǫ2√\n2r3\nb3sin3(θ)(e−3iϕ−e3iϕ)+O(ǫ2)/bracerightbigg\n, (A2)15\n 0 0.5 1 1.5 2 2.5\n 0 0.5 1 1.5 2density ρ~\nx~  (π) 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0  0.1  0.2  0.3 \n 0 0.5 1 1.5 2density ρ~\nx~  (π)\n 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=1\nb~=1.5\n 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0  0.1  0.2  0.3 \n 0 0.5 1 1.5 2density ρ~\nx~  (π)\n 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=1\nb~=1.25\n 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0  0.1  0.2  0.3  0.4\n 0 0.5 1 1.5 2density ρ~\nx~  (π)\n 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=1\nb~=1.0\n 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0  0.1  0.2  0.3  0.4  0.5\n 0 0.5 1 1.5 2density ρ~\nx~  (π)\n 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=1\nb~=0.75\n 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0 0.5 1 1.5 2density ρ~\nx~  (π)\n 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=1\nb~=0.5\n 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0 0.2 0.4 0.6 0.8 1 1.2\n 0 0.5 1 1.5 2density ρ~\nx~  (π)\n 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=1\nb~=0.25\n 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=1\nb~=2.0\nFIG. 6: (Color on-line) The results of one and two clusters on the Fermi gas core in a one-dimension box. Left: The coefficien ts\nf(˜k) of the Fourier transformation. Middle: The density ρχ\nα1(˜x) of the first αlocated around ˜ x= 0 (red thin lines), the PB\neffectN4\nPB(˜s) for the second α(black solid lines). NPB(˜s) is also shown (blue dashed lines). Right: The density ρχ\ns(˜x) (black\nsolid lines). The ˜ sfor the center position of α2is chosen to be ˜ sjwithj=˜kF, which is the closest ˜ sjto ˜x=˜L/2 =πamong\nthe allowed ˜ svalues. The density ρχ\nα1(˜x) is also shown for comparison(red thin lines). The results f or the cluster size ˜b= 2.0,\n1.5, 1.25, 1.0, 0.75, 0.5, 0.25 are shown.\nand its multipole decomposition at a certain radius r=R0is\nρ(R0,θ,ϕ) =8\nπ1/2b3e−R2\n0\nb2/braceleftbigg\n(1+4\n3R2\n0\nb2)Y0\n0(θ,ϕ)−4\n3√\n5R2\n0\nb2Y0\n2(θ,ϕ)+ǫ8√\n35R3\n0\nb3/parenleftbiggY−3\n3(θ,ϕ)√\n2−Y+3\n3(θ,ϕ)√\n2/parenrightbigg\n+O(ǫ2)/bracerightbigg\n.\n(A3)\nIn a similar way, the BB 4 α-cluster wave functions with a regular tetrahedral configuration can be rewritten as\nΦBB\n4α(ǫ)��/productdisplay\nτσ{φ00Xτσφ10Xτσ(φ1−1+ǫφ2+1)Xτσ(φ1+1+ǫφ2−1)Xτσ}. (A4)\nHere it is assumed that the inter-cluster distance in the tetrahedr al configuration is small, i.e., ǫis small. The density16\n 0  0.4  0.8  1.2  1.6  2\n 0 0.5 1 1.5 2density ρ~\nx~  (π) 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0  0.1  0.2  0.3 \n 0 0.5 1 1.5 2density ρ~\nx~  (π)\n 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=2\nb~=1.5\n 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0  0.1  0.2  0.3 \n 0 0.5 1 1.5 2density ρ~\nx~  (π)\n 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=2\nb~=1.25\n 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0  0.1  0.2  0.3 \n 0 0.5 1 1.5 2density ρ~\nx~  (π)\n 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=2\nb~=1.0\n 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0  0.1  0.2  0.3  0.4\n 0 0.5 1 1.5 2density ρ~\nx~  (π)\n 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=2\nb~=0.75\n 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0 0.5 1 1.5 2density ρ~\nx~  (π)\n 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=2\nb~=0.5\n 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0  0.2  0.4  0.6  0.8 \n 0 0.5 1 1.5 2density ρ~\nx~  (π)\n 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=2\nb~=0.25\n 0 0.2 0.4 0.6 0.8 1\n 0 0.5 1 1.5 2NnPB, ρ~\nx~,  s~  (π) 0 0.2 0.4 0.6 0.8 1\n 0 5 10 15 20f(k~)\nk~k~\nF=2\nb~=2.0\nFIG. 7: Same as Fig. 6 but for ˜kF= 2. The blue dashed lines indicate the the density ρχ\ns(˜x) for the smallest allowed ˜ svalue,\n˜sjwithj= 1. The results for the cluster size ˜b= 2.0, 1.5, 1.25, 1.0, 0.75, 0.5, 0.25 are shown.\nis\nρ(r) =4\nπ3/2b3e−r2\nb2/braceleftbigg\n1+2r2\nb2sin2(θ)+ǫ√\n2r3\nb3sin2(θ)(e2iϕ+e−2iϕ)+O(ǫ2)/bracerightbigg\n, (A5)\nand its multipole decomposition at a radius r=R0is\nρ(R0,θ,ϕ) =8\nπ1/2b3e−R2\n0\nb2/braceleftBigg\n(1+2R2\n0\nb2)Y0\n0(θ,ϕ)+ǫ/radicalbigg\n32\n105R3\n0\nb3(Y−2\n3(θ,ϕ)√\n2+Y+2\n3(θ,ϕ)√\n2)/bracerightBigg\n. (A6)\nY−2\n3(θ,ϕ)/√\n2+Y+2\n3(θ,ϕ)/√\n2 in theǫterm can be also transformed to\nY−2\n3(RΩθ′,RΩϕ′)√\n2+Y+2\n3(θ′,ϕ′)√\n2=√\n5\n3Y0\n3(θ,ϕ)+√\n2\n3Y−3\n3(θ,ϕ)−√\n2\n3Y+3\n3(θ,ϕ), (A7)17\n 0 0.5 1 1.5 2\n 0 1 2 3 4 5 6 7 8b~\nk~\nFBEC-likeDW/ExcFermi gas\n∆k~=0.001\n∆k~=0.05BCS-like\nFIG. 8: Diagram of structure transitions between the Fermi g as state, BCS-like state, DW/Exc-like state, and BEC-like s tate\nin the schematic 1D-cluster model. The criterion for the bou ndary are listed in I. The conditions of N4\nPB(˜L/2) = 0.1 and 0.8\nare shown by the solid lines, while the conditions of ∆ ˜k= 0.001,0.005,0.01,0.05 are shown by the dashed lines.\nby a Ω rotation RΩ.\nBy using the creation and annihilation operators, a†\nlm,χandalm,χfor theφlmXτσstate withχ=τσ, the ΦBB\n3α(ǫ)\nand ΦBB\n4α(ǫ) states are expressed as\n|ΦBB\n3α(ǫ)∝an}bracketri}ht=/productdisplay\nχa†\n00,χ(a1−1,χ+ǫa†\n2+2,χ)(a1+1,χ−ǫa†\n2−2,χ)|−∝an}bracketri}ht, (A8)\n|ΦBB\n4α(ǫ)∝an}bracketri}ht=/productdisplay\nχa†\n00,χa†\n10,χ(a1−1,χ+ǫa†\n2+1,χ)(a1+1,χ+ǫa†\n2−1,χ)|−∝an}bracketri}ht. (A9)\nIn the particle and hole representation, they are rewritten as\n|ΦBB\n3α(ǫ)∝an}bracketri}ht=/productdisplay\nχ(1+ǫa†\n2+2,χb†\n1+1,χ)(1−ǫa†\n2−2,χb†\n1−1,χ)|0∝an}bracketri}htF, (A10)\n|0∝an}bracketri}htF≡/productdisplay\nχ/parenleftBig\na†\n00,χa1−1,χa1+1,χ/parenrightBig\n|−∝an}bracketri}ht, (A11)\nand\n|ΦBB\n4α(ǫ)∝an}bracketri}ht=/productdisplay\nχ(1+ǫa†\n2+1,χb†\n1+1,χ)(1+ǫa†\n2−1,χb†\n1−1,χ)|0∝an}bracketri}htF, (A12)\n|0∝an}bracketri}htF≡/productdisplay\nχ/parenleftBig\na†\n00,χa†\n10,χa1−1,χa1+1,χ/parenrightBig\n|−∝an}bracketri}ht. (A13)\nHere the hole operator b†\nlm,χis defined to be b†\nlm,χ=al−m,χ. Equations (A10) and (A12) indicate that the 3 α- and\n4α-cluster wave functions contains the DW-type 1 p-1hcorrelations carrying finite angular momenta.\nAppendix B: Weak coupling regime of one-dimensional cluste r model\nWe consider the weak-coupling case of the 1D-cluster model descr ibed in IV. Hereafter, we take r0= 1 and consider\nthe case that all the parameters equal to the dimensionless param eters, for instance, b=˜b,k=˜k, ands= ˜s. We\ndefine the creation and annihilation operators a†\nkχanda†\n−kχfor the single-particle states φkXχandφkXχ, where\nφkandφ−kare momentum kand−kplane waves written as φk(x) =e+ikx/√\n2Landφ−k(x) =e−ikx/√\n2Lin the\ncoordinate representation. The core state |C∝an}bracketri}htwhere allk<kFstates are occupied by core nucleons is written as\n|C∝an}bracketri}ht=/productdisplay\nχ/productdisplay\n|k|<kFa†\nkFχa†\n−kFχ|−∝an}bracketri}ht, (B1)\nwhere|−∝an}bracketri}htis the vacuum.18\nIn the case of weak coupling, the coefficient f(k) of the momentum kand−kplane waves in Eq. (9) is dominant\nfork=kFand it rapidly decreases as kincreases. In the limit, |f(kF)| ≫ |f(kF+1)| ≫ |f(kF+2)|···, the leading\nterm of the 2 α-cluster state in the 1D-cluster model is the Fermi gas state with t he surface momentum kF, which is\nthe uncorrelated state. For the correlated system, the deviatio n from the Fermi gas state is important. We take into\naccount the next leading term, k=kF+1 components, and ignore the higher momentum components for k≥kF+2.\nWe setf(kF) = 1 andf(kF+1) =ǫwithǫ<<1 and take the leading term of ǫ. Then the state consisting of two α\nclusters around x= 0 andx=swritten by Eqs. (9) and (14) on the Fermi gas core is expressed as\n|ψα1ψs\nα2C∝an}bracketri}ht ∝/productdisplay\nχ/braceleftBig\n(c∗\n0−c0)a†\nkFχa†\n−kFχ\n+ǫ(c0−c1)a†\nk′\nFχa†\nkFχ+ǫ(c∗\n0−c∗\n1)a†\n−k′\nFχa†\n−kFχ\n+ǫ(c∗\n0−c1)a†\nk′\nFχa†\n−kFχ+ǫ(c0−c∗\n1)a†\n−k′\nFχa†\nkFχ\n+ǫ2(c∗\n1−c1)a†\nk′\nFχa†\n−k′\nFχ+···/bracerightBig\n|C∝an}bracketri}ht, (B2)\nwherek′\nF=kF+1,c0(s) =e−ikFs, andc1(s) =e−ik′\nFs. The first term is the Fermi gas state with the Fermi surface\natkF. Its coefficient c∗\n0−c0= 2isin(kFs) vanishes for s=πm/kF(mis an integer) because of Pauli principle. In\nother words, the area around s=π2m/2kF(m= 0,···,2kF) is forbidden for the α2cluster center position, while\nthe area around sj=π(2j−1)/2kF(j= 1,···,2kF) is allowed.\nIn the particle-hole representation with respect to the Fermi sur face atkF, the second and third terms correspond\nto the 1p−1hstates with the DW-type correlation with the wave number k′\nF+kF= 2kF+ 1, and the forth and\nfifth terms are those with the Exc-type correlation with the wave n umberk′\nF−kF= 1, and the last term is the 2 p-2h\nstate. We rewrite the coefficients in |ψα1ψs\nα2C∝an}bracketri}htas\n|ψα1ψs\nα2C∝an}bracketri}ht=n2\n0/productdisplay\nχC0(s)/braceleftBig\na†\nkFχa†\n−kFχ\n+ǫCDW(s)a†\nk′\nFχa†\nkFχ+ǫC∗\nDW(s)a†\n−k′\nFχa†\n−kFχ\n+ǫCExc(s)a†\nk′\nFχa†\n−kFχ+ǫC∗\nExc(s)a†\n−k′\nFχa†\nkFχ\n+ǫ2C2(s)a†\nk′\nFχa†\n−k′\nFχ+···/bracerightBig\n|C∝an}bracketri}ht, (B3)\nC0(s)≡c∗\n0−c0=−2isin(kFs), (B4)\nCDW(s)≡c0−c1\nc∗\n0−c0=e−is−1\n2, (B5)\nCExc(s)≡c∗\n0−c1\nc∗\n0−c0=e−is+1\n2, (B6)\nC2(s)≡c∗\n1−c1\nc∗\n0−c0= cos(is). (B7)\nBy using the hole operator b†\nkχ=a−kχwe obtain\n|ψα1ψs\nα2C∝an}bracketri}ht=n2\n0/productdisplay\nχC0(s){1\n−ǫCDW(s)a†\nk′\nFχb†\nkFχ+ǫC∗\nDW(s)a†\n−k′\nFχb†\n−kFχ\n+ǫCExc(s)a†\nk′\nFχb†\n−kFχ−ǫC∗\nExc(s)a†\n−k′\nFχb†\nkFχ\n+ǫ2C2(s)a†\nk′\nFχa†\n−k′\nFχb†\n−kFχb†\nkFχ+···/bracerightBig\n|HF∝an}bracketri}ht, (B8)\nwhere the Hartree-Fock vacuum |HF∝an}bracketri}ht ≡/producttext\nχa†\nkFχa†\n−kFχ|C∝an}bracketri}ht. Whensis the allowed sjclose toL/2 =π, i.e., the\nα2center is located around the middle of the box, e−isapproaches −1, and therefore, the DW correlation becomes\ndominant and the Exc correlation is minor, while they are opposite for sjclose to 0. In both cases, the correlation\ndisappears in the ǫ→0 limit, and the system goes to the Fermi gas limit.19\nAppendix C: Transition between DW/Exc-like and BCS-like st ates\nWe extend the model in the weak coupling limit described in the previous section by superposing all allowed sj\nstates for the α2wave function,\nψα2=/summationdisplay\njF(sj)ψsj\nα2, (C1)\n|Ψ∝an}bracketri}ht=/summationdisplay\njF(sj)|ψα1ψsj\nα2C∝an}bracketri}ht. (C2)\nIn principle, one should determine the coefficients F(sj) following the energy variation to properly take into account\ndynamics of relative motion between clusters, α1andα2. However, we here do not perform energy optimization but\nadopt a simple ansatz for F(sj) as follows.\nAs described before, if the sj∼L/2 (sj∼0) components are main, the DW (Exc) correlation is dominant. In mo re\ngeneral, in case that the effective interaction between αclusters is repulsive, the amplitudes may be relatively smaller\nin the region around sj∼0 (and also around sj∼Ldue to the periodic boundary) than that around sj=L/2 =π\nresulting in the DW dominance. On the other hand, if the interaction b etweenαclusters is attractive, the amplitudes\nconcentrate on the region around sj∼0, which may cause the Exc dominance. In both cases, spatial corr elation\nbetweenα1andα2remains and the system shows density oscillation and the symmetry b reaking of translational\ninvariance.\nNext we construct the state where there is no spatial correlation betweenα1andα2(no inter-cluster correlation)\nand the translational invariance is completely restored by the supe rposition and the KG= 0 (total momentum of\nc.m.m.) projection of the |ψα1ψsj\nα2C∝an}bracketri}ht. The state with no spacial correlation between α1andα2can be constructed\nby the projection to the relative momentum kα1−kα2= 0 state. Here kα1,2stands for the momentum of c.m.m. of\nα1,2. In the weak coupling regime, this is performed by superposing all allo wedsjstates with an equal weight as\nψα2=/summationdisplay\nj1\n(−2i(−1)j)nψsj\nα2. (C3)\nTaking averages of CDW(j),CExc(j) andC2(j) with respect to j, we obtain\n|ψα1ψα2C∝an}bracketri}ht ∝/productdisplay\nχ/braceleftBig\n1−ǫ\n2(a†\nk′\nFχb†\nkFχ−a†\n−k′\nFχb†\n−kFχ)\n−ǫ\n2(a†\nk′\nFχb†\n−kFχ−a†\n−k′\nFχb†\nkFχ)+···/bracerightBig\n|HF∝an}bracketri}ht, (C4)\nThis corresponds to the intrinsic state for no correlation between clusters. In the ground state of the finite system, the\ntotal momentum KG=kα1+kα2should be zero. With the projection onto KG= 0, we finally obtain the symmetry\nrestored state,\nPKG=0|ψα1ψα2C∝an}bracketri}ht ∝\n\n1+ǫ2\n4/summationdisplay\nχαχβ(a†\nk′\nFχαa†\n−k′\nFχβ−a†\nk′\nFχβa†\n−k′\nFχα)(b†\nkFχαb†\n−kFχβ−b†\nkFχβb†\n−kFχα)+···\n\n|HF∝an}bracketri}ht.(C5)\nIt is found that the state contains 2 p-2hconfigurations and it is consistent with a BCS-like state, where two p articles\nform a (k′\nF,−k′\nF)χαχβpair and two holes do a ( kF,−kF)χαχβpair. 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Brink, International School of Physics “Enrico Fe rmi”, XXXVI, p. 247 (1966)."    },    {        "title": "1208.5710v2.First_Principle_Local_Density_Approximation_Description_of_the_Electronic_Properties_of_Ferroelectric_Sodium_Nitrite.pdf",        "content": "First Principle Local Density Approximation Description of the Electronic\nProperties of Ferroelectric Sodium Nitrite\nC. E. Ekuma, M. Jarrell, and J. Moreno\nDepartmentof Physics & Astronomy, and Center for Computation & Technology, Louisiana\nState University (LSU)\nBaton Rouge, Louisiana 70803, USA\nL. Franklin, G. L. Zhao, J. T. Wang, and D. Bagayoko\nDepartment of Physics, Southern University and A&M College in Baton Rouge (SUBR) Baton\nRouge, Louisiana 70813, USA\nThe electronic structure of the ferroelectric crystal, NaNO 2, is studied by means of first-\nprinciples, local density calculations. Our ab-initio, non-relativistic calculations employed a\nlocal density functional approximation (LDA) potential and the linear combination of atomic\norbitals (LCAO). Following the Bagayoko, Zhao, Williams, method, as enhanced by Ekuma and\nFranklin (BZW-EF), we solved self-consistently both the Kohn-Sham equation and the equation\ngiving the ground state charge density in terms of the wave functions of the occupied states. We\nfound an indirect band gap of 2.83 eV, from W to R. Our calculated direct gaps are 2.90, 2.98,\n3.02, 3.22, and 3.51 eV at R, W, X, \u0000, and T, respectively. The band structure and density of\nstates show high localization, typical of a molecular solid. The partial density of states shows\nthat the valence bands are formed only by complex anionic states. These results are in excellent\nagreement with experiment. So are the calculated densities of states. Our calculated electron\neffective masses of 1.18, 0.63, and 0.73 m oin the \u0000-X,\u0000–R, and \u0000-W directions, respectively,\nshow the highly anisotropic nature of this material.\nPacs Numbers: 77.84.-s, 71.20.-b, 71.20.Mq, 71.20.Nr\nI. INTRODUCTION AND MOTIVATION\nSodium nitrite is one of the ABO 2group with several interesting properties such as ferroelec-\ntricity, paraelectricity and piezoelectricity. The ground state of NaNO 2has one of the simplest\nstructures in the ABO 2group. Since the discovery of its ferroelectric property in 1958, by\nSawada et al. [1], there has been increased interest in diverse properties of NaNO 2, ranging from\nstructural changes to dielectric, electronic, thermal, electrical and elastic properties, as well as\nits non-linear optical spectra [2]. Despite this noted interest in ferroelectric NaNO 2, there has\nbeen very little study of the electronic band structure of the crystal [3].\nOne of the earliest electronic band gap measurements of ferroelectric NaNO 2was by Asao\net al. [4] who found band gaps of 2.3, 2.6, 3.1, and 3.7 eV for different current directions in\ntheir resistivity work. Also, using the same method, Takagi and Gesi [5] found a band gap of\n2.4 eV. These authors did not specify whether the gaps in question are direct or indirect. The\ninfrared absorption spectra study of Sidman [6] at a temperature of 77 Kgave a band gap of\n1arXiv:1208.5710v2  [cond-mat.mtrl-sci]  13 Oct 20123.22 eV while that of Verkhovskaya and Sonin [7], at 293.15 K, led to a value of 3.14 eV. The\nsemi-empirical LCAO work of El-Dib and Hassan [8], for room temperature (293.15 K), found\na band gap of 3.24 eV. The scanning electron microscopy, diffuse reflectance spectroscopy, and\nelectrical measurements study of Balabinskaya et al. [9], at a temperature of 300 K, reported\nan absorption edge of 3.0 eV.\nOne of the earliest theoretical studies of the band structure of ferroelectric NaNO 2was the\nX\u000b(\u000b= 0.75) exchange calculation of Kam et al. [10] who found a direct band gap of 3.45\neV at \u0000and an indirect band gap of 2.0 eV. The Full-potential linear muffin-tin orbital (FP-\nLMTO) method of Ravindran et al. [2] found an indirect band gap of 2.2 eV. The generalized\ngradient approximation (GGA) work of Wang et al. [11] utilized a full-potential linearized\naugmented plane wave (FL-APW) method to calculate a band gap of less than 2.1 eV. The\nLDA approach of Zhuravlev and Korabel’nikov [12], using the nonlocal Troullier–Martins (TM)\npseudopotentials and the Slater exchange potential with a correlation correction, led to a direct\nband gap of 2.50 eV at W. Zhuravlev and Poplavnoi [13], also using nonlocal Troullier–Martins\n(TM) pseudopotentials and the Slater exchange ( \u000b= 1) potential with a correlation correction,\nfound a minimum forbidden band gap of 3.07 eV. The LDA result of Jiang et al. [14], using\nan orthogonalized linear combination of atomic orbitals (OLCAO) method, found an indirect\nband gap of 2.95 eV from S to Rsymmetry points. The periodic Hatree-Fock (PHF) method\nof McCarthy [15] overestimated the gap by a factor of 2-3. Henkel et al. [16] employed their\nplane-wave and Gaussian basis set and an X \u000bexchange and reported a direct band gap of 2.70\neV. These results suffice to see that theoretical calculations have not resolved the under- or\noverestimation of the band gap of ferroelectric NaNO 2.\nIn light of the preceding overview, the aim of this study is to attempt to obtain the mea-\nsured, fundamental band gap and other electronic properties of ferro-NaNO 2with ab-initio,\nself-consistent LDA calculations. The confirmation of our predictions of the band gaps and\nother properties for BaTiO 3[17], TiO 2[18], CdS [19], c-Si 3N4[20], c-InN [21], and SrTiO 3[22]\nis a basis for the above presumption. Further, the mathematical rigor of the method [18-24]\nsuffices to expect it to lead to much better results.\nII. METHOD OF CALCULATION\nOur calculations employed the Ceperley and Alder [25] local density functional potential as\nparameterized by Vosko, Wilk, and Nusair [26]. We implemented the linear combination of\natomic orbitals (LCAO), using Gaussian functions for the radial parts. We utilized a program\npackage developed at the Ames Laboratory of the US Department of Energy (DOE), in Ames,\nIowa [27]. Our calculations are non-relativistic and were performed using low temperature (120\nK) lattice constants. The distinctive feature of our approach resides in the implementation of the\nBagayoko, Zhao, and Williams (BZW) method, as enhanced by Ekuma and Franklin (BZW-EF),\nconsisting of concomitantly solving self-consistently two coupled equations as explained below.\nOne of these equations is the Schrödinger type equation of Kohn and Sham [28], referred to as\nthe Kohn-Sham (KS) equation. The second equation, which can be thought of as a constraint on\nthe KS equation, is the one giving the ground state charge density in terms of the wave functions\nof the occupied states.\n2The essentials of the method follow. Beginning with a minimum or small basis set capable of\naccounting for all the electrons in the system under study, one performs ab-initio, self-consistent\ncalculations. This minimum or small basis set is constructed using orbitals resulting from self-\nconsistent calculations for the atomic or ionic species in the solid. As noted below, these species\nare Na1+, N1+, and O2-for NaNO 2. Subsequently, this basis set is augmented with one orbital\nand self-consistent calculations are done. This new orbital and others that may be added later\nare outputs of the self-consistent studies of the ionic species. The occupied energies of the two\ncalculations for the solid are compared numerically and graphically. These occupied energies\nfrom Calculations I and II are generally different. A third calculation is carried out after adding\nanother “ionic” orbital to the basis set. We note that adding an orbital means augmenting the\nsize of the basis set by 2, 6, 10, or 14, depending on the s,p,d, orfcharacter of it, respectively.\nAgain, the occupied energies from Calculations II and III are compared. This process continues\nuntil the occupied energies from a calculation, i.e., N, are found to be identical to those from\nCalculation(N+1)immediatelyfollowingit, withinourcomputationaluncertaintiesof50meVor\nless. At that point, the calculations are completed and the results from Calculation N represent\nthe physical description of the system under study. The basis set for Calculation N is referred\nto as the optimal basis set. Calculation (N+1) and others, with larger basis sets that contain\nthe optimal one, provided linear dependency is avoided, reproduce the occupied energies from\nCalculation N. However, by virtue of the Rayleigh theorem [19,23], these calculations produce\nsome unoccupied eigenvalues that are lower than those from Calculation N, due to a non-trivial\nbasis set and variational effect [23].\nAll the self-consistent calculations known to us carry out the iterative procedure that involves\nboththeKohn-Shamequationandtheequationgivingthegroundstatechargedensity. However,\nthe methodical increase of the size of the basis set, as done in the BZW-EF method, entails\nchanges in the radial and angular characteristics of the basis set as well as an increase in its size.\nIt is in this sense that the BZW-EF method solves self-consistently the two equations in question\nto obtain, in a verified fashion, the minima of the occupied energies. The difference between\nour method and single trial basis set calculations may be best understood by noting that any\nserious radial, angular, or basis size deficiencies that may exist in a single trial basis set cannot\nbe remedied by the iterative process noted above. In the case of the BZW-EF method, such\ndeficiencies are corrected as the methodical augmentation of the basis set is carried out. The\nattainmentoftheminimaoftheoccupiedstatessignifiesthatthereisnodeficiencylefttocorrect.\nWhen a larger basis set that contains the optimal one is utilized to carry out a calculation, the\ncharge density, the potential, and the Hamiltonian resulting from self-consistency, as well as the\noccupied eigenvalues, are the same as the corresponding ones obtained with the optimal basis set.\nHence, the Rayleigh theorem applies to the outputs of this calculations whose occupied energies\nare the same as those obtained with the optimal basis set. However, some unoccupied energies\nfrom the larger basis set are generally lower than (or equal) to their corresponding ones obtained\nwith the optimal basis set. The lowering of these unoccupied energies cannot be ascribed to\nphysical interactions, as the Hamiltonian did not change. It results from the non-trivial and\nwell-defined basis set and variational effect.\nThe enhancement of the original BZW method is in the methodical increment of the basis set.\nThis enhancement leads to adding the polarization ( p, d, or f ) orbitals, for a given principal\nquantum number, before adding the spherically symmetric sorbital (see Table 1). These addi-\ntional unoccupied orbitals are needed to accommodate the reorganization of the electron cloud,\n3including possible polarization, in the crystal environment. For valence electrons in molecules\nto solids, polarization has primacy over spherical symmetry.\nWhile there are no specific rules governing the order of adding orbitals, our experience has\nbeen that orbitals often need to be added to the heaviest element before the others. Among\nthese others, preliminary studies of charge transfer serve as a significant guide. In the case of\nNaNO 2, even though oxygen is a little heavier than nitrogen, the neon configuration oxygen\napproaches after gaining 2 electrons intimate more stability than that of N1+. The ultimate\naim of the addition of orbitals (i.e., basis functions) is to allow an optimal reorganization of the\nelectronic cloud in the solid environment as compared to atomic or ionic ones.\nComputational details germane to a replication of our work follow. Ferroelectric sodium\nnitrite (ferro-NaNO 2) has an orthorhombic structure [29]. It is in the space group C20\n2v– IMM2,\nwith space group number 44 and Patterson symmetry Immm[30]. The ferro-NaNO 2unit cell\ncontains eight atoms: four (4) cations and four (4) anions whose positions are as indicated\nbetween parentheses: Na: (0,0.5881,0), (0.5,0.0881,0.5); N: (0,0.1224,0), (0.5,0.6224,0.5); and O:\n(0,0,0.1962), (0.5,0.5,0.6962), (0,0,0.8038), (0.5,0.5,0.3038).\nWe carried out five (5) different calculations utilizing five (5) different basis sets in search of\ntheoptimalone. Table2containsthebasissetsutilizedinthefive(5)self-consistentcalculations.\nMethodical increases of the basis set led to calculation IV as the one with the optimal basis set;\ni.e., the calculation yielding the minima of all the occupied energies. Calculation V does not\nlower any occupied energies as compared to Calculation IV. Hence, the electronic structure and\nrelated properties presented here were obtained with basis set IV, the optimal basis set.\nOur self-consistent computations were performed utilizing experimental lattice constants of\n3.518 Å, 5.535 Å and 5.382 Å for a,b, andc, respectively, as obtained at 120 K[31]. Neutral\ncharge calculations (Na0, N0and O0) were carried out. Calculated charges were found to be +1\nfor Na, +1 for N and -2 for O. Then, we performed self-consistent calculations for Na1+, N1+and\nO2-to get the input quantities for the calculations for NaNO 2. The radial parts of the atomic\nwave functions were expanded in terms of Gaussian functions, employing a set of even-tempered\nGaussian exponents.\nThe self-consistent atomic calculations provided trial atomic potentials for Na, N, and O,\nrespectively. These potentials were used to construct the initial potential for ferro-NaNO 2.\nWe used 16 Gaussian functions for the sandpstates and 14 for the dstates for Na and\nN, respectively. We utilized 17 Gaussian functions for the sandpstates and 15 for the d\nstates for O. A mesh of 48 kpoints, with proper weights in the irreducible Brillouin zone,\nwas employed in the self-consistent calculations. A total of 121 weighted k-points were used in\nband structure calculations and a total of 147 weighted k-points were employed to generate the\nenergy eigenvalues for the electronic density of state calculation. The computational error for\nthe valence charge was about 0.073343 for 52 electrons, a little more than 10-3per electron. The\nself-consistent potentials converged to a difference around 10-5after about 60 iterations.\nIII. CALCULATED ELECTRONIC STRUCTURE\nOur calculated, ab-initio, self-consistent bands for ferro-NaNO 2(Calculation IV) are as shown\nin Fig. 1 and the comparison plot of the electron energy bands for basis set IV (solid lines) and\nbasis set V (dashed lines) are as shown in Fig. 2. As is apparent in this graph, the eigenvalues\n4of the occupied states totally converged, within computational errors of about 0.050 eV, showing\nclearly that Calculation IV is the one with the optimal basis set.\nFig. 1. The calculated electronic energy bands of ferro-NaNO 2, from Calculation IV.\nFig. 2. The calculated electronic energy bands of ferro-NaNO 2, from Calculations IV (full\nlines) and V (dashed lines). The occupied energies from Calculations IV and V are practically\nidentical.\n5Fig. 3.The calculated density of states (DOS) of ferro-NaNO 2, obtained with the BZW optimal\nbasis set (i.e. Calculation IV).\nFig. 4. The calculated partial density of states (pDOS) of ferro-NaNO 2as obtained with the\nBZW optimal basis set (i.e. Calculation IV).\n6Table 1. Search for the optimal basis sets (Orbital added is in bold), as per the BZW method,\nfor the description of the valence states of ferroelectric sodium nitrite (ferro-NaNO 2). The\noptimal basis set is that from Calculation IV. Na is Sodium, N is Nitrogen, and O is Oxygen.\nThe maximum of the Valence band occurred at W,while the minimum of the conduction band\noccurred at R.NaNO 2is an indirect band gap material.\nBasis\nSetNa, N\nand O 2\nCoreNa ValenceN\nValenceO2ValenceTotal No.\nof Valence\norbitalsBand Gap (eV)\nDirect at\n\u0000Indirect\nI1s 2s2p3s 2s2p 2s2p 34 2.40 2.38\nII1s 2s2p3s 3p2s2p 2s2p 40 2.70 2.53\nIII1s 2s2p3s3p 2s2p3s2s2p 42 3.18 2.93\nIV1s 2s2p3s3p 3d2s2p3s 2s2p 52 3.22 2.83\nV1s 2s2p3s3p3d 2s2p3s 2s2p3s 56 3.24 2.83\nTable 2. The Calculated Eigenvalues (in eV) at the High Symmetry Points for ferro-NaNO 2.\nTheeigenvaluesareobtainedbysettingtheenergyatthetopofthevalenceband, whichoccurred\natW,equal to zero.\nX\n-25.11 -25.11 -21.03 -21.03 -21.01 -21.01 -20.97 -20.97\n-20.05 -20.05 -9.24 -9.24 -6.73 -6.73 -6.21 -6.21\n-6.08 -6.08 -2.51 -2.51 -1.80 -1.80 -0.01 -0.01\n3.01 3.01 10.08 10.08 10.36 10.36 11.38 11.38\n11.79 11.79 13.96 13.96 14.12 14.12 14.32 14.32\n17.49 17.49 18.90 18.90 19.82 19.82 24.92 24.92\n30.26 30.26\n\u0000\n-25.11 -25.09 -21.13 -21.03 -21.02 -21.01 -20.89 -20.89\n-20.13 -20.07 -9.68 -9.28 -6.34 -6.32 -6.3 -6.16\n-6.00 -5.89 -2.04 -1.76 -1.70 -1.33 -0.51 -0.10\n3.12 3.51 6.66 8.59 9.31 9.61 10.31 10.91\n12.53 13.59 13.86 13.97 13.97 14.67 14.78 16.26\n716.47 18 18.84 20.17 20.43 20.86 20.96 21.97\n22.06 31.43\nR\n-25.15 -25.15 -21.00 -21.00 -20.99 -20.99 -20.92 -20.92\n-20.18 -20.18 -9.44 -9.44 -6.74 -6.74 -6.14 -6.14\n-6.08 -6.08 -2.38 -2.38 -1.47 -1.47 -0.07 -0.07\n2.83 2.83 9.18 9.18 10.12 10.12 11.02 11.02\n12.02 12.02 13.4 13.4 14.46 14.46 15.62 15.62\n15.73 15.73 17.96 17.96 22.35 22.35 24.19 24.19\n24.99 24.99\nW\n-25.15 -25.15 -21.02 -21.02 -21.00 -21.00 -20.98 -20.98\n-20.06 -20.06 -9.28 -9.28 -6.84 -6.84 -6.13 -6.13\n-6.08 -6.08 -2.41 -2.41 -1.79 -1.79 0 0\n2.98 2.98 10.31 10.31 10.39 10.39 10.93 10.93\n11.59 11.59 13.38 13.38 13.50 13.50 15.72 15.72\n17.68 17.68 18.34 18.34 19.96 19.96 24.09 24.09\n26.91 26.91\nT\n-25.14 -25.14 -21.08 -21.08 -21.01 -21.01 -20.89 -20.89\n-20.11 -20.11 -9.53 -9.53 -6.31 -6.31 -6.15 -6.15\n-6.06 -6.06 -1.91 -1.91 -1.52 -1.52 -0.23 -0.23\n3.28 3.28 7.33 7.33 9.74 9.74 10.95 10.95\n12.71 12.71 13.11 13.11 14.43 14.43 16.59 16.59\n17.99 17.99 18.98 18.98 19.60 19.60 20.71 20.71\n25.00 25.00\nTable 2 shows the energies at the high symmetry points. Figs. 3 and 4 show the calculated,\ntotal (DOS) and partial (pDOS) density of states derived from the bands in Fig. 1. The\ncalculated peaks in the valence band DOS closest to the Fermi energy are at -0.25 \u00060.1 eV\nand -2.4 \u00060.2 eV. We found a third peak at -5.8 \u00060.1 eV which agrees rather well with the\nexperimental value of -5.8 eV reported by Kamada et al. [32]. In the conduction band, we\nfound relatively broad peaks whose centers are located at 8.25, 9.50, 10.50, 11.25, and 12.00\neV, respectively. The widths of these broad peaks are about 0.5 eV. Caution is required in\nusing the conduction band density of states at energies above 6 eV. As explained by Jin et al.\n[33] and Bagayoko et al. [24, 34], BZW-EF calculations are concerned with correctly solving\n8equations describing the ground states. Hence, while the low-laying excited states from these\ncalculations have been found to agree with measurement, upper excited states are not expected\nto be correctly described. Specifically, the calculated optical properties of wurtzite InN [33] agree\nwith experiment up to energies of 5.5 to 6.0 eV. The comparison of the bands from Calculations\nIV and V in Fig. 2 suggests that our calculated, excited states for NaNO 2could be meaningful\nup to 20 eV. This situation could be a feature of molecular solids.\nOurcalculatedelectroneffectivemassesinthe \u0000-X,\u0000–R,and \u0000-Wdirectionsare1.22,0.64,and\n0.74 m o, respectively. These values clearly corroborate the anisotropic nature of this material.\nOur calculations were performed at lattice constants for which experimental data are available\nfor comparison purposes. However, the question arises as to what results could be obtained with\na fully optimized crystal structure (i.e., equilibrium lattice structure). The optimization led\nto equilibrium lattice parameters of 3.529 Å, 5.560 Å and 5.391 Å for a,bandc, respectively.\nThe difference between our former (120 K experimental lattice parameter) and the latter (with\noptimized lattice parameters) eigenvalues is approximately 0.02 eV, with the energies obtained\nwith the experimental lattice constants being lower. The two results are practically the same as\ntheir difference is smaller than our computational uncertainty of 50 meV (0.05 eV).\nIV. DISCUSSION\nAn added dimension of the validity of the work reported here stems from the rigor of the BZW-\nEFmethod. Ourcalculationssolvedself-consistentlythesystemofequationsdefiningLDA.They\nare totally ab-initio and do not entail the use of two different theories; the sophisticated work of\nZakharov et al. [35] utilized LDA wave functions as input in their quasi-particle calculations.\nUnlike single trial basis set (STBS) DFT and other calculations, LDA BZW-EF results agree\nwith experiment and point to much more physical content of LDA eigenvalues than previously\nthought. As explained by Zhao et al. [23], STBS calculations do not avoid a well-defined “ basis\nset and variational effect that has plagued electronic structure calculations since their inception,\nwith emphasis on DFT calculations.\nIndeed, BZW-EF calculations begin with the minimum basis set that is just large enough to\naccount for all the electrons in the system, i.e., NaNO 2for this work. It then augments the basis\nset methodically [22-24] and carries out successive, self-consistent calculations with increasingly\nlarger basis sets. Except for the first calculation, the occupied eigenvalues of a calculation\nare compared numerically and graphically to those of the calculation immediately preceding\nit. Ultimately, the occupied eigenvalues of two consecutive calculations, say N and (N+1), are\nfound to be identical-within computational uncertainties of about 50 meV. The results from\ncalculation N provide the physical description of the system and the corresponding basis set is\nthe optimal basis set. While Calculation (N+1) yields the same occupied energies as Calculation\nN, some of the unoccupied energies from it are generally lower than corresponding ones from\nCalculation N. This extra-lowering of the unoccupied eigenvalues, for calculations with basis sets\nlarger than the optimal one, is due to a basis set and variational effect. It is a mathematical\nartifact stemming from the Rayleigh theorem. For basis sets larger than the optimal one, the\nresulting charge density and potential are identical to those from Calculation N [23]. So, the\nabove extra-lowering is not due to a physical interaction.\nFrom the density of states plots, our calculated peaks of -0.25 \u00060.1 eV and -2.4 \u00060.2 eV in\n9the valence band DOS agree with the x-ray photoelectron results of Calabrese and Heyes, [36]\nand the theoretical calculations of Ravindran et al. [3] and Zhuravlev and Korabel’nikov [12].\nThe observed sharp peak at -0.25 eV arises mainly from the hybridization between the O 2p,\nN2p, and N2s. The peak at -2.4 \u00060.2 eV arises from the O 2porbitals. These observations\nare in agreement with that of Zhuravlev and Korabel’nikov [12] that the upper valence bands\nare formed by complex anion states. In the conduction bands, our calculated peaks of 3.25 eV,\n8.25 eV, 9.5 eV, 10.50 eV and 11.25 eV compare favorably with the room temperature vacuum\nultraviolet (VUV) data from measured values of Yamashita and Kato [37] and Ashida et al. [38],\nrespectively.\nA distinct feature of our calculated band structure in Figure 1 is the flat nature of the top of\nthe valence band as well as that of the bottom of the conduction band. This singular feature\nmeans that the band structure of ferro-NaNO 2can well be described by localized molecular\nlevels of an ionic crystal. This observation is similar to that of Ravindran et al. [2], Wang et\nal. [11] and of Jiang et al. [14]. Above the Fermi energy, as can be seen from the DOS, many\nconduction bands exist and some are highly dispersed. From the pDOS for the upper valence\nand the conduction bands, we can infer that the Na states are high up in the conduction bands\nand do not partake in the formation of the molecular solid.\nThe top-most valence bands and bottom-most conduction bands are flat. Consequently, the\nexact maximum and minimum values of the valence and conduction bands, respectively, cannot\nbe easily identified by merely looking at the plots. From our outputs, before setting the top of\nthe valence bands equal to zero, we found that the maximum of the valence band (VBM) is at\ntheWpoint while the minimum of the conduction band (CBM) is at R, respectively, giving a\nfundamental, indirect energy band gap of 2.83 eV from Calculation IV. Our calculated direct\ngaps are 2.90, 2.98, 3.02, 3.22, and 3.51 eV at R, W, X, \u0000, andT, respectively. The scanning\nelectron microscopy measurements of Balabinskaya et al. [9] reported a direct band gap of 3.0\neV.\nThe extent to which our electronic structure calculations take temperature into account is\npractically limited to the use of lattice constants obtained at a given temperature. With the\nlattice constants obtained at 120 K, our calculated direct gap of 3.22 eV compares favorably\nwith the infrared absorption result of 3.22 eV obtained by Sidman [6] at a low temperature (77\nK). Predictably, it is larger than the room temperature (293.15 K) measurement of 3.14 eV of\nVerkhovskaya and Sonin [7].\nThe positions of our VBM (at W) and CBM (at R) are the same as those from the work\nof Ravindran et al. [11] that found a calculated, indirect band gap of 2.2 eV. Jiang et al. [14]\ncalculated an indirect gap of 2.95 eV. However, these authors obtained their VBM at S. In our\ncalculations, only the 1sstates of Na, N, andO, respectively, were in the core. Jiang et al. [14]\nplaced1s, 2s, and2pelectrons of sodium in the core.\nOur calculated electron effective masses of 1.18, 0.63, and 0.73 m oin the \u0000-X,\u0000–R, and \u0000-\nW directions, respectively, show the highly anisotropic nature of this material. We know of\nno experimental or calculated electron effective masses for NaNO 2, except the semi-empirical,\ntemperature dependent result of El-Dib and Hassan [8] who found an electron effective mass of\n0.61 m oin the \u0000–R direction. Our result of 0.63, from \u0000–R, agrees with their finding.\n10V. CONCLUSION\nThe above results and related discussions are expected to add to our understanding of ferro-\nNaNO 2. The noted agreement between our calculated results and experiment will hopefully\nmotivatefurtherexperimentalandtheoreticalstudiesofthismaterial. Ourcalculatedeigenvalues\nat high symmetry points should aid in some comparison with future studies.\nThis work and similar ones from this group [17-24] indicate that LDA can correctly describe\nandpredictelectronicandrelatedpropertiesofsemiconductors,providedonemethodicallysearch\nfor the optimal basis set that minimizes the occupied energies. The predictive capability of such\ncalculations is expected to inform and to guide the design and fabrication of semiconductor\nbased devices. For example, for binary to quaternary systems, theoretical studies of variations\nof the concentration of an element (or more) have the potential to impact positively industrial\nactivities.\nAcknowledgments\nThis work was funded in part by the National Science Foundation and the Louisiana\nBoard of Regents (Award Nos. EPS-1003897, and NSF (2010-15)-RII-SUBR, 0754821, and\nHRD-1002541), the Louisiana Optical Network Initiative (LONI, SUBR Award No. 2-10915).\nCEE wishes to thank Govt. of Ebonyi State, Nigeria.\nReferences\n1. S. Sawada, S. Nomura, S. Fuji, and I. Yoshida, Phys. Rev. Letts. 1, 320 (1958).\n2. P. Ravindran, A. Delin, B. Johansson, and O. Eriksson, Phys. Rev B 59, 1776 (1999).\n3. M. Kamada, M. Yoshikawa, and R. Kato, J. Phys. Soc. Jpn. 39, 1004 (1975).\n4. Y. Asao, I. Yoshida, R. Ando, and S. Sawada, J. Phys. Soc. Jpn., 17, 442 (1962).\n5. Y. Takagi and K. Gesi, J. Phys. Soc. Jpn, 22, 979 (1967).\n6. J.W. Sidman, J. Am. Chem. Soc. 79, 2669 (1957).\n7. K.A. 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Blase, M. Cohen, and S. Louie, Phys. Rev. B 50, 10780 (1994).\n36. A. Calabrese and R. G. Hayes, J. Electron Spectrosc. Relat. Phenom. 6(1), 1 (1975).\n37. A. Yamashita and R. Kato, J. Phys. Soc. Jpn. 29(6), 1557 (1970).\n38. M. Ashida, O. Ohta, M. Kamada, M. Watanabe, and R. Kato, J. Electron Spectrosc.\nRelat. Phenom. 79, 55 (1996).\n13"    },    {        "title": "1208.6266v1.Composite_fermion_state_of_spin_orbit_coupled_bosons.pdf",        "content": "arXiv:1208.6266v1  [cond-mat.quant-gas]  30 Aug 2012Composite fermion state of spin-orbit coupled bosons\nTigran A. Sedrakyan1, Alex Kamenev1, Leonid I. Glazman2\n1William I. Fine Theoretical Physics Institute and Departme nt of Physics,\nUniversity of Minnesota, Minneapolis, MN 55455, USA and\n2Department of Physics, Yale University, New Haven, Connect icut 06520, USA\n(Dated: May 22, 2018)\nWe consider spinor Bose gas with the isotropic Rashba spin-o rbit coupling in 2D. We argue that at\nlow density its groundstate is a composite fermion state wit h a Chern-Simons gauge field and filling\nfactor one. The chemical potential of such a state scales wit h the density as µ∝n3/2. This is a\nlower energy per particle than µ∝nfor the earlier suggested groundstate candidates: a conden sate\nwith broken time-reversal symmetry and a spin density wave s tate.\nPACS numbers: 71.10.Pm, 67.85.Fg, 64.70.Tg\nI. INTRODUCTION\nPrecise control over interactions of ultracold atoms\nwith laser fields opened an opportunity of fabricating\nsyntheticnon-Abeliangaugefields1–3andspin-orbit(SO)\ncouplings4,5of both Rashba6and Dresselhaus7type. In\nanalogy with previously known vector condensates8–12,\nan important tool for synthesizing such systems is the\ncontrolled Raman coupling between oriented sequence\nof initially degenerate hyperfine states of87Rb. Inter-\nactions of spatially varying laser field with hyperfine\natomic levels effectively produces synthetic non-Abelian\ngaugefields,13,16–18originatingfromtheBerryphase,and\nSO coupling4,5,14,15,19–22of momentum to the internal\nisospindegreesoffreedom. BosonswithSOcoupling, dis-\ncussed and studied in the context of cold-atom physics\nin Ref. [15], were realized very recently by Spielman’s\ngroupatNIST4. Depending ontheparticularexperimen-\ntalscheme, i.e. thechoiceofatomicstatesandasequence\nof optically induced transitions between them, one may\nin principle realize various SO Hamiltonians for the pro-\njected low-energy states. Low energy properties of such\nSO bosonic systems have been discussed in Refs. [15, 23–\n33].\nWhile fermions with SO coupling were extensively\nstudied over the last decades, see e.g. Refs. [34–37] for\nreviews, relatively little is known about SO bosons. Yet,\nthey offer a number of fundamental problems which, in\nsome respects, are more challenging than their fermionic\ncounterparts. Indeed, in many instances SO coupling\nleads to single-particle dispersion relations which exhibit\nmultiple minima or even degenerate manifold of mini-\nmal energy states. Fermionic many-body systems form a\nunique Fermi sea state on top of such dispersion relation.\nOn the other hand, bosons tend to occupy the lowest en-\nergystates and thus face macroscopicdegeneracyoftheir\nnon-interacting groundstate. It is entirely the effect of\ncollisions (i.e. boson-boson interactions) which lifts this\ndegeneracy and selects a true many-body groundstate.\nInthis respectthe problemissomewhatsimilartofrac-\ntional quantum Hall effect (FQHE). There the fermionic\nkinetic energy is degenerate due to Landau quantiza-\ntion and the nature of the groundstate is determined bythe interactions. One of the most remarkable concepts,\nwhich emerged in FQHE studies, is that of composite\nparticles45–50. For example, FQHE states with filling\nfractionsν=p/(2p+1), where p= 1,2..., were under-\nstood as integer ν=pquantum Hall states of composite\nfermion particles. The latter are obtained by binding the\noriginalfermionswithfluxtubescarryingexactly twoflux\nquanta. Technically such a binding is achieved by assign-\ning Chern-Simons (CS) phase factor to the many-body\nwavefunction of composite particles.\nThe goal of this paper is to show that a very similar\ntransformationplaysamajorrolein understandingofthe\ngroundstate of 2D bosons with Rashba SO coupling. We\nargue here that their groundstate wavefunction may be\napproximated by that of the Fermi gas at integer filling\nfactorν= 1, dressed by the Chern-Simons phase with\noneflux quantum attached to every fermion. Such phase\nfactor transforms the integer quantum Hall fermionic\nwavefunction into a bosonicone. Fermionization of the\nbosonic system allows the latter to minimize its interac-\ntion energy. This is due to the fact that fermions with\nthe same spin can’t be at the same spatial point and thus\ncannot interact through a short-ranges-waveinteraction.\n(Consequently the amplitude for two fermions with al-\nmost parallel spins to be at the same point is small as\nthe angle between their spins.) As the result the interac-\ntion energy per particle in a Fermi sea is smaller than in\na Bose-condensate of the same density. For low enough\ndensitysuchreductionoftheinteractionenergywinsover\nthe associate increase of the kinetic energy.\nA very similar physics is behind the so-called Tonks-\nGirardeau limit of spinless 1D Bose gas38–40. At a small\ndensity (which in 1D is the same as strong interactions)\nthe bosonic wavefunction approaches symmetrized wave-\nfunction of non-interacting Fermi gas. If spin degree of\nfreedom is present in 1D, the groundstate is known to\nbe fully spin-polarized41–43again leading to the Tonks-\nGirardeau construction in low density limit. We thus\nnotice that 2D Rashba bosons share features of both 2D\nquantum Hall systems as well as 1D Bose gases. The\ndeep connection with the latter is due to the fact that\nsingle-particle density of states for particles with Rashba\nSO coupling behaves as ǫ−1/2at small energy. This is2\ntypical for 1D systems, making particles with Rashba\nSO coupling “1D-like”, irrespective of their actual spa-\ntial dimensionality.\nTechnically blending FQHE and Tonks-Girardeau\nideas with Rashba SO coupling presents one with a num-\nber of challenges. Indeed, the standard way of intro-\nducing Chern-Simons transformation45,47essentially re-\nlies on the spinless nature of particles. Here we suggest\na way to generalize it for spinor particles with strong SO\ncoupling. It achievesthe goaloffermionizationofbosons,\nbut fails toeliminate completely the interactionsbetween\nthe composite spinlessfermions. Still the fermionic in-\nteraction energy can be parametrically smaller than the\nbosonic one. The crucial observation here is that the\nresidual fermion-fermion interactions appear to be pro-\nportionaltotheanglebetweenmomentaoftwoscattering\nparticles. Therefore, confining the Fermi sea of compos-\nite fermions to a small fraction of the momentum space,\nallowsoneto lowertheir interactionenergy. Similarideas\nwere recently put forward by Berg, Rudner and Kivelson\n(BRK)44inthecontextofFermigaswithRashbaSOcou-\npling. They called the resulting time-reversal symmetry\nbrokenstatea nematic. Herewe adopttheir construction\nfor thecomposite fermions.\nThe paper is organized as follows: in section II we in-\ntroduce Rashba SO coupling on a single-particle level.\nWe also discuss earlier mean-field theories for bosonic\nmany-body groundstates along with BRK nematic state\nfor many-body fermionic system. In section III we intro-\nduce fermionizationofspinorbosonsand evaluatethe en-\nergy of the resulting composite fermion state. In section\nIV we discuss the results: present an emerging ground-\nstate phase diagram of Rashba bosons, list possible ex-\nperimental consequences and generalizations of our ap-\nproach. Some technical details of the calculations are\nrelegated to the Appendix.\nII. RASHBA SPIN-ORBIT COUPLING\nA. Single particle spectrum\nThe single-particle Hamiltonian with the Rashba spin-\norbit coupling in two dimensions takes the form\nH0=−∇2\nr\n2m+ivˆz·[σ×∇r], (1)\nwherevis spin-orbit coupling constant having dimen-\nsionality of velocity, ∇r= (∂x,∂y) andσ= (σx,σy) is\nvector of Pauli matrices acting on two component spinor\nψ(r) = (ψ(r,↑),ψ(r,↓)). In 2D the Rashba term in\nEq. (1) may be transformed to another form ivσ·∇r\nbyπ/2 rotation in the spin space.\nIn addition to translational and rotational symmetries\nthe Hamiltonian H0commutes with the two discrete Z2\nsymmetry operations: time-reversal ˆTand 2D parity ˆP.They may be defined in the following way\nˆT/parenleftbigg\nψ(r,↑)\nψ(r,↓)/parenrightbigg\n=/parenleftbigg¯ψ(r,↓)\n−¯ψ(r,↑)/parenrightbigg\n(2)\nand\nˆP/parenleftbigg\nψ(z,↑)\nψ(z,↓)/parenrightbigg\n=/parenleftbigg\n−iψ(¯z,↓)\niψ(¯z,↑)/parenrightbigg\n, (3)\nwhere bar stands for complex conjugated and we intro-\nduced the complex 2D coordinate as z=x+iy. Notice\nthat the parity operation in 2D is defined as y→−y\nandx→xandσymultiplication. Indeed, reflection of\nboth coordinates xandyis equivalent to πrotation. Of\ncourse, one could as well define parity as xreflection and\nσxmultiplication. It is equivalent to the Poperation (3)\ncombined with the πrotation.\nDiagonalization of the Hamiltonian (1) yields the fol-\nlowing single-particle spectrum:\nεk,γ=k2\n2m+γvk. (4)\nHere index γ=±1 labels the two brunches of the\nspectrum depicted in Fig. 1. The corresponding eigen-\nfunctions are plane waves in coordinate space multiply-\ning coordinate-independent spinor, whose form depends,\nhowever, on the momentum kas\nψk,γ(r,s) =1√\n2V/parenleftbigg\n−iγe−iarg(k)\n1/parenrightbigg\nseikr.(5)\nwhere arg( k) is the angle between the momentum vec-\ntorkand thekx-axis,Vis the system’s volume. The\nspin is directed in the x−yplane and is rotated rel-\native to the momentum direction by γπ/2. Hereafter\nwe consider SO energy scale ǫ0=mv2/2 as the largest\nenergy in the problem. One thus expects the ground-\nstate of an interacting system to be confined to the\npart of the Hilbert space projected on the lower branch\nγ=−1. Clearly the two spin components of all such\nstates are connected by the following unitary transfor-\nmationψk,−(r,↑) =ie−iarg(k)ψk,−(r,↓). For a generic\nwavefunction belonging to the lower branch the relation\nbetween its up and down components acquires the form\nψ(r,↑) =/integraldisplay\ndr′R(r−r′)ψ(r′,↓), (6)\nwhere the spin-raising kernel\nR(r−r′) =−1\n2πe−iarg(r−r′)\n(r−r′)2(7)\nis the Fourier transform of ie−iarg(k). Importantly, the\nR-operator in real space has the unitarity property,/integraltext\ndr¯R(r−r1)R(r−r2) =δ(r1−r2), which is the direct\nconsequence of the unitarity of the spin-rising transfor-\nmation in the momentum space representation.\nThe most notable feature of the dispersion relation (4)\nis that its groundstate is degenerate along the circle in3\nFIG. 1. (Color online) Dispersion relation (4)ofparticles with\nRashba SO coupling. The degenerate groundstate k=k0=\nmvis shown by full circle.\nthe momentum space k=k0=mv. As a result the\nmany-body groundstate of Nnon-interacting bosons is\nhighly degenerate (not so for fermions, though). Indeed,\nany occupation of the states along the groundstate circle\nk=k0yields exactly the same kinetic energy −Nk2\n0/2m.\nHereafter we shall measure the energy from that value,\ntaking it for the origin of the energy axis. It is therefore\nonly the interactions, which may break the degeneracy\nand select a true groundstate. The situation is similar to\na partially filled Landau level in the context of FQHE.\nThere too the kinetic energy is fully degenerate and the\ngroundstate is solely determined by the interparticle in-\nteractions.\nB. Bosons with Rashba dispersion\nLetusconsider Nparticleswiththe s-waveshort-range\ninteractions of the form\nHint=1\n2mN/summationdisplay\ni,jδ(ri−rj)/bracketleftbig\ng0+g2σ(i)\nzσ(j)\nz/bracketrightbig\n,(8)\nwhereg0is the spin-isotropic dimensionless interaction\nconstant, while g2is the spin-anisotropicinteraction con-\nstant.\nOne can now evaluate interaction energy of certain\nsimpleN-body Bose states. One such state is a Bose-\nEinstein condensate in a single state belonging to a de-\ngeneratemanifoldofthe single-particlegroundstates, i.e.\na state with momentum k, such that k=k0andγ=−1,\nΨ(0)\nB=N/productdisplay\ni=1ψk,−(ri,si). (9)\nThis wavefunction is obviously symmetric with respect\nto the permutation of any two pairs ( ri,si)↔(rj,sj).\nSuch a state is not symmetric under time-reversal trans-\nformation ˆT, Eq. (2), because of unequal population of kand−kstates. We shall call it thus time-reversal sym-\nmetry broken (TRSB) state. Note that this state does\nnotbreak the parity ˆP, Eq. (3). To see it most clearly\nonemaychoosemomentum kdirectionto be alongthe x-\naxis. Calculating the expectation value of the interaction\nenergy (8) over TRSB state, one finds\nE(0)\nint=N2\n2mVg0. (10)\nSince the kinetic energy in the state (9) is zero, the in-\nteraction energy (10) coincides with the total one.\nOne may consider now a Bose condensate built on a\ncoherent superposition of say two states k1andk2both\nbelonging to the degenerate manifold k=k0:\nΨ(φ)\nB=N/productdisplay\ni=11√\n2[ψk1,−(ri,si)+ψk2,−(ri,si)].(11)\nThe corresponding interaction (and thus total) energy is\nfound to be\nE(φ)\nint=N2\n2mV/bracketleftbigg\ng0+g0\n2cos2φ\n2+g2\n2sin2φ\n2/bracketrightbigg\n,(12)\nwhereφ= arg(k1)−arg(k2) is the angle between the\ntwo states of the degenerate manifold. It is clear that,\nprovidedthespin-isotropicinteractionisrepulsive g0>0,\nthe only way the state (11) may be energetically more\nfavorablethan the state (9) is if g2<0. In the latter case\nthe most favorable choice of the two states corresponds\ntoφ=π, i.e.k2=−k1, with the interaction energy\nE(π)\nint= (N2/2mV) [g0+g2/2]. Such a state represents\na spin density wave (SDW) with a uniform total density\nandthetwospincomponentsoscillatingharmonicallyout\nof phase. It is symmetric with respect to both time-\nreversal and parity symmetries, Eqs. (2), (3), but breaks\nthe rotational symmetry. In either of these states the\nkinetic energy is zero.\nIt was conjectured23,24that TRSB Ψ(0)\nB, (9), and the\nSDW state Ψ(π)\nB, (11), are the many-body groundstates\nof the Rasba interacting bosons for g2>0 andg2<0\ncorrespondingly. It was later suggested27that the tran-\nsition between TRSB and SDW states is shifted towards\npositiveg2due to a ceratin admixture of coherently oc-\ncupied BCS-like pairs of kand−kstates. We notice\nthat for both of these states the chemical potential scales\nlinearly with the density n=N/V,\nµB=∂Eint/∂N∝n. (13)\nIt is instructive to compare this scaling of the bosonic\nchemical potential with the corresponding fermionic one.\nC. Fermions with Rashba dispersion\nUnlike bosons, the non-interacting Fermi gas exhibits\nthe unique ground-state. It is given by the rotationally4\nsymmetric Fermi sea with the Fermi surface consisting of\ntwo concentric circles with the radii kF±=k0±2πn/k0,\nhereafter the small density n≪k2\n0is assumed. The\ncorresponding Fermi energy measured from the bottom\nof the spectrum is\nµ(0)\nF=(πn)2\n2mk2\n0. (14)\nNotice the fact that µ(0)\nF∝n2, which is usually the\nfeature of 1D Fermi gas. Here it happens because the\nsingle-particle density of states exhibits 1D-like behav-\niorν(ǫ)∝ǫ−1/2close to the bottom of the Rashba cir-\ncle. The interesting observation is that at small enough\ndensityn/lessorsimilarg0k2\n0the chemical potential of interacting\nbosons appears to be bigger than that of the free Fermi\ngas with the same dispersion relation (4). Before making\nconclusions from this observation one needs to consider\nthe interaction energy (8) of the Fermi sea state.\nSpinless (or fully spin-polarized) fermions do not in-\nteract through short-range interactions. In our case the\nparticles have spin, which is locked to their orbital mo-\nmenta. As a result the symmetric Fermi sea, described\nabove, contains all spin directions in x-yplane with equal\nweights. Since two fermions with opposite spins interact\nthrough the short range interaction (8), one expects the\naverage interaction energy of the symmetric Fermi sea to\nbe of the same order as bosonic one ∝g0N2/mVand\nthusµF∝nsimilarly to the bosonic condensates. It was\nrecentlynoticedbyBerg,RudnerandKivelson44thatlow\ndensity Rashba fermions may have parametrically lower\ngroundstate energy if they form a nematic state.\nTo motivate the idea, let us consider two-fermion\nstate as a Slater determinant built on single par-\nticle states (5) with momenta k1,2close to the\nRashba circle Ψ F= [ψk1,−(r1,s1)ψk2,−(r2,s2)−\nψk1,−(r2,s2)ψk2,−(r1,s1)]/√\n2. The interaction energy\n(8)ofsuchstateisgivenby Eint= sin2(φ/2)(g0+g2)/2m,\nwhereφ= arg(k1)−arg(k2). Therefore the interac-\ntion energy between fermions tends to zero if arg( k1)→\narg(k2), essentially because the spins are aligned in this\nlimit and fermions with the same spin can not interact\nthrough the short-range interaction potential. The BRK\nnematic state takes advantage of this observation.\nTo describe such a state qualitatively let us imagine\nthat the many-body fermionic state Ψ Fis constructed\nas a Slater determinant of states with the angular di-\nrections confined to the angular segment of the momen-\ntum (and spin) space of size Θ ≪2πand momenta close\nto the spin-orbit circle ||kj|−k0|< k(Θ)\nF, Fig. 2. The\ncorresponding Fermi momentum is found from the con-\ndition 2k(Θ)\nFk0Θ = (2π)2n. As a result the correspond-\ning kinetic energy per particle Ekin/N∝[k(Θ)\nF]2/m∝\nn2/(mΘ2k2\n0). On the other hand, the interaction energy\nper particle is Eint/N∝(g0+g2)nΘ2/m, with the factor\nΘ2originating from sin2(φ/2)∼Θ2. One can now min-\nimize the sum of kinetic and interaction energy over Θ\nto find Θ∝[n/k2\n0(g0+g2)]1/4, which is indeed small aslong as\nn≪n0=k2\n0(g0+g2). (15)\nWith this Θ the chemical potential of short-range inter-\nactingfermions with Rashba spin-orbit coupling is found\nto beµF∝n3/2√g0+g2/mk0, which is the result of\nBRK44. In the small density limit (15) the latter is big-\nger than that of non-interacting Rashba fermions (14),\nbut is advantageous over bosonic TRSB and SDW states\n(13):µ(0)\nF< µF< µB. Notice also the non-analytic de-\npendence of µFon the interaction strength, indicating\nthe non-perturbative nature of this result. We show in\nSection IIID that the composite fermion nematic state\nmaybedescribedquantitativelywithinHartree-Fockself-\nconsistent mean-field treatment, Fig. 2.\n40kyk\nxk\nFIG. 2. (Color online) Hartree-Fock Fermi surfaces of BRK\nnematic states with three different densities. The absolute\nvalue of the momenta are restricted to the vicinity of the\nRashba circle k=k0, while their angular directions are re-\nstricted to the angle Θ ∝[n/k2\n0(g0+g2)]1/4.\nD. Can bosons be fermions?\nWe have arrived thus to the conclusion that at low\ndensity (15) the chemical potential and groundstate en-\nergy of fermionic many-body system is smaller than the\ncorresponding bosonic one. The similar situation seem-\ninglyhappens in a 1D system of spinless particles with\nshortrangeinteractionpotential ( g0/m)δ(xi−xj). There\ntoo the mean-field treatment of bosons suggests that\nµB∼g0n. On the the other hand, spinless fermions\n(which arenot affected byshort-rangeinteractionsdue to\nthe Pauli principle) exhibit µF∼n2. It thus seems that\nat a small enough density the fermionic groundstate en-\nergyissmallerthanthebosonicone. Theactualsituation\nis, of course, very different40. At small density bosonic\nmany-bodygroundstatewavefunctionΨ Bapproachesthe5\nP\nnn~P2~nP\n0n\nFIG. 3. (Color online) Equations of state µ(n) of 1D quantum\ngases with short-range interactions: spinless Fermi gas (l ong-\ndashed); mean-field approximation for Bose gas (dashed); ex -\nact result40for bosons (full). Notice that bosonic ground-\nstate energy E=/integraltext\nµdnisalwayssmaller than the fermionic\none, although the mean-field treatment suggests otherwise f or\nn/lessorsimilarn0=g0.\nsymmetrized fermionic one\nΨB(x1,...,x N) =N/productdisplay\ni,jsign(xi−xj)ΨF(x1,...,x N),\n(16)\nwhere Ψ Fis the fermionic Slater determinant occupying\na finite portion of the momentum space −πn≤k≤πn.\nIt is important to notice that, although sign( xi−xj) is\nundefined at xi=xj, the fermionic part cancels at all\nsuch points making Ψ B(x1,...,x N) well-defined in the\nentire space. Due to the same observation there is no in-\nteraction energy cost for the short-range repulsion. The\ncorresponding kinetic energy per particle ∝(πn)2/2mis\nsmallinthelowdensitylimit. Thisistheso-calledTonks-\nGirardeau limit38,39,42, where bosons redress themselves\nas fermions. This allows them to take advantage of the\nwavefunction, which is nullified at all points where any\ntwo particles approach each other. As a result they avoid\npaying short-range interaction energy cost, while corre-\nsponding kinetic energy cost appears to be worth the\nbargain at low density. The corresponding 1D equations\nof state are depicted in Fig. 3.\nIt is thus tempting to speculate that our spinfull 2D\nsystem may benefit from a similar construction. Namely,\nat low density bosons may want to redress themselves\nas fermions to take advantage of their lower interac-\ntion energy. The blueprints of the boson-fermion cor-\nrespondence in 2D are provided by FQHE studies45,47,\nwherethe correspondenceisachievedbyascribingChern-\nSimons phase factor to many-body wavefunctions. Such\nfactor takes care of the proper symmetry of wavefunc-\ntions. It comes with the price however: the gauge mag-\nnetic field which has a form of delta-functional flux tubes\nattached to every particle. It is believed that (under\nproper conditions) this magnetic field may be substi-tuted by a uniform one, making the problem analytically\ntractable. In the next section we develop a similar strat-\negy for the chiral spinfull boson-fermion correspondence.\nIII. COMPOSITE FERMION STATE\nA. Na¨ ıve fermionization attempt\nOur goal is to construct a variational groundstate for\nRashba bosons based on the composite fermion idea. We\nargue here that in the small density limit such a state,\ninspired by the Tonks-Girardeau limit (16), is advanta-\ngeous over both TRSB (9) and SDW (11) bosonic states.\nThe two main differences with the Tonks-Girardeau case\nare: (a) the 2Dnatureofourproblemand(b) thespinfull\nnature of the particles. It is still tempting to straight-\nforwardly generalize Eq. (16) to the 2D spinfull case by\ntaking the variational ground state of the Bose gas in the\nfollowing form:\nΨB(r1,s1,···rN,sN) (17)\n=/productdisplay\ni<jeiarg(ri−rj)ΨF(r1,s1,···rN,sN),\nwhere Ψ F(r1,s1,···rN,sN) is an N-particle fermionic\nwavefunction. The Chern-Simons phase eiarg(ri−rj)=\n(zi−zj)/|zi−zj|, withzjlabeling complex spatial co-\nordinates,zj=xj+iyj, is antisymmetric with respect\nto exchange of any two coordinates. As a result, many-\nbody wavefunction, Ψ B(r1,s1,···rN,sN), is symmetric\nunder permutation of pairs ri,siandrj,sjof coordi-\nnates and spins, si=↑,↓, if the fermionic wave function,\nΨF(r1,s1,···rN,sN), is antisymmetric with respect to\nthe same permutations. One way of writing the latter is\nto take it as N×NSlater determinant of e.g. single-\nparticle spinor wave functions, ψkj,−(ri,si), Eq. (5),\nwherei,j= 1,2,...,N.\nWhile being probably the most straightforward way of\nguessing a spinfull composite fermion state, Eq. (17) has\na number of fatal drawbacks. First, one might na¨ ıvely\nexpect that similarlyto Eq.(16), this ansatzmapsthe in-\nteractingbosonicsystemontoasystemofnon-interacting\nfermions. A closer look shows that this is not the case.\nIndeed, although the Slater determinant implies that the\nfermionic wave function has zeros at coinciding spatial\npointsand spins , however for coinciding points and dif-\nferent spins the wavefunction is not nullified. As a result\nthe composite fermions with opposite spins still do in-\nteract, despite the short-range nature of the interaction\npotential.\nEven more serious problem with the trial wavefunc-\ntion (17) is that it is not well-defined for coinciding\npoints and opposite spins. This is due to the fact that\nthe Chern-Simons phase is singular at coinciding points,\nwhile the fermionic part Ψ Fis not vanishing if spins are\nopposite. This ambiguity leads to logarithmic divergent\ncontributions to the average kinetic energy of the state6\n(17). Indeed, consider the part of the kinetic energy\n(2m)−1/integraltextdri|∇riΨB|2, where the gradient operators act\non the Chern-Simons phases. This leads to the kinetic\nenergy contribution of the form\n1\n2m/integraldisplay\ndri/summationdisplay\nj,j′(ri−rj)·(ri−rj′)\n|ri−rj|2|ri−rj′|2/vextendsingle/vextendsingleΨF(r1,s1,···rN,sN)/vextendsingle/vextendsingle2.\n(18)\nThe diagonal terms j=j′in this double sum lead to the\nintegrals of the form/integraltextdri/|ri−rj|2, which exhibit loga-\nrithmic behavior when ri≈rj. If particles iandjhave\nthe same spin, Ψ F= 0 atri=rjcutting the logarithmic\ndivergenceof the integralat small distances ∼|ki−kj|−1\n(at large distances it is cut at a typical interparticle dis-\ntancen−1/2due to the random sign of the numerator in\nEq. (18)). However for opposite spins Ψ F∝ne}ationslash= 0 atri=rj\nand the integral (18) diverges in all such points. The re-\nsult is logarithmical divergent chemical potential, mak-\ning the trial wavefunction (17) essentially useless. One\nshould thus lookfor an alternative wayto introduce com-\nposite fermion state for spinfull Rashba particles, which\navoids logarithmic divergent terms in the kinetic energy.\nB. Fermionization\nThe idea for an alternative scheme comes from the ob-\nservation that at small density all relevant energy scales\nare much smaller than SO energy ǫ0=mv2/2. There-\nfore one would like to have a many-body state which is\nprojected onto the Hilbert subspace of the lower spin-\norbit branch γ=−1, Eq. (4). On the single particle\nlevel such a projection is achieved by ensuring the rela-\ntion (6) between up and down components of the spinor.\nOne can straightforwardly generalize it for a many-body\nwavefunction. Tothis end one should specify a fully sym-\nmetric in the coordinate space wavefunction of the min-\nimal spin\nΨ↓...↓(r1,...,rN) = Ψ(r1↓,...,rN↓).(19)\nThen all other spin components of the fully projected\nwavefunction may be uniquely determined from the min-\nimalspin componentby successiveapplicationofthe spin\nraising non-local operator R, Eq. (7), e.g.\nΨ(r1↑,...,rN↓) =/integraldisplay\ndr′\n1R(r1−r′\n1)Ψ↓...↓(r′\n1,...,rN),\nΨ(r1↑,...,rN↑) =/integraldisplay\ndr′\n1...dr′\nNR(r1−r′\n1)...R(rN−r′\nN)\nΨ↓...↓(r′\n1,...,r′\nN). (20)\nIt is easy to see that all the components defined this way\nare symmetric with respect to simultaneous interchange\nofri,siandrj,sj.\nOne can now fermionize such a wavefunction by writ-\ningthe spatiallysymmetric minimal spincomponent (19)\nas a product of Chern-Simons phase and fully antisym-\nmetricspinlessfermionic wavefunction. The latter willbe shown to describe the anisotropic nematic state. It is\ntherefore convenient to incorporate the same anisotropy\nin the Chern-Simons phase too. We thus define rescaled\ncoordinates ˜ x=αx, ˜y=y/αand˜r= (˜x,˜y), where the\ndensity-dependent scaling parameter αwill be specified\nin Section IIIE. The fermionized minimal spin compo-\nnent is then written as:\nΨ↓...↓(r1,...,rN)=1√\n2N/productdisplay\ni<jeiλarg(˜ri−˜rj)ΨF(r1,...,rN),\n(21)\nwhere we have introduced chirality factor λ=±1, which\ndefines the direction of the Chern-Simons flux relative\nto the spin chirality. The many-body fermionic wave-\nfunction Ψ F(r1,...,rN) is antisymmetric with respect to\npermutation of any two of its spatial arguments. Thanks\nto the Chern-Simons phase factor the left hand side of\nEq. (21) is fully symmetric both in coordinate and spin\nspaces. Notice that, unlike the earlier attempt, Eq. (17),\nthe wavefunction (21) is everywhere well-defined. This\nis the case because the Chern-Simons phase multiplies\nthe fully antisymmetric function of spinless fermions,\nwhich cancels if any of its spatial arguments coincide.\nThis became possible by adding the Chern-Simons fac-\ntor to a fully spin-symmetric minimal spin component\nonly. The higher spin components are then built up from\nthe fermionized minimal spin bosonic function (21) by a\nsuccessive application of the spin raising kernel, Eq. (20).\nAsaresult all2Nspincomponentsofthe N-bodybosonic\nwavefunction (20), (21) are well-defined functions in the\nentire 2N-dimensional coordinate space. This is unlike\nthe earlier attempt, see Eq. (15), which resulted in the\nlogarithmic divergences in the kinetic energy.\nOf course, there is a complimentary construction\nwhere one fermionizes the maximal spin component\nΨ↑...↑(r1,...,rN) and builds the lower spin components\nby successive action of spin lowering operator. These\ntwo states are related by the parity transformation (3)\nacting on the coordinates and spins of all Nparticles.\nNamely, acting with such parity operator on the state\n(20), (21) built from the minimal spin with chirality λ,\noneobtainsthestate built fromthe maximalspin compo-\nnentwithchirality −λandparitytransformed, yj→−yj,\nfermionic wavefunction Ψ F. Indeed, the parity operation\n(3) interchanges spins and transforms zj→¯zjresulting\ninλ→−λcorrespondence. Since the parity operator ˆP\ncommutes with the full Hamiltonian H0+Hint, these two\nstates are degenerate.\nLet us now discuss the average kinetic energy of the\nstate with the wavefunction (19), (20). Evaluating the\nexpectation value of the single-particle operator incorpo-\nrating kinetic and spin-orbit parts (1), one finds\nEkin= 2N/integraldisplayN/productdisplay\nj=1drjdr′\nj¯Ψ↓...↓(r1,...,rN)N/summationdisplay\nj=1ˆK(rj−r′\nj)\n×Ψ↓...↓(r′\n1,...,r′\nN), (22)\nwhere the non-local operator of the kinetic energy acting7\non the minimal spin component is\nˆK(r−r′) =−δr−r′∇2\nr′\n2m−v\n2/bracketleftbig¯R(r′−r)∂−\nr′−∂+\nrR(r−r′)/bracketrightbig\n,\n(23)\nhere we employed notations ∂±\nr=∂x±i∂y. This form\nis obtained by expressing the spin-up states in terms of\nthe spin-down states with the help of Eq. (6) and us-\ning unitarity of the spin-raising operator. The Fourier\ntransformation of the kernel (23) results in the lower\nbranch,γ=−1, of the single particle spectrum Eq. (4)\nˆK(k) =k2/2m−v|k|=εk,−, which is, of course, ex-\npected for the projected wavefunction in the form (19),\n(20). It is exactly the reason to build the higher spin\nstates with the help of the spin-raising operator (20) to\nensure that the kinetic energy belongs to the lower spin-\norbit branch.\nThenon-analyticbehaviorofthespectrum(4) at k= 0\ntranslates into the non-local behavior of the kinetic en-\nergy kernel (23) in the coordinate representation. This\nnon-locality complicates the way the kinetic energy oper-\nator acts on the Chern-Simons phase in Eq. (21). On the\nother hand, the low energy part of the Hilbert space is\nlocated close to the Rashba circle |k|=k0, i.e. far away\nfrom thek= 0 singularity. Below we discuss variational\nchoices for the many-body fermionic wavefunction Ψ F,\nwhich explicitly include only momentum components lo-\ncalized around|k|=k0circle. For those components\nthe non-analyticity at k= 0 and thus non-locality of\nthe kinetic energy in the coordinate space are not essen-\ntial. Therefore the single-particle kinetic energy spec-\ntrum near its minimum may be approximated by an an-\nalytic function of momentum (and thus local differential\noperator in the coordinate representation) as, e.g.,\nK(k) =−ǫ0+(|k|2−k2\n0)2/(8mk2\n0).(24)\nIts action on Chern-Simons phase results into the substi-\ntution of the gradient operators by the covariant deriva-\ntives with the gauge vector potential (see below).\nThe important observation at this stage is that there\narenologarithmical divergent contributions to the ki-\nnetic energy, which were fatal for the earlier fermioniza-\ntion attempt (17). Indeed, the kinetic energy kernel (22)\nacting on the Chern-Simons phase (21) results, among\nothers,inthetermsimilarto(18). However,thistimethe\nfermionic wavefunction Ψ Fdoes not contain spin indices\nand cancels any time ri=rj. As a result, all the loga-\nrithmic integrals are cut off at small distances by a scale\nbuilt into the fermionic wavefunction Ψ F(r1,...,rN). At\nlarge distances they are cut off at a typical interparticle\ndistance. As we discuss in Section IIIE, these two length\nscales can be made of the same order by an appropriate\nchoice of the rescaling parameter αin the Chern-Simons\nphase (21). This makes the logarithm to be a number of\norder one, showing that the localized flux-tube structure\nof the Chern-Simons magnetic field (see below) does not\nstronglyaffectthe kineticenergyofthe fermionstateΨ F.\nIt suggests the mean-field substitution of the flux tubesbya uniform magneticfield, analogousto thoseemployed\nin FQHE literature, e.g.45,47.\nNext we discuss inter-particle interactions, excluding\nfor a while the Chern-Simons term in the wavefunction\n(21). Following BRK, we employ Hartree-Fock approxi-\nmation to minimize the total energy of interacting par-\nticles having spectrum (24) and find that the chemical\npotential scales as µ∼n3/2. Finally, we include the\nChern-Simons term within the Hartree-Fock approxima-\ntion for interactions. We argue that it does not affect\nthe parametric dependence µ(n)∝n3/2, although does\naffect the spectrum of single-particle excitations.\nC. Interactions\nWe focus now on the average energy of the short-\nrange interactions (8) in the many-body state specified\nby Eqs. (21) and (20). Since the minimal spin compo-\nnent (21) cancels at coinciding spatial points, it does not\ncontribute to the interaction energy. On the other hand,\nthe higher spin components, built according to Eq. (20),\ndo not vanish at coinciding points and thus lead to a\nnon-zero interaction energy. At the first glance this ob-\nservation makes the interaction energy of the composite\nfermion state Eqs. (21) and (20) to be of the same or-\nder as that of the bosonic condensate (9) or (11), mak-\ning the entire construction useless. We show below that\nthis is not the case thanks to the antisymmetric nature\nof ΨF(r1,...,rN). The latter leads to the Hartree mi-\nnusFock structure ofthe two-particleinteractions, which\nnearly cancels the interaction energy for particles with\nnearly collinear momenta (and thus spin) directions. On\nthe other hand, the bosonic condensate (9) admits only\nthe Hartree contribution, while (11) leads to Hartree plus\nFock structure, not exhibiting the cancelation.\nTo derive the effective two-body interactions in the\ncomposite fermion state it is sufficient to consider two\nparticles in such state built of two single-particle states,\ne.g.ψkj(r) =eikj·r/√\nVwithj= 1,2 as\nΨF(r1,r2) =1√\n2/bracketleftBig\nψk1(r1)ψk2(r2)−ψk1(r2)ψk2(r1)/bracketrightBig\n,\nΨ↓↓(r1,r2) =1\n2eiλarg(˜r1−˜r2)ΨF(r1,r2),\nΨ↑↓(r1,r2) =/integraldisplay\ndr′\n1R(r1−r′\n1)Ψ↓↓(r′\n1,r2), (25)\nΨ↓↑(r1,r2) =/integraldisplay\ndr′\n2R(r2−r′\n2)Ψ↓↓(r1,r′\n2),\nΨ↑↑(r1,r2) =/integraldisplay\ndr′\n1dr′\n2R(r1−r′\n1)R(r2−r′\n2)Ψ↓↓(r′\n1,r′\n2).\nThe average interaction energy according to Eq. (8) is\ngiven by\nEint=1\n2m/integraldisplay\ndr/bracketleftBig\n(g0+g2)|Ψ(λ)\n↑↑(r,r)|2(26)\n+ (g0−g2)|Ψ(λ)\n↑↓(r,r)|2+(g0−g2)|Ψ(λ)\n↓↑(r,r)|2/bracketrightBig8\nand therefore one needs to evaluate the wavefunction\nfor higher spin components at coinciding spatial points.\nThey are given by:\nΨ↑↑(r,r) =1\n2Veir(k1+k2)Fk1,k2, (27)\nΨ↑↓(r,r) = Ψ↓↑(r,r) =1\n2Veir(k1+k2)Gk1,k2,\nwhereFk1,k2andGk1,k2areinteractionform-factorseval-\nuated in the Appendix. In the approximation where\n|k1|≈|k2|≈k0and arg(k1,2)≪2πwe found:\nFk1,k2=ic/parenleftbig\narg(k1)−arg(k2)/parenrightbig\n, (28)\nGk1,k2=d/parenleftbig\narg(k1)−arg(k2)/parenrightbig\n.\nHerec=cλ(α) andd=dλ(α) are numerical factors\nweakly dependent on CS chirality λand anisotropy α.\nFor example,|c+(1)|≃1.22;|c−(1)|≃0.85;c±(0) = 0\nandd+(1) = 0;|d−(1)| ≃1.41;|d±(0)| ≃0.90. A\nvery important observation is that the interaction en-\nergy tends to zero if arg( k1)→arg(k2). The relative\nminus sign in Eqs. (28) is entirely due to the compos-\nite fermion nature of the wavefunction (25). Indeed\nthe two-particle fermionic wavefunction (25) is zero if\narg(k1) = arg( k2) due to Pauli blocking. This offers\na possibility to construct a trial many-body wavefunc-\ntion which takes advantage of the fact that the particles\nwith nearly collinear spins (and thus momenta!) inter-\nact only weakly. As discussed in section IIC, the similar\nconstruction for fermions with the Rashba SO coupling\nwas recently employed by BRK44, who showed that the\noptimal trial state is a nematic one.\nD. Hartree-Fock theory\nThe secondary quantized interaction Hamiltonian for\nprojected spinless fermions takes the form\nˆHint=1\n32m/summationdisplay\nk′\n1+k′\n2=k1+k2Mk1,k2\nk′\n1,k′\n2c†\nk′\n2c†\nk′\n1ck1ck2,(29)\nwhere the interaction matrix elements\nMk1,k2\nk′\n1,k′\n2= (g0+g2)¯Fk′\n1,k′\n2Fk1,k2+2(g0−g2)¯Gk′\n1,k′\n2Gk1,k2.\n(30)\ndepend on all four (two incoming and two outgoing) mo-\nmenta and therefore interactions can’t be reduced to the\ndensity-density form. Moreover in the coordinate repre-\nsentation such interactions acquire essentially non-local\n|r−r′|−2form, which is a consequence of the projec-\ntion on the lower SO branch51. In principle the projec-\ntion, Eqs. (19) and (20), generates also three- and more-\nparticle interactions. One may check however that in the\ndilute limit (15) their effect is negligibly small in com-\nparison with the two-particle term kept in Eq. (29).\nOne can now treat the interaction term in Eq. (29)\nin the Hartree-Fock approximation by pairing one cre-\nation and one annihilation fermionic operator ∝an}b∇acketle{tc†\nk′ck∝an}b∇acket∇i}ht=/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\n(b)(a)k1k2 k2\n1k\n(c) k k kk\nFIG. 4. (Color online) (a) Two-body composite fermion inter -\naction with the matrix element Mk1,k2\nk′\n1,k′\n2, Eq. (30). (b) Hartree\nand (c) Fock ways of pairing the operators in the interaction\nHamiltonian (29).\nδ(k′−k)nk′. The Hartree and Fock ways of pairing are\nillustrated in Fig. 4a,b, correspondingly. This way one\nobtains the mean-field single-particle Hamiltonian\nˆHHF=/summationdisplay\nkKHF(k)c†\nkck, (31)\nwhere the Hartree-Fock kinetic energy is given by\nKHF(k) =εk,−+1\n16m/summationdisplay\nk′/bracketleftBig\nMk,k′\nk,k′−Mk′,k\nk,k′/bracketrightBig\nnk′,(32)\nandtheself-consistencyconditionisimposedbyrequiring\nnk′=f((K(k′)−µF)/T) is the equilibrium occupation\nnumber of the state k′determined by its totalenergy\nK(k′) and the chemical potential µF. The latter is to be\nfound from particle conservation/summationtext\nk′nk′=n.\nIn the limit of small density (15) we anticipate the\nnematic state, discussed qualitatively in section IIC.\nChoosing its center to be along the positive x-direction,\nonemayapproximatearg( k)≈ky/k0≪2π. We cannow\nemploy the form-factors (28) as well as the lower branch\ndispersionεk,−in the form of Eq. (24), to find for the\nHartree-Fock energy\nKHF(k)≈E0+(kx−k0)2\n2m+k2\ny\n2my,(33)\nwhere\nmy=m4k2\n0\ngn≫m (34)\nandE0=g/summationtext\nk′(k′\ny)2nk′/(4mk2\n0) with the effective inter-\naction constant g= (g0+g2)|c|2+ 2(g0−g2)|d|2. We\nnote here that myis essentially insensitive to the distri-\nbutionnk, as long as it covers a small fraction of the\nRashba circle in the momentum space. It is clear that\nthe Fermi surface, corresponding to the dispersion rela-\ntion (33) is an ellipse elongated along the ydirection and9\ncentered around ( k0,0), Fig. 2. One can now find the\nchemical potential of interacting fermions at T= 0 by\nsolving/summationtext\nK(k)<µF=n. This way we find for the chemi-\ncal potential of the interacting composite fermions\nµF=5π\n4√gn3/2\nmk0(35)\nandE0=µF/5, in agreement with the estimate in the\nend of Section IIIC.\nE. Chern-Simons gauge field\nSofarwehavebeendiscussingtheinteractionenergyof\nthe composite fermions. The latter are related to bosons\nthrough the Chern-Simons phase according to Eq. (21).\nWe need to discuss now the role of this phase. At the\nHartree-Fock level the system is effectively described by\nnon-interacting quasiparticles with the anisotropic dis-\npersion relation Eq. (33). In the coordinate representa-\ntion the momentum operators are given by kx,y→i∂x,y.\nTo bring it into a more familiar form one may shift the\nmomentum origin to ( k0,0) by the gauge factor eik0xand\nrescale the variables as ˜ x=αxand ˜y=y/α, where\nα= (m/my)1/4. In the rescaled coordinates the Hartree-\nFock Hamiltonian is isotropic with the cyclotron mass\nmc=√mmy. Upon acting on the Chern-Simons phase\nthe rescaled momentum operator results in a vector po-\ntentialA(˜r), which is included by replacing components\nofi∇˜rby the corresponding components of i∇˜r−A(˜r),\nHHF=N/summationdisplay\nj=11\n2mc/bracketleftbig\n(i∂˜xj−Ax(˜rj))2+(i∂˜yj−Ay(˜rj))2/bracketrightbig\n,\n(36)\nwhere the vector potential is given by52\nAα(˜rj) =/summationdisplay\ni/negationslash=jǫαβ(˜rj−˜ri)β\n|˜rj−˜ri|2, (37)\nwhile the corresponding Chern-Simons magnetic field di-\nrected perpendicular to the 2D plane is given by\nB(˜rj) = curlA(˜rj) = 2π/summationdisplay\ni/negationslash=jδ(˜rj−˜ri) = 2π/summationdisplay\ni/negationslash=jδ(rj−ri).\n(38)\nTo obtain the isotropic form of the Hamiltonian acting\nin thefermionic space, it is important that the Chern-\nSimons phase was defined with the rescaled coordinates\n˜r, Eq. (21). We can now specify the value of the rescal-\ning parameter α= (m/my)1/4∝(n/n0)1/4/lessorsimilar1. Notice\nthat sincemyis itself weakly dependent on α(through\ninteraction form-factors cand) the above definition of\nαis actually a self-consistent equation. Moreover, in-\nclusion of the magnetic field affects the wavefunctions of\nthe quasiparticles, e.g. a homogeneousfield results in the\nLandau quantization. That, in turn, modifies the matrix\nelements of interactions Mk1,k2\nk′\n1,k′\n2definingmy. However,because of the form of interaction Fk1,k2andGk1,k2the\neffective mass appears to depend only weakly on the spe-\ncific form of the wave functions, as long as those com-\nposed of plane waves with wave vectors in the vicinity of\nthe point (k0,0). This appears to be the case even for a\nquantizing magnetic field corresponding to a fully occu-\npied single Landau level. Therefore we use Eq. (34) for\nmyin the following and apply the effective description\nEq. (36) even for a quantizing magnetic field.\nThe importance of bringing the effective fermionic\nHamiltonian to the isotropic form (36) is to argue that\nthe kinetic energy does notcontain large logarithms. As\nexplained below Eq. (18), the kinetic energy in presence\nof the singular magnetic field (38) contain logarithmic\nintegrals. At small distance they are cut by the scale of\nthe correlation hole in the fermionic wavefunction Ψ F,\nwhile at large distance they are cut by the average dis-\ntance between the particles. In the isotropic fermionic\nstate described by the Hamiltonian (36) both of these\ntwo length scales are given by k−1\nF∼n−1/2. As a result,\nthe logarithms contain a number rather than a density-\ndependent parameter. Notice that without rescaling of\nCS phase such logarithms would contain log(1 /α), mak-\ningthekineticenergyoftheorder n3/2log(n0/n). Rescal-\ning avoids this large extra factor in the kinetic energy.\nKeeping the kinetic energy to be ∝n3/2by the\nfermionic nature of the state along with the careful\nrescaling of CS phase, suggests to employ the mean-field\ntreatment45,47of CS magnetic field. It substitutes the\ncollection of flux lines (38) by a uniform magnetic field\nwith the same total flux. The latter is given by 2D den-\nsity of particles (it is important that the rescaling of CS\nphase is area-preserving and thus not affecting the total\nflux),\nB(˜rj)→B= 2πn. (39)\nThe corresponding mean-field vector potential may be\nchosen as Ax(˜rj) =−πn˜yjandAy(˜rj) =πn˜xj. In\nthis approximation the Hamiltonian (36) represents a\none-body problem of a particle with the cyclotron mass\nmc=√mmy∝m/radicalbig\nn0/nin a constant magnetic field\n(39). The corresponding Landau spectrum is εl=\nωc(l+ 1/2), wherel= 0,1,...and the cyclotron fre-\nquencyωc= 2πn/mc. Since by construction we intro-\nduced exactly one flux quantum per particle, the corre-\nsponding filling factor is ν= 1. The N-body ground-\nstate wavefunction Ψ F(˜r1,...,˜rN) is thus given by the\nSlater determinant built from the single-particle wave-\nfunctions of the fully occupied lowest Landau level (LLL)\n(the center of mass shift in the direction of the ne-\nmatic order produces an additional multiplicative factor\nexp{ik0/summationtext\njxj}). In terms of rescaled complex coordi-\nnates ˜zj= ˜xj+i˜yj=αxj+iyj/αthe corresponding\nfermionic wavefunction takes the form50:\nΨF=CNN/productdisplay\ni<j(¯˜zi−¯˜zj)e−πnN/summationtext\nj|˜zj|2/2\neik0N/summationtext\njxj\n,(40)10\nwhereCNis normalization factor. The Chern-Simons\nphase may be also written in terms of the rescaled com-\nplex coordinates as/producttext(˜zi−˜zj)/|˜zi−˜zj|. As a result\nthe minimal-spin component of the bosonic wavefunction\n(21) takes the simple form\nΨ↓...↓= 2−N/2CNN/productdisplay\ni<j|˜zi−˜zj|e−πnN/summationtext\nj|˜zj|2/2\neik0N/summationtext\njxj\n.\n(41)\nThe higher spin components are obtained by acting with\nthe spin raising operators on the minimal spin compo-\nnent, Eq. (20). Notice that the spin-raising operator (7)\nis to be written in non-rescaled original coordinates rjto\nensure that the wavefunction is projected on the lower\nSO branch. The wavefunction (41) is independent of\nthe CS chirality λ. It depends on the single density-\nand interaction-dependent parameter α= (m/my)1/4∝\n(n/n0)1/4, which specifiesthe anisotropy. This wavefunc-\ntion describes an ellipsoidal droplet of the gas elongated\nalong thexdirection with the ratio of xandyaxes given\nbyα2. The average density in this droplet is n, while its\nsize depends on the number of particles N. The state\n(41) clearly breaks rotation symmetry. It also breaks\ntime-reversal symmetry as well as parity. Indeed, the\nparity operation transforms it into the state descendant\nfrom the maximal spin component, which is clearly a dif-\nferent, degenerate state. The average energy per particle\nfortheHamiltonian(36)insuchvariational N-bodystate\nis given by\nµF=E0+ε0=3π\n4√gn3/2\nmk0. (42)\nIt indeed confirms the expectation that the composite\nfermion function of Eqs. (20) and (21) yields the energy\nlower than TRSB and SDW bosonic states.\nIV. DISCUSSION OF THE RESULTS\nA. Phase Diagram\nAs we have seen, the composite fermion energy per\nparticle isµF∼µB/radicalbig\nn/n0, whereµB, Eq. (13), is the\nchemical potential of bosonic TRSB and SDW states.\nTherefore at small density n/lessorsimilarn0=gk2\n0the compos-\nite fermion (CF) groundstate is energetically favorable\nover bosonic states. On the other hand, at larger density\nn/greaterorsimilarn0the bosonic states, discussed earlier15,23–33, have\nlower energy. We expect thus to observe quantum phase\ntransitionsasfunctionsofthechemicalpotential µaswell\nas anisotropy of the interactions g2. The corresponding\nphase diagram is schematically depicted in Fig. 5.\nTofindoutaboutthenatureofthetransitionswerecall\nthat all three phases break rotational symmetry. In ad-\ndition TRSB breaks time-reversal, while CF breaks both\ntime-reversaland parity symmetries. We expect thus the\nfirst order transition between CF and SDW, where the0P\n2gcPSDW TRSB \nCF \nFIG. 5. (Color online) Phase diagram of Rashba SO bosons.\nThe full lines are first order transitions. Insets: schemati c\nrepresentation of occupation factors in momentum space (re d\ndots are coherent bosonic condensates, blue ellipse is CF\nFermi sea).\ntwo discrete symmetries ˆTandˆP, Eqs. (2), (3), are bro-\nken simultaneously. Also the transition between SDW\nand TRSB, taking place close to g2= 0, is to be of the\nfirst order. Indeed, to avoid density wave modulation in\nSDW phase the populations of two opposite states on the\nRashba circle must be exactly equal. Therefore the tran-\nsition into TRSB phase with only single populated state\nmust be of the first order.\nTransition from TRSB into CF phase breaks the par-\nity symmetry and could be of the first or second order.\nOne can investigate stability of the bosonic TRSB state\nagainst introduction of the small CF component. To this\nend one can write a variational wavefunction, which con-\ntains superposition of TRSB condensate, Eq. (9), with\nNbparticles and nematic CF fraction with Nf≪Nb\nparticles and Nb+Nf=N,\nΨ(r1,...,rN)∝/summationdisplay\nPΨ(0)\nB(r1,...,rNb)ΨCF(rNb+1,...,rN),\n(43)\nwherePstays for a permutation of arguments between\nNbandNfsets and we have suppressed spin indices for\nbrevity. It is important to realize that the bosonic con-\ndensate and a small nematic CF fraction prefer to be on\ntheoppositesidesofthe Rashbacircle, Fig.5. Indeed this\nway the spin parts of respective wavefunctions are (al-\nmost) orthogonal, minimizing the exchange energy. The\nresidual exchange interactions, due incomplete orthogo-\nnality, lead to CF effective mass myin the direction tan-\ngential to the Rashba circle. The energy per unit volume\nfor the state (43) is given by\nE=3πn2\nf\n4mc+(n−nf)2\n2mg0+(n−nf)nf\nm[g0+g2],(44)\nwherenf=Nf/Nand the fermionic cyclotron mass11\n0nP\n2 / 3nn\nFncnBncP0VE/2 / 5n2n\nn\ncp\u0010\nb) a) \nFIG. 6. (Color online) (a) Specific energy vs. density for\nCF and Bose condensate phases. The two graphs intersects\nat the critical density nc. In the range nF< n < n Bthere\nis the phase separation. (b) Chemical potential vs. density .\nMaxwell construction is shown. Compare it to1D case, Fig. 3.\nmc=√mmy=mk0//radicalbig\n(g0−g2)n/2 originates from\nthe exchange interactions of the CF fraction with the\nbosonic condensate (in the limit nf≪nthe interactions\nbetween composite fermions are less important). This\nmean-field expression shows that for g2>0 TRSB state,\ni.e.nf= 0, is always the energy minimum and thus it is\nstable against small CF fraction. An additional energy\nminimum develops at nf=nforn<2n0/3π. Although\none should not take Eq. (44) as quantitatively accurate\nbeyondnf≪nlimit, it is true that at small enough\ndensity the CF phase costs less energy than the bosonic\ncondensates. Together with the local stability of the con-\ndensate it suggests the firstorder transition into the CF\nphase.\nFor the first order transition the equation of state im-\nplies the range of density where the Bose condensate\n(TRSW or SDW) fraction separates from CF fraction.\nThis isillustratedon Fig.6a. In the region nF<n<n B,\ndetermined by the common tangential to EF(n) andEB(n), it is energetically favorable to spatially separate\nthe Bose condensate with the fixed density nBfrom the\nCF liquid with fixed density nF. Changing the total den-\nsity within this window results in the change between the\nrelative fraction of the two, not changing their individual\ndensities. In terms of the chemical potential vs. den-\nsity it means a constant critical chemical potential µc,\nfound from the Maxwell construction, within the window\nnF<n<n B, Fig. 6b. Alternatively, it means a range of\na constant critical quantum pressure pc, Fig. 6a, on the\npressure vs. volume T= 0 isotherm. To minimize the\ninterfacial energy TRSB and CF fractions prefer to have\nopposite momenta and spin. This allows avoiding paying\nthe exchange interaction energy thanks to orthogonality\nof the spinors. One expects thus to find antiferromag-\nnetically ordered mixture of the TRSB Bose condensate\nand CF fractions.\nB. Composite fermion state in a trap\nConsider an axially symmetric trap created by a har-\nmonic potential V(r) =mω2\n0r2. In the Thomas-Fermi\napproximation the local density n(r) is found from the\nconditionV(r) +µ(n(r))−µ= 0, where µ(n) repre-\nsents microscopic equation of state and µis the macro-\nscopic chemical potential, found from the condition\n2π/integraltextR\n0rdrn(r) =N, whereV(R) =µ. For CF equation\nof stateµ(n)∝√gn3/2/mk0, this program yields:\nµ∝ω0N3/5/parenleftbigggω0\nk2\n0/m/parenrightbigg1/5\n, R=/radicalbiggµ\nmω2\n0∝N3/10.\n(45)\nHereRrepresents the spatial extent of the N-particle\ngroundstate, while µisthe typicalkinetic energyperpar-\nticle measured upon trap release. These results should\nbe compared with those for the bosonic condensate with\nthe equation of state µ(n)∝gn. The latter yields53\nµ∝ω0(gN)1/2andR∝N1/4. The CF scaling of the\nchemical potential in the harmonic trap µF∝N3/5is\nvalid as long as it less than the corresponding bosonic\nresultµB∝N1/2. Equating the two of them, one finds\nthe condition for the number of particles within the trap\nbelow which the CF groundstate prevails\nN <N 0=g3/parenleftbiggk2\n0/m\nω0/parenrightbigg2\n. (46)\nThis is exactly the condition of having the density in the\nmiddle of the trap less than the critical one n(0)< nc.\nThe density profile acquires the shape n(r)∝(R2−\nr2)2/3. As opposed to the Bose condensate profile33\nn(r)∝(R2−r2), it exhibits infinite slop at the outer\nedger≈R.\nIn experiments of Refs. [1] and [4] synthetic gauge field\nand SO coupling had been engineered with the help of\nλ= 804.1nm Raman lasers, leading to the typical SO\nmomentum k0= 2π/λand energy scale k2\n0/m≃7kHz.12\nTaking a trap frequency ω0≃30Hzas in e.g. Ref. [55],\none obtains k2\n0/mω0≃0.23×103andN0∼g3×105.\nThe effective coupling constant gin 2D gas of Rb with\nas= 55˚Awas estimated to be56g≃0.2, leading to\nN0≃103, which is achievable with modern detection\ntechniques. Interestingly enough, studying much smaller\nparticle number experimentally became feasible with the\ndevelopment of single-atom detection technology57,58.\nSince the CF phase has filling factor one, we expect it\nto be gaped in the bulk of the trap. At the surface it\nsupports a chiral edge mode, which is similar to ν= 1\nquantum Hall edge. In this sense CF state of Rashba SO\nbosons is an interacting topological insulator .\nAt a larger particle number N >N 0the middle of the\ntrap has the density exceeding the critical one and thus\nit reverts to one of the Bose states (TRSB or SDW). The\ndensity decaystowardsthe edgesofthe trap, reaching nB\nat some radius. Here the phase separation, Fig. 6, takes\nplace and the density discontinuously drops down to nF.\nTheouterperipheryofthetrapcontains“vaporized”low-\ndensity CF phase, while its inner core is filled with the\n“liquid”high-densityBosecondensatephase. The overall\nscalingofthechemicalpotentialwiththeparticlenumber\napproaches the Bose one, µ∼N1/2.\nC. Rotating systems\nThe time-reversal and parity broken CF state is chi-\nral. It is not immediately obvious from the minimal\nspin component wavefunction (41), since it depends on\nthe absolute values |˜zi−˜zj|only. However, the spin-\nraising kernel (7) is certainly chiral and so are all the\nhigher spin components of the many-body bosonic wave-\nfunction (20). The degenerate state descending from the\nmaximal spin component (which has exactly the same\nform as Eq. (41)) has the opposite chirality. The CF\ngroundstate spontaneously breaks the Z2symmetry be-\ntween them. Rotation of the system serves as an ex-\nplicit symmetry breaking perturbation, enforcing one of\nthe sates vs. another. Indeed, rotation with the angu-\nlar frequency Ω may be viewed as an external magnetic\nfield54Brot= 2mcΩˆz, which adds up to CS magnetic\nfieldB=±2πnˆz. As a result the total flux is bigger or\nsmaller than one flux quantum per particle, depending\non whether CS magnetic filed is parallel or antiparallel\nwith the rotation direction. This either creates mcΩ/π\nholes per unit area in LLL, or promotes the same num-\nber of particles to the next Landau level. At least from\nthe standpoint of the kinetic energy the former alterna-\ntive requires twice less energy than the latter. One ex-\npects thus that the symmetry is broken in a way to add\nconstructively CS and rotational fields to create holes in\nLLL. Due to presence of these holes, one expects gapless\nbulk excitations in the rotating system. (Alternatively in\nanalogy with FQHE, excitations may have gaps substan-\ntially reduced compared to the Ω = 0 case.)D. Fractional Hall states?\nWe have employed Chern-Simons phase with one flux\nquantum per particle to convert composite fermions into\nbosons, i.e. λ=±1 in Eq. (21). One could also attach\nthree flux quanta by choosing λ=±3. This choice leads\nto the effective magnetic field B= 6πnand thus to ν=\n1/3 filling factor. The CF variationalgroundstate is then\ngiven by the Laughlin59state, i.e. ( ¯˜zi−¯˜zj)→(¯˜zi−¯˜zj)3\nin Eq. (40). Upon multiplication on CS phase with λ= 3\nit leads to the following expression for the minimal spin\ncomponent\nΨ↓...↓∝N/productdisplay\ni<j|˜zi−˜zj|3e−3πnN/summationtext\nj|˜zj|2/2\neik0N/summationtext\njxj\n,(47)\nnotice the factor of 3 πnin the exponent which ensures\nthe correct total density n. The kinetic energy of such\nstate is factor of three higher than that ascending from\nEq. (41). On the other hand, the correlation holes are\nwider, which may lead to a gain in the interaction energy\n(especially since the interactions due to higher spin com-\nponents are essentially non-local). While at the moment\nwe are not aware of a model where the fractional state is\nadvantageous, it is worth pointing out that such a model\nis in principle possible.\nNotice that symmetric minimal spin component may\nbe written with an arbitrary integer exponent pas∝\n/producttextN\ni<j|˜zi−˜zj|pe−pπnN/summationtext\nj|˜zj|2/2\neik0N/summationtext\njxj\n. However only odd\np’s may be traced to CF construction. This seems to\nindicate that states with even p’s yield larger average\nenergy. Thisisindeedthecasefor p= 0, whichisnothing\nbut TRSB condensate with µB∝n. It is not clear at\nthe moment how to demonstrate this statement for p=\n2,4,....\nE. Conclusions\nWe have shown that the low-density phase of Rashba\nSO bosons may be described as the composite fermion\nstateinthequantizingmagneticfiled. Suchastateisvery\ndifferent from both TRSB Bose condensate and SDW\nstate discussed before. In particular its equation of state\nµ(n)∝n3/2leads to a different scaling of the kinetic\nenergy and atomic cloud size with the number of parti-\ncles in shallow traps. It also implies different profile of\nthe cloud density. The excitation spectrum is predicted\nto be gaped in the bulk of trap with the gapless chiral\nsurface mode at its edge. For deeper traps we predict\nthe phase separation between denser condensate phase\nin the middle of the trap and dilute CF phase at its edge\nwith the first order density jump at the interface between\nthem.13\nV. ACKNOWLEDGMENTS\nWe are grateful to E. Altman, N. Cooper, G. Juzeliu-\nnas, A. Lamacraft and A. Vishwanath for useful discus-\nsions. LG thanks Aspen Center for Physics for hospi-\ntality. This work was supported by DOE contract DE-\nFG02-08ER46482.\nAppendix A\nHere we give details of calculations for the compo-\nnents of the two-particle wavefunction, Ψ ↑↑(r,r) and\nΨ↑↓(r,r) = Ψ ↓↑(r,r) in Eq. (26) and derive Eqs. (27),\n(28) of the main text.\n1. Calculation of Ψ↑↑(r,r)and the interaction\nform-factor Fk1,k2\nFrom Eq. (25) we have\nΨ↑↑(r,r) = (A1)\n1\n2/integraldisplay\ndr′\n1dr′\n2R(r−r′\n1)R(r−r′\n2)eiλarg(˜r′\n1−˜r′\n2)ΨF(r′\n1,r′\n2),\nwhere the fermionic part of the wave function is given by\nΨF(r1,r2) =1√\n2V/parenleftBig\neik1r1+ik2r2−eik2r1+ik2r2/parenrightBig\n(A2)\nand anisotropic vectors ˜r′\n1and˜r′\n2in the Chern-Simons\nfactoreiλarg(˜r′\n1−˜r′\n2)are defined as ˜r′\nj= (αx′\nj,y′\nj/α),j=\n1,2, whereαis the anisotropy parameter.\nTo represent the integrand in Eq. (A1) in a convenient\nform, we convert it to the momentum space. Fourier\ntransformation yields\neiλarg˜ r=−2πi/integraldisplaydk\n(2π)2eiλarg˜k\n˜k2eikr,(A3)\nwith˜k= (kx/α, αk y). Fourier image of R(r) isR(k) =\nie−iargk. SubstitutingnowEqs.(A2)and(A3)into(A1),\nwe obtain\nΨ↑↑(r,r) =−iei(k1+k2)r\n4V√\n2π/integraldisplay\ndkeiλarg˜k\n˜k2(A4)\n×/parenleftBig\ne−iarg(k1+k)−iarg(k2−k)−e−iarg(k2+k)−iarg(k1−k)/parenrightBig\nIn order to evaluate Fk1k2from (A4), it is convenient to\nintroduce the following notations\neiλarg˜k=kx/α+iλαky\n|˜k|=/parenleftbigg˜z\n|˜z|/parenrightbiggλ\n,\ne−iarg(k1+k)=¯w1+ ¯z\n|w1+z|, w1=k1,x+ik1,y,\ne−iarg(k2−k)=¯w2−¯z\n|w2−z|, w2=k2,x+ik2,y,\nz=kx+iky. (A5)The complex variable ˜ z=kx/α+iαkycan be conve-\nniently recast as ˜ z=z\n2(α+ 1/α) +¯z\n2(1/α−α). Em-\nploying integration variables zand ¯zand using complex\nrepresentation of vectors k1andk2, one obtains\nFk1k2=i√\n2/integraldisplaydzd¯z\n4π1\n|˜z|2/parenleftBigg\n˜z\n|˜z|/parenrightBiggλ/radicalbigg¯w1¯w2\nw1w2\n×\n¯z2\n¯w1¯w2−1+¯z(¯w1−¯w2)\n¯w1¯w2/radicalBig\n|¯z2\n¯w1¯w2−1+¯z(¯w1−¯w2)\n¯w1¯w2|2−(w1←→w2)\n.(A6)\nIn Eq. (A6) the short hand notation ( w1←→w2) stands\nfor the same first expression in square brackets but with\ninterchanged variables w1andw2.\nUpon introducing new dimensionless variables\nν=z√w1w2, τ=w1−w2√w1w2, (A7)\nEq. (A6) yields\nFk1k2=−i√\n2/radicalbigg¯w1¯w2\n|w1w2|/integraldisplaydνd¯ν\n4π(A8)\n×/parenleftBigg\n˜ν\n|˜ν|/parenrightBiggλ\n1\n|˜ν|2/parenleftBigg\n¯ν2−1+ ¯ν¯τ/radicalbig\n|¯ν2−1+ ¯ν¯τ|2−(τ→−τ)/parenrightBigg\n,\nwhere ˜ν=ν\n2(α+1/α)+¯ν\n2(1/α−α) and parameter τcan\nbe rewritten in terms of original momenta k1andk2as\nfollows\nτ=/radicalbigg\nk1\nk2ei(arg(k1)−arg(k2))/2−/radicalbigg\nk2\nk1e−i(arg(k1)−arg(k2))/2.\n(A9)\nThe integral on the right hand side of (A8) is convergent,\ntherefore one can expand the integrand in ¯ τand then\nperform the integration. Keeping only linear in ¯ τterms\nin this expansion one finds\nFk1,k2=ic/radicalbigg¯w1¯w2\n|w1w2|¯τ≃ic/parenleftBig\ne−iarg(k1)−e−iarg(k2)/parenrightBig\n,\n(A10)\nwhere\ncλ(α) =/integraldisplaydνd¯ν√\n24π/parenleftBigg\n˜ν\n|˜ν|/parenrightBiggλ\n1\n|˜ν|2(|ν|2−1)(¯ν2−1)(ν+ ¯ν)\n|¯ν2−1|3,\n(A11)\nThe functions|c±(α)|are plotted in Fig. 7.\n2. Calculation of Ψ↑↓(r,r)and the interaction\nform-factor Gk1k2\nFrom Eq. (25) one finds\nΨ↑↓(r,r) =1\n2/integraldisplay\ndr′\n1R(r−r′\n1)eiλarg(˜ r′\n1−˜ r)ΨF(r′\n1,r),\n(A12)14\n/s45/s53 /s48/s48/s53\n/s124/s100\n/s43/s44/s45/s124\n/s124/s99\n/s43/s44/s45/s124\n/s32/s32/s100 /s99\n/s108/s111/s103\nFIG. 7. (Color online) Interaction coefficients |cλ(α)|and\n|dλ(α)|. Full lines correspond to λ=−1 and dotted lines –\ntoλ= +1 chirality.\nwhere Ψ F(r′\n1,r) is defined in Eq. (A2), while spin-rising\noperatorRis given by Eq. (7). After substituting these\nexpressions into (A12) we obtain\nΨ↑↓(r,r) =ei(k1+k2)r\n2VGk1k2, (A13)\nGk1k2=−1√\n2π/integraldisplay\ndr′eiλarg(˜ r−˜ r′)−iarg(r−r′)\n|r−r′|2\n×/parenleftBig\ne−ik2(r−r′)−e−ik1(r−r′)/parenrightBig\nwhich shows that Gk1k2is a difference of two functions\nGk1k2=˜Ik1−˜Ik2. Introducing ( r−r′)x+i(r−r′)y=reiφ, andβ= log(α), we will have for λ= +1 chirality\neiarg(˜ r−˜ r′)=cosh[β]eiφ+sinh[β]e−iφ\n|cosh[β]eiφ+sinh[β]e−iφ|(A14)\n=cosh[β]eiφ+sinh[β]e−iφ\n/radicalbig\ncosh[2β](1+tanh[2 β]cos[2φ]).\nImportantly, the form-factor corresponding to the λ=\n−1 chirality can be obtained by interchanging cosh[ β]\nwith sinh[β] in the numerator of Eq. (A14). Then we\nobtain\n˜Ik=1\nπ/radicalbig\n2cosh[2β](A15)\n×/integraldisplaydr\nrdφ(cosh[β]+sinh[β]e−2iφ)e−ikrcos[φ−arg(k)]\n/radicalbig\n1+tanh[2β]cos[2φ].\nWenotethattheintegral ˜Ikhasalogarithmicaldivergent\ncontribution coming from r→0, but it cancels out in the\ndifference ˜Ik1−˜Ik2and therefore in Gk1k2. In order to\nfind the coefficient dλ(α) in Eq. (28) we need to evaluate\nintegral (A15) in linear overarg( k)≪2πapproximation.\nUpon expanding integrand over arg( k) and integrating\noverrwe arrive to the following expression\ndλ(α) =isλ\nπ/radicalbig\n2cosh[2β]/integraldisplay2π\n0dφ1−cos[2φ]/radicalbig\n1+tanh[2β]cos[2φ],\n(A16)\nwheres+= sinh[β] ands−= cosh[β]. The function\n|d±(α)|is plotted vs ln αfor both chiralities λ=±1 in\nFig. 7.\n1Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J.\nV. Porto, and I. B. Spielman, Phys. Rev.Lett. 102, 130401\n(2009).\n2Y.-J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V. Porto,\nand I. B. Spielman, Nature 462, 628 (2009).\n3Y-J. Lin, R. L. Compton, K. Jimenez-Garcia, W. D.\nPhillips, J. V. Porto, and I. B. Spielman, Nature Phys.\n7, 531 (2011).\n4Y.-J. Lin, K. Jimenez-Garcia, and I. B. 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Th is state dependence is successfully\ntaken into account by the density dependence, leading to the energy density functional. Recent\nresults for photoabsorption cross sections in spherical an d deformed Nd isotopes are shown.\n1. Introduction\nThe nucleus is a self-bound quantum system which presents a r ich variety of phenomena. It\nis composed of fermions of spin 1 /2 and isospin 1 /2, called nucleons (protons and neutrons),\ninteracting with each other through a complex interaction w ith a short-range repulsive core\n[1]. Remarkable experimental progress in production and st udy of exotic nuclei requires us to\nconstruct a theoretical model with higher accuracy and reli ability. Extensive studies have been\nmade in the past, to introduce models and effective interactio ns to describe a variety of nuclear\nphenomena and to understand basic nuclear dynamics behind t hem [1, 2]. Simultaneously,\nsignificant efforts have been made in the microscopic foundati on of those models. For light\nnuclei, the “first-principles” large-scale computation, s tarting from the bare nucleon-nucleon\n(two-body & three-body) forces, is becoming a current trend in theoretical nuclear physics.\nAlthough the ab-initio-type approaches have recently show n a significant progress, they are\nstill limited to nuclei with the small mass number. In contra st, the density functional model\nis a leading theory for describing nuclear properties of hea vy nuclei and perhaps the only\ntheory capable of describing all nuclei and nuclear matter w ith a single universal energy density\nfunctional. In the nuclear physics, it is often called the se lf-consistent mean-field model, because\nof a historical development based on the Brueckner-Hartree -Fock theory and introduction of\nthe effective interaction. I briefly review basic properties o f nuclei and discuss whether those\nproperties can be understood by a simple independent-parti cle model. Then, I recapitulate\ndevelopments in the microscopic many-body theory leading t o the nuclear density functional\nmodel with a Skyrme energy functional. Here, the saturation property plays a key role to\nunderstand the nuclear force and the effective interaction.\nIntensivestudiesin nuclear density functional modelsinr ecent years have producednumerous\nresults and new insights into nuclear structure [3, 4]. Howe ver, it is impossible to review all of\nthem in this short paper. Thus, I will present a result of our r ecent study with the time-\ndependent density functional approach, on the photoabsorp tion cross sections in the rare-earth\nnuclei [5].\n1This work is supported by Grant-in-Aid for Scientific Resear ch in Japan (Nos. 21340073 and 20105003).2. Independent particle model for nuclei\nNuclei are known to be well characterized by the saturation p roperty. Namely, they have an\napproximately constant density ρ0≈0.17 fm−3, and a constant binding energy per particle\nB/A≈16 MeV.2In this section, I show that the nuclear saturation property has a great impact\non nuclear models. Especially, it is inconsistent with the i ndependent-particle model of nuclei\nwith a “naive” average (mean-field) potential.\nThere are many evidences for the fact that the mean-free path of nucleons is larger than\nthe size of nucleus. In fact, the mean free path depends on the nucleon’s energy, and becomes\nlarger for lower energy [1]. Therefore, it is natural to assu me that the nucleus can be primarily\napproximated by the independent-particle model with an ave rage one-body potential. The\ncrudest approximation is the degenerate Fermi gas of the sam e number of protons and neutrons\n(Z=N=A/2). The observed saturation density of ρ0≈0.17 fm−3gives the Fermi\nmomentum, kF≈1.36 fm−1, that leads to the Fermi energy (the maximum kinetic energy) ,\nTF=k2\nF/2M≈40 MeV.\nFirst, I show that the independent-particle model with a con stant attractive potential V <0\ncannot describe the nuclear saturation property. It follow s from the simple arguments. The\nconstancy of B/Ameans that it is approximately equal to the separation energ y of nucleons, S.\nIn the independent-particle model, it is estimated as\nS≈B/A≈ −(TF+V). (1)\nSince the binding energy is B/A≈16 MeV, the potential Vis about −55 MeV. It should be\nnotedthattherelatively smallseparationenergyisthecon sequenceofthesignificantcancellation\nbetween kinetic and potential energies. The total (binding ) energy is given by\n−B=A/summationdisplay\ni=1/parenleftbigg\nTi+V\n2/parenrightbigg\n=A/parenleftbigg3\n5TF+V\n2/parenrightbigg\n, (2)\nwhere we assume that the average potential results from a two -body interaction. The two kinds\nof expressions for B/A, Eqs. (1) and (2), lead to TF≈ −5V/4≈70 MeV, which is different from\nthe previously estimated value ( ∼40 MeV). Moreover, it contradicts the fact that the nucleus\nis bound (TF<|V|).\nTo reconcile the independent-particle motion with the satu ration property of the nucleus,\nthe nuclear average potential should be state dependent. Al lowing the potential Videpend on\nthe statei, the potential Vshould be replaced by that for the highest occupied orbital VFin\nEq. (1), and by its average value /angbracketleftV/angbracketrightin the right-hand side of Eq. (2). Then, we obtain the\nfollowing relation:\nVF≈ /angbracketleftV/angbracketright+TF/5+B/A. (3)\nTherefore, the potential VFis shallower than its average value.\nWeisskopf suggested the momentum-dependent potential V, which can be expressed in terms\nof an effective mass m∗[6]:\nVi=U0+U1k2\ni\nk2\nF. (4)\nActually, if the mean-field potential is non-local, it can be expressed by the momentum\ndependence. Equation (4) leads to the effective mass, m∗/m= (1 +U1/TF)−1. Using Eqs.\n(1), (3), and (4), we obtain the effective mass as\nm∗\nm=/braceleftbigg3\n2+5\n2B\nA1\nTF/bracerightbigg−1\n≈0.4. (5)\n2This is the extrapolated value for the infinite nuclear matte r without the surface and the Coulomb energy.\nThe observed values for finite nuclei are B/A≈8 MeV.Quantitatively, this value disagrees with the experimenta l data. The empirical values of the\neffective mass vary according to the energy of nucleons, 0 .7/lessorsimilarm∗/m/lessorsimilar1, however, they are\nalmost twice larger than the value in Eq. (5). As far as we use a normal two-body interaction,\nthis discrepancy should be present in the mean-field calcula tion with any interaction, because\nEq. (5) is valid in general for a saturated self-bound system . Therefore, the conventional\nmodels cannot simultaneously reproduce the most basic prop erties of nuclei; the binding energy\nand the single-particle property. This suggest the importa nce of the state-dependent effective\ninteraction, which will be discussed in Sec. 4.\n3. Nucleon-nucleon interaction (nuclear force)\nThesaturationpropertyofnucleardensityindicates theba lancebetween attractive andrepulsive\ncontributions to nuclear binding energy. One source of such repulsive effects is the nucleonic\nkinetic energy of the Fermi gas. However, its contribution p er particle is proportional to ρ2/3,\nwhich is not strong enough to resist against the collapse cau sed by the attractive force between\nnucleons. Therefore, the nucleonic interaction must conta in a repulsive element. Indeed, the\nphase-shift analysis on the nucleon-nucleon scattering at high energy ( E >250 MeV) reveals\na short-range strong repulsive core in the nucleonic force. The radius of the repulsive core\nis approximately c≈0.5 fm. This strong repulsive core prevents the nucleons appro aching\ncloser than the distance c, which produces a strong two-body correlation, ρ(2)(/vector r1,/vector r2)≈0 for\n|/vector r1−/vector r2|<c. Theattractive partoftheinteraction hasalonger range, w hichcanbecharacterized\nby the pion’s Compton wave length λπ, and is significantly weaker than the repulsion. Thus,\na naive application of the mean-field calculation fails to bi nd the nucleus, since the mean-field\napproximation cannot take account of such strong two-body c orrelations.\nAt first sight, this seems inconsistent with the experimenta l observations. As I mentioned in\nSec. 2, there are many experimental evidences for the indepe ndent-particle motion in nuclei. We\nmay intuitively understand that it is due to the fact that the nucleonic density is significantly\nsmaller than 1 /c3. Therefore, the collisions by the repulsive core rarely occ ur and the system\ncan be approximately described in terms of the independent- particle motion. Furthermore, the\neffects of the Pauli principle hinder the collisions, since th e nucleons cannot be scattered into\noccupiedstates. Althoughtherepulsive-corecollisions a reexperiencedbyonly asmall fraction of\nnucleons ( ∼ρ0c3), each collision carries a large amount of energy. Therefor e, the repulsive core\nprovides an important contribution to the total energy and a re responsible for the saturation.\nAnother important factor for the independent particle moti on is the strong quantum nature\ndue to the weakness of the attractive part of the nuclear forc e. The importance of the quantum\nnature can be measured by the magnitude of the zero-point kin etic energy compared to that\nof the interaction. If the attractive part of the nuclear for ce were much stronger than the unit\nof/planckover2pi12/Mc2, the quantum effect would disappear and each nucleon would sta y at the bottom of\nthe interaction potential (cf. Fig. 2-36 in Ref. [1]). Then, the nucleus would crystallize at low\ntemperature. In reality, the attraction of the nuclear forc e is so weak that it barely produces\nmany-nucleon bound states at the relatively low density. In Sec. 2, I have shown that, in\nnuclear binding energy, there is a strong cancellation betw een the positive kinetic energy and\nthe negative potential energy. The nucleonic kinetic energ y plays an important role in many\nphenomena in nuclei, which can be described in terms of the in dependent-particle motion.\n4. Density-dependent Hartree-Fock method and energy density functional\nThenuclearmattertheorypioneeredbyBruecknergivesahin tforasolutionfortheinconsistency\nbetween the nuclear saturation and the independent-partic le model. Details of the theory can\nbe found at Refs. [7, 2]. The independent-particle motion un der the presence of the interaction\nwith a repulsive core was qualitatively discussed in Sec. 3. The Brueckner theory may providea first step toward the quantitative treatment to understand the saturation property and the\nindependent-particle motion in nuclei.\nThe basic ingredient of the Brueckner theory is a two-body sc attering matrix of particle 1\nand 2 inside nucleus caused by the nuclear force v,\nG(ω)≡v+vQ\nω−Q(T1+T2)QG(ω), (6)\nwhereTiis the kinetic energy of particle i,Qis the Pauli-exclusion operator to restrict the\nintermediate states, and ωis called a starting energy that depends on energies of parti cle 1 and\n2. This is called G-matrix [8]. The G-matrix renormalizes high-momentum components in the\nbare nuclear force and becomes an effective interaction in nuc lei under the independent-pair\napproximation. The G-matrix reflects an underlying structure of the independent many-nucleon\nsystem through the operator Qand the starting energy ω. Inevitably, the G-matrix becomes\nstate (structure) dependent.\nSince the short-range singularity is renormalized in the G-matrix, we can calculate the total\nenergy in the independent-particle (mean-field) model, ana logous to Eq. (2).\n−B=A/summationdisplay\ni=1\n\nTi+1\n2A/summationdisplay\nj=1¯Gij,ij(ωij)\n\n(7)\nwhereωij=ǫi+ǫj, defines the self-consistency condition for the Brueckner’ s single-particle\nenergies, and ¯Gij,ij≡Gij,ij−Gij,ji. This is called Brueckner-Hartree-Fock (BHF) theory. The\nvalidity of the BHF theory is measured by the wound integral κ=/angbracketleftψ−φ|ψ−φ/angbracketright, where|φ/angbracketrightis\nan unperturbed two-particle wave function and |ψ/angbracketrightis a correlated two-particle wave function in\nnucleus.κis known to be of the order of 15 %. The BHF calculation was succ essful to describe\nthe nuclear saturation, however, could not reproduce simul taneouslyB/Aandρ0, known as a\nproblem of the Coester band [9]. Its applications to finite nu clei also quantitatively failed to\nreproduce the energy, radius, and density in the ground stat e.\nThese problems are somewhat miraculously solved by the dens ity-dependent Hartree-Fock\n(DDHF) theory by Negele [10]. Starting from a realistic G-matrix, first, the local density\napproximation is introduced, using the expressions for the Pauli operator\n/angbracketleft/vector r1/vector r2|Q|/vector r′\n1/vector r′\n2/angbracketright=/braceleftbig\nδ(/vector r1−/vector r′\n1)−ρ(/vector r1−/vector r′\n1)/bracerightbig/braceleftbig\nδ(/vector r2−/vector r′\n2)−ρ(/vector r2−/vector r′\n2)/bracerightbig\n, (8)\nand the average single-particle energy ǫ[ρ(/vector r)]. Then, a short-range part of the G-matrix, which\nis not fully understood, is phenomenologically added to the energy expression to quantitatively\nfit the saturation property, and finally, the total energy is t reated variationally. This procedure\nis called the density matrix expansion (DME) [11]. The state dependence of the G-matrix is\nnow replaced by the density dependence. The final result for t he energy is of the form\nE[ρ] =/integraldisplay\nd/vectorRH(/vectorR), H(/vectorR) =H[ρ(/vectorR)] =H[ψ∗,ψ], (9)\nwhich is completely analogous to the Hamiltonian density of the Skyrme energy functional [12].\nThe essential aspect of the DDHF comes from the density depen dence and the variational\ntreatment. The variation of the total energy with respect to the density contains re-arrangement\npotential,∂Veff[ρ]/∂ρ, which appear due to the density dependence of the effective fo rceVeff[ρ].\nThese terms turn out to be crucial to obtain the saturation co ndition. Now, the expression for\nthe total energy, Eq. (2), should be modified to include the re -arrangement effect. This resolvesthe previous issue, then provides a consistent independent -particle description for the nuclear\nsaturation.\nIn summary, the failure in the mean-field description of nucl ei using phenomenological\neffective interactions can be traced back to the missing state (structure) dependence. The\nDDHF takes into account the state dependence in terms of the d ensity dependence. The energy\nfunctional obtained by the DME is essentially identical to t he Skyrme energy functional. This\nprovides a foundation for the nuclear energy functional.\n5. Applications to giant resonances\nThe giant resonance is a typical collective motion in nuclei , which exhausts a major part of\nthe sum-rule value. They are also related to the basic proper ties of nuclear matter, such as\nincompressibility K∞, effective mass m∗, etc. For instance, the nuclear matter incompressibility\nis extracted from properties of the giant monopole resonanc es [13],\nK∞= 210±30 MeV. (10)\nThis gives a restriction on the density dependenceof the phe nomenological short-range repulsive\npart of the energy functional. The effective mass deduced from the analysis on the giant\nquadrupole resonances is [13]\nm∗/m≈0.8∼1. (11)\nThese values are consistent with the DDHF in Sec. 4, but incon sistent with the naive mean-field\nvalue of Eq. (5).\n-1000100200300400500600700800\n828486889092Q0 (e2fm2)\nNNd\nFigure 1. Calculated (open\nsquares) and experimental (filled)\nintrinsic quadrupole moment for Nd\nisotopes [5].The giant resonances can be reasonably described by\na small-amplitude approximation of the time-dependent\nversion of the DDHF. In this approach, again, the re-\narrangement terms, such as ∂Veff[ρ]/∂ρand∂2Veff[ρ]/∂ρ2\nshould be consistently taken into account. The theory\ncan be regarded as the time-dependent density-functional\ntheory, founded by Runge and Gross [14].\nA modern energy functional for nuclei is a functional\nof many kinds of density, such as kinetic τ(/vector r) and spin-\norbit density /vectorJ(/vector r). In addition, to describe superfluid\nnuclei with pairing correlation, we need to add the pair\n(abnormal) density κ(/vector r). These densities are collectively\ndenoted as ˜ ρin the followings. Variation of the\ntotal energy, E[˜ρ], leads to the Hartree-Fock-Bogoliubov\nequation:\nH[Ψ,Ψ∗]|Ψµ/angbracketright=Eµ|Ψµ/angbracketright, (12)\nwhereEµand|Ψµ/angbracketrightare quasi-particle energies and states, respectively [2]. |Ψµ/angbracketrightis composed\nof two components; the upper |Uµ/angbracketrightand the lower one |Vµ/angbracketright. The solution of Eq. (12) defines\nthe normal density ρ(/vector r) =/summationtext\nµVµ(/vector r)V∗\nµ(/vector r), the pair density κ(/vector r) =/summationtext\nµUµ(/vector r)Vµ(/vector r), and other\ndensities at the ground state. Since h[Ψ,Ψ∗] depends on these densities, Eq. (12) must be\nsolved in a self-consistent way. Minimization of the energy density functional may lead to a\nspontaneous breaking of symmetry. An example is given in Fig . 1 for Nd isotopes [5]. The\nintrinsic quadrupole moment calculated with the Skyrme fun ctional of SkM* is compared with\nthe experimental data. At N= 82, the nucleus at the ground state is spherical Q0= 0, while\nforN= 86∼92, the deformation gradually develops. The observed groun d-state deformations\ndeduced from the transition probability B(E2;2+→0+) are nicely reproduced. Note that there\nare no adjustable parameters in this calculation.0100200300\n152\nNdσabs (mb)(a)\n0100150\nNd\n0100148\nNd\n0100146\nNd\n0100144\nNd\n0100\n510152025\nE (MeV)142\nNd\nFigure 2. Calculated (lines)\nand experimental (sym-\nbols) photoabsorption cross\nsections for Nd isotopes [5].For a study of the giant resonances, we need to extend the\nenergy functional to include the time-odd densities, such a s the\nspin density /vector s(/vector r) and the current density /vectorj(/vector r). Now, these\ndensities are time dependent. The time-dependent version o f\nEq. (12) is\ni∂\n∂t|Ψµ(t)/angbracketright=H[Ψ(t),Ψ∗(t)]|Ψµ(t)/angbracketright. (13)\nAssuming the oscillation with a fixed frequency, this equati on\nis linearized with respect to the fluctuation of the densitie s\naround those at the ground state. This leads to the matrix\nform of the equation identical to the quasi-particle random -\nphase approximation (QRPA) [2].\n/summationdisplay\nγδ/parenleftbigg\nAαβ,γδBαβ,γδ\n−Bαβ,γδ−Aαβ,γδ/parenrightbigg/parenleftbigg\nXγδ\nYγδ/parenrightbigg\n=/planckover2pi1ω/parenleftbigg\nXαβ\nYαβ/parenrightbigg\n.(14)\nTheQRPA matrix, AandB, are calculated in thequasi-particle\nbasis, then the normal modesof excitation andtheir energie s are\nobtained from Eq. (14).\nThecalculated photoabsorptioncrosssections forNdisoto pes\nare shown in Fig. 2. For spherical nuclei ( N= 82 and 84),\nthe photoabsorption has a single peak for the photon energy o f\nE≈15 MeV. The increase of the neutron number results in the\nbroadening of the peak, which well agree with the experiment al\ndata. Thisis duetotheincreaseoftheground-statedeforma tion\nshown in Fig. 1. The calculated energy-weighted sum-rule va lues for these isotopes are about\n40 % larger than the classical Thomas-Reiche-Kuhn sum-rule value, because of the momentum\nand isospin dependence of the mean field (Kohn-Sham) potenti al.\n6. Summary\nThe nuclear density functional approaches were developed a s the mean-field theory with the\ndensity-dependent effective interactions. The density-dep endent Hartree-Fock theory succeeded\nto describe both the total energy and single-particle prope rties, simultaneously. In contrast, the\nnormal mean-field calculation with a phenomenological two- body interaction fails. The density-\ndependenceisakey ingredienttounderstandthenuclearsat urationandtheindependent-particle\nmotion. Numerical applications were shown for giant resona nces in shape transitional nuclei.\nReferences\n[1] Bohr A and Mottelson B R 1969 Nuclear Structure, Vol. I (New York: W. A. Benjamin)\n[2] Ring P and Schuck P 1980 The nuclear many-body problems (New York: Springer-Verlag)\n[3] Bender M, Heenen P H and Reinhard P G 2003 Rev. Mod. Phys. 75121–180\n[4] Lunney D, Pearson J M and Thibault C 2003 Rev. Mod. Phys. 751021–1082\n[5] Yoshida K and Nakatsukasa T 2011 Phys. Rev. C 83021304\n[6] Weisskopf V F 1957 Nucl. Phys. 3423–432\n[7] Day B D 1967 Rev. Mod. Phys. 39719;Rev. Mod. Phys. 50495\n[8] Bethe H A and Goldstone J 1957 Proc. Roy. Soc. (London) A238551\n[9] Day B D 1981 Phys. Rev. Lett. 47226;Phys. Rev. C. 241203\n[10] Negele J W 1970 Phys. Rev. C 11260–1321\n[11] Negele J W and Vautherin D 1972 Phys. Rev. C 51472–1493\n[12] Vautherin D and Brink D M 1972 Phys. Rev. C 5626–647\n[13] Blaizot J P 1980 Phys. Rep. 64171–248\n[14] Runge E and Gross E K U 1984 Phys. Rev. Lett. 52997–1000"    },    {        "title": "1209.6353v1.Does_the_derivative_of_the_energy_density_functional_provide_a_proper_quantitative_formulation_of_electronegativity_.pdf",        "content": " 1arXiv:1209 \n \n \n \nDoes the derivative of the energy density functiona l provide a \nproper quantitative formulation of electronegativit y? \n \n \nTamás Gál \n \n \nDepartment of Theoretical Physics, University of Pé cs,  \nPécs, Hungary \n \n \n \n \nAbstract:  It is pointed out that the derivative of the energ y density functional does not \nprovide a valid local electronegativity measure, in  spite of its appealing property of becoming \nconstant for ground-state equilibrium systems. \n \n  2 Density functional theory [1] has been considered as a natural background for defining \nchemical reactivity indicators since Parr et al. [2 ] identified the Lagrange multiplier in the \ncentral Euler-Lagrange equation of DFT, \n            µδδ=)(][\nrnnEvv  ,       (1) \nas the negative of electronegativity, and proposed a very a ppealing interpretation of minus the \nderivative of the energy density functional as a generall y local electronegativity measure, \nwhich equalizes when an electron system reaches its ground-stat e equilibrium. This would \nthen provide a formal background for the electronegativit y equalization principle [3]. \nHowever, so far, the question as to whether \n)(][\nrnnEvvδδ−  can indeed be considered as a measure \nof electronegativity even when the given system is not in its ground state (e.g., two molecules \nbefore interaction) has not been examined. In the following, we will examine this question, \nconcluding a negative answer.  \n The electronegativity as defined by the negative of the chem ical potential \n      \n)(rvNE\nv\n\n\n∂∂=µ  ,      (2) \nappearing as the Lagrange multiplier in Eq.(1), characterize s the change of the ground-state \nenergy induced by a change in the number of electrons in a fixed external potential setting. \nThe main feature of this electronegativity/chemical potential  concept, thus, is that it describes \nenergy change due to electron number change. Consequently,  its local, non-equilibrium \ngeneralization, by \n         )()(][)( rvrnnFrvvv+ =δδµ  ,      (3) \nshould also characterize energy changes as the electron number  changes – but locally.  \n Consider a functional ][nFN that equals ][nF  for ) (rnv’s of a given N, but otherwise \nis different from it. This implies that the derivat ive of ] [nFN with respect to ) (rnv, at ) (rnv’s \nof N, differs from that of ] [nF  by some constant (only), \n         crnnF\nrnnFN N N+ =)(][\n)(][\nv vδδ\nδδ .      (4) \nWith ][nFN, then, we have a local quantity \n            cr r + =′ )( )(vvµ µ  .      (5)  3][nFN cannot contain information about how the energy (i ts internal component) behaves \nwhen the electron number changes, since ][nFN may even be constant with respect to changes \nin the density that go out of the ) (rnNv domain. Consequently, ) (rvµ′ cannot be a general, \nlocal chemical potential, which characterizes E vs N locally. But then ) (rvµ  cannot \ncharacterize E vs N locally either, as ) (rvµ′ and ) (rvµ  differ only by a constant. It will  \ncharacterize something local, but that is not the N-dependence of E; in other words, it is not \nµ NE∂∂= / that has been “localized” in Eq.(3). Similar argum ent holds for the local \nhardness concept of \n        \nv NvNrr \n\n\n∂∂=],)[ ()(vv µη  ,      (6) \ni.e. [4] \n     ∫′′′= r drfrnrnFrvvvvv)()()()(2\nδ δδη  ,     (7) \nwhich becomes constant for ground-state densities [ 5]. That is, on Eq.(6), a hardness \nequalization principle [6] cannot be based. \n In conclusion, density functional derivatives do n ot provide a good basis for  \nnon-equilibrium generalizations of ground-state, co nstant reactivity indices, such as \nelectronegativity and hardness.  \n \nReferences \n \n  [1] R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules   \n        (Oxford University Press, New York, 1989). \n  [2] R. G. Parr, R. A. Donnelly, M. Levy, and W. E . Palke, J. Chem. Phys. 68, 3801 (1978). \n  [3] R. T. Sanderson, Science 114 , 670 (1951). \n  [4] S. K. Ghosh, Chem. Phys. Lett. 172 , 77 (1990). \n  [5] M. K. Harbola, P. K. Chattaraj, and R. G. Par r, Isr. J. Chem. 31 , 395 (1991). \n  [6] D. Datta, J. Phys. Chem. 90 , 4216 (1986). \n \n "    },    {        "title": "1210.2281v1.Engineering_arbitrary_pure_and_mixed_quantum_states.pdf",        "content": "arXiv:1210.2281v1  [quant-ph]  8 Oct 2012Engineering arbitrary pure and mixed quantum states\nAlexander Pechen∗\nSteklov Mathematical Institute of Russian Academy of Scien ces, Gubkina str. 8, Moscow 119991, Russia\nDepartment of Chemical Physics, Weizmann Institute of Scie nce, Rehovot 76100, Israel\nThis work addresses a fundamental problem of controllabili ty of open quantum systems, meaning\nthe ability to steer arbitrary initial system density matri x into any final density matrix. We show\nthatundercertain general conditions openquantumsystems are completely controllable andpropose\nthe first, to the best of our knowledge, deterministic method for a laboratory realization of such\ncontrollability which allows for a practical engineering o f arbitrary pure and mixed quantum states.\nThe method exploits manipulation bythe system with a laser fi eld anda tailored nonequilibrium and\ntime-dependent state of the surrounding environment. As a p hysical example of the environment\nwe consider incoherent light, where control is its spectral density. The method has two specifically\nimportant properties: it realizes the strongest possible d egree of quantum state control — complete\ndensity matrix controllability which is the ability to stee r arbitrary pure and mixed initial states\ninto any desired pure or mixed final state, and is “all-to-one ”, i.e. such that each particular control\ncan transfer simultaneously all initial system states into one target state.\nControlled manipulation by atoms and molecules us-\ning external controls is an active field of modern re-\nsearch with applications ranging from selective creation\nofatomicormolecularexcitations outto controlofchem-\nical reactions or design of nanoscale systems with desired\nproperties. The external control may be either coherent\n(e.g., a tailored laser pulse [1]) or incoherent (e.g., a spe-\ncially adjusted or engineered environment [2, 3] or quan-\ntum measurements commonly used with and sometimes\nwithout feedback [4]).\nA challenging topic in quantum control is to pro-\nvidepracticalmethodsforengineeringarbitraryquantum\nstates [5]. The interest to this topic is driven by funda-\nmental connections to quantum physics as well as by po-\ntential applications to quantum state measurement [5]\nand quantum computing with mixed states and non-\nunitary quantum gates [6]. Various recipes for engineer-\ningarbitraryquantumstatesoflightwereproposed[7,8].\nFor matter, engineering arbitrary open system’s quan-\ntum dynamics with coherent control, quantum measure-\nmentsand feedbackwasshownto be achievable[9]. How-\never, the problem of deterministic engineering of arbi-\ntrary quantum states of matter has generally been re-\nmained unsolved.\nThis works proposes the first, to the best of the au-\nthor’s knowledge, deterministic method for engineering\narbitrary pure and mixed density matrices for a wide\nclass of quantum systems. The method uses a combina-\ntion of incoherent control by engineering the state of the\nenvironment(forwhichweconsiderappropriatelyfiltered\nincoherent radiation) on the time scale of several orders\nof magnitude of the characteristicsystem relaxation time\nτrelfollowed by fast (e.g., femto-second) coherent laser\ncontrol to produce arbitrary pure and mixed states for a\nwide class of quantum systems. The method is determin-\nistic in the sense that it does not use real-time feedback\n∗apechen@gmail.com; http://www.mathnet.ru/eng/person1 7991andcanbe appliedtoan ensembleofsystemswithout the\nneed for an individual addressing of each system. Impor-\ntant is that the suggested scheme requires the ability to\nmanipulate by both Hamiltonian and non-Hamiltonian\naspects of the dynamics; a fundamental result of Altafini\nshows that varying only the Hamiltonian is not sufficient\nto produce arbitrary states of a quantum system [10].\nEngineered environments were suggested for improv-\ning quantum computation and quantum state engineer-\ning [11], making robust quantum memories [12], prepar-\ning many-body states and non-equilibrium quantum\nphases [13], inducing multiparticle entanglement dynam-\nics [14]. This work exploits incoherent control by engi-\nneered environment [2] in a combination with coherent\ncontrol for producing arbitrary pure and mixed density\nmatrices. While incoherent processes were used in var-\nious circumstances, as for example in cold molecule re-\nsearch, to prepare a pure state needed for a full control\nover the system’s pure states [15], their use in the pro-\nposed method serves for a more general goal of engineer-\ning arbitrary pure and mixed quantum states.\nThe method has two special properties. First, it im-\nplements complete density matrix controllability — the\nstrongest possible degree of state control for quantum\nsystems meaning the ability to prepare in a controllable\nway any density matrix starting from any initial state.\nSecond, the produced controls are all-to-one — any such\ncontrolc∗transfers all pure and mixed initial states into\nthe same final state and thus can be optimal simultane-\nously for all initial states [16]. This property has no ana-\nlogforpurelycoherentcontrolofclosedquantumsystems\nwith unitary evolution, where different initial states in\ngeneral require different optimal controls. While an ab-\nstract theoretical construction of all-to-one controls was\nprovided [16], the problem of their physical realization\nhas remained open. The suggested method provides a\nsolution for such a physical realization.2\nI. COHERENT AND INCOHERENT\nCONTROLS\nThe dynamics of a controlled n-level quantum system\nisolated from the environment is described by density\nmatrixρtsatisfying the equation\n˙ρt=−i[H0+u(t)V,ρt], ρ t=0=ρi.(1)\nHereH0=/summationtextεi|i/an}bracketri}ht/an}bracketle{ti|is the free system Hamiltonian\n(with eigenvalues ε1< ε2< ... < ε nand eigenvectors\n|i/an}bracketri}ht) andVis the interaction Hamiltonian describing cou-\nplingofthe systemtothecontrolfield u(t)(e.g., ashaped\nlaser pulse)l; commonly V=−µ, whereµis the dipole\nmoment of the system. The evolution is unitary, ρt=\nUtρ0U†\nt, where the unitary evolution operator Utsatis-\nfies the Schr¨ odinger equation ˙Ut=−i[H0+u(t)V]Ut.\nThe unitary nature of the evolution induced by the field\nu(t) implies preservation of coherence in the system such\nthat for example pure states will always remain pure; the\ncorresponding control is called coherent.\nIf the system interacts with the environment, then its\nevolution in the absence of coherent control ( u(t) = 0) is\ndescribed by the master equation for the reduced density\nmatrixρt[17, 18]\n˙ρt=−i[H0+Heff,ρt]+L(ρt), ρ t=0=ρi(2)\nwhere the superoperator Land the effective Hamiltonian\nHeffdescribe the influence of the environment. The ef-\nfective Hamiltonian Heffrepresents spectral broadening\nof the system energy levels and typically commutes with\nH0. For a Markovian environment the superoperator L\nhas the form L(ρ) =/summationtext\ni(2LiρL†\ni−L†\niLiρ−ρL†\niLi) with\nsome matrices Li[17, 19]. The explicit form of these ma-\ntrices is determined by the state of the environment and\nby the details of the microscopic interaction between the\nsystem and the environment.\nThe matrices Liare usually considered as fixed and\nhaving deleterious effect on the ability to control the sys-\ntem — open quantum systems subject to the Markovian\nevolution (2) with constant Lare uncontrollable [10].\nHowever, the assumption of constant Lis too restric-\ntive, since Lcan be manipulated by adjusting the state\nof the environment through its temperature, pressure, or\nmore generally through its distribution function (spec-\ntral density). Control through adjusting the state of the\nenvironment in general does not preserve quantum co-\nherence in the controlled system and for this reason is\ncalled incoherent [3].\nThis work considers incoherent radiation as an exam-\npleofaMarkoviancontrolenvironmentandtheschemeis\nanalyzed below for this case. Other environments, either\nMarkovian or non-Markovian, can also be used as inco-\nherent controls; however, the ability to use a particular\nenvironmentforengineeringarbitraryquantumstatesus-\ning the proposed scheme requires a separate analysis in\neach case. The state of the environment formed by in-\ncoherent photons is characterized at a time moment tbyspectral density nk(t) of photons with momentum k; in\ngeneral, spectral density of photons with polarization α\ncanalsobeexploited. Forthepurposeofthisworkthedi-\nrectional dependence of spectral density is not necessary\nand its dependence only in the photon energy ω=|k|is\nused;nωis assumed to be constant over the frequency\nrange of significant absorption and emission for each sys-\ntem transition frequency ωij=εj−εi. The spectral\ndensitynω(t) can be experimentally manipulated for ex-\nample by filtering.\nThe evolutionofthe system in the environmentformed\nby incoherent radiation with spectral density nωis de-\nscribed by the master equation (2) with superoperator\nL=Lnωof the form [20, 21]\nLnω(ρ) =/summationdisplay\ni<jAij/bracketleftbig\n(nωij+1)LQij(ρ)+nωijLQji(ρ)/bracketrightbig\n(3)\nwhereAij≥0 are the Einstein coefficients for sponta-\nneous emission, ωij=εi−εjare the system transition\nfrequencies, Qij=|i/an}bracketri}ht/an}bracketle{tj|is the transition operator for the\n|j/an}bracketri}ht → |i/an}bracketri}httransition, and LQ(ρ) = 2QρQ†−Q†Qρ−ρQ†Q.\nThe intensities u(t) andnωof the coherent and incoher-\nent light are the coherent and incoherent controls, re-\nspectively. Radiative energy density per unit angular\nfrequency interval is ρω=/planckover2pi1ωnω(ω2/π2c3).\nII. ENGINEERING ARBITRARY QUANTUM\nSTATES\nLetρibe any (mixed or pure) initial state of the sys-\ntem and ρf=/summationtextpi|φi/an}bracketri}ht/an}bracketle{tφi|be an arbitrary (mixed or\npure) target (final) state. Without loss of generality we\nassumep1≥p2≥...≥pn. The system is assumed to be\ngeneric in the sense that all its transition frequencies are\ndifferent (in particular, its spectrum is non-degenerate)\nand all Einstein coefficients Aijare non-zero. The goal\nis to find a combination of coherent and incoherent fields\ntransforming all ρiintoρf.\nThe scheme to steer ρiintoρfconsists of two stages.\nIn the first stage, the system evolves on the time scale\nof several orders of magnitude of the relaxation time,\nt≈aτrel(whereacan be chosen in the range 2 ÷10\ndepending on the required degree of accuracy) under the\naction of a suitable optimal incoherent control n∗\nωinto\nthe state ˜ ρf=/summationtextpi|i/an}bracketri}ht/an}bracketle{ti|diagonal in the basis of H0and\nhaving the same spectrum as the final state ρf. The state\n˜ρfhas the same purity as ρf; ˜ρfis mixed if ρfis mixed\nand ˜ρfis pure if ρfis pure. In the second stage, the\nsystem evolves on the fast (e.g. femto-second) time scale\nundertheactionofasuitablecoherentlasercontrolwhich\nrotates the basis of H0to match the basis of ρf.\nThe first stage exploits incoherent control with any n∗\nω\nsuch that n∗\nωij=pj/(pi−pj) to prepare the system in\nthe state ˜ ρf. Coherent control is switched off ( u(t) = 0)\nduring this stage and the system dynamics is described\nby the master equation (2) which for off-diagonal matrix3\nelements ρln=/an}bracketle{tl|ρ|n/an}bracketri}htof the density matrix takes the\nform\n˙ρln=−(iαln+Γln)ρln,Γln=/summationdisplay\nj(Wjl+Wjn)≥0\nwhereαln=/an}bracketle{tl|H0+Heff|l/an}bracketri}ht−/an}bracketle{tn|H0+Heff|n/an}bracketri}htandWij=\nAij(2nωij+1). Off-diagonalelementsdecayexponentially\nfor Γln>0,ρln∼exp(−Γlnt). Diagonal elements satisfy\nthe Pauli master equation\n˙ρnn=/summationdisplay\nj(Wnjρjj−Wjnρnn)\nFor generic systems, all nωijcan be independently ad-\njusted, and the detailed balance condition Wnjρjj=\nWjnρnnimpliesn∗\nωij=pj/(pi−pj). The master equa-\ntion for spectral density nωsuch that n∗\nωij=pj/(pi−pj)\nhas ˜ρfas the stationary state and exponentially fast\ndrives the system to ˜ ρf[22]. The case when some\npi=pj/ne}ationslash= 0 formally requires infinite density at the cor-\nresponding transition (e.g. infinite temperature of the\nenvironment is required for creating equally populated\nstate ˜ρf=1\nnI), but for practical applications a reason-\nable degreeof accuracyallowingfora finite n∗\nωijis always\nsufficient. The first stage when necessary can be divided\ninto twoparts, diagonalizationofthe density matrix with\nanynωproducing positive and sufficiently large Γ ln, fol-\nlowed by the evolution produced by any n∗\nωsuch that\nn∗\nωij=pj/(pi−pj) to produce ˜ ρf.\nThe second stage implements coherent dynamics with\nunitary evolution transforming the basis {|i/an}bracketri}ht}into{|φi/an}bracketri}ht}\nand hence steering ˜ ρfintoρf. For successful realization\nof this stage we assume that any unitary evolution op-\nerator of the system can be produced with available co-\nherent controls when L= 0 on a time scale sufficiently\nshorter than τrel, i.e. that the system is unitary control-\nlable when decoherence effects are negligible. Incoherent\ncontrolis switched offduring this stage by setting nω= 0\nand the dynamics is well approximated by the unitary\nevolution (1).\nNecessary and sufficient conditions for unitary control-\nlability were obtained by Ramakrishna et al. [23]. Suf-\nficient conditions for complete controllability of n-level\nquantum systems subject to a single control pulse that\naddresses multiple allowed transitions concurrently were\nestablished [24]. Ramakrishna et al. showed that a nec-\nessary and sufficient condition for unitary controllability\nof a quantum system with Hamiltonian H=H0+u(t)V\nis that the Lie algebra generated by the operators H0\nandVhas dimension n2. We assume that the system\nsatisfies this condition when L= 0 and that any Ucan\nbe produced on a time scale sufficiently shorter than the\ndecoherence time scale, e.g. by using time-optimal con-\ntrol [25]. Under these assumptions a coherent field u∗(t)\nexists which produces a unitary operator Utransforming\nthe basis {|i/an}bracketri}ht}into{|φi/an}bracketri}ht}and therefore steering ˜ ρfinto\nρf. This field implements the second stage of the scheme\nto finish evolving ρiintoρf. The second stage can beimplemented also on the long time scale using dynamical\ndecoupling [26] to effectively decouple the system from\nthe environment.\nThere exist other methods to prepare arbitrary quan-\ntum states. An example is the method of engineering\narbitrary Kraus maps proposed by Lloyd and Viola [9]\nwhich in particular can be used to produce arbitrary\nquantum states. Another option is to cool an ensemble\nof systems to a pure state |φ/an}bracketri}htand then apply randomly\nand independently to each system a control producing a\nunitary operator Uitransforming |φ/an}bracketri}htinto|φi/an}bracketri}ht. Imple-\nmenting each Uiwith probability piwill effectively pro-\nduce the systems in the state ρf=/summationtextpi|φi/an}bracketri}ht/an}bracketle{tφi|. Both\nthese methods require independent addressing of indi-\nvidual systems in the ensemble that in general is hard\nto realize. The proposed method is free of this short-\ncoming and can be applied to an ensemble without the\nneed to independently address each system. This ad-\nvantage should significantly simplify practical realization\nwhen individual addressing of quantum systems is hard\nto realize. We remark that the first control stage re-\nquires individual addressing of each transition frequency.\nIf the system is non-generic, in particular, is degenerate\nthen the method may work if the degenerate states can\nbe selectively addressed by using polarization of the in-\ncoherent radiation. For degenerate systems (harmonic\noscillator) another scheme was proposed [5].\nDetermining the correct form of the master equation\nfor systems with time dependent Hamiltonians in general\nis a non-trivial problem [27, 28]. However, the proposed\nscheme does not use the master equation with time de-\npendent Hamiltonian u(t)Vand therefore the problem\nof choosing the correct form of the master equation with\ntime dependent Hamiltonianisnotrelevantforthis work.\nThe scheme in its first step exploits incoherent control\nwhen the coherent control is switched off and therefore\nthe system Hamiltonian is time independent. In this case\nthe system dynamics under Markovian approximation is\ngoverned by a master equation of the form (2). The sec-\nond step exploits fast coherent control on a short time\nscale when incoherent control is switched off and the de-\ncoherence effects are negligible. In this case L ≈0 and\nthe dynamics is well approximated by (1).\nIf the quantum system is sufficiently simple and all its\nrelevant parameters are known, then methods of optimal\ncontrol theory (OCT) can be used to find optimal fields\nn∗\nωandu∗(t) as shown on Fig. 1, subplot (a). If the sys-\ntem is complex and (or) some of its relevant parameters\nare unknown then various adaptive learning algorithms\ncan be used to implement the proposed scheme for engi-\nneeringarbitraryquantumstates(Fig.1, subplot(b)). In\nthis case, the creation of a desired diagonal mixed state\n˜ρfduring the first stage of the control scheme can be\nrealized with learning algorithms [2]. The second stage\ncan be implemented using learning algorithms for coher-\nent control [29]. Learning algorithms were shown to be\nefficient for both cases and therefore they will be efficient\nwhen used in a successive combination as required in the4\nFIG. 1. (Color online). Implementing complete density ma-\ntrixcontrollability with (a)optimal control theory(OCT) and\n(b) learning control. In (a), methods of OCT and a theoret-\nical model of the system dynamics are used to find optimal\nincoherent and coherent controls n∗\nkandu∗(t). In (b), the\nobjective J=/bardblρT−ρf/bardbl, where ρTis the output state, is\nmeasured (block with letter “M“) at the end of each iteration\nand the result is used to design a new control, ideally with\nbetter performance. The cycle is repeated until obtaining a\nsatisfactory output.\nproposed scheme.\nIII. COMPLETE DENSITY MATRIX\nCONTROLLABILITY\nDepending on a particular problem, various notions of\ncontrollability for quantum systems are used [10, 16, 23,\n30–32], including unitary controllability, pure state, den-\nsity matrix and observable controllability [30], and com-\nplete density matrix controllability [16]. Unitary con-\ntrollability means the ability to produce with available\ncontrols any unitary evolution operator U. Density ma-\ntrix controllability means the ability to transfer one into\nanother arbitrary density matrices with the same spec-\ntrum (i.e., kinematically equivalent density matrices); a\nparticular case is pure state controllability as the ability\nto transfer one into another arbitrary pure states. Com-\nplete density matrix controllability means the ability to\nsteer any initial (pure or mixed) density matrix ρiinto\nany (pure or mixed) final density matrix ρf, irrespective\noftheir relativespectra [16]. This notion is different from\ndensity matrix controllability [30], where only kinemati-\ncallyequivalentdensitymatricesarerequiredtobe acces-sible one from another, and is the strongestamongall de-\ngrees of state controllability for quantum systems; com-\nplete density matrix controllability of a quantum system\nimplies in particular its pure state and density matrix\ncontrollability. The suggested scheme allows for transfer-\nring one into another arbitrary density matrices thereby\napproximately realizing complete density matrix control-\nlability of quantum systems — the strongest possible de-\ngree of their state control.\nIV. ALL-TO-ONE CONTROLS\nAn all-to-onecontrol c∗is a controlwhich steersall ini-\ntial states into one final state. Such controls can be opti-\nmal simultaneously for all initial states [16] and their im-\nportanceismotivated by the following. Let J(c)≡J(ρT)\nbe an arbitrary control objective determined by the sys-\ntem density matrix ρTevolving from the initial state\nρ0to the final time Tunder the action of the control\nc(e.g.,J= Tr[ρTO] for some Hermitian observable O\norJ=/bardblρT−ρf/bardbl). In general, optimal controls — con-\ntrols minimizing the objective — are different for differ-\nent initial states. However, if c∗is an all-to-one control\nsteering all states into a state ρfminimizing J, thenc∗\nwill optimize the objective for any initial system state.\nWhile an abstract theoretical construction of all-to-one\ncontrolswasprovided[16] in terms ofspecial Krausmaps\nwhose definition is also provided below, their physical re-\nalizations has remained as an open problem. The pro-\nposed control scheme provides a physical realization of\nall-to-onecontrols and the correspondingKraus maps for\nany pure or mixed state ρf. The all-to-one property is\nachieved during the first stage, where n∗\nωproduces the\nsame density matrix ˜ ρfindependently of the initial state.\nThe density matrix representing the state of an n-level\nquantum system is a positive, unit-trace n×nmatrix.\nWe denote by Mn=Cn×nthe set of all n×ncomplex\nmatrices, and by Dn:={ρ∈ Mn|ρ=ρ†,ρ≥0,Tr(ρ) =\n1}the set of all density matrices. A map Φ : Mn→ Mn\nis positive if Φ( ρ)≥0 for any ρ≥0 inMn. A map Φ :\nMn→ Mnis completely positive (CP) if for any l∈N\nthe map Φ ⊗Il:Mn⊗Ml→ Mn⊗Mlis positive ( Ilis\nthe identity map in Ml). A CP map is trace preserving\nif Tr[Φ(ρ)] = Tr(ρ) for any ρ∈ Mn. Completely positive\ntrace preserving maps are referred to as Kraus maps;\nthey representthe reduced dynamicsofquantum systems\ninitially uncorrelated with the environment [18, 33].\nAny Kraus map Φ can be represented using\nthe operator-sum representation (OSR) as Φ( ρ) =/summationtextl\ni=1KiρK†\ni, where{Ki}is a set of complex n×nma-\ntrices satisfying the condition/summationtextl\ni=1K†\niKi=Into ensure\ntrace preservation [18, 33]. The OSR is not unique: any\nKraus map Φ can be represented using infinitely many\nsets of Kraus operators.\nThe all-to-one Kraus map for a given final state ρf=/summationtext\nipi|φi/an}bracketri}ht/an}bracketle{tφi|is defined as a Kraus map Φ ρfsteering all\ninitial states into ρf, i.e., such that Φ ρf(ρ) =ρffor all5\nρ∈ Dn[16]. If ρfis a density matrix maximizing a\ngiven objective Jandc∗is a control producing the map\nΦρf, then this control will be simultaneously optimal for\nall initial states, i.e., the same c∗will maximize the ob-\njective for any initial system state. An OSR for a uni-\nversally optimal Kraus map Φ ρfcan be constructed by\nusingKij=√pi|φi/an}bracketri}ht/an}bracketle{tφj|as the Kraus operators. Indeed,\nfor anyρ,/summationtextn\ni,j=1KijρK†\nij=ρf.\nAll-to-one Kraus maps were constructed theoretically\nin [16]. The two stage control scheme described in the\npresent paper provides an approximate physical realiza-\ntion ofall-to-one Krausmaps Φ ρffor allρf(therefore any\nall-to-oneKrausmap can be producedusing this scheme)\nfor generic systems unitary controllable on the fast with\nrespect to τreltime scale.\nV. EXAMPLE: CALCIUM ATOM\nThe scheme is illustrated below with an example of a\ntwo-level atom whose all relevant parameters are known\nand it is easy to analytically understand and visualize\nthe controlled dynamics. We consider calcium upper\nand lower levels 41P and 41S as two states |1/an}bracketri}htand|0/an}bracketri}ht\nof the two-level system. For this system the transition\nfrequency is ω21= 4.5×1015rad/s, the radiative life-\ntimet21= 4.5 ns, the Einstein coefficient A21= 1/t21≈\n2.2×108s−1, and the dipole moment µ12= 2.4×10−29\nC·m [34]. The method works equally well for any ini-\ntial and target states, we take for the sake of definiteness\nρ0=|0/an}bracketri}ht/an}bracketle{t0|andρf=1\n4|0/an}bracketri}ht/an}bracketle{t0|+3\n4|1/an}bracketri}ht/an}bracketle{t1|.\nThe system is generic, all its relevant parameters are\nknownandwecananalyticallyfindoptimalcontrols. The\ngoal of the first (incoherent) stage is to prepare the mix-\nture ˜ρf=3\n4|0/an}bracketri}ht/an}bracketle{t0|+1\n4|1/an}bracketri}ht/an}bracketle{t1|. This goal is realized by ap-\nplying to the system incoherent radiation with spectral\ndensity satisfying nω21= 1/2 during time Tof several\nmagnitudesofdecoherencetime t21; wechoose T= 50ns.\nThe goal of the second (coherent) stage is to rotate the\nstate produced at the end of the first stage to transform\nit intoρf. This goalcan be realizedforexample byapply-\ning a resonant π-pulseE(t) =Ecos(ω12t). Other meth-\nods of coherent control which produce the same unitary\ntransformation can be used as well. The electric field\namplitude which makes the Rabi frequency equal to the\nradiative decay rate is E≈103V/m [34]. The pro-\nposed scheme requires the duration of the second stage\nto be significantly shorter that the decay time. This can\nbe satisfied by choosing E/greaterorapproxeql104V/m. We take reso-\nnant electromagnetic field E(t) =Ecos(ω12t) of ampli-\ntudeE= 107V/m acting on the system during the time\nintervalTf−T= 1310 fs. The Rabi frequency for the\nfieldofsuchamplitudeisΩ R≈1/2320fs−1, thusthefield\nacts as a π-pulse transforming the state ˜ ρfintoρf. The\nresults of the numerical simulation are shown on Fig. 2.\nThe time interval 1310 fs is much less than t21and de-\ncoherence effects are negligible during the second stage.\nFor illustrative purpose of better visualization of the tra-jectory during the second stage we provide in Fig. 3 sim-\nulation results for E= 109V/m and Tf−T= 13.1 fs.\nThe method can be applied for producing arbitrary pure\nor mixed target density matrices from any pure or mixed\ninitial state thereby implying complete density matrix\ncontrollability of this system.\nFIG. 2. (Color online). Application of the control scheme\nfor engineering a desired mixed state of two Ca energy lev-\nels. Subplot (a) shows the behavior of the objective vs. time\nduring the two stages of the control. The timescale is not\nuniform, the duration of the first stage is T= 50 ns and of\nthe second stage is Tf−T= 1310 fs. Solid line is for /bardblρt−ρf/bardbl\nand dotted line for /bardblρt−˜ρf/bardbl; the latter quantity almost com-\npletely vanishes during the first stage of the control. Subpl ot\n(b) shows the corresponding evolution of the Bloch vector.\nThe initial system state corresponds to the south pole, the\nvertical blue line represents incoherent evolution during the\nfirst stage and the green line shows coherent rotations durin g\nthe second stage under the action of resonant field of ampli-\ntudeE= 107V/m. The evolved state approaches the target\nstate indicated by the red marker.\nFIG. 3. (Color online). 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The sta-\ntionary density is the average expected height in the stationary state\nof a \fnite-volume model; the transition density is the critical point\nin the in\fnite-volume counterpart. These two critical densities were\ngenerally assumed to be equal, but this has turned out to be wrong\nfor deterministic sandpile models. We show they are not equal in a\nquenched version of the Manna sandpile model either.\nIn the literature, when the transition density is simulated, it is\nimplicitly or explicitly assumed to be equal to either the so-called\nthreshold density or the so-called critical activity density . We properly\nde\fne these auxiliary densities, and prove that in certain cases, the\nthreshold density is equal to the transition density. We extend the\nde\fnition of the critical activity density to in\fnite volume, and prove\nthat in the standard in\fnite volume sandpile, it is equal to 1. Our\nresults should bring some order in the precise relations between the\nvarious densities.\nKeywords: Sandpile model, phase transition, critical density, transition\ndensity, Manna model, activity density.\nAMS Subject Classification: 60K35, 60K37, 60J05, 60G10.\n1 Introduction\nThis paper aims to bring some order in a long-standing discussion on the\nequality or inequality of various critical densities in sandpile models. By\nmeans of introduction and context, we \frst informally describe the archety-\npal Abelian sandpile model (ASM or `classical' sandpile model). On In=\n1arXiv:1211.4760v1  [math-ph]  20 Nov 2012f0;1;:::;ngwe initially assign a height (a non-negative integer) to each site\nx2In. The dynamics is as follows. Each time step starts with an addition:\nthe height of a randomly chosen site is increased by 1. Then the con\fgu-\nration is stabilized by so called topplings : a site with height 2 or more is\nunstable and may topple. In a toppling, its height decreases by 2 and the\nheight of both neighbors increases by 1. Sites 0 or nhave only one neighbor;\nwhen these sites topple, one particle leaves the system. The con\fguration is\nstabilized when there are no more unstable sites to topple. It is not hard to\nshow that in the stationary state of this model, the expected height is 1 \u00001\nn+2\nfor every site; see for example [24, 29]. We call this expected height, in the\nlimit asn!1 , the stationary density and denote it by \u001as. (So in this case,\n\u001as= 1.)\nNext consider a related model, the in\fnite volume model. On the in\fnite\nlineZ, start with (independent) Poisson- \u001anumber of particles at each site.\nAt each discrete time step, we simply topple all unstable sites simultane-\nously. (There are alternatives, like toppling after exponential waiting times,\nindependently for each site, see [19] for a detailed discussion of this.) It is\nknown [19] that for \u001a <1, only \fnitely many topplings per site are needed\nto obtain at a limiting stable con\fguration, whereas for \u001a > 1, every site\ntopples in\fnitely many times. Hence, this model has a phase transition at\n\u001a= 1 and we call this the transition density \u001atr.\nIn a widely cited series of papers [1, 2, 3, 4, 5], Dickman, Mu~ noz, Vespig-\nnani and Zapperi developed a theory of self-organized criticality (SOC) as a\nrelationship between driven dissipative systems like the ASM, and systems\nwith particle conservation like the in\fnite volume model. Loosely speaking,\nthe following heuristic connection (the `density argument') is o\u000bered: in the\npresence of activity of the ASM, that is, when topplings take place, we should\ntypically have \u001a>\u001atr, and particles are leaving the system. In the absence of\nactivity, we typically have \u001a<\u001atr, and there is addition of particles. Hence\nthe density of particles always changes in the direction of \u001atrand therefore\nthe only candidate-value for the stationary density is \u001atr.\nThis would be a very attractive explanation for the observation that the\nstationary state of the two-dimensional ASM shows power law behaviour of\nmany spatial and temporal quantities, (apparently) without any tuning of\na parameter. However, it requires that \u001as=\u001atr. In the one-dimensional\ncase this is true. At \frst, simulations and exact calculations for the two-\ndimensional ASM seemed to con\frm that \u001asand\u001atrare equal up to three\ndecimals. However, more accurate simulations show a di\u000berence in the fourth\ndecimal for the two densities on Z2. Moreover, on several other graphs they\nare provably di\u000berent [18]. It is at present believed that this di\u000berence is a\nconsequence of the existence of so called toppling invariants [11].\n2The more recent literature (we give references in Section 4) on the density\nargument tends to focus on the Manna model, a sandpile model with stochas-\ntic topplings, for which it is believed that toppling invariants play no role.\nThe Manna model is de\fned as above, with the important di\u000berence that\nduring a toppling, particles go to randomly chosen neighbors, independently\nof each other, and independent of earlier topplings of the same or other sites.\nIn other words, the topplings in the Manna model are subject to annealed\ndisorder.\nSimulating \u001asfor a sandpile model is straightforward, since this density is\nde\fned for a model in \fnite volume. However, \u001atris de\fned in in\fnite volume,\nwhich makes simulations much more di\u000ecult. In reality, when people try to\nsimulate\u001atr, what they really simulate is the so called critical activity density\nor the threshold density , which we de\fne and study below. It is then tacitly\nassumed (sometimes also explicitly) that these densities are equal to \u001atr.\nThe goal of the present paper is to carefully distinguish between the var-\nious densities, and to investigate the density conjecture in quenched versions\nof the Manna model, that is, in sandpile models in which the toppling rules\nper site are random but \fxed throughout the evolution of the process. It\nturns out that we can explicitly compute \u001asand\u001atrin these quenched Manna\nmodels, and that in general they are notequal. We also study the relation\nbetween the various densities in the original (annealed) Manna model.\nWe end this introduction with a precise de\fnition of the sandpile models\nwhich we consider. We distinguish between\n1. Driven models, that is, models with addition of particles and topplings,\nalways on the interval In=f0;1;:::;ngwith dissipation at the open\nboundaries;\n2. In\fnite volume models on Z, without additions, and only topplings;\n3. Fixed-energy models, that is, models with topplings but without addi-\ntions, on the n-cycleCn=f0;2;:::;ng, where 0 and nare neighbors.\nWe abuse notation and view In;CnandZas graphs with nearest neighbor\nconnections. A typical con\fguration will be denoted \u0011and for any vertex\n(site)x,\u0011(x)2f0;1;2;:::gis interpreted as the height at site x. We call a\nsitestable if its height is 0 or 1. If the height of a site is 2 or more, then we call\nthis site unstable . If all sites are stable, then we say that the con\fguration\nis stable.\nUpon an addition at sitex, the height of site xincreases by 1, and can\nhenceforth become unstable. An unstable site may topple . With a toppling\nof sitex, the height of site xdecreases by 2, and the height of the neighboring\n3sites may increase, in such a way that the total increase is at most 2. We\ndenote the neighbors of site xasx\u00001 andx+ 1, where on the graph Cn,\nwe interpret this modulo n+ 1. OnIn, 0 andnonly have 1 neighbor.\nWe consider three possibilities for the height increase of the neighbors in a\ntoppling, namely\n1. The height of site x\u00001 (if present) increases by 2, and the height of\nsitex+ 1 does not change. We call this a lefttoppling (l).\n2. The height of site x+ 1 (if present) increases by 2, and the height of\nsitex\u00001 does not change. We call this a right toppling (r).\n3. The height of both sites x\u00001 andx+ 1 (if present) increase by 1. We\ncall this a symmetric toppling (s).\nIn this paper, we introduce quenched sandpile models , which we de\fne\nas follows. On a graph G, we let\u0017be a probability measure on fl;r;sgG,\nan element of which we call a toppling environment . We choose a toppling\nenvironment \u0010according to \u0017. Given a toppling environment \u0010, we allow\nat sitexonly topplings of type \u0010(x). We say that \u0010(x) is the label at site\nx. The combination of a con\fguration \u0011and a toppling environment \u0010is\ncalled a state, and denoted ( \u0011;\u0010). We call this state stable if \u0011is a stable\ncon\fguration.\nIf a state is unstable then topplings can lead to new unstable sites. We\nsay we stabilize a state (\u0011;\u0010) if we keep toppling unstable sites, according to\nthe (\fxed) toppling environment, until there are no unstable sites anymore.\nIt may happen though that we never obtain a stable con\fguration this way;\nin this case we say that the state is not stabilizable. If a stable con\fguration\ncan be reached, then the order in which we perform topplings has no e\u000bect\non the \fnal stable con\fguration; this is the famous Abelian property of the\nsandpile (see [19] for some technical details on this fact in the case of in\fnite\nvolume). When l- andr-labels are not allowed, then this model reduces to\nthe classical sandpile model.\nThe Manna model is the annealed version of the above quenched sandpile\nmodel. One convenient way to describe the Manna model [30], is that instead\nof a single label, we have a stack of labels at every site. Every stack consists\nof an in\fnite sequence of i.i.d. random labels, which are swith probability\n1/2,lwith probability 1/4 and rwith probability 1/4, independent of all\nother stacks. When a site topples, it topples according to the \frst label in\nits stack, and then removes that label from the stack.\nThis paper is organized as follows: In Section 2, we de\fne and state our\nresults on the two main critical densities, namely the stationary and the\n4transition density. In Section 3, we discuss the additional critical densities\nthat have been introduced in the literature to simulate \u001atr. We state a number\nof new results on these densities. Finally, we give the proofs of all our results\nin a separate Section 5.\n2\u001asand\u001atrfor the quenched sandpile models\n2.1 Driven sandpiles on Inand the stationary density\nDriven sandpile models are de\fned on In. Let\u0016be a probability measure on\nf0;1gNand\u0017a translation-invariant probability measure on fl;r;sgN. We\nuse\u0016to draw an initial con\fguration, and \u0017to draw a toppling environment.\nMore precisely, to implement the process on In, the initial con\fguration \u00110\nnis\nthe restriction of a drawing from \u0016to the \frstn+1 coordinates, and likewise,\nthe toppling environment \u0010nis the restriction to the \frst n+ 1 coordinates\nof a drawing from \u0017.\nDynamics is as follows. Given \u00110, we add a particle at a randomly cho-\nsen site (uniformly over all sites) and stabilize the resulting con\fguration.\nThe result is denoted \u00111\nn. Repeating this process leads to to a sequence of\ncon\fgurations ( \u0011t\nn),t= 0;1;:::.\nWe are interested in the `average' number of occupied sites in the process\nonInin the limit as n!1 . For given nand\u0010, the process is a Markov\nchain, but it turns out that for certain toppling environments, this Markov\nchain is periodic so that we do not have a well de\fned unique stationary\ndistribution. Nevertheless, the `average' hinted at above often exists in an\nunambiguous way. We denote by \u001at\nn=\u001at\nn(\u0016;\u0017) the expected fraction of sites\ninInwhose height is 1 at time t. Here, expectation is with respect to \u0016\u0002\u0017.\nWe say that \u0017isdirected if either\u0017(\u0010(0) =r) = 0 or\u0017(\u0010(0) =l) = 0.\nTheorem 1. Suppose that \u0017is directed. Let fr=\u0017(\u0010(0)6=s). Then\n\u001as=\u001as(\u0016;\u0017) := lim\nn!1lim\nt!1\u001at\nn (2.1)\nexists and is equal to\n1\u0000fr\n2:\nIn particular, \u001asdoes not depend on \u0016. Note that in the classical case, fr= 0,\nand Theorem 1 reduces to the well known result which we described in the\nintroduction.\nThe quantity \u001asis called the stationary density .\n52.2 In\fnite volume sandpiles on Zand the transition\ndensity\nIn\fnite volume sandpiles in this article are de\fned on Z. We choose the\ninitial state \u0011according to \u0016\u001a, a Poisson product measure with parameter\n\u001a. The toppling environment \u0010is chosen as before, according to a stationary\nand ergodic probability measure \u0017onfl;s;rgZ. There are no additions in\nthis model, only topplings. At any discrete time t, we simply topple all\nunstable sites once. This implies that for every t, the distribution of \u0011tis\ntranslation-invariant.\nWe de\fne the transition density as\n\u001atr=\u001atr(\u0017) = supf\u001a: (\u0011;\u0010) is stabilizable ( \u0016\u001a\u0002\u0017)\u0000a.s.g:\nAs mentioned in the introduction, it was recently proved for the classical\nsandpile model on the bracelet graph [18] (with proper interpretations of \u001as\nand\u001atr) that\u001as6=\u001atr. The current belief is that this inequality holds in\ngeneral for classical sandpile models, and that the one-dimensional case is an\nexception.\nIn this paper, we prove that \u001as6=\u001atrfor quenched sandpiles, as long as \u0017\nis directed and satis\fes\n\u0017(\u0010:\u0010(x) =\u0010(x+ 1) =s) = 0 and\u0017(\u0010(x) =\u0010(x+ 2) =s) = 0: (2.2)\nfor allx. In words, this condition means that between every pair of slabels\nthere must be at least two rlabels. Note that the transition density is\nnot interesting for non-directed models: indeed, between r- and anl-labelled\nsites, particles cannot escape, and since it is possible with positive probability\nthat too many particles are trapped in between, the model will not stabilize\na.s., for any value of \u001a>0.\nTheorem 2. Suppose that \u0017is directed and satis\fes (2.2). Let fr=\u0017(\u0010(0) =\nr). Then we have\n(a)\u001as= 1\u0000fr\n2.\n(b)\u001atris the unique solution of \u001a\u00001\n2(1 +e\u00002\u001a) =1\u0000fr\n2(1 +e\u00002\u001a)2.\nIn particular, \u001as6=\u001atrfor\u0017satisfying (2.2).\n63 Simulating \u001atr: more critical densities\n3.1 Fixed-energy sandpile models and the threshold\ndensity\nThe \fxed-energy sandpile models in this article are de\fned on Cn. They\nplay an important role in simulating \u001atr. In the initial state, \u0011nis chosen\naccording to \u0016\u001a, a Poisson product measure with parameter \u001a, and\u0010nis\nchosen according to \u0017. The \fxed-energy model evolves in time by topplings.\nIf the initial state is stabilizable, then every site topples only \fnitely many\ntimes.\nWe de\fne the threshold density as\n\u001ath=\u001ath(\u0017) = lim\nn!1supf\u001a: (\u0016\u001a\u0002\u0017)((\u0011n;\u0010n) is stabilizable)\u00151=2g:\nFor the classical \fxed-energy sandpile model, the threshold density is equal\nto 1.\nNote that the de\fnition of \u001athis very similar to that of \u001atr. It is often\ntacitly assumed that they are equal for the classical sandpile on general\ngraphs: in [18], this assumption is explicitly stated. In the literature, only\nfor the classical sandpile in dimension 1 a proof is available. Note that for\na quenched sandpile model, if both l-sites andr-sites exist, \u001ath= 0, since it\nis always possible to have too many particles between a r-site and an l-site,\nfrom which particles can never escape. Hence, again, only directed models\nare of interest. We prove the following.\nTheorem 3. Suppose that \u0017is directed and satis\fes (2.2). Then \u001atr=\u001ath.\n3.2 Parallel chip \fring and the activity density\nConsider \fxed-energy models with parallel toppling order. These cellular\nautomata are called parallel chip \fring models [28]. We are interested in the\nactivity density \u001aa(\u0016\u001a;\u0017;n), which loosely formulated, is the eventual average\nfraction of unstable sites when parallel chip \fring is run on a \fnite graph,\nCnin our case. The next step is to take the limit as n!1 :\n\u001aa(\u0016\u001a;\u0017) = lim sup\nn!1\u001aa(\u0016\u001a;\u0017;n):\nFor the classical sandpile on Cn, the graph of \u001aaversus\u001ahas been shown\n[6] to take the shape of a `staircase' with three stairs (see Figure 1). More\nspeci\fcally, the classical model on Cnbehaves in the following particularly\nsimple way. If the total number of particles is strictly in ( n;2n), then every\n7Figure 1: The staircase graph for the classical sandpile model on Cn. The\nhorizontal axis is \u001a, the vertical axis is the activity density. The graph is the\nsame for the \fxed-energy and the in\fnite volume model; see [6] and Theorem\n4.\ninitial con\fguration will eventually have period 2. Thus, as t!1 , the total\nnumber of unstable sites in one time step will eventually be close to n=2.\nIf the total number of particles is larger than 2 n, then eventually all sites\nwill be unstable. Finally, it was already known that if the total number of\nparticles is smaller than n, then every initial con\fguration stabilizes, so that\neventually the total number of unstable sites is 0. Thus, the staircase of the\nclassical sandpile model on Cnhas three stairs, and it does not even depend\non the precise initial distribution of the particles. Only if the total number\nof particles is equal to nor 2n, there are multiple possibilities for the limiting\nfraction of unstable sites.\nFor the \rower graph [18], the corresponding plot has \fve stairs. In the\ncase of the two-dimensional torus, it is conjectured, based on numerical obser-\nvations [8], that the plot is a devil's staircase. Levine proved the occurrence\nof a devil's staircase for parallel chip \fring on the complete graph, with a spe-\ncial class of initial con\fgurations [21]. To show that the plot need not always\nbe a staircase, we include a simulation for the sandpile model of Theorem 2\n(see Figure 2).\nThe activity density has been used to simulate \u001atr, under the assumption\nthat\u001atris equal to the so called critical activity density \u001acade\fned as\n\u001aca=\u001aca(\u0017) := supf\u001a:\u001aa(\u0016\u001a;\u0017) = 0g:\nOne can prove this for a number of speci\fc graphs that have an in\fnite\nvolume counterpart, for instance Cn(where both densities are trivially equal\nto 1), or, less obvious, for the bracelet graph [18]. The \rower graph or the\ncomplete graph however do not have an in\fnite volume counterpart.\nThere is an in\fnite-volume analog of the activity density. Let \u000b\u001a(n;t) be\n8Figure 2: Simulation of the the quenched restricted model on I1000, with\nfs= 1=4. There are 10 trials for each of 50 values of \u001a. In each trial, a new\nrandom initial state was generated. Numerically, the predicted value for \u001atr\n(Theorem 2) in this case is 0.556...\nthe fraction of active sites in [ \u0000n;n] aftertiterations of the process, starting\nwith\u0016\u001a. For \fxed t, the limit\u000b\u001a(t) := limn!1\u000b\u001a(n;t) exists by ergodicity,\nand is a\u0016\u001a-a.s. constant. The following result extends [6] to in\fnite volume.\nTheorem 4. In the in\fnite volume classical sandpile model on Zwe have\nthe following.\n1. For\u001a<1,limt!1\u000b\u001a(t) = 0:\n2. For\u001a>2,limt!1\u000b\u001a(t) = 1:\n3. For 1<\u001a< 2,limT!11\nTPT\nt=1\u000b\u001a(t) =1\n2:\nIn words, below \u001a= 1, the fraction of unstable sites converges to 0, above\n\u001a= 2 it converges to 1, and in between the fraction converges, in a weak\nsense, to 1=2. The in\fnite-volume analog of the critical activity density is\ntherefore equal to 1, and in the classical sandpile in one dimension, all critical\ndensities are equal.\n4 Critical densities for the Manna model\nIt is clear how the stationary density should be de\fned for the Manna model,\nand it is conjectured that it is strictly less than 1: Sadhu et al [31] claim\nthat for the Manna model, \u001as= 0:953:::, based on extrapolation of exact\n9calculations for n\u001412. Theorem 1 perhaps supports the plausibility of the\nconjecture that \u001as<1 for the Manna model.\nAs far as\u001atris concerned, Dickman et al. [17] have estimated \u001atrfor the\nManna model as 0 :94887, in fact estimating the critical activity density on Cn.\nHuynh et al. [20] performed extensive simulations for the Manna model on\nmany one- and two-dimensional lattices. Their estimates con\frm the result\nof [17], and in general give a value close to 0.9 for one-dimensional lattices,\nand close to 0.7 for two-dimensional lattices (note that the table with particle\ndensities is not included in the \frst version of this paper). Sidoravicius and\nRolla [30] proved that \u001atr\u00151=4. Sadhu et al. [31] compare their value for \u001as\nof the Manna model to the estimate for \u001atrof [17], implying that they should\nbe equal (See also Conjecture 6 below).\nIt is not hard to show that for the Manna model, \u001ath= 1. For complete-\nness, we state and prove this now.\nProposition 5. For the Manna model on Z,\u001ath= 1.\nProof. We make use of the following fact for the classical sandpile model on\nCn: If the total number of particles is less than n, then the model will stabilize\nafter \fnitely many topplings per site, irrespective of the initial position of\nthe particles. This can be deduced in the following manner: if the model\nstabilizes, then at least one site does not topple. Then the neighbors of that\nsite topple at most once, and so on.\nNow for the Manna model on Cn, we choose \u001a < 1\u0000\u000f, with\u000f > 0\narbitrarily small. Then the probability that the total number of particles is\nmore thann(1\u0000\u000f=2), tends to 0 as n!1 . If the total number of particles is\nless thann, then the model stabilizes a.s. To see this, note that one possibility\nfor stabilizing is if there is some time tsuch that starting from t, all unstable\nsites topple according to an slabel until the model stabilizes. Since all labels\nare i.i.d. and only \fnitely many slabels are needed, this event occurs a.s.\nTherefore, the Manna model on Cnstabilizes with probability tending to 1\nasn!1 for all\u001a<1, so that\u001ath= 1.\nWe conclude that the critical activity density should be at least 1. How-\never, the simulations for the Manna model arrive at a value strictly less than\n1. The discrepancy results from a subtlety in choosing the observation time\nwindow. It is observed that there is a density value strictly smaller than 1,\nsuch that for \u001abelow this value, the total number of topplings before stabi-\nlization is relatively small, whereas for \u001aabove this value, the total number\nof topplings is much larger. It is this value that is taken to be an estimate\nfor\u001atr, rather than \u001athas de\fned in (3.1). In the second case, before even-\ntually stabilizing, the models is observed to settle into a `quasi-stationary'\n10state, where the fraction of active sites is approximately constant for a long\ntime; see [16], or Figure 3 of [13] for numerical simulations of this state. It\nis believed that this quasi-stationary state of the \fxed-energy Manna model\nre\rects the true stationary state of the in\fnite volume Manna model. There-\nfore, the observation time window is chosen such that the model is observed\nin the quasi-stationary state, rather than large enough so that the model has\nstabilized.\nThus the \fndings of [17] and [14] indicate the following conjecture for the\nManna model, which we now state explicitly. We stress that the conjectured\nrelation between the three densities is markedly di\u000berent from what is known\nfor other models. For instance, for the classical sandpile model on the bracelet\ngraph it is known [18] that \u001as6=\u001atr=\u001ath, and combining Theorems 2 and\n3 of this paper gives the same relation for the directed quenched sandpile\nmodels satisfying (2.2).\nConjecture 6. For the Manna model on Z,\n\u001as=\u001atr6=\u001ath(= 1):\nWith kind permission of Lionel Levine, we give an example to illustrate\nquasi-stationarity in the Manna model. In this model, there is a value of \u001a\nsuch that below this value, the total number of topplings scales as O(nlogn),\nwhereas for \u001aabove this value the total number of topplings increases expo-\nnentially in n.\nExample 7. Consider the \fxed-energy Manna model on the complete graph\nKn, with self-loops. A site is active if there are at least two particles, and\nin a toppling each of the two particles chooses a new position uniformly at\nrandom from its neighbors, which in this case means from all sites.\nWe can view this sandpile model as an Ehrenfest urn model, with a\nstopping criterion. We say that the nsites ofKnrepresentnballs in an\nurn, of two colors. One color (green) represents a site with an odd number\nof particles, and one color (red) a site with an even number of particles. In\nan Ehrenfest urn model, a move consists of picking one ball at random from\nthe urn, removing it, and instead adding a ball of the opposite color.\nNow consider a toppling of an active site in the sandpile model. We\nimagine that we \frst (temporarily) remove the two particles from the active\nsite, and then one by one add them to their new positions. Removing the\nparticles does not change any of the parities. Putting back one particle\nchanges the parity of one site chosen uniformly at random from the graph,\n11and likewise for the other particle. Thus, a toppling of an active site is the\nequivalent of two moves in the urn model.\nThe sandpile model is stable if and only if each particle is on a di\u000berent\nsite. Therefore, if the particle number is m, then the stopping time for the\naccording urn model is the \frst time that the number of green balls is m(if\nm>n , then the model never stabilizes).\nNow suppose that nis large. Then typically, the number of particles is\nclose ton\u001a. In the initial con\fguration, typically the number of sites with an\nodd number of particles, is close ton\n2(1\u0000e\u00002\u001a) (ntimes the probability that\na Poisson(\u001a) number is odd), which is strictly less than n\u001a. In an Ehrenfest\nurn model, from any starting point the fraction of green balls typically tends\nto its equilibrium value of 1 =2.\nTherefore, if \u001a<1=2, then the model meets the stopping criterion while\ntending to equilibrium. It is known that the expected total number of moves\nto reach equilibrium for the \frst time, starting from no green balls, is of the\nordernlogn[22], only slightly more than linearly increasing with n. However,\nwhen\u001a > 1=2, then the model does not meet the stopping criterion while\ntending to equilibrium. On the contrary, it meets the stopping criterion only\nin a large deviation from equilibrium. This will typically take an exponen-\ntially large number of moves. Meanwhile, the number of green balls \ructuates\nround its equilibrium number, which we can describe as quasi-stationary be-\nhavior. Therefore, there is a sharp transition in the total number of topplings\nneeded to stabilize, at \u001a= 1=2. If we monitor the quasi-stationary behavior,\nthen we \fnd 1/2 as an estimate for \u001atr. Actually, the threshold density is\n\u001ath= 1.\nTo \fnd the stationary density, we say that one site is the sink. Then both\ntopplings and additions are equivalent to one or more moves in the Ehrenfest\nurn model, this time without stopping criterion. Therefore, \u001as= 1=2. We\nconclude that this model behaves as conjectured for the Manna model on Z.\n5 Proofs\n5.1 Proof of Theorem 1\nOur main tool for the proof of Theorem 1 the following lemma. Consider the\nsetA=f\u0011:\u0011(x) = 1 for all 0\u0014x<ng, which contains two elements which\nwe denote by \u0011f= (1;1;:::; 1) and\u0011e= (1;1;:::; 1;0).\nLemma 8. Consider the driven sandpile model on Inwith toppling environ-\nment\u0010given by\u0010(0) =rand\u0010(x) =sfor all other x2In. Then the set\nAde\fned above is absorbing. More precisely, denoting the addition site by\n12a, transitions between \u0011fand\u0011eare as follows. If n\u0000ais even, then the\ncon\fguration \rips from \u0011fto\u0011eor vice versa; if n\u0000ais odd, then the con-\n\fguration does not change upon addition at aand subsequent stabilization.\nSince obviously, Ais reachable from any con\fguration, the lemma implies\nthat there are only two recurrent con\fgurations: the `full con\fguration' \u0011f\nin which all sites have height 1, or the `non-full con\fguration' \u0011ein which all\nsites have height 1, except site n.\nProof. We use the well known description [29] which gives the result of one\naddition and subsequent stabilization in a particle con\fguration \u0011in the\ndriven model on In: Letabe the addition site. Let ibe maxf\u00001;i\u0014a:\n\u0011(i) = 0g, that is,iis the closest empty site to the left of a, and if there is\nno such site, then i=\u00001. Similarly, let jbe minfn+ 1;j\u0015a:\u0011(j) = 0g.\nThe resulting con\fguration after the addition and subsequent stabilization\nis denoted\u0018. Then only sites in ( i;j) have toppled, and \u0018is as follows: sites\niandjhave gained 1 particle and site i+j\u0000ahas lost 1 particle. Note that\nif\u0011(a) = 0, then the above description simply states that site awill become\nfull. If\u0011(a) = 1, then at most three sites change height.\nNow we return to the sandpile model with \u0010(0) =r, and let\u00112A.\nWe will show that if n\u0000ais odd, then we end up in \u0011after addition and\nsubsequent stabilization. If n\u0000ais even however, then we \rip to the other\nelement inA.\nWe \frst note that if the particle con\fguration is non-full and a=n,\nthen it is obvious that the result is the full con\fguration. Now suppose that\nais a full site. After the addition, we stabilize in steps. In the \frst step,\nwe perform only s-topplings (that is, we do not topple site 0) until all sites\nexcept site 0 are stable. This step is equivalent to stabilization in the driven\nclassical sandpile on [1 ;n], with sinks at sites 0 and n+1. In the second step,\nwe topple site 0 once, and subsequently again only perform s-topplings until\nall sites except site 0 are stable. We repeat the second step until the whole\ncon\fguration is stable.\nTo \fnd the particle con\fguration after the \frst step, we apply the above\ndescription with i= 0 andj=j1wherej1=nif\u0011=\u0011e, andj1=n+ 1 if\n\u0011=\u0011f. Then after the \frst step, site 0 now has two particles, and all other\nsites are full except for site j1\u0000a, which is now empty.\nTo \fnd the particle con\fguration after the second step, we apply the\nabove description twice, because toppling site 0 is equivalent to emptying\nsite 0 and making two additions to site 1. Therefore, after the second step\nsite 0 has again two particles, and all other sites are full except site j1\u0000a\u00002,\nwhich is now empty.\n13We repeat the second step ktimes, with ksuch that either j1\u0000a\u00002k= 0\norj1\u0000a\u00002k= 1. In the \frst case, we have reached a stable con\fguration,\nnamely\u0011f. This will be the case if j1=nandn\u0000ais even, orj1=n+ 1\nandn\u0000ais odd. In the second case, all sites will topple one more time, so\nthat the \fnal con\fguration is \u0011e. This will be the case if j1=n+1 andn\u0000a\nis even, orj1=nandn\u0000ais odd. This \fnishes the proof.\nThe idea of the proof of Theorem 1 in words is as follows: we think of\nalmost the entire state of being built up of `blocks' in which the leftmost site\nisr, and all other sites are s. Each of these blocks is like the model in Lemma\n8: after a transient period, either all sites are full, or only the rightmost site\nis empty. Moreover, we will \fnd that at stationarity, these con\fgurations are\nequally likely. Therefore, since the proportion of rsites isfr, the proportion\nof empty sites at stationarity is fr=2.\nProof of Theorem 1. Let (\u0011;\u0010) be a state on Inof the quenched directed\nmodel. Let Rnbe the number of rsites in\u0010n, and callXi,i= 1;:::;Rnthe\npositions of the rsites, in increasing order. We de\fne B0= [0;X 1\u00001] and\nBi= [Xi;Xi+1\u00001] fori= 1;:::;Rn. We refer to the Bi's as a `block' and\ndenote bylibe the number of sites in Bi.\nThe block B0behaves as a classical sandpile, implying that the only\nrecurrent (sub) con\fgurations on B0are those with at most one site with\nheight 0. For all other blocks, Lemma 8 implies that there are two recurrent\nsub-con\fgurations, namely all lisites are full (a `full' block), or all sites are\nfull except the rightmost site Xi+1\u00001 (a `non-full' block). In fact, according\nto Lemma 8, an addition to the a-th site (counting from the left) in Biwith\nli\u0000aodd will leave the con\fguration unchanged, whereas an addition to\nthea-th site in Biwithli\u0000aeven will \rip the block, that is, change its\ncon\fguration from full to non-full or vice versa.\nAfter a random but a.s. \fnite time, all blocks B1;:::;BRnare either full\nor non-full, and in B0there is at most one empty site. We will now argue\nthat for each block B1;:::;BRn, the probability that it is full converges to\n1=2 as time tends to in\fnity.\nConsider the e\u000bect of one addition. Suppose we add at a site in Bifor\nsomei\u00151. As noted before, either Bi\rips, or it does not, depending only\non the location of the addition and its relative position in Bi. In particular,\nwhether or not the block \rips does not depend on the con\fguration. If Bi\n\rips, then either zero or two particles leave Bi. They have the e\u000bect of an\neven number of additions to site Xi+1. Since one addition at Xi+1has either\nno e\u000bect, or results in a \rip, two additions have the e\u000bect of no change at all.\nTherefore, the con\fguration of block Bi+1does not change, and as a result,\n14two or zero particles leave Bi+1intoBi+2. Continuing this way, we conclude\nthat ifBi\rips, then no other block \rips. If Bidoes not \rip, then one particle\nleavesBi, and it has the e\u000bect of one addition to site Xi+1. Therefore, Bi+1\nwill \rip ifliis odd, otherwise, one particle leaves Bi+1and has the e\u000bect of\none addition to site Xi+2. We conclude that if Bidoes not \rip, then `the \frst\nodd block downstream' \rips. In other words, block Bj\rips, where ljis odd,\nand there is no i<k<j such thatlkis odd. If there exists no such j, then\nno block \rips, so that the addition did not change the con\fguration.\nIn the interval [0 ;X 1\u00001] there is at most one empty site, but it is not\nnecessarily site X1\u00001. If the addition site is in B0, then the new con\fguration\ndepends on the current con\fguration. However, at most two particles leave\nthis interval: either one site \rips from 0 to 1, one site \rips from 1 to 0, or\ntwo sites \rip (one from 0 to 1 and one vice versa). In the \frst two cases,\nzero or two particles leave B0, so that the con\fguration downstream is not\na\u000bected. In the third case, one particle leaves B0, so that the \frst odd block\ndownstream \rips.\nThus we can partition all sites, except the sites in [0 ;X 1\u00001], into deter-\nministic non-empty sets Si,i= 1;:::;Rn, such that if there is an addition\nto a site in Si, thenBiand onlyBi\rips. We can summarize the e\u000bect of\nan addition as follows: pick a random block, such that for all i= 1;:::;Rn,\nblockBiis chosen with probabilityjSij\nn+1, and \rip it. With probabilityX1\nn+1,\n\rip either a site in [0 ;X 1\u00001], or two in [0 ;X 1\u00001] and the \frst odd block.\nWith probability(n+1\u0000X1\u0000P\nijSij)\nn+1, \rip nothing.\nIt is clear from the above description that for each block B1;B2;:::;BRn,\nthe probability that it has \ripped an even number of times until time t\n(inclusive, say) converges to 1 =2 ast!1 . Hence, the expected number\nof sites with height 0 in block Bi, converges to 1 =2 ast!1 , for alli=\n1;:::;Rn. BlockB0can be viewed as a classical sandpile model on X1sites\n(with a slower addition rate of course). Therefore, the expected number of\nsites inB0with height 0 converges to 1 =(x+ 1) ast!1 .\nWriting Pfor\u0016\u0002\u0017andEfor the corresponding expectation, we now\nwrite\n\u001at\nn=1\nn+ 1X\nk;xE nX\ni=0\u0011t(i)jRn=k;X 1=x!\nP(Rn=k;X 1=x);\n15which is a \fnite sum. When t!1 , this converges to\nlim\nt!1\u001at\nn=X\nk;x\u0012\n1\u0000k\n2(n+ 1)\u00001\n(x+ 1)(n+ 1)\u0013\nP(Rn=k;X 1=x)\n= 1\u00001\n2(n+ 1)E(Rn)\u0000X\nk;x1\n(x+ 1)(n+ 1)P(Rn=k;X 1=x):\nSince1\nn+1E(Rn) tends to\u0017(\u0010:\u0010(0) =r) =frasn!1 and the last term\ntends to 0, we obtain\nlim\nn!1lim\nt!1\u001at\nn= 1\u0000fr=2:\n5.2 Proof of Theorem 2 and Theorem 3\nProof of Theorem 2. Part (a) follows from Theorem 1, hence we need only\nto prove (b). Recall that the initial con\fguration \u0011consists of a Poisson- \u001a\nnumber of particles at every site. It will be convenient to also consider the\nparity of the particle numbers at each site. To this end, let \u00110=\u0011mod 2,\nand call\u00110(x) the parity ofx. We de\fne E(x) = (\u0011(x)\u0000\u00110(x))=2, and call\nE(x) the number of excess pairs of particles atx. TheE(x) will play a crucial\nrole in what follows. When a site xis unstable, it topples, which means that\nit loses two particles. If \u0010(x) =r, the two particles can be thought of as\nstaying together, while when \u0010(x) =s, the two particles are separated, since\nthey are sent to di\u000berent neighbors. It is for this reason that we need to\nanalyze in detail how topplings of an s-site take place, since only around\nsuchs-sites it is possible that the excess pairs can be absorbed.\nConsider a site xwith\u0010(x) =s. For the time being, we restrict our\nattention tofx\u00001;x;x + 1g. We topple unstable sites in fx\u00001;x;x + 1g,\nuntil we reach the \fnal stable con\fguration \u0018onfx\u00001;x;x + 1g. Since we\nwork in the restricted model, we have that \u0010(x\u00001) =\u0010(x+ 1) =r. First,\nobserve that topplings will never change the parity of x.\nFurthermore, the sum of the parities of x\u00001 andx+1, computed mod 2,\ndoes not change either under topplings. Finally, note that if xtopples at all,\nthen\u0018(x\u00001) can not be 0, and hence \u0018(x\u00001) = 1. Hence, we have the\nfollowing table, dealing with the parities before versus parity/con\fguration\nafter stabilization of fx\u00001;x;x + 1g, under the assumption that xtopples\nat least once.\n16parity before stabilization con\fguration after stabilization\n000 101\n001 100\n010 111\n100 100\n011 110\n101 101\n110 110\n111 111\nIt follows from the table that the set fx\u00001;x;x + 1gcan absorb an excess\npair if and only if the original parities are 000 or 010. We call a site xwith\n\u0010(x) =sahole, if\u00110(x\u00001) =\u00110(x+1) = 0. In all other cases, all excess pairs\nthat are present in, or enter fx\u00001;x;x + 1gfrom the left, will after some\ntopplings infx\u00001;x;x + 1gresult in excess pairs leaving at the right of the\nset. For every excess pair, this requires at most two topplings per site. This\nobservation suggests that in order to decide whether or not a con\fguration\nis stabilizable, we need to compare the spatial density of excess pairs to the\ndensity of holes. Or, in other words, we need to compare the probability that\na given site xis a hole with the expected number of excess pairs at x. Below,\nwe formally prove that this is the correct picture, but \frst we compute these\nquantities.\nIt is easy to compute the (conditional) probability for a site xto be a\nhole, given that \u0010(x) =s. Indeed, since the original con\fguration is product\nmeasure with Poisson- \u001amarginals, the probability that a site has parity 0 is\n1\n2(1 +e\u00002\u001a). It follows that the probability of a site xbeing a hole is\n1\u0000fr\n4(1 +e\u00002\u001a)2: (5.1)\nA similar and related computation shows that the expected number of excess\npairs is equal to\n1\n2(\u001a\u00001\n2(1 +e\u00002\u001a)): (5.2)\nComparison of (5.1) and (5.2) then \fnishes the proof.\nIt remains to show that it is indeed the case that the relation between\nthe probability of a hole and the expected number of excess pairs determines\nwhether or not stabilization occurs. To this end, we \frst assume that (5.1)\n<(5.2), that is, the spatial density of holes is smaller than the expected\nnumber of excess pairs per site. Suppose that stabilization occurs under\nthis assumption. First we note that if the con\fguration stabilizes, all excess\npairs must be absorbed by a hole. Indeed, if there is an excess pair that is\n17never absorbed, with positive probability, then by ergodicity there must be\nin\fnitely many such pairs to the left of the origin, which implies that the\norigin will topple in\fnitely many times. However, a pairing in which every\nexcess pair is paired to a unique hole is only possible if the spatial density\nof the holes is at least as large as the spatial density of the excess pairs; this\nfollows, for instance, from the well known mass transport principle, see e.g.\n[10]. This violates the assumption, and we are done.\nFor the other direction, we assume that (5.1) >(5.2). De\fne a random\nwalkS(x);x2Zas follows. First we put S0= 0. Writing h(x) = 1 if there\nis a hole at x, andh(x) = 0 otherwise, we put\nS(x) =S(x\u00001) +E(x)\u0000h(x):\nBy assumption, Shas negative drift. We say that xisextremal ifS(y)>S(x)\nfor ally < x . The collection of extremal points has a well de\fned spatial\ndensity. In particular, there are in\fnitely many extremal points in both\ndirections. It is now simple to see that any excess pair will be absorbed before\nreaching the \frst extremal point to the right. In particular, no excess pair will\ntravel across an extremal point. From this it follows that the con\fguration\nstabilizes.\nProof of Theorem 3. The proof of this result is very similar to the proof of\nTheorem 2. Again, we have to compare the number of excess pairs to the\nnumber of holes. For the same reason as before, each hole can accommodate\none excess pair. What follows from this is that \u001athhas the same numerical\nvalue as\u001atr.\n5.3 Proof of Theorem 4\nThe main di\u000eculty of the proof is that we work in in\fnite volume, so we\ncannot use as a starting point that the dynamics becomes periodic.\nThe cases\u001a<1 and\u001a>2 are not di\u000ecult. The case 1 <\u001a< 2 however\nis rather tedious. Our strategy is to generalize the well-known notion of a\nforbidden sub-con\fguration (FSC) [14, 25], special local con\fgurations which\ndo not appear in the stationary state of the system. We will show that in\nin\fnite volume with parallel toppling order, there are a number of additional\nlocal con\fgurations which, essentially, do not appear in the stationary state\nof the in\fnite-volume system in the sense that their spatial density tends to 0\nast!1 . Once we have proved that, it is easy to show that in all remaining\npossibilities, all sites have period 2, and this will conclude the proof.\nIn the proof, the following lemma plays an important role. In [8] it was\nstated and proved for Z2, but their proof works for general graphs. We state\n18a version suitable for our purposes. We say that two con\fgurations \u0011and\u0018\naremirror images of each other, if \u0011(x) = 3\u0000\u0018(x) for allx.\nLemma 9 ([8]).Let two height con\fgurations \u0011and\u0018be mirror images of\neach other. Then after performing, for each, a parallel toppling time step,\nthe two resulting con\fgurations are again mirror images of each other.\nProof of Theorem 4. The case\u001a<1 follows from the fact that the system is\n(a.s.) stabilizable when \u001a <1. This is a well-known fact, proved in several\npapers, see for example [24]. We remark that in [25], it is moreover proved\nthat for\u001a= 1 the system is not stabilizable.\nNext, consider the case \u001a>2. We have that E\u001a(3\u0000\u0011(0))<1. Therefore,\nwhen\u0011is chosen according to \u0016\u001a, the mirror image \u0016 \u0011of\u0011is a.s. stabilizable.\nSo when we start the dynamics from both \u0011and \u0016\u0011, then for every site x,\n\u0016\u0011t(x)2f0;1geventually. According to Lemma 9, \u0011t(x)2f2;3geventually,\nproving this case. (Note that when a site x, and its two neighbours, are all\nunstable, then after toppling all unstable sites, the height of xis the same as\nbefore. So it is possible to have a well de\fned limiting con\fguration, despite\nthe fact that topplings continue to take place.)\nSuppose now that 1 < \u001a < 2. In this case, when we choose \u0011according\nto\u0016\u001a, neither\u0011nor \u0016\u0011is stabilizable a.s. Therefore, again using Lemma 9,\nfor every site x, there is a (random) time T <1such that if t > T ,xhas\nbeen stable for at least one time step, and also has been unstable for at least\none time step. Hence, for any \fnite subset WofZ, there is a (random) time\nTW<1such that before time TW, every site in Whas been both stabel and\nunstable for at least one time step. It is well known (see [14] and also [25])\nthat this implies that for t > TW, the total number of particles in Wis at\nleast equal to the number of internal edges in W. A local con\fguration in W\nwhich does notsatisfy this property is called a forbidden sub-con\fguration\n(FSC). It also follows from [14] that both \u0011t(x) and \u0016\u0011t(x) are at most 3, for\nallx2Wandt>TW.\nComparing the dynamics from both \u0011and its mirror \u0016 \u0011, we have from\nLemma 9 that if \u0011t(x) is unstable, then \u0016 \u0011t(x) is stable, and vice versa. Since\n\u0016\u0011eventually has no FSC in W, we see that in \u0011, also mirrors of FSC's no\nlonger appear eventually. For instance, since 00 is a FSC, and therefore does\nnot appear anymore eventually, neither does e.g. 33 occur in the long run,\non any given position, and similarly for all other FSC's.\nApart from the absence of FSC's, we claim that also the patterns 01 ;10\nand 111 are very rare, in the sense that the spatial density of these \fnite-\ndimensional patterns converges to 0 as ttends to in\fnity. Using Lemma 9\nor the above this implies that also 23 ;32 and 222 have small spatial density\n19fortlarge. We prove this claim at the end of this proof, but \frst show that\nit implies the desired result.\nIt is easy to check from the absence of the above mentioned local con\fgu-\nrations, that apart from local perturbations which occur with spatial density\ntending to 0, the only possible con\fgurations are those in which we alternat-\ningly choose elements from f0;1;11gandf2;3;22g. Any \fnite-dimensional\ncon\fguration which is built up from alternatingly choosing elements from\nf0;1;11gandf2;3;22ghas period 2. If the full con\fguration would be like\nthat (that is, if the FSC's, 01 ;10;111;222;23 and 32 would not occur at all)\nthan it would follow that\n\u000b\u001a(t) +\u000b\u001a(t+ 1) = 1; (5.3)\nfor allt. Indeed, if that were the case, then the con\fguration would be\nperiodic with period 2, with each site alternating between a stable and an\nunstable state. However, we have to deal with the rare perturbations of the\nFSC's and the con\fgurations 01 ;10;111;222;23 and 32. This however is not\na real complication. For any \u000f >0, we \fndTso large that the sum of the\nspatial densities of all the problematic sub-con\fgurations is at most \u000f, for all\nt > T . It is clear that for these values of t, the sum in (5.3) is close to 1.\nThis is enough to prove the result.\nIt remains to prove the claim that the patterns 01, 10 and 111 have spatial\ndensities which converges to 0 as t!1 . We start with 01 and 10. We write\n\u0016t\n\u001afor the probability measure describing the state of the process at time t,\nhence\u00160\n\u001a=\u0016\u001a. Since the initial measure \u00160\n\u001ais ergodic, and \u0016t\n\u001ais a factor\nof\u00160\n\u001a, it follows that also \u0016t\n\u001ais ergodic. Hence both 01 and 10 have a well-\nde\fned spatial density at time t, which is an a.s. constant, depending on t.\nBy symmetry, these two densities must be the same. (We do not need that\nthey are the same, but we do need that if one is positive, so is the other.)\nNow let, as before, Wbe a \fnite interval, and suppose that for some t, the\ncon\fguration in Wcontains the patterns 01 and 10 (in that order). When\n\u0011t(x) = 0 and\u0011t(x+ 1) = 1, we will say that the pattern 01 occurs fromx\n(the leftmost vertex of the pattern) and similarly for other local patterns.\nSuppose 01 occurs from xat a certain time s, wheresis much larger than\nTW, how large exactly will be decided later. We are interested in the last\ntime before time sthat the con\fguration at ( x;x+ 1) was di\u000berent from 01;\nwe call this the previous con\fguration. So the previous con\fguration is not\nnecessarily the con\fguration at time s\u00001. Instead, it is the last con\fguration\nat (x;x+ 1) which was di\u000berent from the current one. The 0 at xcan only\nhave been obtained through a toppling at x, and since x+ 1 has only one\nparticle, site x+ 2 must have been stable. Hence the previous con\fguration\n20fromxmust have been either 200 or 201. Since 00 is an FSC, assuming\nthat the previous con\fguration is at a time still larger than TW, 00 is not\nallowed, so 201 is the only option. This means that the pattern 01 cannot be\ncreated after time TW: if it occurs from x, then the previous con\fguration\nfromxmust have been 201, implying that 01 occurs from x+ 1. A similar\n(symmetric) argument is valid for the occurrence of 10.\nIt follows that when we go backwards in time, as long as we remain larger\nthanTW, the pattern 01 `travels' to the right, and the pattern 10 `travels'\nto the left. This implies that at some time in the past they must have met,\nand we must have either the pattern 0110 or the pattern 01 \u000310 inW, where\n\u0003is arbitrary. All this, again, is only valid if we assume that when we go\nbackwards in time, we never get to a time smaller than Tbefore one of the\ntwo patterns 0110 or 01 \u000310 occurs. It is clear that we can take sso large\nthat this is the case with overwhelming probability, since the model does not\nstabilize.\nBut now observe that 0110 is an FSC, and therefore this is not a possibil-\nity, since FSC's do not occur in Wafter timeTW. Furthermore, the previous\ncon\fguration of 01 \u000310 is an FSC whenever the \u0003is 0 or 1, and therefore these\nare not possible. If \u0003= 2, then the 2 must have arisen from the previous\ncon\fguration through a toppling of (at least) one of its neighbours, but then\nthe 0 next to the toppled site is impossible. Similarly for the possibility that\n\u0003= 3. We conclude that all cases lead to impossibilities, and therefore, on\nany \fnite interval W, we cannot have the patterns 01 and 10 together, after\nsu\u000eciently long time. This implies that the (common) density of 01 and 10\nmust tend to zero, as time tends to in\fnity.\nNext, we focus on the pattern 111, and again we ask the question what the\nprevious con\fguration could have been. (Recall that the previous con\fgura-\ntion is not the con\fguration one time unit earlier, but the last con\fguration\non that position that was di\u000berent from 111.) If the previous con\fguration\ncontains a 0, then exactly one of the neighbours of this 0 must be unstable.\nIndeed, after the toppling of all unstable sites, this is the only way to go\nfrom a 0 to a 1. But this implies that the other neighbour of the 0 is a 0 or\na 1, which means that the previous con\fguration contains either an FSC, or\nthe con\fguration 01 (or 10), which we know by the previous paragraph has\ndensity tending to 0.\nIf the previous con\fguration contains a 1, then the neighbours of this 1\ncan not be unstable, and by the same argument as above, the neighbours of\nthis 1 cannot be 0 or 1 (after time TW). Hence, the only possibility is that\nthe previous con\fguration would be 111, but this is a contradiction, since\nby de\fnition, the previous con\fguration must be di\u000berent from the current\none. It remains to rule out all previous con\fgurations which consist of only\n212's and 3's. It is obvious however that it is not possible to reach 111 through\nsuch a previous con\fguration: if all sites are unstable then the central site\nreceives two particles and is again unstable.\nWe summarize that if the pattern 111 occurs from xat a given time\nlarger than TW, then either it was present from xat timeTWor one of\nthe patterns 01 or 10 was present in the previous con\fguration. The \frst\npossibility requires that the central site has not toppled since time TW, which\nhas probability tending to 0 as time tends to in\fnity. The second possibility\ngives us that the spatial density of the pattern 111 tends to 0, just as we\nshowed for the patterns 01 and 10.\nAcknowledgment: We thank Lionel Levine, David Wilson and Alexander\nHolroyd for stimulating discussions.\nReferences\n[1] R. Dickman and A. Vespignani and S. Zapperi: Self-organized critical-\nity as an absorbing-state phase transition, Phys. Rev. E 57, 5095{5105\n(1998).\n[2] A. Vespignani, R. Dickman, M. A. 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Sidoravicius: Activated Random Walkers:\nFacts, Conjectures and Challenges, Journal of Statistical Physics ,138,\n126{142 (2010).\n24"    },    {        "title": "1212.3383v1.Nuclear_density_probed_by_Kaon_Nucleus_systems_and_Kaon_Nucleus_interaction.pdf",        "content": "arXiv:1212.3383v1  [nucl-th]  14 Dec 2012Nuclear Physics A00 (2018) 1–4Nuclear\nPhysicsA\nNucleardensityprobedbyKaon-Nucleussystemsand\nKaon-Nucleusinteraction\nJ.Yamagata-Sekiharaa, N. Ikenob, H. Nagahirob, D. Jidoc, S. Hirenzakib\naInstitute ofParticle and Nuclear Studies, High EnergyAcce lerator Research Organization (KEK),Ibaraki 305-0801, Ja pan\nbDepartment of Physics, Nara Women’s University, Nara 630-8 506, Japan\ncYukawa Institute for Theoretical Physics, Kyoto Universit y, Kyoto 606-8502, Japan\nAbstract\nEffectivenuclear densitiesprobedbykaon- andanti-kaon-nuc leus systemsarestudiedtheoreticallybothforboundandlo wenergy\nscatteringstates. As for the anti-kaon bound states, we inv estigate kaonic atoms. We findthat the e ffective density depends on the\natomic states significantly and we have the possibility to ob tain the anti-kaon properties at various nuclear densities by observing\nthe several kaonic atom states. We also find the energy depend ence of the probed density by kaon and anti-kaon scattering s tates.\nWe find that the study of the e ffective nuclear density will help to find the proper systems to investigate the meson properties at\nvarious nuclear densities.\nc∝circlecopyrt2012 Publishedby ElsevierLtd.\nKeywords:\n1. Introduction\nMeson-Nuclear systems are very important and useful object s to extract the meson properties at finite density,\nwhich may have close connectionsto the chiral symmetrybrea kingand its partial restoration in the nucleus[1]. The\neffectivenucleardensitiesprobedbypionicatomsareknownto bearound∼0.6ρ0foralmosteverypionicstate[2,3]\nand the change of the chiral order parameter ∝angbracketleft¯qq∝angbracketrightin nuclei is only determined at this specific density [1]. Thu s, we\nareveryinterestedinthedensityprobedbyvariousmeson-n ucleussystemstofindoutthepropermesonicsystemsto\nexplorethe meson propertiesat various nuclear densities, which will be important to investigatethe aspects of QCD\nsymmetriesat variousnucleardensitiesbeyondthe lineard ensityapproximation[4].\n2. Formulation\nWe calculate the effective density of boundstates for anti-kaon(kaonic atoms) , and scattering states of anti-kaon\nandkaonwith nucleus. Asfor the kaonicatoms,we alreadydot he similar analysisas reportedin Ref. [5], wherewe\ndefinetheoverlappingprobability( Sbound(r))ofthekaonicatomdensity( |R(r)|2)withthenucleondensitydistribution\n(ρ(r))to obtaintheeffectivedensity,as\nSbound(r)=|R(r)|2ρ(r)r2. (1)\n1J. Yamagata-Sekihara etal. /Nuclear Physics A00 (2018) 1–4 2\nSince the integrationof the Sbound(r) functiongivesthe shift of anti-kaonbindingenergyas ∆B.E.∝/integraldisplay\nd3rSbound(r)\nin the first orderperturbationtheoryforthe changeof the po tentialterm dependingonthe density linearly,we define\nthenucleondensityatthepeakpositionof Sbound(r)astheeffectivenucleardensityforboundstates,whichisexpected\ntobemostsensitivepartof ρtothe mesonbindingenergy.\nForscatteringstates,we newlydeterminethe di fferentoverlappingprobability( Sscat(r))as,\nSscat(r)=/summationdisplay\nl(2l+1)|Rl(r)|2ρ(r)r2Pl(cos(θ0)). (2)\nInthiscase,wedefinethe Sscat(r)functionsoastohavethepeakatthedensitywhichismostse nsitivetoadifferential\ncross section, namely the change of the di fferential cross section is related to the Sscat(r) as,∆/parenleftbiggdσ\ndΩ/parenrightbigg\n∝/integraldisplay\nd3rSscat(r)\ninsteadofthe bindingenergyfortheboundstates. Here,we c onsiderthescatteringwith12Catθ0=0.\nThemesonwavefunction R(r)inEqs.(1)and(2)areobatinedbysolvingtheKlein-Gordon equationwithmeson-\nnucleusoptical potential. In thisarticle, we use two kinds of the optical potentialsfor anti-kaon,which are the chira l\nunitary potential [6] and the phenomenological potential [ 7, 8]. For K+scattering, we use the potential given in\nRef.[9]. Theall potentialsusedin thisarticleareshownin Fig.1.\nFigure 1. Anti-kaon optical potentials for12C obtained by (a) chiral unitary approach [6] and (b) phenome nological fit [7, 8] are shown as a\nfunction of the radial coordinate r. The kaon-12C potential (c) obtained in Ref. [9] is also shown. The upper a nd lower panels show the real and\nimaginary part of the potential, respectively. The kaon-nu cleus potential in Ref. [9] doesn’t have imaginary part. Som e of the potential have the\nenergy dependence as shown in the figures.\n3. Numerical Results\nAsforthekaonicatoms,weshowthecalculatedanti-kaonden sities(|R(r)|2)andthenucelondensitydistributions\n(ρ(r)),andthe overlappingprobabilities( Sbound(r))inFig. 2[5]. Thenucleardensityat the peakpositionof Sbound(r)\nshownin the lowerpanelscorrespondsto the e ffectivenucleardensity. We can see that the e ffectivedensitiesprobed\nby atomic kaons depend on the atomic orbits significantly. Co mparing the results of two potentials, we find the\ndifferencesofthestructureoftheoverlappingprobabilities Sbound(r). Thesedifferencesarerelatedtothepropertiesof\nthedeeplyboundkaonicnuclearstateforbothpotentialcas es,sincethekaonicnuclearstatesexistforbothpotential s.\nInthe phenomenologicalpotential,the kaonicnuclearstat esare deeperthanthosein the chiralunitarypotential. The\n2J. Yamagata-Sekihara etal. /Nuclear Physics A00 (2018) 1–4 3\nwavefunctionsoftheatomicandnuclearstatesshouldbeort hogonal. Owingtotheorthogonalitythewavefunctions\nofatomicstateshavenodesintheregionofthenuclearsize[ 10]. Theeffectivedensitiesobtainedfromtheoverlapping\nprobabilities will have information on the di fference of the potentials, which might be extracted from expe rimental\ndata.\nFigure 2. Overlapping probabilities (lower frame) of the an ti-kaon wave functions (upper frame ) with the nucleon densi ties (middle frame) in the\natomic kaon bound states in12C calculated with the (left) chiral unitary and (right) phen omenological optical potentials.\nFor scattering states, we show the calculated results of the energy dependence of the e ffective density in Figs. 3\n(anti-kaon) and 4 (kaon). From these results, we find the e ffective density for anti-kaon dicreases for higher anti-\nkaonenergy. It wouldbenaturalthat thee ffectivedensityforhigherenergycouldbecomesmaller thant hat forlower\nenergysincetheimportantpartialwavesbecomelargerforh igherenergyingeneral,andthewavefunctionwithlarger\nangularmomentumwill observemore outer part of the nucleus due to the centrifugalbarrier. This is the case for the\nanti-kaonscattering. Fortheanti-kaon,theincreaseofth eeffectivedensityforeachpartialwaveismoderate,because\nthe anti-kaon exists inside the nucleus with small probabil ity due to the large absorption. In contrast, the e ffective\ndensity for kaonincreases for higherkaonenergy. This is be cause the potential of kaon in Ref. [9] is weak repulsive\nand does not have imaginary part. It means that the kaon with h igher energy can penetrate into the nucleus without\nanyabsorption. Thismakesthee ffectivedensityofeachpartialwaveincreaserapidly. Thisi sthereasonthatthekaon\nsystemhasdifferenteffectivedensitiesfromthoseofthe anti-kaonsystem.\n4. Summary\nWe have calculated the e ffective nuclear density probed by anti-kaon and kaon. The e ffective nuclear density is\ndefinedfromthe overlappingprobabilityofthe mesondensit yand thenucleondensity distribution. As forthebound\nstates,weconsiderthekaonicatoms. Wefindthee ffectivedensityprobedbyanti-kaoninatomicorbitdependso nthe\nquantum numbers of the atomic states. For the scattering sta tes, we have investigated the energy dependence of the\neffectivedensity. Fromthisstudy,wefindthee ffectivedensityofanti-kaondecreasesforhigheranti-kaon energydue\nto the centrifugalbarrier and the strong absorptioninside the nucleus. We also find the kaon for higher energycould\nprobehighernucleardensity,whichistheoppositetendenc ytotheanti-kaonsystem.\n3J. Yamagata-Sekihara etal. /Nuclear Physics A00 (2018) 1–4 4\nFigure 3. Calculated e ffective nuclear densities defined in Eq.(2) (solid line) of an ti-kaon scattering for12C with chiral unitary approach (left) and\nphenomenological fit (right) as plotted as the functions of t he anti-kaon kinetic energy [MeV]. The contributions from a first few partial waves in\nEq.(2) arealso shown in thefigure.\nFigure 4. Sameas Fig.3 except for the kaon scattering case.\nReferences\n[1] K. Suzuki, M. Fujita, H. Geissel, H. Gilg, A. Gillitzer, R . S. Hayano, S. Hirenzaki and K. Itahashi et al., Phys. Rev. Lett. 92(2004) 072302\n[nucl-ex/0211023].\n[2] R.Seki and K. Masutani, Phys.Rev. C 27(1983) 2799.\n[3] N. Ikeno, R. Kimura, J. Yamagata-Sekihara, H. Nagahiro, D. Jido, K. Itahashi, L. -S. Geng and S. Hirenzaki, Prog. Theo r. Phys.126(2011)\n483 [arXiv:1107.5918 [nucl-th]].\n[4] D.Jido and S.Goda, arXiv:1108.6144 [nucl-th].\n[5] J.Yamagata and S.Hirenzaki, Eur. Phys.J.A 31,255 (2007) [nucl-th /0612036].\n[6] A.Ramos and E.Oset, Nucl. Phys.A 671, 481 (2000) [nucl-th /9906016].\n[7] C.J.Batty, E.Friedman and A.Gal, Phys.Rept. 287,385 (1997).\n[8] J.Mares, E.Friedman and A. Gal, Phys.Lett. B 606, 295 (2005), Nucl. Phys.A 770, 84 (2006).\n[9] E.Osetand A. Ramos,Nucl. Phys.A 679, 616 (2001) [nucl-th /0005046].\n[10] J. Yamagata, H. Nagahiro, Y. Okumura and S. Hirenzaki, P rog. Theor. Phys. 114, 301 (2005) [Erratum-ibid. 114, 905 (2005)]\n[nucl-th/0503039].\n4"    },    {        "title": "1301.2544v1.Spin_polarization_of_electrons_in_quantum_wires.pdf",        "content": "1 \n \n Spin polarization of  electrons in quantum wires  \n \nA.A. Vasilchenko \n \nKuban State Technological University, Krasnodar, Russia  \n \nThe total energy of a quasi -one-dimensional electron system is calculated using d ensity \nfunctional theory . It is shown that  spontane ous ferromagnetic state in quantum wire occurs at \nlow one -dimensional electron density.  The critical electron density below which electrons are in \nspin-polarized state is estimated analytically.  \n \nThe low-dimensional electron systems in semiconductors  are one of the most \nactual and intensively developing areas in condensed matter  physics for  the last \ndecades . The phenomena as integer and f ractional quantum Hall effects, Wigner \ncrystallization, metal -dielectric transition, conductivity quantization, persisten t \ncurrent oscillations, Coulomb blockade h ave been discovered  in the low-dimension \nelectron systems . The quantum wires (QW)  properties have been studied  less than \nproperties  of quantum wells and  quantum dots . The phenomena like Wigner \ncrystallization, spon taneous electron polarization in the zero magnetic field , \"0.7 \nanomaly\"  of conductivity  may take place in QW . These  phenomena are still very \nfar from a complete theoretical explanation.  \nThe study of  electron spin polarization in quasi -one-dimensional channels in \nzero magnetic field is of special interest. The results of e xperiment s [1] point out \non presence of a spin polariz ed state  in a narrow QW. Ferromagnetic state in QW \nmay take place  due to exchange and correlation interaction of electrons. Previously \neffects of electron spin polarization in QW have been studied using  Hartree -Fock  \nmethod [2, 3], one -dimensional Hubbard model [4, 5] , density functional theory [6, \n7]. At present,  the density  functional theory is considered as one of the most \npowerful meth ods (excluding the exact many -body Hamiltonian diagonalization ) \ntake into account of  electron -electron interaction.  \nIn this work the density functional theory have been  used to study  transition of \nelectrons in QW to the spin-polarized state in zero magneti c field . \nLet us consider a single quasi -one-dimensional channel.  In the direction of the \nz axis an electron density is given  by function, along the x axis movement of 2 \n \n electrons  is quantized, and along the y axis electrons can move free.  Inside a QW \nelect rons are confined  by a positively charged background with the two-\ndimensional density np (np differs from zero at \n2/ax , where a is width of QW).  \nAccording to the density functional theory  the total energy of the electron \nsystem is the  functional of the electron density n(x): \n][ ] )()[(21][ ][ nExdnxnxV nTnExc p H  \n,                                       (1) \nwhere T[n] is kinetic energy of non -interacting electrons, Exc[n] is exchange -\ncorrelati on energy. The second term in Eq. (1) is a Coulomb energy of el ectrons.  \nBelow  we use the atomic system of units, in which energy is represented in \nunits of \nBkaeRy22\n , and a length in Bohr ’s radius  \n22\nemka\neB , where me  effective \nmass of an electron , k  dielectric con stant . \nAs a rule, the c orrelation energy can be neglected, and only exchange  energy \nis taken into account in calculations [8]: \n21 21 213 16/\ns/ /\nx g /n )n(  \n,                                                                     (2) \nwhere gs is spin factor.  \nFrom Eq. (1) we obtain a Kohn -Sham  equation s \n)( )()()(\n22\nx E x x Vdxx d\nii i effi   \n,                                                             (3) \nwhere  \n      \n)( )( )( xVxVxVx H eff  ,                                                                                 (4) \n\n 1 1 1ln))( (4)( dxxx xn n xVp H\n,                                                            (5) \ndnnn dxVx\nx))(()(\n.                                                                                  (6) \nFurther we consider  that only the lower  energy level is populated , then the \nelectron densit y is defined by  \n)( )(2\n0x N xn\n,                                                                                       (7) \nwhere  N = np a is linear density of electrons.  3 \n \n We have developed and realized an efficient algorithm for numerical solution \nof the Schrödinger equation derived from the Kohn -Sham  system of equation  (2)–\n(7). The algorithm has been tested . Test results show that the developed  algorithm  \nis more efficient compar ed with the stand ard diagonalizati on methods.  The \ncomputer program  of imp lementation of the developed algorithm to  solve Kohn -\nSham nonlinear equations  allows to perf orm computations for a QW with  low \nelectron densities (small quantum well with one bound state), that is extremely \nimportant for studying  of transition to the spin-polarized state and  a system with  \nstrongly inhomogeneous electron gas.  \nKohn -Sham n onlinear system of equations has  been  solved numerically using  \niterations metho d. Calculations results of the potentials and the wave function of \nelectrons are s hown in  Fig. 1. It is seen that Coulomb potential contribution to the \neffective potential is much less, than the exchange  potential . \n \nFIG. 1. Wave function, Coulomb and effective potential s along the x direction in a QW  for a=5, \nn+=0.05, g s=1. \n  \nTransition to the spin-polari zed state occurs  at low electron densities, when \nonly the lowest  energy  level E0 (for not very big а) is populated.  In this case 4 \n \n kinetic energy has the form  \n))()( (\n32\n0 03\n22\n  dxx x V EN N\ngTeff\ns\n.                                                        (8) \nFrom Eqs. (2) and (8) it is seen that at high densities N a state with \nunpolarized electrons ( gs = 2) is always  energetically favorable .  \nThe total energy of electrons for gs = 1 and gs = 2 have been  calculated.  The \nresults of calculations  show  that the state with  completely polarized electrons is \nenergetically favorable at densities np < nc or N < Nc (Fig. 2) . Note, that for narrow \nQW the critical density nс can be high, but the value of Nc is changed weakly . \nFrom the results presented i n Fig. 2 we can also estimate a magnitude of a critical \ndensity for two -dimensional case. At d = 7  we have the value of  \n040, nc\n(approximately  41010 cm for GaAs), which corresponds to the upper \nboundary of transition in to the spin-polarized state of the  two-dimensional electron \ngas. \n \nFIG. 2. The p hase diagram of transition to the spin -polarized state  in a QW.  \n \nResults of the calculations showed that a contrib ution of Coulomb interaction \ninto a total energy is much less than a contribution of kinetic and exchange \nenergies. Neglecting the Coulomb contribution , we estimate the total energy of a \nsystem. We take the trial wave function of electrons as 5 \n \n \n2/1 4/12 2\n0)()2/ exp()(bb xx. \nThen the total energy has the form  \n4/32/3 2/3\n2 23 28\n32\n2 3 \ns s bgN\nbN\ngNE \n.                                                           (9) \nThe first and the second  terms correspond to kinetic energy, the third term \ncorrespond s to exchange energy. From Eq. (9) w e find the value of b, at which a \nminimum of the total energy is achieved : \n3/12/1\n2 43\n\nNgbs\n.                                                                                  (10) \nUsing   Eqs. (9) and (10) we obtain for the value of the critical densi ty Nc =0.3. \nThis value correlates well with exact values presented in Fig. 2. \nWe can also estimate the value of the critical density, representing exchange \npotential as  \n)21() (8\n) ()2/ exp( 8)(22\n2/1 4/32/1\n2/1 4/32 2 2/1\nbx\nbgN\nbgb x NxV\ns sx    .     \nFor this  potential we  obtain  \n3/1 2/1\n2 2Ngbs . \nAlso  this value of b is one and a half times less , than b given by (10), the critical \nelectron density varies insignificantly, and Nc = 0.27.  \nIn summary , the critical electron density below which all electrons are in a \nspin-polarized state in a QW is found . An estimation of the value of the critical \ndensity is obtained analytically, and this estimation is in good agreement with the \nresults of numerical calculation s. \nThis work was  supported by Russian Foundation for Basic Researc h. \n \n          Electronic address:  a_vas2002@mail.ru  \n[1] R. Crook , J. Prance, K.J. Thomas, S.J. Chorley, I. Farrer, D.A. Ritchie, M. \nPepper, C.G. Smith, Scienc e 312, 1359 (2006)  \n[2] F.E. Orlenko, S.I. Chelkak , E.V. Orlenko , G.G. Zegrya , Zh. Eskp. Teor. Fiz.  \n137, 1 (2010) . 6 \n \n [3] N.T. Bagraev, I.A. Shelykh, V.K. Ivanov, a nd L.E. Klyachkin, Phys. Rev. B \n70, 155315 (2004). \n[4] L. Bartosch, M. Kollar, P. Kopietz, Phys. Rev. B 67, 092403 (2003).  \n[5] K. Yang, Phys. Rev. Lett. 93, 066401 (2004).  \n[6] C. K. Wang, K. F. Berggren, Phys. Rev. B 57, 4552 (1998).  \n[7] A. Ashok, R. Akis,  D. Vasileska, D.K. Ferry, Mol. Simulations 31, 797 (2005).  \n[8] B. Tanatar, D.M. Ceperley, Phys. Rev. B 39, 5005 (1989) . \n "    },    {        "title": "1302.3034v1.Photon_gas_with_hyperbolic_dispersion_relations.pdf",        "content": "arXiv:1302.3034v1  [physics.optics]  13 Feb 2013Photon gas with hyperbolic dispersion relations\nMorteza Mohseni\nPhysics Department, Payame Noor University, Tehran 19395-369 7, Iran\nE-mail:m-mohseni@pnu.ac.ir\nAbstract. We investigate the density of states for a photon gas confined in a\nnonmagnetic metamaterial medium in which some components of the p ermittivity\ntensor are negative. We study the effect of the resulting hyperbo lic dispersion relations\non the black body spectral density. We show that for both of the p ossible wave-vector\nspace topologies, the spectral density vanishes at a certain freq uency. We obtain the\npartition function and derive some thermodynamical quantities of t he system. To\nleading order, the results resemble those of a one or two dimensiona l photon gas with\nenhanced density of states.\nKeywords : hyperbolic dispersion, metamaterials, photon gas\n1. Introduction\nThe density of states is a basic factor governing the physical prop erties of many\nstatistical systems. This can be seen, for instance, by considerin g the effect of electronic\ndensity of states on conductivity or other properties of condens ed matter systems.\nSimilarly, therearemanypropertiesofopticalsystems whichdepen donphotonicdensity\nof states. This fact has been used in designing photonic devices with desired properties\nby altering their photonic density of states through a variety of te chniques [1, 2, 3, 4],\nand more recently, by using metamaterials [5]. The later method relie s on the properties\nof metamaterials, which, roughly speaking, are media with negative in dices of refraction\nin certain frequency ranges.\nThe propagation of electromagnetic waves in media with negative per mittivity and\nor permeability was studied decades ago by Veselago [6]. This results , depending on the\nsigns of the permittivity and the permeability, in the emergence of so -called evanescent\nor backward-propagating electromagnetic waves (for a brief rev iew, see e.g., Ref. [7]\nand for a more extensive review see Ref. [8]). Following the experimen tal realization of\nsuch media [9, 10, 11, 12, 13], the subject has attracted a lot of int erest in both science\nand technology. One can mention in this regard, on the practical sid e, the construction\nof various microwave devices ranging from antennas to band-pass filters and lenses, see\ne.g., Ref. [14] and references therein. On the other hand, from th e more formal point of\nview, metamaterials are considered as arenas in which one can mimic ce rtain theoreticalPhoton gas with hyperbolic dispersion relations 2\nmodels such as dynamic spacetime, string theory D-branes and non commutativity [15],\nspacetimes with a metric signature transition [16], Schwarzschild (an ti-) de Sitter black\nholes [17], and motion around celestial bodies [18] .\nThepresentworkaimstostudythethermodynamicsofaphotonga sconfinedwithin\na metamaterial medium, which usually obeys hyperbolic dispersion rela tions [19]. This\nis partly motivated by the recent interest in relativistic gases with mo dified dispersion\nrelations. Such modifications have been considered within several c ontexts, namely, in\nRefs. [20, 21, 22, 23, 24], in connection with Lorentz invariance viola ting models within\nthe context of quantum field theory; in Ref. [25], in the context of q uantum gravity; in\nRef. [26] in the context of high energy astrophysics, in Refs. [27, 2 8] in the framework\nof noncommutative field theories, in Ref. [29], in the context of mode ls with minimal\nlength; and in Ref. [30], in the framework of the Hoˇ rava gravity. An other motivation\nbehind this work is the interest in developing different probes into the properties of\nmetamaterials. As an example, spontaneous emission near hyperbo lic metamaterials\nhas been considered in Refs. [31, 32].\nThe hyperbolic dispersion relation associated with nanostructured metamaterials\nand its application in engineering photonic density of states has been discussed in Ref.\n[5]. Here, we study the statistical mechanics of a photon gas with su ch dispersion\nrelations for different wave-vector space topologies and obtain se veral new results. In\nparticular, we show that they exhibit interesting properties such a s the vanishing of\ndensity of statesat certain frequencies andbehaving like lower dime nsional usual photon\ngases but with enhanced density of states.\nIn the next sections, after a brief review of the dispersion relation for the\npropagationof electromagnetic fields in ananisotropic medium, we sh ow how hyperbolic\ndispersion relations could arise as a result of negative permittivities. We then obtain an\nexpression for the photonic density of states in a rather simple mod el. We apply this\nto study the black body radiation and show that this results in novel properties such\nas exhibiting the behaviour of a photon gas in lower dimensional ordina ry media. We\nstudy the thermodynamics of such media by calculating the partition function in the\ngrand canonical ensemble. We conclude by discussing the results.\n2. Hyperbolic dispersion relations\nThe propagation of electromagnetic waves is governed by the well-k nown Maxwell\nequations\n∇.D=−ρ, (1)\n∇×E=−∂B\n∂t, (2)\n∇.B= 0, (3)\n∇×H=J+∂D\n∂t, (4)Photon gas with hyperbolic dispersion relations 3\ninwhich D=ǫ·EandB=µ·H. Ingeneral, neglectingmediumlosses, thepermittivity ǫ\nandthepermeability µaresymmetrictensors. Here, weconsiderananisotropicdielectric\nmedium with\nǫ=\nǫx0 0\n0ǫy0\n0 0ǫz\n,µ=\nµ00 0\n0µ00\n0 0µ0\n (5)\nwhereǫx,y,zare all constant, and seek plane wave solutions of the above equat ions in the\nabsence of the electric charge ρand the current J. Thus, we set\nE(r,t) =E0eik·r−iω(k)t(6)\nwhere the real part is to be taken. Inserting this back into the Max well equations (2)\nand (4) and making use of Eq. (1), one can show that the following re lation holds\nk2\nx\nk2−ω2\nc2εx+k2\ny\nk2−ω2\nc2εy+k2\nz\nk2−ω2\nc2εz= 1 (7)\nin which εi=ǫi\nǫ0(i=x,y,z), andc=1√ǫ0µ0is the speed of light in vacuum. If we\nconfine ourselves to the special case of a uniaxial medium with εx=εy, we can obtain\nthe following simpler relation\nk2\nx+k2\ny\nεz+k2\nz\nεx=ω2\nc2. (8)\nFor ordinary media, εxandεzare positive and can be constant. However, for\nmetamaterials, either of these can have negative values. Also, in ge neral, metamaterials\nare always highly dispersive media, i.e., they exhibit both temporal (fr equency\ndependence) and spatial (wave-vector dependence) dispersion s [33]. Thus, the explicit\nforms of the permittivity components depend on the structure of the medium under\nconsideration. Since the spatial dispersion effects are usually much weaker than the\ntemporal ones, here, for convenience, we consider a dissipationle ss medium and ignore\nthe spatial dispersion. In the presence of dissipation, which is the c ase for the metal-\ndielectric metamaterials, thepermittivity components arecomplex a ndtheabove simple\nhyperbolic dispersion get replaced by hyperbolic-like dispersion relat ions [34]. However,\nwe expect the results obtained here to be qualitatively valid for low-lo ss media as well.\nWe consider a medium with\nεz= 1 (9)\nεx= 1−Ω2\nω2(10)\nin which Ω is some cutoff frequency. With this choice, we obtain negativ e values for εx\nfor frequencies under the cutoff value. This characterizes a nonm agnetic metamaterial\nwith negative xxandyycomponents of the permittivity. Thus, Eq. (8) reduces to\nk2\nx+k2\ny−ω2\nΩ2−ω2k2\nz=ω2\nc2. (11)Photon gas with hyperbolic dispersion relations 4\nAn example of such dispersion relations has been used in Ref. [35] (s ee also Ref. [36])\nto study the physics of vacuum in a strong magnetic field in which the p ermittivity\ncomponents are\nεxx=εyy=1+α\n1−α\nand\nεzz=α/parenleftbigg\n1−ω2\ns\nω2/parenrightbigg\nwhereωsis a constant frequency proportional to the inverse London pene tration depth\nand 0≤α≤1 is also a constant.\nIn this work, we confine ourselves to the case of nonmagnetic meta materials. To\ntake metamaterials with a magnetic response into account, one wou ld be faced with a\nmore complicated picture, in which a variety of mode-dependent disp ersion relations\n(classified in Ref. [37]) should be considered.\n3. The density of states\nTo study the statistical mechanics of the system, we first need to calculate the density\nof states. This can be obtained, as usual, from the wave-vector s pace volume of the shell\nenclosed by surfaces defined by ωandω+dω. Obviously, this volume is infinite, giving\nrise to a divergent density of states. However, in practice, there is a bound on the values\nofkwhich is imposed by the physical properties of the medium. An example of this is\ndiscussed in Ref. [5] in which the metamaterial under study is fabrica ted via a special\nnanopatterning with a characteristic patterning scale a. This puts an upper cutoff value\nonkwhich is of the order of2π\na. Thus, we need to calculate the above volume taking\nthe constraint k < k cinto account, kcbeing the cutoff value. The volume inside the\nhyperbolic surface described by Eq. (11) (depicted in Fig. 1) and th e planes kz=±k0,\nwherek0≡Ω2−ω2\ncΩ, equals\nV=2π\n3c3Ω3ω2(Ω2−ω2)(4Ω2−ω2)\nwhere we have used kcc= Ω. From this, we obtain\ndV\ndω=4π\n3c3Ω3ω(4Ω4−10Ω2ω2+3ω4).\nBasically, this gives the density of states in the interval [ ω,ω+dω]. However,\nthere are some exotic properties associated with this relation, nam ely, the appearance\nof negative values. This is because of the presence of a fixed cutoff value for k. Incertain\nregions, by increasing ω, the value of k0decreases so the net effect is a decrease in the\nenclosed volume. The behaviour ofdV\ndωis depicted in Fig. 2.\nThe number of states in the interval between ωandω+dωmay be obtained by\nmultiplying a factor of2V\n8π3by the absolute value ofdV\ndωdω. Here,Vis the volume of\nthe medium and the factor2\n8is inserted to account for two polarization states of eachPhoton gas with hyperbolic dispersion relations 5\n−k_0k_0\nkxkykz\nFigure 1. The surface spanned by the dispersion relation k2\nx+k2\ny−ω2\nΩ2−ω2k2\nz=ω2\nc2in\nthe wave-vectorspace. The density of states is given (up to a pro portionality constant)\nby the volume enclosed by the hyperboloid and the planes kz=±k0.\n00.20.40.60.81−1.5−1−0.500.51\nων′\nFigure 2. The function ν′=dV\ndω(in units of4πΩ2\n3c3) in terms of ω(in units of Ω).\nphoton, and to include the positive octant only. Thus, the density o f statesρ(ω) is given\nby\nρ(ω)dω=V\n3π2c3Ω3ω|4Ω4−10Ω2ω2+3ω4|dω. (12)\nForω >Ω, the medium obeys the usual spherical dispersion relation, and we have\nρ(ω)dω=V\nπ2c3ω2dω.\nAn immediate effect of the above modified relation for the density of s tates can\nbe seen by studying the black body radiation. To see this, we conside r a metamaterial\nmedium of the type described above and a gas of photons in equilibrium with the\nmedium at a temperature T=1\nkBβ(kBbeing the Boltzmann constant). The energy\ndensity of the radiation emitted inside the medium at this temperatur e is given by the\nfollowing expression\nu(ω)dω=1\nV/planckover2pi1ωρ(ω)dω\neβ/planckover2pi1ω−1(13)Photon gas with hyperbolic dispersion relations 6\nwhereu(ω) is the spectral density. For the usual spherical dispersion relat ion, this\nleads to the well-known Stefan-Boltzmann law. However, using the h yperbolic density\nof states, Eq. (12), the result would be different. In practice, we need not to use the\nwhole expression Eq. (12) here, but rather a simplified version would be sufficient. To\nsee this, we first rewrite the above expression in the following form\nu(x) =1\n3π2c3/planckover2pi13β4|4X4−10X2x2+3x4|\nX3x2\nex−1(14)\nwherex=β/planckover2pi1ωandX=β/planckover2pi1Ω. Now, we note that for a medium with nanoscale\ncharacteristic patterning length a∼10−8m, for room temperatures we have X∼103,\nwhich means that we can practically confine ourselves to the region x≪X. Thus, we\nobtain\nu(x) =4X\n3π2c3/planckover2pi13β4x2\nex−1(15)\nThe functional form of this expression resembles its counterpart for a usual two-\ndimensional medium. This might be explained from the dispersion relatio n Eq. (11)\nwhere in the left-hand side we have something like the norm of a vecto r computed\nwith respect to a metric with two spacelike components and a timelike o ne. Also,\ncompared with a two dimensional photon gas, here the spectral de nsity contains an\nextra4X\n3c/planckover2pi1βfactor, which is of the order of inverse characteristic length a−1, leading to a\nhuge enhancement of the density. The above relation is plotted in Fig . 3 which shows\nradical enhancement of the spectral density compared with a usu al photon gas in 3+1\nMinkowskian space.\n0246810020406080100\nxu(x)\nFigure 3. Plot of the spectral density (in units of1\nπ2/planckover2pi13β4c3) forX= 100 (upper)\ncompared to the result of the usual spherical dispersion relation ( lower).\nIn spite of the above arguments, one may still be interested in stud ying the\nbehaviour of the spectral density for situations where, say, X∼101, corresponding\nto a microscale patterning length a∼10−6m. This is depicted in Fig. 4. Qualitatively,\nthis looks like the previous case.Photon gas with hyperbolic dispersion relations 7\n02468100510152025\nxu(x)\nFigure 4. Spectral density (in units of1\nπ2/planckover2pi13β4c3) forX= 10.\n4. The partition function\nThe thermodynamics of the above described photon gas can be der ived from the\npartition function in the appropriate limit. Considering a grand canon ical ensemble\nfor the photon gas, we can write the grand partition function as fo llows\nlnQ ≡PV\nkBT=−/integraldisplay∞\n0ρ(ω)ln(1−e−β/planckover2pi1ω)dω (16)\nInserting Eq. (12) into the above expression, we get\nlnQ=/braceleftbigg−V\n3π2c3Ω3/parenleftbigg/integraldisplayαΩ\n0−/integraldisplayΩ\nαΩ/parenrightbigg\nω(4Ω4−10Ω2ω2\n+3ω4)−V\nπ2c3/integraldisplay∞\nΩω2/bracerightbigg\nln(1−e−β/planckover2pi1ω)dω (17)\nwhereα≈0.67 is the root of 4 −10x2+ 3x4= 0. Now, using our previous argument\non largeness of X=β/planckover2pi1Ω for a nanopatterned metamaterial, we can ignore the last two\nintegrals in the above expression. This leaves us with\nlnQ=−VΩ\n3π2c3β2/planckover2pi12/integraldisplayαX\n0f(x)dx (18)\nin which\nf(x) =x/parenleftbigg\n4−10x2\nX2+3x4\nX4/parenrightbigg\nln(1−e−x).\nIn view of the largeness of X, this can be approximated as\nlnQ ≈VΩ\n6π2c3β2/planckover2pi12/integraldisplay∞\n04x2−5x4\nX2+x6\nX4\nex−1dx. (19)\nInserting the values of the Bose-Einstein integrals above, this red uces to\nlnQ ≈4VΩ\n3π2c3β2/planckover2pi12/parenleftbigg\nζ(3)−15\nX2ζ(5)+90\nX4ζ(7)/parenrightbigg\n(20)\nwhereζis the Riemann zeta function.Photon gas with hyperbolic dispersion relations 8\nThe internal energy is given by\nU=−∂\n∂βlnQ\n≈8VΩ\n3π2c3β3/planckover2pi12/parenleftbigg\nζ(3)−30\nX2ζ(5)+270\nX4ζ(7)/parenrightbigg\n(21)\nwhich resembles the energy of a usual photon gas in two dimensions t ogether with\ncorrection terms of the form X−2andX−4which are very small for a metamaterial\nmedium with nanoscale patterning length. Comparing the last two equ ations, one\nreaches the following relation\nP=1\n2U\nV/parenleftbig\n1−O(X−2)/parenrightbig\n(22)\ninwhich higher order correction terms areneglected. Other therm odynamical quantities\nsuch as entropy can also be obtained from the above partition func tion. The result of\nthese calculations is consistent withtheresult obtained intheprevio us section according\nto which thephotongasunder consideration behaves like anordinar y photongasin 2+1\ndimensions with some small corrections.\nIt would also beinteresting to consider a metamaterial with εx=εy>0 andεz<0.\nThis can be modelled by interchanging εxandεzin Eqs. (9) and (10). The relevant\ndispersion relation is then\nk2\nx+k2\ny−/parenleftbiggΩ2\nω2−1/parenrightbigg\nk2\nz=−Ω2−ω2\nc2. (23)\nIn this case, the wave-vector space configuration is again a hyper boloid, but with a\ndifferent topology, namely a two sheeted hyperboloid depicted in Fig. 5.\n−k_0k_0\nkxkykz\nFigure 5. The surface spanned by the dispersion relation k2\nx+k2\ny−/parenleftig\nΩ2\nω2−1/parenrightig\nk2\nz=\n−Ω2−ω2\nc2.\nIn this case, we are interested in the volume between the surfaces spanned by the\nrelation Eq. (23) and the cutoff planes kz=±k0=ω\nΩc√\n2Ω2−ω2. Again, this volumePhoton gas with hyperbolic dispersion relations 9\ncan decrease or increase with increasing values of ω. The behaviour of the derivative of\nthis volume with respect to frequency is shown in Fig. 6.\n00.20.40.60.81−0.2−0.100.10.20.3\nων′\nFigure 6. The function ν′=dV\ndω(in units of4πΩ2\n3c3) in terms of ω(in units of Ω) for\nthe surface k2\nx+k2\ny−/parenleftig\nΩ2\nω2−1/parenrightig\nk2\nz=−Ω2−ω2\nc2.\nRepeating the same steps as in the previous sections, for ω <Ω we reach\nρ(ω) =V\n3π2c3Ω3√\n2Ω2−ω2/vextendsingle/vextendsingle−Ω6+Ω4ω2+5Ω2ω4\n−3ω6+(Ω5−3Ω3ω2)√\n2Ω2−ω2/vextendsingle/vextendsingle/vextendsingle (24)\nThis can be approximated as\nρ(ω)dω≈V\n3π2c3/parenleftigg√\n2−1√\n2Ω2+√\n2−6\n2ω2/parenrightigg\ndω (25)\nThis shows a constant density of states corrected with a ω2term. One may interpret\nthis as the density of states of a photon gas in 1+1 dimensions for lar ge values of Ω.\n5. Discussion and conclusions\nWe have considered a photonic gas with modified dispersion relation pr opagating in a\nnonmagnetic metamaterial medium with some negative components o f the permittivity\ntensor. From the hyperbolic dispersion relations, we obtained the d ensity of states,\nwhich in the case where the xxandyycomponents of the permittivity tensor are\nnegative, shows some interesting features such as vanishing at a c ertain frequency.\nFor the case where only the zzcomponent is negative, the surface spanned by the\ndispersion relation in the wave-vector space is still hyperbolic but wit h a different\ntopology and the density of states can decrease or increase with f requency. We studied\nthe thermodynamics of the system and showed that it resembles a p hotonic gas in an\nordinary one or two dimensional medium but with enhanced density of states. The\nresults are of interest in studying the behaviour of physical syste ms in spacetimes\nwith non- Lorentzian metric signatures. The relevant calculations a re performed underPhoton gas with hyperbolic dispersion relations 10\nthe assumption that the medium dielectric coefficients have a plasma- like frequency\ndependence but are independent of the wave-vector. It would be interesting to consider\nmore general situations where these are functions of both the fr equency and the wave-\nvector.\nAcknowledgements\nI would like to thank the Abdus Salam ICTP where part of this work was done. I also\nthank two anonymous referees of Journal of Optics for comment s.\nReferences\n[1] Moreau E, Robert I, Gerard J M, Abram I, Manin L, Thierrt-Mieg V 2001Appl. Phys. Lett. 79\n2865\n[2] Pelton M, Santori C, Vuckovic J, Zhang B, Solomon G S, Plant J, Ya mamoto Y 2002 Phys. Rev.\nLett.89233602\n[3] Hughes S 2004 Opt. Lett. 292659\n[4] Lodahl P, van Driel A F, Nikolaev I S, Irman A, Overgaag K, Vanma ekelbergh D, Vos W L 2004\nNature430654\n[5] Jacob Z, Kim J Y, Naik G V, Boltasseva A, Narimanov E E, Shalaev V M 2 010Appl. Phys. B\n100215\n[6] Veselago V G 1968 Soviet Phys. Usp. 10509\n[7] Padilla W J, Basov D N, Smith D R 2006 Materials Today 928\n[8] Ramakrishna A A 2005, Rep. Prog. Phys. 68449\n[9] Smith D R, Padilla W J, Vier D C, Nemat-Nasser S C, Schultz S 2000 Phys. Rev. Lett. 844184\n[10] Shelby R A, Smith D R, Schultz S 2001 Science29277\n[11] Eleftheriades G V, Iyer A K, Kremer P C 2002 IEEE Trans. Microwave Theory Techniques 50\n2702\n[12] Grbic A, Eleftheriades G V 2004 Phys. Rev. Lett. 92117403\n[13] Grbic A, Eleftheriades G V 2002 J. Appl. Phys. 925930\n[14] Tan Y S, Seviour R 2009 Eur. Phys. Lett. 8734005\n[15] Miao R X, Zheng R, Li M 2011 Phys. Lett. B 696550\n[16] Smolyaninov I I, Narimanov E E 2010 Phys. Rev. Lett. 105067402\n[17] Mackay T G, Lakhtakia A 2011 Phys. Rev. B 83195424\n[18] Genov D A, Zhang S, Zhang X 2009 Nature Phys. 5678\n[19] Jacob Z, Alekseyev L V, Narimanov E 2006 Optics Express 148247\n[20] Colladay D, Kosteleck´ y V A 1998 Phys. Rev. D 58116002\n[21] Lehnert R 2004 J. Math. Phys. 453399\n[22] Kosteleck´ y V A, Russell N 2010 Phys. Lett. B 693443\n[23] Ghosh S 2006 Phys. Rev. D 74084019\n[24] Romero J M, S´ anchez-Santos O, Vergara J D 2011 Phys. Lett. A 3753817\n[25] Jacobson T, Liberati S, Mattingly D, Stecker F W 2004 Phys. Rev. Lett. 93021101\n[26] Jacobson T, Liberati S, Mattingly D 2003 Phys. Rev. D 67124011\n[27] Guralnik Z, Jackiw R, Pi Y S, Polychronakos A P 2001 Phys. Lett. B 517450\n[28] Castorina P, Iorio A, Zappal` a D 2011 Eur. Phys. J. C 711653\n[29] Mania D, Maziashvili M 2011 Phys. Lett. B 705521\n[30] Visser M 2011 J. Phys.: Conf. Ser. 314012002\n[31] Iorsha I, Poddubnya A, Orlova A, Belova P, Kivshar Y S 2012 Phys. Lett. A 376185Photon gas with hyperbolic dispersion relations 11\n[32] Noginov M A, Li H, Barnakov Y A, Dryden D, Nataraj G, Zhu G, Bo nner C E, Mayy M, Jacob\nZ, Narimanov E E 2010 Opt. Lett. 351863\n[33] Shapiro M A, Shvets G, Sirigiri J R, Temkin R J 2006 Opt. Lett. 312051\n[34] Mackay T G, Lakhtakia A, Depine R A 2005 Microwave Opt. Technol. Lett. 48363\n[35] Smolyaninov I I 2012 Phys. Rev. D 85114013\n[36] Smolyaninov I I 2011 Phys. Rev. Lett. 107253903\n[37] Depine R A, Inchaussandague M E, Lakhtakia A 2006 J. Opt. Soc. Am. A 23949"    },    {        "title": "1302.6568v1.Electronic_and_optical_properties_of_carbon_nanodisks_and_nanocones.pdf",        "content": "Electronic and optical properties of carbon nanodisks and nanocones\nP. Ulloa\u0003and M. Pacheco\nDepartamento de F\u0013 \u0010sica, Universidad T\u0013 ecnica Federico Santa Mar\u0013 \u0010a, Casilla 110-V, Valpara\u0013 \u0010so, Chile\nL. E. Oliveira\nInstituto de F\u0013 \u0010sica, Universidade Estadual de Campinas-UNICAMP, Campinas-SP, 13083-859, Brazil\nA. Latg\u0013 e\nInstituto de F\u0013 \u0010sica, Universidade Federal Fluminense, 24210-340, Niter\u0013 oi-RJ, Brazil\n(Dated: December 2, 2018)\nA theoretical study of the electronic properties of nanodisks and nanocones is presented within the\nframework of a tight-binding scheme. The electronic densities of states and absorption coe\u000ecients\nare calculated for such structures with di\u000berent sizes and topologies. A discrete position approxi-\nmation is used to describe the electronic states taking into account the e\u000bect of the overlap integral\nto \frst order. For small \fnite systems, both total and local densities of states depend sensitively on\nthe number of atoms and characteristic geometry of the structures. Results for the local densities\nof charge reveal a \fnite charge distribution around some atoms at the apices and borders of the\ncone structures. For structures with more than 5000 atoms, the contribution to the total density of\nstates near the Fermi level essentially comes from states localized at the edges. For other energies\nthe average density of states exhibits similar features to the case of a graphene lattice. Results for\nthe absorption spectra of nanocones show a peculiar dependence on the photon polarization in the\ninfrared range for all investigated structures.\nI. INTRODUCTION\nCarbon nanocones (CNCs) have been observed by mi-\ncroscopy techniques[1, 2], showing an ordered atomic\nstructure. Large progress has been made on synthe-\nsis, characterization and manipulation of CNCs and car-\nbon nanodisks (CNDs) [3{8]. From the technological\npoint of view, applications such as microscopy probes\nand electron-emitter devices may be envisaged by con-\nsidering the electric current established through the cone\napex when electric \felds are applied[9].\nThere are di\u000berent theoretical schemes to describe\nthe electronic properties of cone-like structures. Mod-\nels based on the Dirac equation[10, 11] give a conve-\nnient insight of properties in the long wavelength limit.\nHowever, for \fnite-size graphenes, the longest stationary\nwavelength occurs in the border and a correct descrip-\ntion of the states near the Fermi level is given in terms\nof edge states[12, 13]. Ab initio models [14{16] are able\nto predict detailed features, but they are restricted to\nstructures composed of a few hundred atoms due to their\nconsiderable computational costs. Calculations based on\na single\u0019orbital are able to describe the relevant elec-\ntronic properties[17, 18]. In that spirit, we calculate\nthe electronic structure and optical spectra of CNDs and\nCNCs within a tight binding approach. CNC structured\nsystems generated by pentagonal and heptagonal defects\nwere previously studied using a Green function recursive\nmethod [18, 19]. It was shown that, for cones generated\nwith an odd disclination number nw, it is not possible\n\u0003Electronic address: pablo.ulloa@usm.clto de\fne A and B graphene sublattices. In this case,\ntherefore, the electron-hole symmetry is broken.\nThe total number NCof carbon atoms in a cone struc-\nture may be estimated by dividing the cone surface area\nby half of the hexagonal cell's surface,\nNC= [4\u0019=(3p\n3)](1\u0000nw=6)(rD=aCC)2; (1)\nwhere the disclination number nwcorresponds to the in-\nteger number of \u0019=3 wedge-sections eliminated from the\ndisk structure, and rDis the cone generatrix [see Figure\n1]. The nanocone disclination angle is given by nw\u0019=3.\nFor example, for nw= 1 andrD= 1\u0016m, the CNC has\n\u0019108atoms. By extracting an integer number nwof\u0019=3\nsections from a carbon-disk [cf. Figure 1], it is possible to\nconstruct up to 5 di\u000berent closed cones. For nw= 1, the\ncone angle is 2 \u00121= 112:9\u000e, corresponding to the \rattest\npossible cone. In this case, h=rC= 0:66 andh=rD= 0:55.\nIn this work, \fnite-size systems (from two hundred up\nto \fve thousand atoms) are studied by performing di-\nrect diagonalizations of the stationary wave equation in\nthe framework of a \frst-neighbor tight-binding approach.\nEach carbon atom has three nearest neighbours, except\nthe border atoms for which dangling bonds are present.\nThe overlap integral sis considered di\u000berent from zero.\nAs we will show later, this has important e\u000bects on the\ncone energy spectrum.\nAlthough some of the graphene electronic properties\nare present in the CNCs, deviations are always mani-\nfested as a consequence of the di\u000berent atomic arrange-\nments, the \fnite-size of the nanocones, and also the pos-\nsible point symmetry of the distinct cones. In the ab-\nsence of external \felds, the calculated density of states\n(DOS) shows a peak at the Fermi energy and the local\ndensity of states (LDOS) shows that electron states arearXiv:1302.6568v1  [cond-mat.mes-hall]  26 Feb 20132\nFigure 1: (Color online) Pictorial view of (a) a carbon disk\ncomposed of six wedge-section of \u0019=3 central angle, then (b)\nthe remotion of a nw=f1;:::; 5gwedge-sections from the\ndisk, and (c) by folding it is constructed a cone. Geometrical\nelements: generatrix rD, heighthc, base radius rc, and apex\nopening angle 2 \u0012, where sin \u0012= 1\u0000nw=6.\nlocalized at the cone base. On the other hand, the sym-\nmetries observed in the LDOS at di\u000berent energies allow\na systematic description of the electronic structure and\nselection rules of optical transitions driven by polarized\nradiation. Unlike the nanodisk, the presence of topolog-\nical disorders in nanocones involve a deviation from the\nelectrical neutrality at the apex and at the zigzag edges.\nII. THEORETICAL FRAMEWORK\nIn what follows, we present results for nw=f0;1;2g,\ncorresponding to CND and CNCs whose disclination an-\ngles are 60\u000eand 120\u000e. For those systems, the sp2hy-\nbridization may be neglected. The electronic wave func-\ntion may be written as\nj\ti=NCX\nj=1Cjj\u0019ji; (2)\nwhere thej\u0019jidenote the atomic orbitals 2 pat site~Ri.\nNote that the overlapping between neighbouring orbitals\nprevents the set consisting of the ket j\u0019jito be an or-\nthogonal basis. Only in the ideal case of zero overlap\ns= 0, the coe\u000ecients C0\njinj\ti=PNC\nj=1C0\njj\u00190\njimight\nbe considered equal to the discrete amplitude probability\nh\u00190\njj\tito \fnd an electron at the j-th atom (described by\nthe one electron state j\ti). We use the s6= 0 basis,j\u0019ji,\nto construct the eigenvalue equation and the j\u00190\njibase\nto calculate the properties related to discrete positions.\nOf course, to relate both bases it is required to know the\nh\u00190\njj\u0019jiprojection.\nWe de\fne a NC\u0002NCmatrix \u0001(1)relating the nearest\nneighbouring atomic sites i;j,\n\u0001(1)\nij=\u001a\n1;(i;j) are n.n.\n0;otherwise\u001b\n: (3)\nSimilarly,\n\u0001(0)\nij=\u001a\n1;(i=j)\n0;otherwise\u001b\n: (4)TheSoverlap matrix elements are then given by\nSij=h\u0019ij\u0019ji= \u0001(0)\nij+s\u0001(1)\nij: (5)\nThe hopping matrix elements of the tight-binding Hamil-\ntonian ^H=^H(at)+^Vare\nh\u0019ij^Vj\u0019ji=t\u0001(1)\nij; (6)\nwheretis the hopping energy parameter. Assuming the\neigenvalue equation ^H(at)j\u0019ji=\"2pj\u0019ji, the atomic ma-\ntrix elements are\nh\u0019ij^H(at)j\u0019ji=\"2ph\n\u0001(0)\nij+s\u0001(1)\niji\n; (7)\nand\nh\u0019ij^Hj\ti=\"h\u0019ij\ti;1\u0014i\u0014NC: (8)\nThe resulting equation system may be written as a\ngeneralized eigenvalue problem H~C=\"S~C, where the\ncolumn vector ~Ccontains the coe\u000ecient Cj,\nh\n\"2p(\u0001(0)+s\u0001(1)) +t\u0001(1)i\n~C=\"h\n\u0001(0)+s\u0001(1)i\n~C:\n(9)\nThe general solution may be expressed in terms of the\nauxiliar variables ~C(0) and\"(0), which satisfy\nt\u0001(1)~C(0) =\"(0)~C(0): (10)\nAs~C(0) also satis\fes Eq. (9), we obtain\n\"= [\"2p+ (1 +s\"2p=t)\"(0)]=[1 +s\"(0)=t]: (11)\nThe orthogonality condition for the electronic states\nh\tkj\tli=~CkyS~Cl=\u000ek;l (12)\nimplies that\n~C=~C(0)q\n~C(0)yS~C(0): (13)\nFor the calculation of the DOS we use a Lorentzian dis-\ntribution\nDOS(\") = 2NCX\nj=1\u000e(\"\u0000\"j)\u00192NCX\nj=1\u0000=\u0019\n(\"\u0000\"j)2+ \u00002:(14)\nThe LDOS is calculated in terms of the discrete am-\nplitude probability, h\u00190\nij\tji,\nLDOS(~Ri;\") = 2NCX\nj=1jh\u00190\nij\tjij2\u000e(\"\u0000\"j); (15)\nwhere\nh\u00190\nij\tji=h\n\u0001(0)+s\n2\u0001(1)i\n~Cj; (16)3\nas it is shown in Appendix A.\nThe local density of charge (LDOC) related to the \u0019\nelectrons is calculated by assuming that the other 5 elec-\ntrons and the 6 protons of the carbon atom act as a net\ncharge +e. Assuming zero temperature and the indepen-\ndent electron approximation, only the states 1 \u0014j\u0014nF\nwill be occupied, where\nnF=\u001a\nNC=2; NCis even\n(NC+ 1)=2; NCis odd:(17)\nTaking into account that the states below nFcon-\ntribute with\u00002eand the fact that the nFstate con-\ntribution depends on the parity of the number of atoms\nin the system, the LDOC is written as\nLDOC(~Ri) =e2\n41\u00002nF\u00001X\nj=1jh\u00190\nij\tjij2\u0000\rjh\u00190\nij\tnFij23\n5\n(18)\nwith\r=0 and 1, for NCeven and odd, respectively.\nOptical absorption coe\u000ecients \u000b\u000f(!) are calculated by\nconsidering perpendicular ( ^ \u000f?= ^\u000fx;^\u000fy) and parallel ( ^ \u000fk=\n^\u000fz) polarizations, in relation to the cone axis,\n\u000b^\u000f(!)/1\n!X\ni;j\f\fh\tij^\u000f\u0001~ pj\tji\f\f2\u000e\u0002\n\"j\u0000\"i\u0000~!\u0003\n;(19)\nwith\"i;jcorresponding to the energies of occupied and\nunoccupied states, respectively.\nThe oscillator strength may be written in terms of the\nspatial operators (^ x, ^yand ^z) [20], i.e.,\nh\tij^pzj\tji=me\ni~(\"j\u0000\"i)h\tij^zj\tji; (20)\nwhereh\tij^zj\tjiis calculated to \frst order in s, using\n(A9) of Appendix A,\nh\tij^zj\tji=~Ciyh\nz+s\n2\u0010\n\u0001(1)z+z\u0001(1)\u0011i\n~Cj:(21)\nIII. RESULTS AND DISCUSSION\nA. Electronic Density of States\nIn what follows, we present numerical results for sys-\ntems composed of up to 5000 atoms. In the limiting\ncase ofNC!1 , the energy spectrum is in the range\nfrom\"min=\u00003jtj=(1 + 3s) to\"max= +3jtj=(1\u00003s),\nthe van Hove singularities occur at \"v\nvH=\u0000jtj=(1 +s),\n\"c\nvH= +jtj=(1\u0000s), and the Fermi energy is at \"F= 0.\nA \u0000 =jtj=100 broadening and an overlap s= 0:13 are\nassumed. Di\u000berent plots in Figure 2 show the density\nof states averaged over the NCatoms and the LDOS for\na CND [Figures 2(a) and 2(d)], a single-pentagon CNC\n[Figures 2(b) and 2(e)], and for a 2-pentagon CNC [Fig-\nures 2(c) and 2(f)], for NC= 258;245, and 246, respec-\ntively. All results are shown in an energy range around\n\"2p= 0.\nFigure 2: (color online) DOS and LDOS for a NC= 258\nnanodisk [(a) and (d)], a NC= 245 1-pentagon nanocone [(b)\nand (e)], and a NC= 246 2-pentagon nanocone [(c) and (f)].\nLDOS curves (black-1, red-2, blue-3) match, for each system,\nthe corresponding atoms shown in Figure 3. Vertical lines in\neach panel indicate the position of the Fermi energy.\nFigure 3: Pictorial view of (a) a nanodisk center, (b) a\n1-pentagon nanocone apex, and (c) a 2-pentagon nanocone\napex. Atoms with di\u000berent colors-numbers indicate di\u000berent\npoint symmetries for each system.\nFigure 4: (color online) DOS and LDOS for a NC= 5016\nnanodisk [(a) and (d)], a NC= 5005 1-pentagon nanocone\n[(b) and (e)], and a NC= 5002 2-pentagon nanocone [(c) and\n(f)]. LDOS curves (black-1, red-2, blue-3) match, for each\nsystem, the corresponding atoms shown in Figure 3. Vertical\nlines in each panel indicate the position of the Fermi energy.4\nFigure 5: (Color online) LDOS in arbitrary units for a 5016-atom nanodisk [panels (a) to (e)], a 5005-atom nanocone with one\npentagon at the apex [panels (f) to (j)], and a 5002-atom nanocone with 2 pentagons at apex [panels (k) to (o)]. The considered\nenergies are (a,f,k) \"min, (b,g,l)\"v\nvH, (c,h,m)\"F, (d,i,n)\"c\nvH, and (e,j,o) \"max. The LDOS is measured with respect to the mean\nLDOS which is equal to the DOS at the considered energy.\nAs expected for small \fnite systems, the DOS, LDOS,\nand the position of the Fermi energy depend on the num-\nber of atoms considered in the numerical calculation and\non their characteristic geometries [21{23]. A remarkable\ndi\u000berence between CND and CNCs structures is the exis-\ntence of a \fnite DOS above the Fermi level for nanocones.\nThis clear metallic character of the DOS for nanocones\nis more robust for the 2-pentagon CNC [22, 24]. One\nmay remark that this feature comes out from a symme-\ntry break induced by the presence of topological defects\nin the CNC lattices, which generates new states above\nthe Fermi energy, not present in the CND structure. The\ncontributions to the DOS coming from the apex atoms\nstates are apparent in the LDOS of Figures 2(e) and 2(f).\nAlso notice that for the 2-pentagon case, in which there\nis a large topological disorder, the LDOS spectra exhibit\nsigni\fcant di\u000berences depending on the point symmetry\nof the considered atom (cf. Figure 3).\nFor increasing number of atoms, the total DOS for\nthe di\u000berent nanostructures are very similar to the corre-\nsponding DOS of a graphene layer, except for the edges\nstates which show up as a peak at the Fermi energy, as\nshown in Figures 4(a), 4(b), and 4(c). It is interesting\nto note that the apex atomic states do not contribute to\nthe total DOS near the Fermi energy but mainly near the\ngraphene-like van Hove peaks. Notice that in the case of\ntwo-pentagon nanocones the LDOS at the tip exhibits a\nrobust metallic character.\nTo analyse the \fnite-size e\u000bects and the role played by\nthe di\u000berent symmetries of the cone-tip sites, we depict\nLDOS contour plots for the three studied structures by\nconsidering some characteristic energies: the minimum\nenergy, the resonant peak below the Fermi energy, theFermi energy, the resonant peak above the Fermi energy,\nand the maximum energy. Figure 5 illustrates the ex-\nample of a CND with 5016 atoms (top row), a single-\npentagon CNC with 5005 atoms (middle row), and a\n2-pentagon CNC with 5002 atoms (bottom row). The\nelectronic states corresponding to energies at the band\nextrema have the largest wavelength compared to the\ncharacteristic size of the system. In this way, the de-\ntails of the lattice become less important and the states\nexhibit azimuthal symmetry. An interesting feature for\nthe nanocones is that at these energies the apex corre-\nsponds to a node for the maximum energy and an antin-\node for the minimum energy, respectively. On the other\nhand, the states at the Fermi energy are localized at the\ncone border, mainly at the zigzag edges as it is clearly\nshown in Figures 5(c), 5(h), and 5(m). For the states\nwhose energy is near to the van Hove peaks, the LDOS\nre\rect the symmetries of each system, i.e., for CND the\n2\u0019=6-rotation symmetry and 12 specular planes [cf. Fig-\nures 5(b) and 5(d)], for a single-pentagon CNC there is a\n2\u0019=5-rotation symmetry and 5 specular planes [cf. Fig-\nures 5(g) and 5(i)], and for a two-pentagon CNC, there\nis a\u0019=2 rotation symmetry and 2 specular planes [cf.\nFigures 5(l) and 5(i)].\nB. Electric Charge Distribution\nThe electric charge per site, in terms of the fundamen-\ntal chargee, was obtained using Eq. (18). Results for\nthe electric charge distribution for CNDs indicate that\nall the atomic sites preserve the charge neutrality, i.e.,\nLDOC=0. For the CNCs, however, the atoms at the5\nsites 1 2 3 max\n1-pentagon -0.071e +0.014e -0.059e +0.042e\n2-pentagon -0.055e -0.067e -0.066e +0.076e\nTable I: LDOC (fundamental charge units) at some relevant\natoms in the cone apices shown in Figure 3(b) and (c). The\nmaximum (max) value occurs at the zigzag edge of each sys-\ntem.\nFigure 6: (Color online) Electric charge distribution in neutral\nCNCs, for a single-pentagon structure with 245 atoms (a) and\ntwo-pentagon system with 246 atoms (b).\napex acquire negative charge and the atoms around the\ncone base exhibit positive charges at the zigzag edges.\nAsNCincreases, the LDOCs at the apices, for the two\nstudied CNC structures, tend to the asymptotic values\nshown in table I, which are in good agreement with the\nvalues reported by Green method calculations [18, 19].\nFigure 6 depicts the LDOC for the two types of CNC\nstructures, showing that the nonequilibrium of the charge\ndistribution is restricted to the apex and edge regions:\nelectric neutrality is found at all the other surface sites.\nThe values found for the LDOC at the apex regions are\nfound to be independent of the size of the cones whereas\nthis is not true for the edge states. When the number\nof atoms of the CNC structure is even, the edge-states\nLDOC exhibits the same symmetry of the cone. For odd\nNC, the Fermi level is occupied by a single electron and\nthen the LDOC at the edge states re\rect the broken sym-\nmetry.\nC. Absorption Spectra\nWe have also calculated the absorption coe\u000ecient for\nthe CND and CNC structures, for di\u000berent photon po-\nlarizations. Figure 7 shows the results for the absorption\ncoe\u000ecients \u000bxand\u000by, for polarization perpendicular to\nthe cone axis, and \u000bzfor parallel polarization. Calcu-\nlated results are shown for a nanodisk composed of 5016\natoms, a single-pentagon nanocone with 5005 atoms, and\na 2-pentagon nanocone with 5002 atoms. For the case of\nlarge CNDs, the spectra present the general features ob-\nserved for the absorption of a graphene monolayer. In\nthe infrared region the absorption coe\u000ecient of a gra-\nphene monolayer is expected to be strictly constant [25],\nwhereas for higher energies the spectrum shows a strong\ninterband absorption peak coming from transitions near\nthe M point of the Brillouin zone of graphene [26]. The\nFigure 7: (Color online) Absorption coe\u000ecient for x(black\ncurves),y(red curves) and z(blue lines) polarizations for (a)\na nanodisk with 5016 atoms, (b) a single-pentagon nanocone\ncomposed of 5005 atoms, and (c) a two-pentagons nanocone\nwith 5002 atoms. The photon energies are given in units of\n~!=t.\nmain di\u000berence for a \fnite CND is a departure from a\ncompletely frequency-independent behavior for low ener-\ngies, where the absorption coe\u000ecient shows oscillations\nas a function of the photon energy instead of a constant\nvalue. This is a consequence of the border states that\nare manifested as a peak in the total DOS at the Fermi\nenergy [27, 28]. For CNCs, the general behaviour is the\nsame as for nanodisks, except for the dependence of the\nabsorption on the photon polarization, in particular for\nlow energies. Furthermore, the main absorption peaks\nfor di\u000berent polarizations occur when the photon energy\nis equal to the energy between the two DOS van Hove-\nlike peaks (cf. Figure 4). Notice that the overlap integral\ns6= 0 leads to an energy shift of the main resonant ab-\nsorption peak given by \u000e\u00192s2jtj=(1\u0000s2)\u0019100 meV.\nThis is a signi\fcant value for actual experimental mea-\nsurements.\nConcerning the di\u000berent polarization directions, one\nshould notice that, as occurs in C6vsymmetric systems,\n\u000bz= 0 and\u000bx=\u000byfor the nanodisk. On the other\nhand, the absorption coe\u000ecients for the di\u000berent cones\nstudied (single and two pentagons) are \fnite for parallel\npolarization, and it depends on the structure details: as\n\u000bzincreases for a two-pentagon CNC structure, \u000bx;yde-\ncreases. Due to the lack of \u0019=2- rotation symmetry, one6\nFigure 8: (Color online) Absorption coe\u000ecient for x(black\ncurves),y(red curves) and z(blue lines) polarizations for (a)\na nanodisk with 258 atoms, (b) a single-pentagon nanocone\ncomposed of 245 atoms, and (c) a two-pentagon nanocone\nwith 246 atoms. The photon energies are given in units of\n~!=t. Curves in each panel are vertically shifted, for better\nvisualization of di\u000berent polarization results.\nshould expect, in principle, di\u000berent results for x- and\ny-polarizations for any nanocone. However, such di\u000ber-\nence is observable just for the absorption coe\u000ecient of\nthe two-pentagon CNC system, mainly in the range of\nlow photon energies. The fact that \u000bx=\u000by, for the case\nof 1-pentagon CNC structure, may be explained using\nsimilar symmetry arguments applied to C6vsymmetry\ndots[28], extended to the C5vsymmetric cones. In the\ncase of a 2-pentagon CNC, the apex exibits a C2vsym-\nmetry, preventing the cone to be a C4vsymmetric sys-\ntem. As the apex plays a minor role, \u000bxand\u000bywill be\nslightly di\u000berent. A large di\u000berence between the \u000bzand\nthe\u000bx;yCNC absorption spectra occurs in the limit of\nlow radiation energy. The \u000bzcoe\u000ecient goes to zero as\n~!!0 whereas\u000bx;yshows oscillatory features. The be-\nhaviour of the absorption for parallel polarization is due\nto the localization of the electronic states at the atomic\nsites around the cone border. As the spatial distribution\nof those states are restricted to a narrow extension along\nthe z coordinate, the zdegree of freedom is frozen for low\nexcitation energies.\nThe dependence of the absorption spectra on the ge-\nometrical details of the di\u000berent structures is more no-\nticeable for \fnite-size nanostructures. This can be seenin Figure 8 which depicts the absorption coe\u000ecients for\nthe CND composed of 258 atoms, the single-pentagon\nCNC with 245 atoms, and the 2-pentagon CNC with\n246 atoms. The degeneracy of the x- and y- polariza-\ntion spectra is apparent for the smaller one-pentagon na-\nnocone, as expected due to symmetry issues. On the\nother hand, the symmetry reduction for the 2-pentagon\nstructure leads to a rich absorption spectra, exhibiting\npeaks at di\u000berent energies and with comparable weights\nfor distinct polarizations. In that sense, one may pro-\npose absorption experiments as an alternative route to\ndistinguish between di\u000berent nanocone geometries.\nIV. CONCLUSIONS\nHere we have presented a theoretical study on the\nelectronic properties of nanodisks and nanocones in the\nframework of a tight-binding approach. We have pro-\nposed a discrete position approximation to describe the\nelectronic states which takes into account the e\u000bect of\nthe overlap integral to \frst order. While the j\u0019ibase\nkeeps the phenomenology of the overlap between neigh-\nboring atomic orbitals, the j\u00190ibase allows the construc-\ntion of diagonal matrices of position-dependent opera-\ntors. A transformation rule was set up to take advantage\nof these two bases scenarios.\nWe have investigated the e\u000bects on the DOS and\nLDOS, of the size and topology of CND and CNC stru-\ntures. We have found that both total and local density\nof states sensitively depend on the number of atoms and\ncharacteristic geometry of the structures. One impor-\ntant aspect is the fact that cone and disk borders play\na relevant role on the LDOS at the Fermi energy. For\nsmall \fnite systems the presence of states localized in the\ncone apices determines the form of the DOS close to the\nFermi energy. The observed features indicate that small\nnanocones could present good \feld-emission properties.\nThis is corroborated by the calculation of the LDOC,\nthat indicates the existence of \fnite charges at the apex\nregion of the nanocones. For large systems, the contri-\nbution to the DOS near the Fermi level is mainly due to\nstates localized in the edges of the structures wheres for\nother energies, the DOS exhibits similar features to the\ncase of a graphene lattice.\nThe absorption coe\u000ecient for di\u000berent CNC structures\nare calculated and we have found a peculiar dependence\non the photon polarization in the infrared range for the\ninvestigated systems. The symmetry reduction of the 2-\npentagon nanocones causes the formation of highly struc-\ntured absorption spectra, with comparable weights for\ndistinct polarizations. The breaking of the degeneracy for\ndi\u000berent polarizations is found to be more pronounced for\nsmall nanocones. Absorption experiments may be used\nas natural measurements to distinguish between di\u000berent\nnanocone geometries.7\nAcknowledgements\nThis work was supported by Fondecyt grant 1100672\nand USM internal grant 11.11.62. P. Ulloa thanks DGIP\nand Mecesup PhD scholarships and the warm hospital-\nity of the Departamento de Fisica da Materia Conden-\nsada da Universidade Estadual de Campinas and Insti-\ntuto de F\u0013 \u0010sica da Universidade Federal Fluminense. Spe-\ncial thanks to Professor Patricio Vargas for his helpful\nadvices.\nAppendix A\nA discrete position scheme in terms of the j\u00190\njistates\nwas used to represent functions of the position given in\nterms of the atomic base, since they satisfy the same\nproperties of the position states, i.e., orthogonality\nh\u00190\nij\u00190\nji= \u0001(0)\nij; (A1)\nand completeness\nNCX\nk=1j\u00190\nkih\u00190\nkj=^1 (A2)\nin aNC-dimensional subspace. The identity operator\nmay also be constructed using the s6= 0 base as\n^1 =X\nk;lj\u0019ki(S\u00001)klh\u0019lj; (A3)\nwith theS\u00001\u0019\u0001(0)\u0000s\u0001(1)+O(s2) matrix being dif-\nferent from the NC\u0002NCidentity matrix \u0001(0).\nWe takej\u00190ias the discrete position state and assume\nthat the matrix elements fR\nijof position-dependent func-tionsf(~R) are known in the s= 0 representation,\nfR\nij=h\u00190\nij^fj\u00190\nji=f(~Rj) \u0001(0)\nij: (A4)\nDi\u000berently from the fRmatrices,fmatrices in the s6= 0\nrepresentation\nfij=h\u0019ij^fj\u0019ji (A5)\nare not diagonal. However, by performing the similarity\ntransformation\nh\u0019ij^fj\u0019ji=X\nk;lh\u0019ij\u00190\nkih\u00190\nkj^fj\u00190\nlih\u00190\nlj\u0019ji; (A6)\nwe may obtain the unknown fmatrix in terms of the\nknownfRmatrix, provided the transformation rule be-\ntween the \u00190and\u0019bases is known. 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B 77 (23) (2008) 235411."    },    {        "title": "1302.7217v1.Number_and_spin_densities_in_the_ground_state_of_a_trapped_mixture_of_two_pseudospin_1_2_Bose_gases_with_interspecies_spin_exchange_interaction.pdf",        "content": "arXiv:1302.7217v1  [cond-mat.quant-gas]  28 Feb 2013Number and spin densities in the ground state of a trapped mix ture of two\npseudospin-1\n2Bose gases with interspecies spin-exchange interaction\nJinlong Wang and Yu Shi∗\nDepartment of Physics and State Key Laboratory of Surface Ph ysics, Fudan University, Shanghai 200433, China\nWe consider the ground state of a mixture of two pseudospin-1\n2Bose gases with interspecies spin\nexchange in a trapping potential. In the mean field approach, the ground state can be described\nin terms of four wave functions governed by a set of coupled Gr oss-Pitaevskii-like (GP-like) equa-\ntions, which differ from the usual GP equations in the existen ce of an interference term due to\nspin-exchange coupling between the two species. Using thes e GP-like equations, we calculate such\nground state properties as chemical potentials, density pr ofiles and spin density profiles, which are\ndirectly observable in experiments. We compare the cases wi th and without spin exchange. It is\ndemonstrated that the spin exchange between the two species lowers the chemical potentials, tends\nto equalize the wave functions of the two pseudospin compone nts of each species, and thus homog-\nenizes the spin density. The novel features of the density an d spin density profiles can serve as\nexperimental probes of this novel Bose system.\nPACS numbers: 03.75.Mn, 03.75.Gg\nI. INTRODUCTION\nMulticomponent Bose-Einstein condensation (BEC) has been an act ive subject of research in recent years. People\nhave considered BEC of a mixture of two different spinless species [1 –8] and of spinor gases such as spin-1 [9–13] and\npseudospin-1\n2gases [14–16]. More recently, a mixture of two distinct species of ps eudospin-1\n2gases with interspecies\nspin exchange was investigated theoretically [17–23]. In such a mixt ure, there are Naatoms of species aandNbatoms\nof speciesb, while each atom has a pseudospin degree of freedom with basis stat eσ=↑,↓. The particle number of\neach species Nα=Nα↑+Nα↓, (α=a,b), is conserved, but the particle number of each pseudospin compo nent of\neach species Nασ, (σ=↑,↓), is not conserved because of the spin-exchange coupling betwee n the two species. Note\nthat for a pseudospin-1\n2gas, the total spin of each species is always a constant Sα=Nα/2, while its z-component is\nSαz= (Nα↑−Nα↓)/2.\nThe scattering between any two atoms are now pseudospin-depen dent. For the scattering between an atom of\nspeciesαincoming with pseudospin σ1and outgoing with pseudospin σ4and an atom of species βincoming with\npseudospin σ2and outgoing with pseudospin σ3, the scattering length is denoted ξαβ\nσ4σ3σ2σ1. Correspondingly, the\neffective interaction is gαβ\nσ4σ3σ2σ1δ(r−r′), wheregαβ\nσ4σ3σ2σ1≡2π¯h2ξαβ\nσ4σ3σ2σ1/µαβ, whereµαβis the reduced mass of\nthe two atoms. For convenience, we define the shorthands gαα\nσσ≡gαα\nσσσσfor the intraspecies scattering of the same\npseudospin σ,gαα\nσ¯σ≡2gαα\nσ¯σ¯σσfor the intraspecies scattering of different pseudospins σ/ne}ationslash= ¯σ,gab\nσσ′≡gab\nσσ′σ′σfor the\ninterspecies scattering without spin exchange, ge≡gab\nσ¯σσ¯σfor the interspecies spin-exchange scattering. We shall also\nuse the shorthands ξαα\nσσforξαα\nσσσσ,ξαα\nσ¯σfor 2ξαα\nσ¯σσ¯σ,ξab\nσσ′forξab\nσσ′σ′σ. In this paper, all these scattering lengths ξ’s and\nthus the effective interaction strengths g’s are considered to be positive. Thus the many-body Hamiltonian den sity\nis [17, 18]\nˆH(r) =/summationdisplay\nασˆψ†\nασ[−1\n2mα∇2+Uασ(r)]ˆψασ+1\n2/summationdisplay\nασσ′gαα\nσσ′|ˆψασ|2|ˆψασ′|2\n+/summationdisplay\nσσ′gab\nσσ′|ˆψaσ|2|ˆψbσ′|2+ge(ˆψ†\na↑ˆψ†\nb↓ˆψb↑ˆψa↓+ˆψ†\na↓ˆψ†\nb↑ˆψb↓ˆψa↑),(1)\nwhereUασ(r) is the external trapping potential, ˆψασ≡ˆψασ(r) is the Bose field operator for species αwith pseudospin\nσ(α=a,b,σ=↑,↓),σandσ′may or may not be equal. Expanded in terms of an orthonormal set o f single-particle\norbital basis states {φασ,i(r)},\nˆψασ(r) =/summationdisplay\niˆaασ,iφασ,i(r), (2)\n∗yushi@fudan.edu.cn2\nwhere ˆaασ,iis the annihilation operator correspondingto φασ,i(r). InˆH, the first two summations of interaction terms\nare the density-density interactions without spin exchange, as st udied in previous models of Bose mixtures, the last\nterm is the spin-exchange interaction, which causes spin correlatio n or entanglement between the two species, and is\nalso responsible for the novel features discussed in the present p aper.\nUnder the usual single orbital-mode approximation, Bose statistics and energetics governs that all atoms of each\nspeciesαand with the pseudospin state σoccupy the lowest-energy single-particle orbital mode hereby den oted as\nφασ(r), hence in the expansion (2) of the field operator ˆψασ(r), one only needs to consider one term ˆ aασφασ(r), where\nˆaασis the annihilation operator corresponding to φασ(r). Consequently, in each term of the many-body Hamiltonian/integraltext\nd3rˆH(r), there is an integration of a product of single-particle wave funct ions, which now becomes an effective\ncoefficient. Therefore, under the single orbital-mode approximatio n, the details of φασ(r) are not needed in describing\nthe many-body ground state in terms of creation and annihilation op erators or, equivalently, the collective spin\noperators, although such simplification is lost when one goes beyond single orbital-mode approximation. In a broad\nparameter regime, the two species are quantum entangled in the pa rticle numbers of the two pseudospin states or\ncollective spins, that is, the two species do not undergo BEC separa tely, hence the ground state was dubbed entangled\nBEC.\nHowever, these four wave functions and the correspondingeleme ntary excitations are important physical properties.\nFor a uniform system, φασ(r) is simply the constant 1 /√\nΩ, where Ω is the volume of the system. For convenience, we\nwritethemeanfieldvalueoftheBosefieldoperator /an}b∇acketle{tˆψασ/an}b∇acket∇i}htasψασeiγασ, whereψασ>0. Thisistheso-calledcondensate\nwavefunction. The condensatewavefunction for pseudospin σcomponent ofspecies αisψασ=ηασ/radicalbig\nN0ασ/√\nΩ, where\nN0\nασis the correspondingparticlenumber in the many-body groundstat e ofthe system, determined by the many-body\nHamiltonian, ηασ=±is a sign. To minimize the energy, the signs of three components can b e chosen to be + while\nthat of the other one is chosen to be −.\nIn a trapping potential, which is an experimental necessity for BEC o f cold atoms, the wave function ψασ(r)\nis dramatically different from a constant. Moreover, the details of ψασ(r) provide experimentally very important\ninformation, as their modular square is just the particle density, wh ich is directly measurable and is a key observable.\nIn this paper, we considersuch a pseudospin-1\n2mixture in a trapping potential, and find some interesting properties ,\nespecially the density and spin density profiles of the four lowest-en ergy orbital modes {φασ}. There had been many\ncalculations on such properties in other types of BEC mixtures [3, 24 ], which demonstrated that a trapping potential\nbrings significant features absent in a homogeneous system.\nThe GP-like equations can be obtained by using the Euler-Lagrange e quation\nd\ndt/parenleftbigg∂L\n∂˙ψασ/parenrightbigg\n−∂L\n∂ψασ= 0, (3)\nwithL=i/summationtext\nασψ∗\nασ∂tψασ−/an}b∇acketle{tˆH/an}b∇acket∇i}ht, and then substituting i∂tasµασ. One obtains\n/parenleftbigg\n−¯h2\n2mα∇2+Uασ(r)/parenrightbigg\nψασ+gαα\nσσ|ψασ|2ψασ+gαα\nσ¯σ|ψα¯σ|2ψασ+gα¯α\nσσ|ψ¯ασ|2ψασ\n+gα¯α\nσ¯σ|ψ¯α¯σ|2ψασ−geψ∗\n¯α¯σψ¯ασψα¯σ=µασψασ. (4)\nwheregeterm is due to the spin-exchange interaction, and is a new feature a bsent in previous models of Bose\nmixtures. The minus sign comes from the requirement that the phas esγασof the four components should satisfy\ncos(γa↑−γa↓−γb↑+γb↓) =−1 for the minimization of the energy. For ge>0, this spin-exchange interaction is like\nan attractive interaction in some way, counteracting the other int eraction terms if the latter is repulsive. However, it\nis a new effect, as it depends on the wave functions rather than the densities.\nWhengeterm is negligible, the system behaves like the usual Bose mixtures wit h repulsive interactions. When\ngeterm is dominant, the system behaves in a way similar to an attractive mixture with intraspecies interac-\ntion negligible. Moreover, to minimize the spin-exchange interaction e nergyge/an}b∇acketle{tˆψ†\na↑ˆψ†\nb↓ˆψb↑ˆψa↓+ˆψ†\na↓ˆψ†\nb↑ˆψb↓ˆψa↑/an}b∇acket∇i}ht=\n−2geψa↑ψa↓ψb↓ψb↑cos(γa↑−γa↓−γb↑+γb↓) =−2geψa↑ψa↓ψb↓ψb↑,ψa↑ψa↓ψb↓ψb↑should be maximized, as a kind of\ninterference effect, which means that for each species α=a,b,ψα↑(r) =ψα↓(r), hence the density profiles for the two\npseudospin components of each species tends to be the same. To s ee this, note that each wave function can be taken\nto be real and positive [25]. The other terms in the Hamiltonian of cou rse break this equality, as will be studied later\nin this paper.\nBelow we shall describe the finding that the larger the interspecies s pin exchange geis, the stronger the overlap\nbetween the density profiles of the two pseudospin components of each species is. Hence density profiles are very good\nexperimental probes of the underlying interspecies correlations. On the other hand, by comparing the experimental\nandtheoreticalresultsonthe number densityand spindensity pro files, onemayestimatethe spin-exchangeinteraction3\nstrengthge. Experimentally, by studying the the effect of geon the density profiles, one can obtain the information\non such a mixture.\nUp to now, there is not yet a report on experimental studies of suc h a spin-exchange mixture between different\nspecies. However, the interspecies spin-exchange interaction is d etermined by the difference between the interspecies\ntriplet and singlet scattering lengths, which has been found to be qu ite a few nanometer (nm) [26]. In this paper, the\ntheoretical investigation using this parameter value clearly indicate s interesting new features. Hence our work also\nprovides some motivation and methodology for experimental explor ation of such a mixture.\nIn Sec. II, the numerical method is described. In Sec. III, we calc ulate the ground state properties, comparing the\ncases with and without spin exchange, and demonstrating the expe rimentally observable effects of interspecies spin\nexchange. Then we make a summary in Sec. IV.\nII. NUMERICAL METHOD\nWe assume the trapping potential to be\nUασ(r) =1\n2Mαω2\nα(ρ2+λ2z2), (5)\nwhereρ=/radicalbig\nx2+y2,λrepresents the trap anisotropy, Mαandωαare the mass of α-atom and trap frequency,\nrespectively. Uα↑=Uα↓. For a magnetic trap, Mαω2\nα=γαµBB0, withγαbeing thegfactor ofα-atom,B0\nbeing the central magnetic field multiplied by a normalized factor. In o rder that the parameter values are close to\nthe experimental data, we imagine species aas87Rb and species bas23Na, thenMaω2\na/Mbω2\nb=γa/γb= 1, i.e.\nUaσ=Ubσ. Defineκ≡ωb/ωa/radicalbig\nMa/Mb=/radicalbig\n87/23. In our calculation, we use the parameter values ωa= 2π×75Hz,\nωb=κωaandλ=√\n8. The values of scattering lengths are set to be ξaa\n↑↑= 8 nm,ξaa\n↓↓= 6.7 nm,ξaa\n↑↓= 3.7 nm,ξbb\n↑↑= 4\nnm,ξbb\n↓↓= 2 nm,ξbb\n↑↓= 1.7 nm,ξab\n↑↑= 1.2 nm,ξab\n↓↓= 0.67 nm,ξab\n↑↓= 1.9 nm. We vary the value of ξeand study its\neffect on the number densities and the spin densities.\nWe expand ψασin terms of Nbasiseigenfunctions of the non-interacting Schr¨ odinger equation in an anisotropic\nharmonic potential (5), that is,\nψασ(r) =Nbasis/summationdisplay\nr=0Ar\nασRmr\nnr(ρ)Φmr(ϕ)Zwr(z), (6)\nwhereRmrnr(ρ), Φmr(ϕ) andZwr(z) correspond to the cylindrical coordinates ρ,ϕandz, respectively, Ar\nασis the\nexpanding coefficient under the condition/summationtextNbasis\nr(Ar\nασ)2=Nασ. For the ground state, only eigenfunctions with\nmr= 0 are relevant.\nTherefore, GP-like equation (4) is transformed to the following non linear equation\n(El\nασ−µασ)Al\nασ+gαα\nσσ/summationdisplay\nijkAi\nασAj\nασAk\nασI(iα,jα,kα,lα )\n+gαα\nσ¯σ/summationdisplay\nijkAi\nα¯σAj\nα¯σAk\nασI(iα,jα,kα,lα )+gα¯α\nσσ/summationdisplay\nijkAi\n¯ασAj\n¯ασAk\nασI(i¯α,j¯α,kα,lα)\n+gα¯α\nσ¯σ/summationdisplay\nijkAi\n¯α¯σAj\n¯α¯σAk\nασI(i¯α,j¯α,kα,lα)−ge/summationdisplay\nijkAi\n¯α¯σAj\n¯ασAk\nα¯σI(i¯α,j¯α,k¯α,l¯α) = 0, (7)\nwhereEl\nασ= ((2nl+ml+1)+(wl+1/2)λ)¯hωα,\nI(iα,jα,kβ,lβ )≡1\n2π/integraldisplay∞\n0Rmi\nni(η2\nραρ2)Rmj\nnj(η2\nραρ2)Rmknk(η2\nρβρ2)Rmlnl(η2\nρβρ2)ρdρ\n×/integraldisplay+∞\n−∞Zwi(ηzαz)Zwj(ηzαz)Zwk(ηzβz)Zwl(ηzβz)dz,\nwheremr(r=i,j,k,l) should be 0. There are various algorithms to determine the solution s of the nonlinear equation\nset, such as fixed-point iteration and Newton method. We use Broy den method to obtain the solutions of Eq. (7)\nbecause of its high speed and precision.4\n/s48 /s50/s48/s48/s48 /s52/s48/s48/s48 /s54/s48/s48/s48 /s56/s48/s48/s48 /s49/s48/s48/s48/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s32/s97/s44\n/s32/s97/s44\n/s32/s98/s44\n/s32/s98/s44/s32/s47\n/s78\nFIG. 1: Reduced chemical potential µασ/¯hωαvarying with the atom number Nof each species, at a generic parameter point\nwithout interspecies spin exchange, i.e. ξe= 0.µασas well asµaσ/¯hωa−µbσ/¯hωbincrease with N.\nIII. CALCULATIONS\nWe shall use the GP-like equations (4) to study the effect of intersp ecies spin exchange on the number densities\nnασ=|ψασ(r)|2,\nand the spin densities\nSαz(r) =1\n2[|ψα↑(r)|2−|ψα↓(r)|2].\nFor simplicity, we assume that the atom numbers of the two species a re equal, i.e. Na=Nb=N. Although the\ncases ofN≤1000 may not be experimentally realistic, they serve as theoretical demonstration of the dilute limit, in\nwhich the interaction energy is small, especially in comparison with the c ases with larger N.\nA. The case without interspecies spin exchange\nFirst we consider the case of ξe= 0, in which the system reduces to a mixture of four spinless conden sates. The\nresults for this case are summarized in Figures 1, 2, 3, 4, 5, 6. They are all consequences of minimizing the repulsive\ninteractions among the four components, as detailed in the following .\nWe haveµα↑>µα↓under the present parameter values, as indicated in Fig. 1, which sh ows the chemical potential\nµασas a function of N.µασas well asµaσ/¯hωa−µbσ/¯hωbincrease with N. The difference between µα↑andµα↓\nis due to spin dependence of various scattering lengths. This result can be confirmed by a calculation based on the\nThomas-Fermi approximation, which leads to µασ=1\nΩ[/integraltext\nUασdr+gαα\nσσNασ+gαα\nσ¯σNα¯σ+gα¯α\nσσN¯ασ+gα¯α\nσ¯σN¯α¯σ].In a\nsense,µασrepresents the average energy of one atom of species αwith pseudospin σ. We can observe that µασ/¯hωα\ndecreases with N, towards the single particle value 1+ λ/2 = 2.414.\nNow we take a look at the spatial dependence of the atom densities nασ(r) =|ψασ(r)|2.The density profiles\nfor several different values of Nare shown in Fig. 2, where the distributions along ρandzdirections are depicted\nrespectively. These plots display some interesting features. Obvio uslynα↑andnα↓are complementary, because of\nthe normalization condition/integraltext\n(nα↑+nα↓)dr=N. WhenNis small,nα↑andnα↓are close to each other, because\nthe interaction energy is small. But with the increase of N, the difference between nα↑andnα↓increases in order to\nlower the interaction energy. When Nis large enough, two or more peaks may appear in some density profile s, due\nto the inclusion of higher order eigenfunctions in the expansion (6) w hen the interaction energy becomes important.\nThe profiles of the total density of each species, nα=nα↑+nα↓, are shown in Fig. 3. We can see that not all\ncomponents co-exist in every region, in analogy with the two-compo nent BEC [3]. This is in order to minimize the\ntotal energy under the given parameter values.\nThe density profiles are more extended in ρdirection than in zdirection, as exhibited in Fig. 2 and in Fig. 3. The\nreason is that the trapping in zdirection is stronger than that in ρdirection.5\n024600.10.20.30.4\nρ(µm)N=100\n024600.511.5\nρ(µm)N=1000\n0246012345\nρ(µm)N=5000\n02460123456\nρ(µm)N=10000\n024600.10.20.30.4\nz(µm)Density (1014 atoms/cm3)\n  \nN=100\n024600.511.5\nz(µm)  \nN=1000\n0246012345\nz(µm)  \nN=5000\n02460123456\nz(µm)  \nN=10000\na,↑ a,↓ b,↑ b,↓\nFIG. 2: Density nασ=|ψασ|2for each pseudospin component of each species, at a generic p arameter point with ξe= 0, as\ndefined in the text, for N= 100,1000,5000,10000.Nis atom number of each species. The upper plots are profiles al ongρ\ndirection with z= 0. The lower plots are profiles along zdirection with ρ= 0. More than one peak appear in some plots, due\nto the inclusion of higher order eigenfunctions in the expan sion (6) when the interaction energy becomes important.\nWe have also plotted the two-dimensional density profiles with ρandzas the two coordinates, as shown in Fig. 4\nforλ=√\n8, and in Fig. 5 for λ= 1, both with N= 10000. The complementarity between the two pseudospin\ncomponents of each species is very clear. In the former case, as λ/ne}ationslash= 1, the profiles are asymmetric between ρandz\ndirections. In the latter case, as λ= 1, the profiles are symmetric between ρandzdirections.\nNow we consider the spin density Sαz(r) =1\n2[|ψα↑(r)|2−|ψα↓(r)|2].The spin density profiles are shown in Fig. 6\nforN= 100,1000,5000,10000. It can be seen that along ρdirection, the spin density increases from a negative value\nat the center to a positive value at a certain radius, and then gradu ally decreases to zero. Both the radius with the\npositive maximal spin density and the radius where the spin density be comes zero increase with N. This feature can\nbe understood in terms of the difference between the densities of ↑and↓atoms shown in Fig. 2. That the spin density\nis negative in inside regimes while positive in outside regime is because we h ave assumed ξαα\n↑↑> ξαα\n↓↓, consequently\nmore↑atoms tend to stay in the outside regime where the density is lower be cause of trapping potential, while more\n↓atoms tend to stay in the inside regime, in order to lower the total en ergy. Of course the spin density approaches\nzero for large enough value of ρ, as the densities of ↑and↓atoms both approach zero. Along zdirection with ρ= 0,\nthe spin density is mostly negative, also because more ↑atoms than ↓atoms stay in the larger ρregime. This effect\nweakens with the increase of z, because the trapping potential increases, consequently the diff erence between the\nnumbers of ↓and↑atoms decreases. There is only a very small regime where the spin de nsity becomes positive but\nthe values are too small to be visible on the plots. This feature is unlike the that of the profiles along ρdirection, as\nthere must be some regime of ρwith more ↑atoms.\nB. The case with interspecies spin exchange\nNow we come to the effect of interspecies spin exchange, i.e. the cas e ofξe/ne}ationslash= 0. We have chosen ξe=0.53 nm, 1.07\nnm, 2.03 nm, 4.27 nm. As stated in the Introduction, the mean-field s pin-exchange interaction energy\n−2geψa↑ψa↓ψb↓ψb↑ (8)6\n0 5 100123456\nρ(µm)  \nN=100\n0 5 100123456\nρ(µm)N=1000\n0 5 100123456\nρ(µm)N=5000\n0 5 100123456\nρ(µm)N=10000\n0 5 100123456\nz(µm)Density (1014 atoms/cm3)\n  \nN=100\n0 5 100123456\nz(µm)N=1000\na b0 5 100123456\nz(µm)N=5000\n0 5 100123456\nz(µm)N=10000\nFIG. 3: Profiles of the total density nα=nα↑+nα↓, for each species, along ρdirection with z= 0, and along zdirection with\nρ= 0, at a generic parameter point with ξe= 0, as defined in the text, for N= 100,1000,5000,10000. Not all components\nco-exist in every region, in order to minimize the total ener gy under the given parameter values. The density profiles are more\nextended in ρdirection than in zdirection, as the trapping in zdirection is stronger than in ρdirection.\n−505\n−505024\nρ(µm) z(µm)na↑, (1014 atoms/cm3)\n−505\n−505024\nρ(µm) z(µm)na↓, (1014 atoms/cm3)\n−505\n−505024\nρ(µm) z(µm)nb↑, (1014 atoms/cm3)\n−505\n−505024\nρ(µm) z(µm)nb↓, (1014 atoms/cm3)\nFIG. 4: Two-dimensional density profiles on a cross section i ncludingz-axis, with ρandzas the coordinates, in a generic\nparameter point defined in the text, with ξe= 0,λ=√\n8 andN= 10000. The profiles are asymmetric between ρandz\ndirections as λ/negationslash= 1.7\n−505\n−505024\nρ(µm) z(µm)na↑, (1014 atoms/cm3)\n−505\n−505024\nρ(µm) z(µm)na↓, (1014 atoms/cm3)\n−505\n−505024\nρ(µm) z(µm)nb↑, (1014 atoms/cm3)\n−505\n−505024\nρ(µm) z(µm)nb↓, (1014 atoms/cm3)\nFIG. 5: Two-dimensional density profiles on a cross section i ncludingz-axis, with ρandzas the coordinates, in a generic\nparameter point defined in the text, with ξe= 0,λ= 1 andN= 10000. The profiles are symmetric between ρandzdirections\nasλ= 1.\nis an interference term and acts like an attractive interaction amon g the four orbital wave functions in some way, but\nit depends on the wave functions rather than the densities. This int erference effect is manifested in the number and\nspin density profiles.\nThe spin exchange lowers the chemical potential, as evident in Fig. 7, which is the numerical result of µασas a\nfunction of Nfor four values of ξe. It can be seen from the plots that the larger ξeis, the lower µασis. However, in\nall these cases, dµασ/dNremains positive.\nThe interference among the four orbital wave functions is also man ifested in the density profiles, as shown in Fig. 8,\nFig. 9 and Fig. 10. The spin exchange energy (8) is minimized when ψα↑=ψα↓for each species α. It is competitive\nwith the other interactions. If it dominates the energy, in order to lower the energy, the wave functions of the two\npseudospin components of each species tend to be close to each ot her, compared with the case without spin exchange.\nIndeed, this tendency can be observed in Fig. 8 and Fig. 9, which sho w the density profiles for each pseudospin\ncomponent of each species, in ρdirection with z= 0 and in zdirection with ρ= 0 respectively. The larger ξeis,\nthe more dominant the spin-exchange interaction is, then the close rψα↑andψα↓are to each other. When ξeis large\nenough, the overlapping effect becomes very visible. Nevertheless , the overlap is not complete, because of other terms\nin the energy.\nFig. 10 depicts the profile of the total density of each species, whic h can be compared with the plot for the same N\nin Fig. 3, it can be seen that the overlap regime of the two species is als o enhanced by spin exchange, because of the\neffect of spin exchange term (8).\nMoreover, we have also calculated the spin density profile, as shown in Fig. 11 for N= 10000. Spin density\nSαz=1\n2(|ψα↑|−|ψα↓|2) is proportional to the difference between the densities of the two pseudospin components, so\nit is a quantification of the overlap between the two pseudospin comp onents. Evidently, with the increase of ξe, the\nvariation of the spin density with the radius decreases. In other wo rds, both spin density Szαof each species and the\ntotal spin density Szare homogenized by the spin exchange. When ξeis large enough, each spin density tends to\nvanish. The underlying reason is also because ψα↑andψα↓, whose difference gives the spin density of species α, tend\nto be close to each other in order to lower the spin-exchange energ y. Consequently the spin density of each species α\ntend to vanish. In both the case without spin exchange and the cas e with spin exchange, the location on z-axis where\nthe spin density becomes zero is much smaller than that along ρ-direction. Compared with the case without spin\nexchange, another notable feature is that when ξeandNare large enough, in the spin density profile along zdirection\nwithρ= 0, the regime with positive spin density becomes more visible (Fig. 11) . This is because the negative sign of\nthe spin-exchange interaction counteracts the trapping potent ial, even though the trapping is stronger in zdirection\nthan inρdirection.8\n02468−15−10−505x 10−3\nz(µm)Spin density (1014/cm3)\n  \nN=100\n02468−0.4−0.3−0.2−0.100.1\nz(µm)  \nN=1000\n02468−4−3−2−1012\nz(µm)  \nN=500002468−6−4−202\nρ(µm)  \nN=10000\n02468−6−4−202\nz(µm)  \nN=1000002468−4−202\nρ(µm)  \nN=5000\n02468−0.4−0.3−0.2−0.100.1\nρ(µm)  \nN=1000\n02468−15−10−505x 10−3\nρ(µm)  \nN=100\na b total\nFIG. 6: Spin density profiles for N= 100,1000,5000,10000, in a generic parameter point as defined in the main text , with\nξe= 0. The spin density approaches zero for large enough ρorz, as the densities of ↑and↓atoms both approach zero when\nthe trapping potential is large enough.\nIV. SUMMARY\nThis paper concerns the ground state properties of a mixture of t wo species of pseudospin-1\n2Bose gases with inter-\nspecies spin-exchange interaction in a trapping potential. We have n umerically calculated the four orbital condensate\nwave functions, each of which corresponding to a pseudospin comp onent of each species, by using the GP-like equa-\ntions. We set the atom number of each species to be N. Using these wave functions, the number and spin densities\nare obtained. When the spin-exchange scattering length is zero, t his mixture reduces to a mixture of the usual type,\nwith four components. Various features appear as consequence s of minimizing the density-density interaction. For\nexample, with the increase of N, the difference between nα↑andnα↓for each species αincreases.\nIf there exists interspecies spin-exchange scattering, novel fe atures absent in the usual mixtures emerge. As the\nspin-exchange interaction is negative as a consequence of minimizing the energy, it acts like an attractive interaction.\nNevertheless, it depends on the overlap among the four wave func tions. It lowers the chemical potentials and make\nthe densities of the two pseudospin components of each species te nd to be close to each other, and thus the spin\ndensity tends to be homogenized, and even tends to vanish when th e spin-exchange scattering length is so large that\nit dominates over the density-density interaction.\nTherefore as experimentally measurable quantities, the number an d spin density profiles of such a mixture with\ninterspecies spin exchange are effective probes of the novel many -body ground state of this system.\nAcknowledgments\nWe thank Li Ge for useful discussion. This work was supported by t he National Science Foundation of China\n(Grant No. 11074048) and the Ministry of Science and Technology o f China (Grant No. 2009CB929204).9\n/s48 /s50/s48/s48/s48 /s52/s48/s48/s48 /s54/s48/s48/s48 /s56/s48/s48/s48 /s49/s48/s48/s48/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s48 /s50/s48/s48/s48 /s52/s48/s48/s48 /s54/s48/s48/s48 /s56/s48/s48/s48 /s49/s48/s48/s48/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s48 /s50/s48/s48/s48 /s52/s48/s48/s48 /s54/s48/s48/s48 /s56/s48/s48/s48 /s49/s48/s48/s48/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s48 /s50/s48/s48/s48 /s52/s48/s48/s48 /s54/s48/s48/s48 /s56/s48/s48/s48 /s49/s48/s48/s48/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s97 /s97/s97 /s97\n/s78/s101/s61/s48/s46/s53/s51/s32/s110/s109 \n/s32\n/s101/s61/s49/s46/s48/s55/s32/s110/s109 \n/s32\n/s101/s61/s50/s46/s48/s51/s32/s110/s109 \n/s32\n/s101/s61/s52/s46/s50/s55/s32/s110/s109 /s101/s61/s48/s46/s53/s51/s32/s110/s109 \n/s32\n/s101/s61/s49/s46/s48/s55/s32/s110/s109 \n/s32\n/s101/s61/s50/s46/s48/s51/s32/s110/s109 \n/s32\n/s101/s61/s52/s46/s50/s55/s32/s110/s109 \n/s78/s98 /s98/s101/s61/s48/s46/s53/s51/s32/s110/s109 \n/s32\n/s101/s61/s49/s46/s48/s55/s32/s110/s109 \n/s32\n/s101/s61/s50/s46/s48/s51/s32/s110/s109 \n/s32\n/s101/s61/s52/s46/s50/s55/s32/s110/s109 \n/s78\n/s98 /s98/s101/s61/s48/s46/s53/s51/s32/s110/s109 \n/s32\n/s101/s61/s49/s46/s48/s55/s32/s110/s109 \n/s32\n/s101/s61/s50/s46/s48/s51/s32/s110/s109 \n/s32\n/s101/s61/s52/s46/s50/s55/s32/s110/s109 \n/s78\nFIG. 7: Reduced chemical potential µασ/¯hωαvarying with Nfor four different values of ξe. 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Lett. 86, 60008 (2009).10\n024600.10.20.30.4\nρ(µm)N=100,ξe=0.53nm\n024600.511.5\nρ(µm)N=1000, ξe=0.53nm024600.10.20.30.4\nρ(µm)N=100,ξe=1.07nm\n024600.511.5\nρ(µm)N=1000, ξe=1.07nm024600.10.20.30.4\nρ(µm)  \nN=100,ξe=2.03nm\n024600.511.5\nρ(µm)N=1000, ξe=2.03nm024600.10.20.30.4\n  \nN=100,ξe=4.27nm\n024600.511.5\nρ(µm)  \nN=1000, ξe=4.27nm\n024601234\nρ(µm)  \nN=5000, ξe=0.53nm\n024601234\n  \nN=5000, ξe=1.07nm\n024601234\n  \nN=5000, ξe=2.03nm\n024601234\n  \nN=5000, ξe=4.27nm\n02460246\nρ(µm)Density (1014 atoms/cm3)\n  N=10000, ξe=0.53nm\n02460246\nρ(µm)  \nN=10000, ξe=1.07nm\n02460246\nρ(µm)  \nN=10000, ξe=2.03nm\n02460246\nρ(µm)  \nN=10000, ξe=4.27nm\na,↑ a,↓ b,↑ b,↓\nFIG. 8: Density profile nασ=|ψασ|2for each pseudospin component σof each species αalongρdirection on the plane z= 0\nforN= 100,1000,5000,10000 and several values of ξe. Other parameter values are given in the main text. The large rξe, the\nstronger overlap among the four wave functions.\n[20] Y. Shi, Phys. Rev. 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A 78, 010701 (R) (2008).11\n024600.10.20.30.4\nz(µm)N=100,ξe=0.53nm\n024600.511.5\nz(µm)  \nN=1000, ξe=0.53nm0 500.10.20.30.4\nz(µm)  \nN=100,ξe=1.07nm\n024600.511.5\nz(µm)  \nN=1000, ξe=1.07nm024600.10.20.30.4\nz(µm)  \nN=100,ξe=2.03nm\n024600.511.5\nz(µm)N=1000, ξe=2.03nm024600.10.20.30.4\nz(µm)  \nN=100,ξe=4.27nm\n024600.511.5\nz(µm)  \nN=1000, ξe=4.27nm\n024601234\nz(µm)  \nN=5000, ξe=0.53nm\n024601234\nz(µm)  \nN=5000, ξe=1.07nm\n024601234\nz(µm)  \nN=5000, ξe=2.03nm\n024601234\nz(µm)Density (1014 atoms/cm3)\n  \nN=5000, ξe=4.27nm\n02460246\nz(µm)  \nN=10000, ξe=0.53nm\n02460246\nz(µm)  \nN=10000, ξe=1.07nm\n02460246\nz(µm)  \nN=10000, ξe=2.03nm\n02460246\nz(µm)  \nN=10000, ξe=4.27nm\na,↑ a,↓ b,↑ b,↓\nFIG. 9: Density profile nασ=|ψασ|2for each pseudospin component σof each species αalongzdirection on the line ρ= 0\nforN= 100,1000,5000,10000 and several values of ξe. Other parameter values are given in the main text. The large rξe, the\nstronger overlap among the four wave functions.12\n0 5 1002468\nρ(µm)Density (1014 atoms/cm3)\n  \nξe=0.53nm\n0 5 1002468\nρ(µm)ξe=1.07nm\n0 5 1002468\nρ(µm)ξe=2.03nm\n0 5 1002468\nρ(µm)ξe=4.27nm\n0 5 1002468\nz(µm)  \nξe=0.53nm\n0 5 1002468\nz(µm)ξe=1.07nm\n0 5 1002468\nz(µm)ξe=2.03nm\n0 5 1002468\nz(µm)ξe=4.27nm\na b\nFIG. 10: Profiles of the total density nαfor each species αalongρdirection on the plane z= 0 and on the line ρ= 0 for atom\nnumberN= 10000 and several values of ξe. The larger ξe, the closer the profiles of the two species.13\n02468−6−4−202\nz(µm)Spin density (1014/cm3)\n  \nξe=0.53nm\n02468−6−4−202\nz(µm)  \nξe=1.07nm\n02468−6−4−202\nz(µm)  \nξe=2.03nm02468−6−4−202\nρ(µm)  \nξe=4.27nm\n02468−6−4−202\nz(µm)  \nξe=4.27nm02468−6−4−202\nρ(µm)  \nξe=2.03nm\n02468−6−4−202\nρ(µm)  \nξe=1.07nm\n02468−6−4−202\nρ(µm)  \nξe=0.53nm\na b total\nFIG. 11: Spin density profiles in a generic parameter point as defined in the main text, for N= 10000 and various values of\nξe. The larger ξeis, the close the spin density is to 0, even at relatively shor t distances from the origin."    },    {        "title": "1303.4049v1.Localization__disorder_and_boson_peak_in_an_amorphous_solid.pdf",        "content": "arXiv:1303.4049v1  [cond-mat.stat-mech]  17 Mar 2013Localization, disorder and boson peak in an amorphous solid\nLeishangthem Premkumar and Shankar P. Das\nSchool of Physical Sciences,\nJawaharlal Nehru University,\nNew Delhi 110067, India.\nAbstract\nWe demonstrate using the classical density functional theo ry (DFT) model that an intermediate\ndegree of mass localization in the amorphous state is essent ial for producing the boson peak. The\nlocalization length ℓis identified from the width of the gaussian density profile in terms of which\nthe inhomogeneous density n(x) of the solid is expressed in DFT. At a fixed average density, t here\nexists a limiting value ℓ0ofℓsignifying a minimum mass localization in the amorphous sta te. For\nmore delocalized states ( ℓ > ℓ0) occurrence of boson peak is unfeasible .\n0Studies of inelastic scattering of light and neutron from amorphous solids show that\nthe disordered systems are characterized by an excess density o f vibrational modes over that\npredictedbythestandardDebyedistribution. Thisexcessinthede nsityofstates(DOS) g(ω)\nfor a disordered system typically occurs at the THz frequency ( ω) range and is identified as\nthe so called boson peak in the g(ω)/ω2vs.ωplot. The boson peak height decreases and its\nlocation shifts towards the higher frequencies with the increase of pressure or density [1]-[2].\nA number of different theoretical models have been developed to ex plain the anomalous low-\nenergyexcitations[3]-[10]. Itisnowgenerallybelievedthattheboson peakisamanifestation\nof disorder and is key to the understanding of the vibrational stat es of glassy materials. The\norigin of boson peak in the disordered solid has been often linked to tr ansverse sound modes\nfor the system[11]. It has been argued that the boson peak result s from a band of random\ntransverse acoustic vibrational states [7]. Extensive computer s imulations of several glass\nforming systems [12] also indicated subsequently that the origin of t he boson peak is the\ntransverse vibrationalmodesassociatedwithdefective softstr uctures inthedisordered state.\nIn a recent work, Chumakov et al. [13] explain the boson peak in ter ms of (predominantly)\ntransverse sound waves.\nA common aspect of a class of the boson peak models mentioned abov e is that they\nare based on the existence of localized modes in the amorphous solid [ 14]. Longitudinal\nacoustic phonon modes are observed in a liquid by ultrasonic, optical, or inelastic scattering\nexperiments. These hydrodynamic modes occur due to the conser vation laws for the system.\nFor the crystalline state the isotropic symmetry of the liquid is broke n and the constituent\nparticles vibrate around the sites of a lattice with long range order. This symmetry breaking\nleads to the development of the Goldstone modes [15] or the transv erse sound modes in the\ncrystal in addition to the longitudinal sound modes. For the amorph ous solid the symmetry\nis not broken over long length scales like that in a crystal. When the at omic vibration\nwavelength in a liquid approaches the atomic nearest neighbor distan ce, i.e., in the tera\nhertz frequency region, there is a solid like cage formation giving rise to a restoring force for\nthe acoustic transverse modes.\nA typical choice of the order parameter for the crystalline as well a s the amorphous state\nis the inhomogeneous density n(r) with smooth spatial dependence. The density for the\nsolid state is expressed in terms of gaussian profiles of width ℓand centered on the sites of\na chosen lattice {Ri}[17].\n1n(r) =/parenleftBigα\nπ/parenrightBig3\n2/summationdisplay\nRie−α(r−Ri)2, (1)\nwith 1/√α=ℓ. The coarse grained picture for the solid presented above involves two inher-\nent length scales. First, the microscopic length scale σ(say) associated with the interaction\npotential between the constituent particles. For a hard sphere p otential, σcorresponds to\nthe hard sphere diameter. Second, the length ℓwhich represents the degree of localization\nof mass in the system. The homogeneous liquid state is characterize d by the limit ασ2→0\norℓ >> σ. The sharply localized density profiles of the crystalline state corre spond to the\nlimitασ2>>1 orℓ << σ. The ratio ℓ/σis generally referred to as the Lindeman parameter\nfor the crystalline. In the present work we compare the free ener gy contribution due to the\nvibrational modes in the amorphous state to the corresponding de nsity functional result for\nthe ideal gas part fid[n][16] of the free energy for an inhomogeneous density n(r). We show\nthat for a a chosen average density, the characteristic gaussian profile of n(r) signifies a\nminimum localization length ℓ0below which the appearance of boson peak in the reduced\ndensity of states is unfeasible.\nThe equilibrium state of the fluid is identified by minimizing a proper therm odynamic\npotential or free energy with respect to the density n(x). The free energy is expressed as\na sum of two parts which are respectively the free energy of the no n-interacting system Fid\nand the contribution Fexdue to interactions between the particles. Fidis obtained from the\nlogarithm of the partition function ( V/∧3)N/N! for the noninteracting system of Nparticle,\nwhere∧=/planckover2pi1/√2πmkBTdenotes the thermal wave length at temperature T. ThusβFid=\nVn0(ln(n0∧3)−1) for the uniform density fluid n0. For the inhomogeneous density n(x),\nthe expression for the free energy is easily generalized to βFid[n] =/integraltext\ndrn(r)[ln{n(r)∧3}−1].\nFor the uniform density n0changing to a nonuniform one n(x), the ideal gas contribution\nfor theNparticles system changes by an amount n0/integraltext\ndr˜n(r)[ln˜n(r)], where ˜ n(r) =n(r)/n0.\nSince by definition ˜ n(r) is always positive, using the Gibbs inequality {xlnx−x+1} ≥0 for\npositivex, it follows that entropy drops upon localization of the particles. This is a result\nof the restriction of available phase space. If αis larger than a limiting value α0(say) the\ndensityn(r) is well approximated in terms of the gaussian profile centered at th e nearest\nlattice site. The free energy per particle fidfor large ασ2is obtained as,\n2fid[α]≈ −5\n2+3ln/parenleftbigg/radicalbiggα\nπ∧/parenrightbigg\n. (2)\nIn a microscopic approach, vibrational modes with localized density p rofiles contribute to\nthe free energy and the latter is obtained in terms of the density of vibrational states g(ω) of\nthe system. The free energy of the non-interacting system is obt ained asFid=−kBTlnZN\nwhereZNis the partition function for 3N harmonic oscillators in constant NVT e nsemble.\nGoing from discrete to continuum in the frequency spectrum, we ob tain the ideal gas free\nenergy per particle in units of β−1asβFid/N=/integraltext1\n0κ(xmx)¯g(x)dx≡f0[¯g,xm(α)], where\nκ(x) =−1\n4{x+ ln[1−e−x]−2x/(ex−1)}.ωmis the upper cutoff of frequency up to\nwhichg(ω) is nonzero and it is expressed in terms of the dimensionless quantity β/planckover2pi1ωm=\nxm≡xm(α). Thescaled density ofstates ¯ g(x) isobtainedfrom g(ω)as ¯g(x) = (3n0)−1ωmg(ω)\nwherex=ω/ωmis the reduced frequency. The expression for κ(x) indicates the functional\ndependence of the integral f0on the scaled density of states ¯ g. There are 3 n0vibrational\nmodes for anaverage n0number of particles in a unit volume giving rise to the normalization\ncondition/integraltextωm\n0g(ω)dω= 3n0forg(ω). The cutoff frequency ωmis a characteristic property\nof the material and sets the shortest length scale for the vibratio nal modes. For the Debye\ndistribution, g(ω) =gD(ω)≡(9n0/ω3\nD)ω2is nonzero in the range 0 ≤ω≤ωD. The\nDebye frequency ωDis obtained as ωD= (6πn0)1/3c, withcbeing the speed of sound. The\nscaled (Debye) distribution ¯ gDas ¯gD(x) = 3x2wherex=ω/ωD. Considering the case\nxm≡xD=β/planckover2pi1ωD<<1, analysis of the integral f0[¯g,xm(α)] for ¯g(x)≡¯gD(x) obtains the\nfollowing result to leading orders in xD:\nf0[¯gD,xD] =−5\n2+3lnxD+O(x2\nD). (3)\nThis mapping identifies the Debye cutoff frequency ωDin terms of the corresponding width\nparameter αof density functional description as β/planckover2pi1ωD≡xD(α) =/radicalbig\nα/π∧. For a given ∧, a\ncorresponding α0is identified such that for all α≥α0the asymptotic form (2) will represent\nthe ideal gas free energy. The value of ∧/σshould be such that xD(α0)≤xLwhere the\nasymptotic form (3) for f0(x) holds for x < xL.\nIn DFT the free energy fid(α) for the amorphous state is evaluated numerically using\nthen(r) corresponding to the parameter value α. The centers for the gaussian density\nprofiles{Ri}are distributed on a random lattice and the corresponding site-site correlation\n3function g(R) is obtained from the Bernal’s [18] random structure generated th rough the\nBennett algorithm [19]. We approximate the g(R) through the following relation g(R) =\ngB[R(η/η0)1/3] whereηdenotes the average packing fraction and η0is used as a scaling\nparameter for the structure [20] such that at η=η0Bernal’s structure is obtained. Smaller\nthe parameter αinn(x), larger number of sites are needed to be included for an accurate\nevaluation of the fid. The free energy evaluated numerically agrees with the correspon ding\nasymptotic expression (2) for α≥α0. This is shown in the inset of Fig. 1. The difference\n∆fidbetween the two evaluations of fidis shown in Fig. 1 for α < α 0range. This result\ncorresponds to hard sphere system with the fixed average densit yn0σ3= 1.1 or packing\nfractionϕ=πn0σ3/6 =.576. The thermal wavelength is kept constant at ∧/σ= 0.0137.\nThe free energy obtained from the DFT expression fid[n(r)] corresponding to n(r) in the\nrangeα < α 0, deviates completely from the asymptotic result (2). It extends t o the correct\nα→0 limit (ln {n0∧3} −1) for the uniform liquid state. A key observation here is that\nthe free energy curve for small α < α 0matches with the corresponding result obtained\nfrom the microscopic expression f0[¯g,xm(α)] with a modified density of states which is\ndifferent from the Debye distribution. The normalization condition fo rg(ω) is maintained\nwith a corresponding ωm(α) or equivalently xm(α) different from that for the Debye case (=\n/radicalbig\nα/π∧). Therefore we require that the function xm(α)→xD(α) asα→α0. The correction\npart in this functional relation is denoted by C(α), such that xm(α) =xD(α0)+C(α) with\nC(α)→0 asα→α0. Similarly, for the scaled distribution function ¯ g(x) we use a correction\nover the corresponding Debye form ¯ gD(x) = 3x2in terms of the function ∆, such that\n¯g(x,xm(α)) = 3x2(1 + ∆(x,α)). Since for all frequencies x, we must have g→¯gDas\nα→α0, we express ∆ with the separation of variables. Thus ∆( x,α) =B(α)˜∆(x) where\nB(α)→0 asα→α0. Furthermore in order to maintain the normalization condition of g(ω),\nthe function ˜∆(x) for all αmust satisfy/integraltext1\n0x2˜∆(x)dx= 0. The two functions C(α) and\nB(α) are parameterized in terms of two respective polynomials of the se paration parameter\nǫ= 1−α/α0. These functions are determined by requiring that the f0[¯g,xm(α)] obtained\nwith the corresponding ¯ gandxmagree with that of the DFT expression for fidin the range\nα≤α0. We choose the xdependence of the ˜∆(x) so as to have an intermediate peak over the\nwhole frequency range. The relative frequency xat the peak of ˜∆(x) is a parameter which\nis kept fixed through out this work. In order to maintain the positivit y of the density of\nstates the function ˜∆(x) is also kept confined in the range ±1. In Fig. 2 we show the results\n4obtained for the appropriate density of states ¯ g(x) which satisfy the above constraints of\ndensity functional theory. The reduced density of states g(ω)/ω2vs.ω/ωDis shown in Fig.\n2 for three different values of the localization parameter ℓ= 1/√α. The height hBof the\nboson peak increases as ℓgrows and is shown in the inset of Fig. 2. Larger ℓor smaller α\ncorresponds to increased spreading of the density profiles and he nce decreased localization.\nAs the density profiles get increasingly spread out, i.e., asαdecreases, the intensity or\nheight of the boson peak continues to grow while upper cutoff ωmof the frequency keeps\ndecreasing. However, this trend does not continue indefinitely tow ardsα→0. To understand\nthis behavior we note that the upper cutoff ωmsets the shortest wavelength λminfor the\nsound waves. In a valid continuum description of the solid, λmincannot get smaller than\nthe average mean square displacement of the individual particles. I n the density functional\ntheory formulated in terms of gaussian density profiles, the avera ge particle displacement\nisℓ= 1/√αand this is a reasonable estimate for λmin. The corresponding upper limit in\nthe frequency of the sound wave is obtained from the relation ΩL/(2πc) =√αwherecis\nthe speed of sound. In Fig. 3, for n0σ3= 1.1 this limiting value for the upper limit vs.\ncutoff√αis shown as a dashed line. The corresponding upper cutoff of the fre quencyωm(α)\nobtained by matching microscopic expression for the free energy w ith the density functional\nexpression (as described with respect to Fig. 2) is shown as a solid line . We mark with\nan arrow in Fig. 3, the point α=αminwhere the two lines cross. This point depicts the\nmaximum delocalized state for which the boson peak in g(ω) is observed in the vibrational\ndensity of states. For more delocalized states the description in te rms of sound waves breaks\ndown. In the inset of Fig. 3, we show how the point of maximum delocaliz ation obtained in\nthe main figure changes with the density of a hard sphere system.\nMetastable states of strongly heterogeneous density distributio n has been obtained [21–\n23] with free energy minimum intermediate between the uniform liquid a nd crystals with\nlong range order. We regulate the parameter η0to define specific sets of random structures.\nThe lowαor partially delocalized state exists only if total free energy is compu ted with the\nn(x) for lattice points {Ri}on a random structure. Thus disorder is essential for locating\nthe metastable minimum of the free energy in the delocalized region ( α≤α0). This is due to\nthe interaction contribution to the free energy. However the cor responding Fidin the small\nαregion remains the same with {Ri}representing a regular crystalline lattice with long\nrange order instead of the Bernel structure. With mi= exp[−K2\ni/(4α)] the definition (1)\n5for the density n(r) reduces to the form n(r) =n0+/summationtext\nimieiKi.rin terms of an expansion[24]\ninvolving reciprocal lattice vector (RLV) {Ki}. It is useful to note at this point that the\nboson peak in disordered system has been studied in terms of a geom etrically perfect crystal\nhaving random interactions between the neighbors [7, 25].\nFor the hard sphere system with n0σ3= 1.1 we obtain the interaction contribution to the\nfree energy in terms of standard Ramakrishnan-Yussouff functio nal [26] using the solution\nof the Percus-Yevick equation with Verlet-Weiss[27]. For the heter ogeneous glassy state, the\nfree energy barrier fBis obtained as the height of the maximum in the corresponding free\nenergy curve from the metastable minimum at small nonzero α(inset (a) of Fig. 4). For the\noptimum value of αsignifying a metastable state, g(ω) is obtained. In Fig. 4 we show the\nvariation of fBfor the parameter range .62< η0< .68, with the corresponding boson peak\nheighthB. As the free energy barrier fBbecoming weaker, the height of the boson peak hB\ngoes down. In the inset (b) of Fig. 4 is shown the fBvs.ℓ= 1/√αcurve. As the mass\nlocalization gets weaker, i.e.with decreasing(increasing) of α(ℓ) the barrier height fBgets\nsmaller. A lower barrier height implies that the structural degradat ion is less hindered in the\ncorresponding system, i.e., represents a glassy material of higher fragility. The dependence\nof the peak height hBand the peak frequency ωpon the pressure are respectively shown in\nFig. 5 and its inset and conforms to experimental findings.\nAn important characteristic of the boson peak is that it gets weake r, more fragile the\nsystem is. Thislinkbetween thelong-timerelaxationbehavior(fragilit y)oftheglassforming\nmaterial and the vibrational modes in the THz region, follows in a natu ral manner in the\npresent model. Since less fragile or stronger liquids tend to form net work structures, the\ndensity profiles are more localized for them. Therefore increase of fragility is synonymous\nwith decrease of α,i.e., less localized density profiles and lower barriers. Since as discussed\nabove the boson peak height decreases with decreasing α, it also implies weaker boson peak\nfor more fragile systems [28, 29]. Our present approach of DFT is sim ilar to the models of\ndisordered solids in terms of springs. For large values of α, the sharply localized density\nprofiles are interpreted in terms of harmonic oscillators with spring c onstantκwhich is\nproportional to the width parameter α[30]. A natural extension of the present model will\nbe to include fluctuations of the widths αat different sites on the amorphous structure\nas an appropriate description of the heterogeneous glassy state . PK acknowledges CSIR,\nIndia for financial support. SPD acknowledges BRNS, India for fina ncial support under\n62011/37P/47/BRNS.\n[1] R. C. Zeller, and R. O. Pohl, Phys. Rev. B 4, 2029 (1971).\n[2] A. Monaco et.al., Phys. Rev. Lett. 97, 135501 (2006).\n[3] D. A. Parshin, H. R. Schober, and V. L. Gurevich, Phys. Rev . B76, 064206 (2007).\n[4] T. S. Grigera, V. Martin-Mayor, G. Parisi, and P. Verrocc hio, Nature 422, 289 (2003).\n[5] E. Duval, A. Boukenter, and T. Achibat, J. Phys. Cond. Mat t.2, 0227 (1990)\n[6] V. Lubchenko and P. G.Wolynes, Proc. Natl. Acad. Sci. U.S .A.100, 1515 (2003)\n[7] W. Schirmacher, G. Diezemann, and C. Ganter, Phys. Rev. L ett.81, 136 (1998).\n[8] A. P. Sokolov, J. Phys.: Condens. Matter 11, A213 (1999).\n[9] W. Schirmacher, G. Ruocco, and T. Scopigno, Phys. Rev. Le tt.98, 025501 (2007)\n[10] M. T. Dove, M. J. Harris, A. C. Hannon, J. M. Parker, I. P. S wainson, and M. Gambhir, Phys.\nRev. Lett. 78, 1070 (1997).\n[11] S. P. Das, Phys. Rev. E 59, 3870 (1999).\n[12] H. Shintani, H. Tanaka, Nat. Mater. 7, 870 (2008).\n[13] A. I. Chumakov et al. Phys. Rev. Lett. 106, 225501 (2011).\n[14] Reiner Zorn, Phys. Rev. B 81, 054208 (2010).\n[15] J. Goldstone, Nuovo Cim. 19, 15 (1961).\n[16] S. P. Das, Statistical Physics of Liquids at Freezing and Beyond , Cambridge University Press,\nNewYork, 2011.\n[17] P. Tarazona, Mol. Phys. 52, 871 (1984).\n[18] J. D. Bernal, Proc. R. Soc. London, Ser. A 280, 299 (1964).\n[19] Charles Bennett, J. Appl. Phys. 43, 2727 (1972).\n[20] Hartmut Lwen, J. Phys. C 2, 8477 (1990).\n[21] C. Kaur, and S. P. Das, Phys. Rev. Lett. 86, 2062 (2001).\n[22] K. Kim, and T. Munakata, Phys. Rev. E, 68, 021502 (2003).\n[23] P. Chaudhary, S. Karmakar, C. Dasgupta, H.R. Krishnamu rthy, A.K. Sood, Phys. Rev. Lett.\n95, 248301 (2005).\n[24] Y. C. Shen and D. W. Oxtoby, J. Chem. Phys. 105, 6517 (1996).\n[25] S. N. Taraskin, Y. L. Loh, G. Natarajan, S. R. Elliott, Ph ys. Rev. Lett. 86, 1255 (2001).\n7[26] T.V. Ramakrishnan, and M. Yussouff, 1979, Phys. Rev. B 19, 2775.\n[27] Henderson, D. and E. W. Grundke, J. Chem. Phys. 63, 601 (1975).\n[28] A. P. Sokolov, R. Calemczuk, B. Salce, A. Kisliuk, D. Qui tmann, and E. Duval, Phys. Rev.\nLett.78, 2405 (1997).\n[29] C. A. Angell, Science 267, 1924 (1995)\n[30] T.R. Kirkpatrick, and P.G Wolynes, Phys. Rev. A 35, 3072 (1987).\n80 10 20 30\nα∗-20-16-12βfid\n0.04 0.2 1 5 25\nα∗0.010.1110β∆fid\nα0\nFIG. 1: Inset : ideal gas part of free energy fidper particle from numerical evaluation of the density\nfunctional expression (solid line) and from the asymptotic formula (2) (dashed line). The difference\nbetween the the asymptotic formula and the numerical result forfidis shown in an enlarged form\nin the main figure for the 0 < α < α 0region.\n0 0.2 0.4 0.6 0.8\nω/ωD00.20.40.6g(ω)/ω20.2 0.3 0.4\nl00.40.8hBα∗=13\n17\n21\nFIG. 2: The reduced density of states g(ω)/ω2in units of c3(cis the sound speed) vs. ω/ωD\ncorresponding to ασ2: 13 (solid), 17(dashed), 21(dot-dashed), for a hard sphere system at density\nofρ0σ3= 1.1. Inset shows corresponding boson peak height hBvs. localization length ℓ(= 1/√α).\n90 1 2 3 4 5\n√α∗00.511.522.5ωm/ωD 1 1.04 1.08 1.12\nn0σ32.52.6l0\n(αmin)-1/2l0=\nFIG. 3: For a hard sphere system at density ρ0σ3= 1.1, the ratio ωm/ωD(see text) vs.√ασ(filled\ndots). Dashed line shows the upper limit of the frequency of t he sound waves for localized the\nvibrational modes (see text). αminshown with an arrow is the minimum possible value of ωm/ωD\nfor the boson peak. Inset shows variation of ℓ0(= 1/√αminin units of σ) with density ρ0σ3for\nthe hard sphere system.\n102 3 4 5 6\nfB0.30.40.50.60.70.8hB\n0.24 0.28 0.32\nl246fB1 5 25\nα∗03β∆f fB(a)\n(b)\nFIG. 4: Boson peak height hBvs. the free energy barrier fBfor the heterogeneous metastable\nstates obtained in the α0> αregion corresponding to different structures of a hard sphere system\nofρ0σ3= 1.1. Inset a) definition of fBand b)fBvs. the width ℓof the gaussian density profiles.\n20 25 30\np0.40.8hB\n20 25 30\np0.160.180.2ωp/ωD\nFIG. 5: Ratio of frequency ωpat the boson peak to the Debye frequency ωDvs. pressure Pof the\nsystem ( in units of ( βσ3)−1). Inset shows height hBof the peak vs. pressure P.\n11"    },    {        "title": "1304.1328v1.Density_functional_theory_for_the_spin_1_bosons_in_a_one_dimensional_harmonic_trap.pdf",        "content": "arXiv:1304.1328v1  [cond-mat.quant-gas]  4 Apr 2013Density-functional theory for the spin-1 bosons in a one-di mensional harmonic trap\nHongmei Wang1,2and Yunbo Zhang1,∗\n1Institute of Theoretical Physics, Shanxi University, Taiy uan 030006, P. R. China\n2Department of Physics, Taiyuan Normal University, Taiyuan 030001, P. R. China\nWe propose the density-functional theory for one-dimensio nal harmonically trapped spin-1 bosons\nin the ground state with repulsive density-density interac tion and anti-ferromagnetic spin-exchange\ninteraction. The density distributions of spin singlet pai red bosons and polarized bosons with\ndifferent total polarization for various interaction param eters are obtained by solving the Kohn-\nSham equations which are derived based on the local density a pproximation and the Bethe ansatz\nexact results for homogeneous system. Non-monotonicity of the central densities is attributed to\nthe competition between the density interaction and spin-e xchange. The results reveal the phase\nseparation of the paired and polarized bosons, the density p rofiles of which respectively approach\nthe Tonks-Girardeau gases of Bose-Bose pairs and scalar bos ons in the case of strong interaction.\nWe give the R-P phase diagram at strong interaction and find th e critical polarization, which paves\nthe way to direct observe the exotic singlet pairing in spino r gas experimentally.\nPACS numbers: 03.75.Mn,67.85.Fg,71.15.Mb\nI. INTRODUCTION\nSpinor Bose gases and one-dimensional (1D) system\nare both the fascinating topics in the research of cold\natoms [1–4]. The studies of their crossing point for 1D\nspinor bosons are also attractive in theories [5–12] and\nhave been realized in recent experiments [13–15]. Spinor\ngases are prepared in the optical traps where the in-\nduced electric dipole moment determines the laser-atom\ninteraction and involve an ensemble of Bose atoms con-\ndensed in a coherent superposition of all possible hyper-\nfine states. The early experimentally achieved spinor\ngases include23Na [16, 17] and87Rb [18, 19]. In the\nthree-dimensional (3D) case, the spin-dependent spin-\nexchange interactions are much weaker than the spin-\nindependent short-range density-density interactions, for\nexample, the ratio of them are c2/c0(Na) = 0 .03 [20]\nandc2/c0(Rb) =−0.005 [21] respectively. The ground-\nstate wavefunction is represented by a spinor wavefunc-\ntion which minimizes the free energy [22–24] and the\nspin-exchange interactions give rise to a rich variety of\nphenomena such as spin domains [17], textures [22], spin\nmixing dynamics [25–28], and fragmentation of conden-\nsate [29–31] etc. On the other hand, 1D systems can be\nrealized by confining the cold atoms in strong anisotropic\ntraps where the motion of atoms is effectively 1D [13–\n15, 32–36]. The interaction among the atoms can be\ntuned in the whole regime of interaction strength via the\nidea of Feshbach resonance as well as the confinement-\ninduced resonance (CIR) [37–39]. These experimental\ndevelopments have provided unprecedented opportuni-\nties for testing the theory of 1D exactly solvable many-\nbody models [7, 40–46].\nMany theories have studied the 1D spinor gases. Un-\nder the mean-field theory, Zhang and You checked the\n∗Electronic address: ybzhang@sxu.edu.cnvalidity of a Gaussian ansatz for the transverse profile in\nthe weak interaction regime and a Thomas-Fermi ansatz\n(TFA) in the strong interaction regime [5]. Hao et al.[6]\nmodified the Gross-Pitaevskii (GP) equations based on\nthe solution of Lieb-Lininger (L-L) [40] model to reveal\nthat the total densities of the 1D spinor bosons exhibit\nthe Tonks-Girardeau (TG) [47] properties of 1D scalar\nbosonswhendensity-densityinteractionisstrongenough.\nThe detailed TG and super Tonks-Girardeau gas (STG)\nproperties of 1D spinor bosons have been investigated\nparticularly by Deuretzbacher et al.[8] and Girardeau et\nal.[9] with the method of Bose-Fermi mapping.\nIf the spin-exchange interaction can be modulated to\nthe order of density-density interaction, the competition\nbetween these two kinds of interaction must be consid-\nered. Cao et al.find that the 1D homogeneous spinor\nbosons under c0=c2can be exactly solved with Bethe\nansatz(BA) method [7]. Essler et al.showitslow-energy\ndegrees of freedom are equivalent to a spin-charge sepa-\nration theory of the U(1) Tomonaga-Luttinger liquid de-\nscribingthechargesectorandtheO(3)nonlinear σmodel\ndescribing the spin sector [10]. By means of the thermo-\ndynamical Bethe ansatz (TBA) method [11, 12], Lee et\nal.and Kuhn et al.give the ground state phase diagram\nand investigate the universal thermodynamics and quan-\ntum criticality of the trapped 1D spinor bosons for the\nstrong interaction situation.\nSo far, a method is not available for the 1D trapped\nspin-1 bosons in the entire region of interaction from\nweakto strong. In thispaper, wedeveloptheHohenberg-\nKohn-Sham density functional theory (DFT) to inves-\ntigate the ground-state properties of 1D harmonically\ntrapped spin-1 bosons. DFT has been widely used for\ntreatingelectron systems with long-rangeCoulomb inter-\naction [48, 49]. It also has been successfully generalized\nto cold atom systems with short-range contact interac-\ntion [50–53]. For 1D cold-atom systems, the method of\nDFT based on BA results has been developed to solve\nthe ground state problem of bosons [51, 54, 55], fermions2\n[56, 57] and Bose-Fermi mixtures [58].\nHere we apply this method to study how the ground\nstate of 1D trapped spin-1 bosons evolves along with the\ninteraction strength from weak to strong. We derive the\nKohn-Sham(KS) equationsby combingthe BAsolutions\nand Local Density Approximation (LDA). The ground\nstate densities and energies for different interactions are\nobtained by solving these equations iteratively. The pa-\nper is organized as follows. We introduce our theory in\nSec. II and show the numerical results in Sec. III. The\ntheory part includes the model, the BA equation for ho-\nmogeneous system and the KS equations for trapped sys-\ntem. In the results part, we first show the case that all\nbosons are fully paired and then for the partially paired\ncase.\nII. THEORY\nA. Model\nWe consider Nspin-1 bosons of mass mconfined in\nextremely anisotropic crossed optical dipole traps with\ndelta-function type density-density interaction and spin\nexchange interaction between atoms. The trap is charac-\nterized by the radial and axial angular frequencies ω⊥\nandωwith corresponding harmonic oscillator lengths\na⊥=/radicalbig\n/planckover2pi1/mω⊥anda=/radicalbig\n/planckover2pi1/mωrespectively. When ω⊥\n≫ω, the radial Thomas-Fermi radius is small enough\nsuch that only axial spin domains could form [14, 15]. In\nfirst quantized form, the Hamiltonian for 1D spin-1 gas\ncan be written as\nH=N/summationdisplay\ni=1/parenleftbigg\n−/planckover2pi12\n2m∂2\n∂x2\ni+1\n2mω2x2\ni/parenrightbigg\n+/summationdisplay\ni<j/bracketleftbig\nc1D\n0+c1D\n2Si·Sj/bracketrightbig\nδ(xi−xj),(1)\nwhereSi,jare spin-1 operators, c1D\n0andc1D\n2are 1D in-\nteraction parameters which can be expressed through in-\nteraction parameters g1D\nSin total spin S= 0,2 channels\nasc1D\n0=/parenleftbig\ng1D\n0+2g1D\n2/parenrightbig\n/3 andc1D\n2=/parenleftbig\ng1D\n2−g1D\n0/parenrightbig\n/3.\nIn experiments, g1D\nScan be tuned with a⊥and 3D\ns-wave scattering length a3D\nSaccording to g1D\nS=\n2/planckover2pi12a3D\nS/ma2\n⊥/parenleftbig\n1−Aa3D\nS/a⊥/parenrightbig\nwith constant A= 1.0326\n[37]. Thus c1D\n0andc1D\n2may be manipulated in wider\nrange comparing with 3D spinor system.\nThe number of atoms N+,N0andN−corresponding\nto spin states s= +1,0,−1 are not conserved because\nthe scattering between two atoms of spin s=±1 can\nproduce two atoms of spin s= 0 and vice versa, whereas\nthe total number of atoms N=N++N0+N−and total\nspin in the zcomponent Sz=N+−N−are conserved\nyet. Therefore we may consider the system is composed\nof two parts of atoms, particle I and particle II with total\nparticle numbers N=N1+ 2N2and total polarizationP=N1/N. The number of particle I is N1=Szwhere\nall atoms have parallel spin forming the ferromagnetic\nphase. The number of particle II is 2 N2whereN2pairs\nof atoms are formed between two spin states s=±1 or\nbetween two spin states s= 0. Here we have supposed\nthatN+/greaterorequalslantN−and that N0is even. The sign of c1D\n0\ndetermines that the interactions between the bosons are\nrepulsive or attractive, while the sign of c1D\n2determines\nthat the spin exchange interaction in the pairs are ferro-\nmagnetic or anti-ferromagnetic.\nB. Bethe ansatz equations for homogeneous system\nWith the particles confined in a finite 1D tube with\nlengthLinsteadofaharmonictrapasintheHamiltonian\n(1), the system has been exactly solved by Cao et al.[7]\nwith BA method under a special condition\nc1D\n0=c1D\n2=g >0,\ni.e., the repulsive density-density interaction equals the\nantiferromagnetic spin-exchange interaction. They gave\nthe BA equations\neikjL=N/productdisplay\ni=1,i/negationslash=j˜e4(kj−ki)2N2/productdisplay\nα=1˜e−2(kj−Λα),(2)\nN/productdisplay\ni=1˜e2(Λα−ki) =−2N2/productdisplay\nβ=1˜e2(Λα−Λβ),(3)\nwith ˜en(x) = (x+inc′)/(x−inc′),c′=c/4 andc=\n2mg//planckover2pi12.{ki}in the equations is the set of quasi-\nmomentum and {Λα}is the set of the spin rapidity. The\nground state energy of the system is\nE=/planckover2pi12\n2mN/summationdisplay\ni=1k2\ni. (4)\nFor positive c1D\n2, the spin-exchange interaction is an-\ntiferromagnetic. 2 N2particles form spin singlet bound\npairs between two spin s=±1 atoms or between two\nspins= 0 atoms, whereas N1particles are polarized.\nThe equations (2) and (3) have N1real solutions for ki\n(i= 1,···N1) andN2pairs conjugate complex solutions\nor string solutions for kαand Λ α(α= 1,···,2N2), i.e.\nkα=λl±ic′,\nΛα=λl±ic′, (5)\nwithλl(l= 1,···,N2) real numbers [7, 11]. Inserting\n(5) into (2) (3) and adopting the thermodynamic limit,\ni.e., the length of gas L→ ∞and particle numbers\nN,N1,N2→ ∞with densities n=N/L,n1=N1/L3\nandn2=N2/Lfinite, we easily arrive at the BA integral\nequations\n2πρ1(k) = 1+2/integraldisplayB\n−Bdk′ρ1(k′)a4(k,k′)\n+2/integraldisplayQ\n−Qdλρ2(λ)(a5−a1)(k,λ),(6)\nπρ2(λ) = 1+/integraldisplayB\n−Bdkρ1(k)(a5−a1)(k,λ)\n+/integraldisplayQ\n−Qdλ′ρ2(λ′)(a6+a4−a2)(λ,λ′),(7)\nwhere\n(an−am)(x1,x2) =an(x1,x2)−am(x1,x2),(8)\nand\nan(x1,x2) =n|c′|\n(nc′)2+(x1−x2)2, (9)\nwithρ1(k) andρ2(λ) densities of kandλ. The integral\nboundaries BandQare determined by the conditions\nn1=/integraldisplayB\n−Bdkρ1(k), (10)\nn2=/integraldisplayQ\n−Qdλρ2(λ). (11)\nFrom (4), we get the ground state energy per unit length\nE\nL=/planckover2pi12\n2m/integraldisplayB\n−Bdkk2ρ1(k)\n+/planckover2pi12\n2m/integraldisplayQ\n−Qdλ2λ2ρ2(λ)−ǫbn2,(12)\nwhere the binding energy of the pair in defined as\nǫb=/planckover2pi12\n2mc2\n8. (13)\nTo solve the BA equations, we introduce the dimen-\nsionless L-L interaction parameter γ=c/2n=mg//planckover2pi12n\nand polarization parameter p=n1/n. Letk=Bx,\nλ=Qy,B=c/β1andQ=c/β2, the densities\nof quasi-momentum and spin rapidity turn out to be\nρ1(k) =g1(x) andρ2(λ) =g2(y). The integral BA\nequations (6) and (7) are translated into\n2πg1(x) = 1+2\nβ1/integraldisplay1\n−1dx′g1(x′)b4/parenleftbiggx\nβ1,x′\nβ1/parenrightbigg\n(14)\n+2\nβ2/integraldisplay1\n−1dyg2(y)(b5−b1)/parenleftbiggx\nβ1,y\nβ2/parenrightbigg\n,πg2(y) = 1+1\nβ1/integraldisplay1\n−1dxg1(x)(b5−b1)/parenleftbiggx\nβ1,y\nβ2/parenrightbigg\n(15)\n+1\nβ2/integraldisplay1\n−1dy′g2(y′)(b6+b4−b2)/parenleftbiggy\nβ2,y′\nβ2/parenrightbigg\n,\nwith\n(bn−bm)(x1,x2) =bn(x1,x2)−bm(x1,x2),(16)\nand\nbn(x1,x2) =n/4\n(n/4)2+(x1−x2)2.(17)\nThe normalization conditions (10) and (11) are now\nβ1=2γ\np/integraldisplay1\n−1g1(x)dx, (18)\nβ2=4γ\n1−p/integraldisplay1\n−1g2(y)dy. (19)\nFrom (12) the ground state energy per atom for the ho-\nmogeneous 1D spinor gas is\nεhom(n,γ,p) =E\nN=/planckover2pi12n2\n2me(γ,p),(20)\nwith\ne(γ,p) =e1(γ,p)+e2(γ,p)+eb(γ,p) (21)\nand\ne1(γ,p) =8γ3\nβ3\n1/integraldisplay1\n−1x2g1(x)dx, (22)\ne2(γ,p) =16γ3\nβ3\n2/integraldisplay1\n−1y2g2(y)dy, (23)\neb(γ,p) =−γ2(1−p)\n4. (24)\nHere (22) and (23) can be solved numerically with the\ncombination of integral equations (14), (15) and the nor-\nmalization (18), (19). The chemical potentials are taken\nas the derivatives of (20) as\nµhom\n1(n,γ,p) =∂(nεhom)\n∂n1=/planckover2pi12n2\n2mf1(γ,p),(25)\nµhom\n2(n,γ,p) =∂(nεhom)\n∂n2=/planckover2pi12n2\n2mf2(γ,p),(26)\nwhere\nf1(γ,p) = 3e−γ∂e\n∂γ+(1−p)∂e\n∂p,(27)\nf2(γ,p) = 2/parenleftbigg\n3e−γ∂e\n∂γ−p∂e\n∂p/parenrightbigg\n.(28)\nWe see that, when p= 1, the system is in a pure\nferromagnetic phase with spin-polarized bosons. Here4\ne(γ,p= 1)coincideswith e(γ) in the L-L modelofscalar\nbosons [40] with interaction parameter 2 γ. Whenp= 0,\nall the bosons form pairs and the system is in a pure\nantiferromagnetic phase. In the limiting case of γ= 0,\nthe system reduces to free bosons with e(γ= 0,p) = 0.\nWhenγis very strong, the integral equations (14) and\n(15) give g1(x)≈1/2πandg2(x)≈1/π, then\ne(γ→+∞,p)≈π2p3\n3+π2(1−p)3\n48−γ2(1−p)\n4.(29)\nThe energy per unit length in the strong interaction case\nE\nL≈/planckover2pi12\n2m/parenleftbiggπ2n3\n1\n3+π2n3\n2\n6−c2n2\n8/parenrightbigg\n,(30)\nis composed of three parts: the energy density of N1free\nfermions with mass min Ref. [58], the energy density\nofN2free composite fermions with mass 2 m, and the\nbinding energy N2ǫb/Lof the composite fermions. This\nshows that stronger interactions favors the forming of\nboson-boson pairs. The chemical potentials in this case\nare\nµhom\n1(γ→+∞,p)≈/planckover2pi12\n2mπ2n2\n1, (31)\nµhom\n2(γ→+∞,p)≈/planckover2pi12\n2m/parenleftbiggπ2\n2n2\n2−c2\n8/parenrightbigg\n.(32)\nIn Fig. 1, we plot the interaction dependence of energy\ndensitye(γ,p) for various polarization pand compare it\nwith the case of scalar bosons. The upper figure is for\np= 0, i.e., all the bosons form the pairs. Clearly we have\ne1= 0 and e2(γ) (red dashed line), which is obtained\nnumerically from the combination of (15), (19) and (23),\nincreases linearly along with γforγ≪1 and approaches\nslowly to a constant value π2/48 forγ≫1.e2(γ) and\nthe inverse parabola function eb(γ) =−γ2/4 together\ndictate that e(γ) is not monotonic and has a maximum\nvalue 0.1169 atγ= 0.4. The lower figure is for e(γ,p)\nwithp= 0,0.2,0.4,0.6,0.8,1 corresponding to the real\nlines from bottom to top. It shows that the energy func-\ntione(γ,p)increasesalongwith pforthesameparameter\nγ. Forp= 1, i.e., pure ferromagnetic phase case, e(γ)\nincreasesmonotonouslyto constant π2/3whichisexactly\nthe asymptotic value of scalar bosons (gray dash-dotted\nline).\nC. Kohn-Sham equations for trapped system\nNow we consider the spin-1 bosons in a harmonic\ntrapVext(x) =mω2x2/2 by means of the DFT the-\nory based on the theorems of Hohenberg, Kohn and\nSham. The theory enables us to deal with the energy\nand the density profile of inhomogeneous system in the\nground state. According to the Hohenberg-Kohn the-\norem I of DFT, the ground-state density of a bound\nsystem of interacting particles in some external poten-\ntial determines this potential uniquely. Denote now the/s48 /s49 /s50 /s51 /s52 /s53/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s32/s32/s101 /s40 /s32/s112 /s61/s48/s41/s32/s101\n/s50\n/s32/s101\n/s98\n/s32/s101\n/s32/s112 /s61/s49\n/s32/s112 /s61/s48/s46/s56\n/s32/s112 /s61/s48/s46/s54\n/s32/s112 /s61/s48/s46/s52\n/s32/s112 /s61/s48/s46/s50\n/s32/s112 /s61/s48\n/s32/s32/s101 /s40 /s44/s32/s112 /s41\n/s115/s112/s105/s110/s111/s114/s32/s103/s97/s115\n/s32/s32\n/s115/s99/s97/s108/s97/s114/s32/s103/s97/s115/s32/s32 /s32/s32\nFIG. 1: (Color Online) The function e(γ,p) for the ground\nstate energy of 1D homogeneous spin-1 bosons. In the upper\npanel,p= 0. In the lower panel the real lines are for p=\n0,0.2,0.4,0.6,0.8,1 from bottom to top and the dash-dotted\nline is for scalar bosons.\nspace dependent densities of particle I and particle II as\nn1(x) and 2n2(x), respectively. The total density is then\nn(x) =n1(x)+2n2(x) and the number of the particles\nare conserved separately according to\n/integraldisplay\nn1(x)dx=N1, (33)\n/integraldisplay\nn2(x)dx=N2. (34)\nThe ground-state energy is a functional of the densi-\ntiesE0[n1(x),n2(x)]. It can be decomposed as ki-\nnetic energy functional of a reference noninteracting\nsystemTref[n1(x),n2(x)], external potential energy\nfunctional Eext[n1(x),n2(x)] and KS energy functional\nEKS[n1(x),n2(x)] involving the interactions, i.e.,\nE0[n1,n2] =Tref[n1,n2]+Eext[n1,n2]\n+EKS[n1,n2]. (35)\nWe introduce two orthogonal and normalized Bose or-\nbitalfunctionals φ1(x) andφ2(x) toexpressthedensities5\nas\nn1(x) =N1φ∗\n1(x)φ1(x), (36)\nn2(x) =N2φ∗\n2(x)φ2(x). (37)\nThe kinetic energy functional is written as\nTref[n1,n2] =−N1/integraldisplay\ndxφ∗\n1(x)/planckover2pi12\n2md2\ndx2φ1(x)\n−2N2/integraldisplay\ndxφ∗\n2(x)/planckover2pi12\n2md2\ndx2φ2(x)(38)\nand the external potential energy functional is simply\nEext[n1,n2] =/integraldisplay\ndxVext(x)n(x).(39)\nNote for each part of bosons we introduce a single or-\nbital functional assuming that the bosons are in a quasi-\ncondensate state. This is different from the system of\nfermions for which the number of orbital functional is\ndecided by the number of fermions. In Ref. [58] we have\nindicated the validity of single Bose orbital functional in\nDFT. It gives the density profile of 1D bosons varying\nfrom a standard Gaussian shape for weak interaction to\na half-ellipse profile for strong interaction. The density\nprofile for strong interaction is consistent with that of\nnoninteracting fermions except the density oscillations.\nIn the limit of large particle number, the difference be-\ntween the oscillating and non-oscillating profiles becomes\nimperceptible.\nEKS[n1,n2] includes all the contribution of the inter-\naction energies. Sometimes it is partitioned as Hartree-\nFock energy (i.e., the mean field approximation of the\ninteraction energy) and exchange correlation energy [48,\n57, 58]. Following the way in [50, 51, 54, 55], we here\ntreated it as an entity with the LDA, that is, the system\ncan be assumed locally equilibrium at each point xin the\nexternaltrap, with localenergyperatomprovidedbythe\nhomogenous interactional system. Thus the interaction\nenergy functional EKS[n1,n2] can be formulated as\nEKS[n1,n2]≈/integraldisplay\ndxn(x)εhom[n1(x),n2(x)] (40)\nwherethedensities n1(x),n2(x)aretakenatpoint xand\nεhom[n1(x),n2(x)] takes the form of Eq. (20). Note\nthat both the L-L parameter γ(x) =mg//planckover2pi12n(x) and\nthepolarizationparameter p(x) =n1(x)/n(x)arespace\ndependent now. With the explicit form of three terms in\n(35), the ground state energy functional are\nE0[n1(x),n2(x)]\n=N1/integraldisplay\ndxφ∗\n1(x)/bracketleftbigg\n−/planckover2pi12\n2md2\ndx2+1\n2mω2x2/bracketrightbigg\nφ1(x)\n+2N2/integraldisplay\ndxφ∗\n2(x)/bracketleftbigg\n−/planckover2pi12\n2md2\ndx2+1\n2mω2x2/bracketrightbigg\nφ2(x)\n+/integraldisplay\ndxn(x)εhom[n1(x),n2(x)]. (41)Hohenberg-Kohn theorem II guarantees that the\nground-state density distributions are determined by\nvariationally minimizing E0with respect to n1(x) and\nn2(x) [48]. That is equivalent to minimize the free-\nenergy functional F=E0−N1ǫ1−2N2ǫ2with respect\ntoφ∗\n1andφ∗\n2, where the Lagrange multipliers ǫ1,ǫ2are\nintroduced to conserve N1and 2N2. Then we can get the\nKS equations\n/parenleftbigg\n−/planckover2pi12\n2md2\ndx2+1\n2mω2x2+µhom\n1[n1(x),n2(x)]/parenrightbigg\nφ1(x)\n=ǫ1φ1(x), (42)\n/parenleftbigg\n−/planckover2pi12\n2md2\ndx2+1\n2mω2x2+1\n2µhom\n2[n1(x),n2(x)]/parenrightbigg\nφ2(x)\n=ǫ2φ2(x), (43)\nwhere the chemical potentials µhom\n1, µhom\n2of the homo-\ngeneous gas for densities n1,n2atxare given by Eqs.\n(25) and (26). With the eigenvalues of (42) and (43), the\nground state energy can be expressed as\nE0=N1ǫ1+2N2ǫ2+/integraldisplay\ndxn(x)εhom(x)\n−/integraldisplay\nn1(x)µ1(x)dx−/integraldisplay\nn2(x)µ2(x)dx.(44)\nWith the exactly solved εhomfor different n1,n2atx,\nwe can solve the KS equations (42) and (43) together\nwith the definition of orbital functional (36) and (37) to\nfind the density distributions n1(x) andn2(x) and then\ncalculate the ground-state energy E0from Eq. (44).\nWe now discuss the KS equations for the limiting\ncases of weak and strong interaction. When there is\nno interaction, KS equations correctly reduce to the 1D\nSchr¨ odinger equation of simple harmonic oscillator. The\nBose density profiles take the standard Gaussian shape\nn1,2(x) =N1,2\na√πexp/parenleftbig\n−x2/a2/parenrightbig\n. (45)\nWhen the interaction is strong, the kinetic energies in\n(42) and (43) can be ignored, we have the TFA equations\n1\n2mω2x2+µ1(x) =µ0\n1, (46)\n1\n2mω2x2+µ2(x)\n2=µ0\n2\n2. (47)\nThe values of µ0\n1andµ0\n2are fixed by the normalization\nconditions (33) and (34). Based on (46) and (47) Kuhn\net al.give the ground state phase diagram and find that\nthe singlet pairs and unpaired bosons may form a two-\ncomponentLuttingerliquidin thestrongcouplingregime\n[12]. When the interaction approaches infinitely strong,\nwith the limit values of chemical potential (31, 32) and\nthe normalization conditions (33, 34), we can solve (46,6\n/s45/s53 /s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56\n/s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s52/s46/s52/s52/s46/s54/s52/s46/s56/s53/s46/s48/s32/s120 /s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s97 /s41/s32\n/s32/s32/s32/s32/s32/s115/s112/s105/s110/s45/s49/s32/s98/s111/s115/s111/s110/s115\n/s32/s32/s32/s32/s32/s85/s61/s48/s126/s49/s50/s46/s53\n/s32\n/s32/s32/s32/s32/s32/s102/s101/s114/s109/s105/s111/s110\n/s32/s32/s110 /s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32/s49/s47 /s97 /s41/s32\n/s32/s102/s101/s114/s109/s105/s111/s110\n/s32/s85/s61/s49/s50/s46/s53\n/s32/s85/s61/s50/s53\n/s32/s85/s61/s49/s53/s48/s110 /s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32/s49/s47 /s97 /s41\n/s120/s32 /s40/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s97 /s41\nFIG. 2: (Color Online) Density distribution of 1D trapped\nspin-1 bosons with N= 30 and P= 0. In the upper figure,\nfive black solid lines from top to bottom are respectively for\nspinor bosons with U= 0,0.1,0.5,2,12.5. Red dotted line\nis for non-interacting Fermions with mass 2 m. The lower\nfigure shows the details of the peak density for even stronger\ninteractions U= 12.5,25,150. We observe a slightly rising of\nthe peak after U >12.5.\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56\n/s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s52/s46/s56/s52/s52/s46/s56/s54/s52/s46/s56/s56/s52/s46/s57/s48/s52/s46/s57/s50/s52/s46/s57/s52\n/s40/s98/s41\n/s32/s32/s110/s40/s48/s41\n/s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32/s49/s47 /s97 /s41\n/s85/s40/s97/s41\n/s32/s110/s40/s48/s41\n/s32/s32\n/s85\nFIG. 3: Non-monotonicity of the central density n(0)of 1D\ntrapped spin-1 bosons with increasing UforN= 30 and\nP= 0. The exists a minimum value of n(0)atU= 12.5\nwhich is shown in the inset.47) to obtain the following half-ellipse-like density pro-\nfiles\nn1,2(x)≈/radicalBig\n2N1,2−(x/a1,2)2\nπa1,2, (48)\nwherea1=aanda2=/radicalbig\n/planckover2pi1/2mω. We see that the den-\nsity of particle I is just that of N1noninteracting har-\nmonically trapped fermions with mass m. The density\nof particles II is that of N2noninteracting fermions with\nmass 2m. It shows that for infinitely strong interaction,\nthe property of particles II approaches the TG gas of\nBose-Bose pairs. Resorting to the Bose-Fermi mapping\nmethod [8, 9, 47], we maymap the densities ofpairedand\nunpaired components exactly to those of non-interacting\nFermions\nn1,2(x) =1\na1,2√πexp/parenleftbig\n−x2/a2\n1,2/parenrightbigN1,2−1/summationdisplay\nl=0H2\nl(x/a1,2)\n2ll!,\n(49)\nwhereHl(x) is the Hermite polynomials.\nIII. NUMERICAL RESULTS\nWe introduce the interacting parameter U=g/a/planckover2pi1ω.\nThespace-dependentLieb-Linigerparameterisexpressed\nasγ(x) =U/an(x). For the 1Dspin-1 bosons with given\nN,PandU, we present the numerical results for densi-\ntiesn1andn2of particle I and II obtained by solving the\nKS equations (42) and (43) with the iteration method.\nThe ground state energy E0can be obtained via the rela-\ntion (44). The numerical results are summarized in Figs.\n2-10.\nA.P= 0system\nWe first study the system with N= 30 and P= 0. In\nthis fully paired case, the length, density and energy are\nin units of a, 1/aand/planckover2pi1ωrespectively. Fig. 2 provides an\nunderstanding of how the density profiles change along\nwith the interaction parameter for U= 0,0.1,0.5,2,12.5\nin the upper figure and U= 12.5,25,150 in the lower\nfigure. With the increasing of U, the total density pro-\nfile varies from a standard Gaussian-like shape charac-\nterizing the distribution of non-interacting Bose gas to\na half-ellipse shape indicating the distribution of non-\ninteracting Fermi gas. Interestingly we find that the den-\nsity profile of Nspin-1 bosons with mass mfor strongly\ninteracting case e.g. U= 150 overlaps that of N/2\nnoninteracting fermions with mass 2 m, except the emer-\ngence of the density oscillation in the Fermionic case. It\nmeans that for strong interaction all atoms in the anti-\nferromagnetic spin-1 bosons are paired with each other,\nbehaving like the TG gas of Boson pairs.7\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48 /s49/s52/s48/s53/s55/s46/s53/s53/s56/s46/s48/s53/s56/s46/s53/s53/s57/s46/s48/s53/s57/s46/s53\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s32/s32/s69\n/s48/s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s41\n/s85/s32 /s32/s84/s114/s101/s102\n/s32/s32/s32/s32/s32 /s32/s69\n/s101/s120/s116/s32/s32/s32/s32/s32/s32 /s32/s69\n/s75/s83\n/s32 /s32/s69\n/s48/s61 /s84/s114/s101/s102\n/s32/s43 /s69\n/s101/s120/s116/s43 /s69\n/s75/s83/s40/s97/s41\n/s40/s99/s41\n/s32/s32/s69\n/s101/s120/s116/s32\n/s85/s32/s32/s69\n/s48\n/s85/s40/s98/s41\nFIG. 4: (Color Online) (a) Evolution of the ground state en-\nergyE0and all contributedenergy terms of 1Dtrappedspin-1\nbosons with increasing UforN= 30 and P= 0. (b) the de-\ntails of energies in the weak interaction regime. (c) the det ails\nof external potential energy.\nWe surprisingly notice that the density profile does\nnot change with the interaction parameter monotoni-\ncally. The full tendency of the central density n(0)(den-\nsity at the trap center x= 0) can be seen in Fig. 3\nand the inset shows in detail the non-monotonicity of\nn(0). We find that central density firstly decreases to\na minimum value n(0)\nmin= 4.858/aatU= 12.5 before\nreaching the constant central density of non-interacting\nfermions for large U. The result can be understood as\nthe competition between the repulsivedensity-densityin-\nteraction and the anti-ferromagnetic spin-exchange in-\nteraction with equal strength. The repulsive density-\ndensityinteractiontendstoincreasethedistancebetween\nbosons while the anti-ferromagnetic spin-exchange inter-\naction leads essentially attraction between s=±1 or\ns= 0 bosons. For U <12.5, the prevailing repulsions\namong bosons tend to broaden them to wider space area\nand reduce the density of bosons in the trap center to a\nminimum. The anti-ferromagneticeffect is prominent for\nU >12.5 which contracts the bosons slightly.\nFig. 4 describes the evolution of the ground state en-\nergyE0and all contributed energy terms in Eq. (35)\nas a function of U. We can see the whole trend in (a)\nand the details in the mean field regime in (b). It shows\nthat the kinetic energy Trefdecreases slowly to a con-\nstant energy as the result of interactions restraining the\nmovement of atoms. However, the external potential en-\nergyEextincreasesduetothe atomsoccupywiderregimeof the trap. In correspondence to the non-monotonicity\nofn(0),Eextshows non-monotonicity too, which can be\nseen clearly in Fig. 4(c). Close to the critical inter-\naction value of U= 12.5 forn(0)\nminin Fig. (3c), Eext\nreaches its maximum Emax\next= 59.092/planckover2pi1ωatU= 11.5.\nBut the non-monotonicity of n(0)doesn’t develop visible\neffects on other energy terms. Analogous to the ground\nstateenergyfunctional e(γ,p= 0)forhomogeneousspin-\n1 bosons (see the upper panel of Fig. 1), the KS energy\nEKS, i.e. the interaction energy, increases linearly in\nweak interaction regime and approaches its peak value\nEmax\nKS= 43.033/planckover2pi1ωatU= 1.4, corresponding to an axial\nL-L interaction parameter γ(0)= 0.222 (γat the trap\ncenter), followed by a monotonously decreasing in strong\ninteraction regime. The contributions from Tref,Eext\nandEKStogether establish the ground state energy E0\nto increase to a maximum Emax\n0= 85.994/planckover2pi1ωatU=\n1.9, corresponding to γ(0)= 0.324, and then decrease\nmonotonously. We notice that the value γ(0)= 0.324\nforEmax\n0are close to γ= 0.4 for the maximal ein the\nupper panel of Fig. 1. We understand that the compe-\ntition between the repulsion among paired bosons and\nthe binding energy of the pair gives the peak of E0at\nU= 1.9 for the same reason as in the homogeneous case,\nwhile that between the repulsive density-density interac-\ntionandtheanti-ferromagneticspin-exchangeinteraction\ngives the non-monotonicity of n(0)atU= 12.5.\nB.P/negationslash= 0system\nForP= 1, none of the bosons can form pair and\nthe system reduces to the scalar Bose gas with density-\ndensity interaction 2 g, whose ground state properties\nhave been studied by means of DFT in Ref. [58]. With\nthe increasing of U, the density distribution and energy\nof the system approach those of a single-component TG\ngas.\nHere we study the interesting situation of a partially\npolarized spinor gas with total polarization 0 < P <\n1. In this case the length, density and energy are in\nunits of N1/2a,N1/2/aand/planckover2pi1ωrespectively. In Fig. 5\nwe illustrate the density of polarized particles n1(blue\nsolid lines), density of paired bosons 2 n2(red dashed\nlines) and the total density n(black dotted lines) for\nvarious polarization P= 0.1,0.2,0.4,0.6,0.8 (the fig-\nures in the columns from left to right) and interaction\nU= 0.1,2,10,50 (the figures in the rows from up to\ndown). Horizontal view shows that the density profile\nn1expands gradually with the increasing of Paccompa-\nnied by the corresponding shrinking of 2 n2. These two\ndensities together result in a slightly decreasing of the\npeak density n(0)for increasing P. An analysis of ver-\ntical scope for increasing interaction parameter U(note\nthat we use different vertical axis scale in the weak in-\nteraction case U= 0.1) tells us that the density 2 n2\nchanges smoothly from the Gaussian distribution of non-\ninteracting Bosegas to the half-ellipse distribution of TG8\n/s45/s49 /s48 /s49 /s45/s49 /s48 /s49 /s45/s49 /s48 /s49 /s45/s49 /s48 /s49/s48/s49/s50\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s49 /s48 /s49/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s32/s32/s32\n/s32\n/s32 /s32/s32/s32/s32 /s110\n/s49\n/s32/s32/s50 /s110\n/s50\n/s32/s32/s110/s80 /s61/s48/s46/s52/s80 /s61/s48/s46/s54 /s80 /s61/s48/s46/s56\n/s85 /s61/s48/s46/s49\n/s85 /s61/s50\n/s85 /s61/s49/s48\n/s85 /s61/s53/s48/s110\n/s49/s44/s32/s50 /s110\n/s50/s32/s97/s110/s100/s32 /s110 /s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s78/s49/s47/s50\n/s47/s97 /s41\n/s32/s32\n/s32/s32\n/s32 /s32/s80 /s61/s48/s46/s49\n/s32/s32\n/s120/s32 /s40/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s78/s49/s47/s50\n/s97 /s41/s32\n/s32/s80 /s61/s48/s46/s50\n/s32 /s32\n/s32/s32\n/s32 /s32/s32/s32 /s32/s32\n/s32/s32\n/s32/s32 /s32/s32\n/s32\n/s32/s32/s32\n/s32\n/s32\nFIG. 5: (Color Online) Density distributions of 1D trapped s pin-1 bosons at P= 0.1,0.2,0.4,0.6,0.8 (panels from left to right)\nandU= 0.1,2,10,50 (panels from top to bottom). Blue solid lines denote the de nsities for polarized particles n1, red dashed\nlines denote the densities for paired bosons 2 n2, and black dotted lines denote the total densities n.\ngas of Bose-Bose pairs. The density peak is located in\nthe center of the trap for all interaction strengths. The\nbehavior of density n1is, however, a little complicated.\nThough n1shows Gaussian distribution for weak inter-\naction (U= 0.1), bosons of Particle I tend to occupy\nwider space and start to be excluded from the trap cen-\nter for intermediate interaction ( U= 2,10). The single\npeak profile of n1changes into the double-peak distribu-\ntion. This reminds us the partially phase separation of\nBose-Fermi mixture with equal mass and equal repulsive\ninteraction [44, 58]. In our case, the phase separation of\npaired and unpaired bosons occurs in the system, i.e. the\ncore area of the 1D spinor gas is filled with the mixture\nof paired and unpaired bosons and in the outer region we\nfind either polarized bosons for P >0.2 or paired bosons\nforP <0.2. For even stronger interaction ( U= 50), the\npeak ofn1returns back to the trap center and the den-\nsity profile n1approaches the half-ellipse distribution of\nTG gas. Nevertheless the evidence of phase separation\nbecomes moreprominent in the stronglyinteractioncase.\nAs a result the total density nshows the overall Gaus-\nsian distribution showing a fully mixing phase of n1and\n2n2at weak interaction ( U= 0.1). When the interaction\nincreases, we find a bi-modal distribution of the total/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s32/s68/s70/s84/s32/s32/s50 /s110\n/s50\n/s32/s84/s70/s65/s32/s32/s50 /s110\n/s50\n/s32/s70/s101/s114/s109/s105/s111/s110/s32/s32/s50 /s110\n/s50\n/s32/s32/s110\n/s49/s32/s97/s110/s100/s32/s50 /s110\n/s50/s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s78/s49/s47/s50\n/s47/s97 /s41\n/s120 /s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s78/s49/s47/s50\n/s97 /s41/s32/s68/s70/s84/s32/s32 /s110\n/s49\n/s32/s84/s70/s65/s32/s32 /s110\n/s49\n/s32/s70/s101/s114/s109/s105/s111/s110/s32/s32 /s110\n/s49\nFIG. 6: (Color Online) Comparison of the density distribu-\ntion of 1D trapped spin-1 bosons at P= 0.4 in the case of\nstrong interaction. The results for both n1and 2n2are from\nDFT for U= 50 (solid lines), TFA for U= +∞(dashed\nlines), and N= 30 non-interacting fermions (dotted lines),\nrespectively. Clear evidence is observed for the phase sepa -\nration and in this case the surrounding wings are composed\nof polarized particles. The radius of vanishing density fro m\nDFT is obviously larger than the TFA result.9\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s32/s32/s67/s101/s110/s116/s114/s97/s108/s32/s100/s101/s110 /s115/s105/s116/s105/s101/s115/s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s78/s49/s47/s50\n/s47/s97 /s41\n/s85/s32/s80 /s61/s48\n/s32/s80 /s61/s48/s46/s50\n/s32/s80 /s61/s48/s46/s52\n/s32/s80 /s61/s48/s46/s54\n/s32/s80 /s61/s48/s46/s56\n/s32/s80 /s61/s49/s110\n/s49/s40/s48/s41\n/s32/s32\n/s32/s50 /s110\n/s50/s40/s48/s41\n/s85\n/s32/s32/s32/s110/s40/s48/s41\n/s85\nFIG. 7: (Color Online)Evolutionofcentral densities n(0)\n1,n(0)\n2\nandn(0)(panels from left to right) of trapped spin-1 bosons\nwith increasing interaction parameter U. Non-monotonicity\nis seen for all partially polarized cases 0 < P <1.\ndensity with particle II imposed on the top of particle I,\ni.e. the mixed core of particles I and II is surrounded by\ntwo wings composed of solely polarized bosons for large\nPand large U(see the several right-down panels of Fig.\n5). Take the P= 0.4 case as an example. We compare\nthe density plots for the two components in Fig. 6. The\nsolid linesareour DFT results fromthe iterationsolution\nof the KSE Eqs. (42) and (43), while the dashed lines are\nanalytical TFA results Eq. (48) for infinitely strong in-\nteraction. Alsoshownarethe densitiesofnon-interacting\nfermions Eq. (49) according to the Bose-Fermi mapping\n(dotted lines) for N= 30. Weidentify readilythe density\noscillations with 12 peaks in the polarized component n1\nin the TG limit and 9 peaks in the paired component n2\nrepresenting the TG gas of 9 pairs of bosons.\nAn interesting observation is the non-monotonicity of\nthe central density value n(0)which already occurs in the\nfully paired case P= 0. We take a close look at the rise-\nand-fall of the peak values for polarized bosons, paired\nbosons, and the total density. Fig. 7 shows the evolution\nof central densities with increasing interaction parame-\nterUfor different P. For the central density of polarized\nparticles n(0)\n1, we find it is a constant zero at P= 0 since\nthere are no Particle I in the gas and a monotonically de-\ncreasing curve at P= 1 corresponding to the scalar Bose\ngas. In all partially polarized case 0 < P <1, the peak\nvalues rapidly decrease to a minimum n(0)\n1minand then\ngradually approach to a constant for strong interaction.\nThe minimum is seen to move toward the stronger inter-\naction direction with increasing Pand finally disappear\nforP= 1. On the contrary, the central density of paired\nparticles n(0)\n2is a constant zero at P= 1 since there are\nno paired bosons in the gas and a seemingly monotoni-\ncallydecreasingcurveat P= 0correspondingtothefully\npaired bosons. Yet we know from the result of P= 0 in\nFig. 3 there indeed exists a minimum due to the compe-\ntition between the density interaction and spin exchange.\nThe competition persists here for all partially polarized\ncases, leading to similar shallow low-lying areas, whose/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/s32/s32/s67/s101/s110/s116/s114/s97/s108/s32/s100/s101/s110/s115/s105/s116/s105/s101/s115/s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s78/s49/s47/s50\n/s47/s97 /s41\n/s80/s84/s70/s65/s32/s32 /s32/s110\n/s49/s40/s48/s41\n/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s50 /s110\n/s50/s40/s48/s41\n/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s110/s40/s48/s41\n/s68/s70/s84/s32 /s32/s32 /s110\n/s49/s40/s48/s41\n/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s50 /s110\n/s50/s40/s48/s41\n/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32 /s110/s40/s48/s41\nFIG. 8: (Color Online) Central densities n(0)\n1,n(0)\n2andn(0)\nas a function of Pfor 1D trapped spin-1 bosons in the strong\ninteraction case. The results of DFT for U= 50 denoted\nby lines are in agreement with those in TFA for U= +∞\ndenoted by symbols.\nminimum is seen to move toward the weaker interaction\ndirection with increasing Pas shown in the middle panel\nof Fig. 7. The total central density comes from the com-\nbination of these two terms n(0)=n(0)\n1+ 2n(0)\n2. The\nright panel shows that the minimum hollow for the total\ncentral density remains for all partially or fully polarized\ncases 0≤P <1 and the tail of n(0)in strong interaction\nlimitfirstlygoesupslightlyfollowedbyanearlylinearde-\ncreasing for increasing P. This non-monotonicity of the\ntotal central density in strong interaction case ( U= 50)\nis depicted in detail in Fig. 8 for increasing P, together\nwith the monotonic behavior of the central density for\npolarized bosons (down-ward)and that for paired bosons\n(up-ward). The results from DFT are compared with the\nanalytical curves from the TFA. We find that in the case\nofU= 50 the central densities for all polarized situation\narealreadyverycloseto the limiting value in TFA, which\nare obtained from eq. (48) as\nn(0)\n1=√\n2NP\nπa,\n2n(0)\n2=2/radicalbig\n2N(1−P)\nπa,\nn(0)=√\n2N(√\nP+2√\n1−P)\nπa.(50)\nWe maygofurther into the detailed quantum phasesof\nthe spinor gas by defining the radii of the vanishing den-\nsities for the two components. Fig. 9 shows the phase\ndiagram as a function of the global polarization Pfor\nU= 50, where the axial radii of the ensemble of the\npolarized and paired components are extracted from the\nnumerical result of the density profiles from the KSE.\nWithout loss of generality, we set the threshold values\nof the vanishing scaled density as 0 .02 in the numeri-\ncal simulation. The intersection of these two radii gives\nthe boundaries which divided the phase plane into three10\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s32/s32/s82/s97/s100/s105/s117/s115/s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32 /s78/s49/s47/s50\n/s97 /s41\n/s80/s84/s70/s65/s32/s32/s32 /s32/s82\n/s49\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s82\n/s50\n/s86\n/s70\n/s83\n/s77/s68/s70/s84/s32 /s32/s32/s82\n/s49\n/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32 /s82\n/s50\nFIG. 9: (Color Online) R-P phase diagram of 1D trapped\nspin-1 bosons in terms of the scaled density radii Ras the\nfunctions of total polarization P. Three quantum phases are\nidentified: spin-singlet-paired phase (S), ferromagnetic spin-\naligned phase (F) and mixed phase of the pairs and unpaired\nbosons (M). V stands for vacuum. The results of DFT for\nU= 50 (n1, 2n2<0.02 are considered as vanishing scaled\ndensities) and analytical results in TFA for U= +∞are\ncompared.\nquantum phases: spin-singlet-paired bosons S, ferromag-\nnetic spin-aligned bosons F, mixed phase of the pairs and\nunpaired bosons M, while V stands for the vacuum. At\nlow polarization a partially polarized region forms at the\ntrap center, the radius of which increases with increasing\npolarization. At a critical polarization Pc, the partially\npolarized region extends to the edge of the cloud. When\nthe polarization increases further, the edge of the cloud\nbecomes fully polarized. This process can be evidently\nseen in the lowest panels in Fig. 5. Together shown in\nFig. 9 are the theoretical results in TFA. According to\neqs. (48), the radii of the vanishing densities are calcu-\nlated as\nR1=√\n2NPa,\nR2=/radicalbigg\nN(1−P)\n2a (51)\nWhen the two radii equal to each other, we find the crit-\nical polarization is Pc= 0.2 and the critical radius is\nRc= 0.63. The DFT results for the radii are appar-\nently larger than the TFA estimation for both polarized\nand fully paired bosons, which can be understood easily\nfrom the extension of the tail of the density profile into\nouter region (see Fig. 6). Energetically this extension\nis due to the kinetic term neglected in TFA. The criti-\ncal polarization is in agreement with the TFA crossing at\nslightly higher polarization Pc∼0.23 and larger radius\nRc∼0.73.\nThe phenomena of phase separation at strong interac-\ntion is consistent with the results of Ref. [12]. There\nthey gave the ground state phase diagram according\nto the evolvement of Thomas-Fermi radii of n1andn2/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s45/s53/s48/s53/s49/s48/s49/s53\n/s32/s32/s69\n/s48/s47/s78 /s32/s40/s117/s110/s105/s116/s115/s32/s111/s102/s32/s32 /s41\n/s85/s32/s80/s61/s49\n/s32/s80/s61/s48/s46/s56\n/s32/s80/s61/s48/s46/s54\n/s32/s80/s61/s48/s46/s52\n/s32/s80/s61/s48/s46/s50\n/s32/s80/s61/s48\nFIG. 10: (Color Online) The ground state energy per atom\nof 1D trapped spin-1 bosons as functions of Ufor different P.\nThe lines are for P=0,0.2,0.4,0.6,0.8,1 from bottom to top.\nalong with Pby the means of TBA for strong interac-\ntion case. Our results, on the other hand, are valid for\nsystems in the whole interaction regime. Interestingly\nwe found the double-peak structure of the density of po-\nlarized bosons at intermediate interaction, which is elu-\nsive from the method in [12]. In a seminal experiment\non spin-imbalanced 1D two-component Fermi gas [13],\nphase separation of the two-spin mixture of ultracold6Li\natoms trapped in an array of 1D tubes is reported. The\npartially polarized core is surrounded by wings which are\ncomposed of either a paired or a polarized Fermi gas de-\npending on the polarization P. The pair mechanism in\nspinor gas is challenged by the density repulsive inter-\naction, which makes the phase separation more compli-\ncated.\nFinally, just as e(γ,p) shows the interaction depen-\ndence for various polarizations in the homogeneous case\nandE0in Eq. (44) gives the change with interaction in\nthe fully paired case, we illustrate in Fig. 10 the evo-\nlution of ground state energies E0per atom along with\ninteraction Uand total polarization Pin all partially\npolarized cases. The black solid line for P= 0 here re-\npeats the result of E0in Fig. 4. The fully polarized\nP= 1 system is equivalent to the 1D trapped repul-\nsive scalar bosons in [58] and E0increases monotonously\nand approach the ground state energy of non-interacting\nfermion system. The energy for partially polarized sys-\ntem with 0 < P < 1 interpolates between these two\nextremes, i.e., the positive energy terms including the\nkinetic, potential and density-density repulsive interac-\ntions together compete with the negative binding energy\nin the anti-ferromagnetically paired bosons, giving rise\nto the non-monotonically dependence of the energy on\nthe interaction parameter U. In the weak interaction\ncase, the repulsive interaction is prominent so that E0in-\ncreases along with U. In the strong interaction case, E0\ndecreases because the repulsive energy slowly increases\nto a constant whereas the binding energy goes downward11\nparabolically. LargerpolarizationdestroystheBose-Bose\npairs one by one, which diminishes the effect of pairing\nbinding energy and finally leads to the monotonic behav-\nior ofE0forP= 1.\nIV. CONCLUSION\nIn conclusion, using DFT we study the density distri-\nbution and energy of the 1D harmonically trapped spin-\n1 bosons in the ground state. We numerically solve the\nKSEs based on LDA and the solution of Bethe ansatz.\nThe results show that the competition between the re-\npulsive density-density interaction and antiferromagnetic\nspin-exchange interaction results in complicated density\ndistributions and energy evolutions along with the inter-\naction parameter. We found a non-monotonic behavior\ninthecentraldensitiesofbothspin-singletpairedandpo-\nlarized bosons. Some polarized bosons are repelled out\nof the trap center in the intermediate interaction region,\nshowing the double-peak structure of density profiles.\nThe total density exhibits a bi-modal distribution with\npaired bosons imposed on the top of polarized bosons.\nThe phenomena of phase separation occurs for stronginteraction with the partially polarized core surrounded\nby wings which are composed of either paired bosons or\npolarized bosons depending on the polarization P. We\ngive the R-P phase diagram at strong interaction and\nfind that the critical polarization Pcin DFT is slightly\nlarger than the TFA result. Although we treat with an\nintegrable model with equal repulsive density interaction\nand antiferromagnetic spin-exchange, the results do shed\nsome light on the relativistic spinor gases. We speculate\nthat the new quantum phases investigated here could be\nprobed in experiment by in situimaging, analogously to\nthe 1D trapped Fermi gas [13].\nAcknowledgments\nThis work is supported by the NSF of China under\nGrant Nos. 11234008, 11104171 and 11074153, the Na-\ntional Basic Research Program of China (973 Program)\nunder Grant Nos. 2010CB923103, 2011CB921601, and\nthe Program for New Century Excellent Talents in Uni-\nversity (NCET). We thank Gao Xianlong, Junpeng Cao\nand Xiwen Guan for helpful discussions.\n[1] Y. Kawaguchi and M. Ueda, Phys. Rep. 520, 253 (2012).\n[2] D. M. Stamper-Kurn and M. Ueda, arXiv:1205.1888.\n[3] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and\nM. Rigol, Rev. Mod. Phys. 83, 1405 (2011).\n[4] X.-W. Guan, M. T. Batchelor and C. Lee,\narXiv:1301.6446.\n[5] W. Zhang and L. You, Phys. Rev. A 71, 025603 (2005).\n[6] Y. Hao, Y. Zhang, J. Q. Liang, and S. Chen, Phys. Rev.\nA73, 053605 (2006).\n[7] J. Cao, Y. Jiang, and Y. Wang, Europhys. 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Otherwise it\ncan be obtained using a Laplace transform method. Since the L aplace transform of the microcanon-\nical density of states is the canonical partition function, it factorizes for a system of noninteracting\nparticles and the solution of the problem is straightforwar d. The results are illustrated on several\nclassical examples.\nPACS numbers: 05.20.-y, 05.20.Gg, 01.40.Fk\nI. INTRODUCTION\nIn statistical mechanics, an isolated macroscopic sys-\ntem at equilibrium is characterized by its microcanonical\nentropy, given by the Boltzmann formula\nS=kBlnΩ, (1)\nwherekBis the Boltzmann constant and Ω the number\nof accessible microstates in phase space, i.e., the number\nof equiprobable microstates compatible with the fixed\nvalues of the macroscopic external parameters such as\nthe volume V, the number of particles N, the energy E,\netc. For a classical system in Ddimensions, the phase\nspace has 2 sdimensions corresponding to the compo-\nnents of the generalized momenta pand coordinates q\nsuch that s=DN. Each microstate corresponds to a 2 s-\ndimensional cell in phase space with a volume hs, where\nhis the Planck constant. The accessible volume is such\nthat\nE < H(p,q)< E+δE, (2)\nwhere the arbitrary uncertainty δE≪Eaffecting the\nvalue of the total energy is introduced in order to obtain\na 2sdimensional accessible volume, thus leading to a\nnonvanishing value of Ω.\nThe number of accessible microstates is proportional\ntoδEand can be written as\nΩ =nN(E)δE, (3)\nwhere the coefficient of proportionality nN(E) is the mi-\ncrocanonical density of states which plays a central role\nin the statistical physics of isolated systems.\nThe calculation of the density of states amounts to de-\ntermine the volume enclosed by the hypersurface of con-\nstant energy E=H(p,q) in the 2 sdimensional phase\n†Laboratoire associ´ e au CNRS UMR 7198.space, a quite difficult task in general. In textbooks the\nproperties of the microcanonical ensemble are often illus-\ntrated by the study of some classical systems involving\nnoninteractingparticlesforwhichtheHamiltonianissep-\narableand the constant energysurface simple enough. In\nthe case of the ideal gas one needs the area of a hyper-\nsphere and for a collection of harmonic oscillators the\narea of a hyperellipsoid which can be transformed to a\nhypersphere through rescaling. The area of the hyper-\nsphere is usually obtained by comparing the values, cal-\nculated either in Cartesian or in spherical coordinates, of\nthe integral over the 2 s-dimensional space of exp( −r2),\nwhereris a radius vector (see, e.g., Refs 1–5). Other\nindependent particle problems have also been treated6,7\nby making use of Dirichlet’s integral formula.8\nThe purpose of the present paper is to propose an-\nother approach to noninteracting particle systems, based\non a recursion relation for the microcanonical density of\nstates. Assumingthatthesingle-particledensityofstates\nis known, the particles are added one by one. Summing\ntheir contribution to the total density of states, a recur-\nsive integral equation is obtained which can be solved\nusing a Laplace transform method. When the single-\nparticle Hamiltonian is itself separable, involving a sum\nof contributions from each of the Ddimensions, the same\nrecipe can be used to evaluate the single-particle density\nof states in Ddimensions.\nThe recursive approach to the calculation of the N-\nparticle microcanonical density of states is presented in\nSec. II. Some illustrative examples are discussed in Sec.\nIII. Summary and conclusion are given in Sec. IV.\nII. MICROCANONICAL DENSITY OF STATES\nWe consider an isolated classical D-dimensional sys-\ntem containing Nnoninteracting distinguishable parti-\ncles. Let Np(E) denote the p-particle integrated den-\nsity of states, i.e., the number of microstates with en-\nergy smaller than E≥0 in a system with pparti-\ncles. The p-particle density of states, np(E), is such that2\ndNp(E) =np(E)dE.\nThe number ofmicrostates with energysmaller than E\nin a system with p+1 particles is obtained by first com-\nbining the microstates with energy smaller than E−ǫin\na system with pparticles with the microstates on the en-\nergy interval [ ǫ,ǫ+dǫ] associated with the added particle\nand then integrating over ǫ. Doing so, one obtains\nNp+1(E) =/integraldisplayE\n0Np(E−ǫ)n1(ǫ)dǫ. (4)\nTaking the derivative of both sides and noticing that\nNp(0) = 0, leads to a recursion relation for the p-particle\ndensity of states\nnp+1(E) =dNp+1(E)\ndE=/integraldisplayE\n0np(E−ǫ)n1(ǫ)dǫ,(5)\nTheexpressionontherightisaconvolutionproductthus,\ntaking the Laplace transform of both sides, one obtains\nzp+1(s) =/integraldisplay∞\n0e−sEnp+1(E)dE\n=/integraldisplay∞\n0e−sEdE/integraldisplayE\n0np(E−ǫ)n1(ǫ)dǫ\n=/integraldisplay∞\n0n1(ǫ)e−sǫdǫ/integraldisplay∞\nǫe−s(E−ǫ)np(E−ǫ)dE\n=zp(s)z1(s). (6)\nIterating this result, the Laplace transform of the N-\nparticle density of states is given by\nzN(s) = [z1(s)]N, (7)\nwhen the particles are identical. Otherwise the power is\nreplaced by a product over the different contributions.\nTheN-particle density of states is obtained by taking\nthe inverse Laplace transform\nnN(E) =1\n2πi/integraldisplayγ+i∞\nγ−i∞esEzN(s)ds, (8)\nwhereγis a vertical contour in the complex plane chosen\nso that all singularities of zN(s) are on the left of it.\nOne may notice that with 1 /s=kBT,zN(s) is the\nN-particle canonical partition function and Eq. (7) sim-\nply states that fornoninteractingparticles the N-particle\ncanonicalpartitionfunction factorizesandis givenby the\nNth powerofthe single-particlecanonicalpartition func-\ntionz1(s).\nLet us assume that the single-particle density of states\nvaries as a power of the energy\nn1(E) =C1Eκ1−1, E≥0. (9)\nThe Laplace transform is then given by\nz1(s) =C1/integraldisplay∞\n0e−sEEκ1−1dE\n=C1\nsκ1/integraldisplay∞\n0e−ttκ1−1dt=C1Γ(κ1)\nsκ1,(10)where Γ( x) is the Euler Gamma function. For the N-\nparticle system we have\nzN(s) =[C1Γ(κ1)]N\nsNκ1. (11)\nThe inverse Laplace transform can be deduced from\nEq. (10) and leads to the N-particle density of states\nnN(E) = [C1Γ(κ1)]NENκ1−1\nΓ(Nκ1). (12)\nIn this simple case one may also proceed as follows:\nUsing Eq. (9) and the power-law expression of the p-\nparticle density of states\nnp(E) =CpEκp−1. (13)\nin the integral equation (5) leads to\nnp+1(E) =C1Cp/integraldisplayE\n0ǫκ1−1(E−ǫ)κp−1dǫ\n=C1CpEκ1+κp−1/integraldisplay1\n0tκ1−1(1−t)κp−1dt\n=C1CpΓ(κ1)Γ(κp)\nΓ(κ1+κp)Eκ1+κp−1. (14)\nTheintegralin thesecondline isthe EulerBetafunction9\nwhich is written in terms of Gamma functions in the last\nline. Comparing with the form of np+1(E) resulting from\nEq. (13) we deduce the following recursion relations for\nthe amplitude and the exponent:\nCp+1=C1CpΓ(κ1)Γ(κp)\nΓ(κ1+κp), κ p+1=κ1+κp.(15)\nIterating these relations, one easily obtains\nCp=[C1Γ(κ1)]p\nΓ(pκ1), κ p=pκ1,(16)\nand the form of the N-particle density of states given in\nEq. (12) is recovered.\nFor a mixture with Naparticles of type aandNbpar-\nticles of type b, we have a straightforward generalization\nof Eq. (12), namely\nnNa,Nb(E) =/integraldisplayE\n0nNa(ǫ)nNb(E−ǫ)dǫ\n= [CaΓ(κa)]Na[CbΓ(κb)]NbENaκa+Nbκb−1\nΓ(Naκa+Nbκb).(17)\nIn order to calculate the microcanonical density of\nstates, one needs the form of the single-particle density\nof states.\nWhen the single-particle Hamiltonian is the sum of\nequivalent (kinetic and potential) contributions coming\nfrom each of the Ddimensions of the system, the num-\nber of dimensions plays the same role as the number of3\nparticles did previously and a recursion relation similar\nto Eq. (5) can be written for the single-particle density\nof states in q+1 dimensions\nn(q+1)\n1(E) =/integraldisplayE\n0n(q)\n1(E−ǫ)n(1)\n1(ǫ)dǫ(18)\nwhere the upper index now refers to the dimension of\nthe system while the lower one refers to the number of\nparticles.\nAs before the convolution product leads to a form sim-\nilar to (7)\nz(D)\n1(s) = [z(1)\n1(s)]D, (19)\nfor the Laplace transform of the single-particle density of\nstates in Ddimensions.\nWhen the single-particle density of states in one di-\nmension behaves as\nn(1)\n1(E) =C(1)\n1Eκ(1)\n1−1, E≥0,(20)\nthe single-particledensityofstatesin Ddimensionsreads\nn(D)\n1(E) =/bracketleftBig\nC(1)\n1Γ(κ(1)\n1)/bracketrightBigDEDκ(1)\n1−1\nΓ/parenleftBig\nDκ(1)\n1/parenrightBig.(21)\nin analogy with Eq. (12).\nWhen the system is anisotropic with Didimensions\n(i=a,b) for which the one-dimensional single-particle\ndensityofstatesis n(i)\n1(E)withLaplacetransform z(i)\n1(s),\nEq. (19) has to be replaced by\nz(Da,Db)\n1(s) = [z(a)\n1(s)]Da[z(b)\n1(s)]Db.(22)\nIII. SOME EXAMPLES\nA. Boltzmann Ideal gas\nWe consider an ideal gas of Npoint particles of mass\nm, confined inside a cubic box of volume VD=LD. The\nsingle-particle Hamiltonian is\nH(p) =p2\n2m. (23)\nIn the two-dimensional phase space of the single-particle\nproblem there are two segments of constant energy, E,\nsuch that p(E) =±√\n2mEand 0< q < L . Hence, as\nshown in Fig. 1, the microstates with energy lower than\nEare located inside a rectangle with sides 2√\n2mEand\nL. Since each microstates corresponds to an area h, the\nPlanck constant, we obtain\nN(1)\n1(E) =2L\nh√\n2mE, n(1)\n1(E) =L\nh/radicalbigg\n2m\nE,(24)Lp\nq(2mE)1/2\n−(2mE)1/20\nFIG. 1: Rectangular domain of accessible microstates with\nenergy lower than Ein the phase space of a free particle with\nmassm, confined on a segment with length L.\nfor the one-dimensional single-particle densities of states.\nThusC(1)\n1=√\n2mL/h,κ(1)\n1= 1/2 and Eq. (21) imme-\ndiately gives the single-particle density of states in D\ndimensions:\nn(D)\n1(E) =VD/parenleftbigg2πm\nh2/parenrightbiggD/2ED/2−1\nΓ(D/2).(25)\nThe microcanonical density of states of the ideal gas\nwithNparticles of mass min a volume VDfollows from\nEq. (12) with κ1=D/2,C1Γ(κ1) =VD(2πm/h2)D/2,\naccording to Eq. (25), so that\nn(D)\nN(E) =VN\nD/parenleftbigg2πm\nh2/parenrightbiggND/2END/2−1\nΓ(ND/2)(26)\nand the integrated density of states is given by\nN(D)\nN(E) =VN\nD/parenleftbigg2πm\nh2/parenrightbiggND/2END/2\nΓ(ND/2+1),(27)\nin agreement with the result obtained by standard\nmethods.1,2,5For undistinguishable particles the densi-\nties in Eqs (26) and (27) must be divided by N!, the\nnumber of permutations between the Nparticles.\nWith an idealgasconsistingof Naparticlesofmass ma\nandNbparticles of mass mb, such that Na+Nb=N,\nEq. (17) leads to\nn(D)\nNa,Nb(E) =VN\nD/parenleftbigg2π\nh2/parenrightbiggND/2/parenleftBig\nmNa\namNb\nb/parenrightBigD/2END/2−1\nΓ(ND/2)\n(28)\nfor the microcanonical density of states of the ideal gas\nmixture in Ddimensions. The integrated density of4\nstates is given by\nN(D)\nNa,Nb(E)=VN\nD/parenleftbigg2π\nh2/parenrightbiggND/2/parenleftBig\nmNa\namNb\nb/parenrightBigD/2END/2\nΓ(ND/2+1),\n(29)\nin agreement with Ref. 6. A division of the densities by\nNa!Nb! is needed when the two species consist of undis-\ntinguishable particles.\nB. Classical harmonic oscillators\np\nq−βα\n−αβ\nFIG. 2: Elliptic Domain of accessible microstates with ener gy\nlower than Ein the phase space of a one-dimensional har-\nmonic oscillator with mass mand angular frequency ω. The\nsemiaxes are α=√\n2mEandβ=/radicalbig\n2E/m/ω.\nWe consider a collection of Nnoninteracting classical\nharmonicoscillatorsin Ddimensionswhich forsimplicity\nare assumed to be isotropic.\nFor a single oscillator in one dimension the Hamilto-\nnian reads\nH(p,q) =p2\n2m+mω2\n2q2. (30)\nThe line of constant energy Ein the two-dimensional\nphase space is an ellipse with semiaxes α=√\n2mEand\nβ=/radicalbig\n2E/m/ωenclosing an area παβ= 2πE/ωshown\nin Fig. 2. Each microstate corresponding to an area h,\nwe obtain the integrated denstity of states as\nN(1)\n1(E) =2π\nhωE, n(1)\n1(E) =2π\nhω.(31)\nThusC(1)\n1= 2π/hωandκ(1)\n1= 1. According to Eq. (21)\nthe density of states for one oscillator in Ddimensions is\nn(D)\n1(E) =/parenleftbigg2π\nhω/parenrightbiggDED−1\n(D−1)!(32)so thatκ1=D,C1Γ(κ1) = (2π/hω)D, and, according\nto Eq. (12), the microcanonical density of states for N\noscillators in Ddimensions reads\nn(D)\nN(E) =/parenleftbigg2π\nhω/parenrightbiggNDEND−1\n(ND−1)!(33)\nand one obtains\nN(D)\nN(E) =/parenleftbigg2π\nhω/parenrightbiggNDEND\n(ND)!(34)\nfor the integrated density of states in agreement with the\nknown result.6,10,11The generalization to a collection of\nanisotropic oscillators is immediate: It amounts to re-\nplaceωDby/producttextD\ni=1ωi.\nC. Ideal gas in a gravitational field\nWe consider a classical ideal gas with Npoint parti-\ncles of mass minDdimensions. The gravitational field\nwith acceleration gacts downwards along the vertical di-\nrection. The gas is confined inside a vessel with height l\nin the vertical direction and section SD=LD−1in the\ntransverse directions. This is an example for which the\nsingle-particle density of states is nota simple power of\nEso that we shall use the Laplace transform method.\nE/mgp\nq(2mE)1/2\n−(2mE)1/20\nFIG. 3: Parabolic domain of accessible microstates with en-\nergy lower than Ein the phase space of a particle with mass\nm, moving in the one-dimensional half-space q >0, in a gravi-\ntational field with uniform acceleration gdirected downwards.\nWhen the motion is restricted to 0 < q < l, the domain of\naccessible microstates is restricted to the part of the shad ed\nregion with q < l < E/mg whenE > mgl .\nLet us calculate the single-particle density of states\nstarting from Eq. (22) since the system is anistropic.\nFor each of the Da=D−1 transverse dimensions,\nthe single-particle density of states is given by Eq. (24)5\nso thatC(a)\n1=√\n2mL/h,κ(a)\n1= 1/2 and according to\nEq. (10)\nz(a)\n1(s) =L√\n2m\nhΓ(1/2)\ns1/2=L√\n2πm\nhs1/2.(35)\nIn the remaining dimension ( Db= 1) we have to cal-\nculate the one-dimensional density of states for a particle\nin a gravitational field. The Hamiltonian reads\nH(p,q) =p2\n2m+mgq, 0< q < l. (36)\nThe line of constant energy in phase space is the seg-\nment of parabola with equation p=±√\n2m√E−mgq\nin the region 0 < q < l (see Fig. 3). Thus the number\nof microstates with energy lower than Eis obtained by\ndividing by hthe surface between the parabola and the p\naxis (restricted to the region where q < lwhenE > mgl )so that\nN(b)\n1(E) =2√\n2m\nh/integraldisplaymin(E\nmg,l)\n0/radicalbig\nE−mgqdq\n=2√\n2m\nhmg/integraldisplayE\nmax(E−mgl,0)√\ntdt\n=4√\n2m\n3hmg/bracketleftBig\nE3/2−(E−mgl)3/2Θ(E−mgl)/bracketrightBig\n,(37)\nwhere Θ( x) is the Heaviside step function such that\nΘ(x) = 0 when x <0 and Θ( x) = 1 when x≥0. The\nderivative with respect to Eleads to\nn(b)\n1(E) =2√\n2m\nhmg/bracketleftBig\nE1/2−(E−mgl)1/2Θ(E−mgl)/bracketrightBig\n.(38)\nTaking the Laplace transform of this density of states, one obtain s\nz(b)\n1(s) =2√\n2m\nhmg/bracketleftbigg/integraldisplay∞\n0E1/2e−sEdE−e−smgl/integraldisplay∞\nmgl(E−mgl)1/2e−s(E−mgl)dE/bracketrightbigg\n=2√\n2m\nhmgΓ(3/2)\ns3/2/parenleftbig\n1−e−smgl/parenrightbig\n=√\n2πm\nhmg/parenleftbig\n1−e−smgl/parenrightbig\ns3/2. (39)\nThe Laplace transform of the N-particle density of states in Ddimensions then follows from Eqs (7) and (22) together\nwith Eqs (35) and (39)\nz(D−1,1)\nN(s) = [z(a)\n1(s)]N(D−1)[z(b)\n1(s)]N=/parenleftbiggSD\nmg/parenrightbiggN/parenleftbigg2πm\nh2/parenrightbiggND/2/parenleftbig\n1−e−smgl/parenrightbigN\nsN(D+2)/2\n=/parenleftbiggSD\nmg/parenrightbiggN/parenleftbigg2πm\nh2/parenrightbiggND/2N/summationdisplay\nk=0(−1)k/parenleftbiggN\nk/parenrightbigge−skmgl\nsN(D+2)/2. (40)\nThe inverse Laplace transform is obtained using the transformatio n formula\ne−sa\nsµ←→(E−a)µ−1\nΓ(µ)Θ(E−a),(µ >0). (41)\nfor each term in the sum. It leads to the following expression for the density of states\nn(D−1,1)\nN(E)=/parenleftbiggSD\nmg/parenrightbiggN/parenleftbigg2πm\nh2/parenrightbiggND/2N/summationdisplay\nk=0(−1)k/parenleftbiggN\nk/parenrightbigg(E−kmgl)N(D+2)/2−1\nΓ[N(D+2)/2]Θ(E−kmgl), (42)\nand the integrated density of states is given by\nN(D−1,1)\nN(E)=/parenleftbiggSD\nmg/parenrightbiggN/parenleftbigg2πm\nh2/parenrightbiggND/2N/summationdisplay\nk=0(−1)k/parenleftbiggN\nk/parenrightbigg(E−kmgl)N(D+2)/2\nΓ[N(D+2)/2+1]Θ(E−kmgl), (43)\nin agreement with the result of Ref. 12.\nWhenl→∞, only the first term contributes to the\nsum and one recovers the result of Ref. 7. This resultcan be obtained more directly since the deviation of the\nsingle-particle density of states from a simple power law6\ncoming from the last term in Eq. (38) disappears in this\nlimit.\nHere too, one has to divide the densities by N! when\nundistinguishable particles are considered.\nIV. SUMMARY AND CONCLUSION\nWe have shown how, using the integral equation (5),\none can construct the N-particle density of states itera-\ntively, starting from the single-particle density of states\nand, in the same way, using the integral equation (18),\nhow to deduce the single-particle density of states in D\ndimensions from the single-particle density of states in\none dimension for a microcanonical system of noninter-\nacting particles with a separable Hamiltonian.\nThe solutions of the integral equations are easily ob-\ntained when the single-particle density of states in onedimension follows a power law. This has been illustrated\non two classical examples: The Boltzmann gas and a col-\nlection of classical harmonic oscillators. In other cases,\nthe density of states may be obtained using a Laplace\ntransform method. Since the Laplace transform of the\nmicrocanonical density of states is given by the canonical\npartition function of the system, it factorizes for nonin-\nteracting particles. As a consequence it is easily deduced\nfrom the form of the single-particle densities of states for\neach dimension as shown for an ideal gas in a vessel of\nfinite height under the influence of the gravitationalfield.\nAcknowledgments\nIt is a pleasure to thank Dragi Karevski for helpful\ndiscussions.\n∗Electronic address: Loic.Turban@univ-lorraine.fr\n1R. Kubo, Statistical Mechanics , 2nd printing (North-\nHolland Publishing Company, Amsterdam, 1967), pp. 35–\n36.\n2K. Huang, Statistical Mechanics , 2nd edition (John Wiley\n& Sons, Inc., New York, 1987), pp. 138–140.\n3B. Diu, C. Guthmann, D. Lederer, and B. Roulet,\n´El´ ements de Physique Statistique (Hermann, Paris, 1989),\npp. 82–84.\n4C. Garrod, Statistical Mechanics and Thermodynamics\n(Oxford U. P., New York, 1995), pp. 47–48.\n5G. F. Mazenko, Equilibrium Statistical Mechanics (John\nWiley & Sons, Inc., New York, 2000), pp. 33–36.\n6C. Fernandez-Pineda, J. I. Mengual, and A. Diez de los\nRios, “Dirichlet’s integral formula and the evaluation of\nthe phase volume,” Am. J. Phys. 47, 814–817 (1979).\n7F. L. Rom´ an, J. A.White, andS. Velasco, “Microcanonicalsingle-particle distributions for an ideal gas in a gravita -\ntional field,” Eur. J. Phys. 16, 83–90 (1995).\n8E. T. Whittaker and G. N. Watson, A Course of Modern\nAnalysis, 4th edition (Cambridge U. P., London, 1927), p.\n258.\n9M. Abramowitz and I. A. Stegun, Handbook of Mathemat-\nical Functions with Formulas, Graphs and Mathematical\nTables, 9th printing (Dover, New York, 1972), p. 258.\n10L. A. Beauregard, “N oscillators in the microcanonical en-\nsemble,” Am. J. Phys. 33, 745 ( 1965).\n11R. Kubo, Statistical Mechanics , 2nd printing (North-\nHolland Publishing Company, Amsterdam, 1967), pp. 78–\n79.\n12F. L. Rom´ an, A. Gonz´ alez, J. A. White, and S. Velasco,\n“Microcanonical ensemble study of a gas column under\ngravity,” Z. Phys. B 104, 353–361 (1997)."    },    {        "title": "1304.7405v1.Nuclear_state_densities_of_odd_mass_heavy_nuclei_in_the_shell_model_Monte_Carlo_approach.pdf",        "content": "arXiv:1304.7405v1  [nucl-th]  27 Apr 2013Nuclear state densitiesof odd-massheavynucleiin the shel lmodelMonte Carlo\napproach\nC.¨Ozena,b,,Y. Alhassidb, H.Nakadac\naFaculty ofEngineering and Natural Sciences, Kadir Has Univ ersity, Istanbul 34083, Turkey\nbCenter for Theoretical Physics, Sloane Physics Laboratory , Yale University,\nNew Haven, CT 06520, USA\ncDepartment ofPhysics, Graduate School ofScience, Chiba Un iversity, Inage, Chiba, 263-8522, Japan\nAbstract\nWhile the shell model Monte Carlo approach has been successf ul in the microscopic calculation of nuclear state\ndensities, it has been di fficult to calculate accurately state densities of odd-even he avy nuclei. This is because the\nprojection on an odd number of particles in the shell model Mo nte Carlo method leads to a sign problem at low\ntemperatures, making it impractical to extract the ground- state energy in direct Monte Carlo calculations. We show\nthat the ground-state energy can be extracted to a good preci sion by using level counting data at low excitation\nenergiesand the neutronresonancedata at the neutronthres holdenergy. Thisallows usto extendrecent applications\nof the shell-model Monte Carlo method in even-even rare-ear th nuclei to the odd-even isotopic chains of149−155Sm\nand143−149Nd. We calculate the state densities of the odd-even samariu m and neodymium isotopes and find close\nagreementwith thestate densitiesextractedfromexperime ntaldata.\nKeywords: nuclearshell model,quantumMonteCarlo,leveldensity,ba ck-shiftedBetheformula\n1. Introduction\nReliablemicroscopiccalculationofnuclearstateden-\nsities of heavy nuclei is a challenging task because it\noften requires the inclusion of correlations beyond the\nmean-field approximation. Correlation e ffects can be\ntaken into account in the context of the configuration-\ninteraction(CI)shellmodelapproach,butthesizeofthe\nrequired model space in heavy nuclei is prohibitively\nlarge for direct diagonalization of the CI shell model\nHamiltonian. Thislimitationcanbeovercomeinpartby\nusingtheshellmodelMonteCarlo(SMMC)method[1,\n2, 3, 4, 5]. TheSMMCmethodenablesthecalculations\nofstatisticalnuclearproperties,andinparticularoflev el\ndensitiesinverylargemodelspaces[6, 7, 8, 9,10].\nFermionic Monte Carlo methodsare often hampered\nby the so-called sign problem,which leads to large and\nuncontrollable fluctuations of observables at low tem-\nperatures [1, 2, 11, 12]. Statistical and collective prop-\nertiesofnucleicanbereliablycalculated[13,6]byem-\nploying a class of interactions that have a good Monte\nCarlo sign in the grand-canonical formulation [2, 13].\nThe projection on an even number of particles keeps\nthe good sign of the interaction, allowing accurate cal-culations for even-even nuclei. However, the projec-\ntion on an odd number of particles leads to a new sign\nproblem,makingitdi fficulttocalculatethermalobserv-\nablesatlowtemperaturesforodd-evenandodd-oddnu-\nclei. In particular, the ground-state energy of the odd-\nparticle system cannot be extracted from direct SMMC\ncalculations. A method to extract the ground-state en-\nergyoftheodd-particlesystemfromtheimaginary-time\nGreen’s functions of the even-particle system was re-\ncently introduced in Ref. [14] and applied successfully\nto medium-mass nuclei. However, the application of\nthis method to heavy nuclei is computationally inten-\nsive and requiresadditionaldevelopment. In this work,\nwe describe a practical method that enables us to de-\ntermine the ground-state energy using a complete set\nof experimentally known low-lying levels and the neu-\ntronresonancedataat theneutronthresholdenergy. We\nthenextendtherecentSMMCstatedensitycalculations\nof even-even nuclei in the rare-earth region [15, 16]\nto include odd-even isotope chains of149−155Sm and\n143−149Nd. We find close agreement with the state den-\nsitiesthat areextractedfromexperimentaldata.\nPreprintsubmitted to Physics Letters B August13,20182. Choice ofModel Space andInteraction\nIn rare-earth nuclei we use the model space of\nRefs. [15, 16] spanned by the single-particle orbitals\n0g7/2, 1d5/2, 1d3/2, 2s1/2, 0h11/2and 1f7/2for protons,\nand the orbitals 0 h11/2, 0h9/2, 1f7/2, 1f5/2, 2p3/2, 2p1/2,\n0i13/2, and 1g9/2for neutrons. We have chosen these\norbitalsby the requirementthat their occupationproba-\nbilitiesinwell-deformednucleibebetween0.9and0.1.\nTheeffectofotherorbitalsisaccountedforbytherenor-\nmalization of the interaction. The bare single-particle\nenergiesintheshell-modelHamiltonianareobtainedso\nas to reproduce the single-particle energies in a spheri-\ncal Woods-Saxon plus spin-orbitpotential in the spher-\nicalHartree-Fockapproximation.\nTheeffectiveinteractionconsistsofmonopolepairing\nandmultipole-multipoleterms[15]\n−/summationdisplay\nν=p,ngνP†\nνPν−/summationdisplay\nλχλ: (Oλ;p+Oλ;n)·(Oλ;p+Oλ;n):,(1)\nwhere the pair creation operator P†\nνand the multipole\noperatorOλ;νaregivenby\nP†\nν=/summationdisplay\nnljm(−)j+m+la†\nαjm;νa†\nαj−m;ν,(2a)\nOλ;ν=1√\n2λ+1/summationdisplay\nab/angbracketleftja||dVWS\ndrYλ||jb/angbracketright[a†\nαja;νטaαjb;ν](λ)\n(2b)\nwith ˜ajm=(−)j+maj−m. In Eq. (1), :: denotes normal\nordering and VWSrepresents the Woods-Saxon poten-\ntial. The pairing strengths is expressed as gν=γ¯gν,\nwhereγis a renormalization factor and ¯ gp=10.9/Z,\n¯gn=10.9/Nare parametrized to reproduce the exper-\nimental odd-even mass di fferences for nearby spher-\nical nuclei in the number-projected BCS approxima-\ntion [15]. The quadrupole, octupole and hexadecupole\ninteraction terms have strengths given by χλ=kλχfor\nλ=2,3,4 respectively. The paprameter χis deter-\nmined self-consistently[13] and kλare renormalization\nfactorsaccountingfor core polarizatione ffects. We use\ntheparametrizationofRef.[16]for γandk2\nγ=0.72−0.5\n(N−90)2+5.3, (3a)\nk2=2.15+0.0025(N−87)2,(3b)\nwhile the otherparametersare fixedat k3=1 andk4=\n1.3. Ground-stateEnergy\nBecause of the sign problem introduced by the pro-\njection on an odd number of particles, the thermal en-\nergyE(β) of the odd-even samarium and neodymium\nisotopes can in practice be calculated only up to β≡\n1/T∼4−5 MeV−1. For comparison, SMMC cal-\nculations in the neighboring even-even samarium and\nneodymiumnuclei, for which there is no sign problem,\nwere carried out up to β=20 MeV−1[16]. System-\natic errors introduced by the discretization of β[2], are\ncorrected by calculating E(β) for the two time slices of\n∆β=1/32MeV−1and∆β=1/64MeV−1andthenper-\nforming a linear extrapolation to ∆β=0. Forβ/greaterorapproxeql3\nMeV−1, the dependence of E(β) on∆βis weaker and\nan average value was taken instead. The calculations\nforβ>2 MeV−1were carried out using a stabilization\nmethodoftheone-bodycanonicalpropagator[15]fora\ngivenconfigurationoftheauxiliaryfields.\nSinceourcalculationsofthethermalenergy E(β)are\nlimited toβ∼4−5 MeV−1, we cannot obtained a re-\nliable estimate of the ground-state energy E0in direct\nSMMC calculations. The Green’s function method of\nRef.[14]wasusedsuccessfullyin medium-massnuclei\ntocircumventtheoddparticle-numbersignproblemand\nextract accurate ground-state energies. However, this\nmethodbecomescomputationallyintensiveinheavynu-\nclei and requires additional development. Here we ex-\ntractE0byperformingaone-parameterfitoftheSMMC\nthermalexcitationenergy Ex(T)=E(T)−E0totheex-\nperimentalthermalenergy. Thelatteriscalculatedusing\nthe thermodynamical relation Ex(β)=−dlnZ(β)/dβ,\nwhereZis the experimentalpartitionfunctionin which\nthe energy is measured relative to the ground state (see\nbelow).\nWe demonstrate the method for147Nd. Fig. 1 shows\nthe thermal excitation energy (left panel) and the par-\ntition function (right panel) as a function of temper-\nature. The SMMC results (open circles) are shown\ndown toT=0.25 MeV, below which the statistical er-\nrors become too large due to the odd particle-number\nsign problem. We calculated the experimental parti-\ntion function (right panel, dashed line) from Z(T)=/summationtext\ni(2Ji+1)e−Ex,i/TwhereEx,idenote the excitation en-\nergies of the experimentally known energy levels. The\nincompleteness of the level counting data above a cer-\ntain excitation energy leads to a saturation of the ex-\nperimentalpartitionfunctionandthe correspondingav-\nerage experimental thermal energy (left panel, dashed\nline) above T∼0.15 MeV. In order to determine the\nground-state energy E0from a fit of the SMMC ther-\nmal energy to the experimental data, it is necessary to\n20 0.5 1\nT [MeV]1103106Z(T)\n0 0.25 0.5\nT [MeV]01.53Ex(T)147Nd147Nd\nFigure1: Thermalexcitation energy Ex(leftpanel)andpartition func-\ntionZ(right panel) versus temperature Tfor147Nd. The SMMC re-\nsults (open circles) are compared with the results deduced d irectly\nfrom experimentally known levels (dashed lines), and from E q. (4)\n(solid lines). The ground-state energy E0is obtained by a one-\nparameter fit of the SMMC thermal energy to the solid line on th e\nleft panel.\nhavearealisticestimateoftheexperimentalthermalen-\nergy above T∼0.25 MeV where SMMC results are\navailable. This can be accomplished by calculating an\nempiricalpartitionfunction\nZ(T)=N/summationdisplay\ni(2Ji+1)e−Ex,i/T+/integraldisplay∞\nENdExρBBF(Ex)e−E/T,\n(4)\nwherethediscretesumiscarriedoutoverasuitablycho-\nsen complete set of Nexperimental levels up to an ex-\ncitationenergy EN,andthecontributionofhigher-lying\nlevels is included e ffectively through the integral over\nan empirical level density with a Boltzmann weight.\nTheempiricalstatedensityisparametrizedbytheback-\nshiftedBethe formula(BBF) [17]\nρBBF(E)=√π\n12a−1/4(E−∆)−5/4e2√a(E−∆),(5)\nwhereais the single-particle level density parameter\nand∆is the backshift parameter. For these parameters\nwe use the values in Ref. [18] determined from level\ncounting data (at low excitation energies) and the s-\nwave neutron resonance data (at the neutron resonance\nthreshold) for the given nucleus of interest. The solid\nlines in Fig. 1 are calculated using Eq. (4) for the em-\npirical partition function. The number of the low-lying\nstatesN(determining EN)arechosensuchthatthesolid\nand dashed curves for the experimental partition func-\ntion merge smoothly at su fficiently low temperatures.\nThe values of aand∆and the suitably chosen values\nofNare tabulated in Table 1 for the nuclei of interest.\nThe ground-stateenergy E0is determinedby fitting the\nSMMC thermal excitation energy (open circles) to the\nsolidcurveintheleftpanelofFig.1. Table1alsoshows\nthefittedvaluesof E0forthecorrespondingneodymium\nandsamariumisotopes.1031061091012ρ [MeV-1]\n0 36912\nEx [MeV]11031061091012\n36912143Nd145Nd\n147Nd149Nd\nFigure 2: State densities of odd-mass143−149Nd isotopes versus exci-\ntation energy Ex. The SMMC densities (open circles) are compared\nwith experimental results. The histograms at low excitatio n energies\narefromlevelcountingdata,thetriangleistheneutron res onancedata\nand the dashed line is the BBF state density [Eq. (5)] that is e xtracted\nfrom theexperimental data.\nWe emphasize that only a single parameter E0is ad-\njusted to fit the empirical partition function. Neverthe-\nless, we see that boththe SMMC partitionfunctionand\nthermal energy fit very well the corresponding experi-\nmental quantities over a range of temperatures. This is\nan evidence that our SMMC results are consistent with\ntheexperimentaldata.\n4. StateDensities\nTheaveragestatedensityisdeterminedinthesaddle-\npoint approximation to the integral that expresses the\nstatedensityasaninverseLaplacetransformofthepar-\ntitionfunction. Thisaveragedensityisgivenby\nρ(E)≈1√\n2πT2CeS(E), (6)\nwhereS(E) is the entropy and Cis the heat capacity\nin the canonical ensemble. In SMMC we first calculate\nthethermalenergyasanobservable E(β)=/angbracketleftH/angbracketrightandthe\nintegrate the thermodynamicrelation −dlnZ/dβ=E(β)\nto find the partition function Z(β). The entropy and the\nheatcapacityare thencalculatedfrom\nS(E)=lnZ+βE;C=−β2∂E/∂β (7)\n3Nucleus a(MeV−1)∆(MeV) N E N(MeV) E0(MeV)\n143Nd 15.951 0.759 3 1.228 −191.609±0.020\n145Nd 16.792 0.135 4 0.658 −210.010±0.028\n147Nd 18.276 0.058 4 0.190 −227.870±0.029\n149Nd 18.552 -0.644 3 0.138 −245.440±0.021\n149Sm 19.086 -0.196 8 0.558 −244.894±0.026\n151Sm 18.999 -0.771 8 0.168 −264.968±0.020\n153Sm 17.848 -1.053 7 0.098 −284.796±0.020\n155Sm 17.067 -0.834 7 0.221 −304.245±0.025\nTable 1: The values of aand∆in the BBF [see Eq. (5)] as determined from the level counting data at low excitation energies and the neutron\nresonance data [18], the number Nof a complete set of experimentally known levels and its corr esponding energy ENused in Eq. (4), and the\nextracted ground-state energies E0for theodd-mass143−149Nd and149−155Sm isotopes.\nand substituted into Eq. (6) to yield the state den-\nsity. The SMMC state densities for the odd-mass\nneodymiumand samarium isotopes are shown by open\ncircles in Fig. 2 and Fig. 3, respectively. We com-\nparetheseSMMCdensitieswiththelevelcountingdata\n(histograms) and with the neutron resonance data (tri-\nangles). We also show the empirical BBF level densi-\nties (dashed lines). For the neodymiumnuclei (Fig. 2),\nwe find excellent agreement among the SMMC results\n(open circles), the experimental state densities (both at\nthe level counting and at the neutron resonance ener-\ngies) and the BBF state densities (dashed lines). In\nthe case of the samarium nuclei (Fig. 3), a similarly\ngood agreement is observed for149Sm and151Sm. We\nalso find overall good agreement for153Sm and155Sm,\nalthough we observe some discrepancies between the\nSMMC and the experimental state densities at the neu-\ntronresonanceenergy. Sincetheneutronresonancedata\npoint is used to determine the parameters of the BBF\nstate densities, similar discrepancies are observed be-\ntweentheSMMCandthe BBF state densities.\n5. Conclusion\nWehaveextendedtheSMMCcalculationsintherare-\nearth region to include the odd-mass isotopic chains of\n149−155Sm and143−149Nd. The sign problem introduced\nby the projection on an odd number of particles makes\nit difficult to calculate directly the ground-state energy.\nWe circumventthisproblemin practicebycarryingout\naone-parameterfitoftheSMMCthermalexcitationen-\nergyto the experimentalaverage thermal excitation en-\nergyasdeterminedfromtheneutronresonancedataand\nlevel counting data at low excitation energies. We then\ncalculate the state densities of the even-odd samarium\nand neodymium isotopes and find them to be in good1031061091012ρ [MeV-1]\n0 36912\nEx [MeV]11031061091012\n36912149Sm151Sm\n153Sm155Sm\nFigure 3: State densities ofodd-mass149−155Smisotopes versus exci-\ntation energy Ex. Symbols and lines as in Fig. 2.\noverall agreement with state densities extracted from\navailableexperimentaldata. Thismethodrequiressu ffi-\ncient level counting data at low excitation energies and\nthe neutron resonance data point at the neutron reso-\nnance threshold. The good overall agreement between\nthe SMMC and experimental densities is an evidence\nthatthetheoryisconsistentwith thedata.\nThisworkissupportedinpartbytheU.S.DOEgrant\nNo. DE-FG-0291-ER-40608 and as Grant-in-Aid for\nScientific Research (C) No. 25400245 by the JSPS,\nJapan. Computational cycles were provided by the\nNERSC high performance computing facility at LBL\nand by the facilities of the Yale University Faculty of\n4Arts and Sciences High Performance Computing Cen-\nter.\nReferences\n[1] C. W. Johnson, S. E. Koonin, G. H. Lang and W. E. Ormand,\nPhys.Rev. Lett. 69(1992) 3157.\n[2] G. H. Lang, C. W. Johnson, S. E. Koonin and W. E. Ormand,\nPhys.Rev. C 48(1993) 1518.\n[3] Y. Alhassid, D. J. Dean, S. E. Koonin, G. Lang, and W. E. Or-\nmand, Phys.Rev. Lett. 72(1994) 613.\n[4] S.E.Koonin, D. J.Dean, K.Langanke, Phys.Rep. 278(1997) 1.\n[5] Y.Alhassid, Int. J.Mod. Phys.B 15(2001) 1447.\n[6] H.Nakada and Y.Alhassid, Phys.Rev. Lett. 79(1997) 2939.\n[7] W.E.Ormand, Phys.Rev. C 56(1997) R 1678.\n[8] Y. Alhassid, S. Liu, and H Nakada, Phys. Rev. Lett. 83(1999)\n4265.\n[9] Y. Alhassid, S. Liu, and H Nakada, Phys. Rev. Lett. 99(2007)\n162504.\n[10] C. ¨Ozen, K. Langanke, G. Martinez-Pinedo, and D. J. Dean,\nPhys.Rev. C 75(2007) 064307.\n[11] S.R. White, D.J. Scalapino, R.L. Sugar, E.Y. Loh, J.E. G uber-\nnatis, and R.T.Scalettar, Phys.Rev. B 40(1989) 506.\n[12] W.Von der Linden, Phys.Rep. 220(1992) 53.\n[13] Y. Alhassid, G. F. Bertsch, D. J. Dean, and S. E. Koonin, P hys.\nRev. Lett. 77(1996) 1444.\n[14] A. Mukherjee and Y. Alhassid, Phys. Rev. Lett. 109(2012)\n032503.\n[15] Y.Alhassid,L.FangandH.Nakada,Phys.Rev.Lett. 101(2008)\n082501.\n[16] C.¨Ozen, Y. Alhassid, H Nakada, Phys. Rev. Lett. 110(2013)\n042502.\n[17] W. Dilg, W. Schantl, H. Vonach, M. Uhl, Nucl. Phys. A 217\n(1973) 269.\n[18] C.¨Ozen and Y.Alhassid, to bepublished.\n5"    },    {        "title": "1305.1039v2.Convergence_of_the_density_of_states_and_delocalization_of_eigenvectors_on_random_regular_graphs.pdf",        "content": "arXiv:1305.1039v2  [math-ph]  8 May 2014CONVERGENCE OF THE DENSITY OF STATES AND\nDELOCALIZATION OF EIGENVECTORS\nON RANDOM REGULAR GRAPHS\nLEANDER GEISINGER\nAbstract. Consider a random regular graph of fixed degree dwithnvertices. We study\nspectral properties of the adjacency matrix and of random Sc hr¨ odinger operators on such\na graph as ntends to infinity.\nWe prove that the integrated density of states on the graph co nverges to the integrated\ndensity of states on the infinite regular tree and we give unif orm bounds on the rate of con-\nvergence. This allows to estimate the number of eigenvalues in intervals of size comparable\nto log−1\nd−1(n). Based on related estimates for the Green function we deriv e results about\ndelocalization of eigenvectors.\n1.Introduction\nIn his seminal work from 1955, Wigner showed that the density of states of random\nmatrices withindependentidentically distributedentrie s converges –as thesize ofthematrix\ntends to infinity – to a universal deterministic probability measure, the semicircle law [49].\nThis result was gradually improved and today it is known that universality of spectral\nproperties of Wigner matrices goes far beyond that. Even the local eigenvalue statistics,\nstudied via the eigenvalue gap distribution, is given by uni versal laws [19–21,44]. These\nlaws can be calculated explicitly from Gaussian ensembles a nd are characterized by local\nrepulsion of the eigenvalues. We refer to the textbooks [7,8 ,36] and the review [13] for\ngeneral results about random matrices and further referenc es.\nOne field where random matrices arise is the study of random gr aphs. Along with a\ngraph of nlabeled vertices one considers the adjacency matrix which i s the symmetric n×n\nmatrix with entry ijequal to 1, if vertex iis connected by an edge to vertex j, and equal\nto 0 otherwise. The spectrum of this matrix bears informatio n about the geometry of the\ngraph, however determining spectral properties is often a d ifficult task.\nFor Erd˝ os-R´ enyi graphs, where every edge is chosen indepe ndently with probability p\n(see [10] for details concerning random graph models), univ ersality of the local eigenvalue\nstatistics for the corresponding adjacency matrix has rece ntly been proved under suitable\nconditions on p[17,18]. Two of the steps towards universality are proving l ocal convergence\nof the density of states and delocalization of eigenvectors . The former refers to the fact that\nthe average number of eigenvalues in small intervals conver ges to a universal law. (Ideally\nthis holds for intervals of size comparable to 1 /n, up to logarithmic corrections.) The latter\nmeans that eigenvectors are typically uniformly distribut ed over the whole graph. Let us\nDate: June 12, 2018.\nAMS 2000 subject classifications: primary 60B20; secondary 05C80, 35P20. Keywords and phrases: Ran-\ndom regular graph, random Schr¨ odinger operator, density o f states, delocalization of eigenvectors, local\nspectral distribution.\n12 LEANDER GEISINGER\nemphasizethat thementioned results, for Wigner matrices a s well as for Erd˝ os-R´ enyi graphs,\nrely on independence of the entries.\nIn this note we investigate the spectrum on random regular gr aphs with fixed degree d.\nIn a regular graph of degree deach vertex is connected to dother vertices. Thus in the\ncorresponding adjacency matrix each row and each column con tainsdentries that are 1 and\nn−dentries that are 0. Such a graph, or equivalently such a matri x, is chosen at random\nwith uniform probability. These random matrices are sparse : only few entries are non-zero\nso that moments of the distribution of the entries decay slow ly. More importantly the entries\nlack independence. Therefore it is not clear how to apply the methods developed to study\nWigner matrices and Erd˝ os-R´ enyi graphs.\nWhile there are results concerning extreme eigenvalues at t he edge of the spectrum [23,42]\nnot much seems to been known about eigenvalues in the bulk of t he spectrum. However, it\nis conjectured that the local statistics is universal and go verned by repulsion of eigenvalues,\nsee for example [16,17,26,37].\nIn this article a small step is made in this direction by study ing convergence of the density\nof states and delocalization of eigenvectors. First we anal yze the density of states. The\nanalogue of Wigner’s result for random regular graphs was pr oved by McKay [33]: The\ndensity of states of the adjacency matrix of a random regular graph converges in distribution\nto a probability measure, known as the Kesten-McKay law.\nIn Theorem 2.1 we refine this result by giving uniform bounds o n the rate of convergence.\nThis allows us to deduce local convergence of the number of ei genvalues in intervals of\nsize comparable to log−1\nd−1(n). Our approach is based on the fact that a random regular\ngraph coincides locally with a regular tree. This explains t he rate log−1\nd−1(n) since this\napproximation typically works well up to distances compara ble to logd−1(n). While this\nrate is far from the desired 1 /n, our results are strong enough to deduce statements about\nconvergence of the Green function and delocalization of eig envectors.\nOur findings are similar to results recently obtained by Dumi triu and Pal [15] and by\nTran, Vu, and Wang [46]. They also study spectral properties of random regular graphs\nand consider the case where the degree dis not fixed but tends to infinity together with\nthe number of vertices n. The main results also include delocalization of eigenvect ors and\nlocal convergence of the number of eigenvalues. Again the si ze of the allowed intervals\nis comparable to log−1\ndn−1(n). Let us remark that our methods – even though they are\ntailored for regular graphs with fixed degree – also extend to graphs with growing degree. In\nTheorem 2.2 we recover results from [15] and [46] and we sligh tly improve them in a certain\nsense that is made precise below.\nAfter this prelude, the main purpose of this article is to der ive similar results for ran-\ndom Schr¨ odinger operators and to deduce delocalization of eigenvectors. Motivated by re-\ncent physical and numerical considerations about eigenval ue statistics [9,32] we study the\nAnderson model of random Schr¨ odinger operators on random r egular graphs: A random\nSchr¨ odinger operator consists of the adjacency matrix per turbed by a random potential,\nthat means one adds independent identically distributed en tries on the diagonal of the ma-\ntrix. Again we explore the density of states by comparing wit h the Anderson model on the\ninfinite regular tree. The strength of our approach lies in th e fact that it depends only on\nlocal properties of the graph, therefore it extends to gener al local operators such as random\nSchr¨ odinger operators.DENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 3\nThe Anderson model on the tree is one of the most studied model s of random Schr¨ odinger\noperators, see [47] for an overview of results and reference s. It is a natural question in\nwhat way spectral properties on the infinite tree extend to th e corresponding finite-volume\noperator on a random regular graph. However, for Schr¨ oding er operators the analysis of\nspectrum and eigenvectors is more challenging since the cor responding spectral measure\non the infinite tree is not purely absolutely continuous but i t also contains a pure-point\ncomponent.\nThis should influence the behavior of eigenvectors of the fini te-volume operator. In spec-\ntral regimes that correspond to pure-point spectrum of the i nfinite-volume operator one\nexpects to find exponential localization while eigenvector s with eigenvalues within the abso-\nlutely continuousspectrumareexpected tobedelocalized. Inturn, propertiesof eigenvectors\nare conjectured to determine whether the local eigenvalue s tatistics is Poisson or governed\nby level repulsion [9,32,47] .\nAgain we are able to make a small step in this direction. In The orem 2.3 we show that\nthe mean density of states of a random Schr¨ odinger operator on a random regular graph\nconverges to the density of states on the infinite tree. We giv e bounds on the rate of\nconvergence and show that the mean number of eigenvalues in i ntervals of size comparable\nto log−1\nd−1(n) converges. Finally, in Theorem 5.3 we apply these results a nd combine them\nwith recent results about random Schr¨ odinger operators on the infinite tree [3] to prove that\ntypically eigenvectors on a random regular graph with eigen values within the absolutely\ncontinuous spectrum of the infinite-volume operator are not localized.\nThe article is organized as follows. In the next section we ex plain the relevant nota-\ntion concerning random regular graphs and spectral measure s. Then we give the precise\nstatements of the main results about convergence of the dens ity of states.\nIn Section 3 we provide the main tools: The fact that a random r egular graph coincides\nlocally with a tree implies that low moments of the spectral m easure on the graph agree\nwith the respective moments on the tree. Therefore also the e xpectation of polynomials\nof low degree (typically up to logd−1(n)) is the same. Thus to obtain uniform estimates\non the rate of convergence of the density of states one needs t o approximate the Heaviside\nfunction by polynomials. As was noticed by Chebyshev, Marko v, and Stieltjes orthogonal\npolynomials are well suited for this purpose. In Theorem 3.1 we use this approach to prove\na deterministic estimate for the density of states valid for all regular graphs.\nIn Section 4 we study the distribution of cycles in a random re gular graph and we show\nthat it is justified to approximate a random regular graph loc ally by a tree. We use that to\ncomplete the proof of the results from Section 2.\nIn the final section we apply the developed techniques to prov e convergence of the Green\nfunction and to deduce results about delocalization of eige nvectors.\n2.Notation and results\nConsider the set Gn,dof simple regular graphs with nvertices and degree d≥3. LetPn,d\ndenote the uniform probability measure on this set and let En,ddenote the expectation with\nrespect to Pn,d. We study this ensemble as ntends to infinity. We say that an event happens\nasymptotically almost surely if the probability Pn,dof the event tends to one as ntends to\ninfinity.4 LEANDER GEISINGER\nWe assume that the degree dis fixed unless stated otherwise. However, the methods that\nare employed here are not limited to this case. In particular in Theorem 2.2 we consider the\ncase where d=dntends to infinity with n.\n2.1.The adjacency matrix. First we study the spectrum of the adjacency matrix Anof\na random regular graph Gn∈ Gn,d. The adjacency matrix is the self-adjoint operator on the\nHilbert space ℓ2(Gn) defined by\n(Anφ)(x) =/summationdisplay\ny∈Gn:d(x,y)=1φ(y), φ∈ℓ2(Gn), x∈Gn.\nHere the distance d(x,y) of two vertices x,y∈Gnis the length of the shortest path connect-\ningxandy. The adjacency matrix is the discrete Laplace operator on Gnwith the diagonal\nterms removed.\nLet (λj)n\nj=1denote the eigenvalues of Anand let ( ϕj)n\nj=1be the corresponding ℓ2(Gn)-\nnormalized eigenvectors. If an eigenvalue has multiplicit y higher than one we repeat the\nvalue according to its multiplicity and we choose the eigenv ectors as an orthonormal basis\nof the eigenspace. Since Gnhasnvertices, Anis a symmetric nbynmatrix and we obtain\nneigenvalues and neigenvectors. To study the distribution of the eigenvalues we introduce\nthe following spectral measures.\nFor a vertex x∈Gnwe write δx∈ℓ2(Gn) for its characteristic function: δx(x) = 1 and\nδx(y) = 0 for y/\\e}atio\\slash=x. Also, for a set I⊂RletχIdenote its characteristic function: χI(t) = 1\nfort∈IandχI(t) = 0 for t /∈I. We write |I|for the length of the set. The local spectral\nmeasure µn,xwith respect to a vertex x∈Gnis given by\nµn,x(I) = (δx,χI(An)δx)ℓ2(Gn)=/summationdisplay\nλj∈I|ϕj(x)|2. (2.1)\nWithNI(Gn) we denote the counting measure that counts the number of eig envalues λjin\nthe setI,\nNI(Gn) =n/summationdisplay\nj=1χI(λj) = Tr[χI(An)]\nand we remark the identity\nNI(Gn) =/summationdisplay\nx∈Gnµn,x(I). (2.2)\nOur main tool in the analysis of the spectral distribution of Anis the fact that a typical\nregular graph locally resembles a tree in the sense that it co ntains large regions without\ncycles. (A cycle is a closed path without repetitions of vert ices and edges other than the\nstarting and ending vertex.) So we also consider the infinite regular tree Tdof degree d. On\nthe tree the adjacency matrix ATdis again defined by\n(ATdφ)(x) =/summationdisplay\ny∈Td:d(x,y)=1φ(y), φ∈ℓ2(Td), x∈ Td.\nSince the degree dis finiteATdis a bounded and self-adjoint operator with domain ℓ(Td). In\nanalogy with the local spectral measure (2.1) we define the lo cal density of states measure\nσ0(I) = (δx,χI(ATd)δx)l2(Td)(2.3)DENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 5\nwhich is independent of x∈ Td. On the tree this can be calculated explicitly and is given by\nthe Kesten-McKay measure:\nσ0(dλ) =d\n2π/radicalbig\n4(d−1)−λ2\nd2−λ2χ(−2√d−1,2√d−1)(λ)dλ. (2.4)\nThe fact that a random regular graph is locally identical to a tree has already been used\nby McKay to determine the limiting distribution of eigenval ues of the adjacency matrix.\nAssume that for each fixed k≥3 the number of cycles of length kinGnis of order o(n)\nasntends to infinity. Then it is shown in [33] that the measure N(·)(Gn)/nconverges in\ndistribution to σ0. This local convergence was generalized in [39] to the much r icher class\nof self-adjoint graphs that includes trees and, in particul ar, regular trees. Since we aim at\ncomparing with results on regular trees we restrict ourselv es here to local convergence of\nrandom regular graphs to regular trees.\nWe refine the result of McKay by giving an estimate on the rate o f convergence. Note\nthat the measure σ0is supported in the interval [ −2√\nd−1,2√\nd−1] and that its density is\nbounded by γd, where\nγd=d\n4π1/radicalbig\nd2−4(d−1)ifd≤6 and γd=√\nd−1\ndπifd≥7.\nTheorem 2.1. The local density of states of the adjacency matrix of a random regular graph\nsatisfies the estimate\nsup\nt∈R/bracketleftBigg\n1\nn/summationdisplay\nx∈Gn|µn,x((−∞,t])−σ0((−∞,t])|/bracketrightBigg\n≤Cγd√\nd−1log−1\nd−1(n)\nasymptotically almost surely for any constant C >8π.\nIn particular, the estimate\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnNI(Gn)−σ0(I)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δ|I| (2.5)\nholds asymptotically almost surely, for all δ >0and all intervals I⊂Rsatisfying\n|I| ≥2Cγd√\nd−1\nδlog−1\nd−1(n).\nRemark. As mentioned in the introduction this result can be seen as a g eneralization of\nrecent results found in [15,46], where the spectral distrib ution of the adjacency matrix is\nanalyzed on random regular graphs with degree d=dnthat tends to infinity as n→ ∞. In\nfact from [15, Lemma 10] one can also derive a result for fixed d egreed, namely that (2.5)\nholds if the size of the interval Iis larger than log(log( n))log−1(n).\nTo emphasize that our approach can be applied in various situ ations let us now briefly\nconsider the case where the degree d=dndepends on the number of vertices and tends to\ninfinity as n→ ∞, as studied in [15,46]. To confine the spectrum to a finite inte rval one\nconsiders the rescaled operator\n˜An=1\n2√dn−1An6 LEANDER GEISINGER\nonℓ2(Gn) and the correspondingly rescaled operator ˜ATdnonℓ2(Tdn). Then the local density\nof states measure ˜ σ0(I) = (δx,χI(˜ATdn)δx)ℓ2(Tdn)is given by\n˜σ0(dλ) =2\nπdn(dn−1)\nd2n−4(dn−1)λ2/radicalbig\n1−λ2χ(−1,1)(λ)dλ. (2.6)\nAsntends to infinity this measure converges to the semicircle me asure\nσsc(dλ) =2\nπ/radicalbig\n1−λ2χ(−1,1)(λ)dλ. (2.7)\nIn the same way as above we define the local spectral measure ˜ µn,x(I) = (δx,χI(˜An)δx)ℓ2(Gn)\nand the counting measure\n˜NI(Gn) = Tr/bracketleftBig\nχI(˜An)/bracketrightBig\n=/summationdisplay\nx∈Gn˜µn,x(I) (2.8)\nand we obtain the following local semicircle law.\nTheorem 2.2. Letdn→ ∞asn→ ∞withdn≤(n/ln(n))1/3. Then the local density of\nstates of the operator ˜Ansatisfies the estimate\nsup\nt∈R/bracketleftBigg\n1\nn/summationdisplay\nx∈Gn|˜µn,x((−∞,t])−σsc((−∞,t])|/bracketrightBigg\n≤C/parenleftbiggln(dn−1)\nln(n)+1\ndn/parenrightbigg\nasymptotically almost surely for any constant C >8.\nIn particular, the estimate\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nn˜NI(Gn)−σsc(I)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δ|I| (2.9)\nholds asymptotically almost surely, for all δ >0and all intervals I⊂Rsatisfying\n|I| ≥2C\nδ/parenleftbiggln(dn−1)\nln(n)+1\ndn/parenrightbigg\n.\nRemark. This theorem is similar to recent results from [15,46]. In [1 5, Theorem 2] it is\nshown that (2.9) holds with probability at least 1 −o(1/n), ifdn= lnγ(n) and the size of the\nconsidered interval Iis comparable to ln−βγ(n) for 0< γ <1 and comparable to ln−β(n) for\nγ≥1 both with β <1. Theorem 2.2 above improves this in the sense that one can co nsider\nslightly smaller intervals I.\nIn [46, Theorem 1.6] a similar estimate is shown to hold with p robability at least 1 −\nO(exp(−cn√dnln(dn))) if the size of the considered interval Iis comparable to ln1/5(dn)\nd−1/10\nn. As far as the size of the interval is concerned Theorem 2.2 ab ove gives better bounds\nifdnis less than ln10(n). Fordnlarger than that the result in [46] is stronger.\n2.2.Random Schr¨ odinger operators. Now we study the distribution of eigenvalues of a\nrandom Schr¨ odinger operator on a random regular graph Gn∈ Gn,dwith fixed degree d≥3.\nWe define the operator\nHn(V) =An+V\nonℓ2(Gn). The operator Vdenotes a random potential, a multiplication operator,\n(Vφ)(x) =ωxφ(x), φ∈ℓ2(Gn), x∈Gn,DENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 7\nwhere (ωx)x∈Gnstands for a collection of independent identically distrib uted real random\nvariableswithdensity ρ. Weassumethat ρhasboundedsupportsuchthatsupp ρ⊂(−ρ0,ρ0)\nfor some 0 < ρ0<∞and we write /bardblρ/bardbl∞= supt∈Rρ(t).\nFor random Schr¨ odinger operators we apply the same notatio n as above for eigenvalues\nand eigenvectors. Here the eigenvalues ( λj)n\nj=1and eigenvectors ( ϕj)n\nj=1are random objects\neven for a fixed graph Gnsince they depend on the potential V. So forx∈Gnwe consider\nthe random spectral measure\nµn,x(I;V) = (δx,χI(Hn(V))δx)ℓ2(Gn)=/summationdisplay\nλj∈I|ϕj(x)|2(2.10)\nthat corresponds to (2.1). As in (2.2), with NI(Gn;V) we denote the random variable that\ncounts the number of eigenvalues λjin the set I,\nNI(Gn;V) =n/summationdisplay\nj=1χI(λj) = Tr[χI(Hn(V))] =/summationdisplay\nx∈Gnµn,x(I;V). (2.11)\nIn the same way we define the corresponding objects on the tree Td: First the operator\nHTd(V) =ATd+V\nonℓ2(Td). Here V= (ωx)x∈Tddenotes again a collection of independent identically dis-\ntributed real random variables with density ρ. We write PandEfor the probability and\nexpectation with respect to the distribution of ( ωx)x∈Td. We recall that the density of the\nrandom potential has finite support. Hence, HTdis a bounded self-adjoint operator with\ndomainℓ2(Td).\nForx∈ Tdwe define the local density of states measure\nµTd,x(I;V) = (δx,χI(HTd(V))δx)ℓ2(Td).\nThis measure depends on the potential Vand thus on the vertex x∈ Td. To obtain an\ninvariant measure we take the expectation with respect to th e random potential and set\nσρ(I) =E[µTd,x(I;V)]. (2.12)\nBy translation invariance of the operator HTd(V) the measure σρis independent of x∈ Td\nand thus depends only on the density ρ. We refer to Appendix A.2, where we state selected\nproperties of µTd,xandσρ.\nTo identify the random potential on the graph Gnwith the random potential on the tree\nTdwe consider the tree as the universal cover of the graph. To co nstruct the universal cover\none starts with an arbitrary vertex o∈Gnand the set of non-backtracking walks in Gnthat\nstart ato. This is the set of finite sequences ( xj)k\nj=1such that x1=o,xjis adjacent to xj+1\ninGnforj= 1,...,k−1, andxj−1/\\e}atio\\slash=xj+1forj= 2,...,k−1. Two such walks are said\nto be adjacent if their lengths differ by one and if they agree ex cept for the last vertex of\nthe longer walk. The set of non-backtracking walks in Gnstarting at owith this notion of\nadjacency is isomorphic to the tree Tdand forms the universal cover of the graph Gn.\nThe graph Gninduces an equivalence relation on its universal cover and t hus on the tree:\nTwo vertices of Tdare equivalent if the corresponding non-backtracking walk s inGnhave\nthe same endpoints. Hence, the graph Gncan be recovered from the universal cover as the\nset of equivalence classes. This induces a map\nι:Td→Gn8 LEANDER GEISINGER\nwhich is onto and preserves the local geometry in the sense ex plained in the following.\nFor a vertex x∈Gnandk∈NletBk(x) ={y∈Gn:d(x,y)≤k}denote the k-\nneighborhood of x, including all edges of Gnthat are incident with at least one vertex y\nwithd(x,y)< k. IfBk(x) is acyclic then Bk(x) is a finite tree of depth k. To compare the\ngraphGnlocally to the tree we define\nR(x) = max{k∈N:Bk(x) is acyclic }. (2.13)\n(SinceGndoes not contain double edges we have R(x)≥1 for all x∈Gn.) Let ˆx∈ Tdbe\nan arbitrary vertex from the preimage of xunder the map ιand let\nιˆx:BR(x)(ˆx)⊂ Td→BR(x)(x)⊂Gn (2.14)\nbe the restriction of ιto the neighborhood BR(x)(ˆx) ={ˆy∈ Td:d(ˆx,ˆy)≤R(x)}. By (2.13)\nthe map ιˆxis an isomorphism from BR(x)(ˆx) toBR(x)(x).\nGiven a realization of the random potential V= (ωˆx)ˆx∈Tdon the tree, this map generates\na realization of the potential on the graph: For x∈Gnwe choose ˆ x∈ Tdas above and for\ny∈BR(x)(x) we set ωy=ωˆy, where ˆy∈BR(x)(ˆx) is the unique preimage of yunderιˆx.\nFory /∈BR(x)(x) we set ωy=ωˆy, where ˆy∈ Tdis an arbitrary vertex from the preimage\nofyunderι. This procedure yields a collection of independent and iden tically distributed\nrandom variables ( ωy)y∈Gnwith density ρ. In fact, this realization of the random potential\ndependson x∈Gnandon thechoice of preimages. However, byindependenceof t herandom\nvariables ( ωy)y∈Tdthis dependence disappears after taking expectations. Thu s we denote the\nresulting random potential on Gnagain by V.\nWith this construction the local geometry and therandompot ential in BR(x)(x)⊂Gnand\nBR(x)(ˆx)⊂ Tdcoincide. By induction in k∈N, it is easy to see that ( δx,Hn(V)kδx)ℓ2(Gn)\nand (δˆx,HTd(V)kδˆx)ℓ2(Td)only depend on the local geometry and on the random potential\ninBm(x) andBm(ˆx) respectively, where m=k/2 forkeven and m= (k+1)/2 forkodd.\nIn particular, it follows that\n(δx,Hn(V)kδx)ℓ2(Gn)= (δˆx,HTd(V)kδˆx)ℓ2(Td) (2.15)\nfor allk= 0,1,...,2R(x). This identity is a key ingredient in the proof of the follow ing\nresult.\nTheorem 2.3. Assume that the density ρof the random potential Vsatisfies /bardblρ/bardbl∞<∞\nandsupp(ρ)⊂(−ρ0,ρ0)with0< ρ0<∞. Then the local density of states of the operator\nHn(V)on a random regular graph satisfies the estimate\nsup\nt∈R/bracketleftBigg\n1\nn/summationdisplay\nx∈GnE|µn,x((−∞,t];V)−µTd,ˆx((−∞,t];V)|/bracketrightBigg\n≤C/bardblρ/bardbl∞/parenleftBig\n2√\nd−1+ρ0/parenrightBig\nlog−1\nd−1(n)\nasymptotically almost surely for any constant C >4π.\nIn particular, the estimate\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnE[NI(Gn;V)]−σρ(I)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δ|I|\nholds asymptotically almost surely, for all δ >0and all intervals I⊂Rsatisfying\n|I| ≥2C/bardblρ/bardbl∞/parenleftbig\n2√\nd−1+ρ0/parenrightbig\nδlog−1\nd−1(n).DENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 9\nIn Section 5 we apply the developed methods to deduce estimat es for the Green function.\nWe establish convergence of the imaginary part of the Green f unction on a random regular\ngraph to the respective quantity on the infinite regular tree , see Corollary 5.1. Based on\nthese bounds we prove statements about delocalization of ei genvectors. In particular, in\nTheorem 5.3 we show that eigenvectors of the operator Hn(V) with eigenvalues within the\nabsolutely continuous spectrum of the infinite-volume oper atorHTd(V) are not localized.\nSince these results require more notation and since they ask for some discussion we state\nthese results in Section 5.\n3.A deterministic estimate for the integrated density of stat es\nIn this section we fix a graph Gn∈ Gn,dand a vertex x∈Gn. We derive an estimate for\nthe difference between the spectral measure on Gnand the respective measure on the tree\nTdthat depends on the local geometry of the graph.\nRecall the definition of R(x) from (2.13) and set R(x)∗=R(x) forR(x) odd and R(x)∗=\nR(x)−1 forR(x) even. As in Theorem 2.3 we write ˆ x∈ Tdfor a vertex from the preimage\nofx∈Gnunder the universal cover.\nTheorem 3.1. For allGn∈ Gn,dand allx∈Gnthe local density of states of Ansatisfies\nsup\nt∈R[|µn,x((−∞,t])−σ0((−∞,t])|]≤4πγd√\nd−11\nR(x)∗.\nMoreover, under the conditions of Theorem 2.3, the local dens ity of states of Hn(V)satisfies\nsup\nt∈R[E|µn,x((−∞,t];V)−µTd,ˆx((−∞,t];V)|]≤2π/bardblρ/bardbl∞/parenleftBig\n2√\nd−1+ρ0/parenrightBig1\nR(x)∗\nfor allGn∈ Gn,dand allx∈Gn.\nRemark. For the adjacency matrix Anthere is a variant of this result. In [41] it is shown\nthat there is a constant C >0 such that, for all m∈N,\nsup\nt∈R[|µn,x((−∞,t])−σ0((−∞,t])|]≤C\n1\nm+m6/parenleftBigg2m−2/summationdisplay\nk=1Wk(x,Gn)2/parenrightBigg1/2\n,\nwhereWk(x,Gn) is related to the number of closed non-backtracking walks o f length kin\nGnthat start at x. Form < R(x) the second summand is zero. More generally, Wk(x,Gn)\ncan be estimated in terms of the number of cycles in Gnand the resulting bounds are similar\nto Theorem 2.1.\nIn the remainder of this section we prove Theorem 3.1. The pro of is based on a general\nestimate for measures on the real line that we give in the next subsection. In Subsection 3.2\nwe show how the theorem can be deduced from Proposition 3.2.\n3.1.A general estimate. The following result is related to the classical moment prob lem\nand the Chebyshev-Markov-Stieltjes inequality; it is base d on an approximation of the Heav-\niside function by orthogonal polynomials. We refer to the bo oks [5,30,43] for background\ninformation regarding this approach.10 LEANDER GEISINGER\nProposition 3.2. Letσbe a measure on the real line with bounded density wand with\nsupport in the finite interval (−w0,w0). LetN∈Nand assume that µis another measure\non the real line such that, for all k= 0,...,2N,/integraldisplay\nRλkσ(dλ) =/integraldisplay\nRλkµ(dλ). (3.1)\nThen the estimate\nsup\nt∈R|σ((−∞,t])−µ((−∞,t])| ≤2π\nN∗/bardblw/bardbl∞w0\nholds with N∗=NforNodd and with N∗=N−1forNeven.\nProof.First we note that for t≤ −w0we have\n|σ((−∞,t])−µ((−∞,t])| ≤ |µ((−∞,−ω0])|=|σ((−∞,−w0])−µ((−∞,−w0])|\nand fort≥w0(by (3.1) with k= 0),\n|σ((−∞,t])−µ((−∞,t])|=σ((−∞,ω0])−µ((−∞,t])≤ |σ((−∞,w0])−µ((−∞,w0])|.\nHence, we can assume t∈(−w0,w0).\nForn∈N0, letPndenote the orthonormal polynomial of degree nwith respect to the\nmeasure σ. We claim that\n|σ((−∞,t])−µ((−∞,t])| ≤1/summationtextN\nn=0Pn(t)2. (3.2)\nCombining this bound with Lemma 3.3 below proves the proposi tion.\nTo establish (3.2) let us first assume that tis a zero of the polynomial PN. We remark\nthat these zeros are real and simple [5] and we denote them by λ1< λ2<···< λN. Then\nthis assumption means that t=λjfor an index j∈ {1,...,N}. We construct a polynomial\nR2N−2of degree 2 N−2 that satisfies\nR2N−2(λ1) =···=R2N−2(λj) = 1, R 2N−2(λj+1) =···=R2N−2(λN) = 0,\nand\nR′\n2N−2(λi) = 0\nfor alli/\\e}atio\\slash=j. These 2 N−1 assumptions determine the polynomial R2N−2uniquely and we\nsee that R2N−2(λ)≥χ(−∞,λj](λ) for allλ∈R. In the same way we construct a polynomial\nQ2N−2of degree 2 N−2 by changing only the condition at λjtoQ2N−2(λj) = 0. Then we\ngetQ2N−2(λ)≤χ(−∞,λj](λ) for allλ∈R. Hence, we can estimate\nσ((−∞,λj]) =/integraldisplay\nRχ(−∞,λj]σ(dλ)≤/integraldisplay\nRR2N−2(λ)σ(dλ),\nµ((−∞,λj]) =/integraldisplay\nRχ(−∞,λj]µ(dλ)≥/integraldisplay\nRQ2N−2(λ)µ(dλ) (3.3)\nand\nσ((−∞,λj])−µ((−∞,λj])≤/integraldisplay\nRR2N−2(λ)σ(dλ)−/integraldisplay\nRQ2N−2(λ)µ(dλ).(3.4)\nToboundtheright-handsideweinvoke theGaussianquadratu reformulafromLemmaA.1\nin Appendix A.1: With M=N−1 ands= 0 we find\n/integraldisplay\nRR2N−2(λ)σ(dλ) =N/summationdisplay\nk=1R2N−2(λk)/summationtextN−1\nn=0Pn(λk)2=j/summationdisplay\nk=11/summationtextN−1\nn=0Pn(λk)2.DENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 11\nBy assumption, the first 2 Nmoments of the measures σandµagree, so we also get\n/integraldisplay\nRQ2N−2(λ)µ(dλ) =/integraldisplay\nRQ2N−2(λ)σ(dλ) =N/summationdisplay\nk=1Q2N−2(λk)/summationtextN−1\nn=0Pn(λk)2=j−1/summationdisplay\nk=11/summationtextN−1\nn=0Pn(λk)2.\nCombining these identities with the estimate (3.4) yields t he upper bound\nσ((−∞,λj])−µ((−∞,λj])≤1/summationtextN−1\nn=0Pn(λj)2=1/summationtextN\nn=0Pn(λj)2,\nwhere we used the assumption PN(λj) = 0 in the last step. The lower bound is proved in\nthe same way by exchanging the roles of σandµin (3.3) and (3.4). This proves (3.2) if tis\na zero of PN.\nIt remains to prove (3.2) if tis not a zero of PN. Fors∈Rwe define a polynomial of\ndegreeN+1 by\nˆPN+1(λ) =PN+1(λ)+sPN(λ).\nSincePN(t)/\\e}atio\\slash= 0 we can choose sin such a way that ˆPN+1(t) = 0. Thus we can argue in the\nsame way as before, with Nreplaced by N+1.\nThis establishes (3.2) and completes the proof. /square\nThe proof of Proposition 3.2 is based on the following estima te of the so-called Christoffel\nnumbers (/summationtextN\nn=0Pn(t)2)−1.\nLemma 3.3. Assume that σis a measure on the real line with bounded density wand with\nsupport in the finite interval (−w0,w0). LetPn,n∈N0, denote the orthonormal polynomial\nof degree nwith respect to σ.\nThen for all N∈Nand allt∈(−w0,w0)the estimate\n1/summationtextN\nn=0Pn(t)2≤2πw0/bardblw/bardbl∞\nN∗\nholds with N∗=NforNodd and with N∗=N−1forNeven.\nProof.We fixN∈Nandt∈(−w0,w0). Below we construct a polynomial S(t)\n2N−2of degree\nless or equal than 2 N−2 with the properties\nS(t)\n2N−2(λ)≥0 (3.5)\nfor allλ∈R,\nS(t)\n2N−2(t) = 1, (3.6)\nand/integraldisplayw0\n−w0S(t)\n2N−2(λ)dλ≤2πw0\nN∗. (3.7)\nTo estimate/summationtextN\nn=0Pn(t)2in terms of this polynomial let us first assume that tis a zero\nofPN. Letλ1< λ2<···< λNdenote the zeros of PNand assume that t=λjfor an index\nj∈ {1,...,N}. Then the assumption PN(λj) = 0 together with (3.5) and (3.6) implies\n1/summationtextN\nn=0Pn(t)2=1/summationtextN−1\nn=0Pn(λj)2=S(t)\n2N−2(λj)\n/summationtextN−1\nn=0Pn(λj)2≤N/summationdisplay\nk=1S(t)\n2N−2(λk)\n/summationtextN−1\nn=0Pn(λk)2.12 LEANDER GEISINGER\nWe combine this estimate with the quadrature formula from Le mma A.1 and get\n1/summationtextN\nn=0Pn(t)2≤/integraldisplay\nRS(t)\n2N−2(λ)σ(dλ). (3.8)\nIft∈(−w0,w0) is not a zero of PNwe define, for s∈R, a polynomial\nˆPN+1(λ) =PN+1(λ)+sPN(λ).\nSincePN(t)/\\e}atio\\slash= 0 we can choose ssuch that ˆPN+1(t) = 0. Now we can argue similarly as\nabove: Again let λ1<···< λN+1denote the zeros of ˆPN+1so thatt=λjfor an index\nj∈ {1,...,N+1}. Now we apply Lemma A.1 directly to\n1/summationtextN\nn=0Pn(t)2=S(t)\n2N−2(λj)\n/summationtextN\nn=0Pn(λj)2≤N+1/summationdisplay\nk=1S(t)\n2N−2(λk)\n/summationtextN\nn=0Pn(λk)2\nand we obtain (3.8). We have shown that the estimate (3.8) is v alid for all t∈(−w0,w0).\nNow we combine this estimate with (3.7) and arrive at\n1/summationtextN\nn=0Pn(t)2≤/integraldisplay\nRS(t)\n2N−2(λ)σ(dλ)≤ /bardblw/bardbl∞/integraldisplayw0\n−w0S(t)\n2N−2(λ)dλ≤2πw0/bardblw/bardbl∞\nN∗\nwhich is the claimed estimate.\nIt remains to construct the polynomial S(t)\n2N−2. LetTmdenote the Chebysheff polynomial\nof degree m∈N0that is defined by the equation Tm(cosθ) = cos(mθ). ForN∈N, letn∈N\ndenote the largest integer satisfying 4 n≤2N−2. Then, for x∈R, we set\nF2N−2(x) =1\n2n+1+2\n(2n+1)22n/summationdisplay\nm=1(2n−m+1)(−1)mT2m(x).\nThis defines a polynomial of degree 4 n≤2N−2.\nLet us note the following relation to the Fej´ er kernel. For x∈(−1,1) writex= cosθwith\nθ∈(0,π) and calculate\nF2N−2(x) =F2N−2(cosθ)\n=1\n2n+1+2\n(2n+1)22n/summationdisplay\nm=1(2n−m+1)(−1)mcos(2mθ)\n=1\n21\n(2n+1)2/parenleftBigg\nsin2/parenleftbig\n(2n+1)(θ−π\n2)/parenrightbig\nsin2/parenleftbig\nθ−π\n2/parenrightbig+sin2/parenleftbig\n(2n+1)(θ+π\n2)/parenrightbig\nsin2/parenleftbig\nθ+π\n2/parenrightbig/parenrightBigg\n.\nThis identity shows that F2N−2(0) =F2N−2(cos(π/2)) = 1, that F2N−2(x)≥0 forx∈\n(−1,1), and that/integraldisplay1\n−1F2N−2(x)dx≤π\n2n+1≤π\nN∗.\nTo see that F2N−2(x) is non-negative for all x∈Rnote that it vanishes together with its\nderivative at the points xk= cos(π/2+kπ/(2n+1)),k= 1,2,...,n. Together with the\ncondition F2N−2(0) = 1 these are 2 n+1 conditions. Since F2N−2is by definition an even\npolynomial of exact degree 4 nthese conditions determine the polynomial uniquely and sho w\nthat it is non-negative.DENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 13\nThus, for t∈(−w0,w0) andλ∈R, we set\nS(t)\n2N−2(λ) =F2N−2/parenleftbiggλ−t\n2w0/parenrightbigg\nand the properties (3.5), (3.6), and (3.7) follow directly f rom the properties of F2N−2. This\ncompletes the proof. /square\n3.2.Proof of Theorem 3.1. Let us first consider the adjacency matrix An. For this\noperator the claim follows directly from Proposition 3.2. W e only have to show that the\nmeasures µn,xandσ0satisfy the conditions of the proposition with N=R(x): Recall that\nthe measure σ0is supported in the finite interval ( −2√\nd−1,2√\nd−1) and that its density\nis bounded by γd. Hence it remains to establish that the k-th moments of σ0andµn,xagree\nfork= 0,...,2R(x).\nForx∈Gnwe pick a vertex ˆ x∈ Tdfrom the preimage of xunder the universal cover and\nconsider the map ιˆxfrom (2.14). It maps the neighborhood BR(x)(ˆx)⊂ Tdisomorphically\ntoBR(x)(x)⊂Gnand as in (2.15) we find ( δx,Ak\nnδx)ℓ2(Gn)= (δˆx,Ak\nTdδˆx)ℓ2(Td)fork=\n0,1,...,2R(x). By definition of the measures µn,xandσ0, see (2.1) and (2.3), this implies\n/integraldisplay\nRλkµn,x(dλ) = (δx,Ak\nnδx)ℓ2(Gn)= (δˆx,Ak\nTdδˆx)ℓ2(Td)=/integraldisplay\nRλkσ0(dλ)\nfork= 0,1,...,2R(x). Hence, these measures satisfy the conditions of Proposit ion 3.2 and\nthe proof of the first statement is complete.\nTo prove the second claim we have to argue a bit more carefully . For the random operator\nHn(V) the spectral measure µTd,x(I;V) is not necessarily absolutely continuous, hence we\ncannot apply Proposition 3.2 directly. (We note that σρhas bounded density by the Wegner\nestimate (A.2), so we can apply the proposition to the measur esσρandE[µn,x]. This\nimmediately yields a similar estimate for the difference of E[µn,x] andσρ, however, we want\nto prove a stronger statement.)\nFrom identity (2.15) we see that\n/integraldisplay\nRλkµn,x(dλ;V) = (δx,Hn(V)kδx)ℓ2(Gn)= (δˆx,HTd(V)kδˆx)ℓ2(Td)=/integraldisplay\nRλkµTd,ˆx(dλ;V)\nis valid for all k= 0,...,2R(x). Hence, we can apply estimate (3.2) to the measures µn,x\nandµTd,ˆxand obtain\n|µn,x((−∞,t];V)−µTd,ˆx((−∞,t];V)| ≤1\n/summationtextR(x)\nn=0Pn(t)2, (3.9)\nwherePndenotes the orthonormal polynomial of degree nwith respect to the random\nmeasure µTd,ˆx.\nFrom the general property (A.1) we learn the the support of th is measure is almost surely\ncontained in the interval ( −2√\nd−1−ρ0,2√\nd−1+ρ0). Hence, in the same way as in the\nbeginning of the proof of Proposition 3.2 we can reduce the pr oblem to t∈(−2√\nd−1−\nρ0,2√\nd−1+ρ0). For such testimate (3.8) is valid and combining it with (3.9) we find tha t\nthe bound\n|µn,x((−∞,t];V)−µTd,ˆx((−∞,t];V)| ≤/integraldisplay\nRS(t)\n2R(x)−2(λ)µTd,ˆx(dλ;V)14 LEANDER GEISINGER\nholds almost surely. Hence, recalling definition (2.12), th e Wegner estimate (A.2), and the\nbound (3.7), we obtain\nE|µn,x((−∞,t];V)−µTd,ˆx((−∞,t];V)| ≤/integraldisplay\nRS(t)\n2R(x)−2(λ)σρ(dλ)≤2π/bardblρ/bardbl∞(2√\nd−1+ρ0)\nR∗(x).\nThis proves the second claim and completes the proof of Theor em 3.1.\n4.Asymptotically almost sure bounds on the rate of convergenc e\nIn this section we combine the deterministic estimates of Th eorem 3.1 with bounds on\nthe number of cycles in random regular graphs to prove the res ults from Section 2.\n4.1.Acyclic regions in random regular graphs. Here we collect information about\ncycles in random regular graphs based on results from [34,35 ]. We establish the fact that\ntypically at most sites the graph looks locally like a tree. T his is made precise in Lemma 4.2\nbelow.\nFor a set Awe write |A|for the number of elements, in particular for a subset F⊂Gn\nof a graph, |F|denotes the number of vertices. For a graph Gn∈ Gn,dandk∈NletC(k)\ndenote the set of cycles of length kinGn.\nLemma 4.1. For3≤k≤nd/4−2d2one has\nEn,d[|C(k)|]≤(d−1)k\n2k/parenleftbigg\n1+8\nn/parenleftbigg\nd+k\n2d/parenrightbigg/parenrightbiggk\n.\nProof.LetGnbe an arbitrary graph from Gn,dand lete(Gn) denote the set of edges of Gn.\nWe also introduce Kn, the complete graph of nvertices and the set e(Kn) of edges of Kn.\nWe consider a cycle c⊂Knof length kand its set of edges e(c). To estimate the expectation\nof|C(k)|we use the following relation to the number of cycles cof length kinKn:\nEn,d[|C(k)|] =/summationdisplay\nc⊂KnPn,d[e(c)⊂e(Gn)]≤ |{c⊂Kn}|max\nc⊂KnPn,d[e(c)⊂e(Gn)].(4.1)\nFrom [35, Theorem 3] (see also [34, Theorem 2.10]) it follows that fork≤nd/4−2d2one\nhas\nPn,d[e(c)⊂e(Gn)]≤dk(d−1)k\n2k/parenleftbigg2\nnd−4d2−2k/parenrightbiggk\n=(d−1)k\nnk/parenleftbiggnd\nnd−4d2−2k/parenrightbiggk\n.\nMoreover, a counting argument shows that\n|{c⊂Kn}|=n!\n2k(n−k)!≤nk\n2k.\nWe combine these estimates with (4.1) and get\nEn,d[|C(k)|]≤(d−1)k\n2k/parenleftbiggnd\nnd−4d2−2k/parenrightbiggk\n.\nA simple estimate using the fact that k≤nd/4−2d2yields the claim. /square\nRemark. A similar argument leads to a lower bound on En,d[|C(k)|] with the same leading\nterm (d−1)k/2k. In this sense the estimate in Lemma 4.1 is sharp.\nThe following result is a simplified version of [15, Lemma 4]. It quantifies how well one\ncan approximate a graph Gn∈ Gn,dby a tree.DENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 15\nLemma 4.2. Fork∈NletFn(k)⊂Gndenote the set of vertices x∈Gnsuch that Bk(x)\nis acyclic. Then for all ǫ >0andk≤n/4d−2d2we have\nPn,d/bracketleftbigg\n1−|Fn(k)|\nn> ǫ/bracketrightbigg\n≤1\n2nǫ(d−1)2k+1/2\n√\nd−1−1/parenleftbigg\n1+8\nn/parenleftbigg\nd+k\nd/parenrightbigg/parenrightbigg2k\n.\nProof.We apply the Markov inequality, namely that for all ǫ >0,\nPn,d/bracketleftbigg\n1−|Fn(k)|\nn> ǫ/bracketrightbigg\n=Pn,d[|Gn\\Fn(k)|> nǫ]≤1\nnǫEn,d[|Gn\\Fn(k)|].(4.2)\nHence, we have to estimate En,d[|Gn\\Fn(k)|].\nConsider c∈ C(m), i.e. a cycle c⊂Gnof length m∈N. Fork∈NsetNc(k) ={x∈Gn:\nc⊂Bk(x)}. Fork < m/2 the set Nc(k) is empty. For k≥m/2 it is included in the set of\nvertices that are at distance less or equal than k−m/2 fromc. Hence, we have |Nc(k)|= 0\nfork < m/2 and\n|Nc(k)| ≤m(d−1)k−m/2\nfork≥m/2. For each vertex x∈Gn\\Fn(k) the neighborhood Bk(x) contains at least one\ncycle so that Gn\\Fn(k)⊂/uniontext\ncNc(k). Hence,\n|Gn\\Fn(k)| ≤/summationdisplay\nm≥3/summationdisplay\nc∈C(m)|Nc(k)| ≤2k/summationdisplay\nm=3m(d−1)k−m/2|C(m)|.\nWe combine this estimate with Lemma 4.1 and conclude that\nEn,d[|Gn\\Fn(k)|]≤1\n22k/summationdisplay\nm=3(d−1)k+m/2/parenleftbigg\n1+8\nn/parenleftBig\nd+m\n2d/parenrightBig/parenrightbiggm\n≤1\n2(d−1)2k+1/2\n√\nd−1−1/parenleftbigg\n1+8\nn/parenleftbigg\nd+k\nd/parenrightbigg/parenrightbigg2k\n.\nInserting this into (4.2) finishes the proof. /square\n4.2.Proof of the main results. With Lemma 4.2 at hand we can deduce the results of\nSection 2 from Theorem 3.1. Here we give the proofs of Theorem 2.3 and Theorem 2.2. The\nproof of Theorem 2.1 is similar.\nProof of Theorem 2.3. Recall the definition of R(x) from the beginning of Section 3 and the\ndefinition of Fn(k) from Lemma 4.2. We can always estimate R(x)≥1 and for x∈Fn(k)\nwe have R(x)≥k. Thus Theorem 3.1 implies that, for all k∈N,\nsup\nt∈R/bracketleftBigg\n1\nn/summationdisplay\nx∈GnE|µn,x((−∞,t];V)−µTd,ˆx((−∞,t];V)|/bracketrightBigg\n≤2π/bardblρ/bardbl∞/parenleftBig\n2√\nd−1+ρ0/parenrightBig1\nn/parenleftbigg1\nk−1|Fn(k)|+|Gn\\Fn(k)|/parenrightbigg\n≤2π/bardblρ/bardbl∞/parenleftBig\n2√\nd−1+ρ0/parenrightBig/parenleftbigg1\nk−1+1\nn|Gn\\Fn(k)|/parenrightbigg\n. (4.3)\nWe apply Lemma 4.2 to estimate the second term. For ǫ >0 let Ω(ǫ,k) denote the event\n{|Gn\\Fn(k)| ≤nǫ/(k−1)}and note that by Lemma 4.2\nPn,d[Ω(ǫ,k)]≥1−k−1\n2nǫ(d−1)2k+1/2\n√\nd−1−1/parenleftbigg\n1+8\nn/parenleftbigg\nd+k\nd/parenrightbigg/parenrightbigg2k\n. (4.4)16 LEANDER GEISINGER\nNow we choose k=κlogd−1(n)+1 with κ <1/2. Then we see that for fixed ǫ >0 the event\nΩ(ǫ,κlogd−1(n)+1) holds asymptotically almost surely. With this choice o fkestimate (4.3)\nshows that the bound\nsup\nt∈R/bracketleftBigg\n1\nn/summationdisplay\nx∈GnE|µn,x((−∞,t];V)−µTd,ˆx((−∞,t];V)|/bracketrightBigg\n≤(1+ǫ)2π\nk−1/bardblρ/bardbl∞/parenleftBig\n2√\nd−1+ρ0/parenrightBig\n≤(1+ǫ)2π\nκlogd−1(n)/bardblρ/bardbl∞/parenleftBig\n2√\nd−1+ρ0/parenrightBig\nholds on the event Ω( ǫ,κlogd−1(n) + 1), so the bound holds asymptotically almost surely.\nThis proves the first part of the theorem. In view of (2.11) and (2.12) the second statement\nis a direct consequence of the first. /square\nProof of Theorem 2.2. Note that the measure ˜ σ0, given in (2.6), is supported in ( −1,1) and\nthat its density is bounded by\n˜γd=d√\nd−1\n2π1/radicalbig\nd2−4(d−1)ifd≤6 and ˜ γd=2\nπd(d−1)\nd2ifd≥7.\nThus we can argue as in Section 3.2 to obtain the estimate\nsup\nt∈R[|˜µn,x((−∞,t])−˜σ0((−∞,t])|]≤2π˜γd\nR(x)∗. (4.5)\nHence, on the event Ω( ǫ,k) ={|Gn\\Fn(k)| ≤nǫ/(k−1)}we find that\nsup\nt∈R/bracketleftBigg\n1\nn/summationdisplay\nx∈Gn|˜µn,x((−∞,t])−˜σ0((−∞,t])|/bracketrightBigg\n≤2π˜γdn/parenleftbigg1\nk−1+1\nn|Gn\\Fn(k)|/parenrightbigg\n≤(1+ǫ)˜γdn2π\nk−1.\nThe probability of Ω( ǫ,k) is bounded by (4.4). So we can again choose k=κln(n)/ln(dn−\n1) + 1 with κ <1/2 and we note that the assumption dn≤(n/ln(n))1/3ensures that the\ncondition of Lemma 4.2 is satisfied. We deduce that the event Ω (ǫ,κln(n)/ln(dn−1) +1)\nholds, for all ǫ >0, asymptotically almost surely. Thus the estimate\nsup\nt∈R/bracketleftBigg\n1\nn/summationdisplay\nx∈Gn|˜µn,x((−∞,t])−˜σ0((−∞,t])|/bracketrightBigg\n≤4(1+2ǫ)\nκln(dn−1)\nln(n)\nholds, for all ǫ >0, asymptotically almost surely. Here we used the fact that ˜ γdntends to\n2/πasdntends to infinity. It remains to note that\nsup\nt∈R[|˜σ0((−∞,t])−σsc((−∞,t])|]≤(1+ǫ)2\nπdn(4.6)\nfordnlarge enough. Thus applying the triangle inequality yields the first claim. By (2.8)\nthe second claim follows from the first. /squareDENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 17\n5.Estimates for the Green function and delocalization of eige nvectors\nHere we apply the results of the previous sections to compare Green functions on the\nfinite graph Gnwith the respective Green functions on the infinite tree. The n we deduce\ndelocalization ofeigenvectors fortheadjacencymatrixan dforrandomSchr¨ odingeroperators.\nInparticular, weshowthat eigenvectors oftheoperator Hn(V)witheigenvalues intheregime\nof absolutely continuous spectrum of the infinite-volume op eratorHTd(V) are not uniformly\nlocalized as ntends to infinity.\n5.1.Convergence of the Green function. Let us first introduce some notation. For\nz∈C+andx∈Gnwe consider diagonal elements of the Green function, the Sti eltjes\ntransform of the local spectral measures:\nΓn(x,z) = (δx,(An−z)−1δx)ℓ2(Gn)=/integraldisplay\nR(λ−z)−1µn,x(dλ),\n˜Γn(x,z) = (δx,(˜An−z)−1δx)ℓ2(Gn)=/integraldisplay\nR(λ−z)−1˜µn,x(dλ),\nΓn(x,z;V) = (δx,(Hn(V)−z)−1δx)ℓ2(Gn)=/integraldisplay\nR(λ−z)−1µn,x(dλ;V).(5.1)\nIn the same way, we define the corresponding Green function on the infinite tree Td. One\ncan either use resolvent expansions and the geometric struc ture of the tree or the explicit\nrepresentations of themeasures σ0andσsc(see (2.4) and(2.7)) to derivethat, for any ˆ x∈ Td,\nΓTd(ˆx,z) = ΓTd(z) =/integraldisplay\nR(λ−z)−1σ0(dλ) =−z(d−2)−d/radicalbig\nz2−4(d−1)\n2(z2−d2)(5.2)\nand lim n→∞˜ΓTdn(ˆx,z) = Γsc(z) with\nΓsc(z) =/integraldisplay\nR(λ−z)−1σsc(dλ) =−1\n2/parenleftBig\nz−/radicalbig\nz2−4/parenrightBig\n.\nHere we specify the root of a complex number as the one with pos itive imaginary part. As in\nTheorem 2.3, for given x∈Gn, we choose ˆ x∈ Tdfrom the preimage of xunder the universal\ncover.\nCorollary 5.1. Let(εn)n∈Nbe a sequence of positive numbers such that εn=o(1)asn→ ∞.\nThen the following estimates hold asymptotically almost sur ely.\n(1)Assume that (zn)n∈Nis a sequence of complex numbers with ℑzn≥C(εnlogd−1(n))−1\nfor a constant C >16πγd√\nd−1. Then we have\n1\nn/summationdisplay\nx∈Gn|ℑΓn(x,zn)−ℑΓTd(zn)| ≤εn.\n(2)Letdn→ ∞asn→ ∞withdn≤(n/ln(n))1/3and assume that (zn)n∈Nis a\nsequence of complex numbers with ℑzn≥Cε−1\nn/parenleftbig\nlog−1\ndn−1(n)+d−1\nn/parenrightbig\nfor a constant\nC >16. Then we have\n1\nn/summationdisplay\nx∈Gn|ℑ˜Γn(x,zn)−ℑΓsc(zn)| ≤εn.18 LEANDER GEISINGER\n(3)Let the random potential Vsatisfy the conditions of Theorem 2.3 and assume that\n(zn)n∈Nis a sequence of complex numbers with ℑzn≥C(εnlogd−1(n))−1for a con-\nstantC >8π/bardblρ/bardbl∞/parenleftbig\n2√\nd−1+ρ0/parenrightbig\n. Then we have\n1\nn/summationdisplay\nx∈GnE|ℑΓn(x,zn;V)−ℑΓTd(ˆx,zn;V)| ≤εn.\nProof.Let us show how the first statement can be deduced from Theorem 2.1. The other\ntwo statements follow in the same way from Theorem 2.2 and The orem 2.3 respectively.\nForz∈C+writez=E+iη, withE∈Randη >0, and\ngE,η(λ) =ℑ(λ−z)−1=η\n(E−λ)2+η2. (5.3)\nWe note that/integraldisplay\nR/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂gE,η\n∂��(λ)/vextendsingle/vextendsingle/vextendsingle/vextendsingledλ=2\nη. (5.4)\nBy (5.1) we have\nℑΓn(x,zn) =/integraldisplay\nRgEn,ηn(λ)µn,x(dλ) =−/integraldisplay\nRµn,x((−∞,λ])dgEn,ηn(λ)\nand by (5.2)\nℑΓTd(zn) =−/integraldisplay\nRσ0((−∞,λ])dgEn,ηn(λ).\nCombining these identities with Theorem 2.1 and with (5.4) y ields the claim. /square\nRemark. The third statement of the corollary gives a partial answer t o a question raised\nin [24], whether\nE|Γn(0,E+iηn;V)−ΓTd(0,E+iηn;V)| →0\nfor a sequence of positive numbers ηnthat is of order o(1) asn→ ∞. At least for the\nimaginary part of the Green function, the third statement of Corollary 5.1 can beinterpreted\nin this way, if the vertex 0 is chosen at random with uniform pr obability from Gn.\n5.2.Delocalization of eigenvectors. In this subsection we analyze how the behavior of\neigenvectors of the finite-volume operator Hn(V) is related to spectral properties of the\ninfinite-volume operator HTd(V). In the regime of absolutely continuous spectrum of the\ninfinite-volume operator the corresponding generalized ei genfunctions are delocalized: They\nare not square-summable and in particular not localized to a bounded set (see for example\n[22] for the behavior of eigenfunctions of the adjacency mat rix on an infinite regular tree\nand [4,29] for random Schr¨ odinger operators).\nIn finite volume the spectrum is of course always pure point an d eigenvectors are square-\nsummable. Inthefollowingweshowthatonecannevertheless findremnantsofdelocalization\nfor eigenvectors of the finite volume operator.\nLet us first consider the rescaled adjacency matrix ˜Anon a random regular graph with\ndegreedntending to infinity as n→ ∞. In this case the spectral measure in the limit of\ninfinite volume n→ ∞is given by the semicircle measure σscdefined in (2.7). This measure\nis purely absolutely continuous with bounded density. Thus eigenvectors of ˜Anare expected\nto be delocalized for large n. Proposition 5.2 shows that this is justified. Theresult is s imilar\nto [15, Theorem 3] and the remark after Theorem 2.2 applies. W e include the statement andDENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 19\nits proof because it serves as an illustration for the more in volved result about eigenvectors\nof random Schr¨ odinger operators that is given in Theorem 5. 3 below.\nThe proof relies on the fact that the Stieltjes transform of a n absolutely continuous mea-\nsure has a uniformly bounded imaginary part. In this way abso lute continuous spectrum is\nrelated to boundedness of the imaginary part of the Green fun ction: Since the semicircle\ndistribution is absolutely continuous with bounded densit y we find, for all E∈Randη >0,\nℑΓsc(E+iη) =/integraldisplay\nRgE,η(λ)σsc(dλ)≤2\nπ/integraldisplay\nRgE,η(λ)dλ= 2. (5.5)\nThis uniform estimate is a crucial ingredient in the proof of the following proposition.\nProposition 5.2. Letdnsatisfy the conditions of Theorem 2.2 and let Λnbe a deterministic\nsubset from nvertices with |Λn| ≤ln(n).\nLetGn∈ Gn,dnbe a random regular graph of degree dn. Then for any ℓ2(Gn)-normalized\neigenvector φof the operator ˜Anthe estimate\n/summationdisplay\nx∈Λn|φ(x)|2≤C/parenleftbiggln(dn−1)\nln(n)+1\ndn/parenrightbigg\n|Λn|\nholds asymptotically almost surely with a uniform constant C >0.\nThe same estimate holds asymptotically almost surely if one first chooses Gn∈ Gn,dat\nrandom and then the subset Λ n⊂Gnat random with uniform probability.\nProof of Proposition 5.2. Let (ϕj)n\nj=1and (λj)n\nj=1denote the eigenvectors and eigenvalues\nof the operator ˜An. Then for any m∈ {1,...,n}andη >0 we estimate\n|ϕm(x)|2≤ηn/summationdisplay\nj=1η\n(λm−λj)2+η2|ϕj(x)|2=ηℑ˜Γn(x,λm+iη), (5.6)\nwhere in the last step we used (5.1). To derive an upper bound o nℑ˜Γn(x,λm+iη) we\ncompare with the Stieltjes transform of the semicircle dist ribution:\n/vextendsingle/vextendsingle/vextendsingleℑ˜Γn(x,λm+iη)−ℑΓsc(λm+iη)/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRgλm,η(λ)d˜µn,x(λ)−/integraldisplay\nRgλm,η(λ)dσsc(λ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR(˜µn,x((−∞,λ])−σsc((−∞,λ]))dgλm,η(λ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/integraldisplay\nR|˜µn,x((−∞,λ])−σsc((−∞,λ])|/vextendsingle/vextendsingleg′\nλm,η(λ)/vextendsingle/vextendsingledλ.\nInserting estimates (4.5), (4.6), and (5.4) yields\n/vextendsingle/vextendsingle/vextendsingleℑ˜Γn(x,λm+iη)−ℑΓsc(λm+iη)/vextendsingle/vextendsingle/vextendsingle≤C/parenleftbigg1\nR(x)∗+1\ndn/parenrightbigg/integraldisplay\nR/vextendsingle/vextendsingleg′\nλm,η(λ)/vextendsingle/vextendsingledλ\n≤2C\nη/parenleftbigg1\nR(x)∗+1\ndn/parenrightbigg\nfor allx∈Gnandη >0 with a constant C >0 independent of xandn. We combine this\nbound with (5.5) and obtain\nℑ˜Γn(x,λm+iη)≤2+2C\nη/parenleftbigg1\nR(x)∗+1\ndn/parenrightbigg\n. (5.7)20 LEANDER GEISINGER\nRecall the definition of Fn(k)⊂Gnfrom Lemma 4.2 and the fact that R(x)≥kfor\nx∈Fn(k). Let us assume that Λ n⊂Fn(k). Under this assumption we combine (5.7) with\n(5.6) and take the limit η↓0. This yields\n/summationdisplay\nx∈Λn|ϕm(x)|2≤2C/parenleftbigg1\nk−1+1\ndn/parenrightbigg\n|Λn| (5.8)\nfor any eigenfunction ϕm,m∈ {1,...,n}, on the event {Λn⊂Fn(k)}.\nNow we estimate the probability of this event with the help of Lemma 4.2. We remark\nthat, for |Λn|<|Fn(k)|,\nPn,d[Λn⊂Fn(k)] =/parenleftbiggn−|Λn|\n|Fn(k)|−|Λn|/parenrightbigg/parenleftbiggn\n|Fn(k)|/parenrightbigg−1\n=(n−|Λn|)!|Fn(k)|!\n(|Fn(k)|−|Λn|)!n!.\nFor a parameter 0 < τn<1−|Λn|/nwe introduce the event Ω( τn,k) ={|Fn(k)|> n(1−τn)}\nand estimate\nPn,d[Λn⊂Fn(k)]≥ Pn,d[Λn⊂Fn(k)|Ω(τn,k)]Pn,d[Ω(τn,k)].\nThe first factor is bounded below by\n(n−|Λn|)!(n(1−τn))!\n(n(1−τn)−|Λn|)!n!≥/parenleftbiggn(1−τn)−|Λn|\nn/parenrightbigg|Λn|\n=/parenleftbigg\n1−τn−|Λn|\nn/parenrightbigg|Λn|\nand from Lemma 4.2 we obtain that the second factor is bounded below by\n1−1\n2nτn(d−1)2k√\nd−1√\nd−1−1/parenleftbigg\n1+8\nn/parenleftbigg\nd+k\nd/parenrightbigg/parenrightbigg2k\n.\nNow we choose τncomparable to 1 /√nandk=κlogdn−1(n) + 1 with κ <1/4. Then\nboth lower bounds tend to 1 as n→ ∞and we get\nPn,d/bracketleftbig\nΛn⊂Fn/parenleftbig\nκlogdn−1(n)+1/parenrightbig/bracketrightbig\n= 1−o(1)\nasn→ ∞. Inserting this choice of kin the bound (5.8) proves the claim. /square\nRemark. The same methods yield a similar statement for the adjacency matrixAnon\na random regular graph with fixed degree. However, for this op erator better results have\nrecently been derived in [11]: Brooks and Lindenstrauss als o investigate delocalization on\nregular graphs by comparing with the regular tree. They use t he explicit representation of\nspherical generalized eigenfunctions on the tree and estim ate the norms of certain propaga-\ntion operators (also build from orthogonal polynomials). F rom these estimates they deduce\ninformation about eigenvectors on regular graphs. This dir ect comparison of eigenvectors\nleads to delocalization bounds that decay not logarithmica lly but with a small power of n.\nResultsaboutthebehaviorofeigenvectors onregulargraph srelatedtoquantumergodicity\nhave also been obtained in [6], where delocalization is test ed by averaging an observable.\nBoth results rely on explicit formulas for eigenfunctions o n trees that are not available for\nrandom Schr¨ odinger operators.\nLet us now study eigenvectors of random Schr¨ odinger operat ors on random regular graphs\nwith fixed degree d. In this case the analysis is more complicated since the spec tral mea-\nsure of the corresponding infinite volume operator is not pur ely absolutely continuous: The\nspectrum of a random Schr¨ odinger operator on an infinite tre e can consist of different com-\nponents, including absolutely continuous spectrum but als o pure-point spectrum. We referDENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 21\nto [47] for an overview of spectral properties of the operato rHTd(V), see also Appendix A.2,\nwhere we state selected results.\nExistence of pure-point spectrum and exponential localiza tion of the corresponding eigen-\nfunctions of HTd(V) was proved in [1]. Therefore one cannot expect that all eige nvectors of\nthe finite volume operator Hn(V) on a random regular graph are delocalized. The existence\nof absolutely continuous spectrum of HTd(V) was also established, first in [29] and later\nin [2,25] and the regime where absolutely continuous spectr um can be found was recently\nextended in [4]. A relevant criterion for absolutely contin uous spectrum is positivity of the\nimaginary part of the Green function. Thus one defines\nσac(HTd) =/braceleftbigg\nλ∈R:P/bracketleftbigg\nlim\nη↓0ℑΓTd(x,λ+iη;V)>0/bracketrightbigg\n>0/bracerightbigg\n.\nThis set is independent of x∈ Td, deterministic and forms the support of the absolutely\ncontinuous component of the spectrum (almost surely with re spect to the random potential),\nsee [3,4]. In Theorem 5.3 we show that eigenvectors of the fini te volume operator Hn(V)\nwith eigenvalues in σac(HTd) are typically delocalized for large n.\nAn important ingredient in the proof of delocalization for t he adjacency matrix alone is\nthe fact that the limiting spectral measure is absolutely co ntinuous with uniformly bound-\nedness of the density. This allows for estimate (5.5). For ra ndom Schr¨ odinger operators the\ndensity of the limiting spectral measure µTd,xis given by lim η↓0ℑΓTd(x,λ+iη;V). This\nlimit exists almost everywhere and it is finite if λlies within the absolutely continuous\nspectrum [4,45]. However, even for compact intervals I⊂σac(HTd) it is not clear whether\nsupλ∈I,η>0ℑΓTd(x,λ+iη;V) has finite expectation. Therefore we can not treat single ei gen-\nvectors but we have to select a suitable combination as follo ws.\nFor a given realization of the random potential Von the tree Td, a graph Gn∈ Gn,d, and\na vertex x0∈Gnwe identify the potential on the graph with the potential on t he tree as\ndescribed in Section 2.2. Let ( λj)n\nj=1and (ϕj)n\nj=1denote the eigenvalues and correspond-\ningℓ2(Gn)-normalized eigenvectors of the operator Hn(V) and let I⊂Rbe bounded and\nmeasurable. For j= 1,2,...,nwe define non-negative coefficients\ncj(x0,I) =|ϕj(x0)|2ifλj∈Iandcj(x0,I) = 0 ifλj/∈I (5.9)\nand we note that/summationtextn\nj=1cj(x0,I)≤1.\nIn Lemma 5.4 below we prove the following estimate under the a ssumption I⊂σac(HTd):\nFor all ˆx∈ Tdandη >0 we have\nE\nn/summationdisplay\nj=1cj(x0,I)ℑΓTd(ˆx,λj+iη;V)\n≤C|I| (5.10)\nwith a constant C >0 depending only on the degree dand on the density of the random\npotential. In the proof of Theorem 5.3 we use this estimate in the same way as we used (5.5)\nin the proof of Proposition 5.2.\nWe remark that an eigenvector ϕjofHn(V) that is localized close to the vertex x0leads\nto a coefficient cjof order 1. The following result shows that this can not happe n for large n\nfor eigenvectors with eigenvalues within the absolutely co ntinuous spectrum of HTd. Recall\nthat we write Br(x0) ={y∈Gn:d(x0,y)≤r}and that |Br(x0)|denotes the number of\nvertices in this neighborhood.22 LEANDER GEISINGER\nTheorem 5.3. Assume that the density of the random potential is bounded an d has bounded\nsupport. Let I⊂σac(HTd)be bounded and measurable. For a random regular graph Gn∈ Gn,d\nchoose a vertex x0∈Gnat random with uniform probability. Let (ϕj)n\nj=1denote the ℓ2(Gn)-\nnormalized eigenvectors of Hn(V)and letcj(x0,I)be as defined in (5.9).\nThen for any sequence (εn)n∈Nof positive numbers with εn=o(1)asn→ ∞the estimate\n/summationdisplay\nx∈Br(x0)n/summationdisplay\nj=1cj(x0,I)|ϕj(x)|2≤|Br(x0)|\nεn/radicalbig\nlogd−1(n)\nholds asymptotically almost surely (with respect to the ran dom potential Vand the choice of\ngraphGn∈ Gn,dand vertex x0∈Gn) for all r≤ln(ln(n)).\nProof.First we fix a graph Gnand a vertex x0∈Gnand we choose x∈Br(x0). For a given\nrealization of the potential Von the tree Tdwe consider the corresponding potential on the\ngraphGnas explained in Section 2.2.\nAs in (5.6), we have for all j∈ {1,...,n}and allη >0\n|ϕj(x)|2≤ηℑΓ(x,λj+iη;V)\nand therefore\nE\nn/summationdisplay\nj=1cj|ϕj(x)|2\n≤ηE\nn/summationdisplay\nj=1cjℑΓn(x,λj+iη;V)\n (5.11)\nfor allη >0. Here and in the remainder of the proof we write cj=cj(x0,I) for short.\nTo derive an upper bound on ℑΓn(x,λj+iη;V) we compare with the Green function on\nthe treeTd. Consider the map ιˆx0given in (2.14) and let ˆ x∈ Tdbe the preimage of x∈Gn\nunder this map. Recall the definition of gE,ηfrom (5.3). For η >0 and any eigenvalue λj∈I\nwe have\n|ℑΓn(x,λj+iη;V)−ℑΓTd(ˆx,λj+iη;V)|\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRgλj,η(λ)µn,x(dλ;V)−/integraldisplay\nRgλj,η(λ)µTd,ˆx(dλ;V)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/integraldisplay\nR|µn,x((−∞,λ];V)−µTd,ˆx((−∞,λ];V)|sup\nξ∈I/vextendsingle/vextendsingleg′\nξ,η(λ)/vextendsingle/vextendsingledλ.\nNote that a coefficient cjis non-zero only if the corresponding eigenvalue λjlies inIand\nthat the sum of the coefficients cjis bounded by one. Hence, we find\nE\nn/summationdisplay\nj=1cj|ℑΓn(x,λj+iη;V)−ℑΓTd(ˆx,λj+iη;V)|\n\n≤/integraldisplay\nRE[|µn,x((−∞,λ];V)−µTd,ˆx((−∞,λ];V)|] sup\nξ∈I/vextendsingle/vextendsingleg′\nξ,η(λ)/vextendsingle/vextendsingledλ\nand Theorem 3.1 yields the upper bound\nE\nn/summationdisplay\nj=1cj|ℑΓn(x,λj+iη;V)−ℑΓTd(ˆx,λj+iη;V)|\n\n≤2π/bardblρ/bardbl∞/parenleftBig\n2√\nd−1+ρ0/parenrightBig1\nR(x)∗/integraldisplay\nR/parenleftBigg\nsup\nξ∈I/vextendsingle/vextendsingleg′\nξ,η(λ)/vextendsingle/vextendsingle/parenrightBigg\ndλ≤C/parenleftbigg\n1+|I|\nη/parenrightbigg1\nηR∗(x),(5.12)DENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 23\nwhere we used the fact that the integral of the supremum is bou nded by a constant times\nη−1(1+|I|η−1). Hereρdenotes the density of the random potential and we use that /bardblρ/bardbl∞<\n∞and that supp ρ= [−ρ0,ρ0] withρ0<∞. To shorten notation Cdenotes various positive\nconstants that may depend only on dandρ.\nCombining (5.11) and (5.12) with (5.10) yields, for η >0 small enough,\nE\nn/summationdisplay\nj=1cj|ϕj(x)|2\n≤C/parenleftbigg\n|I|η+1\nR(x)∗+|I|\nηR(x)∗/parenrightbigg\n≤C|I|/parenleftbigg\nη+1\nηR(x)∗/parenrightbigg\n.\nApplying the Markov inequality we deduce that for any 0 < δn<1\nP\n/summationdisplay\nx∈Br(x0)n/summationdisplay\nj=1cj|ϕj(x)|2≤C|I|\nδn/summationdisplay\nx∈Br(x0)/parenleftbigg\nη+1\nηR(x)∗/parenrightbigg\n≥1−δn.(5.13)\nIt remains to estimate R(x)∗. Recall the definition of Fn(k)⊂Gnfrom Lemma 4.2 and\nassume that x0∈Fn(k). Sincex0is chosen from Gnat random with uniform probability we\ncan estimate the probability of this event with the help of Le mma 4.2. Indeed, we have, for\nany 0< τn<1,\nPn,d[x0∈Fn(k)]≥ Pn,d[x0∈Fn(k)||Fn(k)| ≥n(1−τn)]Pn,d[|Fn(k)| ≥n(1−τn)]\n≥(1−τn)Pn,d[|Fn(k)| ≥n(1−τn)].\nBy Lemma 4.2 the latter probability is bounded below by\n1−1\n2nτn(d−1)2k√\nd−1√\nd−1−1/parenleftbigg\n1+8\nn/parenleftbigg\nd+k\nd/parenrightbigg/parenrightbigg2k\n.\nLet us now choose k=κlogd−1(n)+1 with κ <1/4 andτncomparable to 1 /√n. Then the\nlower bound tends to 1 as n→ ∞and we get\nPn,d/bracketleftbig\nx0∈Fn/parenleftbig\nκlogd−1(n)+1/parenrightbig/bracketrightbig\n= 1−o(1) (5.14)\nasn→ ∞.\nBy assumption, r≤ln(ln(n))< κlogd−1(n)+1 fornlarge enough. Hence, for x∈Br(x0)\nwe have\nBκlogd−1(n)+1−r(x)⊂Bκlogd−1(n)+1(x0).\nThusx0∈Fn(κlogd−1(n) + 1) implies that these neighborhoods are acyclic. In parti cular\nwe findx∈Fn(κlogd−1(n)−r+ 1) so that R(x)∗≥κlogd−1(n)−r. We choose η=\n(κlogd−1(n)−r)−1/2and arrive at\n/summationdisplay\nx∈Br(x0)/parenleftbigg\nη+1\nηR(x)∗/parenrightbigg\n≤2|Br(x0)|/radicalbig\nκlogd−1(n)−r≤4|Br(x0)|/radicalbig\nκlogd−1(n)\nfornlarge enough.\nWith this choice of parameters (5.13) reads as\nP\n/summationdisplay\nx∈Br(x0)n/summationdisplay\nj=1cj|ϕj(x)|2≤C|I|\nδn|Br(x0)|/radicalbig\nκlogd−1(n)\n≥1−δn24 LEANDER GEISINGER\nand this estimate is valid for nlarge enough on the event {x0∈Fn(κlogd−1(n)+1)}. Finally,\nwe choose δncomparable to C|I|εn/√κsuch that\nP\n/summationdisplay\nx∈Br(x0)n/summationdisplay\nj=1cj|ϕj(x)|2≤|Br(x0)|\nεn/radicalbig\nlogd−1(n)\n≥1−C|I|√κεn (5.15)\nfornlarge enough on the event {x0∈Fn(κlogd−1(n)+1)}. Relations (5.14) and (5.15) show\nthat the claimed estimate holds asymptotically almost sure ly and the proof is complete. /square\nThe proof of Theorem 5.3 relies on the following estimate for the Green function on the\ntree. For this estimate it is essential that I⊂σac(HTd).\nLemma 5.4. Under the conditions of Theorem 5.3 the bound (5.10)holds for all ˆx∈ Td\nandη >0with a constant C >0depending only on the degree dand on the density of the\nrandom potential.\nProof.The proof is based on estimate (A.3), the fact that within the absolutely continuous\nspectrum the imaginary part of the Green function has finite i nverse moments. This was\nrecently proved in [3]. To apply this result we rely on recurs ion properties of the Green\nfunction on trees and on results from rank-one perturbation theory.\nWe fix a graph Gnand a vertex x0∈Gn. Recall the definition of the coefficients cj(x0,I)\nfrom (5.9). Again we write cj=cj(x0,I) for short. By (2.10) we have, for any ˆ x∈ Td,\nn/summationdisplay\nj=1cjℑΓTd(ˆx,λj+iη;V) =/summationdisplay\nλj∈I|ϕj(x0)|2ℑΓTd(ˆx,λj+iη;V)\n=/integraldisplay\nIℑΓTd(ˆx,λ+iη;V)µn,x0(dλ;V).\nWe use the Stieltjes inversion formula, see for example [45, Theorem 3.21], to write the\nspectral measure µn,x0in terms of the Green function and we obtain\nn/summationdisplay\nj=1cjℑΓTd(ˆx,λj+iη;V) = lim\nǫ↓01\nπ/integraldisplay\nIℑΓTd(ˆx,λ+iη;V)ℑΓn(x0,λ+iǫ;V)dλ.(5.16)\nToestimate theexpectation oftheright-handside, weuseth efollowingresults fromrank-one\nperturbation theory [1,14,40]. This allows to analyze the d ependence on the single random\nvariable ωx0.\nFirstwerewrite ℑΓn(x0,λ+iǫ;V): Forarealization oftherandompotential V= (ωx)x∈Gn\nwe denote by ˆVthe same collection of random variables with ωx0replaced by zero. Then\nHn(V) =Hn(ˆV)+ωx0δx0,\nwhereδx0(x0) = 1 and δx0(x) = 0 for x/\\e}atio\\slash=x0. The resolvent identity yields, for z∈C+,\n1\nHn(V)−z−1\nHn(ˆV)−z=1\nHn(V)−zωx0δx01\nHn(ˆV)−z\nand thus\nΓn(x0,z;V) =Γn(x0,z;ˆV)\n1+ωx0Γn(x0,z;ˆV)=1\nωx0−Ξx0(z)\nwith\nΞx0(z) =−Γn(x0,z;ˆV)−1.DENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 25\nIt follows that\nℑΓn(x0,λ+iǫ;V) =ℑΞx0(λ+iǫ)\n(ωx0−ℜΞx0(λ+iǫ))2+ℑΞx0(λ+iǫ)2. (5.17)\nWe emphasize that Ξ x0is independent of ωx0. We also note that the limit Ξ x0(λ) =\nlimǫ↓0Ξx0(λ+iǫ) exists almost everywhere with respect to Lebesgue measure , see for exam-\nple [45, Theorem 3.23].\nNext, we estimate ℑΓTd(ˆx,λ+iη;V): If we remove the vertex ˆ xfrom the tree Td, it is\ndecomposed into ddisjoint infinite rooted trees that are rooted at the the near est neighbors\nof ˆx. (We remark that these trees are no longer regular, since the degree at the root equals\nd−1.) LetN(ˆx) ={y∈ Td:d(ˆx,y) = 1}denote the set of nearest neighbors of ˆ x. For\ny∈N(ˆx), letTydenote the rooted tree with root at y. In the same way as on the regular\ntree we define the bounded self-adjoint operators HTy(V) with domain ℓ2(Ty). Foru∈Ty\nandz∈C+we write\nΓTy(u,z;V) =/parenleftBig\nδu,/parenleftbig\nHTy(V)−z/parenrightbig−1δu/parenrightBig\nℓ2(Ty)\nfor the respective Green function. The trees Tywithy∈N(ˆx) are not connected to each\nother, hence the random variables Γ Ty(y,z;V),y∈N(ˆx), are independent.\nEmploying the resolvent equation one can derive the recursi on formula\nΓTd(ˆx,z;V) =1\nωˆx−z−/summationtext\ny∈N(ˆx)ΓTy(y,z;V).\nTaking the imaginary part, we see that for all z∈C+\nℑΓTd(ˆx,z;V) =|ΓTd(ˆx,z;V)|2\n/summationdisplay\ny∈N(ˆx)ℑΓTy(y,z;V)+ℑz\n.\nApplying the recursion formula one also gets that\n|ΓTd(ˆx,z;V)|2≤1/parenleftBig\nℑ/parenleftBig\nωˆx−z−/summationtext\ny∈N(ˆx)ΓTy(y,z;V)/parenrightBig/parenrightBig2=1/parenleftBig/summationtext\ny∈N(ˆx)ℑΓTy(y,z;V)+ℑz/parenrightBig2\nand we obtain\nℑΓTd(ˆx,z;V)≤1/summationtext\ny∈N(ˆx)ℑΓTy(y,z;V). (5.18)\nLet again ˆ x0∈ Tddenote the vertex corresponding to x0∈Gnunder the universal cover\nand let us for the moment assume that ˆ x/\\e}atio\\slash= ˆx0. Then there is one unique vertex y∗∈N(ˆx)\nsuch that the correspondingrooted tree Ty∗contains ˆ x0. LetN+(ˆx) ={y∈N(ˆx) : ˆx0/∈Ty}\ndenote the subset of all other nearest neighbors of ˆ x. In particular, we find that N+(ˆx)\ncontains d−1 vertices. For y∈N+(ˆx) the tree Tydoes not contain ˆ x0, thus the collection\nof random variables/parenleftbig\nΓTy(y,λ+iη;V)/parenrightbig\ny∈N+(ˆx)\nis independent of ωˆx0, the value of the random potential Vat ˆx0. If ˆx= ˆx0, then all random\nvariables Γ Ty(y,λ+iη,V) withy∈N(ˆx) are independent of ωˆx0and we can continue the\nproof in the same way with N+(ˆx) replaced by N(ˆx).26 LEANDER GEISINGER\nWe note that, for z∈C+, the imaginary part of the Green function is positive. Hence , in\nview of (5.18) we have\nℑΓTd(ˆx,z;V)≤1/summationtext\ny∈N+(ˆx)ℑΓTy(y,z;V). (5.19)\nCombining (5.16), (5.17), and (5.19) we arrive at the bound\nn/summationdisplay\nj=1cjℑΓTd(ˆx,λj+iη;V)\n≤lim\nǫ↓01\nπ/integraldisplay\nI1/summationtext\ny∈N+(ˆx)ℑΓTy(y,λ+iη;V)ℑΞx0(λ+iǫ)\n(ωx0−ℜΞx0(λ+iǫ))2+ℑΞx0(λ+iǫ)2dλ.\nRecall that ωˆx0=ωx0and that the random variables Ξ x0(λ+iǫ) and Γ Ty(y,λ+iη;V)\nwithy∈N+(ˆx) are independent of ωˆx0. Hence, we condition on the random potential\nV= (ωy)y∈Tdat all other vertices and take the expectation with respect t oωˆx0only. By\ndominated convergence, we find\nE\nn/summationdisplay\nj=1cjℑΓTd(ˆx,λj+iη;V)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(ωy)y/negationslash=ˆx0\n\n≤lim\nǫ↓01\nπ/integraldisplay\nR/integraldisplay\nI1/summationtext\ny∈N+(ˆx)ℑΓTy(y,λ+iη;V)ℑΞx0(λ+iǫ)\n(v−ℜΞx0(λ+iǫ))2+ℑΞx0(λ+iǫ)2dλρ(v)dv.\nNow we estimate ρ(v)≤ /bardblρ/bardbl∞and apply Fubini’s theorem to perform the integration in v.\nSince/integraldisplay\nRℑΞx0(λ+iǫ)\n(v−ℜΞx0(λ+iǫ))2+ℑΞx0(λ+iǫ)2dv=π\nwe obtain\nE\nn/summationdisplay\nj=1cjℑΓTd(ˆx,λj+iη;V)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(ωy)y/negationslash=ˆx0\n≤ /bardblρ/bardbl∞/integraldisplay\nI1/summationtext\ny∈N+(ˆx)ℑΓTy(y,λ+iη;V)dλ.\nTaking now the expectation with respect to ( ωy)y/negationslash=ˆx0yields\nE\nn/summationdisplay\nj=1cjℑΓTd(ˆx,λj+iη,V)\n≤ /bardblρ/bardbl∞/integraldisplay\nIE/bracketleftBigg\n1/summationtext\ny∈N+(ˆx)ℑΓTy(y,λ+iη,V)/bracketrightBigg\ndλ.\nFinally, we use that the set N+(x) contains d−1 elements. Jensen’s inequality tells us that\n/summationdisplay\ny∈N+(ˆx)ℑΓTy(y,λ+iη;V)≥(d−1)/productdisplay\ny∈N+(ˆx)ℑΓTy(y,λ+iη;V)1/(d−1)\nand using the fact that the random variables ℑΓTy(y,λ+iη;V),y∈N+(ˆx), are independent\nwe conclude\nE\nn/summationdisplay\nj=1cjℑΓTd(ˆx,λj+iη;V)\n≤/bardblρ/bardbl∞\nd−1/integraldisplay\nI/productdisplay\ny∈N+(ˆx)E/bracketleftBig\nℑΓTy(y,λ+iη;V)−1/(d−1)/bracketrightBig\ndλ.\nHence, applying (A.3) yields the claim of the lemma. /squareDENSITY OF STATES AND EIGENVECTORS ON RANDOM REGULAR GRAPHS 27\nAppendix A.Auxiliary results\nIn the appendix we collect some known results that are used in the previous sections. We\nindicate where proofs can be found in the literature.\nA.1.Orthonormal polynomials and Gaussian quadrature. InSection 3werepeatedly\nused the following quadrature formula for polynomials. A pr oof and further references can\nbe found for example in [5, Chapter 1.4.1].\nLemma A.1. Letσbe a measure on the real line with finite moments and let (Pn)n∈N0\ndenote the orthonormal polynomials with respect to σ. For arbitrary M∈Nands∈Rset\nˆPM+1=PM+1+sPMand letλ1< λ2<···< λM+1denote the zeros of ˆPM+1.\nThen the identity\n/integraldisplay\nRR(λ)σ(dλ) =M+1/summationdisplay\nk=1R(λk)/summationtextM\nn=0Pn(λk)\nholds for any polynomial Rof degree less or equal than 2M.\nA.2.Spectrum and Green function on the infinite tree. Herewemention someresults\nabout the spectrum of the random Schr¨ odinger operator HTd(V) defined in Section 2.2\nand about the Green function Γ Td(x,z;V) defined in Section 5.1. We refer to the books\n[12,28,38,45] for more information and further references .\nRecall that the random potential Vis defined as a multiplication operator\n(Vφ)(x) =ωxφ(x), φ∈ℓ2(Td), x∈ Td,\nwhere (ωx)x∈Tdare independent identically distributed real random varia bles with density\nρ. Hence one can refer to the theory of ergodic operators to det ermine the spectrum of\nHTd. In [27,31] it is shown that the spectrum corresponds almost surely to the set-sum of\nthe spectrum of the adjacency matrix and the support of ρ. On the tree the spectrum of\nATdis given by ( −2√\nd−1,2√\nd−1). So under the assumption supp( ρ) = [−ρs,ρs] with\nρs<∞the spectrum of HTd(V) is almost surely given by the deterministic set [ −2√\nd−1−\nρs,2√\nd−1+ρs]. In particular, the spectral measure µTd,xsatisfies, for all x∈ Td,\nsupp(µTd,x) = [−2√\nd−1−ρs,2√\nd−1+ρs] (A.1)\nalmost surely and this implies supp( σρ) = [−2√\nd−1−ρs,2√\nd−1+ρs].\nIt was noticed by Wegner [48] that regularity of the distribu tion ofωximplies regularity\nof the density of states measure: Under the assumption /bardblρ/bardbl∞<∞one has\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubledσρ\ndλ/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n∞≤ /bardblρ/bardbl∞. (A.2)\nAlong with the spectrum also the spectral components, the pu re-point spectrum, the\nsingular continuous spectrum, and the absolutely continuo us spectrum form almost surely\ndeterministic sets. It is the subject of extensive research to determine the location of these\nspectral components and we refer to [47] for an overview of re sults and further references.\nOne useful criterion for absolutely continuous spectrum is that the imaginary part of the\ngreen function does not vanish. So one considers the set\nσac(HTd) =/braceleftbigg\nλ∈R:P/bracketleftbigg\nlim\nη↓0ℑΓTd(x,λ+iη;V)>0/bracketrightbigg\n>0/bracerightbigg28 LEANDER GEISINGER\nthat also forms a deterministic set that does not depend on x∈ Td. For almost every\nrealization oftherandomness σac(HTd)isthesupportoftheabsolutelycontinuouscomponent\nof the spectrum [3,4]. Within this set the imaginary part of t he Green function has finite\ninverse moments. 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Sodin, Random matrices, nonbacktracking walks, and orthogonal po lynomials , J. Math. Phys. 48\n(2007), no. 12, 123503, 21.\n[42] S. Sodin, The Tracy-Widom law for some sparse random matrices , J. Stat. Phys. 136(2009), no. 5,\n834–841.\n[43] G´ abor Szeg˝ o, Orthogonal polynomials , fourth ed., American Mathematical Society, Providence, R .I.,\n1975, American Mathematical Society, Colloquium Publicat ions, Vol. XXIII.30 LEANDER GEISINGER\n[44] Terence Tao and Van Vu, Random matrices: universality of local eigenvalue statist ics, Acta Math. 206\n(2011), no. 1, 127–204.\n[45] Gerald Teschl, Mathematical methods in quantum mechanics , Graduate Studies in Mathematics, vol. 99,\nAmerican Mathematical Society, Providence, RI, 2009, With applications to Schr¨ odinger operators.\n[46] L. V. Tran, V. H. Vu, and K. Wang, Sparse random graphs: Eigenvalues and eigenvectors , Random\nStructures Algorithms 42(2013), no. 1, 110–134.\n[47] S. Warzel, Surprises in the phase diagram of the Anderson model on the Be the lattice , Preprint:\narXiv:1212.4367 (2012).\n[48] Franz Wegner, Bounds on the density of states in disordered systems , Z. Phys. B 44(1981), no. 1-2,\n9–15.\n[49] Eugene P. Wigner, Characteristic vectors of bordered matrices with infinite d imensions , Ann. of Math.\n(2)62(1955), 548–564.\nLeander Geisinger, Princeton University, Princeton, NJ 08 544, USA\nE-mail address :leander.geisinger@gmail.com"    },    {        "title": "1305.5699v1.Mean_field_limit_of_bosonic_systems_in_partially_factorized_states_and_their_linear_combinations.pdf",        "content": "arXiv:1305.5699v1  [math-ph]  24 May 2013Mean field limit of bosonic systems in partially factorized s tates and their\nlinear combinations.\nMarco Falconi\nDipartimento di Matematica, Universit` a di Bologna\nPiazza di Porta San Donato 5 - 40126 Bologna, Italy∗\n(Dated: October 6, 2018)\nWe study the mean field limit of one-particle reduced density matrices , for a bosonic\nsystem in an initial state with a fixed number of particles, only a fract ion of which occupies\nthe same state, and for linear combinations of such states. In the mean field limit, the time-\nevolved reduced density matrix is proved to converge: in trace nor m, towards a rank one\nprojection (on the state solution of Hartree equation) for a single state; in Hilbert-Schmidt\nnorm towardsamixed state, combinationofprojectionsondifferen t solutions(corresponding\nto each initial datum), for states that are a linear superposition.\nI. INTRODUCTION.\nThe mathematics of mean field limit of quantum systems has bee n widely investigated in the\nlate 35 years. The first to put on a sound mathematical basis th e concept of mean field limit\nof many-boson systems were Ginibre and Velo [5], developing an idea by Hepp [8]. Actually, in\ntheir work they performed the classical limit h→0, and used the formalism of Fock space: they\nshowed that, in the limit, bounded functions of annihilatio n and creation operators converge in\nsome sense to bounded functions of the solution of classical equation corresponding to the system\n(Hartree equation); also, the quantum evolution between h-dependent coherent states converges\nwhenh→0 to the evolution of quantum fluctuations around the classic al solution. Their result\nwas extended by Rodnianski and Schlein [15]: they studied th e convergence of reduced density\nmatrices of many-bosons systems in the mean field limit, and a lso provided a bound on the rate of\nconvergence. To do that they proved the convergence of norma l ordered products of time evolved\ncreation and annihilation operators (averaged on initial s tates that depend suitably on the number\nof particles). Further improvements were made by Chen and Le e [2] and Chen et al. [3]. The Fock\nspace method has also been used to study the mean field limit of other bosonic systems, such as\nthe Nelson model with cut off, or with a different scaling of the po tential [see 1, 4, 6, 11, for further\n∗m.falconi@unibo.it2\ndetails].\nAnother method has been widely used to study mean field limit o f quantum systems, and it\nis based on a hierarchy of equations called BBGKY. This metho d has been very successful but\nhas some limitations: in particular, due to its abstract arg ument, it does not give information on\nthe rate of convergence of reduced density matrices [see 7, f or a review of BBGKY methods, and\ndetailed references]. Recently, yet another simple method has been developed by Pickl [9, 13]: it\navoids the technicalities of BBGKY hierarchies and to intro duce the formalism of Fock spaces (in\nparticular the use of Weyl operators).\nTo sum up, we review briefly the results of the works cited abov e. Letϕbe a one-particle\nnormalized state; then ϕ⊗nis a state with nparticles, all in the same state ϕ, andC(√nϕ)Ω (C\nis the Weyl operator, Ω the vacuum of Fock space) a coherent st ate with an average number of\nparticlesn. Now let Tr 1ρϕ⊗n(t) and Tr 1ρC(√nϕ)Ω(t) be the corresponding one-particle reduced\ndensity matrices evolved in time by quantum dynamics. In the mean field limit n→ ∞, the\nfollowing convergences in trace norm are proved, for suitab le bosonic systems (Pickl [13] proves\nconvergence in operator norm):\n(1) Tr/vextendsingle/vextendsingleTr1ρϕ⊗n(t)−|ϕt/an}b∇acket∇i}ht/an}b∇acketle{tϕt|/vextendsingle/vextendsinglen→∞−→0,Tr/vextendsingle/vextendsingle/vextendsingleTr1ρC(√nϕ)Ω(t)−|ϕt/an}b∇acket∇i}ht/an}b∇acketle{tϕt|/vextendsingle/vextendsingle/vextendsinglen→∞−→0\nwhereϕtis the solution of the classical equation with initial datum ϕ. A bound on the rate of\nconvergence of order n1/2[e.g. 1, 9, 15] is given; for some systems, using quantum fluct uations, it\ncan be improved to order n−1[e.g. 3, 4].\nA. Basic notions on symmetric Fock spaces.\nWe would like to extend the convergence results above to othe r types of states (that include\nϕ⊗nas a particular case). We will use the Fock space method, so we recall some basic concepts of\nFock spaces.\nLetHbe a separable Hilbert space, with scalar product\n/an}b∇acketle{tf,g/an}b∇acket∇i}htH=/integraldisplay\ndx¯f(x)g(x).\nFrom H, construct the symmetric Fock space Fs(H) as follows. Define:\nH0=C,Hn:=Sn/parenleftBig\nH⊗···⊗ H/parenrightBig\nwithSnorthogonal symmetrizer on the n-th fold tensor product of H, i.e.\nHn=/braceleftBig\nφn(x1,...,x n)/vextendsingle/vextendsingle/vextendsingleφnis invariant for any permutation of variables/bracerightBig\n.3\nThen\nFs(H) :=∞/circleplusdisplay\nn=0Hn.\nLetφ= (φ0,φ1,...,φ n,...) be a vector of Fs(H). Then Fs(H), endowed with the norm\n/vextenddouble/vextenddoubleφ/vextenddouble/vextenddouble:=/parenleftBig∞/summationdisplay\nn=0/vextenddouble/vextenddoubleφn/vextenddouble/vextenddouble2\nHn/parenrightBig1/2\nis a (separable) Hilbert space. The vector Ω = (1 ,0,0,...) plays a special role in Fock spaces and\nit is often called the vacuum.\nThe basic operators of Fock spaces are the annihilation and c reation operators. They are the\nadjoint of one another and are defined as following. Let f∈H,φ= (φ0,φ1,...,φ n,...); then\na(f) =/integraldisplay\ndxf(x)a(x)\na∗(f) =/integraldisplay\ndxf(x)a∗(x) ;\nwith\n(a(x)φ)n(x1,...,x n) =√\nn+1φn+1(x,x1,...,x n)\n(a∗(x)φ)n(x1,...,x n) =1√nn/summationdisplay\ni=1δ(x−xj)φn−1(x1,...,ˆxj,...,x n),\nwhere ˆximeans such variable is omitted. The a#satisfy the following commutation properties:\n[a(x),a∗(x′)] =δ(x−x′),[a(x),a(x′)] = [a∗(x),a∗(x′)] = 0.\nFrom annihilation and creation operators we can construct a n unitary operator, called the Weyl\noperator that plays a crucial role in our treatment of mean fie ld limits. Let α∈H; then the Weyl\noperator, denoted by C(α), is defined as follows:\nC(α) = exp/parenleftbig\na∗(α)−a(¯α)/parenrightbig\n.\nFor allα∈H,C(α) is unitary on Fs(H). In addition to its unitarity, we will use the following\nproperties:\ni.C∗(α)a(x)C(α) =a(x)+α(x), and therefore C∗(α)a∗(x)C(α) =a∗(x)+ ¯α(x);\nii.C(α)C(β) =C(α+β)e−iIm/an}bracketle{tα,β/an}bracketri}ht;\niii.C(α) =e−/bardblα/bardbl2/2exp{a∗(α)}exp{−a(α)}.4\nAnother useful class of operators are the ones defined by the s o-called second quantization\n[see e.g. 14, Chapter X.7]. Let Abe a self-adjoint operator on H, with domain of essential self-\nadjointness D. Wedefinethesecondquantization of A, denoted bydΓ( A), asthefollowing operator\nofFs(H): letφ= (φ0,φ1,...,φ n,...)∈Fs(H), then\n(dΓ(A)φ)n(x1,...,x n) =n/summationdisplay\ni=1A(xi)φn(x1,...,x n),\nwhereA(xi)denotestheoperator Aactingonthesubspaceof Hncorrespondingtothe i-thvariable.\ndΓ(A) is essentially self-adjoint on the domain\nDA=/braceleftBig\nφ∈Fs(H)/vextendsingle/vextendsingle/vextendsingle∃˜n,∀n≥˜nφn= 0 ;∀m∈Nφm∈D⊗···⊗D/bracerightBig\n.\nThe last notion we introduce is that of number operator N. For alln∈N, everyφn∈Hnis\nan eigenvector of Nwith eigenvalue n. Precisely, let φ= (φ0,φ1,...,φ n,...)∈Fs(H), thenNis\ndefined as\n(Nφn)n(x1,...,x n) =nφn(x1,...,x n).\nNis a self-adjoint operator with domain\nD(N) =/braceleftBig\nφ∈Fs(H),∞/summationdisplay\nn=0n2/vextenddouble/vextenddoubleφn/vextenddouble/vextenddouble2<∞/bracerightBig\n;\nit satisfies the following properties:\ni.N= dΓ(1) =/integraltext\ndxa∗(x)a(x).\nii. Letf∈H; then for all φ∈D(N1/2):\n/vextenddouble/vextenddoublea#(f)φ/vextenddouble/vextenddouble≤/vextenddouble/vextenddoublef/vextenddouble/vextenddouble\nH/vextenddouble/vextenddouble(N+1)1/2φ/vextenddouble/vextenddouble.\nFurthermore, let EN(λ) be the spectral family of the operator N. Then for all E-measurable\noperator valued function g, we have\ng(N)a(x) =a(x)g(N−1)\ng(N)a∗(x) =a∗(x)g(N+1) ;\non suitable domains.\niii. Letf∈H⊗H; then for all φ∈D(N):\n/vextenddouble/vextenddouble/integraldisplay\ndxdyf(x,y)a#(x)a#(y)φ/vextenddouble/vextenddouble≤/vextenddouble/vextenddoublef/vextenddouble/vextenddouble\nH⊗H/vextenddouble/vextenddouble(N+1)φ/vextenddouble/vextenddouble\n/vextenddouble/vextenddouble/integraldisplay\ndxdyf(x,y)a∗(x)a(y)φ/vextenddouble/vextenddouble≤/vextenddouble/vextenddoublef/vextenddouble/vextenddouble\nH⊗H/vextenddouble/vextenddoubleNφ/vextenddouble/vextenddouble.5\nB. Main results.\nThe quantum system we would like to study describes nnon-relativistic interacting bosons;\nits dynamics is dictated by the following Hamiltonian of L2\ns(R3n): letVbe a real and symmetric\nfunction of R3(other assumptions on the potential will be specified later) , then\n(2)Hn=H0n+Vn=n/summationdisplay\nj=1−∆xj+1\nnn/summationdisplay\ni<jV(xi−xj).\nThis operator can be written in the language of second quanti zation on Fs(L2(R3)) as:\n(3)H=/integraldisplay\ndx(∇a)∗(x)∇a(x)+1\n2n/integraldisplay\ndxdyV(x−y)a∗(x)a∗(y)a(x)a(y) ;\nHnandHagree on Hn. Let nowφ∈Hnwith/vextenddouble/vextenddoubleφ/vextenddouble/vextenddouble\nHn= 1; we denote by φ(t) the time evolution\nofφand by\nρφ(t) =|φ(t)/an}b∇acket∇i}ht/an}b∇acketle{tφ(t)|\nthe corresponding density matrix, with integral kernel\nρφ(t,x1,...,x n;y1,...,yn) =φ(t,x1,...,x n)¯φ(t,y1,...,yn).\nAlso, we denote by Tr 1ρφ(t) the one-particle reduced density matrix, with integral ke rnel\nTr1ρφ(t,x;y) =/integraldisplay\ndx1···dxn−1φ(t,x,x1,...,x n−1)¯φ(t,y,x1,...,x n−1).\nIn the mean field limit n→ ∞, using suitable initial states, the one-particle reduced d ensity\nmatrix is expected to converge in some sense to the solution o f Hartree equation:\n(4)i∂tϕt=−∆ϕt+(V∗|ϕt|2)ϕt.\nAs discussed above, such convergence has already been prove d for factorized ( ϕ⊗n) and coherent\n(C(√nϕ)Ω) states. In this paper we prove that mean field limit conver gence can be obtained for\na wider class of states. First of all we consider states with nparticles, only a fraction of which is\nfactorized in the same state ϕ[see also 12, where these states are used to construct an isom orphism\nbetween Hnand the truncated Fock space orthogonal to ϕ]. The precise definition is the following:\nDefinition 1 (θn,m).LetHbeaHilbert space; Fs(H) =/circleplustext∞\nn=0Hnthecorrespondingsymmetric\nFock space. We also denote by D⊆Ha subspace of H.\nNow letϕ∈Dsuch that /an}b∇acketle{tϕ,ϕ/an}b∇acket∇i}htH= 1 andψm∈Hmsuch that/vextenddouble/vextenddoubleψm/vextenddouble/vextenddouble\nHm= 1 and\n(5)/an}b∇acketle{tϕ,ψm/an}b∇acket∇i}htH1=/integraldisplay\ndx¯ϕ(x)ψm(x,x1,...,x m−1) = 0.6\nWe define\nθn,m:=cn,mSn(ϕ⊗n−m⊗ψm)∈Hn;\nwhere:\nϕ⊗j=ϕ⊗···⊗ϕ/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nj,\nSj:H⊗···⊗ H/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nj−→Hjorthogonal projector (symmetrizer),\ncn,m=/parenleftbiggn\nm/parenrightbigg1/2\n(such that/vextenddouble/vextenddoubleθn,m/vextenddouble/vextenddouble= 1).\nThe convergence result about θ-vectors is formulated in the following theorem:\nTheorem 1. Suppose there exists D>0such that the operator inequality\nV2(x)≤D(1−∆x)\nis satisfied. Let θn,msatisfy definition 1 with ϕ∈H1(R3). Also, let ϕtbe the C0(R,H1(R3))\nsolution of Hartree equation with initial datum ϕ(0) =ϕ. Then∀t∈R:\nTr/vextendsingle/vextendsingleTr1ρθn,m(t)−|ϕt/an}b∇acket∇i}ht/an}b∇acketle{tϕt|/vextendsingle/vextendsingle≤2K1eK2|t|1√nem/2(m+1)7;\nwhere theKi,i= 1,2, are positive and depend only on Dand/vextenddouble/vextenddoubleϕ/vextenddouble/vextenddouble\nH1.\nRemark. For technical reasons (see lemma 2below), theresult above h olds only if m≤√7+3n−3.\nHowever since we are considering the limit n→ ∞, a stricter bound for mhas to be imposed if we\nwant convergence to hold. In particular,\nTr−lim\nn→∞Tr1ρθn,m(t) =|ϕt/an}b∇acket∇i}ht/an}b∇acketle{tϕt|\nwhenever exists 0 ≤a<1 such that, for large n,\nm∼alnn.\nAlso, we study the mean field limit for superpositions of ϕ⊗n,θn,morC(√nϕ)Ω states. Such\nlinear combinations have to satisfy the following definitio n:\nDefinition 2 (Φ, Θ, Ψ) .LetHbe a Hilbert space; Fs(H) =/circleplustext∞\nn=0Hnthe corresponding\nsymmetric Fock space. We also denote by D⊆Ha subspace of H. Furthermore, assume:7\ni. (αi)i∈N,(βi)i∈N,(γi)i∈N∈l1.\nii. (ϕ(i))i∈Nsuch thatϕ(i)∈D,∀i∈Nand supi∈N/vextenddouble/vextenddoubleϕ(i)/vextenddouble/vextenddouble\nD=M <+∞. Also we ask either\na)/vextenddouble/vextenddoubleϕ(i)/vextenddouble/vextenddouble\nH= 1,ϕ(i)andϕ(j)linearly independent for all i/ne}ationslash=j; or\nb)ϕ(i)/ne}ationslash=ϕ(j)for alli/ne}ationslash=j.\niii. To each vector of ( ϕ(i))i∈Nsatisfying iia we associate ( ϕ(i))⊗n∈Hnandθ(i)\nn,misatisfying\ndefinition 1, such that mi≤mjifi≤j,mi≤mfor alli∈N. To each vector of ( ϕ(i))i∈N\nsatisfying iib we associate the Weyl operator C(√nϕ(i)).\nThen define\nΦ =/summationdisplay\ni∈Nαi(n)(ϕ(i))⊗n,Θ =/summationdisplay\ni∈Nβi(n)θ(i)\nn,miΨ =/summationdisplay\ni∈Nγi(n)C(√nϕ(i))Ω.\nThe suites ( αi(n))i∈N, (βi(n))i∈Nand (γi(n))i∈Nare chosen such that/vextenddouble/vextenddoubleΦ/vextenddouble/vextenddouble=/vextenddouble/vextenddoubleΘ/vextenddouble/vextenddouble=/vextenddouble/vextenddoubleΨ/vextenddouble/vextenddouble= 1.\nRemark 1.In particular we have\nαi(n) =αi/parenleftBig/summationdisplay\ni,j∈N¯αiαj/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}htn/parenrightBig−1/2\n,\nβi(n) =βi/parenleftBig/summationdisplay\ni,j∈N¯βiβj/an}b∇acketle{tθ(i)\nn,mi,θ(j)\nn,mj/an}b∇acket∇i}ht/parenrightBig−1/2\n,\nγi(n) =γi/parenleftBig/summationdisplay\ni,j∈N¯γiγj/an}b∇acketle{tC(√nϕ(i))Ω,C(√nϕ(j))Ω/an}b∇acket∇i}ht/parenrightBig−1/2\n.\nFor alli∈N,αi(n),βi(n) andγi(n) are convergent when n→ ∞, and their absolute value is\nuniformly bounded in nrespectively by Kα|αi|,Kβ|βi|andKγ|γi|, where the constants depend\nonly on the l1-norm of the respective suite. Further details are discusse d in section IVB.\nRemark 2.Since Ψ states do not belong to a fixed particle subspace, we de fine the integral kernel\nof the reduced density matrix to be\nTr1ρΨ(t,x;y) =1\n/an}b∇acketle{tΨ(t),NΨ(t)/an}b∇acket∇i}ht/an}b∇acketle{tΨ(t),a∗(y)a(x)Ψ(t)/an}b∇acket∇i}ht.\nWith a linear superposition of states as initial condition, the mean field limit is not a pure\nstate. The reduced density matrix converges to a linear comb ination of projections on the solution\nof Hartree equation corresponding to each initial datum ϕ(i); in the topology induced by the\nHilbert-Schmidt norm (/vextenddouble/vextenddouble·/vextenddouble/vextenddouble\nHS):8\nTheorem 2. Suppose there exists D>0such that the operator inequality\nV2(x)≤D(1−∆x)\nis satisfied. Let Φ,ΘandΨsatisfy definition 2 with D=H1(R3). Also, for all i∈N, letϕ(i)\ntbe\ntheC0(R,H1(R3))solution of Hartree equation with initial datum ϕ(i)(0) =ϕ(i). Then∀t∈R, if\nthere is0≤a<1/2such that, for large n,m∼alnn:\nHS-lim\nn→∞Tr1ρΦ(t) =/summationdisplay\ni∈N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleαi/vextenddouble/vextenddouble(αi)i∈N/vextenddouble/vextenddouble\nl2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n|ϕ(i)\nt/an}b∇acket∇i}ht/an}b∇acketle{tϕ(i)\nt|;\nHS-lim\nn→∞Tr1ρΘ(t) =/summationdisplay\ni∈N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβi/vextenddouble/vextenddouble(βi)i∈N/vextenddouble/vextenddouble\nl2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n|ϕ(i)\nt/an}b∇acket∇i}ht/an}b∇acketle{tϕ(i)\nt|;\nHS-lim\nn→∞Tr1ρΨ(t) =/summationdisplay\ni∈N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleγi/vextenddouble/vextenddouble(γi)i∈N/vextenddouble/vextenddouble\nl2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n|ϕ(i)\nt/an}b∇acket∇i}ht/an}b∇acketle{tϕ(i)\nt|.\nIn particular, the following bounds hold:\n/vextenddouble/vextenddoubleTr1ρΦ(t)−/vextenddouble/vextenddouble(αi)i∈N/vextenddouble/vextenddouble−2\nl2/summationdisplay\ni∈N|αi|2|ϕ(i)\nt/an}b∇acket∇i}ht/an}b∇acketle{tϕ(i)\nt|/vextenddouble/vextenddouble\nHS≤K1/summationdisplay\ni<j|¯αi(n)αj(n)|/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsinglen\n+K2eK3|t|1\nn1/4,(6)\n/vextenddouble/vextenddoubleTr1ρΘ(t)−/vextenddouble/vextenddouble(βi)i∈N/vextenddouble/vextenddouble−2\nl2/summationdisplay\ni∈N|βi|2|ϕ(i)\nt/an}b∇acket∇i}ht/an}b∇acketle{tϕ(i)\nt|/vextenddouble/vextenddouble\nHS≤K1/summationdisplay\ni<j/vextendsingle/vextendsingle¯βi(n)βj(n)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tθ(i)\nn,mi,θ(j)\nn,mj/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle\n+K2eK3|t|1\nn1/4em/2(m+1)3;(7)\n/vextenddouble/vextenddoubleTr1ρΨ(t)−/vextenddouble/vextenddouble(γi)i∈N/vextenddouble/vextenddouble−2\nl2/summationdisplay\ni∈N|γi|2|ϕ(i)\nt/an}b∇acket∇i}ht/an}b∇acketle{tϕ(i)\nt|/vextenddouble/vextenddouble\nHS≤K1e−4M2n+K2eK3|t|1\nn1/2; (8)\nwhere theKi,i= 1,2,3, are positive and depend on D,Mand/vextenddouble/vextenddouble(·i)i∈N/vextenddouble/vextenddouble\nl1.\nRemark 3.The first term on the right hand side of equation (6) converges to zero when n→ ∞\nbecause, by definition 2 and Riesz’s lemma,/vextendsingle/vextendsingle/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}ht/vextendsingle/vextendsingle<1 (and|¯αi(n)αj(n)| ≤K2\nα|¯αiαj|). The\nfirst term on the right hand side of equation (7) also converge s to zero if m≤lnn. This is because\n/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tθ(i)\nn,mi,θ(j)\nn,mj/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle≤(m+1)(m!)2nm/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsinglen−2m\n≤Kn2(lnn+1)/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsinglen−2lnn\n−→\nn→∞0 ;\nfurthermore since/vextendsingle/vextendsingle¯βi(n)βj(n)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tθ(i)\nn,mi,θ(j)\nn,mj/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle≤K2\nβ/vextendsingle/vextendsingle¯βiβj/vextendsingle/vextendsinglewe can exchange summation with the\nlimitn→ ∞.\nRemark 4.In this paper we focused attention only on one-particle redu ced density matrices. The\nsame method can be used to calculate the limit of k-particle reduced density matrices (with k>1).\nThe rest of the paper is organized as follows. In section II we analyze the dynamics of classical\n(Hartree) and quantum system. In section III the combinator ial properties of θn,mstates are\nstudied. Finally in section IV we consider the limit n→ ∞, and prove theorems 1 and 2.9\nII. CLASSICAL AND QUANTUM DYNAMICS.\nIn this section we review some properties of quantum and clas sical evolution needed to perform\nthe classical limit.\nThe first proposition concerns the existence and unicity of s olutions of the Hartree Cauchy\nproblem [see 15, Remark 1.3]:\n(9)\n\ni∂tϕt=−∆ϕt+(V∗|ϕt|2)ϕt\nϕ(t0) =ϕ0.\nProposition 1. Under the assumption V2(x)≤D(1−∆x),(9)has a unique solution ϕt∈\nC0(R,H1(R3))for allϕ0∈H1(R3). Furthermore, using also the conservation in time of/vextenddouble/vextenddoubleϕ/vextenddouble/vextenddouble\nL2\nand energy\nE(ϕ) =/integraldisplay\ndx|∇ϕ(x)|2+1\n2/integraldisplay\ndxdyV(x−y)/vextenddouble/vextenddoubleϕ(x)/vextenddouble/vextenddouble2/vextenddouble/vextenddoubleϕ(y)/vextenddouble/vextenddouble2,\nwe obtain the following bound for/vextenddouble/vextenddoubleϕt/vextenddouble/vextenddouble\nH1:\n/vextenddouble/vextenddoubleϕt/vextenddouble/vextenddouble2\nH1≤c/parenleftBig/vextenddouble/vextenddoubleϕ0/vextenddouble/vextenddouble2\nH1(1+/vextenddouble/vextenddoubleϕ0/vextenddouble/vextenddouble2\nL2)+/vextenddouble/vextenddoubleϕ0/vextenddouble/vextenddouble4\nL2+/vextenddouble/vextenddoubleϕ0/vextenddouble/vextenddouble2\nL2/parenrightBig\n,\nfor some positive cthat depends only on D.\nWe can formulate also the following proposition concerning quantum evolution [see 5, 15].\nProposition 2. LetVbe a real symmetric function such that for some 0< Dthe operator\ninequalityV2(x)≤D(1−∆x)is satisfied. Then:\ni.His a self-adjoint operator with domain D(H)⊂Fs(L2(R3)), defined through the direct sum\ndecomposition\nH=∞/circleplusdisplay\nn=1Hn.\nHnis the self-adjoint operator (2)onD(H0n)(defined as a Kato sum).\nii. Letϕ0∈H1(R3),ϕt∈C0(R,H1(R3))the associated solution of (9). Define also\nU(t) = exp(−itH),\nW(t,t0) =C∗(√nϕt)U(t−t0)C(√nϕ0).\nThen for all δ≥0,φ∈D(Nmax{2,2δ+1})the following inequality holds:\n(10)/vextenddouble/vextenddouble(N+1)δW(t,t0)φ/vextenddouble/vextenddouble≤K1(δ)eK2(δ)|t−t0|/vextenddouble/vextenddouble(N+1)max{2,2δ+1}φ/vextenddouble/vextenddouble;\nwhereKi(δ),i= 1,2, are positive and depend only on δand/vextenddouble/vextenddoubleϕ0/vextenddouble/vextenddouble\nH1.10\nIII. PROPERTIES OF θn,mSTATES.\nIn this section we study the partially factorized states θn,mdefined in section I. The technical\nresult formulated in lemma 2 is crucial to perform the mean fie ld limitn→ ∞. Its proof mimics\nthe one performed in the case of completely condensed states , found in [2, Lemma 6.3].\nLemma 1. The following identities hold:\nθn,m=1√\nm!/integraldisplay\ndy1···dymψm(y1,...,ym)a∗(y1)···a∗(ym)ϕ⊗n−m\n=1/radicalbig\n(n−m)!m!/integraldisplay\ndy1···dymψm(y1,...,ym)/parenleftbig\na∗(ϕ)/parenrightbign−ma∗(y1)···a∗(ym)Ω\n=dn,m√\nm!/integraldisplay\ndy1···dymψm(y1,...,ym)PnC(√nϕ)a∗(y1)···a∗(ym)Ω ;\nwherePnis the projector on Hnand\ndn,m=/radicalbig\n(n−m)!\nexp(−n/2)n(n−m)/2(dn,m∼(n−m)1/4em/2for largen).\nProof.First two equalities are easy to prove. To prove the last one, recall the formula\nϕ⊗n−m=dn,mPn−mC(√nϕ)Ω.\nWe then obtain:\nθn,m=dn,m√\nm!/integraldisplay\ndy1···dymψm(y1,...,ym)Pna∗(y1)···a∗(ym)Pn−mC(√nϕ)Ω\n=dn,m√\nm!/integraldisplay\ndy1···dymψm(y1,...,ym)PnC(√nϕ)(a∗(y1)+√n¯ϕ(y1))···\n(a∗(ym)+√n¯ϕ(ym))Ω ;\nnow all the terms containing at least a ¯ ϕvanish by (5).\nLemma 2. Letθn,msatisfy definition 1, with m≤√7+3n−3. Then∀ǫ>0:\n/vextenddouble/vextenddouble(N+1)−(1/4+ǫ)C∗(√nϕ)θn,m/vextenddouble/vextenddouble≤K(ǫ)\ndn,mem/2;\nfor some positive K(ǫ)that depends only on ǫ.\nProof.ConsiderC∗(√nϕ)θn,m; by definition of Weyl operators, we obtain ∀j≤n+m:\n/parenleftBig\nC∗(√nϕ)θn,m/parenrightBig\nj=e−n/2j/summationdisplay\ni=0(−√n)i\ni!/parenleftbig\na∗(ϕ)/parenrightbigi(√n)n−j+i\n(n−j+i)!/parenleftbig\na(¯ϕ)/parenrightbign−j+iθn,m\n=e−n/2(√n)n−jj/summationdisplay\ni=0(−1)ini\ni!(n−j+i)!/parenleftbig\na∗(ϕ)/parenrightbigi/parenleftbig\na(¯ϕ)/parenrightbign−j+iθn,m.11\nBy lemma 1 (second equality) we write\n/parenleftbig\na∗(ϕ)/parenrightbigi/parenleftbig\na(¯ϕ)/parenrightbign−j+iθn,m=1/radicalbig\n(n−m)!m!/integraldisplay\ndy1···dymψm(y1,...,ym)/parenleftbig\na∗(ϕ)/parenrightbigi/parenleftbig\na(¯ϕ)/parenrightbign−j+i\n/parenleftbig\na∗(ϕ)/parenrightbign−ma∗(y1)···a∗(ym)Ω ;\nso ifn−j+i>n−m, i.e.m−j+i>0, then either a(¯ϕ) acts on some a∗(y), or on Ω, yielding\nzero (by (5), and definition of Ω). Therefore we must have i≤j−m, and consequently j≥m. In\nsuch case we obtain\n/parenleftbig\na∗(ϕ)/parenrightbigi/parenleftbig\na(¯ϕ)/parenrightbign−j+iθn,m=/radicalbig\n(n−m)!(j−m)!\n(j−m−i)!θj−m,m;\nthis leads to\n/parenleftBig\nC∗(√nϕ)θn,m/parenrightBig\nj=\n\n0 ifj <m\nj−m/summationdisplay\ni=0(−1)inie−n/2(√n)n−j/radicalbig\n(n−m)!(j−m)!\ni!(j−m−i)!(n−j+i)!θj−m,mifm≤j≤n.\nSo if we set k=j−m, for all 0 ≤k≤n−m:\n/parenleftBig\nC∗(√nϕ)θn,m/parenrightBig\nk+m=e−n/2(√n)n−m−k/radicalbigg\n(n−m)!\nk!/parenleftbiggk/summationdisplay\ni=0(−1)ini k!\ni!(k−i)!(n−m−k+i)!/parenrightbigg\nθk,m.\nNow define\nAk:=/vextenddouble/vextenddouble/parenleftBig\nC∗(√nϕ)θn,m/parenrightBig\nk+m/vextenddouble/vextenddouble\nHk+m,\nand the Laguerre polynomial\n(11)L(α)\nk(x) :=k/summationdisplay\ni=0(−1)ixi(k+α)!\ni!(k−i)!(α+i)!.\nThen we have\nAk=e−n/2(√n)n−m−k/radicalBigg\nk!\n(n−m)!/vextendsingle/vextendsingle/vextendsingleL(n−m−k)\nk(n)/vextendsingle/vextendsingle/vextendsinglefor all 0≤k≤n−m.\nActually, an explicit calculation shows that A0= 1/dn,m,A1=m/√ndn,m. Fork≥2, we use\na sharp estimate for Laguerre polynomials obtained by Krasi kov [10]: let s=√\nk+α+1+√\nk,\nq=√\nk+α+1−√\nkandr(x) = (x−q2)(s2−x); then forα>−1,k≥2 andx∈(q2,s2)\n(12)/vextendsingle/vextendsingle/vextendsingleL(α)\nk(x)/vextendsingle/vextendsingle/vextendsingle</radicalbigg\n(k+α)!\nk!/radicalBigg\nx(s2−q2)\nr(x)ex/2x−(α+1)/2.12\nIn the case of L(n−m−k)\nk(n) we obtain: s=√n−m+1+√\nk,q=√n−m+1−√\nkandr(n) =\n4k(n−m+1)−(k−m+1)2. The condition n∈(q2,s2) implies that m≤√7+3n−3. Therefore\nwe obtain:\nAk</radicalBigg\n4/radicalbig\nk(n−m+1)\n(2/radicalbig\nk(n−m+1)+k−m+1)(2/radicalbig\nk(n−m+1)−k+m−1).\nThen, since 2 ≤k≤n−m:\nAk<(√\n2/2−1/2)−1/2(k+1)−1/4(n−m+1)−1/4.\nLetδ>0, and consider\n/an}b∇acketle{tC∗(√nϕ)θn,m,(N+1)−δC∗(√nϕ)θn,m/an}b∇acket∇i}ht=∞/summationdisplay\nk=0|Ak|2\n(k+m+1)δ,\nsince/parenleftBig\nC∗(√nϕ)θn,m/parenrightBig\nj= 0 for all 0 ≤j <m. Using the bound above, we obtain:\n/an}b∇acketle{tC∗(√nϕ)θn,m,(N+1)−δC∗(√nϕ)θn,m/an}b∇acket∇i}ht ≤/parenleftbigg1\nd2n,m(1+m/√n)+(√\n2/2−1/2)−1\n(n−m+1)−1/2n−m/summationdisplay\nk=21\n(k+1)δ+1/2/parenrightbigg\n+∞/summationdisplay\nk=n−m+1|Ak|2\n(k+1)δ;\nfurthermore, since\n∞/summationdisplay\nk=n−m+1|Ak|2≤/vextenddouble/vextenddoubleC∗(√nϕ)θn,m/vextenddouble/vextenddouble2= 1,\nwe have\n/an}b∇acketle{tC∗(√nϕ)θn,m,(N+1)−δC∗(√nϕ)θn,m/an}b∇acket∇i}ht ≤/parenleftbigg1\nd2n,m(1+m/√n)+(√\n2/2−1/2)−1\n(n−m+1)−1/2/parenleftbig\nHδ+1/2(n−m+1)−1/parenrightbig\n+(n−m+2)−δ/parenrightbigg\n,\nwhereHδ+1/2(n−m+1) is a generalized harmonic number, whose limit for n→ ∞converges only\nifδ+1/2>1, i.e.δ >1/2; and\nlim\nn→∞Hδ+1/2(n−m+1) =ζ(δ+1/2) (ζis the Riemann zeta function).\nThen for all ǫ>0 we can write\n/an}b∇acketle{tC∗(√nϕ)θn,m,(N+1)−(1/4+ǫ)C∗(√nϕ)θn,m/an}b∇acket∇i}ht ≤/parenleftbigg1\nd2n,m(1+m/√n)+(√\n2/2−1/2)−1\n/parenleftbig\nζ(1+ǫ)−1/parenrightbig\n(n−m+1)−1/2+(n−m+2)−(1/2+2ǫ)/parenrightbigg\n;\nthus concluding the proof.13\nIV. MEAN FIELD LIMIT.\nIn this section we investigate the behaviour of the transiti on amplitude of a creation and an\nannihilation operator between time evolved θn,m, Φ and Θ states, when nis large. Hence we are\nable to prove theorems 1 and 2 on the convergence of one-parti cle reduced density matrices.\nA. Proof of Theorem 1.\nFirst of all we consider the transition amplitude Tθn,m(t). Letθn,mandU(t) be as above; define\nTθn,m(t)≡Tθn,m(t,x,y) :=/an}b∇acketle{tU(t)θn,m,a∗(x)a(y)U(t)θn,m/an}b∇acket∇i}ht.\nThen we can formulate the following proposition:\nProposition 3. Letϕt∈C0(R,H1(R3))be the solution of Hartree equation with initial datum\nϕ∈H1(R3). Then∀t∈R,m≤√7+3n−3:\n/vextenddouble/vextenddoubleTθn,m(t)−n¯ϕtϕt/vextenddouble/vextenddouble\nL2(R6)≤K1eK2|t|(1+2√n)em/2(m+1)7;\nwhere theKi,i= 1,2, are positive and depend only on/vextenddouble/vextenddoubleϕ/vextenddouble/vextenddouble\nH1.\nProof.By lemma 1 and property i of Weyl operators, we can write:\nTθn,m(t) =dn,m/an}b∇acketle{tU(t)θn,m,PnC(√nϕt)/parenleftbig\na∗(x)+√n¯ϕt(x)/parenrightbig/parenleftbig\na(y)+√nϕt(y)/parenrightbig\nC∗(√nϕt)U(t)\nC(√nϕ)ψm/an}b∇acket∇i}ht\n=n¯ϕt(x)ϕt(y)+dn,m/an}b∇acketle{t(N+1)−(1/4+ǫ)C∗(√nϕ)θn,m,(N+1)1/4+ǫW∗(t,0)\n/parenleftbig\na∗(x)a(y)+√n(a∗(x)ϕt(y)+a(y)¯ϕt(x))/parenrightbig\nW(t,0)ψm/an}b∇acket∇i}ht.\nNow, letξ∈L2(R6). Then by lemma 2, with ǫ≤1/4:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\ndxdy¯ξ(x,y)/parenleftBig\nTθn,m(t,x,y)−n¯ϕt(x)ϕt(y)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Kem/2/vextenddouble/vextenddouble(N+1)1/4+ǫW∗(t,0)/integraldisplay\ndxdy\n¯ξ(x,y)/parenleftbig\na∗(x)a(y)+√n(a∗(x)ϕt(y)+a(y)¯ϕt(x))/parenrightbig\nW(t,0)ψm/vextenddouble/vextenddouble.\nNow using: (10) two times and the properties of annihilation , creation and number operators listed\nin section IA, we obtain:\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\ndxdy¯ξ(x,y)/parenleftBig\nTθn,m(t,x,y)−n¯ϕt(x)ϕt(y)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤K1eK2|t|(1+2√n)em/2/vextenddouble/vextenddouble(N+1)7ψm/vextenddouble/vextenddouble\n/vextenddouble/vextenddoubleξ/vextenddouble/vextenddouble\nL2(R6).\nThe result is then proved applying Riesz’s lemma on L2(R6).14\nThe trace of Tθn,m(t) is defined as\nTrTθn,m(t) =/integraldisplay\ndxTθn,m(t,x,x) =/an}b∇acketle{tU(t)θn,m,NU(t)θn,m/an}b∇acket∇i}ht=n.\nConsidernow theone-particle reduceddensity matrix Tr 1ρθn,m(t); its integral kernel can bewritten\nas\nTr1ρθn,m(t,x;y) =Tθn,m(t,y,x)\nTrTθn,m(t)=Tθn,m(t,y,x)\nn.\nThen, by proposition 3 we immediately obtain\n/vextenddouble/vextenddoubleTr1ρθn,m(t)−|ϕt/an}b∇acket∇i}ht/an}b∇acketle{tϕt|/vextenddouble/vextenddouble\nHS≤K1eK2|t|/parenleftBig1\nn+2√n/parenrightBig\nem/2(m+1)7,\nwhere/vextenddouble/vextenddouble·/vextenddouble/vextenddouble\nHSstands for the Hilbert-Schmidt norm. Denote now X= Tr1ρθn,m(t)−|ϕt/an}b∇acket∇i}ht/an}b∇acketle{tϕt|. Since\n|ϕt/an}b∇acket∇i}ht/an}b∇acketle{tϕt|is a rank one projection, we have that Tr |X| ≤2/vextendsingle/vextendsingle/vextenddouble/vextenddoubleX/vextenddouble/vextenddouble/vextendsingle/vextendsingle, where/vextendsingle/vextendsingle/vextenddouble/vextenddouble·/vextenddouble/vextenddouble/vextendsingle/vextendsingledenotes the operator\nnorm. Since/vextendsingle/vextendsingle/vextenddouble/vextenddoubleX/vextenddouble/vextenddouble/vextendsingle/vextendsingle≤/vextenddouble/vextenddoubleX/vextenddouble/vextenddouble\nHS, it follows that Tr |X| ≤2/vextenddouble/vextenddoubleX/vextenddouble/vextenddouble\nHS, hence the theorem is proved.\nB. Proof of Theorem 2.\nAs discussed in remark 1 of section IB, an n-dependent normalization has to be performed on\nlinear superposition states. In the following lemma we stud y its behavior at large n.\nLemma 3. Letmbe as in definition 2. Then for all m≤lnn, the following limits hold:\nlim\nn→∞/summationdisplay\ni,j∈N¯αiαj/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}htn=/summationdisplay\ni∈N|αi|2\nlim\nn→∞/summationdisplay\ni,j∈N¯βiβj/an}b∇acketle{tθ(i)\nn,mi,θ(j)\nn,mj/an}b∇acket∇i}ht=/summationdisplay\ni∈N|βi|2\nlim\nn→∞/summationdisplay\ni,j∈N¯γiγj/an}b∇acketle{tC(√nϕ(i))Ω,C(√nϕ(j))Ω/an}b∇acket∇i}ht= lim\nn→∞/summationdisplay\ni,j∈N¯γiγjeinIm/an}bracketle{tϕ(i),ϕ(j)/an}bracketri}hte−n/vextenddouble/vextenddoubleϕ(j)−ϕ(i)/vextenddouble/vextenddouble2\n/2\n=/summationdisplay\ni∈N|γi|2.\nHence|αi(n)|,|βi(n)|and|γi(n)|are uniformly bounded in n.\nProof.We can take the limit n→ ∞for each term of the summation by the dominated con-\nvergence theorem, because the absolute values of the n-dependent scalar products of vectors\nare bounded by 1, and the suites ( αi), (βi) and (γi) are inl1. Then the first limit is ob-\ntained because /an}b∇acketle{tϕ(i),ϕ(i)/an}b∇acket∇i}ht= 1 for all i∈N, while, by definition 2,/vextendsingle/vextendsingle/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}ht/vextendsingle/vextendsingle<1 for all\ni/ne}ationslash=j. The second limit is analogous, since /an}b∇acketle{tθ(i)\nn,mi,θ(i)\nn,mi/an}b∇acket∇i}ht= 1, while for all m≤lnnwe have15\n/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tθ(i)\nn,mi,θ(j)\nn,mj/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle≤Kn2(lnn+1)/vextendsingle/vextendsingle/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}ht/vextendsingle/vextendsinglen−2lnn→0 whenn→ ∞. The last one is obtained\nbecause, using the properties of Weyl operators, we have:\n/an}b∇acketle{tC(√nϕ(i))Ω,C(√nϕ(j))Ω/an}b∇acket∇i}ht=/an}b∇acketle{tΩ,C(−√nϕ(i))C(√nϕ(j))Ω/an}b∇acket∇i}ht=einIm/an}bracketle{tϕ(i),ϕ(j)/an}bracketri}hte−n/vextenddouble/vextenddoubleϕ(j)−ϕ(i)/vextenddouble/vextenddouble2\n/2.\nThe proof of theorem 2 is similar to the one above for theorem 1 ; we carry it out explicitly only\nfor Φ vectors, the other case being analogous. Define the tran sition amplitude\nTΦ(t,x,y) =/an}b∇acketle{tΦ,eitHa∗(x)a(y)e−itHΦ/an}b∇acket∇i}ht.\nLetξ∈L2(R6). Then using property iii of the number operator Nwe obtain\n/vextendsingle/vextendsingle/an}b∇acketle{tξ,TΦ(t)/an}b∇acket∇i}htL2(R6)/vextendsingle/vextendsingle≤(n+1)/vextenddouble/vextenddoubleξ/vextenddouble/vextenddouble\nL2(R6),\nthenTΦ(t)∈L2(R6) for alln∈N. Also, since/vextenddouble/vextenddoubleΦ/vextenddouble/vextenddouble= 1 by definition 2, we have that\nTrTΦ(t) =/integraldisplay\ndxTΦ(t,x,x) =n.\nFurthermore the series\n/integraldisplay\ndxdy¯ξ(x,y)/summationdisplay\ni∈N|��i(n)|2¯ϕ(i)\nt(x)ϕ(i)\nt(y) =/summationdisplay\ni∈N|αi(n)|2/an}b∇acketle{tξ,¯ϕ(i)\ntϕ(i)\nt/an}b∇acket∇i}htL2(R6)\nis absolutely convergent under the hypotheses of definition 2:\n/summationdisplay\ni∈N|αi(n)|2/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tξ,¯ϕ(i)\ntϕ(i)\nt/an}b∇acket∇i}htL2(R6)/vextendsingle/vextendsingle/vextendsingle≤/vextenddouble/vextenddoubleξ/vextenddouble/vextenddouble\nL2(R6)/vextenddouble/vextenddouble(αi(n))i∈N/vextenddouble/vextenddouble2\nl2.\nTherefore we can write, with dn=√\nn!/exp(−n/2)nn/2,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\ndxdy¯ξ(x,y)/parenleftBig\nTΦ(t,x,y)−n/summationdisplay\ni∈N|αi(n)|2¯ϕ(i)\nt(x)ϕ(i)\nt(y)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/summationdisplay\ni∈Ndn|αi(n)|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tC∗(√nϕ(i)\n0)(ϕ(i))⊗n,\nW∗(t,0)/integraldisplay\ndxdy¯ξ(x,y)/parenleftBig√n(¯ϕ(i)\nta(y)+ϕ(i)\nt(y)a∗(x))+a∗(x)a(y)/parenrightBig\nW(t,0)Ω/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+2/summationdisplay\ni<j|¯αi(n)αj(n)|/parenleftBigg\nn/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsinglen/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tξ,¯ϕ(j)\ntϕ(j)\nt/an}b∇acket∇i}htL2(R6)/vextendsingle/vextendsingle/vextendsingle+dn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tC∗(√nϕ(j)\n0)(ϕ(i))⊗n,W∗(t,0)\n/integraldisplay\ndxdy¯ξ(x,y)/parenleftBig√n(¯ϕ(j)\nta(y)+ϕ(j)\nt(y)a∗(x))+a∗(x)a(y)/parenrightBig\nW(t,0)Ω/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightBigg\n≤/parenleftBigg\nK1eK2|t|(√n+1)/parenleftBig/summationdisplay\ni∈N|αi(n)|2+dn/summationdisplay\ni<j|¯αi(n)αj(n)|/parenrightBig\n+2n/summationdisplay\ni<j|¯αi(n)αj(n)|\n/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsinglen/parenrightbigg/vextenddouble/vextenddoubleξ/vextenddouble/vextenddouble\nL2(R6).16\nBy Riesz’s lemma, keeping in mind that, for large n,dn∼n1/4, we can write\n/vextenddouble/vextenddoubleTφ(t)−n/summationdisplay\ni∈N|αi(n)|2ϕ(i)\ntϕ(i)\nt/vextenddouble/vextenddouble\nL2(R6)≤2n/summationdisplay\ni<j|¯αi(n)αj(n)|/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsinglen\n+K2eK3|t|n3/4\n/parenleftBig/vextenddouble/vextenddouble(αi(n))i∈N/vextenddouble/vextenddouble\nl2+/summationdisplay\ni<j|¯αi(n)αj(n)|/parenrightBig\n.\nDividing by Tr TΦ(t) =nwe obtain the corresponding L2(R6)-bound for the integral kernel of the\none-particle reduced density matrix, hence the bound in Hil bert-Schmidt norm. Since αi(n) are\nuniformly bounded, we obtain:\n(13)/vextenddouble/vextenddoubleTr1ρΦ(t)−/summationdisplay\ni∈N|αi(n)|2|ϕ(i)\nt/an}b∇acket∇i}ht/an}b∇acketle{tϕ(i)\nt|/vextenddouble/vextenddouble\nHS≤K1/summationdisplay\ni<j|¯αi(n)αj(n)|/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsinglen\n+K2eK3|t|1\nn1/4.\nConsider now/vextenddouble/vextenddoubleTr1ρΦ(t)−/vextenddouble/vextenddouble(αi)i∈N/vextenddouble/vextenddouble−2\nl2/summationtext\ni∈N|αi|2|ϕ(i)\nt/an}b∇acket∇i}ht/an}b∇acketle{tϕ(i)\nt|/vextenddouble/vextenddouble\nHS, we can write\n/vextenddouble/vextenddoubleTr1ρΦ(t)−/vextenddouble/vextenddouble(αi)i∈N/vextenddouble/vextenddouble−2\nl2/summationdisplay\ni∈N|αi|2|ϕ(i)\nt/an}b∇acket∇i}ht/an}b∇acketle{tϕ(i)\nt|/vextenddouble/vextenddouble\nHS≤/vextenddouble/vextenddouble/summationdisplay\ni∈N/parenleftBig\n|αi(n)|2−/vextenddouble/vextenddouble(αi)i∈N/vextenddouble/vextenddouble−2\nl2|αi|2/parenrightBig\n|ϕ(i)\nt/an}b∇acket∇i}ht/an}b∇acketle{tϕ(i)\nt|/vextenddouble/vextenddouble\nHS\n+/vextenddouble/vextenddoubleTr1ρΦ(t)−/summationdisplay\ni∈N|αi(n)|2|ϕ(i)\nt/an}b∇acket∇i}ht/an}b∇acketle{tϕ(i)\nt|/vextenddouble/vextenddouble\nHS.\nThe first term satisfies the following inequality:\n/vextenddouble/vextenddouble/summationdisplay\ni∈N/parenleftBig\n|αi(n)|2−/vextenddouble/vextenddouble(αi)i∈N/vextenddouble/vextenddouble−2\nl2|αi|2/parenrightBig\n|ϕ(i)\nt/an}b∇acket∇i}ht/an}b∇acketle{tϕ(i)\nt|/vextenddouble/vextenddouble\nHS≤/summationdisplay\ni<j|¯αi(n)αj(n)|/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tϕ(i),ϕ(j)/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsinglen\n.\nACKNOWLEDGMENTS\nThe author would like to thank Professor Giorgio Velo, for ha ving introduced him to the use of\nθn,mstates for mean field limits, and all other interesting discu ssions.\n[1] N.Benedikter, G.deOliveira,andB.Schlein. QuantitativeDerivat ionoftheGross-PitaevskiiEquation.\nArXiv e-prints , August 2012.\n[2] L. Chen and J. O. Lee. Rate of convergence in nonlinear Hartree dynamics with factorized initial data.\nJournal of Mathematical Physics , 52(5):052108, May 2011. doi:10.1063/1.3589962.\n[3] Li Chen, Ji Oon Lee, and Benjamin Schlein. Rate of convergence towards Hartree dynamics.\nJ. Stat. Phys. , 144(4):872–903, 2011. ISSN 0022-4715. doi:10.1007/s10955-0 11-0283-y. URL\nhttp://dx.doi.org/10.1007/s10955-011-0283-y .\n[4] Marco Falconi. Classical limit of the nelson model with cutoff. Journal of Mathematical Physics , 54\n(1):012303, 2013. doi:10.1063/1.4775716. URL http://link.aip.org/link/?JMP/54/012303/1 .17\n[5] J. Ginibre and G. Velo. The classical field limit of scattering theory f or nonrelativistic\nmany-boson systems. I. Comm. Math. Phys. , 66(1):37–76, 1979. ISSN 0010-3616. URL\nhttp://projecteuclid.org/getRecord?id=euclid.cmp/11 03904940 .\n[6] Jean Ginibre, Fabio Nironi, and Giorgio Velo. Partially classical limit of t he Nelson model.\nAnn. Henri Poincar´ e , 7(1):21–43, 2006. ISSN 1424-0637. doi:10.1007/s00023-005-0 240-x. URL\nhttp://dx.doi.org/10.1007/s00023-005-0240-x .\n[7] F. Golse. On the Dynamics of LargeParticle Systems in the Mean Fie ld Limit. ArXiv e-prints , January\n2013.\n[8] Klaus Hepp. The classical limit for quantum mechanical correlation functions. Comm. Math. Phys. ,\n35:265–277, 1974. ISSN 0010-3616.\n[9] Antti Knowles and Peter Pickl. Mean-field dynamics: singular pote ntials and rate of convergence.\nComm. Math. Phys. , 298(1):101–138, 2010. ISSN 0010-3616. doi:10.1007/s00220-0 10-1010-2. URL\nhttp://dx.doi.org/10.1007/s00220-010-1010-2 .\n[10] Ilia Krasikov. Inequalities for Laguerre polynomials. East J. Approx. , 11(3):257–268, 2005. ISSN\n1310-6236.\n[11] JiOon Lee. Rate of convergence towards semi-relativistic hart ree dynamics. Annales Henri\nPoincar´ e , 14(2):313–346, 2013. ISSN 1424-0637. doi:10.1007/s00023-01 2-0188-6. URL\nhttp://dx.doi.org/10.1007/s00023-012-0188-6 .\n[12] M. Lewin, P. Th` anh Nam, S. Serfaty, and J. P. Solovej. Bogoliu bov spectrum of interacting Bose gases.\nArXiv, to appear on Comm. Pure Appl. Math. , November 2012.\n[13] Peter Pickl. A simple derivation of mean field limits for quantum syst ems. Lett.\nMath. Phys. , 97(2):151–164, 2011. ISSN 0377-9017. doi:10.1007/s11005-01 1-0470-4. URL\nhttp://dx.doi.org/10.1007/s11005-011-0470-4 .\n[14] Michael Reed and Barry Simon. Methods of modern mathematical physics. II. Fourier analys is, self-\nadjointness . Academic Press [Harcourt Brace Jovanovich Publishers], New York , 1975.\n[15] Igor Rodnianski and Benjamin Schlein. Quantum fluctuations an d rate of convergence towards mean\nfield dynamics. Comm. Math. Phys. , 291(1):31–61, 2009. ISSN 0010-3616. doi:10.1007/s00220-009 -\n0867-4. URL http://dx.doi.org/10.1007/s00220-009-0867-4 ."    },    {        "title": "1306.5894v1.High_frequency_vibrational_density_of_states_of_a_disordered_solid.pdf",        "content": "arXiv:1306.5894v1  [cond-mat.dis-nn]  25 Jun 2013High-frequency vibrational density of states of a disorder ed solid\nC. Tomaras1,2and W. Schirmacher1,2,3\n1Institut f¨ ur Physik, Universit¨ at Mainz, Staudinger Weg 7 , D-55099 Mainz, Germany\n2Inst. f. funktionelle Materialien, Phys. Dept. E13, TU M¨ un chen, D-85747 Garching, Germany and\n3Dipartimento di Fisica, Universit´ a di Roma “La Sapienza”, P’le Aldo Moro 2, I-00185, Roma, Italy\nWeinvestigatethehigh-frequencybehaviorofthedensityo fvibrationalstates inthree-dimensional\nelasticity theory with spatially fluctuating elastic modul i. At frequencies well above the mobility\nedge, instanton solutions yield an exponentially decaying density of states. The instanton solutions\ndescribe excitations, which become localized due to the dis order-induced fluctuations, which lower\nthe sound velocity in a finite region compared to its average v alue. The exponentially decaying\ndensity of states (known in electronic systems as the Lifshi tz tail) is governed by the statistics of a\nfluctuating-elasticity landscape, capable of trapping the vibrational excitations.\nPACS numbers: 65.60.+a\nI. INTRODUCTION\nThe density of states (DOS) of a disordered system is\na quantity which has been vividly discussed until today\n[1–4]. Lifshitz has been one of the first authors who cal-\nculated the disorder-induced corrections on the DOS, for\nelectron and phonon systems [5]. By a phenomenological\nargument, he explained the occurrence of an exponential\ntail ofthe DOS in the vicinity ofthe band-edges, which is\nrelated to the localization of the one-particle states. As\nthe penetrability of an arbitrarily shaped barrier tends\ntowards zero at large energies [6], one expects the high-\nenergy eigenstates of a system with a random fluctuating\npotential to be localized. The DOS is then proportional\nto the probability of the occurance of wells, capable of\ntrapping the single-particle states, which for weak fluc-\ntuations (e.g. low impurity concentrations ) can always\nbe approximated by an exponential function. The en-\nergy dependence of the exponent depends crucially on\nthe energy of localization, from which some substantial\ncorrection from the set of maximaly crossed diagrams in\nthe perturbation expansion arises, similar to the energy-\nmomentum relation of a wave in a potential well, if the\nlength scaleset bythe disorderprameterisofthe orderof\nthe localization length. Such corrections can be treated\ninafield-theoreticalapproachbymeansofan ǫ-expansion\nof the nonlinear σmodel in the vicinity of the localiza-\ntion energy [7, 8], or by self-consistent mode coupling ex-\npansion techniques [9], which lead to the potential-well\nanalogy [10–13]. These corrections are important for the\nelectronic problem, because the Lifshitz tail mostly con-\nsists of the ground states of such wells, for which the\nirreducible 1-particle self-energy is energy dependent.\nFor phonons the situation is different, because one\ndeals with fluctuating elastic constants instead of poten-\ntials, and localized states appear only at the upper band\nedge [14, 15] for positive values of ω2[16, 17], where\nωis the vibrational excitation frequency. There is an\nextended amount of literature concerning the cross-over\nfrom the Debye type acoustic wave regime to a regime of\ndiffusive, random-matrix type of vibrational excitations(“boson peak” [18–20]), which has been accurately de-\nscribedwithin the σ-model approachbytwo non-crossing\ntechniques, namely the self-consistent Born [21–24] and\ncoherent-potential approximation [15, 25–27]. Instead\nhere we are interested in the behavior at large positive\nfrequencies. In the σ-model approach one finds that the\none-particle self-energyon the localized side becomes fre-\nquency independent, and the system can be assumed to\nbe described by a renormalized Ginzburg-Landau theory\n[28, 29]. The states with the highest values of ω2within\na region of constant elastic modulus, bounded by a mis-\nmatch, carry wave numbers of the order of the Debye\nwavenumber. Hence these states dominate the Lifshitz\ntail. If one assumes further the fluctuations of the elas-\ntic constants to be small and the localization length to\nscalewith the inversefrequency ξ∝ω−1asinelementary\nwave mechanics [6], the DOS should be proportional to\nexp(−aω4−d). Such a behavior of the exponent is sug-\ngested by recent experiments [30, 31], in which a linear\nexponential decay of the DOS in the high-frequency re-\ngion is found.\nIn the remainder of this paper we will reformulate the\nfield theoretic approach within the Keldysh formulation\nof quantum dynamics, in order to include the instanton\ncontribution, from wich the tail of the DOS can be ex-\ntracted.\nII. KELDYSH FORMULATION OF WEAKLY\nDISORDERED PHONONS\nThe quantum dynamics for the displacement-field\nu(x,t) with spatially dependent elastic moduli will\nbe calculated from the Keldysh partition function\n1 =/integraltext\nDueiS, where the action involves the classical\n(symmetric) and quantum (antisymmetric) linear com-\nbinations of the two replicas acting left and right onto\nthe density matrix describing the inital state [32]. The\nreason for naming them classical and quantum is as fol-\nlows: if the action is related to the Lagrangian of a\nsimple non-relativistic one-particle system, the saddle-\npoint solution of the field theory with uq=u+−u−= 0,2\nucl≡u++u−=u(t)yieldsNewton’sequationofmotion\nfor the particle’s trajectory u(t). Quantum corrections\nlike the appearance of a phase or tunneling contributions\narise from finite expectation values and higher correla-\ntions of the antisymmetric u+−u−linear combination,\nwhich is hence named the quantum component.\nThe action may involve arbitrary nonlinear terms,\nwhich in a classic kinetic approximation gives rise to\nphonon thermalization, at least in three dimensions. For\nthe vibrational spectrum of disordered systems the an-\nharmonic interactions are only important below the bo-\nson peak [33, 34]. For our purpose it is hence enough\nto approximate the Keldysh action to quadratic order in\nthe displacement field,\nS=/integraldisplay\ndd+1x/braceleftbig\n−uq∂2\ntucl+2µ(x)uq\nijucl\nji+λ(x)uq\niiucl\nii/bracerightbig\n+/integraldisplay\ndd+1xuq/parenleftbig\nG−1/parenrightbig\nKuq(1)\nwherex= (x,t) anduij=1\n2(∂iuj+∂jui) is the usual\nstrain-tensor, and λandµare Lam´ e’s elastic constants,\nwhich are assumed to fluctuate due to the structural dis-\norder of the material [21–25]. We use units in which the\nmass density equals unity. The Keldysh component GK\ncharacterises the actual state of the system, which can\nbe determined from knowledge of the retarded and ad-\nvanced Green’s function and a proper initial condition.\nIf one assumes that the disorder does not alter the ther-\nmalisation process, the Keldysh component can safely be\nreplaced by a thermal distribution. The formulation (1)\nignores further initial correlations.\nThe advantage of Keldysh’s closed time-contour is the\nabsence of vacuum contributions to the partition func-\ntion, hence one can avarage it directly over an arbitrary\ndistributionofelasticconstants. Thisaveragecanbeper-\nformed formally by characterizing the probability distri-\nbution through its irreducible correlation functions\nK(x1,...,xn)≡δ\nδx1....δ\nδxnln/integraldisplay\ndµP[µ]e(µ,j),(2)\nwhich leads to an action of the form\nSdis=S−S[∝angb∇acketleftµ∝angb∇acket∇ight,∝angb∇acketleftλ∝angb∇acket∇ight] =\n+/integraldisplay\nx1x2Kµ(x1,x2)uq\nij(x1)ucl\nji(x1)uq\nlm(x2)ucl\nml(x2)\n+/integraldisplay\nx1x2Kλ(x1,x2)uq\nii(x1)ucl\nii(x1)uq\njj(x2)ucl\njj(x2)+....\n(3)\nFor the Gaussian theory the structure of the action is\nthe same as for a dissipative quantum system [35], in\nwhich the spectral function of the heat bath is replaced\nby two classical strain fields. Therefore the calculation of\ninstantonsolutionsindisorderedsystemsexhibitastrong\nresemblance to the calculation of thermal activation of a\nsystem attached to some heat bath [35].Within this article we do not want to perform tech-\nnically detailed calculations. We rather want to demon-\nstrate the general procedure of mapping the disordered\nbosonic system onto a disordered fermionic one within\nthe Keldysh prescription. Of course this mapping is\nonly possible between the pair modes of the bosonic and\nfermionic systems.\nIn general the theory can be represented in terms of\nthese 2-point functions using the Fadeev-Popov transfor-\nmation\nF/bracketleftBig\nuα\nij(x1)uβ\nlm(x2)/bracketrightBig\n=/integraldisplay\nQ,ΛeiTr[(Λ−uu†)Q]\n≡/integraldisplay\nDQDΛF[Qαβ\nijlm(x1,x2)]eiΛαβ\nijlm(Qβα\nmlji−uβ\nml(x1)uα\nji(x2)).\nHere Greek indices represent the Keldysh degrees of free-\ndom. The following mapping achieves the same causality\nstructure for the composite phonons as for the electrons\n[35]\nAR≡A/parenleftbigg\n0 1\n1 0/parenrightbigg\n,AL≡/parenleftbigg\n0 1\n1 0/parenrightbigg\nA.\nIf the matrix Ahas the causality structure of a bosonic\nGreen’s function, ARhas the c. s. of a fermionic Green’s\nfunction and ALthe c.s. of a fermionic self energy. The\nadvantageisnow,thatthe phononnonlinear σ-modelcan\nbe developed along the eletronic counterpart. One can\nnow write the disorder-induced nonlinear terms as\nSdis=/summationdisplay\nnTr/bracketleftbig\nKµ\nnQn\nR+Kλ\nn(χQR)n/bracketrightbig\n(4)\nin which χis a characteristic matrix defined according to\nχQijlm=δijδlmQijlm.\nFor sake of simplicity we will only consider a locally\nfluctuating shear modulus , Kλ\nn= 0,Kµ\n2=γ−1δ(x−y),\nKµ\nn/negationslash=2= 0.\nIn absence of anharmonicities one can immediatly in-\ntegrate out the composite field Λ, which reduces the\nFadeev-Popov procedure to the Hubbard Stratonovich\ntransformation. This could be the starting point to for-\nmulate the nonlinear σmodel approach on phonon local-\nization around the weakly disordered SCBA saddle point\n[21].\nHowever, as also discussed by Cardy [29], in the\nlarge energy regime, where the phonon energy is large\ncompared to the energy-fluctuations set by the disor-\nder potential, the semiclassical one-particle approxima-\ntion of the partition function becomes valid. It is more\nconvenient to discuss the Lifshitz-tail in terms of the\none-particle functions, as one avoids the complicated\nrenormalization-group method. One would agree, that\nthe simple one-particle saddle point is sufficient, as long\nas one is interested in the deep localization regime, where\nknowledge of the mobility edge is lost. In contrast, the\nuniversalpropertiesofvibrationsin a glassin the vicinity\nof the mobility edge [16, 17], must be developed from (4)\nwithin the usual Keldysh nonlinear σmodel.3\nIII. INSTANTON SOLUTIONS\nIn the remainder we will use the simple instanton pic-\nture in order to calculate the exponential dependence of\nthe vibrational density of states at high energies.\nThe first step is to express the partition function into\nan infinte product of discrete frequencies 1 =/producttext\nωZ(ω),\nwhereZ(ω) =eis(ω)\nand\ns(ω) =/integraldisplay\nddx/bracketleftbig\nω2/vector uq/vector ucl+gk/vector uq/vector uq+2µuq\nijucl\nji+λuq\niiucl\nii/bracketrightbig\n(5)\n+/integraldisplay\nddxγ/parenleftbig\nuq\nij(−ω,x)ucl\nji(ω,x)/parenrightbig2\nIf the frequency is sufficiently large, this field theory\ncan be solved in the semiclassical saddle-point approxi-\nmationbyextremizingtheactionwithrespecttotheclas-\nsical and quantum components. In analogy to the theory\nof dissipative quantum systems there exist instanton so-\nlutions in which the expectation value of the quantum\ncomponent ∝angb∇acketleftui∝angb∇acket∇ight=ivq\niis finite and purely imaginary.\nFromtheequationofmotion(6)onededucesthatthese\nsolutions describe a situation, in which condensation of\nstrainivq\nij=−vij=−vcl\nijleads to a finite region with a\nsound velocity substancially lower than the average. The\nstates are hence trapped in this finite region. There is a\nsecond instanton equation describing the other possibil-\nity, where the sound velocity is raised. We seek for an\ninstanton ansatz where the saddle-point solution satisfies\n(ω2−(λ+µ[v])∇∇◦−µ[v]∆)/vector v(ω,x) = 0 (6)\nµ[v] =µ(1−γvijvji).\nReinsertion of this equation of motion (6) into (5)\nyields a finite action if(ω), and hence an exponential fac-\ntor of the partition function e−f(ω), which is calculated\nvia formula (12).\nIn order to calculate the density of states and the\nGreen’s function we have to expand (5) with respect to\nthe fluctuations above this instanton saddle-point uα\nij=\nvij+δuα\nij. The irreducible retarded Green’s function is\nby definition just the correlator of ∝angb∇acketleftδucl\niδjuq∝angb∇acket∇ight.Obviously\nthis quantity is proportional to the factor e−f(ω)which\nis nothing than the exponential Lifshitz-tail. The action\n(5) expanded among the instanton saddlepoint reads\ns(ω) =if(ω)+O[δu3\nij,δu4\nij]+/integraldisplay\nddxω2δ/vector uqδ/vector ucl(7)\n+/integraldisplay\nddx/parenleftbigg\n2µ(1−2γ\nµ)δuq\nijvhkvkhδucl\nji+λδuq\niiδucl\nii/parenrightbigg\n(8)\nThe next step is to use the Bosonization-procedure\n(5) to express this action in terms of the pair modes\nQR=∝angb∇acketleftδucl\niδuq\nj∝angb∇acket∇ight.Using a further saddle-point approxi-\nmation in order to determine QRyields to the followingset of equations\n∝angb∇acketleftQR(ω,x)∝angb∇acket∇ight\n≡iγe−f/parenleftBigg\n1\nG−1\n0(ω,x)−∇(Σ+vijvji)ω,x∇T+/vector v/vector vT/parenrightBigg\n(9)\nΣR(ω,x)∝e−fγTr∇1\nG−1\n0(ω,x)−∇(Σ+vijvji)ω,x∇T∇T\n(10)\nf(ω) =/integraldisplay\nx/braceleftbig\nω2|/vector v(ω,x)|2+2µvijvji+λv2\nii/bracerightbig\n(11)\nin which the fields satisfies the exact equation of mo-\ntion\nδ\nδvαs(ω) =δ\nδvα/braceleftbig\n/vector vT(G−1\n0+∇Σ∇T)/vector v+Sdis[v]/bracerightbig\n= 0.\n(12)\nThese equations have a rather simple interepretation;\nthey allow for a self-consistent determination of the in-\nstanton solution in Hartree-approximation and a fur-\nther non-crossing approximation of the self-energy of the\npropagator, which has a frequency and spatially depen-\ndent sound-velocity, due to the finite instanton ampli-\ntude. However, we are mostly interested in a discussion\nofthe exponentialfactorandleavethe numericalsolution\nof this set of equations for future work.\nFrom (9) it becomes clear that within this approx-\nimation the density of states is just the usual one in\nSCBA approximation, multiplied with the instanton fac-\ntore−f(ω). In order to estimate the power-law satisfied\nby the exponent of the DOS we look for a spherically\nsymmetric solution with longitudinal polarisation in 3\ndimensions.\nThe equation for the radial part reads\nω2Ψ(r)+(1−4γ(∂rΨ)2))∆rΨ(r) = 0 (13)\nThe frequency and the disorder parameter can imme-\ndiatly be scaled out according to φ=1\nω√γΨ(ωr), where\nthe spatially dependent part Ψ( y=ωr) satisfies the re-\nduced equation\nΨ(y)+(1−4(∂yΨ)2))∆yΨ(y) = 0 (14)\nFor numerical solution one replaces (14) with the first-\norder system\n∂rΨ(r) =φ(r)\n∂rφ(r) = Ψ(1−4φ(r)2)−1−2r−1φ(r)\nwhich may readily be solved by means of a second-order\nRunge Kutta alogrithm.4\nThe phase can be estimated at large frequencies\nf=/integraldisplay\nddrφ(r)(ω2+∆2\nr)φ(r)\n=1\nγω−d/integraldisplay\nddyΨ(y)(1+∆ y)Ψ(y)\n=/integraldisplay\nq<qDωddq\n(2π)3Ψ(q)(1+q2)Ψ(q)\nω→ ∞:1\nγ/parenleftbiggω\nωD/parenrightbigg4−d/integraldisplayddq\n(2π)3Ψ(q)Ψ(q) =1\nγ/parenleftbiggω\nωD/parenrightbigg4−d\n(15)\nFrom (14) it is clear, that there exists a localized ex-\nponential solution, as long as the state has a single point\nwhere the condensed dimensionless strain exceeds the\ncritical value ∂yΨ(y) =1\n2, which can always be imposed\nas a boundary condition. Note that qis not the Fourier\ncomponent of the spatial variable r, but of the frequency\nscaled variable y=ωr.\nIn contrast to the electronic calculation the phase isdominated by the small-distance behavior of the wave\nfunction. As a result we find the same power law of the\nexponent as predicted by the Lifshitz-argument.\nIV. SUMMARY AND CONCLUSION\nIn this work, we formulated the theory of weakly fluc-\ntuating elastic constants of a glass within the Keldysh-\nformalism, in orderto proposean expressionforthe high-\nfrequency behavior of the density of states of a glass. In\nthree dimensions, where this approach is valid, it yields\nan exponential decaying density of states. This expo-\nnential decay is expected by basic Lifshitz-like consider-\nations, and has also been observed in recent experiments\n[30, 31]. If one measures this exponential decay in an\nexperiment, it is therefore straightforward to extract the\ndisorder parameter γ, which can be compared with the\nvalues extracted from experiments and simulations for\ndescribing the boson peak [21–25].\n[1] B. Velick´ y, Phys. Rev. 184, 614 (1969).\n[2] R. J. Elliott, J. A. Krumhansl, and P. L. Leath, Rev.\nMod. Phys. 46, 465 (1974).\n[3] F. Yonezawa and M. Watabe, Phys. Rev. B 11, 4746\n(1975).\n[4] D. Vollhardt, arxiv: 1004.5069v3 (2010).\n[5] I. Lifshitz, Advances in Physics 13, 483 (1964).\n[6] L. D.Landau andE. M. Lifshitz, Nonrelativistic quantum\nmechanics (Mir, Moscow, 1967).\n[7] S. Hikami, Physical Review B 24, 2671 (1981).\n[8] A. McKane and M. Stone, Annals of Physics 131, 36\n(1981).\n[9] D. Vollhardt and P. W¨ olfle, Physical Review B 22, 4666\n(1980).\n[10] E. Economou, C. Soukoulis, M. Cohen, and A. Zdetsis,\nPhysical Review B 31, 6172 (1985).\n[11] E. Economou and C. Soukoulis, Physical Review B 28,\n1093 (1983).\n[12] E. Economou, C. Soukoulis, and A. Zdetsis, Physical Re-\nview B30, 1686 (1984).\n[13] C. Soukoulis, M. Cohen, and E. Economou, Physical re-\nview letters 53, 616 (1984).\n[14] S. John, H. Sompolinsky, and M. J. Stephens, Phys. Rev.\nB27, 5592 (1983).\n[15] W. Schirmacher, G. Diezemann, and C. Ganter, Physical\nreview letters 81, 136 (1998).\n[16] S. D. Pinski, W. Schirmacher, and R. R¨ omer, Europhys.\nLett.97, 16007 (2012).\n[17] S. D. Pinski, W. Schirmacher, T. Whall, and R, J. Phys:\nCondens. Matter 24, 405401 (2012).\n[18] S. R. Elliott, The Physics of Amorphous Materials\n(Longman, New York, 1984).\n[19] J. Horbach, W. Kob, and K. Binder, Eur. Phys. J. B 19,\n531 (2001).\n[20] T. Nakayama, Rep. Prog. Phys. 65, 1195 (2002).\n[21] W. Schirmacher, EPL (Europhysics Letters) 73, 892(2006).\n[22] W. Schirmacher, G. Ruocco, and T. Scopigno, Physical\nreview letters 98, 25501 (2007).\n[23] W. Schirmacher, B. Schmid, C. Tomaras, G. Viliani,\nG. Baldi, G. Ruocco, and T. Scopigno, physica status\nsolidi (c) 5, 862 (2008).\n[24] A. Marruzzo, W. Schirmacher, A. Fratalocchi, and\nG. Ruocco, Nature Scientific Reports 3, 1407 (2013).\n[25] A. Marruzzo, S. K¨ ohler, A. Fratalocchi, G. Ruocco, and\nW. Schirmacher, Eur. Phys. J. Special Topics 216, 83\n(2013).\n[26] W. Schirmacher, phys. stat. sol. (b) 250, 937 (2013).\n[27] S. K¨ ohler, G. Ruocco, and W. Schirmacher, Phys. Rev.\nB, to appear (2013).\n[28] S.JohnandM.Stephen,Journal ofPhysicsC:SolidState\nPhysics17, L559 (1984).\n[29] J. Cardy, Journal of Physics C: Solid State Physics 11,\nL321 (1978).\n[30] A. Chumakov, I. Sergueev, U. Van B¨ urck, W. Schirma-\ncher, T. Asthalter, R. R¨ uffer, O. Leupold, and W. Petry,\nPhysical review letters 92, 245508 (2004).\n[31] M. Walterfang, W. Keune, E. Schuster, A. T. Zayak,\nP. Entel, W. Sturhahn, T. S. Toellner, E. E. Alp, P. T.\nJochym, and K. Parlinski, Phys. Rev. B 71, 035309\n(2005).\n[32] A. Levchenko and A. 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Freericks1\n1Department of Physics Geogetown University, Washington, DC 20057,USA\n2Stanford Institute for Materials and Energy Sciences,\nSLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305, USA\n3Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA\n(Dated: August 25, 2021)\nCharge-density-wave systems have a static modulation of the electronic charge at low temperatures\nwhen they enter an ordered state. While they have been studied for decades in equilibrium, it is\nonly recently that they have been examined in nonequilibrium with time-resolved studies. Here, we\npresent the exact solution for the nonequilibrium response of electrons (in the simplest model for a\ncharge density wave) when the system is placed under a strong DC electric \feld. This allows us to\nexamine the formation of driven Bloch oscillations and how the presence of a current modi\fes the\nnonequilibrium density of states.\nPACS numbers: 72.10.-d, 72.20.Ht, 71.45.Lr, 71.10.Fd\nI. INTRODUCTION\nCharge-density-wave (CDW) behavior occurs in a wide\nvariety of di\u000berent materials. Peierls1originally showed\nthe instability of a one-dimensional metal to a distor-\ntion with a unit cell twice as large as in the uniform\nphase, that opens an insulating gap at the Fermi level.\nSince then, CDW behavior has been seen in many dif-\nferent materials and in higher dimensions as well. Still,\nthe precise origin of CDW behavior in real materials is\ncontroversial2{4|is it arising from electron-phonon in-\nteractions and a softening phonon or from a purely elec-\ntronic instability in the static charge susceptibility or via\nan instability driven by the electron-phonon coupling?\nWe do not investigate that question here, but instead\nfocus on the behavior in the nonequilibrium state.\nRecently, nonequilibrium photoelectron pump-probe\nexperiments have been carried out for a number of\ndi\u000berent CDW systems, with both valence electron\nphotoemission5{9and core-hole photoemission10. In\nthese experiments, the charge density wave material\nis pumped with an infrared laser pulse and displays\nnonequilibrium melting of the CDW state, which is il-\nlustrated by a \flling in of the gap in the photoemission\nspectrum, while the system still retains its modulation of\nthe electronic charge in the ordered CDW phase. This\nphenomena has already been examined with an exactly\nsolvable model11.\nThe simplest model for a CDW insulator is to start\nwith a system that can be divided into two sublattices,\ncalledAandBand having hopping only between the two\nsublattices. Then we pick an on-site energy to be equal\ntoUon theAsublattice and 0 on the Bsublattice. The\nequilibrium Hamiltonian becomes\nH=−/summation.disp\nijtijc†\nicj+/summation.disp\ni∈A(U−\u0016)c†\nici+/summation.disp\ni∈B(−\u0016)c†\nici:(1)\nHere c†\niandciare the creation and annihilation operators\nfor a spinless fermion at site i. The operators satisfy thecanonical anticommutation relations\n{ci;c†\nj}+=\u000eij; (2)\nand\n{ci;cj}+={c†\ni;c†\nj}+=0 (3)\nwhere the+subscript denotes the anticommutator of the\ntwo operators. In Eq. (1), \u0016is the chemical potential\nandUis the aforementioned site energy. The electrons\nare allowed to hop between nearest neighbors with a hop-\nping matrix −tij, which is a real and symmetric matrix\nthat equals −tforiandjnearest neighbors and van-\nishes otherwise. When this Hamiltonian is diagonalized\n(see below), it forms two bands, so \flling the electrons\nhalfway (one electron per two lattice sites), yields an in-\nsulating phase. Because the site energy is \fxed and never\nvaries, the CDW order is always frozen in, and remains\nfor all \fnite temperatures. While this might seem like an\nextreme limit, it should describe the behavior of exper-\nimentally studied CDW systems for short times, before\nthe phonons are able to relax the system and reduce or\neliminate the e\u000bective site energy (which arose from the\nphonon distortion in the ordered phase). Nevertheless,\nbecause this model can be solved exactly, it provides an\ninteresting limit for other (more accurate) model calcu-\nlations and displays interesting nonequilibrium behavior\nthat has a number of nontrivial results.\nThe organization of this paper is as follows. In Section\nII, we develop the formalism for the exact solution of the\nnonequilibrium problem. In Section III, we present our\nsolutions for the Bloch oscillations and nonequilibrium\ndensity of states, and we conclude in Section IV.\nII. FORMALISM\nIn this section, we will \frst describe how to solve for the\nequilibrium bandstructure of the CDW and then we will\nshow how to employ the Peierls substitution to describe\nthe nonequilibrium solution.arXiv:1308.6060v1  [cond-mat.str-el]  28 Aug 20132\nA. Equilibrium formalism\nThe lattice translational symmetry is broken when U\nis nonzero, resulting in a doubling of the size of the unit\ncell in real space, and a halving of the Brillouin zone in\nreciprocal space. Hence, the conversion from real space to\nmomentum space is more complicated than in a system\nwith one atom per unit cell. The momentum points k\nandk+Qare coupled, where Q=(\u0019;\u0019;::: )due to the\npresence of the CDW order. The transformation from\nreciprocal space to real space becomes\nc†\ni=/summation.disp\nk(e−ik⋅Ric†\nk+e−i(k+Q)⋅Ric†\nk+Q); (4)\nwhere the sum is over the reduced Brillouin zone, and\nRiis the position vector for the ith lattice site. Since\ne−iQ⋅Ris equal to one for lattice sites on the Asublat-\ntice and minus one on the Bsublattice, we have explicit\nexpressions\nc†\ni∈A=/summation.disp\nke−ik⋅Ri(c†\nk+c†\nk+Q); (5)\nc†\nj∈B=/summation.disp\nke−ik⋅Rj(c†\nk−c†\nk+Q): (6)\nThe corresponding annihilation operator identities are\nfound by taking the respective hermitian conjugates.\nIf we write the electronic band structure at U=0 as\n\"k, then we have\n\"k=−/summation.disp\nijtijexp[−ik⋅(Ri−Rj)] (7)\n=−2td\n/summation.disp\nl=1cos(kla)=−lim\nd→∞t∗\n√\ndd\n/summation.disp\nl=1cos(kla);\nwhere we assumed the hopping was only between near-\nest neighbors on a d-dimensional hypercubic lattice and\nsatis\fedt=t∗/slash.left(2√\nd)in the limit of large dimensions in\nthe second line (we will present results in the in\fnite-\ndimensional limit here, but the formalism gives the exact\nsolution in any dimension). Restricting kto the reduced\nBrillouin zone, is equivalent to having \"k≤0. In Eq. (7),\nlis the index for spatial component along an axial di-\nrection,ais the lattice constant, and dis the number of\nspatial dimensions.\nThe Hamiltonian in Eq. (1) can now be written in a\n2×2/ucurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/ucurlyrightck;ck+Q\n/dcurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlyrightbasis via\nH=/summation.disp\nk/ucurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/ucurlyright\nc†\nkc†\nk+Q\n/dcurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlyright/parenleft.alt4U\n2−\u0016+\"kU\n2U\n2U\n2−\u0016−\"k/parenright.alt4/parenleft.alt3ck\nck+Q/parenright.alt3:\n(8)\nThis is diagonalized via the following eigenfunction basis,\nck+=\u000bkck+\fkck+Q; (9)\nck−=\fkck−\u000bkck+Q; (10)with\u000bkand\fksatisfying\n\u000bk=U\n2/radical.alt4\n2/parenleft.alt3\"2\nk+U2\n4−\"k/radical.alt2\n\"2\nk+U2\n4/parenright.alt3(11)\nand\n\fk=−\"k+/radical.alt2\n\"2\nk+U2\n4/radical.alt4\n2/parenleft.alt3\"2\nk+U2\n4−\"k/radical.alt2\n\"2\nk+U2\n4/parenright.alt3: (12)\nThe operators c†\nk+andc†\nk−create electrons in the upper\nand lower bands, respectively. The Hamiltonian matrix\nis then diagonalized as follows:\nH=/summation.disp\nk/parenleft.alt1\"k+c†\nk+ck++\"k−c†\nk−ck−/parenright.alt1: (13)\nHere,\"k+and\"k−are given by\n\"k±=U\n2−\u0016±/radical.alt4\n\"2\nk+U2\n4: (14)\nWe will also want to work with Green's functions. The\nlocal retarded Green's function is de\fned by the following\nformula in equilibrium\nGR\nA;B(t)=−i\u0012(t−t′)/summation.disp\nk/uni27E8{ck(t)±ck+Q(t);c†\nk(0)±c†\nk+Q(0)}+/uni27E9\n(15)\nwhere ^O(t)=eiHtOe−iHtis the operator representa-\ntion in the Heisenberg picture and the angle brack-\nets are a shorthand for the trace over all states\nweighted by the density matrix, which is equal to\nexp[−\fH]/slash.left(Tr exp[−\fH]), with\fthe inverse tempera-\nture (in this work we will start the system from zero\ntemperature or \f→∞). The plus sign represents the A\nsublattice and the minus sign represents the Bsublattice.\nIn order to explicitly determine the Green's function,\nit is more convenient to work in the diagonalized basis,\nwhere the local retarded Green's function on the Asub-\nlattice becomes,\nGR\nA(t)=−i\u0012(t−t′)/summation.disp\nk/uni27E8{(\u000bk+\fk)ck+(t)+(\fk−\u000bk)ck−(t);\n(\u000bk+\fk)c†\nk+(0)+(\fk−\u000bk)c†\nk−(0)}+/uni27E9: (16)\nThe time evolution of the ck+(t)andck−(t)operators is\nfound by solving their equations of motion to yield\nc†\nk+(t)=exp(i\"k+t)c†\nk+(0) (17)\nc†\nk−(t)=exp(i\"k−t)c†\nk−(0): (18)\nThe local equilibrium density of states (DOS) Ai(!)=\n−ImGR\ni(!)/slash.left\u0019then becomes\nAA;B(!)=Re/uni23A1/uni23A2/uni23A2/uni23A2/uni23A2/uni23A3/roottop\n/rootmod/rootmod/rootbot!±U\n2\n!∓U\n2/uni23A4/uni23A5/uni23A5/uni23A5/uni23A5/uni23A6\u001a/uni239B\n/uni239D/radical.alt4\n!2−U2\n4/uni239E\n/uni23A0; (19)3\nFIG. 1. (Color online.) Equilibrium density of states for\nU=0:5 at half \flling. The red curve is the local DOS on the\nAsublattice, the blue curve on the Bsublattice and the black\ncurve is the average local DOS.\nwith\u001a(\u000f)the noninteracting DOS at U=0 (and given\nby\u001a(\u000f)=exp(−\u000f2)/slash.left(t∗√\u0019)in the in\fnite-dimensional\nlimit). The average DOS is found by summing over the\nlocal DOS for each sublattice with weight 1 /slash.left2. Fig. 1\nshows the equilibrium DOS for U=0:5. Note that the\nband gap is equal to the onsite potential U, which shows\nthe system is an insulator for nonzero U. The local DOS\non theAsublattice (red line) has a divergence at !=U/slash.left2,\nwhile the local DOS on the Bsublattice (blue line) has\na divergence at !=−U/slash.left2 (the black curve is the average\nlocal DOS).\nIn equilibrium, the local lesser Green's function, de-\n\fned by\nG<\ni(t)=i/uni27E8c†\ni(0)ci(t)/uni27E9; (20)\nsatis\fesG<\ni(!)=−2if(!)ImGR\ni(!), withf(!)=1/slash.left[1+\nexp(\f!)]the Fermi-Dirac distribution function. At T=\n0,f(!)vanishes for positive frequency and is equal to\none for negative frequency, so the local equilibrium lesser\nfunction at T=0 is simply given by the left hand side of\nFig. 1.\nThe lesser Green's function also gives us the local den-\nsity of electrons on each sublattice\nnA;B=Im[G<\nA;B(t=0)]=/integral.disp∞\n−∞d!\n2\u0019Im/bracketleft.alt1G<\nA;B(!)/bracketright.alt:(21)\nThe equilibrium order parameter for the conduction elec-\ntrons is just the di\u000berence between the electron number\ndensity on the AandBsublattice:\n\n=nB−nA\nnA+nB: (22)\nSince there is a repulsive potential on the Asublattice,\nthere are always more electrons on the Bsublattice than\nFIG. 2. (Color online.) Equilibrium CDW order parameter\nas a function of Uat zero temperature.\non theAsublattice in equilibrium and the order parame-\nter is maximal for \fxed Uwhen the temperature is equal\nto zero. The equilibrium order parameter at zero tem-\nperature is plotted in Fig. 2 as a function of the onsite\npotentialU. Formulas for empirical \fts for small and\nlargeUare also shown.\nB. Nonequilibrium formalism\nIn the case of nonequilibrium, the Hamiltonian be-\ncomes time dependent due to the presence of an elec-\ntric \feld. We include this electric \feld via the Peierls'\nsubstitution12, which is a simpli\fed semiclassical treat-\nment of the electromagnetic \feld that is exact and non-\nperturbative. With the Peierls' substitution, the hopping\nmatrix gains a time dependent phase factor13due to the\n\feld\ntij→tij(t)=tijexp/bracketleft.alt4−ie\n/uni0335hc/integral.dispRj\nRiA(r;t)⋅dr/bracketright.alt4: (23)\nThis result just follows from the phase a particle picks up\nwhen moving under the in\ruence of a vector potential\nand is su\u000ecient to describe the electric \feld when we\nwork in a gauge where there is zero scalar potential and\nonly a time-dependent vector potential. The electric \feld\nE(r;t)is found from the temporal derivative of the vector\npotential A(r;t).\nE(r;t)=−1\nc@A(r;t)\n@t: (24)\nWe will further assume that the electric \feld is spa-\ntially uniform, even when it is time dependent, neglecting\nthe time-dependent magnetic \feld required by Maxwell's\nequations since those e\u000bects are much smaller than the4\nelectric \feld e\u000bects. We remark that the case considered\nhere is one of the few in which Peierls' inclusion of the\nvector potential can be done gauge-invariantly, yet still\nresult in photon-assisted transitions across the band gap.\nAs pointed out in Ref. 14, a Peierls' construction often\nhas limitations in multiband systems. However, since the\ngap in this model results from a site-potential that cou-\nples electrons with the same symmetry, interband tran-\nsitions can be driven by the \feld via the Peierls' substi-\ntution.\nWe choose the spatially uniform \feld to lie along the\ndiagonal direction, so A(t)=A(t)(1;1;:::; 1). The time-\ndependent bandstructure in momentum space for the U=\n0 case then becomes,\n\"k(t)=−/summation.disp\nijtijexp[−i(k−e\n/uni0335hcA(t))⋅(Ri−Rj)]:(25)\nSo the e\u000bect of the Peierls' substitution is to add a time-\ndependent shift to the momentum in the noninteracting\nelectronic band structure:\n\"k(t)=−lim\nd→∞t∗\n√\ndd\n/summation.disp\nl=1cos/bracketleft.alt4a/parenleft.alt4kl−eA(t)\n/uni0335hc/parenright.alt4/bracketright.alt4: (26)\nThe time-dependent Hamiltonian in the Schr odinger pic-\nture then becomes\nHS(t)= (27)\n/summation.disp\nk/ucurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/ucurlyright\nc†\nkc†\nk+Q\n/dcurlyleft/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/udcurlymod/dcurlyright/parenleft.alt4U\n2−\u0016+\"k(t)U\n2U\n2U\n2−\u0016−\"k(t)/parenright.alt4/parenleft.alt3ck\nck+Q/parenright.alt3\nin momentum space. The time-dependent band structure\n\"k(t)can be expanded with the di\u000berence formula of the\ncosine (for the diagonal \feld) via\n\"k(t)=cos/parenleft.alt4eaA(t)\n/uni0335hc/parenright.alt4\"k+sin/parenleft.alt4eaA(t)\n/uni0335hc/parenright.alt4\u0016\"k (28)\nwhich depends on the band structure \"(k)and the pro-\njection of the velocity along the \feld\n\u0016\"k=−lim\nd→∞t∗\n√\ndd\n/summation.disp\nl=1sin(akl): (29)\nIn the Heisenberg picture, we can write the equation of\nmotion for the operators ck(t)andck+Q(t),\nidck(t)\ndt=−[HH(t);ck(t)] (30)\nand\nidck+Q(t)\ndt=−[HH(t);ck+Q(t)]; (31)\nwhereHH(t)is the Heisenberg representation for the\nHamiltonian. If we substitute in the time-dependent\nHamiltonian and evaluate the commutators, we have\nidck(t)\ndt=/summation.disp\nk/bracketleft.alt3/parenleft.alt3U\n2−\u0016+\"k(t)/parenright.alt3ck(t)+U\n2ck+Q(t)/bracketright.alt3(32)idck+Q(t)\ndt=/summation.disp\nk/bracketleft.alt3U\n2ck(t)+/parenleft.alt3U\n2−\u0016−\"k(t)/parenright.alt3ck+Q(t)/bracketright.alt3:\n(33)\nThen we have the time evolution for the annihilation op-\nerators satis\fes\n/parenleft.alt3ck(t)\nck+Q(t)/parenright.alt3=U(k;t;t 0)/parenleft.alt3ck(t0)\nck+Q(t0)/parenright.alt3: (34)\nThe time-evolution operator U(k;t;t′)is a time ordered\nproduct for each momentum\nU(k;t;t′)= (35)\nTtexp/bracketleft.alt4−i/integral.dispt\nt′d\u0016t/parenleft.alt4U\n2−\u0016+\"k(\u0016t)U\n2U\n2U\n2−\u0016−\"k(\u0016t)/parenright.alt4/bracketright.alt4:\nSince there are time-dependent terms inside the expo-\nnential, we must numerically calculate the time evolution\nU(k;t;t′)by employing the Trotter formula:\nU(k;t;t′)=U(k;t;t−\u0001t)U(k;t−\u0001t;t−2\u0001t)/uni22EFU(k;t′+\u0001t;t′):\n(36)\nFor a small time step \u0001 tat timet, we have\nU(k;t;t−\u0001t)= (37)\nexp/bracketleft.alt4−i\u0001t/parenleft.alt4U\n2−\u0016+\"k(t−\u0001t/slash.left2)U\n2U\n2U\n2−\u0016−\"k(t−\u0001t/slash.left2)/parenright.alt4/bracketright.alt4:\nThis exponential can be exactly found since it is a 2 ×2\nmatrix, and we show the result for the case of interest of\nhalf-\flling, where \u0016=U/slash.left2\nU(k;t;t−\u0001t)=cos/uni239B\n/uni239D\u0001t/radical.alt4\n\"2\nk/parenleft.alt3t−\u0001t\n2/parenright.alt3+U2\n4/uni239E\n/uni23A0I\n−i/parenleft.alt4\"k/parenleft.alt1t−\u0001t\n2/parenright.alt1U\n2U\n2−\"k/parenleft.alt1t−\u0001t\n2/parenright.alt1/parenright.alt4\n×sin/parenleft.alt3\u0001t/radical.alt2\n\"2\nk/parenleft.alt1t−\u0001t\n2/parenright.alt1+U2\n4/parenright.alt3\n/radical.alt2\n\"2\nk/parenleft.alt1t−\u0001t\n2/parenright.alt1+U2\n4: (38)\nIn our calculations, we must start from a \fnite minimum\ntime instead of tmin→−∞, so we calculate the time-\nevolution operator from a minimal time t0:U(k;t;t 0).\nFor eachk, we \fnd the two-time evolution operator from\nthe identity\nU(k;t;t′)=U(k;t;t 0)U†(k;t0;t′): (39)\nOnce the time evolution at each time pair is found, we\nthen calculate the nonequilibrium Green's functions to\nobtain the physical properties of the system. Therefore,\none can see that the exact solution in the nonequilibrium\ncase is much more complex than the equilibrium solution\ndue to the change in time of the instantaneous eigenba-\nsis; this then requires signi\fcant numerical resources to\nproperly solve the problem.\nThe contour-ordered single-particle Green's function\nalong the Kadano\u000b-Baym-Keldysh contour (see Fig. 3)5\nFIG. 3. Kadano\u000b-Baym-Keldysh contour. The contour\nevolves from a minimal time to a maximal one, then evolves\nbackwards to the minimal one, and \fnally evolves parallel to\nthe negative imaginary time axis out to a distance given by\n\f=1/slash.leftT.\nis de\fned as\nGc\nij(t;t′)=−i/uni27E8Tcci(t)c†\nj(t′)/uni27E9\n=−i\u0012c(t;t′)/uni27E8ci(t)c†\nj(t′)/uni27E9+i\u0012c(t′;t)/uni27E8c†\nj(t′)ci(t)/uni27E9;(40)\nwhere\u0012cis the generalization of the unit step function to\nthe contour and vanishes if tis beforet′on the contour\nand is equal to one if tis aftert′on the contour. The\ncontour-ordering operator Tcorders the two operators\na(t)andb(t′)according to which is \frst along the contour\n(later along the contour on the left), and we have\nTc{a(t)b(t′)}=/braceleft.alt3a(t)b(t′)t′beforeton contour\n−b(t′)a(t)t′afterton contour\n(41)\nThe minus sign arises because a(t)andb(t′)are fermionic\noperators.\nThe retarded and lesser Green's functions continue to\nbe de\fned by\nGR\nij(t;t′)=−i\u0012(t−t′)/uni27E8{ci(t);c†\nj(t′)}+/uni27E9 (42)\nand\nG<\nij(t;t′)=i/uni27E8c†\nj(t′)ci(t)/uni27E9 (43)\nwhere the operators are in the Heisenberg picture with re-\nspect to the time-dependent Hamiltonian, but they now\ndepend on two times instead of just on the time di\u000ber-\nence.\nAssuming the system starts at a time t0well before the\n\feld is turned on, so the system is in equilibrium, we can\nintroduce the evolution operators for the time-dependent\ncreation and annihilation operators and evaluate the re-\ntarded Green's function as follows\nGR\nii(t;t′)=−i\u0012(t−t′) (44)\n×/uni27E8{ck(t0)U11(k;t;t 0)+ck+Q(t0)U12(k;t;t 0)\n±ck(t0)U21(k;t;t 0)±ck+Q(t0)U22(k;t;t 0);\nc†\nk(t0)U†\n11(k;t0;t′)+c†\nk+Q(t0)U†\n21(k;t0;t′)\n±c†\nk(t0)U†\n12(k;t0;t′)±c†\nk+Q(t0)U†\n22(k;t0;t′)}+/uni27E9:The symbols Uab(k;t;t′)andU†\nab(k;t;t′)represent the el-\nements at row aand column bof the evolution matrices\nU(k;t;t′)andU†(k;t;t′). Evaluating the anticommuta-\ntor, and using the identity of the evolution matrices in\nEq. (39), then shows that the local retarded Green's func-\ntion is just a function of the time-evolution between the\ntwo timestandt′:\nGR\nii(t;t′)=−i\u0012(t−t′)/summation.disp\nk{U11(k;t;t′)+U22(k;t;t′)\n±U12(k;t;t′)±U21(k;t;t′)}: (45)\nThe plus sign is for i∈Asublattice, while the minus sign\nis fori∈Bsublattice because of the phase shift exp (±i/uni20D7Q⋅\nRA)=1 and exp (±i/uni20D7Q⋅RB)=−1 . The retarded Green's\nfunction determines the character of the quantum states\nof the system and does not depend on the history, which\nis why it depends only on the times between tandt′.\nNote that this is not the same as saying it depends on the\ntime di\u000berence t−t′since the retarded Green's function\ndoes change with average time and the evolution operator\nis a complicated function of tandt′(see below).\nNow we can introduce Wigner's average and relative\ntime coordinates15which are de\fned via,\ntrel=t−t′; tave=t+t′\n2: (46)\nThe Fourier transform of the local retarded Green's func-\ntion with respect to the relative time for a \fxed average\ntime\nGR\nii(!;tave)=/integral.dispdtrelei!trelGR\ni(trel;tave) (47)\nyields the transient nonequilibrium local DOS\nAii(!;tave)=−ImGR\nii(!;tave)/slash.left\u0019.\nThis local DOS on each sublattice satis\fes a num-\nber of exact sum-rule relations even in nonequilibrium16\nwhich are proved in Appendix A. This includes the zeroth\nthrough third moment.\nIn addition to the retarded Green's function, we also\nneed the lesser Green's function to calculate the current,\nthe kinetic and potential energy, the CDW order param-\neter, and the \fllings in di\u000berent bands as functions of\ntime. With the Fourier transform of the de\fnition of\nthe lesser Green's function in Eq. (43), we can write the\nmomentum-dependent lesser Green's functions as\nG<\n11(k;t;t′)=i/uni27E8c†\nk(t′)ck(t)/uni27E9 (48)\n=iU†\n11(k;t0;t′)U11(k;t;t 0)/uni27E8c†\nk(t0)ck(t0)/uni27E9\n+iU†\n11(k;t0;t′)U12(k;t;t 0)/uni27E8c†\nk(t0)ck+Q(t0)/uni27E9\n+iU†\n21(k;t0;t′)U11(k;t;t 0)/uni27E8c†\nk+Q(t0)ck(t0)/uni27E9\n+iU†\n21(k;t0;t′)U12(k;t;t 0)/uni27E8c†\nk+Q(t0)ck+Q(t0)/uni27E9;6\nG<\n12(k;t;t′)=i/uni27E8c†\nk+Q(t′)ck(t)/uni27E9 (49)\n=iU†\n11(k;t0;t′)U21(k;t;t 0)/uni27E8c†\nk(t0)ck(t0)/uni27E9\n+iU†\n11(k;t0;t′)U22(k;t;t 0)/uni27E8c†\nk(t0)ck+Q(t0)/uni27E9\n+iU†\n21(k;t0;t′)U21(k;t;t 0)/uni27E8c†\nk+Q(t0)ck(t0)/uni27E9\n+iU†\n21(k;t0;t′)U22(k;t;t 0)/uni27E8c†\nk+Q(t0)ck+Q(t0)/uni27E9;\nG<\n21(k;t;t′)=i/uni27E8c†\nk(t′)ck+Q(t)/uni27E9 (50)\n=iU†\n12(k;t0;t′)U11(k;t;t 0)/uni27E8c†\nk(t0)ck(t0)/uni27E9\n+iU†\n22(k;t0;t′)U12(k;t;t 0)/uni27E8c†\nk(t0)ck+Q(t0)/uni27E9\n+iU†\n12(k;t0;t′)U11(k;t;t 0)/uni27E8c†\nk+Q(t0)ck(t0)/uni27E9\n+iU†\n22(k;t0;t′)U12(k;t;t 0)/uni27E8c†\nk+Q(t0)ck+Q(t0)/uni27E9;\nand\nG<\n22(k;t;t′)=i/uni27E8c†\nk+Q(t′)ck+Q(t)/uni27E9 (51)\n=iU†\n12(k;t0;t′)U21(k;t;t 0)/uni27E8c†\nk(t0)ck(t0)/uni27E9\n+iU†\n22(k;t0;t′)U22(k;t;t 0)/uni27E8c†\nk(t0)ck+Q(t0)/uni27E9\n+iU†\n12(k;t0;t′)U21(k;t;t 0)/uni27E8c†\nk+Q(t0)ck(t0)/uni27E9\n+iU†\n22(k;t0;t′)U22(k;t;t 0)/uni27E8c†\nk+Q(t0)ck+Q(t0)/uni27E9:\nNote how these results do not simplify to depend only\non the times between tandt′, but instead depend on all\ntimes. In addition, these results depend upon the initial\noccupancies of the di\u000berent momentum states, which are\ndetermined by the initial conditions at t=t0and follow\nfrom the equilibrium analysis described above. In partic-\nular, we start the system in equilibrium at T=0 which\ncorresponds to a \flled lower band /uni27E8c†\nk−ck−/uni27E9=1 and an\nempty upper band /uni27E8c†\nk+ck+/uni27E9=0. Converting from the\neigenfunction basis to the kandk+Qbasis then shows\nthat we must take /uni27E8c†\nkck/uni27E9=\f2\nk,/uni27E8c†\nk+Qck+Q/uni27E9=\u000b2\nk, and\n/uni27E8c†\nkck+Q/uni27E9=/uni27E8c†\nk+Qck/uni27E9=−\u000bk\fkas the inital expectation\nvalues for the occupancies. Here, the \u000bkand\fkare the\nequilibrium values in Eqs. (11) and (12) since the system\nstarts in equilibrium before the \feld is turned on.\nThe time-dependent CDW order parameter then fol-\nlows in a simple fashion\n\n(t)=nB(t)−nA(t)\nnB(t)+nA(t)(52)\n=−∑k∶\"k≤0[G<\n12(k;t;t)+G<\n21(k;t;t)]\n∑k∶\"k≤0[G<\n11(k;t;t)+G<\n22(k;t;t)]:\nC. Gauge invariance\nPhysically measurable properties are gauge invariant,\nand depend only upon the \felds, not the scalar and vec-\ntor potentials. While local quantities are always gauge\ninvariant, quantities that depend upon momentum do\ndepend on the gauge, and we need to make a transfor-\nmation from the Green's functions in a particular gaugeto the gauge-invariant Green's functions in order to de-\ntermine those quantities13,17We let G(k;trel;tave)de-\nnote the 2 ×2 matrix for the Green's function in the\ngauge with momentum k, relative time treland aver-\nage timetave. A superscript of Ror<will denote\nthe retarded or lesser Green's function. In the gauge,\nthe reduced Brillouin zone is de\fned by \u000fk<0. When\nwe go to the gauge-invariant Green's function [denoted\nby~G(k;trel;tave)=G(k(trel;tave);trel;tave)], we simply\nmake the transformation\nk→k(trel;tave)=k+/integral.disp1\n2\n−1\n2d\u0015A(tave+\u0015trel) (53)\nwhich means the reduced Brillouin zone for the gauge-\ninvariant Green's function satis\fes \"k(trel;tave)≤0. This\nis an added complication that one has to deal with in\na gauge-invariant formulation for a system with reduced\nspatial symmetry, because if we had the full Brillouin\nzone, it would be identical for the gauge-invariant case\nand the case in a gauge. Instead, the reduced Brillouin\nzone becomes time-dependent for the gauge-invariant for-\nmalism.\nOur strategy for calculating the current is to use the\ngauge-invariant Green's function. We will \frst determine\nthe linear-response formula for the current, and then gen-\neralize it to the nonlinear-response case. This is simple to\ndo in a gauge-invariant formalism because the formula is\nindependent of the vector potential. The current density\noperator is de\fned as the commutator of the Hamiltonian\nand the charge polarization\nj(t)=i[H(t);/summation.disp\njRjc†\nj(t)cj(t)]; (54)\nwhich becomes\nj(t)=−i/summation.disp\ni;\u000eti;i+\u000e(t)/uni20D7\u000ec†\ni(t)ci+\u000e(t) (55)\nwhere\u000edenotes a nearest neighbor translation vector\nfrom sitei. In linear response, we ignore the time-\ndependence of the hopping. Next, we write the linear-\nresponse current by summing only over the Asublattice\nbecause the \feld is uniform and we have current conser-\nvation in the system. With the \feld along the diagonal,\nthe current along each spatial component can be written\nas,\nj\u000b(t)=−it/summation.disp\ni∈A;\u000e\u000e\u000b[c†\ni(t)ci+\u000e(t)+c†\ni+\u000e(t)ci+2\u000e(t)];(56)\nwith the second term coming from the Bsublattice con-\ntributions. We now convert to a momentum-space rep-\nresentation, which gives\nj\u000b(t)=/summation.disp\nk∶\"k≤0∇k\u000b\"k[c†\nk(t)ck(t)−c†\nk+Q(t)ck+Q(t)](57)\nand is easily recognizable as the correct formula for the\nlinear-response current.7\nTo generalize this formula to the nonequilibrium case,\nwe evaluate the expectation value of the current operator\nby using the gauge-invariant Green's function\n/uni27E8j\u000b(t)/uni27E9=/summation.disp\nk∶\"k+A(t)≤0∇k\u000b\"k[~G<\n11(k;t;t)−~G<\n22(k;t;t)]:(58)\nThis formula has been written for the nonequilibrium\ncase since the linear-response form of the current has no\ndependence on the vector potential, and hence is the cor-\nrect form to generalize to nonequilibrium. Using the fact\nthat ~G<(k;trel=0;tave)=G<(k+A(tave);trel=0;tave)\nthen shows that\n/uni27E8j\u000b(t)/uni27E9=/summation.disp\nk∶\"k+A(t)≤0∇k\u000b\"k[G<\n11(k+A(t);t;t)\n−G<\n22(k+A(t);t;t)] (59)\nwhere we changed to using the t,t′representation for the\nGreen's function. Now, we simply shift k→k−A(t)to\nget our \fnal formula for the current\n/uni27E8j\u000b(t)/uni27E9=/summation.disp\nk∶\"k≤0∇k\u000b\"k−A(t)[G<\n11(k;t;t)−G<\n22(k;t;t)]:\n(60)\nOne can also derive the current in the more traditional\nway by working with the formulation entirely in the\nHamiltonian gauge, and the result is identical.\nIII. BLOCH OSCILLATION IN A CDW SYSTEM\nIn a pure material composed of electrons interacting\nwith the periodic lattice potential, the system does not\nsatisfy Ohm's law. Instead, if a \feld is applied to the ma-\nterial, it generates an oscillating current called a Bloch\noscillation18,19. Ohm's law is recovered when one intro-\nduces scattering into the system and the \feld is small\nenough to be in the linear-response regime. As surprising\nas this result might be, it is even more surprising in that\nit is quite di\u000ecult to observe in real materials, because\nthe scattering time, for even the most pure materials,\nis too short for the system to show the Bloch oscillation.\nIt has been observed in semiconductor superlattices20{22,\nin cold atoms in an optical lattice23,24and also in ultra-\nsmall Josephson junctions25.\nIf we go back to the theory we developed in Section\nII, and set U=0 to recover the single-band model, then\none immediately can solve for the evolution operators,\nbecause the 2 ×2 matrix becomes diagonal. If we work\nin the full Brillouin zone, then we \fnd that the current\nsatis\fes\n/uni27E8j\u000b(t)/uni27E9=/integral.dispd\"/integral.dispd\u0016\"\u001a(\")\u001a(\u0016\")f(\") (61)\n×[−\u0016\"cos(A(t))+\"sin(A(t))]\nwhich is proportional to sin (Et), with a temperature-\ndependent amplitude. The current oscillates about zero,\nso the energy added to the system also oscillates about\nzero, and after each Bloch period, given by 2 \u0019/slash.leftE, the\nFIG. 4. (Color online.) Current, upper band occupancy and\nchange in energy for a U=0:01 CDW system placed in a DC\n\feld withE(t)=\u0012(t). Panel (a) shows the current, which\nis close to the known sine wave that occurs at U=0 with a\nfrequency equal to the amplitude of the DC \feld ( !=E0=\n1). Panel (b) shows the upper band electron occupancy as a\nfunction of time. Panel (c) shows the total energy shift from\nthe ground state energy as a function of time.\nsystem returns to the original state it was in before the\n\feld was put on. This is a well-known property of Bloch\noscillations in a noninteracting single-band model. When\nscattering is added into the system, the current oscillates\nabout a net positive value, so that a nonzero amount of\nheat is always added to the system.\nOur goal here, is to study how this situation changes\nwhen we have a two-band model that arises from the\npresence of CDW order. Even though the system is non-\ninteracting, we no longer have any guarantee that the\nsystem can return to its initial state after a Bloch period.\nSince the excitation of electrons across the gap requires\nquantum-mechanical tunneling across the gap, it is not\nobvious that the electrons excited across the gap can all\nbe de-excited at any speci\fc time. Indeed, we will \fnd\nout that this does not occur in general.\nBased on the theory developed in the Sec. II, we can\ncalculate the current with a DC \feld using the nonequi-\nlibrium Green's function technique. The procedure is to\ndiscretize the contour, calculate the relevant evolution\noperators for each momentum point, and use them to\nconstruct the retarded and lesser Green's functions as\nfunctions of momentum and time. From these Green's\nfunctions all relevant physical quantities can be calcu-\nlated. We turn on a spatially uniform DC electric \feld\nof amplitude E0abruptly at time t=0, and calculate the\nsubsequent evolution of the system. To verify the for-\nmalism, we examine the case with U=0:01 andE0=1,\nwhere the DC \feld is much larger than the gap size, so the\nsystem should have similar behavior to the single-band8\nFIG. 5. (Color online.) Dielectric breakdown of the CDW\ninsulator for large electric \felds. The upper band \flling is\nplotted for di\u000berent UandE0values. When E0/uni226BU, as\nshown in panel (a), the upper band electrons are excited as if\nthere is no gap, although we never fully \fll the upper band,\nwhich occurs when n+=0:5. Here, we show two cases with\nthe same ratio of U/slash.leftE0:U=0:05 andE0=1 (red curve,\nlong period) and U=0:5;E0=10 (blue curve, short period).\nNote how the total amount excited depends on more than just\nthe ratio of U/slash.leftE0. WhenU/uni226BE0[panel (b), note change in\nvertical scale], much fewer electrons are pumped to the upper\nband. Here we show a large gap U=5 and smaller \felds\nE0=1 (violet, short period) and E0=0:5 (magenta, long\nperiod). Note how the maximum amplitude decreases as the\ngap increases by comparing panel (a) to panel (b).\nmodel and illustrate similar Bloch oscillations. Fig. 4\nshows the current, the upper band electron occupancy\nand the total energy for this case. The current oscillates\nwith the Bloch frequency and has a small reduction of\nthe amplitude over time due to the dephasing in the sys-\ntem (because there are two bands since U≠0). Note\nhow the current appears to oscillate about zero, but be-\ncause of the decaying amplitude, there is a net current,\nwhen integrated over time, and hence there is a small\nnet transfer of energy into the system. The upper band\nelectron occupancy has a nearly sawtooth nature to it as\nelectrons are excited and then de-excited from the upper\nband. The electrons move between the lower and upper\nbands almost as if they don't feel the presence of the gap.\nThe increase of energy as a function of time appears to\nbe quite similar to the single-band case, but it doesn't\nactually go to zero at the Bloch period; instead it has a\nsmall residual energy gain there.\nWe illustrate the behavior of dielectric breakdown,\nwhere current is induced across the gap due to a large\nelectric \feld, in Fig. 5. When the \feld magnitude is much\nsmaller than the gap, \feld assisted tunneling across the\nFIG. 6. (Color online.) Results for U=1 in a DC \feld\nE(t)=\u0012(t)(E0=1). Panel (a) shows the current, which has\nan initial transient response that settles down to a \\steady\nstate\" and shows complex oscillations that are not given solely\nby the Bloch oscillation frequency. Panel (b) shows the upper\nband \flling as a function of time as the system is driven by the\nelectric \feld. Panel (c) shows the total energy as a function\nof time.\ngap is a rare event that is di\u000ecult to achieve because the\ntunneling process sees a large barrier, while the oppo-\nsite case, where the \feld magnitude is much larger than\nthe gap, readily has \feld-assisted tunneling occur. The\nbottom panel shows the former case, while the top panel\nshows the latter. Because quantum tunneling always oc-\ncurs, no matter how small the \feld amplitude is, there\nis no sharp distinction between the case where Bloch os-\ncillations readily occur and where they are suppressed.\nIt is instead a crossover. But we expect the crossover to\noccur close to the region where E0=U, since the gap is\nalways equal to Uin equilibrium.\nWe next look at what happens at the crossover where\nthe dielectric breakdown occurs U=E0in Fig. 6 with\nU=E0=1. In this case, the current is initially driven\nto a relatively large amplitude and then stabilizes and\nshows complex oscillations. The transient response set-\ntles down rapidly to a more steady state behavior. This\noccurs even though there is no scattering, and hence must\ncome from dephasing e\u000bects. About 40% of the elec-\ntrons are driven to the upper band as the initial \feld\nis switched on. The upper band electron number tran-\nsiently oscillates, but maintains a level around 40% of the\ntotal number of electrons. In this quasi-steady state, the\ncurrent oscillates around zero, so there is no net increase\nin the energy transfered to the system after we absorbed\nthe initial energy that occurred during the transient re-\nsponse. It is clear in this case, that the system will not\nreturn back to the state it initially was in after the Bloch9\nFIG. 7. (Color online.) Fourier transform of the current re-\nsponsej(!)to a DC \feld E=\u0012(t)(E0=1) for di\u000berent U\nvalues.\nperiod. This is one of the main di\u000berences that occurs for\nthe noninteracting model in a two-band system versus a\nsingle-band system.\nThe e\u000bect of scattering on Bloch oscillations driven by\na DC \feld (in the high-temperature disordered phase) has\nbeen studied in the Falicov-Kimball model26,27and in the\nHubbard model28as has photoexcitation29{31. There are\ntwo important energy scales in the problem. The \frst\nis the \feld amplitude E0and the second is the scatter-\ning strength, which is commonly denoted by Uint. In the\ncase of weak scattering, two scenarios are possible for the\nevolution of the local density of states: (i) the delta func-\ntion peaks of the Wannier-Stark ladder (a series of delta\nfunction peaks in the local DOS separated by the Bloch\nfrequency!B=E) broaden due to scattering but remain\nnear the Bloch frequencies or (ii) the delta function peaks\nsplit due to Uintand broaden, but each Wannier-Stark\nminiband has a Mott-like transition. In cases where the\ninteraction is strong enough to drive the system into an\ninsulating phase, more complex behavior can occur. For\nthe Falicov-Kimball model, one still sees remnants of the\nWannier-Stark physics17,32, while they seem to be more\nstrongly suppressed in the Hubbard model33{35.\nOne way to try to identify the important energy scales\nis to look at the oscillation frequencies that arise in the\ncurrent that is driven by the \feld. To do this, we use\na discrete Fourier transform of the current versus time\ntraces, over a range of time where the transient e\u000bects\nhave died o\u000b and the system is in a quasi-steady state\n(which is actually a nearly periodically varying state with\na period given by the Bloch period); the time interval\nwe used is 40 ≤t≤80. These results are plotted in\nFig. 7. We \fnd that when Uis much larger than the\n\feld amplitude E0, the current oscillates at frequencies\nFIG. 8. (Color online.) Local DOS for U=1:5 at di\u000berent\naverage times with E0=1 and on the Asublattice. The \frst\npanel is the equilibrium result ( tave→−∞). Panel (b) has\ntave=0 and corresponds to when the \feld is turned on. Panel\n(c) hastave=10 and hence is in the transient response regime.\nPanel (d) has tave=500 and is approaching the steady state.\nU±E0as shown in the U=5 case. When U<2E0, peaks\nappear around E0±U/slash.left2. For intermediate values, these\ntwo e\u000bects mix as indicated with the U=3 case. Hence\nin this two band system, when the DC \feld is applied,\nthe band gap Uand the electric \feld amplitude are the\ntwo main factors that a\u000bect the electron oscillations and\nthese two factors interact with each other to produce new\noscillation frequencies, that are not simply what might\nhave been predicted by just looking at the two energy\nscales.\nAs we hinted at above, it is also interesting to exam-\nine the nonequilibrium local DOS. We show this DOS\nin Fig. 8 for a number of di\u000berent average times. Even\nthough one might have expected the DOS to instantly\nswitch between the equilibrium and the nonequilibrium\nresults as the \feld is turned on, since the DOS measures\nthe quantum mechanical states of the system, they must\nevolve from one result to the other continuously and this\noccurs slowly in this case because the equilibrium DOS\nhas long tails in the time domain due to the inverse square\nroot singularity at the upper or lower band edge. When\nwe construct the retarded Green's function to describe\nthese two systems and \fx the average time, then for some\nrelative times, we have one time before the \feld being\nturned on and one after. This \\mixed\" Green's function\ninterpolates between the equilibrium and nonequilibrium\nsteady state DOS. If it also has long tails in time, then\nthe evolution from equilibrium to nonequilibrium can be\nslow.\nOne should note that due to the long tail in time for10\nFIG. 9. (Color online.) Local DOS on the Asublattice for\nU=3 at di\u000berent average times with E0=1. The same\naverage times are chosen as shown in Fig. 8. In this case, the\nevolution of the DOS is much more mild, but we de\fnitely\nsee a reduction in the magnitude of the overall gap for long\ntimes and a splitting of the lower band, while the upper band\ndoes not appear to change too signi\fcantly.\nthe equilibrium Green's function, it becomes di\u000ecult to\nthink of the local DOS as being de\fned at a speci\fc av-\nerage time only. The DOS involves the Fourier transform\nwith respect to relative time, so it is de\fned in terms of\ntwo times, and one can have one of the times being long\nafter the \feld is turned on, even if the average time is\nbefore the \feld is turned on. This is illustrated dramat-\nically in panel (b), which has an average time equal to\nthe time the \feld is turned on, but already shows signif-\nicant deviations from the equilibrium DOS in panel (a).\nIn particular, the gap is no longer well de\fned, and there\nis signi\fcant subgap structure. In the transient regime of\npanel (c), we see this evolve even further, and we also see\nthe DOS go negative for some frequencies. This is not\nan artifact of a truncation of the Fourier transform, but\ncommonly occurs for transient DOS in nonequilibrium\ncalculations since there is no Lehmann representation for\nthe transient DOS that allows us to prove nonnegativity\nof the DOS. Finally, in panel (d) we see the emergence\nof the steady state DOS, with its new gap structure oc-\ncuring at the half-odd integers (except for the missing\nsubband at !=−1:5); the separation of the gaps ap-\npears to be governed primarily by E0here. This result is\nsimilar to what one might expect due to Wannier-Stark\nphysics, where the delta function peaks are split due to\nthe CDW gap, but what is surprising here is the center\nof the CDW gap is occurring at half-odd integers here\nrather than at integers which is what the Wannier-Stark\npicture would predict.\nThe caseU=3 andE0=1 is plotted in Fig. 9. It\nshows that the lower band can split and be shifted. Al-\nFIG. 10. (Color online.) Evolution of the \\steady\nstate\"(tave=300) local DOS for the Asublattice and U=1 for\ndi\u000berent DC \feld amplitudes. Red arrows show the positions\nof the electric \feld induced gaps.\nready attave=0, the nonequilibrium local DOS shows\na split lower band that eventually approaches a shift of\nthe (split) band edge by ±E0from its equilibrium loca-\ntion at−U/slash.left2. The upper band density of states doesn't\nshift at all and it shows a large enhancement at the lower\nband edge of the upper band (although, we don't believe\nthe singularity remains, but cannot go far enough out in\ntime to verify this). In this case, there aren't as many\n\feld related gaps observed, perhaps because the DOS is\ntoo small at larger frequencies for those structures to be\nseen. More intriguing is the fact that the structure tends\nto be more broadened at larger Ueven though there still\nis no interaction in the system.\nWe next show the evolution of the local DOS on the\nAsublattice for U=1 and various \feld amplitudes\nin Fig. 10. The arrows indicate the positions of \feld-\ninduced gaps in the spectrum. We \fnd that for all cases\nof the \feld amplitudes, that the DOS shows gaps at half-\nodd integer multiples of E0, just like what we already\nobserved in Fig. 8 when we had U=1:5. We also note\nthat in some cases, there is no miniband centered around\na speci\fc gap that we expect to see. This occurs for\nlarge frequencies for all cases. In addition, the case with\nE0=1 and!=0:5 [panel (b)] shows a pseudogap-like be-\nhavior, with the \\gap\" occurring at just one point. The\n\feld magnitude controls the size of the gap as well, as we\nsee the gap size grow as E0increases. In the E0=2\ncase, only the gap at !=1 is clear, while the other\ngap edges do not show signi\fcant features. One can still\nsee a Wannier-Stark-like ladder in these systems, but the\npeaks no longer occur just at the Bloch frequencies, but\nare shifted, and also broadened, as we expect them to11\nFIG. 11. (Color online.) Imaginary part of the lesser Green's\nfunction (black) at (a) U=1:5 and (b)U=3 compared to\nthe corresponding local DOS (red) on the Asublattice with\nE0=1. The curves where calculated at tave=500. The ratio\nof the black curve to the red curve gives the local distribution\nfunction, which would be Fermi-Dirac-like for a thermalized\nsystem. The distribution function that one would extract\nfrom theU=1:5 case is clearly nonthermal (focus on the\nregion around !=1), since it would not be monotonic. It is\nmore di\u000ecult to judge whether the case with U=3 could be\ndescribed by some e\u000bective thermal distribution, but the last\npeak (near !=2) does not look like it has the right shape to\nbe described by a Fermi-Dirac-like distribution.\nbe. Clearly the behavior of the DOS is complex, and\ndepends in a nontrivial way on the site energy Uand\nthe \feld amplitude E0. There doesn't appear to be any\nsimple way to relate them to Wannier-Stark physics in a\ncoherent fashion.\nThe imaginary part of the Fourier transform of the lo-\ncal lesser Green's function at tave=500 is plotted for\ntwo di\u000berent Uvalues and compared to the local DOS in\nFig. 11. The ratio of the lesser curve to the DOS curve\ngives the local distribution function. If the system was\ndriven to a thermal distribution at long times, then the\ndistribution function would be monotonic and given by\nthe Fermi-Dirac distribution function with some e\u000bec-\ntive temperature. This clearly is not the case for U=1:5,\nsince one can see for the miniband centered around !=1\nthat the distribution function must be increasing for a \f-\nnite range of frequency. It is more di\u000ecult to see whether\nthe largerUcase could be described via the \ructuation-\ndissipation theorem, but the miniband occupancy near\n!=2 does not have the same shape as the local DOS,\nand hence it looks like this would require a nonmonotonic\ndistribution function as well.IV. CONCLUSIONS\nIn this work, we have given details about how one can\nexactly solve for the Bloch-oscillation problem in a lat-\ntice with a basis, as given by the two-sublattice CDW\norder. This is the simplest generalization of the Bloch\noscillation problem to multiple bands and surprisingly, it\nhas much richer behavior than one might have naively\nexpected. The problem can be solved exactly by working\nin a 2×2 basis for the evolution operators for each mo-\nmentum pair kandk+Q. We developed this formalism\nand showed how to calculate relevant observables. We\nalso veri\fed a number of nontrivial exact relations that\nthe solution must satisfy including moment sum rules of\nthe local DOS and an equality of two di\u000berent forms for\nthe average energy as a function of time. This formalism\nworks and is exact in any spatial dimension. For con-\ncreteness, and to compare to work done for interacting\nsystems, we solved the problem in in\fnite dimensions on\nthe hypercubic lattice.\nWe used this formalism to study the Bloch-oscillation\nproblem given by the question of how does this system\nevolve after a large amplitude \feld is turned on at t=0.\nWe found that the current undergoes a transient response\nbefore settling into a steady state-like behavior for longer\ntimes, but the time trace of the current versus time is not\ngoverned solely by the Bloch frequency but shows more\ncomplex structure and behavior.\nWe also examined the local DOS to see how the\nWannier-Stark-ladder physics was modi\fed by the gap\nin the spectrum. We found complex behavior here as\nwell, which is not determined solely by any simple rule\nfor how the system will evolve, although we did \fnd a\npropensity for minibands to form with gaps at half-odd\ninteger multiples of the \feld amplitude.\nFinally, we examined whether the \ructuation-\ndistribution theorem holds at long times, and we found\nthat typically, as one might expect, the systems do not\nfollow a thermal distribution, but show a markedly ather-\nmal distribution of states.\nThe work we presented here will be challenging to ob-\nserve in CDW systems in condensed matter, but they\ncould be seen more easily in cold atom systems on double-\nwell optical lattices, where this model is a natural model\nto describe the behavior of those systems. While the lo-\ncal DOS cannot be measured, analogs to photoemission\nexperiments are possible, as is the possibility of measur-\ning the number of particles in each band as a function of\ntime. It might even be possible to construct the current\nversus time by processing time-of-\right images to deter-\nmine the momentum distribution functions and summing\nthem over momentum (weighted by the velocity) to get\nthe current.12\nACKNOWLEDGMENTS\nThe development of the parallel computer algorithms\nwas supported by the National Science Foundation un-\nder Grant No. OCI-0904597. The data analysis was sup-\nported by the Department of Energy, O\u000ece of Basic En-\nergy Research under Grants No. DE-FG02-08ER46542\n(Georgetown), DE-AC02-76SF00515 (Stanford/SLAC),\nand DE-SC0007091 (for the collaboration). High perfor-\nmance computer resources utilized the National Energy\nResearch Scienti\fc Computing Center supported by the\nDepartment of Energy, O\u000ece of Science, under Contract\nNo. DE- AC02-05CH11231. J.K.F. was also supported\nby the McDevitt bequest at Georgetown.\nAppendix A: Calculation of \frst three moments of\nthe retarded Green's function\nThenthorder moments are de\fned as follows:\n\u0016ii\nn(tave)=/integral.disp+∞\n−∞d!!nAii(!;tave): (A1)\nThis expression is equivalent to\n\u0016ii\nn(tave)=−Im[in@n\n@tn\nrelGR\nii(trel;tave)]trel=0+:(A2)\nWe now verify the exact results for the \frst three\nmoments for our retarded Green's function. In in\f-\nnite dimensions, the moments for the local retarded\nGreen's function in our inhomogeneous system satisfy\nthe following16: (1) the zeroth order moment is 1; (2)\nthe \frst order moment is ±U/slash.left2; (3) the second order mo-\nment is 1 /slash.left2+U2/slash.left4 ; and (4) the third order moment is\n±U/slash.left4±U3/slash.left8. (The A sublattice has a plus sign in the\n\frst and third moments). The zeroth moment follows\ndirectly by just setting t→t′+in the retarded Green's\nfunction. Since the retarded Green's functions are just\nlinear combinations of the elements of the time-evolution\noperators, we can directly verify the moments by cal-\nculating the \frst three derivatives of the time evolution\nmatrix (we set \u0016=U/slash.left2 to make the formulas less cum-\nbersome).\nThe \frst derivative is easy to \fnd:\n@U(k;t;t′)\n@trel=−i\n2/parenleft.alt4\"k(t)U\n2U\n2−\"k(t)/parenright.alt4U(k;t;t′)\n−i\n2U(k;t;t′)/parenleft.alt4\"k(t′)U\n2U\n2−\"k(t′)/parenright.alt4 (A3)\n(the factor of 1 /slash.left2 comes from the fact that t=tave+trel/slash.left2\nandt′=tave−trel/slash.left2). In the limit trel→0+,U(k;t;t)→I\nand the \frst derivative becomes\n@U(k;t;t′)\n@trel/divides.alt0trel→0+=−i/parenleft.alt4\"k(t)U\n2U\n2−\"k(t)/parenright.alt4: (A4)\nWe substitute this result into the de\fnition of the local\nretarded Green's function in Eq. (45), which requires usto sum over momentum. Using the fact that ∑k1=1/slash.left2\nfor a summation over the reduced Brillouin zone, then\nyields\n@GR\nii(t;t′)\n@trel/divides.alt0trel→0+=−/summation.disp\nk/parenleft.alt3\"k(t)−\"k(t)±U\n2±U\n2/parenright.alt3(A5)\nwhich becomes ∓U/slash.left2 and shows that the \frst-order mo-\nment is±U/slash.left2 from Eq. (A2) with n=1.\nThe second derivative of the retarded Green's function\ncan be obtained by di\u000berentiating the \frst derivative\n@2U(k;t;t′)\n@t2\nrel=−i\n2@\n@trel/parenleft.alt4\"k(t)U\n2U\n2−\"k(t)/parenright.alt4U(k;t;t′)\n−i\n2/parenleft.alt4\"k(t)U\n2U\n2−\"k(t)/parenright.alt4@U(k;t;t′)\n@trel\n−i\n2U(k;t;t′)@\n@trel/parenleft.alt4\"k(t′)U\n2U\n2−\"k(t′)/parenright.alt4\n−i\n2@U(k;t;t′)\n@trel/parenleft.alt4\"k(t′)U\n2U\n2−\"k(t′)/parenright.alt4:(A6)\nThe derivative of the U(k)matrix was evaluated in\nEq. (A3). The derivative of the Hamiltonian matrix be-\ncomes\n@\n@trel/parenleft.alt4\"k(t)U\n2U\n2−\"k(t)/parenright.alt4=1\n2E(t)⋅∇k\"k(t)/parenleft.alt31 0\n0−1/parenright.alt3(A7)\nwith a similar equation for the t′matrix, but with\nt→t′and an overall sign change on the right hand\nside. Here E(t)is the time-dependent electric \feld E(t)=\n−dA(t)/slash.leftdt. So the \frst and third terms in Eq. (A6) can-\ncel in the limit t=t′+, and the rest of the terms give\n@2U(k;t;t′)\n@t2\nrel/divides.alt0trel→0+=−/parenleft.alt4\"2\nk(t)+U2\n4/parenright.alt4I: (A8)\nSo the second derivative of the Green's function becomes\n@2GR\nii(t;t′)\n@t2\nrel/divides.alt0trel→0+=i/summation.disp\nk/parenleft.alt4\"2\nk(t)+U2\n4+\"2\nk(t)+U2\n4±0±0/parenright.alt4\n=i/parenleft.alt41\n2+U2\n4/parenright.alt4 (A9)\nin the limit as t→t′+, where we used the relation\n∑k\"2\nk(t)=1/slash.left4 for the in\fnite-dimensional density of\nstates. So using Eq. (A2) with n=2 shows that the\nsecond moment satis\fes \u0016ii\n2=1/slash.left2+U2/slash.left4.13\nThe third derivative of U(k;t;t′)has many more terms,\n@3U(k;t;t′)\n@t3\nrel=−i\n4E(t)⋅∇k\"k(t)/parenleft.alt31 0\n0−1/parenright.alt3@U(k;t;t′)\n@trel\n−i\n4@\n@trelE(t)⋅∇k\"k(t)/parenleft.alt31 0\n0−1/parenright.alt3U(k;t;t′)\n−i\n4E(t)⋅∇k\"k(t)/parenleft.alt31 0\n0−1/parenright.alt3@U(k;t;t′)\n@trel\n−i\n2/parenleft.alt4\"k(t)U\n2U\n2−\"k(t)/parenright.alt4@2U(k;t;t′)\n@t2\nrel\n+i\n4@U(k;t;t′)\n@trelE(t′)⋅∇k\"k(t′)/parenleft.alt31 0\n0−1/parenright.alt3\n+i\n4U(k;t;t′)@\n@trelE(t′)⋅∇k\"k(t′)/parenleft.alt31 0\n0−1/parenright.alt3\n−i\n2@2U(k;t;t′)\n@t2\nrel/parenleft.alt4\"k(t′)U\n2U\n2−\"k(t′)/parenright.alt4\n+i\n4@U(k;t;t′)\n@trelE(t′)⋅∇k\"k(t′)/parenleft.alt31 0\n0−1/parenright.alt3:(A10)\nIn the limit trel→0+, we substitute in the \frst and sec-\nond derivatives at equal times from Eqs. (A4) and (A8)\nand sett=t′in the rest. This yields\n@3U(k;t;t′)\n@t3\nrel=i\n4@\n@t[E(t)⋅∇k\"k(t)]/parenleft.alt31 0\n0−1/parenright.alt3\n−1\n2E(t)⋅∇k\"k(t)/parenleft.alt31 0\n0−1/parenright.alt3/parenleft.alt4\"k(t)U\n2U\n2−\"k(t)/parenright.alt4\n+1\n2E(t)⋅∇k\"k(t)/parenleft.alt4\"k(t)U\n2U\n2−\"k(t)/parenright.alt4/parenleft.alt31 0\n0−1/parenright.alt3\n+i/parenleft.alt4\"2\nk(t)+U2\n4/parenright.alt4/parenleft.alt4\"k(t)U\n2U\n2−\"k(t)/parenright.alt4:(A11)\nSince we will be substituting this result into the formula\nfor the retarded Green's function, where the U11(k)term\nalways appears plus the U22(k)term and similarly for the\nU12(k)andU21(k)terms, the only nonvanishing contri-\nbutions come from the o\u000b diagonal pieces of the last term\nin Eq. (A11). Hence, the third derivative of the retarded\nGreen's function is\n@3Gii(t;t′)\n@t3\nrel/divides.alt0trel→0+=±/summation.disp\nk/bracketleft.alt4U\"2\nk(t)+U3\n4)/bracketright.alt4\n=±/parenleft.alt4U\n4+U3\n8/parenright.alt4: (A12)\nSo the third moment satis\fes \u0016ii\n3=±(U/slash.left4+U3/slash.left8)with\nplus on the Aand minus on the Bsublattices. Hence,\nwe have now analytically veri\fed the results for the \frst\nthree moment sum rules. This is an important check that\nour formalism is correct.Appendix B: Derivation of the formulas for the\nenergy and the heating rate\nAn additional check that we will use for the current\nformula is to verify energy conservation. The total energy\nEtotis found by simply taking the expectation value of\nthe Hamiltonian (at half-\flling with \u0016=U/slash.left2)\nEtot(t)=/summation.disp\nk∶\"k≤0/bracketleft.alt2\"k(t)/uni27E8c†\nk(t)ck(t)/uni27E9+U\n2/uni27E8c†\nk(t)ck+Q(t)/uni27E9\n+U\n2/uni27E8c†\nk+Q(t)ck(t)/uni27E9−\"k(t)/uni27E8c†\nk+Q(t)ck+Q(t)/uni27E9/bracketright.alt2:\n(B1)\nUsing the Green's functions in the gauge, then yields\nEtot(t)=−i/summation.disp\nk∶\"k≤0/bracketleft.alt2\"k(t){G<\n11(k;t;t)−G<\n22(k;t;t)}\n+U\n2{G<\n12(k;t;t)+G<\n21(k;t;t)}/bracketright.alt2:\n(B2)\nThe power pumped into the system is due to the ac-\ncelleration of the electron along the electric \feld. A direct\ncomputation shows that\n@Etot(t)\n@t=/uni27E8j/uni27E9⋅E(t): (B3)\nWe now show this identity analytically, using our results\nfor the current and the total energy. The equation of mo-\ntion for the lesser Green's function follows from deriva-\ntive of the respective evolution operator. We \fnd the\nfollowing four equations:\n@tG<\n11(k;t;t′)=iU\n2G<\n12(k;t;t)−iU\n2G<\n21(k;t;t); (B4)\n@tG<\n12(k;t;t′)=iU\n2G<\n11(k;t;t)−iU\n2G<\n22(k;t;t)\n−2i\"k(t)G<\n12(k;t;t); (B5)\n@tG<\n21(k;t;t′)=−iU\n2G<\n11(k;t;t)+iU\n2G<\n22(k;t;t)\n+2i\"k(t)G<\n21(k;t;t); (B6)\nand\n@tG<\n22(k;t;t′)=−iU\n2G<\n12(k;t;t)+iU\n2G<\n21(k;t;t);(B7)\nin the limit t′→t. Hence, the derivative of the total\nenergy with respect to time becomes\n@Etot(t)\n@t=−i/summation.disp\nk∶\"k≤0[@t\"k(t)]{G11(k;t;t)−G22(k;t;t)}\n(B8)\nsince the terms that don't involve the time derivative of\nthe time-dependent bandstructure all cancel (which also14\nfollows from the Feynman-Hellman theorem). Using the\nfact that@t\"k(t)=E(t)⋅∇k\"k(t)and the \fnal result for\nthe expectation value of the current in Eq. (60) yields\nEq. (B3). This is a second stringent test that the formal-\nism is correct.\nFinally, we examine the \flling in the transient upper\nand lower bands by determining the corresponding oc-\ncupancy. Similar to the equilibrium case, the creation\noperators for the instantaneous upper and lower bands\nare written as,\nc†\nk+(t)=\u000bk(t)c†\nk(t)+\fk(t)c†\nk+Q(t) (B9)\nc†\nk−(t)=\fk(t)c†\nk(t)−\u000bk(t)c†\nk+Q(t) (B10)\nwith the corresponding annihilation operators being the\nhermitian conjugate. Now the electron occupancy in the\nupper band is denoted by n+(t)and that of the lower\nband byn−(t).\nn+(t)=/summation.disp\nk∶\"k≤0/uni27E8c†\nk+(t)ck+(t)/uni27E9 (B11)\n=/summation.disp\nk∶\"k≤0/bracketleft.alt2\u000b2\nk(t)/uni27E8c†\nk(t)ck(t)/uni27E9+\f2\nk(t)/uni27E8c†\nk+Q(t)ck+Q(t)/uni27E9\n+\u000bk(t)\fk(t)/uni27E8c†\nk(t)ck+Q(t)+c†\nk+Q(t)ck(t)/uni27E9/bracketright.alt2\nand\nn−(t)=/summation.disp\nk∶\"k≤0/uni27E8c†\nk−(t)ck−(t)/uni27E9 (B12)\n=/summation.disp\nk∶\"k≤0/bracketleft.alt2\f2\nk(t)/uni27E8c†\nk(t)ck(t)/uni27E9+\u000b2\nk(t)/uni27E8c†\nk+Q(t)ck+Q(t)/uni27E9\n−\u000bk(t)\fk(t)/uni27E8c†\nk(t)ck+Q(t)+c†\nk+Q(t)ck(t)/uni27E9/bracketright.alt2:\nHere we have the time-dependent generalization of the\ninstantaneous eigenvectors with\n\u000bk(t)=U\n2/radical.alt4\n2/bracketleft.alt3\"2\nk(t)+U2\n4−\"k(t)/radical.alt2\n\"2\nk(t)+U2\n4/bracketright.alt3(B13)\nand\n\fk(t)=−\"k(t)+/radical.alt2\n\"2\nk(t)+U2\n4/radical.alt4\n2/bracketleft.alt3\"2\nk(t)+U2\n4−\"k/radical.alt2\n\"2\nk(t)+U2\n4/bracketright.alt3: (B14)\nNote that the momentum-dependent occupancies are just\ngiven by the summand element for each kandk+Qpair.One immediately sees that the \flling satis\fes\nn+(t)+n−(t)=/summation.disp\nk∶\"k≤0/uni27E8c†\nk(t)ck(t)+c†\nk+Q(t)ck+Q(t)/uni27E9(B15)\nas it must. Furthermore, the energy becomes\nEtot(t)=/summation.disp\nk∶\"k≤0/uni23A1/uni23A2/uni23A2/uni23A2/uni23A2/uni23A3nk+(t)/radical.alt4\n\"2\nk(t)+U2\n4\n−nk−(t)/radical.alt4\n\"2\nk(t)+U2\n4/uni23A4/uni23A5/uni23A5/uni23A5/uni23A5/uni23A6; (B16)\nwhich follows from the instantaneous eigenenergies. We\nnow directly show that Eq. (B16) is equivalent to\nEq. (B1). Substituting in the values for n+(t)andn−(t)\nshows that the coe\u000ecient of the /uni27E8c†\nk(t)ck(t)/uni27E9term is\n[\u000b2\nk(t)−\f2\nk(t)]/radical.alt4\n\"2\nk(t)+U2\n4\n=/radical.alt2\n\"2\nk(t)+U2\n4/bracketleft.alt3U2\n4−U2\n4−2\"2\nk(t)+2\"k(t)/radical.alt2\n\"2\nk(t)+U2\n4/bracketright.alt3\n2/bracketleft.alt3\"2\nk(t)+U2\n4−\"k(t)/radical.alt2\n\"2\nk(t)+U2\n4/bracketright.alt3\n=2\"k(t)/bracketleft.alt3\"2\nk(t)+U2\n4−\"k(t)/radical.alt2\n\"2\nk(t)+U2\n4/bracketright.alt3\n2/bracketleft.alt3\"2\nk(t)+U2\n4−\"k(t)/radical.alt2\n\"2\nk(t)+U2\n4/bracketright.alt3\n=\"k(t): (B17)\nThe coe\u000ecient of /uni27E8c†\nk+Q(t)ck+Q(t)/uni27E9is just the nega-\ntive of this and the coe\u000ecients of /uni27E8c†\nk(t)ck+Q(t)/uni27E9and\n/uni27E8c†\nk+Q(t)ck(t)/uni27E9are equal and satisfy\n\u000bk(t)\fk(t)/radical.alt4\n\"2\nk(t)+U2\n4\n=U\n2/bracketleft.alt3−\"k(t)+/radical.alt2\n\"2\nk(t)+U2\n4/bracketright.alt3/radical.alt2\n\"2\nk(t)+U2\n4\n/bracketleft.alt3\"2\nk(t)+U2\n4−\"k(t)/radical.alt2\n\"2\nk(t)+U2\n4/bracketright.alt3\n=U\n2/bracketleft.alt3\"2\nk(t)+U2\n4−\"k(t)/radical.alt2\n\"2\nk(t)+U2\n4/bracketright.alt3\n/bracketleft.alt3\"2\nk(t)+U2\n4−\"k(t)/radical.alt2\n\"2\nk(t)+U2\n4/bracketright.alt3\n=U\n2: (B18)\nWhich proves the result we needed to show and which\nprovides the third stringent test of the formalism.\n1R. 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The\ntopological phase is characterized by a perfectly quantize d conducting plateau, carried by helical\nedge states, in a two-terminal setup. In the presence of diso rder, the bulk of the topological phase\nis either a band insulator or an Anderson insulator. Both of t hem can protect edge states from\nbackscattering. The topological phases are explicitly dis tinguished as topological band insulator or\ntopological Anderson insulator from the ratio of the algebr aic mean density of states to the geo-\nmetric mean density of states. The calculation reveals that topological Anderson insulator can be\ninduced by disorders from either a topologically trivial ba nd insulator or a topologically nontrivial\nband insulator.\nPACS numbers: 71.23.-k, 73.21.-b, 73.43.Nq\nI. INTRODUCTION\nTwo-dimensional (2D) time-reversal invariant insula-\ntors can be classified by Z2topological invariants for\noccupied energy bands1–3. Around the boundary of a\nsystem, a nontrivial topological invariant guarantees the\nexistence of gapless edge states lying in the bulk gap be-\ntween conduction and valence bands. The edge states\nare responsible for the dissipationless transport of elec-\ntrons, which persists even in the presence of weak disor-\nder. This is the picture of quantum spin Hall effect or\n2D topological band insulator (TBI)4–6. Experimental\nobservation of quantum spin Hall effect was reported in\nrecent years7–10. On the other hand, it has been further\nshown that the presence of disorder makes the above pic-\nture more complicated: disorder can induce a topological\nnontrivial phase from a trivial one, named as the topo-\nlogicalAndersoninsulator(TAI)11–22. Asinconventional\ntopological insulators, the dissipationless conductance of\nTAI is also attributed to helical edge states. Neverthe-\nless, there is a remarkable difference: the robust edge\nstates in TBI are protected by an energy band gap while\nthoseinTAI areprotectedbyamobilitygap3,22. Inother\nwords, the TAI is a topological state living in an Ander-\nson insulator (AI), instead of a band insulator (BI).\nInthispaper,wecalculatedthearithmeticandgeomet-\nric mean density of states (DOS) for a two-dimensional\ndisordered topological insulator. The DOS of a system\ndescribes the number of states per interval of energy at\neach energy level that are available to be occupied by\nelectrons. Usually it is an average over the space and\ntime. A zero value of DOS at a finite interval of energy\nmeans that no states can be occupied at that energy in-\nterval, which defines an energy gap. Consider the local\ndensity of states (LDOS) ρ(i) at each site iof the dis-\nordered sample. There are two types of mean DOS over\nsites. The arithmetic mean of the LDOS over the sites ofthe sample\nρave≡≪ρ(i)≫≡1\nN/summationdisplay\ni∈sampleρ(i) (1)\nis just the bulk DOS of the sample where Nis the total\nnumber of sites. The geometric mean\nρtyp≡exp[≪lnρ(i)≫] (2)\ngives the typical value of the site LDOS over the sam-\nple. When electron states are extended over the space,\nthe DOS is almost uniform in the space, and there is\nno distinct difference between the two means. How-\never, when electron states are localized, the local DOS\nis high near some sites, but low near other sites. This\nfact will lead to different mean values of DOS. While\nthe DOS ρaveprovides the information of energy lev-\nels, the ratio ρtyp/ρavereveals the information of local-\nization of the eigenstates: In the thermodynamic limit\n(N→ ∞), ifρtyp/ρave(E)→finite, the states around E\nis extended; while if ρtyp/ρave(E)→0, the states around\nEis localized23.\nAlthough both two-dimensional disordered TBI and\nTAI are characterized by quantized conductance plateau\nin the measurement of two-probe setup, we explicitly dis-\ntinguish TAI from disordered TBIs from the arithmetic\nmean DOS ρave: there is an energy gap ( ρave) for the\ndisordered TBI while ρaveis finite for TAI. Starting from\neither a normal band or an inverted band in the clean\nlimit and tuning disorder on gradually, we find that the\nTAI plays important roles on the route towards final lo-\ncalization. Tuning on disorder from a normal band, a\ndisorder-induced band inversion will trigger a topolog-\nical Anderson insulating phase in which the bulk elec-\ntrons are localized. On the other hand, starting from\nan inverted band, the topological phase can be divided\ninto two well defined parts: TBI and TAI, at weak and\nmedium disorder strength, respectively.2\nII. MODEL AND GENERAL FORMALISM\nWe startwith the Bernevig-Hughes-Zhangmodel5ona\nsquarelattice, which is equivalent to the two-dimensional\nmodified Dirac equation24. The model was proposed\nto describe the quantum spin Hall effect in HgTe/CdTe\nquantum well, and has been studied extensively. In the k\nrepresentation on a lattice with periodic boundary con-\ndition, the Bloch Hamiltonian His written as\nHαβ(k) =/parenleftbigg\nh(k) 0\n0h∗(−k)/parenrightbigg\nαβ(3)\nh(k) =d0I2×2+d1σx+d2σy+d3σz\nd0(k) =−2D/parenleftbig\n2−coskxa−coskya/parenrightbig\nd1(k) =Asinkxa, d2(k) =−Asinkya\nd3(k) =M−2B/parenleftbig\n2−coskxa−coskya/parenrightbig\n.\nHereα,βare the indices of electron spinorbitals within\nthe unit cell, α,β∈ {1,2,3,4} ≡ {|s↑/an}bracketri}ht,|p↑/an}bracketri}ht,|s↓\n/an}bracketri}ht,|p↓/an}bracketri}ht}.σiare the Pauli matrices acting on the spinor\nspace spanned by sandporbitals. The real space\nHamiltonian Hof this model can be obtained from\nHαβ(k) by a straightforward inverse Fourier transfor-\nmation. Throughout this manuscript, we adopt the fol-\nlowing model parameters from the HgTe/CdTe quantum\nwells:A= 73.0 meV nm, B=−27.4 meV, C= 0,\nD=−20.5 meV, and lattice constant a= 5nm11,25.\nThen, the parameter Mis a function of the thickness\nof quantum well: a positive Maccounts for a topologi-\ncally trivial phase while a negative Mfor a topologically\nnontrivial phase in the clean limit as we take a negative\nvalue for B26. The effect of non-magnetic impurities is\nincluded by introducing a term\nVI=/summationdisplay\ni/summationdisplay\nαWUic†\niαciα, (4)\nwhereUiare random numbers uniformly distributed in\n(−0.5,0.5) andWis a single parameter to control the\ndisorder strength. In the following, we will call a definite\nrealization of Uiasonedefinite disorder sample.\nThetwo-terminalconductanceofthe sampleinunitsof\ne2\nhis calculated by the standard non-equilibrium Green’s\nfunction method27\nGL(EF) = Tr(Γ LGRΓRGA), (5)\nwhere the retarded (advanced) Green’s function\nGR(A)(EF) =/parenleftbig\nEF−H−ΣR(A)\nL−ΣR(A)\nR/parenrightbig−1,His the\nHamiltonian of the disordered sample, Γ p=i/bracketleftbig\nΣR\np−ΣA\np/bracketrightbig\n,\nand ΣR(A)\npis the retarded (advanced) self energy from\nleadp(=L,R), i.e., the left and right lead. The topolog-\nical phase of the present quantum spin Hall model can\nbe identified by the quantized conductance GL= 2e2\nh.\nThe single particle local density ofstates (LDOS) of an\nisolated sample at its site ican be calculated explicitly\nby diagonalizing the Hamiltonian of the disordered sam-\nple. After obtaining the energy eigenstates for a specificdisordered sample, |n/an}bracketri}ht, wherenis the index for energy\nlevel, the LDOS is defined as\nρ(i,EF) =/summationdisplay\nn|/an}bracketle{ti|n/an}bracketri}ht|2δ(EF−En). (6)\nIn numerical calculations, the Dirac delta function is ap-\nproximated as δ(E) =η/(π(E2+η2)), with a small, but\nfinite broadening η= +0. With the LDOS in hand, both\nthe arithmetic mean ρaveand geometric mean ρtypcan\nbe calculated as defined in Eqs. (1) and (2).\nIII. NUMERICAL RESULTS\nA. A normal band\nLet us start from the clean system with M= 1 meV,\nwhich is topologically trivial. When disorder strength W\nis tuned on gradually, as first found in Ref.11, the system\ncan evolve into TAI in a very large range in the param-\neter space, which is characterized by perfectly quantized\nconductance plateau 2 e2/has expected to be carried by\nthe helical edge states. This disorder induced conduc-\ntance plateau is contrary to the general intuition that\ndisorder always tends to localize electronic states. Here\nwe reconstruct the conductance diagram on the W−EF\nplane for a definite sample with the open boundary con-\ndition in Fig. 1 (a). To compare with the bulk transport\nmechanism, we also present the conductance diagram for\nthe same sample with the periodic boundary condition\nin the transverse direction in Fig. 1 (b). Notice the con-\nductance GL= 2e2\nhplateau in Fig. 1 (a) corresponds to\na bulk mobility gap with GL∼0 in Fig. 1 (b). This\nconfirms once again that the quantum transport in TAI\nis carried by helical edge states11,12.\nIn order to relate the above transport properties to\nthe developments of band structures of the sample, we\npresent the arithmetic mean DOS ρaveof the same sam-\nple in Fig. 1 (c). The developments of energy levels with\nthe increasingofdisorderstrength Wcan be clearlyillus-\ntrated. At small W, the conduction and valance bands\ntend to get closer until they cross at W1∼0.036meV,\nwhere a band inversion occurs. This gap closing and\nreopening indicates a topological quantum phase tran-\nsition induced by disorder13,22. This can be confirmed\nfrom Fig. 1 (a) that after W1, there emerges the con-\nductance plateau. Soon after the gap re-opening W1in\nFig. 1 (c), one may notice a small inconsistency between\ntheangle-shapedboundaryofconductanceplateau(thick\nblack line) and the opened gap (blue region between the\nbands). Thiscanbesimplyattributedtodifferentbound-\nary conditions in the two calculations: the DOS was cal-\nculated from an isolated sample while the conductance\nwas calculated in the presence of two leads connecting to\nthe sample. This boundary condition sensitivity in weak\ndisorder regime can also be observed in the same region\nFig. 1 (b).3\n/s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s51/s53/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54\n/s87 /s32/s40/s109/s101/s86/s41/s69\n/s70/s32/s40/s109/s101/s86/s41/s48\n/s48/s46/s55/s53/s48/s48\n/s49/s46/s53/s48/s48\n/s50/s46/s48/s48/s53\n/s51/s46/s48/s48/s48\n/s51/s46/s55/s53/s48\n/s52/s46/s53/s48/s48\n/s53/s46/s50/s53/s48\n/s54/s46/s48/s48/s48/s71/s32/s40/s101/s50\n/s47/s104/s41/s99/s111/s110/s100/s117/s99/s116/s97/s110/s99/s101/s58/s32/s111/s112/s101/s110/s32/s98/s111/s117/s110/s100/s97/s114/s121\n/s40/s97/s41\n/s49/s46/s57/s57/s53\n/s84/s65 /s73\n/s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s51/s53/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54\n/s40/s98/s41/s99/s111/s110/s100/s117/s99/s116/s97/s110/s99/s101/s58/s32/s112/s101/s114/s105/s111/s100/s105/s99/s32/s98/s111/s117/s110/s100/s97/s114/s121\n/s71/s32/s40/s101/s50\n/s47/s104/s41\n/s87 /s32/s40/s109/s101/s86/s41/s69\n/s70/s32/s40/s109/s101/s86/s41/s48\n/s48/s46/s55/s53/s48/s48\n/s49/s46/s53/s48/s48\n/s50/s46/s50/s53/s48\n/s51/s46/s48/s48/s48\n/s51/s46/s55/s53/s48\n/s52/s46/s53/s48/s48\n/s53/s46/s50/s53/s48\n/s54/s46/s48/s48/s48/s84/s65 /s73\n/s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s51/s53/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54\n/s87\n/s50/s68/s79/s83\n/s87\n/s51/s40/s99/s41/s97/s118/s101/s32/s40/s97/s46/s117/s46/s41\n/s87 /s32/s40/s109/s101/s86/s41/s69\n/s70/s32/s40/s109/s101/s86/s41/s48/s46/s48/s52/s48/s48/s48\n/s48/s46/s48/s57/s50/s53/s56\n/s48/s46/s50/s49/s52/s51\n/s48/s46/s52/s57/s53/s57\n/s49/s46/s49/s52/s56\n/s50/s46/s54/s53/s54\n/s54/s46/s49/s52/s56\n/s56/s46/s54/s48/s48/s87\n/s49/s66\n/s65 /s84/s65 /s73\n/s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s51/s53/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s68/s79/s83/s32/s114/s97/s116/s105/s111\n/s116/s121/s112/s47\n/s97/s118/s101\n/s40/s100/s41\n/s87 /s32/s40/s109/s101/s86/s41/s69\n/s70/s32/s40/s109/s101/s86/s41/s48/s46/s48/s56/s48/s48/s48\n/s48/s46/s50/s50/s51/s56\n/s48/s46/s51/s54/s55/s53\n/s48/s46/s53/s49/s49/s50\n/s48/s46/s54/s53/s53/s48\n/s48/s46/s55/s57/s56/s56\n/s48/s46/s57/s52/s50/s53\n/s49/s46/s48/s48/s48/s84/s65 /s73\nFIG. 1: (Color online) Phase diagram in the parameter space o f disorder strength Wand Fermi energy EF, for a normal\nband with M= 1meV in the clean limit. (a) Two-terminal conductance of a s tripe sample with an open boundary condition\nin the transverse direction; (b) Two-terminal conductance of a rolled sample with the periodic boundary condition; (c) the\narithmetic mean DOS ρave; (d) the ratio of the geometric mean DOS to arithmetic mean DO S:ρtyp/ρave. Data in (c) and (d)\nare calculated numerically from the isolated sample with th e periodic boundary conditions in both directions, and with energy\nbroadening η= 2×10−4meV. Other model parameters are fixed throughout this work: A= 73.0 meV,B=−27.4 meV,C= 0,\nD=−20.5 meV (see text). The sample size is of 100 ×100 sites, and the lattice constant a= 5nm. The boundary of the\ntopological phase in (c) and (d) is adapted from (a).\nHowever, by comparing Fig. 1 (c) with the GL= 2e2\nhplateau region (enclosed by the thick black line) obtained\nfrom Fig. 1 (a), we can see that this disorder induced\nband inversion is not the whole story of the TAI: there\nis a level merging at W2∼0.14meV after which the en-\nergy levels merge together without another gap reopen-\ning again. This level merging does not affect the quan-\ntized conductance at all. At a finite size (for example\nNx=Ny= 100 here), the full gap regime ( ρave∼0)\nbetween W1toW2(the blue region between W1andW2\nin Fig. 1 (c)) only constitutes a very small portion of the\nconductance plateau. To investigate the fate of this band\ngapped topological insulator regime, i.e. TBI, under the\nthermodynamic limit ( Nx,Ny→ ∞), we plot the size de-\npendence of the averagedgap size at W= 0.1meV(along\nlineABin Fig. 1 (c)) in Fig. 2. This curve is approxi-\nmately linear with Nxin logarithmic scale, therefore the\ngapdecaysslowly(logarithmically)with the increasingof\nsample size Nx, suggesting that this small “TBI” region\nshrinks smaller or even vanishes in the thermodynamic\nlimit.\nMost part of the conductance plateau in Fig. 1 corre-\nsponds to a region with a large bulk DOS ρave. In other\nwords,thesetopologicalstatesdonotliveinabulkgapat\nall. From the plot of ratio ρtyp/ρavein Fig. 1 (d), we can\nsee that this region of TAI is related to extremely local-\nized states with ρtyp/ρave→0. From Fig. 1 (d), it is in-/s49/s48 /s49/s48/s48/s48/s46/s48/s48/s48/s48/s46/s48/s48/s50/s48/s46/s48/s48/s52/s48/s46/s48/s48/s54/s48/s46/s48/s48/s56/s48/s46/s48/s49/s48/s48/s46/s48/s49/s50\n/s32/s87/s61/s48/s46/s48/s49/s32/s109/s101/s86/s44/s32/s97/s108/s111/s110/s103/s32 /s65 /s66 /s32/s105/s110/s32/s70/s105/s103/s46/s32/s49/s32/s40/s99/s41\n/s32/s32/s103/s97/s112\n/s78 /s120\nFIG. 2: (Color online) The energy gap at W= 0.1meV (along\nthe red line ABin Fig. 1 (c)) as a function of lattice size Nx\nfor square samples. The Nxaxis is in logarithmic scale. Each\ndot is a statistical average over 100 disorder samples.\nteresting to note that ρtyp/ρavein the TAI region is even\nsmaller (i.e., more localized) than any other region (even\nwith larger W) in the plot range. This is consistent with\nthe physical picture found in our previous work22, that\ntheseextremelylocalizedstates(Andersoninsulator)cor-\nrespond to flat but densely distributed subbands (there-\nfore contribute to DOS, but not to conductance). These4\nflat subbands have trivial topological numbers, therefore\nthey do not change the topological nature of nearby en-\nergy region. With open boundaries, helical edge states\nappear between adjacent flat subbands and contribute to\nthe conductance 2 e2/h. These subbands are so flat that\ntheir total measure on the energy axis is extraordinarily\nsmall, therefore there is always enough space to occupy\nhelical edge states between them, in the case of an open\nboundary. In other words, the TAI phase lives in a bulk\nstate of Anderson insulator (instead of band insulator),\nas the name suggests. The extremely localized nature of\nthe bulk Anderson insulator also prohibits backscatter-\ning between edge states on opposite edges, therefore we\ncall that the TAI is protected by a mobility gap instead\nof a band gap.\nWith further increasing of disorder strength W, as in\nany lattice systems, the electronic states will finally be\nlocalized completely and the system becomes an Ander-\nson insulator28, which happens after W3∼0.28meV in\nthe present system.\nB. An inverted band\nNow we turn to the case of an inverted band: M=−10\nmeV, which is a TBI in the clean limit. From the\nconductance plateau in Fig. 3 (a) and (b), it seems\nthat this phase diagram is quite simple: the topological\nphase characterized by helical edge states survives ro-\nbustly until a complete collapse at a very large disorder\nW3∼0.28meV. However, a detailed numerical calcula-\ntion demonstrates that the regime of topological phase\nactually also consists of two parts: one is the TBI and\nthe other is TAI. Although the conductance plateau in\nFig. 3 (a) is continuous, the DOS can be used to dis-\ntinguish the two phases very well. In the DOS plot in\nFig. 3 (c), we can see the developments of energy levels\nunder increasing of disorder strength W. Note the con-\nductance and valence bands merge at W2∼0.13meV,\nwhich is much weaker than W3where the conductance\nplateau collapses.\nWe firstconcentrateonthe conductanceplateauregion\nwithW < W 2, i.e., before the band merging, which is ap-\nparently a band insulator. As can be seen in Fig. 3 (c)\nfor a definite size and disorderconfiguration, at weak dis-\norder starting from W= 0, the bulk gap increases slowly\nwith increasing disorder until W1, reflecting the band re-\npelling of topological nontrivial bands under disorder22.\nOn the other hand, as the red curve with square dots in\nFig. 4, the size dependence of the averaged gap at a defi-\nniteW(along line ABin Fig. 3 (c) ) indicates that it is a\nhorizontal line with negligible statistical errors, suggest-\ningarobustregionofTBIindependent ofsamplesizeand\ndisorderconfigurations. After W1, the bulk gapbegins to\nshrink, but the conductance plateau remains unchanged.\nIn other words, TAI with quantized GL= 2e2\nhpersists\nand finite ρavegradually appears when W > W 1. In Fig.\n4, the gap size in this region for a definite W(along lineCDin Fig. 3 (c) ) as a function of sample size is plotted\nas the violet curve: it decays logarithmically. This sim-\nply suggests that in the thermodynamic limit, the region\nof the TBI is even smaller than that plotted in Fig. 3 for\nfinite size Nx=Ny= 100. On the other hand, as dis-\ncussed above, the TBI is always robust and well-defined\nbeforeW1.\nOn the right side of the border between TBI and TAI,\nthe phase possesses the typical characters of TAI: quan-\ntized conductance plateau living in a bulk mobility gap\nwith nonzero DOS from extremely localized states. From\nthe phase diagram in Fig. 3, It is interesting to notice\nthat TAI phase (with quantized GL= 2e2\nh, finiteρave\nand small ρtyp/ρave) still occupies a large portion of the\nconductance plateau. This comes from the localization\nof bulk sates near band edges under disorder. Therefore\nwe emphasize that even tuning disorder on from an in-\nverted band, the TAI also plays a very important role in\nthe whole topological phase. As can be seen from Fig.\n3, the boundary between the two topological phases is\nsharp and clear without any intermediate state, say, a\nmetal phase16.\nIV. SUMMARY AND DISCUSSIONS\nNow we can summarize the route towards localization\nschematically fora disordered quantum spin Hall system\nin Fig. 5. In the clean limit of W= 0, the electronic\nstates form well-defined energy bands, and the band gap\nbetween the conduction and valence band are well de-\nfined. If we start from a conventional band insulator, as\nshown in Fig. 5 (a), weak disorder tends to shrink the\nbulk band gap13,20,22. If the gap is small enough, then\nthis disorderinduced band attraction will eventually lead\nto a band closing at W1, after which edge states emerge\nin the re-opened gap. On the other hand, disorder de-\nstroysthetranslationinvariance,makingtheenergyband\nworse defined and split into more discrete energy levels.\nAround the band edges, these levels constitute to local-\nized states29,30. The calculationof a single impurity indi-\ncates that the in-gap bound states can be induced easily\nby a single impurity in the inverted band regime31. Af-\nterW2, these localized states permeate into the bulk gap,\nleading to a well-defined TAI.\nFrom this picture, we can see the condition for a sta-\nble disorder induced TAI phase. First, the clean limit\ngap should be sufficiently small to guarantee W1< W3.\nSecond, after W1, the in-gap bulk states should be suffi-\nciently localized (or equivalently, sufficiently flat). From\nthe viewpoint of momentum space, the in-gap states\nshould be flat enough for edge states between them to\nshow up. From the view point of real space, these states\nin the bulk should be localized enough to protect he-\nlical edge states from backscattering. This is why we\ncall TAI a mobility gap protected topological insulat-\ning state, compared with TBI, an energy gap protected\none. When we start from topological insulating phase,5\n/s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s40/s97/s41/s99/s111/s110/s100/s117/s99/s116/s97/s110/s99/s101/s58/s32/s111/s112/s101/s110/s32/s98/s111/s117/s110/s100/s97/s114/s121\n/s48\n/s48/s46/s55/s53/s48/s48\n/s49/s46/s53/s48/s48\n/s50/s46/s48/s48/s53\n/s51/s46/s48/s48/s48\n/s51/s46/s55/s53/s48\n/s52/s46/s53/s48/s48\n/s53/s46/s50/s53/s48\n/s54/s46/s48/s48/s48/s49/s46/s57/s57/s53/s71/s32/s40/s101/s50\n/s47/s104/s41\n/s87 /s32/s40/s109/s101/s86/s41/s69\n/s70/s32/s40/s109/s101/s86/s41\n/s84/s66/s73/s84/s65 /s73\n/s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s51/s53/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54\n/s84/s66/s73/s84/s65 /s73/s40/s98/s41/s71/s32/s40/s101/s50\n/s47/s104/s41/s99/s111/s110/s100/s117/s99/s116/s97/s110/s99/s101/s58/s32/s112/s101/s114/s105/s111/s100/s105/s99/s32/s98/s111/s117/s110/s100/s97/s114/s121\n/s87 /s32/s40/s109/s101/s86/s41/s69\n/s70/s32/s40/s109/s101/s86/s41/s48\n/s48/s46/s55/s53/s48/s48\n/s49/s46/s53/s48/s48\n/s50/s46/s50/s53/s48\n/s51/s46/s48/s48/s48\n/s51/s46/s55/s53/s48\n/s52/s46/s53/s48/s48\n/s53/s46/s50/s53/s48\n/s54/s46/s48/s48/s48\n/s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s51/s53/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s68/s79/s83\n/s97/s118/s101/s32/s40/s97/s46/s117/s46/s41\n/s40/s99/s41\n/s87 /s32/s40/s109/s101/s86/s41/s69\n/s70/s32/s40/s109/s101/s86/s41/s48/s46/s48/s51/s55/s48/s48\n/s48/s46/s48/s57/s49/s51/s50\n/s48/s46/s50/s50/s53/s52\n/s48/s46/s53/s53/s54/s50\n/s49/s46/s51/s55/s51\n/s51/s46/s51/s56/s56\n/s56/s46/s51/s54/s49\n/s49/s50/s46/s48/s48/s87\n/s51/s87\n/s49/s68\n/s67/s66\n/s65 /s87\n/s50/s84/s66/s73/s84/s65 /s73\n/s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s51/s53/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s68/s79/s83/s32/s114/s97/s116/s105/s111\n/s116/s121/s112/s47\n/s97/s118/s101\n/s40/s100/s41\n/s87 /s32/s40/s109/s101/s86/s41/s69\n/s70/s32/s40/s109/s101/s86/s41/s48/s46/s48/s56/s48/s48/s48\n/s48/s46/s50/s50/s51/s56\n/s48/s46/s51/s54/s55/s53\n/s48/s46/s53/s49/s49/s50\n/s48/s46/s54/s53/s53/s48\n/s48/s46/s55/s57/s56/s56\n/s48/s46/s57/s52/s50/s53\n/s49/s46/s48/s48/s48/s84/s66/s73/s84/s65 /s73\nFIG. 3: (Color online) Phase diagram in the parameter space o f disorder strength Wand Fermi energy EF, for an inverted\nband with M=−10meV in the clean limit. (a) Two-terminal conductance of a s tripe sample with an open boundary condition\nin the transverse direction; (b) Two-terminal conductance of a rolled sample with the periodic boundary condition; (c) the\narithmetic mean DOS ρave; (d) the ratio of the geometric mean DOS to arithmetic mean DO S:ρtyp/ρave.\n/s49/s48 /s49/s48/s48/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48/s48/s46/s48/s49/s53/s48/s46/s48/s50/s48\n/s32/s32/s103/s97/s112\n/s78 /s120/s32/s87/s61/s48/s46/s48/s50/s53/s32/s109/s101/s86/s44/s32/s97/s108/s111/s110/s103/s32 /s65 /s66 /s32/s105/s110/s32/s70/s105/s103/s46/s32/s51/s32/s40/s99/s41\n/s32/s87/s61/s48/s46/s48/s55/s48/s32/s109/s101/s86/s44/s32/s97/s108/s111/s110/s103/s32 /s67/s68 /s32/s105/s110/s32/s70/s105/s103/s46/s32/s51/s32/s40/s99/s41\nFIG. 4: (Color online) The energy gap at W= 0.1meV (along\nlinesAB(red square dot) and CD(violet circle dot) in Fig.\n3 (c)) as a function of square samples with lattice size Nx.\nTheNxaxis is in logarithmic scale. Each dot is a statistical\naverage over 100 disorder samples.\nas in Fig. 5 (b), weak disorder tend to expel the conduc-\ntion and valance bands22. Before the disorderdecompose\nthe energy bands into sufficiently discrete energy levels\n(W < W 2), the system is still in a well-defined topolog-\nical band insulating phase. After that, the bulk states\nnear the band edges will be localized29,30, and grow into\nthe bulk gap to make a TAI phase.\nIn this work, we try to distinguish the topological in-\nsulating phases in 2D as a band insulator and an An-\nderson insulator. In Fig. 3 we define a border between/s87/s87\n/s69/s69\n/s48/s48\n/s40/s98/s41\n/s87\n/s51/s87\n/s50/s32/s32/s32/s32/s32/s32/s32 /s116/s114/s105/s118/s105/s97/s108/s32/s98/s97/s110/s100/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s110/s111/s116/s114/s105/s118/s105/s97/s108/s32/s98/s97/s110/s100/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s101/s100 /s103/s101/s32/s115/s116/s97/s116/s101\n/s65/s73/s84/s65/s73/s84/s66/s73/s65/s73 /s84/s65/s73\n/s87\n/s51/s87\n/s50/s87\n/s49/s66/s73/s40/s97/s41\nFIG. 5: (Color online) Schematic of the evolution of band (en -\nergy level) structures with disorder strength, starting fr om (a)\na conventional band insulator; (b) a topological band insul a-\ntor. Topological trivial (nontrivial) bands are represent ed by\nblack (red) bars, and the green lines represents the presenc e\nof the helical edge states between the two topologically non -\ntrivial bands.\nTBI and TAI in the inverted band structure. This raises\na question of whether they are physically distinct from\neach other by measuring the mobility gap or any other\nphysical quantities. Usually temperature-dependencies\nof the bulk conductance are different between band in-\nsulators and Anderson insulators. With the help of\nthermal activation and phonons, the typical tempera-\nture dependence of conductivity for the band insulator6\nisσ∼exp[−Eg/(2kBT)] while that of the Anderson in-\nsulator is σ∼exp[−(T0/T)1/3]32.\nThe energy scale of the two-dimensional mobility gap\ncan be determined from capacity measurement as well as\ntransport measurement in Corbino samples.33In the ge-\nometry of Corbino disk, edge transport is stuck around\nthe concentric contacts, and the radial conductance mea-\nsurement mainly probe the bulk properties. The mo-\nbility gap can be revealed by using capacitance-voltage\nand conductance measurement. 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Lett. 78, 4613-4616 (1997)."    },    {        "title": "1310.1246v2.Fermion__N__representability_for_prescribed_density_and_paramagnetic_current_density.pdf",        "content": "arXiv:1310.1246v2  [math-ph]  28 Nov 2013Fermion N-representability for prescribed density and paramagneti c current density\nErik Tellgren,∗Simen Kvaal, and Trygve Helgaker\nUniversity of Oslo, Centre for Theoretical and Computation al Chemistry, N-0315 Oslo, Norway\nTheN-representability problem is the problem of determining wh ether or not there exists N-\nparticle states with some prescribed property. Here we repo rt an affirmative solution to the fermion\nN-representability problem when both the density and parama gnetic current density are prescribed.\nThis problem arises in current-density functional theory a nd is a generalization of the well-studied\ncorresponding problem (only the density prescribed) in den sity functional theory. Given any density\nand paramagnetic current density satisfying a minimal regu larity condition (essentially that a von\nWeiz¨ acker-like the canonical kinetic energy density is lo cally integrable), we prove that there exist\na corresponding N-particle state. We prove this by constructing an explicit o ne-particle reduced\ndensity matrix in the form of a position-space kernel, i.e. a function of two continuous position\nvariables. In order to make minimal assumptions, we also add ress mathematical subtleties regarding\nthe diagonal of, and how to rigorously extract paramagnetic current densities from, one-particle\nreduced density matrices in kernel form.\nI. INTRODUCTION\nThe question of N-representability has been studied\nextensively in quantum chemistry and related fields [1].\nIn particular, it plays an important role in density-\nfunctional theory (DFT). Given prescribed values for\nquantities in a fermionic system, e.g. its electron density\nor its reduced density matrix, one may ask whether or\nnot it can be obtained from a Slater determinant, from a\npureN-particle state, or from a mixed N-particle state.\nRegarding the density, it is well known that any den-\nsity with a finite von Weizs¨ acker kinetic energy may be\nreproduced using a Slater determinant that also has fi-\nnite kinetic energy [2–6]. For a one-particle reduced den-\nsity matrix (1-rdm), Slater-determinantal representabil-\nity is equivalent to idempotency; in general, however,\npure-state N-representability of 1-rdms is a difficult and\nlargely unsolved problem [7–9]. On the other hand, any\n1-rdm with spin-orbital occupation numbers (eigenval-\nues) in the range [0 ,1] and trace Nmay be obtained\nfrom a mixed N-particle state.\nIn this note, we report the solution to the mixed-state\nN-representability problem when both the density and\nparamagnetic current density are prescribed. More pre-\ncisely, we answer the following question: given a den-\nsityρ(r) and a paramagnetic current density jp(r), does\nthere exist a mixed state Γ with the prescribed density\nand current density, written Γ /ma√sto→(ρ,jp)? We answer this\nquestion affirmatively by constructing an explicit 1-rdm,\nfrom which the existence of the N-particle state follows.\nWe note that standard constructions demonstrating\nthe Slater-determinantal N-representability when only\nthe density is prescribed rely on the use of equiden-\nsity orbitals. Also, early work by Ghosh and Dhara [10]\nsketched a construction of such equidensity orbitals that\nreproduce both densities and currents. However, such\nsolutions have limited scope, since the vorticity vanishes\n∗erik.tellgren@kjemi.uio.nowhen orbitals give rise to the same density.\nDuring the preparation of the present work, one of us\n(S. Kvaal) met E.H. Lieb during a trimester at Institut\nHenri Poincar´ e in Paris in July 2013, and became aware\nthat together with R. Schrader, he had shown a Slater\ndeterminant representability result for ( ρ,jp) andN≥4.\nThis work has now been published [11].\nClearly, N-representability via a Slater determinant\nimplies representabiility via a mixed state. However,\nLieb and Schrader’s result requires N≥4, and they\nalso give a counterexample for N= 2, where no Slater\ndeterminant can exist (with continuously differentiable\nand single-valued orbital phase functions). Our result,\nwhile showing a weaker sense of N-representability, has\nno condition on N. Moreover,both the present work and\nRef. [11] have mild regularity and decay assumptions on\n(ρ,jp) that ensure representability, but these are differ-\nent in the two approaches. The techniques of proof are\nalso otherwise significantly different: Lieb and Schrader\nrely on the so-called smooth Hobby–Rice theorem, while\nour approach is by direct construction of a 1-rdm. The\npresent work and the workof Lieb and Schraderare com-\nplementary, offering two different points of view and so-\nlutions to a long-standing problem.\nThe remainder of this paper contains five sections. In\nSection II, we give some background information and es-\ntablish notation. Following a discussion of the relation-\nship between a reduced density matrix and its associated\ndensity and paramagnetic current density in Section III,\nwe construct in Sections IV and IVC a reduced density\nmatrixforaprescribeddensityandparamagneticcurrent\ndensity. Section V contains some concluding remarks.\nFinally, two appendices are also provided. Appendix A\ncontains a brief overview of some mathematical concepts\nand results on Hilbert–Schmidt operators needed for the\nmain results of Section III. Appendix B contains proofs\nof theorems in Section III.2\nII. BACKGROUND\nTheN-representability problem with prescribed den-\nsityρand paramagnetic current density jparises in\ncurrent-density functional theory (CDFT) [12]. In\nCDFT, a magnetic vector potential A, in addition\nthe scalar potential v, enters the (spin-free) N-electron\nHamiltonian. In atomic units,\nH[v,A] =1\n2N/summationdisplay\nk=1(−i∇k+A(rk))2\n+N/summationdisplay\nk=1v(rk)+/summationdisplay\nk<l1\nrkl.(1)\nHererkis the position of electron k, the operator ∇kdif-\nferentiates with respect to rk, andrklis the distance be-\ntween electrons kandl. The corresponding ground-state\nenergy is given by the Rayleigh–Ritz variation principle,\nE[v,A] = inf\nΓTr(ΓH[v,A]) (2)\nwhere the minimization is over all mixed states Γ with a\nfinite canonical kinetic energy\nT[Γ] :=1\n2Tr/parenleftig\n∇Γ∇†/parenrightig\n. (3)\nIntroducing the constrained-search universal functional\nF[ρ,jp] = inf\nΓ/mapsto→ρ,jpTr(ΓH[0,0]), (4)\nwe may rewrite the Rayleigh–Ritz variation principle in\nEq.(2) in the form of a Hohenberg–Kohn variation prin-\nciple,\nE[v,A] = inf\nρ,jp/parenleftig\nF[ρ,jp]+/integraldisplay/parenleftig\nρ(v+1\n2A2)+jp·A/parenrightig\ndr/parenrightig\n.(5)\nThe mixed-state N-representability problem is directly\nrelated to how large the search domain in Eq. (5) needs\nto be: if no Γ /ma√sto→(ρ,jp) exists, then F[ρ,jp] = +∞by\ndefinition.\nIn Kohn–Sham theory, the idea is to express the den-\nsities in Eq.(5) in terms of a single Slater determinant of\nnon-interacting particles and to approximate the kinetic-\nenergy contributions to F[ρ,jp] by the non-interacting\nkinetic energy,\nTs[ρ,jp] = inf\n{φk}N\nk=1/mapsto→ρ,jp1\n2N/summationdisplay\nk=1/an}b∇acketle{t∇φk,∇φk/an}b∇acket∇i}ht,(6)\nwhere the infimum is over an orthonormal set of orbitals\nφkor, equivalently, the corresponding Slater determi-\nnants or idempotent 1-rdms. At this point, the Slater-\ndeterminantal N-representability problem arises.\nIn general, densities ρandjparising from a single or-\nbital have a vanishing paramagnetic vorticity,\nν=∇×jp\nρ= 0, (7)except for possible Dirac-delta singularities at points r\nwhereρ(r) = 0. Consequently, a closed-shell two-particle\nKohn–Sham system can only reproduce paramagnetic\ndensities with vanishing vorticity. In general, therefore\nan extended Kohn–Sham approach with fractional occu-\npation numbers is required (see Refs. [13–16] for work in\nthis direction),\n¯Ts[ρ,jp] = inf\n{nkφk}/mapsto→ρ,jp1\n2∞/summationdisplay\nk=1nk/an}b∇acketle{t∇φk,∇φk/an}b∇acket∇i}ht,(8)\nwhere orthonormality, 0 ≤nk≤1, and/summationtext\nknk=N\nare additional constraints on the infimum. Alternatively,\nsincenkandφkare eigenvalues and eigenvectors of 1-\nrdms, the minimization may equivalently be performed\nover 1-rdms. Here, the mixed-state N-representability\nproblem appears.\nFor a mixed state Γ /ma√sto→(ρ,jp), it is known that\nTW[ρ]+Tp[ρ,jp]≤T[Γ], (9)\nwhere the von Weizs¨ acker kinetic-energy functionals are\ngiven by\nTW[ρ] :=1\n8/integraldisplay\nρ(r)−1|∇ρ(r)|2dr,(10)\nTp[ρ,jp] :=1\n2/integraldisplay\nρ(r)−1|jp(r)|2dr. (11)\nA necessary condition for a finite-kinetic-energy repre-\nsentability is therefore that TW[ρ]+Tp[ρ,jp]<+∞. For\nthe case jp= 0, this is also a sufficient condition. It is\nof interest to know whether this sufficiency generalizes to\njp/ne}ationslash= 0. In this paper, we shall prove sufficiency under\nmild additional conditions on the current density.\nIII. DIAGONALS OF DENSITY OPERATORS\nWe do not explicitly consider spin and therefore take\nas our point of departure an N-electron density matrix\nthat depends only on spatial coordinates, with the spin\ncoordinates integrated out:\nΓ(r1:N,s1:N) =/summationdisplay\nipiΨi(r1:N)Ψ∗\ni(s1:N).(12)\nSuch a density matrix is an element of a Lebesgue space,\nΓ∈L2(R3N×R3N), (13)\nand is symmetric with respect to permutations πof the\ncoordinate labels, ( rk,sk)π/ma√sto→(rπ(k),sπ(k)). The right-\nhand side of Eq. (12) is a convex combination of properly\nnormalized pure states, Ψ i∈L2(R3N), with coefficients\npi≥0 such that/summationtext\nipi= 1. Moreover, each (spin-free)\npure state is either totally symmetric or anti-symmetric.\nIn what follows, it does not matter whether each pure\nstate is required to be anti-symmetric, symmetric, or ei-\nther anti-symmetric or symmetric with respect to index\npermutations πofthe spatial coordinates. For simplicity,\nwe shall occasionally simplify the presentation by taking\nΓ to be a pure state.3\nA. Density matrices\nThe 1-rdm belonging to a pointwise defined Γ on the\nform (12) is given by the convex combination\nDΓ(r,s) :=N/summationdisplay\nipi/integraldisplay\nR3N−3Ψi(r,r2:N)Ψ∗\ni(s,r2:N),dr2:N\n(14)\nand belongs to L2(R3×R3). For pure states Γ = |Ψ/an}b∇acket∇i}ht/an}b∇acketle{tΨ|,\nwemayalternativelywrite DΨ. Duetopermutationsym-\nmetry, the 1-rdm is independent of which N−1 coordi-\nnates that have been integrated out.\nGiven that DΓ∈L2(R3×R3),DΓis by definition\nthekernel of a Hilbert–Schmidt integral operator . Our\ndiscussion makes extensive use of this basic fact about\nthe reduced density matrix. In particular, Γ and DΓ\nare both trace-class operators —that is, Hilbert–Schmidt\noperators for which the matrix trace has a meaningful\ngeneralization. Let {φk} ⊂L2(X) be an orthonormal\nbasis. By definition, Ais a trace-class operator if and\nonly if the trace\nTrA:=/summationdisplay\nk/an}b∇acketle{tφk,Aφk/an}b∇acket∇i}ht (15)\nhas a finite value, independent of the orthonormal basis.\nFortwoHilbert–Schmidtoperators BandC, thekernel\nof the operator product A=B∗Cis easily seen to be\n(B∗C)(x,y) :=/integraldisplay\nXB(x,z)C(z,y)dz,(16)\nwhich is also Hilbert–Schmidt. Importantly, it can be\nshown that Ais (the kernel of) a trace-class operator if\nand only if A=B∗CwithBandCHilbert–Schmidt.\n(Indeed, this is often taken as an alternative definition\nof trace-class operators.) The trace is then given by [17]\nthe integral of the diagonal,\nTrA=/integraldisplay\nX(A∗B)(x,x)dx. (17)\nIfAis diagonable (e.g., symmetric positive semidefinite),\nthen TrAis the sum of the eigenvalues, like in the finite-\ndimensional case. For further information on these oper-\nator classes, see for example the standard textbook [18].\nWe denote by DNthe set of mixed N-electron states\nΓ and by DN,1the set of 1-rdms that belong to some\nmixedN-electron state. The set DN,1has the following\nwell-known characterization:\nTheorem III.1. DN,1consists of those D∈L2(R3×R3)\nwith the following properties:\n1.Dis the kernel of a trace-class operator on L2(R3).\n2.Dis Hermitian:\nD(r,s) =D∗(s,r)for almost all (r,s).\n3.Dis positive semidefinite:\n0≤/integraltext\nφ∗(r)D(r,s)φ(s)dsfor allφ∈L2(R3)4.Dhas no eigenvalues greater than two:\n2≥/integraltextφ∗(r)D(r,s)φ(s)dsfor allφ∈L2(R3)\n5.Dhas eigenvalues that add up to N:\nTrD=N.\nProof.See Ref. [19], Section 2.6.\nThe last three conditions mean that D(r,s) has eigen-\nvalues in the interval [0 ,2]—that is, eigenvalues inter-\npretable as fermion occupation numbers—and that the\nsum of the eigenvalues Tr Dis equal to N, the number\nof particles.\nSinceD∈ DN,1is Hermitian and positive, it is easy to\nshow that there always exists a factorization of the form\nD=G†∗G, meaning that we may write the density\nmatrix in the form\nD(r,s) = (G†∗G)(r,s) =/integraldisplay\nR3G∗(u,s)G(u,r)du,(18)\nwhich plays an important role in the following.\nB. Density\nWe now define the densityρΨassociatedwith the wave\nfunction Ψ as\nρΨ(r) :=DΨ(r,r) =N/integraldisplay\nR3N−3|Ψ(r,r2:N)|2dr2:N.(19)\nForalmostall r, itholdsthatΨ( r,·)∈L2(R3N−3). Using\ntheCauchy–Schwarzinequality,weseefromEq. (19)that\nρΨ(r) =DΨ(r,r) is well defined for almost all r. For a\nmixed state Γ ∈ DN, the density ρΓis defined in the\nsame manner but from DΓ.\nThe following point is subtle but important here. We\nwrite Γ/ma√sto→Dwhenever /ba∇dblDΓ−D/ba∇dblL2(R3×R3)= 0. This\nstatement does not imply that that DΓ=Deverywhere,\nonlythat DΓandDareequalaselements of L2(R3×R3).\nConsequently, DΓandDmay differ at a set of measure\nzero,including the totality of the diagonal . Therefore,\nwe need to examine carefully the validity or meaning of\nthe statement “ ρΓ(r) =D(r,r)” for a state and density\nmatrix related by Γ /ma√sto→D.\nSuppose next that we are able to assign a diagonal\ndiagDtoDinsomeunambiguouswayandletΓ ,Γ′∈ DN\nbe two (possibly distinct) states such that Γ /ma√sto→Dand\nΓ′/ma√sto→D, meaningthat D=DΓ=DΓ′almosteverywhere\ninR3×R3. Is it then true that ρΓ=ρΓ′= diagDalmost\neverywhere in R3? Intuitively, this should be so.\nThefollowingtheorem, whichisprovedinAppendix B,\nresolves the issue:\nTheorem III.2. LetD∈ DN,1, and suppose that G∈\nL2(R3×R3)is such that\nD(r,s) = (G†∗G)(r,s)4\nalmost everywhere in R3×R3. Then, for every Γ∈ DN\nsuch that Γ/ma√sto→D, it holds that\nρΓ(r) = (G†∗G)(r,r)\nalmost everywhere in R3.\nSince the factorization of D=G†∗Gdoes exist follow-\ningthediscussioninSection IIIA, itisindeedmeaningful\nto talk about “the density ρofD” without reference to\na specific Γ /ma√sto→D:\nρD(r) = diagD(r) := (G†∗G)(r,r) a.e. (20)\nIn particular, it follows that\nTrD=/integraldisplay\nρD(r)dr. (21)\nWe emphasize that we only define the diagonal when a\nfactorization is present, and that this diagonal is inde-\npendent of the factorization.\nC. Momentum density\nBefore considering the momentum density, we note\nthat all derivatives that occur in the subsequent treat-\nment are distributional or weak derivatives . A function\nf∈Lp(X), withX⊂Rnopen, is said to have a weak\nderivative g=∂αf∈L1\nloc(X) if, for all smooth, com-\npactly supported “test functions” u∈ C∞\nc(X),\n/integraldisplay\nXg(x)u(x)dx=−/integraldisplay\nXf(x)∂αu(x)dx.(22)\nThus, the weak derivative acts just like the standard\nderivative ∂f/∂x αwhen we apply integration by parts,\ncoinciding with the classical derivative whenever this ex-\nists. Higher-order weak derivatives are defined in a simi-\nlar manner. A standard monograph for weak derivatives\nis Ref. [20].\nBy analogy with the density in Eq. (19), we now define\nthemomentum density cΨof a state Ψ as\ncΨ(r) :=N/integraldisplay\nR3N−3[−i∇rΨ(r,r2:N)]Ψ∗(r,r2:N)dr2:N\n=−i∇rDΨ(r,s)|r=s, (23)\nwhose real part is the paramagnetic current density:\njpΨ(r) = RecΨ(r) (24)\nwith an analogous definitions for a mixed state Γ. We\nnote, however, that this definition may not make sense\nwithout additional assumptions on the wave function Ψ,\nbeyond those needed for the definition of the density. We\nalso observe that the second equality in Eq. (23) needs to\nbe justified further since ∇rDΨ(r,s)|r=sis only defined\npointwise almost everywhere and since integration may\nnot commute with differentiation.\nTo assign unambiguously a momentum density cD(r)\ntoD∈ DN,1, we first introduce the notion of a locally\nfinite kinetic energy :Definition III.1 (Locally finite kinetic energy) .We say\nthatD∈ DN,1has a locally finite kinetic energy if the\nweak derivative ∇1·∇2Dis the kernel of a trace class\noperator over L2(K)for every compact K⊂R3. Like-\nwise, we say that Ψ∈L2(R3N)has a locally finite ki-\nnetic energy if ∇1Ψ∈L2(K×R3N−3)for every compact\nK⊂R3—that is, ∇1Ψ∈L2(R3\nloc×R3N−3). (See Ap-\npendix A2.) A mixed state Γ∈ DNhas a locally finite\nkinetic energy if/summationtext\nipi/ba∇dbl∇1Ψi/ba∇dbl2\nL2(K×R3N−3)is finite for\nevery compact K⊂R3.\nNote that the pure-state definition of locally finite\nkinetic energy follows from that of the mixed state.\nThe various definitions of a locally finite kinetic energy\nare connected, as summarized in the following theorem,\nproved in Appendix B:\nTheorem III.3. ForD∈ DN,1, the following state-\nments are equivalent:\n1.Dhas a locally finite kinetic energy.\n2. There exists a factorization D=G†∗G(a.e.) with\nGHilbert–Schmidt and ∇2G∈L2(R3×R3\nloc).\n3. AnyΓ∈ DNwithΓ/ma√sto→Dhas a locally finite kinetic\nenergy.\nIfDhas a locally finite kinetic energy, then the asso-\nciatedkinetic-energy density is defined as\nτD(r) :=1\n2/ba∇dbl∇2G(·,r)/ba∇dbl2\nL2(R3)\n=1\n2/integraldisplay\nR3[∇2G(u,r)]∗·[∇2G(u,r)]du(25)\n=1\n2diag(∇1·∇2D)(r),\nwhich is finite almost everywhere. From the proof of the\nTheorem III.3 in Appendix B, it follows that τDis in fact\nthe kinetic energy density of any Γ /ma√sto→D. We also see\nthatτD∈L1\nloc(R3) and that the total kinetic energy is\nfinite if and only if τD∈L1(R3).\nFinally, the following theorem (proved in Appendix B)\nstates that, if Dhas a locally finite kinetic energy, then\nthe momentum density of Eq. (23) is also well defined:\nTheorem III.4. LetD∈ DN,1have a locally finite\nkinetic energy and let G∈L2(R3×R3)be such that\nD=G†∗Gand∇2G∈L2(R3×R3\nloc). For each Γ∈ DN\nwithΓ/ma√sto→D, it then holds that cΓ∈L1\nloc(R3)and that\ncΓ(r) =/parenleftbig\n[−i∇2G]†∗G/parenrightbig\n(r,r) a.e.\n= diag(−i∇1D)(r).\nThis result implies that if Dhas locally finite ki-\nnetic energy, then jpΓ∈L1\nloc(R3). Moreover, ∇ρΓ=\n−2ImcΓ∈L1\nloc(R3) as well.5\nD. Summary\nFor easy reference, we collect the main conclusions of\nthis section in a separate theorem:\nTheorem III.5. LetD=G†∗GwithG∈L2(R3×R3)∈\nDN,1,∇2G∈L2(R3×R3\nloc). For every Γ∈ DNsuch\nthatΓ/ma√sto→D, it then holds that the density ρΓ=ρ∈\nL1(R3), the momentum density cΓ=c∈L1\nloc(R3), and\nthe kinetic energy density τΓ=τ∈L1\nloc(R3)are given\nalmost everywhere by the expressions\nρ(r) =/integraldisplay\nR3G∗(u,r)G(u,s)du,\nc(r) =/integraldisplay\nR3[−i∇2G∗(u,r)]G(u,s)du,\nτ(r) =1\n2/integraldisplay\nR3[∇2G∗(u,r)]·[∇2G(u,s)]du.\nIV. A REDUCED DENSITY MATRIX FOR A\nPRESCRIBED DENSITY AND PARAMAGNETIC\nCURRENT DENSITY\nLet a density ρbe given. We assume that the density\nis non-negative and that it belongs to the intersection of\ntwo Lebesgue spaces,\nρ(r)≥0,andρ∈L1(R3)∩Lq(R3),(26)\nfor some q >1. The latter condition amounts to\nN:=/ba∇dblρ/ba∇dbl1=/integraldisplay\n|ρ(r)|dr<+∞, (27)\n/ba∇dblρ/ba∇dblq\nq=/integraldisplay\n|ρ(r)|qdr<+∞,(28)\nwhereNis the number of particles in the density ρ. (For\nsimplicity, we restrict ourselves to states with integral N\nbut note the 1-rdm constructions given below are valid\nalsoforfractional N.) Furthermore,letanarbitrarymea-\nsurable vector-valued function κ:R3→R3be given and\nlet it prescribe a paramagnetic current density by the\nrelation\njp(r) =1\n2ρ(r)κ(r). (29)\nWe now consider the question: does there, for every pair\nofρandjpsatisfying these minimal requirements, exist\naD∈ DN,1that reproduces ρandjp? In short, we seek\na reduced density matrix Dsuch that\n(a)ρ(r) =ρD(r) = (diag D)(r)\n(b)jp(r) = RecD(r) =−i\n2(diag∇1D)(r)+c.c.,\nassuming that Dhas a locally finite kinetic energy for\n(b) to be well defined. We can indeed find such a density\nmatrixD∈ DN,1but shall see that the condition of a\nlocally finite kinetic energy of Dimplies mild additional\nconditions on ρandκ.A. Factorized elements PλandQλ\nOur strategy is to construct explicitly factorized ele-\nmentsPλ=G†\nλ∗GλandQµ=H†\nµ∗HµinDN,1with\na locally finite kinetic energy. Here, λ,µ >0 are real\nparameters that allow some freedom, noting that a con-\nvex combination Dλµ= (Pλ+Qµ)/2 remains in DN,1,\nalso with a locally finite kinetic energy. The flexibility\nof having several independent factorized reduced density\nmatrices PλandQµallows the convex combination to\nreproduce the desired current.\nThetwotermsaredefinedbythefactorizedexpressions\nPλ(r,s) =/radicalbig\nρ(r)ρ(s)/integraldisplay\nR3g∗(u,r)g(u,s)du,(30)\nQµ(r,s) =/radicalbig\nρ(r)ρ(s)/integraldisplay\nR3h∗(u,r)h(u,s)du,(31)\nwhere\ng(u,v) =√\n8λ3/4\nπ3/4e−iv·κ(v)e−2λ(u−v)2,(32)\nh(u,v) =√\n8µ3/4\nπ3/4eiu·κ(v)e−2µ(u−v)2.(33)\nClearly, these operators may be written in the form\nPλ=G†\nλ∗Gλ, Gλ(r,s) =g(r,s)/radicalbig\nρ(s),(34)\nQµ=H†\nµ∗HµHµ(r,s) =h(r,s)/radicalbig\nρ(s),(35)\nIt is straightforwardto verify that Gλ,Hλ∈L2(R3×R3).\nThe integration over umay be performed analytically,\nyielding the alternative expressions\nPλ(r,s) =/radicalbig\nρ(r)ρ(s)e−λ|r−s|2ei(r·κ(r)−s·κ(s)),(36)\nQµ(r,s) =/radicalbig\nρ(r)ρ(s)e−µ|r−s|2\n×e−i\n2(r+s)·(κ(r)−κ(s))−|κ(r)−κ(s)|2/16µ.(37)\nThese operators were found by making the initial ansatz\nφ(r) =/radicalbig\nρ(r)eir·κ(r)for an unnormalized natural orbital.\nThe corresponding paramagnetic current is then almost\ncorrect but contains an extra term that is most easily\ncanceled if the density matrix contains exponential fac-\ntors of the form eir·κ(s). Since the elements of DN,1and\ntheir properties are conveniently described if an explicit\nfactorization is available (see Theorem III.5), Gaussian\nkernels are suitable since since they allow mixed phase\nfactors of the type eir·κ(s)to survive the integration.\nB. The density of PλandQλ\nWe now need to verify that PλandQµare elements of\nDN,1by checking points (1)–(5) of Theorem III.1.\nTheorem IV.1. Letρ∈L1(R3)∩Lq(R3)for some q >\n1,ρ≥0a.e.,/ba∇dblρ/ba∇dbl1=N, and let λ,µ∈Rbe such that\nλ,µ≥2p\nπ(1\n4N/ba∇dblρ/ba∇dblq)2p/36\nwhere1/p+1/q= 1. ThenPλandQµin Eqs.(30)–(33)\nare elements of DN,1, with\nρPλ(r) =ρQµ(r) =ρ(r)\nalmost everywhere. The same is true for any convex com-\nbination θPλ+(1−θ)Qµ∈ DN,1withθ∈[0,1].\nProof.Both operators are Hermitian and positive semi-\ndefinite. From the expressions in Eqs. (36) and (37),\n(diagPλ)(r) = (diag Qλ)(r) =ρ(r) almost everywhere.\nIt follows that Tr Pλ= TrQλ=/integraltext\nρ(r)dr=N.\nIt remains to compute a bound on the largest eigen-\nvalues, demonstrating point (4) of Theorem III.1 for the\ncorresponding parameter values λandµ. For an arbi-\ntrary normalized orbital,\nn2≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nφ∗(r)Pλ(r,s)φ(s)drds/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≤/parenleftbigg/integraldisplay\n|φ(r)|/radicalbig\nρ(r)ρ(s)e−λ|r−s|2|φ(s)|drds/parenrightbigg2\n(38)\n=/parenleftbigg/integraldisplay\n|φ(r)|/radicalbig\nρ(r)/parenleftbigg/integraldisplay/radicalbig\nρ(s)e−λ|r−s|2|φ(s)|ds/parenrightbigg\ndr/parenrightbigg2\n.\nGiven that φ,√ρ∈L2(R3), the Cauchy–Schwarz in-\nequality may be applied twice to give\nn2≤/parenleftbigg/integraldisplay\n|φ(r′)|2dr′/parenrightbigg\n×\n×/parenleftigg/integraldisplay\nρ(r)/parenleftbigg/integraldisplay/radicalbig\nρ(s)e−λ|r−s|2|φ(s)|ds/parenrightbigg2\ndr/parenrightigg\n≤/integraldisplay\nρ(r)/parenleftbigg/integraldisplay\n|φ(s′)|2ds′/integraldisplay\nρ(s)e−2λ|r−s|2ds/parenrightbigg\ndr\n≤/integraldisplay\nρ(r)/parenleftbigg\nsup\nc/integraldisplay\nρ(s)e−2λ|c−s|2ds/parenrightbigg\ndr\n=Nsup\nc/integraldisplay\nρ(s)e−2λ|c−s|2ds (39)\nFinally, exploiting the fact that ρ∈Lq(R3), the integral\noversmaybebounded byinvokingtheH¨ olderinequality,\nn2≤N/ba∇dblρ/ba∇dblqsup\nc/ba∇dble−2λ|c−s|2/ba∇dblp=N/ba∇dblρ/ba∇dblq/parenleftbiggπ\n2pλ/parenrightbigg3/2p\n(40)where 1/p+1/q= 1. This bound is independent of the\ncurrent density. Hence, Pλhas no eigenvalues greater\nthan two if\nλ≥2p\nπ/parenleftbig1\n4N/ba∇dblρ/ba∇dblq/parenrightbig2p/3. (41)\nThese steps hold also for Qµ, showing that it has no\neigenvalues greater than 2 when µ≥2p\nπ(1\n4N/ba∇dblρ/ba∇dblq)2p/3.\nFinally, consider a convex combination Dθ=θPλ+\n(1−θ)Qµ, which belongs to DN,1since this set convex.\nMoreover, diag Ais linear in Asince diag( A+B)(r) =\ndiag(A)(r) + diag( B)(r) almost everywhere. Therefore,\ndiagDθ=θdiagPλ+(1−θ)diagQλ=ρalmost every-\nwhere.\nC. The canonical kinetic energy of PλandQλ\nWe now turn to the question of whether the current\njpcan be reproduced by D. Indeed, a formal calculation\nshows that\n−i\n2∂\n∂rαPλ(r,s))/vextendsingle/vextendsingle\ns=r+c.c. = ρ(r)/parenleftbigg\nκα(r)+r·∂κ(r)\n∂rα/parenrightbigg\n= 2jp;α(r)+ρ(r)r·∂κ(r)\n∂rα(42)\nand\n−i\n2∂\n∂rαQµ(r,s))/vextendsingle/vextendsingle\ns=r+c.c. = −ρ(r)r·∂κ(r)\n∂rα.(43)\nThus, we expect jpDλµ=1\n2jpPλ(r)+1\n2jpQµ(r) =jp(r) to\nhold almost everywhere. To prove this result, it suffices\nto find conditions on ρandκsuch that PλandQµhave\na locally finite kinetic energy.\nThe kinetic energy density of Pλis\nτP(r) =1\n2/ba∇dbl∇2Gλ(·,r)/ba∇dbl2\nL2(R3)=1\n2∇r·∇sPλ(r,s)/vextendsingle/vextendsingle\ns=r\n=|∇ρ(r)|2\n8ρ(r)+1\n2|∇(r·κ(r))|2ρ(r)+λρ(r).\n(44)\nHere, we have used the fact that the integral in\n/ba∇dbl∇2G(·,r)/ba∇dbl2\nL2(R3)can be performed analytically, so that\nthe evaluation at s=rafter the second equality is in fact\nwell-defined. Similarly,\nτQ(r) =1\n2∇r·∇sQµ(r,s)/vextendsingle/vextendsingle\ns=r=|∇ρ(r)|2\n8ρ(r)+1\n2\n3/summationdisplay\nα=1\n3/summationdisplay\nβ=1rβ∂κβ(r)\n∂rα\n2\n+1\n8µ3/summationdisplay\nβ=1|∇κβ(r)|2\nρ(r)+µρ(r).7\nThe total kinetic energy density becomes\nτD(r) =|∇ρ(r)|2\n8ρ(r)+1\n43/summationdisplay\nα=1\n/parenleftbigg∂\n∂rαr·κ(r)/parenrightbigg2\n+/parenleftbigg\nr·∂κ(r)\n∂rα/parenrightbigg2\n+1\n8µ3/summationdisplay\nβ=1/parenleftbigg∂κβ(r)\n∂rα/parenrightbigg2\nρ(r)+1\n2(λ+µ)ρ(r),(45)\nwhere we have used the fact that the kinetic energy den-\nsityislinearinthedensitymatrix. Usingthespecialform\n2|ab| ≤a2+b2of Young’s inequality, with a=κα(r) and\nb=r·∂κ(r)/∂rα, we find that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂\n∂rαr·κ(r)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingleκα(r)+r·∂κ(r)\n∂rα/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≤2|κα(r)|2+2/vextendsingle/vextendsingle/vextendsingle/vextendsingler·∂κ(r)\n∂rα/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n.(46)Hence, the canonical kinetic energy density is bounded\nby\nτD(r)≤|∇ρ(r)|2\n8ρ(r)+1\n2\n|κ(r)|2+3\n2/parenleftbigg\nr·∂κ(r)\n∂rα/parenrightbigg2\n+1\n16µ3/summationdisplay\nβ=1/parenleftbigg∂κβ(r)\n∂rα/parenrightbigg2\nρ(r)+1\n2(λ+µ)ρ(r)\n≤|∇ρ(r)|2\n8ρ(r)+1\n2\n|κ(r)|2+(3\n2r2+1\n16µ)3/summationdisplay\nα,β=1/parenleftbigg∂κβ(r)\n∂rα/parenrightbigg2\nρ(r)+1\n2(λ+µ)ρ(r),(47)\nwhere the second inequality was obtained by using |rβ| ≤\n|r|. A finite canonical kinetic energy of Pλ,QµandDλµ\nis ensured if\nTW[ρ] =/integraldisplay|∇ρ|2\n8ρdr=1\n2/integraldisplay\n|∇√ρ|2dr<∞,(48)\nTp[ρ,jp] =/integraldisplay|jp|2\n2ρdr=1\n8/integraldisplay\nρκ2dr<∞,(49)\nTαβ[ρ,jp] =/integraldisplay\n(1+r2)ρ/parenleftbigg∂κβ(r)\n∂rα/parenrightbigg2\ndr<∞.(50)\nWe remark that, by the definition of the vorticity, ν=\n∇×ρ−1jp=1\n2∇×κ, a consequence of the last condition\nis that\n/integraldisplay\n(1+r2)ρν2dr<∞. (51)\nWe have thus proved the following result:\nTheorem IV.2. Letρandκbe given such that ρ≥0,√ρ∈H1(R3),ρκ2\nα∈L1(R3),(1 +r2)ρ(∂κβ/∂rα)2∈\nL1(R3)for all Cartesian components α,β∈ {1,2,3}.\nThen there exist real constants λ,µ≥0and a 1-rdm\nDwith density ρand current jp=1\n2ρκsuch that thecanonical kinetic energy is bounded by\n1\n2Tr(∇1·∇2D)≤TW[ρ]+4Tp/bracketleftbig\nρ,1\n2ρκ/bracketrightbig\n+1\n2(µ+ν)N\n+/integraldisplay\nρ(r)/parenleftig\n3\n4r2+1\n32µ/parenrightig3/summationdisplay\nα,β=1/parenleftbigg∂κβ(r)\n∂rα/parenrightbigg2\ndr.\nNote that, by a Sobolev inequality,√ρ∈H1(R3) im-\nplies that ρ∈Lq(R3) for allq∈[1,3].\nD. Lifting the global integrability condition\nTheorem IV.2 may be strengthened by replacing the\nglobal integrability conditions on the total kinetic energy\nby local integrability conditions, replacing integrals R3\nby integrals over arbitrary compact sets K⊂R3. A\nlarger class of ρandκare then seen to be reproducible,\nalbeit with merely a locally finite kinetic energy.\nTheorem IV.3. Letρandκbe given such that ρ≥\n0,ρ∈L1(R3)∩Lq(R3),q >1,ρ−1|∇ρ|2∈L1\nloc(R3),\nρκ2\nα∈L1\nloc(R3),(1+r2)ρ(∂κβ/∂rα)2∈L1\nloc(R3)for all\nCartesian components α,β∈ {1,2,3}. Then there exist\na 1-rdm Dwith density ρand current jp=1\n2ρκ.8\nV. DISCUSSION\nWe have provided an explicit construction of a 1-rdm\nthat reproduces a prescribed density and paramagnetic\ncurrent density. This type of N-representability problem\narises in Kohn–Sham CDFT, as it is known that not all\ncurrent densities can be represented by a single Kohn–\nSham orbital. Lieb and Schrader have recently proved\n[11], under some additional assumptions, that there also\nexist current densities that cannot be represented by two\nKohn–Sham orbitals. The question is open for three\norbitals. For four or more orbitals, Lieb and Schrader\nprovide an explicit Slater determinant that reproduces\nany density and paramagnetic current that satisfy mild\nregularity conditions. Our results are complementary\nin that we establish that an extended Kohn–Sham ap-\nproach, where fractional occupation numbers are allowed\neven if there is an integral total number of electrons, is\nflexible enough to represent any density and paramag-\nnetic current density, under minimal regularity assump-\ntions (finite TW,Tp, andTαβ).\nThe generalization from finite total canonical kinetic\nenergy to finite local canonical kinetic energy is of some\nvalue in light of gauge freedom. The kinetic energy\nTp[ρ,jp] is not gauge invariant; on the contrary, it can\nmade to become infinite by applying a gauge transfor-\nmationjp/ma√sto→jp+ρ∇χwith a rapidly growing gauge\nfunction χ. Our results establish that such gauge trans-\nformations do not affect N-representability, as long as χ\nexhibits some minimal regularity.\nThe explicit constructions of density matrices can be\nused to provide orbital-free upper bounds on the canon-\nical kinetic energy Ts[ρ,jp] for an extended Kohn–Sham\nformalism. Combining the above results with the stan-\ndard lower bound TW+Tpon the kinetic energy, we get\nthe following orbital-free bounds on the extended Kohn–\nSham kinetic energy,\nTW+Tp≤¯T[ρ,jp]\n≤TW+4Tp+(λ+µ)N\n+/integraldisplay\nρ(r)/parenleftig\n3\n2r2+1\n16µ/parenrightig\n|∇ακβ(r)|2dr.(52)\nNoting that severalauthors, following Vignale and Ra-\nsolt [21], have discussed CDFT formulations in terms\nof spin-resolved densities ( ρ↑,ρ↓,jp;↑,jp;↓), we also re-\nmark that our 1-rdm construction is easily modified for\nspin-resolved 1-rdms D↑↑andD↓↓. The eigenvectors\nthen correspond to natural spin-orbitals with eigenval-\nues bounded by one rather than by two as in the case of\nnatural spatial orbitals. The modifications to the above\npresentationaretrivial—condition4inTheoremIII.1be-\ncomesthatnooccupationislargerthan1, andthe factors\n1\n4consequently disappear from Eqs. (IV.1) and (41).ACKNOWLEDGMENTS\nThe authors would like to thank E.H. Lieb and\nR. Schraderfor giving one ofus (S. Kvaal) early accessto\ntheir manuscript of Ref. [11] and for interesting discus-\nsions, which spurred the completion of the present work.\nThis work was supported by the Norwegian Re-\nsearch Council through the CoE Centre for Theoreti-\ncal and Computational Chemistry (CTCC) Grant No.\n179568/V30and the Grant No. 171185/V30and through\nthe European Research Council under the European\nUnion Seventh Framework Program through the Ad-\nvanced Grant ABACUS, ERC Grant Agreement No.\n267683.\nAppendix A: The regular representation of\ntrace-class operators\nThe density matrix D(r,s) is an element of L2(R3×\nR3) and also the kernel of a trace-class operator over\nL2(R3). As such, it is not pointwise defined everywhere.\nAt the same time, we wish to makesense of“the diagonal\nD(r,r)” in order to define the density in an unambiguous\nmanner.\nBrislawn [17] has presented a thorough study of trace-\nclass operators and their kernels. The basic tools are\nfound in this reference, but we restate some results for\na self-contained treatment. We begin by clarifying some\npoints concerning Lebesgue spaces that are often glossed\nover but are important here.\n1. Lebesgue spaces\nLetX⊂Rnbe an open set. The Lp(X) norm of a\nmeasurable function f:X→Cis defined by\n/ba∇dblf/ba∇dblp:=/parenleftbigg/integraldisplay\nX|f(x)|pdx/parenrightbigg1/p\n. (A1)\nThe vector space Lp(X) consists of all functions fsuch\nthat/ba∇dblf/ba∇dblp<+∞. The space Lp(X) is not a normed\nspace, since /ba∇dblf/ba∇dblp= 0 if and only if f(x) = 0 for almost\nallx∈X(rather than for all x∈X). On the other\nhand, the set Lp(X) consisting of all equivalence classes\n[f] ={g∈ Lp(X) :/ba∇dblf−g/ba∇dblp= 0}is a normed space. It\nis customary to speak of a function fas an element of\nLp(X) even though, strictly speaking, it is a representa-\ntiveof [f]∈Lp(X).\nThis distinction between fand [f] is not merely aca-\ndemic: twopointwise defined wavefunctions Ψ and Φ de-\nscribethe samephysicalstate ifand only if /ba∇dblΨ−Φ/ba∇dbl2= 0.\nThus, [Ψ] ∈L2(R3N) is the wave function. Similarly, a\nreduced density matrix D∈L2(R3×R3) is not defined\npointwise: its formal diagonal D(r,r) may therefore be\nredefinedwithoutchangingthephysics. If[Ψ] = [Φ], then\nDΨ=DΦalmost everywhere, but if Disgiventhere is9\nnoa prioriway to know how the pointwise values D(r,s)\nare affected by modifying the wave function on a set of\nzero measure.\n2. Locally integrable functions\nA function f∈Lp\nloc(Rn) if and only if f∈Lp(K)\nfor every compact measurable K⊂Rn. We furthermore\nhaveLq\nloc⊂Lp\nlocforq≥p, and\nLp(Rn)⊂Lp\nloc(Rn)⊂L1\nloc(Rn).(A2)\nClearly, L1\nlocis a large class of functions and functions\ninL1\nlocare said to be “locally integrable”. We also need\na slightly more general notion of local integrability as\nfollows:\nDefinition A.1. LetX⊂Rn,Y⊂Rmbe open sets.\nThe setLp(Xloc×Y)is the set of (equivalence classes of)\nall measurable functions u:X×Y→Csuch that for all\ncompact measurable K⊂X,u∈Lp(K×Y). A similar\ndefinition is made for arbitrary products and positions of\nthe subscript “ loc”. In particular, Lp\nloc(X) =Lp(Xloc).\n3. The regular representation\nThe goal of this section is to establish a unique repre-\nsentative ˜fof [f]∈L1\nloc, called the regularrepresentative\noff. This representative will aid in defining the diago-\nnal ofD∈ DN,1. The first step is to introduce the local\naveraging operator Aǫ:\nDefinition A.2 (Local averaging operator Aǫ).Letf∈\nL1\nloc(Rn)andǫ >0. For a box Cǫ= [−ǫ,ǫ]nof Lebesgue\nmeasure |Cǫ|= (2ǫ)n, the (linear) local averaging opera-\ntorAǫ:L1\nloc→L1\nlocis defined by\nAǫf(x) :=1\n|Cǫ|/integraldisplay\nCǫf(x+y)dy. (A3)\nSinceCǫis compact, Aǫf(x) is everywhere finite and is\nindependent of the particular representative fof [f] that\nappears in the integrand. It can be shown that Aǫf(x)\nis continuous both in xand inǫ >0 [22]. We are here\ninterested in the limit ǫ→0 and therefore invoke the\nLebesgue differentiation theorem:\nTheorem A.1 (Lebesgue differentiation theorem) .Let\nf∈L1\nloc(Rn). Then for almost all x∈Rn,\nlim\nǫ→0Aǫf(x) =f(x). (A4)\nProof.See Ref. [22]\nSinceAǫf(x) is independent of the particular f∈[f],\nthis limit determines a unique representative:Definition A.3 (Regular representative) .The regular\nrepresentative ˜foff∈L1\nloc(Rn)is defined by\n˜f(x) := lim\nǫ→0Aǫf(x) (A5)\nwhenever the limit in Eq. (A4)exists.\nSince˜f(x) =f(x) almost everywhere, ˜fandfrepre-\nsent the same element [ f]∈L1\nloc. Moreover, it is easy to\nsee that ˜fis independent of the starting representative f\nand that the set of zero measure(where ˜fis undefined) is\nuniquely given by [ f]∈L1\nloc. Intuitively, ˜fis more regu-\nlar than f, “smoothing out” unnecessary discontinuities,\nand so on.\nRelated to the regular representative is the Hardy–\nLittlewood maximal function and associated inequality:\nDefinition A.4 (Hardy–Littlewood maximal function) .\nForf∈L1\nloc(Rn)andCǫ= [−ǫ,ǫ]nof Lebesgue mea-\nsure|Cǫ|= (2ǫ)n, the Hardy–Littlewood maximal func-\ntionMfis defined by\nMf(x) := sup\nǫ>01\n|Cǫ|/integraldisplay\nCǫ|f(x+y)|dy.(A6)\nThe following theorem is also called the Maximal The-\norem:\nTheorem A.2 (Hardy–Littlewood maximal inequality) .\nIff∈Lp(Rn), thenMf(x)is finite almost everywhere.\nMoreover, there exists a constant Cp(independent of f\nandn) such that\n/ba∇dblMf/ba∇dblp≤Cp/ba∇dblf/ba∇dblp.\nProof.See Ref. [22].\nSince|Aǫf(x)| ≤Mf(x) for allx, we obtain as a corol-\nlary that Aǫis a bounded linear operator from LptoLp.\nUsing this fact, it is straightforward to show that Aǫnot\nonly smoothes f, but also the mode of convergence:\nLemma A.1. Suppose fn→finLp(X). For all ǫ >0,\nAǫfn→Aǫfuniformly (i.e., in L∞(X)).\nProof.We show that, for every ǫ >0, there exists a con-\nstantK(ǫ) such that, for all f∈Lp,\n/ba∇dblAǫf/ba∇dbl∞≤K(ǫ)/ba∇dblf/ba∇dblp.\nWe have\n|Aǫf(x)| ≤1\n|Cǫ|/ba∇dblf/ba∇dblL1(x+Cǫ).\nSinceCǫis bounded in Rn,\n/integraldisplay\nx+Cǫ1×|g(x)|dx≤ |Cǫ|1/q/ba∇dblg/ba∇dblLp(x+Cǫ)(A7)\nwhere 1/q+1/p= 1. Thus,\n|Aǫf(x)| ≤ |Cǫ|1/q−1/ba∇dblf/ba∇dblLp(Rn),\nindependent of x.10\n4. The diagonal of a factorized kernel\nBased on our intuition, we may now hypothesize that,\ngiven an arbitrary reduced density matrix D(r,s)∈\nDN,1, the diagonal of ˜Dis the proper definition of the\ndensity:\nρ(r) =˜D(r,r). (A8)\nThis is indeed true, as we shall show. To this end, a\nslight reformulation and generalization of Theorem 3.5\nin [17] is useful for us. The reformulation states that,\nif an operator kernel is factorized, then the diagonal of\nthe regular representative is given by the diagonal of the\nfactorization, almost everywhere. The proof carries over\nwith only trivial modifications, but since it is important,\nwe rephrase it here.\nTheorem A.3 (Diagonal of factorization) .Let(X,dx)\nand(Y,dy)be open subsets of Euclidean spaces equipped\nwith the standard Lebesgue measures. For P∈L2(X×Y)\nandQ∈L2(Y×X), letC:X×X→Cbe given by\nC(x,x′) = (P∗Q)(x,x′) =/integraldisplay\nYP(x,y)Q(y,x′)dy.(A9)\nThenC∈L2(X×X)(a pointwise representative) and\n˜C(x,x) =C(x,x) (A10)\nfor almost all x∈X. Moreover, the map x/ma√sto→C(x,x) =\n(P∗Q)(x,x)belongs to L1(X).\nProof.We now demonstrate that C∈L2(X×X). For\nalmost all x∈Xand for almost all x′∈X, it holds\nthatP(x,·),Q(·,x′)∈L2(Y). Fromthe Cauchy–Schwarz\ninequality, we obtain\n|C(x,x′)| ≤/integraldisplay\n|P(x,y)||Q(y,x′)|dy\n≤ /ba∇dblP(x,·)/ba∇dblL2(Y)/ba∇dblQ(·,x′)/ba∇dblL2(Y)<+∞(A11)\nfor almost all xand almost all x′and hence also for al-\nmost all ( x,x′)∈X×X. Squaring and integrating, we\nobtain/ba∇dblC/ba∇dbl2\nL2(X×X)≤ /ba∇dblP/ba∇dbl2\nL2(X×Y)/ba∇dblQ/ba∇dbl2\nL2(Y×X)<+∞.\nNext, we demonstrate that the diagonal is in L1(X×\nX). Forǫ >0, letAǫ,iP(x,y) be the averaging oper-\nator acting on the ith argument and let MiP(x,y) be\nthe maximal operator acting on the ith argument. For\nalmost all x,x′,y, we then obtain\n|Aǫ,1P(x,y)Aǫ,2Q(y,x′)| ≤M1P(x,y)M2Q(y,x′).\n(A12)\nBy the Cauchy–Schwarz inequality, we obtain\n/integraldisplay\n|M1P(x,y)M2Q(y,x′)|2dy≤\n/parenleftbigg/integraldisplay\n|M1P(x,y)|2dy/parenrightbigg/parenleftbigg/integraldisplay\n|M2Q(y,x′)|2dy/parenrightbigg\n(A13)where both factors on the right-hand side are finite by\nthe maximal theorem, for almost all xand almost all x′.\nThese bounds justify the use of Fubini’s theorem to write\nAǫC(x,x′) =1\n|Cǫ|2/integraldisplay\nCǫ×Cǫ×YP(x+t,y)Q(y,x′+t′)dtdt′dy\n=/integraldisplay\nYAǫ,1P(x,y)Aǫ,2Q(y,x′)dy,(A14)\nwhich holds for almost every xandx′.\nWe now observe that, for each factor on the right-hand\nside,\nlim\nǫ→0Aǫ,1P(x,y) =P(x,y) a.a.x∈X(A15)\nlim\nǫ→0Aǫ,2Q(y,x′) =Q(y,x′) a.a.x′∈X.(A16)\nThe dominated convergence theorem together with the\nbounds in Eqs. (A12) and (A13) now imply that we can\ntake the limit in Eq. (A14) to get\n˜C(x,x) = lim\nǫ→0AǫC(x,x) =/integraldisplay\nYP(x,y)Q(y,x)dy(A17)\nfor almost all x. We have\n/integraldisplay\nX(P∗Q)(x,x)dx=/an}b∇acketle{tˆQ,P/an}b∇acket∇i}htL2(X×Y),(A18)\nwithˆQ(x,y) =Q∗(y,x). Being an inner product on L2,\nthis expression is finite, completing the proof.\nRemark 1: Although the diagonal of P∗Qis inL1, we\ncannot conclude that P∗Qis trace class—see Ref.[17]\nfor a counterexample. On the other hand, if X=Yin\nTheorem A.3, then P∗Qis by definition trace class and\nit is also true that Tr P∗Q=/integraltext\nX(diagP∗Q)(x)dx.\nRemark2: C=P∗QisthekernelofaHilbert–Schmidt\noperator over L2(X). We see that it is meaningful to de-\nfine the diagonal diag Cof any Hilbert–Schmidt operator\non an explicitly factorized form from the expression\n[diagP∗Q](x) := (P∗Q)(x,x),(A19)\nand the theorem states that this function belongs to\nL1(X), independent of the factorization.\nRemark 3: As a corollary, if P∈L2(Rn\nloc×Rm),Q∈\nL2(Rm×Rn\nloc), thenP∗Q∈L1\nloc(Rn).\nRemark 4: If P(x,y) =Q∗(y,x), thenP∗Qis positive\nsemidefinite. Since diag P∗Qis integrable, it follows\nfrom a theorem in Ref. [17] that P∗Qis trace class over\nL2(X).\nAppendix B: Some proofs from Section III\n1. Proof of Theorem III.2\nFor this proof, we use Theorem A.3 in Appendix A.11\nProof.Let Γ∈ DNbe given. Assume that that\nDΓ(r,s) =D(r,s) almost everywhere in R3×R3. It\nfollows that ˜DΓ(r,r) =˜D(r,r) = (G†∗G)(r,r) for al-\nmost all r, since the regular representative is unique,\nand by using Theorem A.3, with P(r,s) =G(s,r)∗and\nQ(r,s) =G(s,r) (X=Y=R3).\nWe need to show that ρΓ(r) = (G†∗G)(r,r) for al-\nmost all r. Assume that Γ = |Ψ/an}b∇acket∇i}ht/an}b∇acketle{tΨ|. Now, ρΓ(r) =\nρΨ(r) =DΓ(r,r) for almost every r, by definition of\nρΨ(r). Applying Theorem A.3 to P(r,r2:N) = Ψ(r,r2:N)\nandQ(r2:N,r) = Ψ(r,r2:N)∗,X=R3andY=R3N−3,\nwe see that ρΓ(r) =˜DΓ(r,r) = (G†∗G)(r,r).\nWe invite the reader to fill in the details when Γ is a\ngeneral mixed state.\n2. Proof for Theorem III.3\nProof.2⇒1: Let a G∈L2(R3×R3) be given such that\n∇2G∈L2(R3×R3\nloc). Then, for every compact K⊂R3,\nTα(r,s) :=1\n2[∂2,αG]†∗[∂2,αG](r,s)\n=1\n2/integraldisplay\ndu∂2,αG(u,r)∗∂2,αG(u,s) (B1)\nis inL2(K×K) by Theorem A.3. Tαis positive semidefi-\nnite, so by Remark 4 after Theorem A.3, Tαis trace class\noverL2(K).\nBy the definition of the weak derivative and Fubini’s\nTheorem, we easily verify that in fact Tα=1\n2∂1,α∂2,αD\nalmost everywhere. Thus ∇1·∇2Dis trace-class, and D\nhas locally finite kinetic energy.\n1⇒2:\nSinceD∈ DN,1there exists a spectral decomposition\nB(r,s) =/summationdisplay\nkλkφk(r)φk(s)∗, (B2)\nwhere{φk} ⊂L2(R3) is a complete, orthonormal set,\nand where 0 ≤λk≤2 such that/summationtext\nkλk=N. Of course\nB(r,s) =D(r,s) almost everywhere, but they may be\npointwise different.\nLetK⊂R3be compact. Restricted to K×K,∇1·\n∇2D=∇1·∇2B(a.e.) is trace-class, and we compute\n∇1·∇2B(r,s) =/summationdisplay\nkλk∇φk(r)·∇φk(s)∗a.e. (B3)\nLetAk(r,s) =∇φk(r)·∇φk(s)∗. By assumption,\nTr(∇1·∇2B) =/summationdisplay\nkλkTrAk=/summationdisplay\nkλk/ba∇dbl∇φk/ba∇dbl2\nL2(K)<+∞,\n(B4)\nimplying that ∇φk∈L2(K) for every K, hence∇φk∈\nL2\nloc(R3).\nLetGbe given by\nG(r,s) =/summationdisplay\nkλ1/2\nkφk(r)φk(s)∗. (B5)Clearly,G∈L2(R3×R3) andD=G†∗G. Moreover,\n∇2G(r,s) =/summationdisplay\nkλ1/2\nkφk(r)∇φk(s)∗.(B6)\nComputing the L2(R3×K) norm,\n/ba∇dbl∇2G/ba∇dbl2=/summationdisplay\nkℓλ1/2\nkλ1/2\nℓ/an}b∇acketle{tφℓ,φk/an}b∇acket∇i}htL2(R3)/an}b∇acketle{t∇φk,∇φℓ/an}b∇acket∇i}htL2(K)\n=/summationdisplay\nkλk/ba∇dbl∇φk/ba∇dbl2\nL2(K). (B7)\n3⇔1:\nLet Γ be such that DΓ=Da.e. We have,\nDΓ(r,s) =/summationdisplay\nipi/integraldisplay\ndr2:NΨi(r,r2:N)Ψi(s,r2:N)∗.(B8)\nFurthermore,\nTα(r,s) :=1\n2∂1,α∂2,αD(r,s)\n=1\n2/summationdisplay\nipi/integraldisplay\ndr2:N∂1,αΨi(r,r2:N)∂1,αΨi(s,r2:N)∗,\n(B9)\nusing the definition of the weak derivative and Fubini’s\ntheorem. By Theorem A.3,\n˜Tα(r,r) =1\n2/summationdisplay\nipi/integraldisplay\n|∂1,αΨi(r,r2:N)|2dr2:N(B10)\nfor almost all r. For any compact K⊂R3, integration\nyields\n/integraldisplay\nKdr˜Tα(r,r) =1\n2/summationdisplay\nipi/ba∇dbl∂1,αΨi/ba∇dbl2\nL2(K×R3N−3).(B11)\nSinceTαis positive semidefinite, the left hand side is the\ntrace of1\n2∂1,α∂2,αD. Thus, Dhas locally finite kinetic\nenergy if and only if any representing Γ /ma√sto→Dhas locally\nfinite kinetic energy.\n3. Proof of Theorem III.4\nProof.Most of the proof is similar that of Theorem III.3,\nso we skip some details.\nLet Γ be such that DΓ=Dalmost everywhere.\nThe state Γ has a locally finite kinetic energy by The-\norem III.3. By a reasoning similar to that of the proof of\nthis lemma, we obtain\ncΓ,α(r) =cα(r) = [diag( −i∂2,αG)†∗G](r,r) (B12)\nalmost everywhere, independently of Γ. Taking the ab-\nsolute value, integrating over a compact K⊂R3and12\napplying the Cauchy–Schwarz inequality, we obtain the\nbound\n/integraldisplay\nK|cα(r)|dr≤ /ba∇dbl∂2,αG/ba∇dblL2(R3×K)/ba∇dblG/ba∇dblL2(R3×K)\n<+∞. (B13)\n[1] A. J. Coleman, Rev. Mod. Phys. 35, 668 (1963).\n[2] W. Macke, Phys. Rev. 100, 992 (1955).\n[3] T. L. Gilbert, Phys. Rev. B 12, 2111 (1975).\n[4] J. E. Harriman, Phys. Rev. A 24, 680 (1981).\n[5] G. Zumbach and K. Maschke,\nPhys. Rev. A 28, 544 (1983).\n[6] S. K. Ghosh and R. G. Parr,\nJ. Chem. Phys. 82, 3307 (1985).\n[7] M. B. Ruskai and J. E. Harriman,\nPhys. Rev. 169, 101 (1968).\n[8] E. V. Lude˜ na, J. Mol. Struct. THEOCHEM 123, 371\n(1985).\n[9] M. B. Ruskai, J. Phys. A 40, F961 (2007).\n[10] S. K. Ghosh and A. K. Dhara,\nPhys. Rev. A 38, 1149 (1988).\n[11] E. H. Lieb and R. Schrader,\nPhys. Rev. A 88, 032516 (2013), arXiv:1308.2664v1.\n[12] G. Vignale and M. Rasolt,\nPhys. Rev. Lett. 59, 2360 (1987).\n[13] E. Canc` es, J. Chem. Phys, 114, 10616 (2001).[14] E. Cances, K. N. Kudin, G. E. Scuseria, and G. Turinici,\nJ. Chem. Phys. 118, 5364 (2003).\n[15] E. Kraisler, G. Makov, N. Argaman, and I. Kelson,\nPhys. Rev. A 80, 032115 (2009).\n[16] C. R. Nygaard and J. Olsen,\nJ. Chem. Phys. 138, 094109 (2013).\n[17] C. Brislawn, Proc. Am. Math. Soc. 104, 1181 (1988).\n[18] M. Reed and B. Simon, Methods of modern mathemati-\ncal physics I: Functional analysis (Academic Press, San\nDiego, CA, 1980).\n[19] R. Parr and W. Yang, Density Functional Theory of\nAtoms and Molecules (Oxford University Press, New\nYork, 1989).\n[20] L. Evans, Partial Differential Equations (American\nMathematical Society, Providence, R.I., 2000).\n[21] G. Vignale and M. Rasolt,\nPhys. Rev. B 37, 10685 (1988).\n[22] E. Stein, Singular integrals and differentiability proper-\nties of functions (Princeton Univ. Press, Princeton, N.J.,\n1970)."    },    {        "title": "1310.5851v1.Impurity_Effect_on_the_Local_Density_of_States_around_a_Vortex_in_Noncentrosymmetric_Superconductors.pdf",        "content": "arXiv:1310.5851v1  [cond-mat.supr-con]  22 Oct 2013ImpurityEffectontheLocalDensityofStatesarounda\nVortexin NoncentrosymmetricSuperconductors\nYoichi H IGASHI∗,1,3,Yuki N AGAI2, and Nobuhiko H AYASHI3\n1DepartmentofMathematicalSciences,OsakaPrefectureUni versity, 1-1Gakuen-cho,Naka-ku,\nSakai599-8531,Japan\n2CCSE,JapanAtomicEnergyAgency,5-1-5Kashiwanoha,Kashi wa,Chiba277-8587,Japan\n3NanoscienceandNanotechnologyResearchCenter(N2RC),Os akaPrefectureUniversity, 1-2\nGakuen-cho,Naka-ku,Sakai599-8570,Japan\n(ReceivedSeptember30,2013)\nWe numerically study the effect of non-magnetic impurities on the vortex bound states in noncen-\ntrosymmetricsystems.Thelocaldensityofstates(LDOS)ar oundavortexiscalculatedbymeansof\nthequasiclassicalGreen’sfunctionmethod.Wefindthatthe zeroenergypeakoftheLDOSsplitsoff\nwithincreasingtheimpurityscatteringrate.\nKEYWORDS: Noncentrosymmetricsysyem,Vortex core,Localdensity of s tates,Impurity\neffect\n1. Introduction\nMuch attention has been focused on the superconductivity in noncentrosymmetric systems be-\ncauseofitsnovelsuperconductivity duetospin-orbit coup ling (SOC).Recently, activeinvestigations\nhave been conducted on the inhomogenious superconducting s tate in noncentrosymmetric systems\nsuchasthehelicalphaseinamagneticfield[1],thevortexst ate[2,3],andtheexoticsuperconducting\nstate inlocally noncentrosymmetric systems [4].\nIn this study, we investigate the non-magnetic impurity sca ttering effect on the local density of\nstates (LDOS)around asingle vortex. Impurity effects onth e vortex core structure innoncentrosym-\nmetric systems can differ from the simple two-gap systems be cause there is SOC in this system.\nSo unusual phenomena are expected in the LDOS inside a vortex core. There are several previous\nstudies for the impurity effect on the noncentrosymmetric s uperconductivity in the bulk without vor-\ntices[5,6].However,thereisnoresearchontheimpurityef fectinspatiallyinhomogeneous situations\nsuch asvortex state innoncentrosymmetric systems.\nThrough our formulation using the quasiclassical Green’s f unction method, it turned out that the\nform of the impurity self energy around a vortex is quite diff erent from that in the bulk. This comes\nfrom the effect of the superflow on the impurity self energy. I n addition, the Green’s function cannot\nbeseparatedwithrespecttoeachFermisurface(FS)splitdu etotheRashba-typeSOCinthepresence\nof both impurities and vortices. This fact might also influen ce the impurity effect on the vortex core\nstructure.\nIn this paper, we numerically investigated the effect of non -magnetic impurities on the LDOS\naround a vortex. We found that the zero-energy peak (ZEP) of t he LDOS splits off with increasing\ntheimpurity scattering rate. Wegivethe rough physical int erpretation for thenumerical results.\n∗E-mailaddress: higashiyoichi@ms.osakafu-u.ac.jp\n12. Formulation\nThe noncentrosymmetricity of the system induces the anitis ymmetric SOCand permits the mix-\ningoftheCooperpairwavefunctionwithdifferent parity.W eassumetheRashba-typeSOC,whichis\ndescribed as gk=/radicalbig\n3/2(−ky,kx,0)/kFby the orbital vector gk.kFis the Fermi wave number. We\nconsider asingle vortex line along the zaxis situated at the origin ( r= 0). Weassume that the FSis\nspherical. Thelackofspatial inversion symmetryofthesys tem isdetrimental tospin-triplet state[7].\nHowever, the spin-triplet state is not suppressed under the particular situation in which dk||gk[8].\nThenweconsider theparity mixing of the Cooper pair.\nThe parity-mixed pairing state is expressed as ˆ∆(r,˜k) = [Ψ(r)ˆσ0+dk(r)·ˆσ]iˆσy= [Ψˆσ0+\n∆(−˜kyˆσx+˜kxˆσy)]f(r)exp[iφr]iˆσy,where the spin-singlet s-wave component Ψ(r), thedvector\ndk=∆(r)(−˜ky,˜kx,0)with the unit vector ˜k= (˜kx,˜ky,˜kz) = (cosφksinθk,sinφksinθk,cosθk),\nΨ(∆)is the bulk amplitude of the pair potential for singlet (trip let) component, ˆσ= (ˆσx,ˆσy,ˆσz)is\nthevector consisted of thePauli spin matrices and ˆσ0istheunit matrix in thespin space. Weassume\nthat the both components of the pair potential have the same r eal space profile around a vortex and\nsetf(r) =r//radicalbig\nr2+ξ2\n0.Here,ξ0=vF/Tcisthecoherence length, Tcisthesuperconducting critical\ntemperature, vF(=|vF|)is the Fermi velocity. In this paper, wedoes not solve the gap equations for\nthe pair potentials Ψ(r)and∆(r)self-consistently. We conduct the numerical calculation f or zero\ntemperature and take into account the effect of temperature through the energy smearing factor η.\nInordertoobtaintheLDOSandtheimpurityselfenergy,weco nsiderthefollowingquasiclassical\nGreen’s function, which is defined inthe particle-hole spac e.\nˇg(r,˜k,iωn) =−iπ/parenleftBigg\nˆg iˆf\n−iˆ¯f−ˆ¯g/parenrightBigg\n. (1)\nHere,ωnis the Matsubara frequency. ˆ·denotes the 2×2matrix in the spin space and ˇ·denotes the\n4×4matrix in the particle-hole and spin space. The Eilenberger equation with the SOC term and\nimpurity self energy isgiven as [9]\nivF(˜k)·∇ˇg(r,˜k,iωn)+/bracketleftBig\niωnˇτ3−ˇ∆−ˇΣ−αˇgk·ˇS,ˇg(r,˜k,iωn)/bracketrightBig\n=ˇ0, (2)\nwhere\nˇτ3=/parenleftbigg\nˆσ0ˆ0\nˆ0−ˆσ0/parenrightbigg\n,ˇS=/parenleftbiggˆσˆ0\nˆ0ˆσtr/parenrightbigg\n,ˆσtr=−ˆσyˆσˆσy, (3)\nˇgk=/parenleftbigg\ngkˆσ0ˆ0\nˆ0g−kˆσ0/parenrightbigg\n,ˇ∆=/parenleftbiggˆ0 ˆ∆(r,˜k)\n−ˆ∆†(r,˜k)ˆ0/parenrightbigg\n. (4)\nHere,αisthestrengthoftheSOC. gk=/radicalbig\n3/2(−˜ky,˜kx,0),whichisoddin k(i.e.,g−k=−gk).We\nuse the units in which /planckover2pi1=kB= 1. The quasiclassical Green’s function satisfies the normali zation\ncondition ˇg2=−π2ˇτ0withthe4×4unit matrix ˇτ0. Thebracket [···,···]isacommutator.\nThe Eilenberger equation is separated into two equations wi thout spin degree of freedom using\nthebandbasisbothinthecleanvortexstate[3]andinthebul kcleansystem[10,11].Thenormal-state\nHamiltonian in the clean limit, which is a 2×2matrix in the spin space, becomes diagonal in the\nbandbasis. Theoff-diagonal components ofthequasiclassi cal Green’sfunction decay tozeroaround\navortex andthequasiclassical Green’s function intheband basis becomes diagonal inthespin space\n(see the appendix of Ref. [10]). Each diagonal component obe ys the Eilenberger equation, which is\ndefinedonthetwoFSssplitduetotheSOC.Then,wecansolveth eEilenberger equation withoutthe\nSOCterm with respect to each FS in the clean limit. This situa tion is not changed in the presence of\nimpurities inthe bulk.\n2However, the quasiclassical Green’s function is not diagon al in the band basis in the presence of\nboth impurities and vortices. Therefore the Eilenberger eq uation is not separated into two equations\nwithrespect toeachFS.Thus,weusetheorbital basis,inwhi chthespinquantization axisisoriented\nparallel to the zaxis. Under this basis, we transform Eq. (2) into the two matr ix Riccati type differ-\nential equations, which is the first-order differential equ ations [see Appendix]. Then we obtain the\nstable numerical solution solving the Eq. (A ·9) and (A ·10) by means of the adaptive stepsize control\nRunge-Kutta method [13].\nInthispaper, weinvestigate theimpurity scattering inthe Bornlimit(Weset thescattering phase\nshift exactly zero). Inthis limit, the impurity self energy isgiven as [14,15]\nˇΣ(r,E) =Γn\nπ/angbracketleftBig\nˇg0(r,˜k,iωn→E+iη)/angbracketrightBig\n˜k, (5)\n=Γn/parenleftBigg\n−i∝an}bracketle{tˆg∝an}bracketri}ht˜k∝an}bracketle{tˆf∝an}bracketri}ht˜k\n−∝an}bracketle{tˆ¯f∝an}bracketri}ht˜ki∝an}bracketle{tˆ¯g∝an}bracketri}ht˜k/parenrightBigg\n, (6)\nwhereΓnis the impurity scattering rate in the normal state, ˇg0is the quasiclassical Green’s function\nin the clean limit, ∝an}bracketle{t···∝an}bracketri}ht˜kdenotes the average over the FS and ˇΣ≡ {ˆΣij}i,j=1,2. Now we consider\nthe system with the rotational symmetry about a vortex line. In the clean limit, the parity mixing\npair potential has a form ∆I,II(r,φr;θk) = [Ψ(r)±∆(r)sinθk]exp[iφr]around a vortex using the\nband basis [3]. These forms of the pair potentials have the θkdependence only. We assume that the\npairing state does not change with non-magnetic impurities . Under an axial rotation about a vortex\nline, the pair potential has the azimuthal angle dependence in the form of the phase factor in the\nreal space. From Eq. (A ·9) and (A ·10), we can see that the anomalous self energies ˆΣ12,21have the\nsame phase factor as the pair potentials, whereas the normal self energies ˆΣ11,22are invariant under\nthe axial rotation. Thus the anomalous self energies ˆΣ12,21have the following φrdependence [15]:\nˆΣ12,21(r,φr,E) =ˆΣ12,21(r,E)exp[±iφr].The plus (minus) sign corresponds to ˆΣ12(ˆΣ21). In the\nactual numerical calculation, the self energies are discre tely calculated in a radial rdirection. Thus\nthey are linearly-interpolated in this direction. As for th e direction of an azimuthal angle, the self\nenergies have the continuous φrdependence.\nTheLDOSper spin is obtained from\nN(r,E) =−NF\n21\nπ/angbracketleftBig\nIm/bracketleftBig\nTr ˆg(r,˜k,iωn→E+iη)/bracketrightBig/angbracketrightBig\n˜k, (7)\nwhereNFis thedensity of states per spin at the Fermi level inthe norm al state.\n3. ResultandDiscussion\nThroughout this section, we set α′/Tc= 1withα′=/radicalbig\n3/2α. We show in Fig. 1(a) the energy\ndependence of the LDOS at the vortex center for several value s of the impurity scattering rate Γn\nfor thes-wave case ( Ψ/Tc= 1,∆/Tc= 0).Γn/Tcis roughly estimated of the order of ξ0/l.l\nis the mean free path. In the clean limit ( Γn/Tc= 0), the peak of the LDOS is seen at the zero\nenergy. In the presence of impurities, the ZEP splits into tw o peaks. We confirmed that the split\nof the ZEP does not occur for α′/Tc= 0. The splitting width becomes larger and the peak height\ndecreases with increasing the impurity scattering rate Γn. ForΓn= 0.7Tc, the value of the zero\nenergy LDOS at the vortex center equals to the normal state de nsity of states NF. In Fig. 1(b), we\nshow the energy and the spatial profile of the LDOS inside a vor tex core for Γn= 0.3Tc. We see\nclearly that the ZEP of the LDOS splits off at the vortex cente r. Far from the vortex center, the zero\nenergy density of states decays to zero and the fully opened s uperconducting energy gap appears in\nthebulk,whichisunderstood fromtheAnderson’stheoremfo rnon-magnetic impurities[16].Wecan\n30123456789\n-1.5 -1 -0.5 0 0.5 1 1.5N(E,r= 0) / NF\nE/ Tc\n\u0001n/ Tc= 0\n= 0.1\n= 0.2\n= 0.3\n= 0.5\n= 0.7(a)\n0\n1\n2r/ ξ0\n-1.5 -1 -0.5 0 0.5 1 1.5E/ Tc01234N(E,r) / NF(b)\nFig. 1. (Coloronline) s-wavecase.Theenergysmearingfactor ηissetto0.05Tc.(a)Theenergydependence\nof the local density of states at the vortex center for severa l values of the impurity scattering rate Γn. (b) The\nlocaldensityofstateswithina vortexcorefor Γn= 0.3Tc.\n0123456789\n-2-1.5-1-0\u0000500\u0002511\u000352N(E\n,r=0)\n/N\nF\nE\u0004Tc\n\u0005n\n\u0006T\u0007=0\n=0\b1\n=0\t2\n=0\n3\n=0\u000b5\n=0\f7\n( \r\u000e\n0\n1\n2r\u000f \u00100\n-2\u00115-2-1\u00125-1-0\u0013500 \u0014511\u0015522 \u00165E \u0017T\u001801234N(E,r)\n\u0019NF(b)\nFig. 2. (Coloronline) s+p-wavecase.Theenergysmearingfactor ηissetto0.05Tc.(a)Theenergydepen-\ndenceofthelocaldensityofstatesatthevortexcenterfors everalvaluesoftheimpurityscatteringrate Γn. (b)\nThelocal densityofstateswithina vortexcorefor Γn= 0.3Tc.\nsee that the ridge lines of the vortex bound states approach t he gap edges of the s-wave component\nE/Tc=Ψ/Tc=±1.\nAsshowninFig.2(a),theZEPsplitsoffalsointhecaseofthe s+p-wave(Ψ/Tc:∆/Tc= 0.7 :\n1.4).Theridgelinesrelatedtothevortexboundstatesofthequ asiparticles onthesplitFSarebroaden\ndue to impurity scattering [see Fig. 2(b)]. We cannot observ e the two-gap like quasiparticle spectra\nwithinacore,whichisseeninthecleanlimit[3].Withinthe superconducting gapenergy, thereexists\nthe non-zero density of states which comes from excitations in the vicinity of the horizontal line\nnodes. Nodes can appear under Ψ < ∆[11], which corresponds tothe present condition.\nLet us explain about the present physical interpretation of the split ZEP. The condition that the\nGreen’sfunctions areinvariant under k→ −kissatisfied atleast inspatially uniform systems, when\n∆I,IIare invariant under k→ −k. In this situation, ˆΣ0\nij∝ˆσ0. Therefore, the impurity scattering\neffect appears only in the modification of the Matsubara freq uency and the pair potential in the band\nbasis. The Green’s function is still diagonal in the band bas is. That is, the split two bands are not\nmixed by impurity scattering in the bulk within the Born appr oximation. Frigeri et al.,have already\n4pointed out this fact [5].\nHowever, the condition is not satisfied in systems where the s upercurrent flows (e.g., around a\nvortex core), namely the Green’s functions are not invarian t underk→ −kin such a system. Sothe\nsituation at the vortex core is quite different from that in t he bulk. Through a careful examination, it\nturns out that the anomalous self energies ˆΣ0\n12andˆΣ0\n21are not diagonalized in the band basis. They\ncontribute to mixthe two bands split due tothe SOC.If regard ing them as the perturbation by which\nthe transition occurs between the bound state spectra of qua siparticles on the each split FS, it might\nbe considered that the overlap or the crossing of the vortex b ound state spectra is resolved to give a\nenergy gap. This situation is similar to the two bands separa ted by a band gap at the corners of the\nBrillouin-zone.\n4. Conclusion\nIn conclusion, we calculated the local density of states aro und a vortex core in the presence of\nnon-magnetc impurities in noncentrosymmetric systems. We found that the zero-energy peak of the\nvortex bound states splits off with increasing the impurity scattering rate for both s-wave and s+p-\nwave pairing state. In the noncentrosymmetric systems, the impurity effect on vortex bound states\nexhibitsthedifferent onecomparing withthat ofsimpletwo -gapsuperconducting systems. Thespin-\norbit coupling isessential for this phenomena.\n5. Acknowledgments\nWe thank Y. Masaki for discussions from the more microscopic point of view and also thank N.\nNakai and M. Katofor helpful discussions.\nAppendix: Matrix Riccati equation\nIn this appendix, we describe briefly the derivation of the ma trix Riccati equation including the\nRashba-typespin-orbit couplingtermandtheimpurityself energybymeansoftheprojection method\n[12]. Weintroduce the projecters:\nˇP±=1\n2/parenleftbigg\nˇτ0±1\n−iπˇg/parenrightbigg\n. (A·1)\nTheseprojectors satisfy therelations:\nˇP±·ˇP±=ˇP±, (A·2)\nˇP++ˇP−= ˇτ0, (A·3)\nˇP+·ˇP−=ˇP−·ˇP+=ˇ0. (A·4)\nWecan confirm the following relation: ˇP+−ˇP−= ˇg/(−iπ). Then, the quasiclassical Green’s func-\ntion is obtained as ˇg=−iπ/parenleftbigˇP+−ˇP−/parenrightbig\n. Substituting this Green’s function into Eq. (2), wehave th e\nequation for the projectors ˇP±:\n−ivF·∇ˇP±=/bracketleftbig\niωnˇτ3−ˇ∆−ˇΣ−αˇgk·ˇS,ˇP±/bracketrightbig\n. (A·5)\nWedefine the projectors as\nˇP+=/parenleftbiggˆσ0\n−iˆb/parenrightbigg/parenleftBig\nˆσ0+ˆaˆb/parenrightBig−1/parenleftbigˆσ0iˆa/parenrightbig\n, (A·6)\n5ˇP−=/parenleftbigg\n−iˆa\nˆσ0/parenrightbigg/parenleftBig\nˆσ0+ˆbˆa/parenrightBig−1/parenleftbig\niˆbˆσ0/parenrightbig\n, (A·7)\nwhich satisfy Eq. (A ·2) and (A ·4). To satisfy Eq. (A ·3), we need the following relations between ˆa\nandˆb:(ˆσ0+ ˆaˆb)−1ˆa= ˆa(ˆσ0+ˆbˆa)−1,ˆb(ˆσ0+ ˆaˆb)−1= (ˆσ0+ˆbˆa)−1ˆb.Using these relations, the\nquasiclassical Green’s function is obtained as\nˇg=−iπ\n/parenleftBig\nˆσ0+ˆaˆb/parenrightBig−1/parenleftBig\nˆσ0−ˆaˆb/parenrightBig\n2i/parenleftBig\nˆσ0+ˆaˆb/parenrightBig−1\nˆa\n−2iˆb/parenleftBig\nˆσ0+ˆaˆb/parenrightBig−1\n−/parenleftBig\nˆσ0+ˆbˆa/parenrightBig−1/parenleftBig\nˆσ0−ˆbˆa/parenrightBig\n.(A·8)\nSubstituting Eq. (A ·6) and (A ·7) into Eq. (A ·5) and calculating with respect to each component, we\nobtaine the matrix Riccati equations:\nvF·∇ˆa0+2ωnˆa0+ˆa0/parenleftBig\nˆ∆†\n0−ˆΣ0\n21/parenrightBig\nˆa0−/parenleftBig\nˆ∆0+ˆΣ0\n12/parenrightBig\n+iαgk·(ˆσˆa0−ˆa0ˆσ)+i/parenleftBig\nˆΣ0\n11ˆa0+ˆa0ˆΣ0\n22/parenrightBig\n=ˆ0, (A·9)\nvF·∇ˆb0−2ωnˆb0−ˆb0/parenleftBig\nˆ∆0+ˆΣ0\n12/parenrightBig\nˆb0+/parenleftBig\nˆ∆†\n0−ˆΣ0\n21/parenrightBig\n+iαgk·/parenleftBig\nˆb0ˆσ−ˆσˆb0/parenrightBig\n−i/parenleftBig\nˆb0ˆΣ0\n11+ˆΣ0\n22ˆb0/parenrightBig\n=ˆ0.(A·10)\nHere, we define ˆa= ˆa0iˆσy,ˆb=−iˆσyˆb0,ˆΣ11=ˆΣ0\n11,ˆΣ12=ˆΣ0\n12iˆσy,ˆΣ21=−iˆσyˆΣ0\n21,ˆΣ22=\n−ˆσyˆΣ0\n22ˆσy,ˆ∆=ˆ∆0iσy,ˆ∆†=−iσyˆ∆†\n0.\nReferences\n[1] R. P. Kaur, D. F. Agterberg, and M. Sigrist: Phys. Rev. Let t.94(2005) 137002 ; D. F. Agterberg and R.\nP. Kaur: Phys.Rev. B 75(2007)064511;Y. Matsunaga,N. Hiasa, and R. Ikeda: Phys.Re v. B78(2008)\n220508(R);K.AoyamaandM.Sigrist: Phys.Rev.Lett. 109(2012)237007.\n[2] S. K. Yip: J. LowTemp.Phys. 140(2005)67; Y. Nagai,Y. Ueno,Y. Kato, andN. Hayashi:J. Phys. Soc.\nJpn.75(2006)10470;C. K. LiuandS. Yip: Phys.Rev B 82(2010)104501.\n[3] N. Hayashi,Y. Kato,P.A. Frigeri,K. Wakabayashi,andM. Sigrist: PhysicaC 437-438(2006)96.\n[4] D.Maruyama,M.Sigrist,andY.Yanase:J.Phys.Soc.Jpn. 81(2012)034702;T.Yoshida,M.Sigrist,and\nY. Yanase:Phys.Rev.B 86(2012)134514;J. Phys.Soc.Jpn. 82(2013)074714.\n[5] P. A.Frigeri,D.F. Agterberg,I.Milat,andM.Sigrist: E ur.Phys.J. B 54(2006)435.\n[6] V.P.MineevandK.V.Samokhin:Phys.Rev.B 75(2007)184529;V.M.Edelstein:Phys.Rev.B 78(2008)\n094514.\n[7] P. W. Anderson:Phys.Rev.B 30(1984)4000.\n[8] P. A.Frigeri,D.F. Agterberg,A.Koga,andM.Sigrist: Ph ys.Rev.Lett. 92(2004)097001.\n[9] J.W.SereneandD.Rainer:Phys.Rep. 101(1983)221andreferencescitedtherein;N.Schopohl:J.Low\nTemp.Phys. 41(1980)409;J.A.X.Alexander,T.P.Orlando,D.RainerandP. M.Tedrow:Phys.Rev.B\n31(1985)5811; C. T. Rieck, K. Scharnbergand N. Schopohl: J. Lo w Temp.Phys. 84(1991)381; C. H.\nChoi andJ. A.Sauls: Phys.Rev.B 48(1993)13684.\n[10] N. Hayashi,K. Wakabayashi,P.A. Frigeri,andM.Sigris t: Phys.Rev.B 73(2006)024504.\n[11] N. Hayashi,K. Wakabayashi,P.A. Frigeri,andM.Sigris t: Phys.Rev.B 73(2006)092508.\n[12] M.Eschrig:Phys.Rev.B 61(2000)9061;Y.Nagai,Y.Ueno,Y.Kato,andN.Hayashi:J.Phy s.Soc.Jpn.\n75(2006)104701;Y. Nagai:Master Thesis(inJapanese),TheUn iversityofTokyo(2007).\n[13] W. H. Press, S. A. Teukolsky,W. T. Vetterling, and B. P. F lannery:Numerical Recipes in FORTRAN 77 ,\n2nded.(CambridgeUniversityPress, London,1992),p.708.\n[14] Y. Kato:J. Phys.Soc.Jpn. 69(2000)3378.\n[15] N. Hayashi,Y. Kato,andM.Sigrist: J.LowTemp.Phys. 139(2005)79.\n[16] P. W. Anderson: J. Phys. Chem. Solids 11(1959) 26; N. Hayashi and Y. Kato: J. Low Temp. Phys. 130\n(2003)193.\n6"    },    {        "title": "1310.5887v1.Intrinsic_exchange_correlation_magnetic_fields_in_exact_current_density_functional_theory_for_degenerate_systems.pdf",        "content": "Intrinsic exchange-correlation magnetic \felds in exact current-density functional\ntheory for degenerate systems\nJ. D. Ramsden and R. W. Godby\nDepartment of Physics, University of York, Heslington, York YO10 5DD, United Kingdom and\nEuropean Theoretical Spectroscopy Facility (ETSF)\n(Dated: June 7, 2022)\nWe calculate the exact Kohn-Sham (KS) scalar and vector potentials that reproduce, within\ncurrent-density functional theory, the steady-state density and current density corresponding to an\nelectron quasiparticle added to the ground state of a model quantum wire. Our results show that,\neven in the absence of an external magnetic \feld, a KS description of a steady-state system in\ngeneral requires a non-zero exchange-correlation magnetic \feld that is purely mechanical in origin.\nThe KS paramagnetic current density is not, in general, that of the interacting system in any gauge.\nPACS numbers: 71.15.Mb, 73.63.-b, 73.23.-b, 85.35.Be\nI. INTRODUCTION\nTime-dependent density-functional theory1(TDDFT)\nin the Kohn-Sham (KS) scheme2is a powerful and\nin principle exact tool for predicting the dynamics of\nnonequilibrium systems subject to time-dependent scalar\npotentials. The success of the theory lies in the casting of\nproperties of interacting N-particle systems in terms of\nsystems ofNnoninteracting (Kohn-Sham) electrons, and\nthe unique determination of KS potentials by the time-\ndependent charge density and the initial state. In the\nground-state limit, the theory approaches the standard\ndensity-functional theory (DFT) of Hohenberg, Kohn\nand Sham3,4.\nIn the presence of time-dependent vector potentials,\none needs to know how the potentials couple to the time-\ndependent current density. The current density can be\ndecomposed into two parts: the longitudinal part obeying\nr\u0002jL(r;t) =0, and the transverse part obeying r\u0001\njT(r;t) = 0. The longitudinal part is given entirely by\nthe time-dependent density via the continuity equation\n@\n@tn(r;t) +r\u0001jL(r;t) = 0: (1)\nThe transverse part of the current is not given by the\ncontinuity equation and must be calculated directly from\nthe time-dependent wavefunction. The complete current\ndensity is given by\nj(r;t) =jp(r;t) +A(r;t)n(r;t); (2)\nwhere\njp(r;t) =D\n\t(t)\f\f\f^jp(r)\f\f\f\t(t)E\n(3)\nis the paramagnetic current density operator, A(r;t)\nis the time-dependent external vector potential, n(r;t)\nis the time-dependent density and j\t(t)iis the time-\ndependent many-body wavefunction.\nIt has been shown5that, even in the absence of external\nvector potentials in the time-dependent regime, the cor-\nresponding KS system will generally have an exchange-\ncorrelation (XC) vector potential that couples to the fullcurrent density. As such, one needs a time-dependent\ncurrent-density functional theory (TDCDFT) to fully de-\nscribe the dynamics of electronic systems. Such a theory\nexists6and is representable in a KS scheme7in which the\nKS potentials are unique functionals of the initial state\nand the time-dependent physical current density.\nFor ground-state systems, two CDFTs based on the\nphysical current have been proposed8,9but neither ful\fll\nthe requirements of uniqueness and amenability to a KS\nminimisation scheme10,11. The most complete theory of\nground-state current-carrying systems subject to exter-\nnal scalar and vector potentials remains the current- and\nspin-density functional theory (CSDFT) of Vignale and\nRasolt12which takes as its basic variables the ground-\nstate charge and paramagnetic current densities ( n;jp).\nIt has been shown that the ground-state wavefunction,\nand therefore the universal functional\nF[n;jp] =D\n\t\f\f\f^T+^U\f\f\f\tE\n; (4)\nis uniquely de\fned by these quantities. However, at\npresent there exists no current-density functional theory\n(CDFT) for ground-state systems to which TDCDFT ap-\nproaches in the steady, ground-state limit.\nIn the KS scheme for CSDFT, one then constructs\na system of noninteracting electrons having the same\ncharge and paramagnetic current density as the inter-\nacting system it represents. There are two problematic\naspects of such an approach. First, the KS system will\ntypically not have the same physical current as the sys-\ntem it represents and, as such, is not approached by TD-\nCDFT in the ground-state limit. Di\u000berent schemes for\nthe construction of KS systems is the primary focus of\nthis work.\nSecond, and also very pertinent to this study, the KS\npotentials are not determined by the two basic densities\n(in contrast to other KS-DFTs). One area of research in\nDFT that is quickly growing in activity is the calculation\nof exact KS systems from exactly-solvable systems for the\npurpose of advising the construction of better function-\nals, including the exact KS potentials required to repro-\nduce the analytic two-particle wavefunction solutions ofarXiv:1310.5887v1  [cond-mat.mes-hall]  22 Oct 20132\nthe time-dependent Schr odinger equation13,14, the exact\npotentials for time-dependent Hubbard chains15, and the\ntime-dependent potentials required to describe nonequi-\nlibrium quasiparticles described by a nonlocal model self-\nenergy operator16.\nHere, we take a related approach to the steady-state\nregime. We consider a three-dimensional steady-state in-\n\fnite wire whose QPs are, once again, taken to be de-\nscribed by the model self-energy operator employed in\nRef.16. The systems with which we concern ourselves are\nthose in which the Nground-state electrons \fll the va-\nlence band together with the standing state at the bottom\nof the conduction band, yielding a convenient nondegen-\nerate ground state of zero current density. The N+ 1th\nelectron is a QP added to one of the next-lowest-energy,\ndegenerate, current-carrying Bloch states. As such, the\nsystem approximates a quantum wire bridging two elec-\ntron reservoirs held at slightly di\u000berent chemical poten-\ntials.\nDegenerate systems (for which the DFT and VR exis-\ntence proofs are not intended to hold) may have a cur-\nrent even in the absence of an external vector potential,\nmaking an interesting study. The external scalar poten-\ntial of a system subject to no external vector potential is\nuniquely determined by n(r) alone (or, more accurately,\nthe density ensemble of all degenerate states17) and thus\nis amenable to description by DFT. However, the ground-\nstate wavefunction is uniquely de\fned by ( n;jp) together,\nas in VR theory, and thus the system is also amenable to\nstudy within current-density functional theory (CDFT)\nfor spinless systems. A key question is if and how the\nexact KS schemes di\u000ber between DFT, the CSDFT of\nVignale and Rasolt, and a CDFT wherein the KS and\ninteracting systems share the same physical current.\nA second question, central to the study in this paper,\nconcerns the construction of KS systems. Having cho-\nsen the basic variables that characterize a many-electron\nsystem, it does not necessarily follow that the KS rep-\nresentation will have the same values of other variables\nas the interacting system, which is particularly impor-\ntant if those other variables are measurable properties\nof the system. In particular, in order to have a bear-\ning on the TDCDFT, it is necessary that the physical\ncurrent density be reproduced by the KS system at all\ntimes. We investigate the implications of di\u000berent rules\nfor constructing current-carrying KS systems.\nII. THE QUASIPARTICLE CURRENT DENSITY\nThe wave equation governing the added QP is (in\natomic units as throughout)\n\u0000\n\u00001\n2r2+vext(r) +vH(r)\u0000EQP\u0001\n QP(r)\n+Z\nd3r0\u0006(r;r0) QP(r0) = 0 (5)\nwhereEQPis the quasiparticle energy, vext(r) the exter-\nnal potential to which the entire system is subject, vH(r)the Hartree potential, and \u0006 the self-energy operator.\nSince the QP is to be added to the lowest-energy unoccu-\npied state, the QP lifetime is expected to be in\fnite and\ntherefore the self-energy real. Generally the self-energy\nis an energy-dependent operator, however it has been\nshown19that this energy-dependence yields rather small\nquantitative changes to the band structure in ground-\nstate semiconductors and as such is not expected to give\nrise to any qualitatively di\u000berent features in the corre-\nsponding KS potential. As such, we take the self-energy\nto be nonlocal but energy-independent and Hermitian.\nThe electrons are con\fned to the wire by the scalar\npotential\nvext+vH=Hr6(6)\nin cylindrical polar coordinates. Thus, in the ground\nstate, the external and Hartree potentials due to the un-\nderlying lattice are assumed to approximately cancel. (In\nreality, there will be some periodic variation in vext+vH\nin thez-direction. However, this contribution will be the\nsame in both the QP and KS descriptions, and, since we\nare interested in the di\u000berences between the two, such a\nterm can be safely neglected without altering the physics\nbeing addressed.)\nThe spin- and current-independent self-energy opera-\ntor employed18has been shown to approximate the GW\nself-energy of nearly-free electronic materials19and is of\nthe form\n\u0006(r;r0) =f(z) +f(z0)\n2g(jr\u0000r0j) (7)\nwhereg(jr\u0000r0j) = exp\u0010\n\u0000(jr\u0000r0j=w)2\u0011\n=p\u0019win-\ntroduces the nonlocal operation of the self-energy\non the quasiparticle wavefunction, while f(z) =\n\u0000F0[1\u0000cos(2\u0019z=a )] imposes the periodicity along the\nwire of the underlying crystal lattice. As before, we\nchoosea= 4 a.u. and F0= 4:1 eV. The parameter wis\nchosen as 0.5 a.u. to approximate the Wigner-Seitz ra-\ndiusrs= (3n=4\u0019)1=3on the axis of the nanowire. These\nparameters approximately model a one-atom-thick sili-\ncon nanowire. For the con\fning potential, H= 3 eV\na.u.\u00006ensures that the charge and current density go to\nzero smoothly at the edge of the wire and that all occu-\npied electron states lie within the \frst subband.\nWe sample the band structure at the \u0000-point for a sys-\ntem of length Lz= 10a, thus the quasiparticle is normal-\nized to a supercell of 10 unit cells with each unit cell con-\ntributing one spinless electron to the ground-state charge\ndensity, plus one additional electron added to the stand-\ning wave state at the bottom of the conduction band.\nThe QP is added to the lowest right-going unoccupied\neigenstate of the Hamiltonian, yielding a total of 12 elec-\ntrons per supercell.\nThe divergence of the QP current density is given by\nthe continuity equation and the time-dependent form of\nEq. 5 (EQP!i@t):3\nr\u0001j(r;t) =\u0000@\n@tn(r;t) =i \u0003\nQP(r;t)\u0014\n\u00001\n2r2 QP(r;t) +Z\ndr0\u0006(r;r0) QP(r0;t)\u0015\n+ c.c\n=r\u0001j0(r;t) + 2ReZ\ndr0i \u0003\nQP(r;t)\u0006(r;r0) QP(r0;t) (8)\nwhere j0(r;t) =D\n QP(t)\f\f\f^jp(r)\f\f\f QP(t)E\n. Choosing our axes such that the current density is in the positive z-direction\ngives a steady-state QP current density of\nj(r) =j0(r)\u00002Zz\n\u00001dz0Z\ndr00Im \u0003\nQP(r0)\u0006(r0;r00) QP(r00): (9)\n(Note that the paramagnetic current operator acting on\nthe QP wavefunction does notgenerally give the param-\nagnetic current density of the many-body system.)\nOne may note that the second term in Eq. 9 is zero\nfor the homogeneous electron gas (HEG) for a real-valued\nand spherically-symmetric self-energy operator. As such,\nthe QP current would have no explicit dependence on\nthe nonlocal range of the operator, as one would expect\nsince the QP wavefunction varies spatially only in its\nphase. Thus exact KS-DFT should generally be able to\nreproduce the current-density of a degenerate HEG. For\nspatially-varying systems, however, the current density\nwill explicitly depend on the k-dependent self-energy, and\ntherefore the Coulomb strength of the interacting system,\nwith the gradient of the current given by\nr\u0001j(r) =r\u0001j0(r)\n\u00002\nVImX\n\u000b;\fc\u0003\n\u000bc\f\u001b(k\f) exp (i(k\u000b\u0000k\f)\u0001r) (10)\nwhereVis the volume of the supercell and\n QP(r) =X\n\u000bc\u000bexp (ik\u000b\u0001r); (11)\n\u0006(r;r0) =X\n\u000b\u001b(k\u000b) exp (ik\u000b\u0001(r\u0000r0)): (12)\nFor spatially-varying systems, therefore, the current\ndensity depends nonlocally on the system and explicitly\non the interaction strength. It is possible, then, that\ntwo degenerate systems having the same charge density\nand di\u000berent self-energy operators will not have the same\ncurrent density, and therefore not have V-representable\n(i.e. representable by a noninteracting scalar potential\nonly) charge and current densities.\nIII. THE APPLICATION OF DFT TO\nCURRENT-CARRYING SYSTEMS\nFigure 1 shows (a) the charge density of the steady-\nstate system, along with (b) the exact DFT scalar po-\ntential which reproduces it in the KS scheme. Thescalar potential is calculated \frst using the van Leeuwen-\nBaerends20procedure which iteratively multiplies the po-\ntential bynKS=n, and is then further re\fned by iterative\naddition of nKS\u0000n, achieving an accuracy of 0 :005%.\nThe net e\u000bect of the additional DFT potential, besides\nyielding the correct density amplitude along the wire, is\nto make the e\u000bective external potential somewhat less\ncon\fning to re\rect the additional electron-electron re-\npulsion.\nAlso shown is (c) the radial variation of the resul-\ntant QP and DFT current densities. Since we are in\nthe steady-state (indeed ground-state) regime, any spa-\ntial variation in the current density must be due to a\npurely transverse component. Without vector potentials,\nthe current densities of both the QP and DFT repre-\nsentations are purely paramagnetic. In the steady-state\nregime, the continuity equation merely constrains the di-\nvergence of the current density to be zero. Since all of\nthe current is in the axial direction, the radial and az-\nimuthal dependence of the current must be calculated\ndirectly from the wavefunction. The DFT calculation\nof this current density yields an error of 5 :1% in the\ncenter of the wire21due to the di\u000berences in the band\nstructures, and thus the group velocities, of the QP and\nDFT electrons. This re\rects the known fact that exact\nKS-DFT calculations may yield qualitatively inaccurate\nband structures22.\nIt follows that DFT does not generally yield the cor-\nrect current density of a ground-state system subject only\nto an external electric \feld. This is a time-independent\nvariation of the phenomenon observed in time-dependent\nsystems subject only to external scalar potentials demon-\nstrated by D'Agosta and Vignale5. Thus the necessity of\nmoving to a current-density functional theory lies not in\nthe presence of an external vector potential, but in the\nspatial variation of the current-carrying system.\nIt is worthy of note that there are no magnetic phe-\nnomena at all in the model interacting system. The ex-\nternal vector potential is everywhere zero and the 12 elec-\ntrons are spinless. An exact self-energy operator calcu-\nlated self-consistently from Hedin's equations would not\ncontain current-dependent and the model self-energy em-\nployed above introduces no such e\u000bects.4\n048121600.511.522.500.050.10.15n(a.u.)(a)\nz(a.u.) r(a.u.)n(a.u.)\n048121600.511.522.5 0.11101001000 vDFT(eV)(b)\nz(a.u.) r(a.u.)vDFT(eV)\n 0 0.003 0.006 0.009\n 0  0.5  1  1.5  2jz(r) (a.u.)\nr (a.u.)(c)\nQP\nDFT\nVR 0 10 20 30\n 0 0.5 1E (eV)\nkz (π/L)(c)\nQP\nDFT\n 0 4 8 12 16 0 1 2\n-0.005-0.004-0.003AVR (a.u.)(d)\nz (a.u.)r (a.u.)AVR (a.u.)\nFIG. 1: (Color online) The axial and radial dependence of\n(a) the charge density of the system governed by the nonlocal\nself-energy, and (b) the KS scalar potential which reproduces\nthe charge density exactly. The quasiparticle current den-\nsity (c) varies only radially in the steady-state regime. The\ncurrent density predicted by this DFT scalar potential (blue\ndashed) underestimates the true current density (red solid)\nby 5%. The underestimate is due to the di\u000berence in band\nstructures (inset) which have di\u000berent gradients in the region\nof the highest occupied state (denoted by square and circle\nrespectively). In a standard CSDFT approach, the correct\nparamagnetic current can be obtained with the inclusion of a\nKS vector potential (d); however such a potential corresponds\nonly to a gauge transformation: it leaves the charge and phys-\nical current density (circles) unchanged, and thus gives no\nimprovement over the DFT description of the system.IV. THE PARAMAGNETIC CURRENT IN THE\nKS SCHEME\nLet us \frst consider the CSDFT approach of VR, in\nwhich KS systems are constructed to yield the same\n(n;jp) as the interacting systems they represent. The\nHamiltonian for a KS system of spinless electrons sub-\nject to both scalar and vector potentials is\n^H=1\n2[^p+AKS(r)]2+vKS(r): (13)\nWhile the KS wavefunctions are uniquely determined by\n(n;jp), the potentials ( vKS;AKS) are not23and so one\ncannot generally speak of theexact KS vector poten-\ntialper se . Fig. 1(d) shows areverse-engineered vec-\ntor potential that achieves our desired densities (the cor-\nresponding scalar potential is simply that of the DFT\ncalculation). However, this vector potential corresponds\nonly to a gauge transformation of the KS single-particle\nwavefunctions of the form\n KS;k(r) ei\u0015(r) KS;k(r): (14)\nwhere\u0015(r) is the scalar \feld such that A(r) =\u0000r\u0015(r).\nThis vector potential ensures that ( n;jp) of the KS sys-\ntem are those of the interacting system (Fig. 1(c)), but\nno physical quantity has changed as a result and the\nphysical current of the VR KS system is identical to\nthe ordinary DFT prediction. Since the wavefunction\nis uniquely determined by ( n;jp), it follows that an al-\nternative choice of ( vKS;AKS) that yield the same ( n;jp)\nalso corresponds to a gauge-transform of the DFT result.\nThus, as was the case with DFT, the VR formulation of\nCSDFT does not in general reproduce the correct phys-\nical current. For instance, it follows that the VR theory\nis not the limit of time-dependent CDFT in the steady-\ncurrent limit.\nV. THE PHYSICAL CURRENT IN THE KS\nSCHEME\nFor this reason, we consider instead a Kohn-Sham\nscheme that reproduces the physical current density of\nthe ground-state nanowire. Putting aside the question\nof which basic variables uniquely determine the ground-\nstate properties of a current-carrying system, one may\nretain from the approaches of Diener and Pan and Sahni\nthe convention of constructing the auxiliary KS systems\nto have the same ( n;j) as the interacting system. In the\nabsence of an external vector potential, the exact KS vec-\ntor potential arises purely from exchange and correlation\n(XC) and can be calculated iteratively using\nAxc(r) Axc(r) +j(r)\u0000jKS(r)\nn(r): (15)\nGenerally, transverse components to the reverse-\nengineered vector potential will result in a change in the5\ncharge density and therefore necessitate a recalculation\nof the scalar potential. However, the stronger the con-\n\fnement in the transverse direction, the smaller the per-\nturbation of the charge density. As a result, the con-\nverged CDFT scalar potential is almost exactly that of\nDFT (Fig. 1). We have con\frmed that the sensitivity of\nthe charge density to external vector potentials returns\nas one makes the con\fning electric \feld weaker.\nThe XC vector potential and the current density that it\nyields are shown in Fig. 2. One can see that, due to the\nlarge con\fning external potential, the vector potential\nbecomes large in magnitude at large radii where nandjp\n(and thus the e\u000bect of the vector potential) is smallest.\nBecause of the necessary presence of the nonzero KS vec-\ntor potential, the physical current density in the KS sys-\ntem is no longer purely paramagnetic, in contrast to both\nthe QP and DFT systems. Instead, the physical QP cur-\nrent is reproduced by \fnding the correct combination of\nparamagnetic and diamagnetic KS current densities for\na given charge density. Thus in constructing KS func-\ntionals, it is necessary to go beyond the paramagnetic\ncurrent density if one wishes to describe the KS system\nexactly. Furthermore, because of the distribution of the\ncurrent density between paramagnetic and diamagnetic\nparts, 12% of the current density is now carried by the\nNlowest-energy KS electrons, whose counterpart in the\nQP description of the system contributed zero current.\nAs in the time-dependent regime, the vector potential\ncalculated corresponds to an exchange-correlation mag-\nnetic \feld Bxc=r\u0002Axc, also shown in Fig. 2. The\nmanner in which the magnetic \feld \fxes the physical cur-\nrent density while keeping the charge density \fxed can\nbe characterized as the interplay between two distinct\ne\u000bects.\nFirst, the Kohn-Sham magnetic \feld is entirely az-\nimuthal, which for an axial current corresponds to a ra-\ndial, velocity-dependent force which augments the radial\nforce arising from the Kohn-Sham scalar potential24. In\nthe current-carrying region, the ratio of the magnetic and\nelectric radial forces (as measured by uzBxc;\u001e=@rvKS) is\naround 2-3%, comparable to the 5% adjustment to the\ncurrent density that CDFT needs to achieve. The mag-\nnitude of the XC magnetic \feld may be gauged from the\nfact that it is similar in strength to the elementary Biot-\nSavart magnetic \feld that the current density generates,\nin the vicinity of the wire.\nSecond, the vector potential \\tunes\" the band struc-\nture such that the current density (both paramagnetic\nand diamagnetic) of all of the current-carrying electrons\nin the KS system, determined in part by the local gradi-\nent of the band structure, sum to the correct current of\nthe quasiparticle.\nWhat is unusual in this case is that the interacting\nsystem being modelled does not require the existence of\nmagnetic \felds or magnetic interactions at all: the elec-\ntrons are spinless; there are no external magnetic \felds\napplied, and the self-energy operator contains no current-\nor spin-dependence. The XC vector potential is therefore\n0481216 00.511.52 00.0050.010.0150.020.025Axc(a.u.)(a)\nz(a.u.) r(a.u.)Axc(a.u.)\n 0 4 8 12 16 0 0.25 0.5 0.75 -0.0004 0 0.0004 0.0008 0.0012 Bxc,φ (a.u.)(b)\nz (a.u.) r (a.u.)Bxc,φ (a.u.)\n-0.75-0.5-0.25 0 0.25 0.5 0.75\n-0.75-0.5-0.25 0 0.25 0.5 0.75y (a.u.)\nx (a.u.)(c)\n 0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012\n 0 0.003 0.006 0.009\n 0 0.5 1 1.5 2 2.5jz (a.u.)\nr (a.u.)(d)\nQP\nExact KS\nKS Paramagnetic\nKS DiamagneticFIG. 2: (Color online) (a) The intrinsic KS vector potential\nand (b) the magnetic \feld to which it corresponds. These,\nalong with the scalar potential of Fig. 1b, reproduce the\nexact charge and current density of the nanowire. The mag-\nnetic \feld is purely azimuthal and increases with radius (c,\nshown for z= 0). The resulting KS current density (d, red\nsolid) is now that of the QP system (black squares), but now\ncomprises both paramagnetic (green dashed) and diamagnetic\n(blue dotted) components.6\npurely mechanical in nature, even though it enters into\nthe Kohn-Sham equations as a magnetic \feld.\nVI. SUMMARY AND CONCLUSIONS\nIn conclusion, even in the absence of spin and external\nmagnetic \felds, current-carrying systems cannot gener-\nally be represented exactly by density functional theory\nand require a purely XC magnetic \feld that is mechanical\nin nature and depends on the charge and physical current\ndensity of the system. This XC magnetic \feld arises fromthe construction of the KS scheme such that the nonin-\nteracting physical densities are those of the interacting\nsystem, irrespective of the choice of basic variables in\nthe underlying CDFT. We have demonstrated a method\nfor the calculation of the necessary exact XC magnetic\n\felds, \fnding them to be dependent on both the charge\nand current density of the interacting system. Generally,\nthe exact KS representation of a ground-state current-\ncarrying system does notcarry the paramagnetic current\ndensity of the interacting system that it represents.\nWe thank Peter Bokes and Giovanni Vignale for fruit-\nful discussions, and acknowledge funding from EPSRC.\n1E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997\n(1984).\n2R. van Leeuwen, Phys. Rev. Lett. 82, 3863 (1999).\n3P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964).\n4W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 (1965).\n5R. D'Agosta and G. Vignale, Phys. Rev. B 71, 245103\n(2005).\n6S. K. Ghosh and A. K. Dhara, Phys. Rev. A 38, 1149\n(1988).\n7G. Vignale, Phys. Rev. B 70201102(R) (2004).\n8X. Pan and V. Sahni, Int. J. Quant. Chem. 110, 2833\n(2010).\n9G. Diener, J. Phys.: Condens. Matter 3, 9417 (1991).\n10E. I. Tellgren et al, Phys. Rev. A 86, 062506 (2012).\n11, G. Vignale, C. A. Ullrich and K. Capelle, Int. J. Quantum\nChem. 113, 1422 (2013).\n12G. Vignale and M. Rasolt, Phys. Rev. B 37, 10685 (1988).\n13M. Thiele, E. K. U. Gross, and S. K ummel, Phys. Rev.\nLett.100, 153004 (2008).\n14P. Elliott, J. I. Fuks, A. Rubio and N. T. Maitra, Phys.\nRev. Lett. 109, 266404 (2012).\n15C. Verdozzi, Phys. Rev. Lett. 101, 166401 (2008).\n16J. D. Ramsden and R. W. Godby, Phys. Rev. Lett. 109,\n036402 (2012).\n17M. Levy, Phys. Rev. A 26, 1200 (1982).\n18R. W. Godby and L. J. Sham, Phys. Rev. B 49, 1849(1994).\n19R. W. Godby, M. Schl uter and L. J. Sham, Phys. Rev. B\n37, 10159 (1988).\n20R. van Leeuwen and E. J. Baerends, Phys. Rev. A 49, 2421\n(1994).\n21We have con\frmed that, as the nonlocal range, w, of the\nself-energy operator is reduced to zero, the error in the\nDFT current density also goes to zero.\n22R. W. Godby and R. J. Needs, Phys. Rev. Lett. 62, 1169\n(1989).\n23K. Capelle and G. Vignale, Phys. Rev. B 65, 113106\n(2002); K. Capelle, C. A. Ullrich and G. Vignale, Phys.\nRev. A 76, 012508 (2007); E. Engel and R. M. Dreizler,\nDensity Functional Theory: An Advanced Course. New\nYork: Springer-Verlag (2011).\n24In general, the QP will carry a nonzero spin and thus the\nXC magnetic \feld will couple to this via the Stern-Gerlach\nterm in Eq. 13. The Stern-Gerlach interaction will modify\nthe KS charge density from that of the model nanowire\nwithout the inclusion of a corresponding modi\fcation to\nthe scalar potential. Since the interaction depends on both\nthe XC magnetic \feld and the spin, the KS scalar poten-\ntial is necessarily and, in principle, strongly dependent on\nthe current density and the magnetization, as well as the\ncharge density."    },    {        "title": "1310.7019v1.Simulations_of_imaging_of_the_local_density_of_states_by_charged_probe_technique_for_resonant_cavities.pdf",        "content": "Simulations of imaging of the local density of states by charged probe technique for\nresonant cavities\nK. Kolasinski and B. Szafran\nAGH University of Science and Technology,\nFaculty of Physics and Applied Computer Science,\nal. Mickiewicza 30, 30-059 Krak\u0013 ow, Poland\n(Dated: August 11, 2021)\nWe simulate scanning probe imaging of the local density of states related to scattering Fermi\nlevel wave functions inside a resonant cavity. We calculate potential landscape within the cavity\ntaking into account the Coulomb charge of the probe and its screening by deformation of the two-\ndimensional electron gas using the local density approximation. Approximation of the tip potential\nby a Lorentz function is discussed. The electron transfer problem is solved with a \fnite di\u000berence\napproach. We look for stable work points for the extraction of the local density of states from\nconductance maps. We \fnd that conductance maps are highly correlated with the local density of\nstates when the Fermi energy level enters into Fano resonance with states localized within the cavity.\nGenerally outside resonances the correlation between the local density of states and conductance\nmaps is low.\nI. INTRODUCTION\nIn semiconductor systems based on the two-\ndimensional electron gas (2DEG) the linear conductance\nis determined by the scattering properties of the Fermi\nlevel wave functions.1Since relatively recently the elec-\ntron transport properties can be probed with a local per-\nturbation introduced to 2DEG by the charge probe at the\natomic force microscope tip,2,3used as a \roating gate.\nThe scanning gate microscopy2,3allows for visualization\nof magnetic de\rection of electron trajectories,4quantum\ninterference due to the elastic scattering5and Aharonov-\nBohm e\u000bects,6,7charged islands and the edge currents in\nnanostructurized quantum Hall bars,8branching of the\nelectron \row,18tip-induced lifting of the Coulomb block-\nade in quantum dots,9,10etc.\nIn this paper we consider electron transport across a\ncavity side-attached to the semiconducting channel. The\nelectron \row through resonant cavities is a basic prob-\nlem for the quantum transport with a long history, start-\ning from the weak localization e\u000bects,11the relation be-\ntween the classical and quantum modes of transport,12in\nparticular the scars of the classical trajectories on wave\nfunctions,13,14and most recently the pointer states15,16\nthat are robust against decoherence and stable despite\ncoupling to the environment. The spatial distribution\nof the pointer states was extracted by post-treatment\nof the conductance images as obtained by scanning gate\nmicroscopy.2,17\nIn this work we consider a purely coherent electron\ntransport. Our purpose is to determine an extent to\nwhich the details of the local density of states6,7at the\nFermi level can be extracted from the raw conductance\nmaps gathered with the charge perturbation scanning the\nsurface of the structure. The original tip potential as\nseen by the 2DEG is of the long range Coulomb form.\nThe Coulomb potential is screened by deformation of\n2DEG density. Usually for theoretical modeling the tip\npotential is assumed short-range in a form given by aclosed formula.4,6,7,19,20For 2DEG which is not con\fned\nlaterally4,5,18the deformation of the electron gas follows\nthe tip as it scans the surface and the e\u000bective (screened)\ntip potential preserves its form. On the other hand in\nsystems with lateral con\fnement { in the cavity in par-\nticular { the 2DEG deformation cannot freely follow the\ntip, so a local form of the e\u000bective potential can only be\nan approximation of the actual potential. In this work\nthe e\u000bective potential of the tip is evaluated by solving\nthe density functional theory equations.22We \fnd that\nthe tip potential is close to Lorentzian which for small\n2DEG-tip distance is isotropic outside the edges of the\ncavity and of the width which is close to the tip-2DEG\ndistance.\nWe demonstrate that the resolution of the local density\nof states at the Fermi with the charge probe technique de-\npends on the work point de\fned by the electron density.\nThe local density of states is highly correlated to con-\nductance maps when the Fermi energy level enters into\nFano resonance with states localized within the cavity.\nOutside the resonances the maps calculated for weak tip\npotentials agree with the Lipmann-Schwinger perturba-\ntion theory23but are not correlated in a clear way with\nthe local density of states.\nII. THEORY\nWe consider the electron gas \flling the channels in-\ncluding the side-attached square cavity (300 nm \u0002300\nnm), that is depicted in Fig. 1(a). The entire com-\nputational box is taken as large as 1.5 \u0016m. This length\nguarantees, that the potential of the tip is screened be-\nfore it reaches the ends of the channels. The channels\nare taken 50 nm wide. The electron gas is considered\nstrictly two-dimensional. We apply local density approx-\nimation (LDA) for description of the electrostatics of thearXiv:1310.7019v1  [cond-mat.mes-hall]  28 Oct 20132\nsystem,24with the single-electron Hamiltonian\nH=\u0000~2r2\n2me\u000b+U(x;y) (1)\nwhereme\u000bis the electron e\u000bective band mass, and the\npotential is given by,\nU=W+VH+Vxc+Vtip; (2)\nwhereWis the con\fnement potential of the channels\n(we take 0 inside the channels and 200 meV outside), VH\nis the Hartree potential, Vxc{ the exchange correlation\npotential (we apply the parametrization by Perdew and\nZunger21), andVtipis the tip potential. The Coulomb\npotential of the charge at the tip of the probe as seen at\nthe 2DEG level is given by\nVtip(x;y;xt;yt) =eQtip\n4\u0019\u000f\u000f0p\n(x\u0000xt)2+ (y\u0000yt)2+z2\nt;\n(3)\nwhere (xt;yt;zt) is the tip location, and Qtipis the charge\nat the tip.\nFIG. 1. Schematics of the considered system. (a) Geom-\netry within the plane of con\fnement. The square cavity of\nside length 300 nm is side attached to 50 nm wide channel.\nThe length of the computational box is 1.5 \u0016m. The dashed\nstraight lines indicate the paths along which the Lorentzian\n\ft to the tip potential is discussed (see Fig. 3 below). (b)\nSide view: the plane of electron con\fnement at a distance of\nzdto the sheet of ionized donors and ztto the tip.\nIn 2DEG, the carriers are delivered by the donor layer\nwhich is left charged positively. We assume that a ho-\nmogenous sheet of positive charge is present on top of the\narea accessible to electrons [see Fig. 1(b)] at a distance\nofzdfrom the electron gas. The considered model is\ncharge neutral, with the average electron density within\nthe channels matched to the donor density in the doped\nlayer above the 2DEG [see Fig. 1(b)].\nIn order to simulate open in\fnite leads we apply peri-\nodic boundary conditions for the Schroedinger equation\nwith Hamiltonian (1). For calculation of the Hartree po-\ntential we produce copies of the charge present within the\noriginal computational box and place them on its left and\nright sides [see Fig. 1(a)], and then integrate the charge\nwith the Coulomb potential\nVH(x;y) =e2\n4\u0019\u000f\u000f0\u0012\n\u0000Z\ndx0dy0nd(x0;y0;zd)\nj(x;y;0)\u0000(x0;y0;zd)j\n+Z\ndx0dy0n(x0;y0;0)\nj(x;y;0)\u0000(x0;y0;0)j\u0013\n(4)In Eq. (4), nis the electron density, which is determined\nin the following manner. We solve the eigenequation for\nHobtaining eigenfunctions  iwith eigenvalues \"i. Then,\nthe electron density is calculated as\nn(r) = 21X\nif(\"i)j i(r)j2; (5)\nwherefis given by the Fermi-Dirac distribution,\nf(\"i) = [exp ((\"i\u0000EF)=kB\u001c) + 1]\u00001; (6)\nand the factor 2 accounts for the degeneracy of energy\nlevels with respect to the spin. The Fermi energy EFis\ndetermined by the normalization condition\nN= 21X\nlf(\"l): (7)\nIn the calculations we assume the temperature of \u001c=\n0:28 K and material parameters of InAs: me\u000b= 0:023m0\nand\u000f= 15:5.\nOnce the self-consistence of the Schroedinger-Poisson\nscheme is reached, we obtain the Fermi energy and the\npotential distribution which is then used for determina-\ntion of the electron transfer probability T, and thus the\nconductance G=2e2\nhT, according to the Landauer ap-\nproach. Within the leads, far away from the cavity the\npotential depends only on y, so that the Hamiltonian\neigenstates carrying the probability current can be de-\nscribed by the wave vector ( k>0),\n\tk(x;y) =eikx (y); (8)\nwhere is the wave function describing the state of the\ntransverse quantization. In this paper we consider low\nelectron densities, for which the transport occurs in the\nlowest subband only. Then a single value of the wave\nvectorkappears at the Fermi level. In order to de-\ntermine the electron transfer probability we solve the\nSchroedinger equation with the \fnite di\u000berence method\nand the scattering boundary conditions. Let us assume\nthat the electron is incident from regions of negative x\n(see Fig. 1) and let us denote the scattering wave func-\ntion for this case by \t +. In the channels far away from\nthe cavity { i.e. outside the range of the evanescent\nmodes { the wave function acquires the form\n\t+(x;y) = \t k(x;y) +r+\t\u0000k(x;y) (9)\nin the input channel, where r+is the backscattering am-\nplitude. In the output channel one \fnds\n\t+(x;y) =t+\tk(x;y); (10)\nwheret+is the electron transfer amplitude for the elec-\ntron incident from the left. The electron transfer proba-\nbility isT=jt+j2. The solution of the scattering prob-\nlem, i.e. the Hamiltonian eigenstate matched to bound-\nary conditions given by Eqs. (9) and (10) is found by an3\niterative approach which is described in detail in Refs.\n22 and 25.\nBelow we discuss the spatial density of states at the\nFermi level (also known as the local density of states {\nLDOS6). The local density of states is obtained as a sum\nof the scattering probability densities for the Fermi-level-\nelectron incident to the cavity from the left \t +and right\nchannels \t\u0000\nLDOS(x;y) =j\t+(x;y)j2+j\t\u0000(x;y)j2: (11)\nIn order to support the discussion of the simulated\nscanning gate microscopy conductance maps, in the limit\nof weak perturbations we employ the general formulas for\nthe \frst and second order corrections to conductance due\nto the tip developed23using the Lippmann-Schwinger ap-\nproach. In the lowest-subband transport case discussed\nhere, the formulas read\nG(1)(xt;yt) =\u00002me\u000b\n~2k=(r\u0003\n+t\u0000V(\u0000;+)); (12)\nfor the \frst-order and\nG(2)(xt;yt) =\u00002\u0019me\u000b\n~2k<h\njt+V(\u0000;+)j2+jt\u0000V(+;\u0000)j2(13)\n+r\u0003\n+t\u0000(V(\u0000;\u0000)V(\u0000;+)\u0000V(\u0000;+)V(+;+))i\n;\nfor the second-order corrections, with the potential ma-\ntrix element de\fned as\nVa;b(xt;yt) =Z\ndxdy \t\u0003\na(x;y)V(x;y;xt;yt)\tb(x;y)\n(14)\nthat is evaluated from the scattering wave functions com-\ning from left ( a;b= +) or right ( a;b=\u0000) leads. The\nwave functions and the scattering amplitudes used in\nEqs. (13-15) are calculated in the absence of the tip,\nwhich only enters the kernel of Eq. (14). The conduc-\ntance is then approximated by Gpert(xt;yt) =G(1) +\nG(1)(xt;yt)+G(2)(xt;yt), whereG(1) is the conductance\nin the absence of the tip.\nFor quantitative discussion of the similarity between\nmaps of conductance and the local density of states we\ncalculate correlation factors in the following manner. We\n\frst perform normalization of the map\nf=F\u0000Fmin\nFmax\u0000Fmin; (15)\nwhereFmaxandFminare the maximal and minimal val-\nues of the quantity Fwithin the cavity area S. Next, we\ncalculate the correlation factor for dimensionless normal-\nized mapsfandgas\nr=1\nSR\nSdxdy (f(x;y)\u0000hfi)(g(x;y)\u0000hgi)\n\u001bf\u001bg; (16)\nwherehfi =R\nSdxdyf (x;y)=S and\u001b2\nf =R\nSdxdy (f(x;y)\u0000hfi)2=S.\nFIG. 2. (a) Charge density of the electron gas for 100 elec-\ntrons present within the computational box in the absence\nof the tip. (b) Total potential Udistribution correspond-\ning to (a). (c) Bare potential of the tip within 2DEG plane.\nThe probe charge is above the spot marked by the cross for\nQtip=\u00001 [e] at a distance of zt= 15 nm from the plane. (d)\nCharge density for the tip localized like in (c). (e) The po-\ntentialUdistribution corresponding to (d). (f) The e\u000bective\npotential of the tip calculated as the di\u000berence of (e) and (b).\n(g) The e\u000bective potential for zt= 40 nm for yt= 300 nm\nandxt= 850 nm. (h,i,j,k) same as (g) only for xt= 750 nm,\n800 nm, 950 nm, and 1000 nm, respectively.4\nFIG. 3. Parameters of the Lorentz potential [Eq. 17] \ftted\nto the e\u000bective tip potential for Qtip=\u0000jejand the tip-2DEG\ndistance of zt= 15 nm (as in Fig. 2(e)] and zt= 40 nm along\nthe dashed lines plotted in Fig. 1. Plot (a) shows the height\nof the potential U0maximum. Plot (b) shows dy(dash-dotted\ncurves) and dxgiven by the error bars. The \ft was performed\nwithin the cavity area inside a square of 200 nm side length\ncentered on the tip.\nIII. RESULTS AND DISCUSSION\nA. Screening of the tip potential\nThe calculated self-consistent charge and potential dis-\ntributions as obtained for N= 100 electrons within the\nbox in the absence of the tip are plotted in Figures 2(a)\nand 2(b), respectively. The charge density within the\nsystem is distinctly maximal at its edges. In the chan-\nnels near the ends of the computational box Ubecomes\nindependent of x, which is consistent with the boundary\nconditions used for the transport problem [Eq. (8)]. Fig-\nure 2(c) shows the bare potential of the tip localized at\nzt= 15 nm above the 2DEG. The applied e\u000bective charge\nat the tipQtip=\u00001 [e] for the tip radius of R= 5 nm\ncorresponds to the tip potential of V=Qtip=R= 0:288\nV. Figures 2(d) and 2(e) display the charge density and\npotential as perturbed by the tip, respectively. The con-\nsequence of the charge redistribution is the screening of\nthe original tip potential. The screened (e\u000bective) tip\npotential can be calculated as a di\u000berence of perturbed\n[Fig. 2(e)] and unperturbed potentials [Fig. 2(b)]. The\ndi\u000berence is displayed in Fig. 2(f). The e\u000bective poten-tial has a form of a 4 meV peak of '30 nm diameter.\nFor comparison, the e\u000bective potential calculated for the\n2DEG-tip distance increased to 40 nm is displayed in\nFig. 2(g-l). As the tip approaches the edge of the cav-\nity the second elongated maximum is formed along the\nedge [Fig. 2(g)]. In the present model the screening is\nonly due to 2DEG which is missing outside the cavity\nhence the reduced screening at the edge. When the tip\nis displaced to the outside of the cavity [Fig. 2(j,k)] the\nelongated maximum at the edge is preserved.\nFor a quantitative parametrization of the tip poten-\ntial we \ftted to the results of the Schroedinger-Poisson\nscheme the Lorentz function\nVL(x;y;xt;yt) =U0\n1 +(x\u0000xt)2\nd2x+(y\u0000yt)2\nd2y; (17)\nwhereU0is the potential height above the constant po-\ntential background, dxanddyare potential widths in the\ndirections parallel and perpendicular to the channels, re-\nspectively. The results of the \ft for the tip following the\npaths marked by horizontal dashed lines in Fig. 1 are dis-\nplayed in Fig. 3 for parameters of Fig. 2. For stronger\ninteraction ( zt= 15 nm) the height of the tip potential\noscillates. The period of the oscillation is roughly half\nof the Fermi wavelength (here kF= 0:05/nm), which is\nreminiscent of the Friedel oscillations discussed in Ref.\n22. When the tip is near the center of the cavity [cf.\nFig. 2(b) for y= 300 nm and xt= 750 nm] its potential\nis isotropic. Note that, in this case the potential width\ndx=dyis of the order of zt. The height of the tip poten-\ntial [Fig. 3(a)] tends to increase at the edge of the cavity\nwhich is due to the reduced screening by the electron\ngas. The potential becomes anisotropic when the tip ap-\nproaches the edges of the cavity due to the anisotropy of\nthe screening medium. An extreme anisotropy is reached\nfor the tip above the edge: the potential is strongly elon-\ngated along the edge, it weakly penetrates the cavity [cf.\nFig. 2(j)].\nFor the discussion below we use both the results for\nthe Coulomb potential (3) and the anisotropic Lorentz\nansatz (17) with d=dx=dy. There is an essential\ndi\u000berence between these two calculations. Vtipgiven by\nEq. (3) enters the DFT equations, so that the charge\ndensitynreacts to its presence resulting in the screening\nof the tip potential. On the other hand, whenever the\nansatz in form given by Eq. (17) is used, we simply add\nthis potential to the self-consistent potential as obtained\nby DFT in the absence of the tip and use it in the electron\nscattering problem.\nThe present \fnding that the tip potential can be ap-\nproximated by the Lorentz function with the width of\nthe order of the 2DEG - tip distance is consistent with\nthe experiment of Ref. 10 which investigated the form\nof the actual tip potential in the Coulomb blockade mi-\ncroscopy of double quantum dots de\fned within 2DEG.\nThe conclusion10was that the potential is close to the\ncircularly symmetric Lorentz function that for the tip\n'234 nm above the electron gas (200 nm above the5\nsample surface with the electron gas buried 34 nm below\nit) possesses a dip that is 250 nm wide.\nFIG. 4. Results for N= 78 electrons within the computa-\ntional box. (a) The density of states at the Fermi level (local\ndensity of states), calculated as a sum of probability densi-\nties corresponding to scattering wave functions incident with\nkF= 0:039 / nm ( ~2k2\nF=2me\u000b= 2:5 meV) from the left and\nright to the cavity. Transfer probability maps [or conductance\nmaps in units of G0= 2e2=h] obtained for Qtip=\u00001[e] and\nthe tip at the distance of zt= 15 nm (b) and zt= 40 nm (c)\nfrom the 2DEG. (d) Conduction map for Lorentz perturba-\ntion [Eq. 17] with parameters corresponding to plot (b), i.e.\nU0= 3:5 meV and d=dx=dy= 15 nm. (e) Conduction\nmap for Lorentz perturbation with parameters corresponding\nto plot (c), i.e. U0= 0:7 meV and d= 40 nm. Plots (f) and\n(g) show the conductance maps as calculated with the \frst\nand the second order corrections of the Lipmann-Schwinger\nequation theory [Eqs. (12, 14)] obtained with the potential of\nplot (e). The values of r(b;a);r(b;d);::: etc. give the corre-\nlation factors between the maps of panels ( b;a);(b;d);:::, as\ncalculated according to Eq. (16).\nB. Density of states at the Fermi level versus the\nconduction maps\nIn Fig. 4(a) we show the density of states at the Fermi\nlevel obtained for N= 78 electrons within the compu-\ntational box. Figures 4(b) and 4(c) display the corre-\nsponding maps of the transfer probability for zt= 15 nm\nandzt= 40 nm, respectively. Correlation between the\nLDOS and Tmap forzt= 15 nm is evident, although\nthe relative amplitudes of extrema as observed in LDOS\nandTvary. The correlation factor between the Tmap\nwithzt= 15 nm and the LDOS [Figs. 4(a) and (b), re-\nspectively] is r=\u00000:52 (the negative sign is due to the\ncoincidence between maxima of LDOS and minima of G\nmap). The details of the Tmap become less resolved for\nFIG. 5. Results for N= 92 electrons within the computa-\ntional box ( kF= 0:046 /nm, ~2k2\nF=2me\u000b= 3:5 meV) (a) The\ndensity of states at the Fermi level (local density of states),\nand conduction maps obtained for the tip at the distance of\nzt= 15 nm, for Qtip=\u00000:1 [e] (b),Qtip=\u00001[e] (c), and\nQtip=\u00008 [e] (d). Results for a Lorentzian ansatz of the ef-\nfective tip potential are displayed in (e) and (f) for U0= 0:33\nmeV and radii d= 4 nm (e) and d= 40 nm (f). Plots (g) and\n(h) [(i) and (j)] are the \frst and second-order perturbations\ncalculated for parameters of (e) [(f)].\nzt= 40 nm [Fig. 4(c)] as should be expected from the\nincrease of the e\u000bective width of the tip discussed in Fig.\n2(f) and Fig. 2(g). The correlation factor between the\nTmap of Fig. 4(c) and the LDOS of Fig. 4(a) falls to\nr=\u00000:36.\nAs discussed in Fig. 3, the parameters of the e\u000bec-\ntive tip potential vary with the tip position. In order to\nillustrate the importance of the variation in Fig. 4(d)6\nand (e) we presented the Tmaps as obtained with the\nisotropic Lorentz potential [Eq. 17]. For zt= 15 nm\n(zt= 40 nm) we \fxed parameters to U0= 3:5 meV and\nd=dx=dy= 15 nm (U0= 0:7 meV and d= 40 nm)\naccording to the \ft obtained for the tip near the center\nof the cavity [cf. Fig. 3]. The similarity between the\nTmaps with the Lorentz ansatz [Fig. 4(d,e)] and the\nones obtained by the full calculation with the Coulomb\npotential of the tip [Fig. 4(b,c)] is quite distinct and the\ncorrelation factor equals r= 0:61 forzt= 15 nm and\nr= 0:7 forzt= 40 nm. For zt= 40 nm, the results\nwith the Coulomb potential indicate a stronger variation\nalong the edges of the cavity which results from the in-\ncreased height of the actual perturbation near the edges\n(see the discussion above).\nFor the parameters corresponding to zt= 40 nm the\nheight of the perturbation U0= 0:7 meV is distinctly\nsmaller than the kinetic energy of the Fermi level electron\n(~2k2\nF=2me\u000b= 2:5 meV). It is instructive to check how\nthe perturbation theory works for the Lorentz potential\nas compared to the exact results [Fig. 4(e)]. The \frst and\nsecond order Tmaps are displayed in Figs. 4(f) and (g),\nwith the correlation factors reaching r=0.48 and 0.65,\nrespectively.\nThe correspondence of LDOS with the Tmap found\nin Fig. 4 for zt= 15 nm is relatively close. In order to\nsee whether this can be considered a rule, we performed\ncalculations for a varied electron number { N= 92 {\nsee the results of Fig. 5. The result for Qtip=\u00001[e]\n[Fig. 5(c)] exhibits a similar correlation to LDOS [Fig.\n5(a)] with the correlation factor of r= 0:47. Note, that\nthe correspondence has now an inverse character with\nrespect to Fig. 4: in Fig. 5 the maximal Tcorresponds\nto maximal LDOS hence the positive sign of r. Figure\n5(d) indicates that the correspondence of Tto LDOS\nis reduced for the larger charge at the tip. For Qtip=\n\u00008ethe 2DEG beneath is completely depleted, and the\nvolume of the cavity itself, i.e. the space accessible for\nelectrons changes with the varied tip position, hence the\nabrupt variation of the Tmap. The correlation to LDOS\nfactor drops to r= 0:26.\nWe learn from Fig. 5(b) that as the tip charge is re-\nduced further from Qtip=\u0000etoQtip=\u00000:1etheTmap\ndoes not get any closer to LDOS. Only the amplitude of\nthe variation is reduced for smaller jQtipj. We \fnd that in\ngeneral, the e\u000bective width of the tip depends on ztand\nnot onQtip. The experimental literature on quantum\nrings2,6contains the discussion of the in\ruence of the\ntip potential on the conductance patterns in the scan-\nning gate microscopy maps. The conclusion2,6was that\nin the linear perturbation regime (i.e. for relatively weak\nperturbation) the potential at the tip in\ruences only the\namplitude of the map and not the pattern, which is con-\nsistent with the result obtained for Qtipvaried between\n\u0000eand\u00000:1e.\nFor comparison in Figs. 5(e) and (f) we presented re-\nsults forTmaps as obtained for the Lorentz potential of\nthe tip for d= 4 nm and d= 40 nm, respectively for asmall height of potential U0= 0:33 meV (about 10% of\n~2k2\nF\n2meff). The result of Fig. 5(e) for d= 4 nm reproduces\nin a close detail the LDOS of Fig. 5(a) { with the cor-\nrelation factor reaching r= 0:86. The result with d= 4\nnm and small U0is exactly reproduced by the pertur-\nbation calculus [Fig. 5(g-h)]. For the wider tip d= 40\nnm, the second correction plays a more decisive role [Fig.\n5(i,j)], with the values of Tcloser to the results of the ex-\nact calculation [Fig.5(f)] but with a reduced correlation\nof the images due to lower contrast of the map Fig.5(j).\nThe discussed case of d= 40 nm is near the verge of the\napplicability of the perturbation theory. We \fnd that in-\ndependent of Nford= 4 nm and U0of the order of 10%\nof the kinetic Fermi energy the perturbation calculus23\nincluding the second order correction exactly reproduces\nthe exact transfer probability maps with rclose to 1.\n(a)\nFIG. 6. The black line shows the dependence of the transfer\nprobability TonN. The green curve shows the correlation\nfactor between LDOS and the Tmap as obtained for a point-\nlike perturbation d= 4 nm and U0= 0:1~2k2\nF\n2meff. The vertical\ndashed lines show the electron numbers which are discussed\nin the text.\nC.T(N)versus the resolution of the local density\nof states at the Fermi level\nLDOS { that characterizes only the scatterer in the\nabsence of the tip { can be resolved by scanning gate\nmicroscopy maps with the spatial precision that is limited\nby the width of the tip potential. When the tip potential\nis large, the scattering wave functions will be strongly\nin\ruenced by the tip. In the limit of weak and point-like\ntip perturbation the correlation between the LDOS and\ntheTmaps should be as high as possible. For the rest\nof the paper we concentrate on this limit. The limit of\nweak and delta-like perturbation is investigated using a\nshort range Lorentz potential d= 4 nm and small height\nU0= 0:1~2k2\nF=2me\u000b.\nWe \fnd that the resolution of the local density of states\nnear the Fermi level by the Tmaps depends on the elec-\ntron density (or N) which sets the Fermi energy within\nthe system. The dependence of TonNis displayed in\nFig. 6. For the purpose of Fig. 6, we allowed Nto take\non also non-integer values [ Nenters the expression for the\nFermi energy (7)]. Additionally, we plotted the correla-\ntion factor between LDOS and the Tmap as calculated\nwith the Lorentz potential.7\nFIG. 7. The \frst row of plots corresponds to N= 65, second\ntoN= 97, and the third to N= 104 within the system. In\nthe \frst column we show the local density of states, and in\nthe second column the Tmaps as obtained with the Lorentz\nansatz of the e\u000bective tip potential for U0= 0:1~2=k2\nF=2me\u000b\nand widthd= 4 nm. The Fermi wave vectors and the related\nkinetic energies for N= 65, 97 and 104 are: 0.032 / nm, 0.049\n/ nm, 0.052 /nm; 1.7 meV, 4 and 4.5 meV, respectively.\nThe results for N= 78 presented above [Fig. 4(b)]\ncorresponded to a plateau of T(N) in Fig. 6. In this case\nthe correlation factor rbetween the point-like weak per-\nturbation is as small as 0.35. The case of N= 92 in Fig.\n5 corresponds to a neighborhood of a dip of T[see Fig.\n6(a,c)]. This dip results from the Fano interference that\ninvolves a resonant localized state within the cavity. The\ncorrelation factor becomes as large as r= 0:86. When\njdT=dNjis large, so is the amplitude of the T(x;y) vari-\nation as obtained for the Coulomb tip with Qtip=\u0000eat\nzt= 15 nm. For maps plotted at the plateaux of Tthe\namplitude is distinctly smaller [see: Fig. 4(b)].\nWe \fnd as a general rule, that near the Fano reso-\nnances the correlation between the LDOS and Tmaps\nreachesr'0:8 (seeN= 59, 92.3, 96.7, 106.7). For an\nextra illustration the results near another resonance for\nN= 97 are displayed in Fig. 7(c,d). Near the plateaux\nofTthe correlation becomes distinctly lower { see the re-\nsults forN= 78 discussed above or the results presented\nforN= 104 in Fig. 7(e,f). Note, that for N= 104\nthere is an apparent similarity between the LDOS andtheTmap since the extrema of the maps coincide with\none another. However, the maxima of LDOS correspond\nvariably to either minimum or maximum of T{ hence\nthe low value of r=\u00000:42. The case for N= 65 of Fig.\n7(a,b) corresponds to an o\u000b-resonant conditions but for\nlargejdT=dNj. In this case the correlation between the\nLDOS and the conductance map is still signi\fcant.\nThe results presented so far were obtained for a sym-\nmetric structure. In the experiment the asymmetry is\ninevitable and the present study requires generalization\nto a non-symmetric case. The T(N) and the correlation\nfactor of LDOS to conductance maps obtained with the\nLorentz tip potential of width d= 4 nm and a height\nofU0= 0:1~2k2\n2meffare displayed in Fig. 8 for the asym-\nmetric cavity of Fig. 9. We can see that, whenever T\nfalls to zero { as a result of the Fano resonance with the\nstates localized within the cavity { one \fnds an increase\nof the correlation factor rto about 0.8 or higher [see the\nresult of Fig. 9]. On the other hand for \rat maxima\nofT{ when the system is transparent for electrons {\nor in other words { when there are no cavity-localized\nstates for a given energy { the correlation factor drops\nto distinctly lower values. As a representative examples\nwe plotted in Fig. 9 and Fig. 10 the conduction maps\nforN= 77 nadN= 92 electrons for which a dip and a\nmaximum of Tis obtained, respectively. For the dip at\nN= 77 a large correlation of r=\u00000:83 is found between\nthe LDOS and the conductance map. For the maximum\natN= 92 { as seen above for symmetric cavity of Fig.\n7(e-f) { the local extrema of Tand LDOS coincide but\nthe maxima of one of the quantities correspond to alter-\nnately a minimum or maximum of the other, hence the\nlow value ofjrj= 0:12.\nFIG. 8. Same as Fig. 6 but for an asymmetric cavity of Fig.\n9.\nWe checked that the conclusions reached in this work\nhold also when a small external magnetic \feld is applied\nperpendicular to the system. For higher when the Zee-\nman splitting becomes non-negligible as compared to the\nFermi energy the Fano resonances for opposite spin orien-\ntations do not appear at the same values of the electron\ndensities, so indication of conditions for which the LDOS\nis in a reliable manner imaged by conductance maps be-\ncomes di\u000ecult.8\nFIG. 9. The results for N= 77 electrons and the asymmet-\nric cavity obtained with a cut introduced to the right upper\ncorner of the structure. (a) The transfer probability map\nas obtained with the Lorentz tip potential, with the width\nofd= 4 nm, and U0equal to 10% of the kinetic Fermi en-\nergy. (b) The transfer probability obtained with the Lipmann-\nSchwinger equation up to the second correction.23(c) The\nlocal density of states.\nFIG. 10. Same as Fig. 9 only for N= 92 electrons.\nIV. SUMMARY AND CONCLUSIONS\nIn summary, we have studied the correlation between\nthe local density of states (LDOS) calculated from thescattering wave functions of the cavity at the Fermi level\nand the maps of the transfer probability Tin function of\nposition of a charged probe. We solved the Schroedinger-\nPoisson problem including the intrinsic screening of the\ntip potential by the electron gas. Our results indicate\nthat the e\u000bective potential generated by the Coulomb\ncharge of the tip becomes short range when the tip ap-\nproaches 2DEG at a close distance.\nWe studied both the conductance maps calculated with\nthe Coulomb potential of the tip and ansatz Lorentz po-\ntentials including the limit of point-like gentle perturba-\ntion introduced by the tip for which the correlation of\nthe LDOS to the conductance maps should be the clos-\nest. We found as a general rule that the Fermi level wave\nfunctions can be quite precisely mapped by the model\nLorentz tip potential of weak amplitude and short range\nat the resonances involving states localized within the\ncavity. The resonances induce strong backscattering of\nthe electron incoming from the Fermi level and result in\ndips of the transfer probability falling to zero. Then,\nthe correlation factor between LDOS and the electron\ntransfer probability becomes of the order of 0.8. Within\nthe regions of \rat plateaux of T, where no resonant local-\nized states within the cavity are present, the conductance\nmaps have no evident correspondence with the LDOS.\nACKNOWLEDGMENTS\nThis work was supported by National Science Centre\naccording to decision DEC-2012/05/B/ST3/03290 and\nby PL-Grid Infrastructure. Calculations were performed\nin ACK{ CYFRONET{AGH on the RackServer Zeus.\n1S. Datta, Electronic Transport in Mesoscopic Systems\n(Cambridge Univ. Press, Cambridge, 1995).\n2H. Sellier, B. Hackens, M.G. Pala, F. Martins, S. Baltazar,\nX. Wallart, L. Desplanque, V. Bayot and S. Huant, Sem.\nSci. Tech. 26, 064008 (2011).\n3D.K. Ferry, A.M. Burke, R. Akis, R. Brunner, T.E. Day,\nR. Meisels, F. Kuchar, J P Bird, and B R Bennett, Sem.\nSci. Tech. 26, 043001 (2011).\n4K. E. Aidala, R.E. Parott, T. Kramer, E.J. Heller, R.M.\nWestervelt, M.P. Hanson, and A.C. Gossard, Nature\nPhysics 3, 464 (2007).\n5M.A. Topinka, B.J. LeRoy, S.E.J. Shaw, E.J. 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Philbin\u0003\nPhysics and Astronomy Department, University of Exeter, Stocker Road, Exeter EX4 4QL, UK\n(Dated: December 19, 2013)\nIn the coherent state of the harmonic oscillator, the probability density is that of the ground state\nsubjected to an oscillation along a classical trajectory. Senitzky and others pointed out that there\nare states of the harmonic oscillator corresponding to an identical oscillatory displacement of the\nprobability density of any energy eigenstate. These generalizations of the coherent state are rarely\ndiscussed, yet they furnish an interesting set of quantum states of light that combine features of\nnumber states and coherent states. Here we give an elementary account of the quantum optics of\ngeneralized coherent states.\nI. INTRODUCTION\nThe harmonic oscillator has a special signi\fcance in\nquantum mechanics due to its role as a bridge between\nthe non-relativistic theory and quantum \feld theories.\nThe equivalence of electromagnetic waves to a collection\nof non-relativistic harmonic oscillators provides the most\ndirect route from quantum particles to quantum \felds.\nThere are many extra complications and di\u000eculties to be\nencountered in quantum electrodynamics, but at the low\nenergies relevant to quantum optics the direct connection\nto the harmonic oscillator is almost all that is required\nto begin the study of quantum light.1\nGiven that all quantum states of single-mode (one\nwave-vector, one polarization) light are states of the\nquantum harmonic oscillator, we can expect some im-\nportant quantum states of light to have been discovered\nearly in the twentieth century, without their signi\fcance\nbeing fully appreciated at the time. It is nevertheless\nimpressive that what has become known as the coherent\nstate was identi\fed by Schr odinger as early as 1926.2It\nis also remarkable that a natural generalization of the\ncoherent state to a hierarchy of similar states is rarely\ndiscussed in quantum mechanics or quantum optics. The\ncoherent state has a Gaussian probability density whose\npeak follows the sinusoidal trajectory /sin(!t+\u001e) of the\nclassical particle. This leads to the description of the co-\nherent state as a \\displaced\" ground state since its prob-\nability density di\u000bers from that of the ground state only\nthrough its oscillation in time. Senitzky showed in 19543\nthat the obvious generalization of the coherent state, in\nwhich any other energy eigenstate is displaced so that its\nprobability density oscillates according to the classical\ntrajectory, is also a solution for the harmonic oscillator.\nAfter brief attention in the 1950s, these displaced en-\nergy eigenstates were rediscovered in the 1970s.4,5Roy\nand Singh6introduced the designation \\generalized co-\nherent states\" (GCS) for Senitzky's solutions, and their\nproperties in quantum optics were discussed by Oliveira\net al.7Nieto8provides a helpful history of the litera-\nture on GCS, which should be supplemented by Marhic's\nrediscovery,5often cited9{15as the original reference for\nthese states. There have also been a few subsequent re-\ndiscoveries of GCS.16{18So-called \\squeezed\" versions ofGCS, in which the probability densities pulsate in time\nas well as following a sinusoidal trajectory, have also been\nstudied.8,13,19{21\nInterestingly, GCS have also been introduced indepen-\ndently in work on coupled \feld-matter systems, where\nthey provide a useful set of basis states for some calcu-\nlations. As our interest here is in the free oscillator, or\nsingle optical mode, we refer the reader to some relevant\nliterature and the citations therein.22{25\nGCS are as basic and elementary as the coherent state,\nyet they receive scant attention in comparison. They pro-\nvide a facinating set of quantum states of light that com-\nbines features of two stalwarts of quantum-optics courses:\nnumber states and coherent states. GCS are a natural\nand instructive accompaniment to the theory of coherent\nstates, and as such they merit entry into the textbooks of\nquantum optics. Indeed, GCS are simpler than squeezed\nstates, which are standard textbook material, and they\nare just as interesting from the theoretical point of view.\nOur aim here is to set out the basic quantum optics of\nGCS, in a manner accessible to any student with an ele-\nmentary knowledge of single-mode quantum light. Cur-\nrently the best single source for the quantum optics of\nGCS is Oliveira et al. 's analysis,7but some of the for-\nmulae there can be simpli\fed and additional visualisa-\ntions can be given. Moreover, the behaviour of GCS at a\nbeamsplitter, which we discuss below, provides perhaps\nthe most striking demonstration of how GCS combine\nproperties of coherent states and number states.\nII. GENERALIZED COHERENT STATES (GCS)\nWe consider the quantum harmonic oscillator with unit\nmass (m= 1) and Hamiltonian\n^H=1\n2^p2+1\n2!2^x2; (1)\nand throughout we set ~= 1. The following wave\nfunction is a solution of the time-dependent Schr odingerarXiv:1311.1920v2  [quant-ph]  18 Dec 20132\nequation for this oscillator:3,5\n n;\u000b(x;t) =(!=\u0019)1\n4\n2n\n2p\nn!e\u0000!\n2(x\u0000h^xi)2Hn[p!(x\u0000h^xi)]\n\u0002ei[\u0000(n+1\n2)!t+xh^pi\u00001\n2h^xih^pi]; (2)\nwhereh^xiandh^piare the time-dependent expecta-\ntion values of position and momentum in this state\n n;\u000b(x;t) =hxjn;\u000bi:\nh^xi=hn;\u000bj^xjn;\u000bi=r\n2\n!j\u000bjcos(!t\u0000\u0012); (3)\nh^pi=hn;\u000bj^pjn;\u000bi=\u0000p\n2!j\u000bjsin(!t\u0000\u0012): (4)\nIn (2),nis a non-negative integer and Hn(z) are the\nHermite polynomials; the solution (2) also depends on\nan arbitrary constant complex number \u000b=j\u000bjei\u0012that\nappears in (3) and (4). Note that (2) is separated into an\n(x;t)-dependent amplitude and an ( x;t)-dependent phase\nfactor.\nWhen\u000b= 0, the solution (2) reduces to the energy\neigenstates  n;0(x;t) labelled by n(including their time\ndependence). The set of solutions (2) thus di\u000bers from\nthe energy eigenstates in the following manner: the x-\ndependence of the amplitude is displaced by the time-\ndependent term (3) and the phase is displaced by an\namountxh^pi\u00001\n2h^xih^pi. The probability density of (2)\n(the square of the amplitude) therefore maintains ex-\nactly the shape of the static density j n;0(x;t)j2, but\nperforms an oscillation in time that follows a classical\ntrajectory. The dynamics of the states  n;\u000b(x;t) are de-\npicted in Figs. 1{3 for n= 0;1;and 2.\nForn= 0, the solution (2) is the coherent state.\nSch odinger's discovery of  0;\u000b(x;t) was motivated2by\nthe desire to \fnd a quantum state that would repro-\nduce the classical motion at macroscopic scales.26For\nthe coherent state, the Gaussian spread in xof the prob-\nability density becomes smaller as a fraction of the am-\nplitudej\u000bjp\n2=!of its oscillation as j\u000bjincreases. The\nclassical motion at macroscopic energies with trajectory\nx(t) =j\u000bjp\n2=!cos(!t\u0000\u0012) can then be viewed as an ap-\nproximation to a coherent state with large j\u000bj. This pic-\nture is validated in quantum optics by results that show\nthe coherent state to be the closest state to a classical\nplane wave allowed by quantum mechanics.1Note that\nthe same argument concerning the spread of the proba-\nbility density as a fraction of the amplitude of its oscil-\nlation can be made for any GCS, since they all feature\nthe same Gaussian factor e\u0000!(x\u0000h^xi)2=2. But the states\n n;\u000b(x;t) forn > 0 are notclassical-like states. This\nwill become clear in the quantum-optics setting of GCS,\nthrough their measurement properties and behaviour at\na beamsplitter. In fact, the most interesting aspect of\nGCS is how they combine a \\quantum number\" \u000bthat\nimparts classical features to the state with a quantum\nnumbernthat imparts quantum features.\nThe position and momentum operators for the oscilla-\ntor are related to the creation and annihilation operators\nFIG. 1. The coherent state  0;\u000b(x;t) with\u000b= 3 and!= 1.\nTop: the real and imaginary parts of the wave function plotted\nfor one period 0 \u0014t\u00142\u0019=!. Note that the wave function\nacquires an overall sign change after one period 2 \u0019=!; two\nperiods are required for the wave function to repeat because\nof the zero-point energy term !t=2 in the phase. Also shown is\nthe classical trajectory with the same amplitude of oscillation.\nBottom: the probability density.\nby\n^x=1p\n2!\u0000\n^a+ ^ay\u0001\n;^p=\u0000ir!\n2\u0000\n^a\u0000^ay\u0001\n:(5)\nHere we are using the Schr odinger picture and have in-\ncluded in (2) the time dependence of the energy eigen-\nstatesjn;0i, so that we have\n^ajn;0i=pne\u0000i!tjn\u00001;0i; (6)\n^ayjn;0i=p\nn+ 1ei!tjn+ 1;0i: (7)\nThe coherent state j0;\u000biis an eigenstate of ^ a(=p\n!=2(x+!\u00001d=dx ) in the coordinate representation\n(2)):\n^aj0;\u000bi=\u000be\u0000i!tj0;\u000bi; (8)\nan equation usually written for t= 0. The transition\nto single-mode quantum optics1is achieved through re-\nplacement of the position ^ xby the electric-\feld operator\n^E(x) =1p\n2!\u0010\n^aeikx+\u0019=2+ ^aye\u0000ikx\u0000\u0019=2\u0011\n; k =!=c;\n(9)3\nFIG. 2. The \frst generalized coherent state  1;\u000b(x;t). Pa-\nrameters are the same as in Fig. 1.\nwhich is time-independent in the Schr odinger picture.\nThe GCSjn;\u000bithen have electric-\feld probability dis-\ntributions that are equal to the probability distributions\nfor the position of the oscillator but with !treplaced\nby!t\u0000kx\u0000\u0019=2. In Fig. 4 we plot the electric-\feld\nprobability distributions for the \frst three GCS, which\ncorrespond to the position probability distributions in\nFigs. 1{3. The electric \feld of the coherent state j0;\u000bi\nhas the familiar appearance of a plane wave with a noise\nband. For general GCS jn;\u000bi, the electric \feld shows\nn+ 1 noise bands separated by nnodes. In the photon\nnumber statesjn;0ithese noise bands do not oscillate,\ni.e. they show no dependence on the phase kx\u0000!t+\u0019=2.\nMany of the properties of the number states jn;0iare\npreserved by GCS jn;\u000biwith\u000b>0, as we shall see. The\nnumber of nodes in the electric-\feld probability distribu-\ntion is thus the important signature of these properties,\nnot the number of photons in the state; the latter is un-\ncertain for GCS and can be of arbitrarily large average\nvalue for any n.\nIII. GCS AS A BASIS\nUsing the representation (2) one can verify that the or-\nthonormality relation hn;0jm;0i=\u000enmfor number states\nis preserved for GCS with the same complex amplitude\nFIG. 3. The second generalized coherent state  2;\u000b(x;t). Pa-\nrameters are the same as in Fig. 1.\n\u000b:\nhn;\u000bjm;\u000bi=\u000enm: (10)\nFor givenn, however, GCS with di\u000berent complex ampli-\ntudes are not orthogonal, as is familiar for the coherent\nstatej0;\u000bi; the general relation is\nhn;\fjn;\u000bi=e\u0000(j\u000bj2+j\fj2\u00002\u000b\f\u0003)=2Ln\u0000\nj\u000b\u0000\fj2\u0001\n;(11)\nwhereLn(z) are the Laguerre polynomials. The overlap\nbetween two GCS with equal nthus becomes exponen-\ntially small as the di\u000berence j\u000b\u0000\fjin their amplitudes\nincreases, just as for coherent states.\nLadder operators exist for the quantum number nof\njn;\u000biand are in fact given by displaced versions of the\nnumber-state ladder operators ^ aand ^ay. De\fning\n^a\u000b= ^a\u0000\u000be\u0000i!t=)\u0002\n^a\u000b;^ay\n\u000b\u0003\n= 1; (12)\none can show using the representation (2) that\n^a\u000bjn;\u000bi=pne\u0000i!tjn\u00001;\u000bi; (13)\n^ay\n\u000bjn;\u000bi=p\nn+ 1ei!tjn+ 1;\u000bi; (14)\nwhich are the generalizations of (6) and (7). The dis-\nplacement (12) of ^ ais e\u000bected by the displacement\noperator1\n^D\u000b= exp\u0000\n\u000be\u0000i!t^ay\u0000\u000b\u0003ei!t^a\u0001\n; (15)\n^Dy\n\u0000\u000b^a^D\u0000\u000b= ^a\u0000\u000be\u0000i!t= ^a\u000b; (16)4\n00.10.20.30.40.5\n0p2p-100-50050100\n00.10.20.30.4\n0p2p-100-50050100\n00.10.20.3\n0p2p-100-50050100\nFIG. 4. Electric-\feld probability distributions in jn;\u000bifor\nn= 0;1 and 2, with \u000b= 3 and!=c= 1. Darker colours\nrepresent higher probabilities.and GCS are displaced number states:\njn;\u000bi=^D\u000bjn;0i; (17)\nas is veri\fed by showing that (17) and (16) imply (13)\nand (14).\nIt should not be surprising that jn;\u000bifor \fxed\u000bform\na complete basis for single-mode states (just like number\nstates), whereasjn;\u000bifor \fxednform an over-complete\nbasis (just like coherent states). The relevant relations\nare\n1X\nn=0jn;\u000bihn;\u000bj=I; (18)\n1\n\u0019Z\nd2\u000bjn;\u000bihn;\u000bj=I: (19)\nThe completeness relation (18) follows immediately from\n(17), but the over-completeness relation (19) is perhaps\nmost easily veri\fed using the number state expansion of\njn;\u000bibelow (equation (26)).\nIV. EXPECTATION VALUES AND\nUNCERTAINTY RELATIONS\nThe results of the previous section give the electric-\n\feld expectation value and uncertainty in GCS:\nhn;\u000bj^E(x)jn;\u000bi=r\n2\n!j\u000bjcos\u0010\nkx\u0000!t+\u0012+\u0019\n2\u0011\n;\n(20)\n(\u0001E(x;t))2=2n+ 1\n2!; (21)\nwhere\u000b=j\u000bjei\u0012as before. Note from (21) that the\nuncertainty in the electric \feld is identical to that in\nthe related number state (it is independent of \u000b). Note\nthat the expectation value (20) and uncertainty (21) of\nthe electric \feld does not convey the nodal structure of\nthe electric-\feld probability distributions of GCS, as de-\npicted in Fig. 4. The quadrature operators ^Xand ^Y,\nde\fned by\n^E(x) =2p!h\n^Xcos(kx+\u0019=2) + ^Ysin(kx+\u0019=2)i\n;\n(22)\nhave uncertainties in GCS that are also identical to those\nin the related number state, as is easily veri\fed.\nFor the photon number operator ^N= ^ay^awe obtain\nhn;\u000bj^Njn;\u000bi=n+j\u000bj2; (23)\n(\u0001N)2=j\u000bj2=hn;\u000bj^Njn;\u000bi\u0000n; (24)\n\u0001N\nhn;\u000bj^Njn;\u000bi=j\u000bj\nn+j\u000bj2: (25)\nContrary to the electric \feld, we see from (24) that the\nuncertainty in photon number is independent of nand is5\nthus identical to that in a coherent state with the same\ncomplex amplitude \u000b. The relation between \u0001 Nand the\nexpectation value of ^Nis, however, di\u000berent from the\ncoherent state for n>0, as is shown by (25).\nEquation (24) shows the important property of coher-\nent states ( n= 0) that (\u0001 N)2is equal toh0;\u000bj^Nj0;\u000bi.\nThis property that the variance is equal to average value\nis a characteristic of the Poisson distribution, and the\ncoherent state has indeed a Poissonian photon-number\ndistribution1(the photon-number distributions for gen-\neral GCS are given in the next section). Equation (24)\nalso shows that (\u0001 N)2is less thanhn;\u000bj^Njn;\u000bifor\nn > 0, which means that GCS for n > 0 exhibit sub-\nPoissonian \ructuations,1i.e the \ructuations in photon\nnumber in measurements of the states are less than those\nof a Poisson distribution and thus less than those of\na coherent state. The photon-number probability dis-\ntributions for GCS are given below and indeed di\u000ber\ngreatly from the Poisson distribution when n>0. Sub-\nPoissonian \ructuations are a signature of nonclassical\nlight.1It is not surprising that we \fnd by this criterion\nthat GCS for n > 0 are nonclassical; their nonclassical\nnature was already clear in Fig. 4. Note from (25) that\nfor all GCS the fractional uncertainty in photon number\ngoes to zero as the amplitude j\u000bjgoes to in\fnity; the\n\ructuations approach Poissonian behaviour in this limit\nas we then have n+j\u000bj2\u0019j\u000bj2.\nV. NUMBER-STATE EXPANSION AND\nPHOTON STATISTICS\nGCS can be generated from the coherent state by us-\ning the creation operator ^ ay\n\u000b(see (14)), or alternatively\nthey can be generated from number states by using the\ndisplacement operator ^D\u000b(see (17)). Either procedure\nallows the number-state expansion of jn;\u000bito be calcu-\nlated. It not a trivial matter to express the result in\nterms of special functions but one can verify that the\nnumber-state expansion is\njn;\u000bi=1p\nn!e\u0000j\u000bj2=2\n\u00021X\nk=0(\u00001)n+kp\nk!(\u000b\u0003)n\u0000kLn\u0000k\nk\u0000\nj\u000bj2\u0001\njk;0i;\n(26)\nwhereLm\nk(z) are generalized Laguerre polynomials.\nFrom (26) we see that the probability Pk(n;\u000b) of \fnding\nkphotons in the GCS jn;\u000biis given by\nPk(n;\u000b) =k!\nn!e\u0000j\u000bj2j\u000bj2(n\u0000k)\u0002\nLn\u0000k\nk\u0000\nj\u000bj2\u0001\u00032:(27)\nFor the coherent state n= 0, equation (27) is of course a\nPoisson distribution and Pk(1;\u000b) can also be written ina simple form:\nPk(0;\u000b) =e\u0000j\u000bj2j\u000bj2k\nk!; (28)\nPk(1;\u000b) =e\u0000j\u000bj2j\u000bj2(k\u00001)(j\u000bj2\u0000k)2\nk!: (29)\n50100150200k0.010.020.030.04PkH0,aL\n50100150200k0.0050.0100.0150.0200.0250.030PkH1,aL\n50100150200k0.0050.0100.0150.0200.025PkH2,aL\nFIG. 5. Photon probability distributions in jn;\u000biforn= 0;1\nand 2, withj\u000bj= 10.\nThe distributions Pk(n;\u000b) are plotted in Fig. 5 for\nn= 0;1 and 2. The most striking feature of these\ndistributions is a similarity of their general shapes to\nthose of the electric-\feld probability densities (and thus\nto the probability densities of the harmonic-oscillator en-\nergy eigenstates). But note that that there are in general6\nno nodes in the distributions Pk(n;\u000b), just local minima,\nand the relative sizes of the local maxima in Pk(n;\u000b)\nare also not the same as in the electric-\feld probabil-\nity densities. Oliveira et al.7have given a phase-space\ninterpretation of the oscillations in the photon-number\ndistributions.\nVI. SECOND-ORDER COHERENCE\nOptical coherence is an extremely complicated\nsubject27but basic measures of coherence are not dif-\n\fcult to de\fne and calculate. First-order coherence is\nessentially a measure of correlations of the electric \feld\nin the beam, while second-order coherence is a measure\nof intensity correlations. The correlation function of the\nelectric \feld, the average hE((x1;t1)E((x2;t2)iof the\nproduct of the electric \feld at two space-time points, be-\ncomes in quantum optics an expectation value quadratic\nin the annihilation and creation operators (see (9)). Sim-\nilarly, in quantum optics the intensity correlation func-\ntionhI((x1;t1)I((x2;t2)itakes the form of an expecta-\ntion value quartic in ^ aand ^ay. The ordering of ^ aand\n^ayin these expectation values is chosen to correspond\nto expressions for measured intensities.1When the re-\nsulting expectation values are normalized, so that they\ntake the value 1 when all space-time points coincide,\nthey form the quantum measures of \frst- and second-\norder coherence, respectively. The resulting degree of\n\frst-order coherence, denoted by g(1)(x1;t1;x2;t2), sim-\npli\fes greatly for a plane parallel single-mode beam such\nas a GCS. In fact, for such beams it is always the case\nthatjg(1)(x1;t1;x2;t2)j= 1, and the beam is said to be\n\frst-order coherent.1The second-order coherence func-\ntion, denoted g(2)(x1;t1;x2;t2), takes the following sim-\nple form for a plane parallel single-mode beam1\ng(2)(x1;t1;x2;t2) =h^ay^ay^a^ai\nh^ay^ai2: (30)\nThis expression is easily rewritten in terms of the pho-\nton number operator and can then be evaluated for GCS\nusing (23) and (24), as follows:\ng(2)(x1;t1;x2;t2) =h^N2i\u0000h ^Ni\nh^Ni2= 1 +(\u0001N)2\u0000h^Ni\nh^Ni2\n= 1\u0000n\n(n+j\u000bj2)2(31)\nA beam is said to be second-order coherent if g(2)= 1\nin addition tojg(1)j= 1, which for GCS occurs only for\nthe coherent state n= 0. Forn>0, GCS have g(2)<1,\nwhich is another signature of nonclassical light.1In the\nlimitn!1 , or in the limit j\u000bj!1 , GCS approach\nsecond-order coherence g(2)!1.\nFor givenn,g(2)for GCS is minimized by the number\nstates\u000b= 0. Interestingly, for given \u000b>0 the minimum\ning(2)is at the value of nclosest toj\u000bj2; from (23) wesee that this corresponds to an equal contribution from\nnand\u000bto the average photon number in the state. If\nj\u000bj2is an integer this minimum in g(2)for \fxed\u000b>0 is\ng(2)= 1\u00001=(4j\u000bj2).\nVII. BEHAVIOUR AT A BEAMSPLITTER\nThe results so far have shown that GCS combine fea-\ntures of coherent states and number states. This is per-\nhaps most vividly illustrated by the behaviour of the GCS\njn;\u000biat a beamsplitter, where we will \fnd that the \\ \u000b\"\npart of the state behaves like a coherent state and the \\ n\"\npart behaves like a number state. We consider a symmet-\nric beam splitter with input arms 1 and 2, and output\narms 3 and 4 (see Fig. 6). The beamsplitter input-output\nrelations are given by1\n^a3=R^a1+T^a2;^a4=T^a1+R^a2; (32)\n^a1=R\u0003^a3+T\u0003^a4;^a2=T\u0003^a3+R\u0003^a4; (33)\njRj2+jTj2= 1; RT\u0003+TR\u0003= 0; (34)\nwhereRandTare the complex re\rection and trans-\nmission coe\u000ecients. These are the relations that would\nbe satis\fed classically by the complex amplitudes of the\nelectric \feld in the arms of the beamsplitter.1\nˆa4ˆa3ˆa2ˆa1\nFIG. 6. Beamsplitter with input and output arms showing\nthe annihilation operators for the corresponding modes.\nConsider a GCS entering the beam splitter in arm 1,\nwith arm 2 in the vacuum state. The input state is thus\njn;\u000bi1j0;0i2, which can be written\njn;\u000bi1j0;0i2=^D1\u000b1p\nn!\u0010\ne\u0000i!t^ay\n1\u0011n\nj0;0i1j0;0i2;(35)\n^D1\u000b= exp\u0010\n\u000be\u0000i!t^ay\n1\u0000\u000b\u0003ei!t^a1\u0011\n: (36)\nIn (35) the state jn;\u000bi1is built up from the number\nstatejn;0i1, created using nfactors of ^ay\n1, followed by\nthe displacement operator ^D1\u000bfor arm 1, which creates\nthe GCS from the number state as in (17). Recall that we\ninclude the time dependence in the number states and in\nGCS. By substitution for ^ ay\n1using (33) the number state\njn;0i1is expressed in terms of the output modes in the7\nstandard manner:1\njn;0i1=1p\nn!\u0010\ne\u0000i!t^ay\n1\u0011n\nj0;0i1\n=nX\nm=0\u0012n!\nm!(n\u0000m)!\u00131=2\nRmTn\u0000mjm;0i3jn\u0000m;0i4:\n(37)\nIt is also a standard result1that the displacement opera-\ntor (36) for arm 1 becomes a product of displacement\noperators in the output arms when (33) is employed:\n^D1\u000b=^D3R\u000b^D4T\u000b. Using this last result and (37) in\n(35) gives\njn;\u000bi1j0;0i2=nX\nm=0\u0012n!\nm!(n\u0000m)!\u00131=2\nRmTn\u0000m\n\u0002jm;R\u000bi3jn\u0000m;T\u000bi4: (38)\nFor\u000b= 0, the right-hand side of (38) gives the familiar\nsuperposition of number states in the output arms of the\nbeamsplitter, produced when a number-state is sent into\none arm. For n= 0, the result (38) is the familiar product\nstate of independent coherent states in the output arms,\nproduced by a coherent-state input. For general GCS\nwith nonzero \u000bandn, we have in (38) the remarkable\nresult that the quantum number nof GCS behaves at a\nbeamsplitter exactly like a number state with nphotons,\nwhile the \\quantum number\" \u000bbehaves exactly like a\ncoherent state with complex amplitude \u000b. The quantum\nnumbernof GCS in no way measures the number of\nphotons in the state; instead it measures the number of\nnodes in the electric-\feld probability distribution, and\nthe number of local minima in the photon distribution.\nYet thenof GCS produces the same entanglement at\na beamsplitter as that produced by nphotons. Each\nelectric-\feld node in arm 1 has a probability amplitude\nRto be re\rected into arm 3 and a probability amplitude\nTto be transmitted into arm 4, just like a photon. For\nthe case of n= 1 we have from (38)\nj1;\u000bi1j0;0i2=Rj1;R\u000bi3j0;T\u000bi4+Tj0;R\u000bi3j1;T\u000bi4:\n(39)\nThus, if the GCS in the middle plot of Fig. 4 is sent\nthrough a beam splitter and both output arms are mea-\nsured, then one output arm will contain the same state as\nthe input, but with reduced amplitude, while the other\noutput arm will contain a coherent state. Results like\n(39) are particularly interesting because the input GCS in\nthis relation can have arbitrarily large (average) energy,\nyet it produces the same entanglement at the beamsplit-\nter as a single photon.\nVIII. GENERATION OF GCS\nReturning to the case of the quantum harmonic os-\ncillator, let us recall the exact solution for dynamicallygenerating Schr odinger's coherent state from an initial\nground state.28The coherent state is generated from the\nground state by acting on it with the displacement oper-\nator (equation (17) with n= 0). The time-development\noperator ^T(t;t0) takes the statej (t0)iat timet0to the\nstatej (t)iat timet:j (t)i=^T(t;t0)j (t0)i. It fol-\nlows that to generate the coherent state from the ground\nstate, the time-development operator must be equal to\nthe displacement operator, up to phase factors. If the\noscillator is subjected to a classical external force f(t),\nwhich corresponds to adding a term f(t)^xto the Hamil-\ntonian (1), then the dynamics can be solved exactly and\nthe resulting time-development operator is28\n^T(t;t0) =ei\f(t;t0)exp\u0002\n\u0010(t;t0)^aye\u0000i!t\u0000\u0010\u0003(t;t0)^aei!t\u0003\n\u0002e\u0000i^H0(t\u0000t0)=~; (40)\nwhere ^H0is the Hamiltonian without the external driving\ntermf(t)^x(i.e. the Hamiltonian (1)), \u0010(t;t0) is given by\n\u0010(t;t0) =\u0000ip\n2!Zt\nt0dt0f(t0)ei!t0; (41)\nand the c-number real phase \f(t;t0) is given by\n\f(t;t0) =1\n2!Zt\nt0dt0Zt0\nt0dt00f(t0)f(t00) sin[!(t0\u0000t00)]:\n(42)\nIf the initial state j (t0)iis an energy eigenstate,\nthen (40) shows that the driving force gives a time-\ndevelopment operator that is a displacement operator\n(15), up to a c-number phase factor. Thus the oscilla-\ntor is driven into a coherent state by a classical external\nforce ifj (t0)iis the ground state, an important and well-\nknown result in quantum mechanics.28From (17) we also\n\fnd that a classical external force will drive the oscillator\ninto the GCSjn;\u000bi, with\u000bdetermined by the external\nforce and the interaction time, if the initial state is the\nenergy eigenstate jn;0i.\nThe foregoing results apply also to single-mode quan-\ntum light interacting with a classical external current j,\nsince the coupling term in the Hamiltonian is j\u0001^A, where\n^Ais the vector potential operator.28There are di\u000ber-\nences in the formulae due to the spatial dependence of the\nquantum \felds and the fact that the current is coupled\nto^Arather than ^E=\u0000@t^A, but the calculation goes\nthrough as before28and thus a GCS can be generated by\nacting on a number state with a classical external cur-\nrent. This fact was noted by Oliveira et al. ,7who point\nout that the preparation of number states can be carried\nout in a cavity, which can then be driven by a current\nthat is classical up to negligible quantum \ructuations. It\nappears however that quantum-optical GCS have yet to\nbe generated experimentally, though GCS of a quantum\noscillator consisting of a trapped ion have very recently\nbeen reported.298\nIX. CONCLUSIONS\nWe have given a basic description of generalized coher-\nent states (GCS) in quantum optics. These states are an\ninteresting and natural accompaniment to presentations\nof number states and coherent states. GCS combine fea-\ntures of number states and coherent states in a nontriv-\nial manner. The electric-\feld probability distributions ofGCS show the phase oscillation of coherent states but\nalso contain nodes that behave like single photons. This\nallows GCS to have macroscopic energies while still hav-\ning some of the properties of few-photon number states.\nACKNOWLEDGMENTS\nI thank S. Horsley for helpful discussions. I am also\nindebted to E. K. Irish for information on literature.\n\u0003t.g.philbin@exeter.ac.uk\n1R. Loudon, The Quantum Theory of Light , 3rd. ed. (Ox-\nford University Press, 2000).\n2E. Schr odinger, \\Der stetige Ubergang von der Mikro-\nzur Makromechanik,\" Naturwissenschaften 14, 664{666\n(1926).\n3I. R. Senitzky, \\Harmonic oscillator wave functions,\" Phys\nRev.95, 1115{1116 (1954).\n4M. Boiteux and A. Levelut, \\Semicoherent states,\" J.\nMath. Phys. A: Math., Nucl. Gen. 6, 589{596 (1973).\n5M. F. Marhic, \\Oscillating Hermite-Gaussian wave func-\ntions of the harmonic oscillator,\" Lett. Nuovo Cim. 22,\n376{378 (1978).\n6S. M. Roy and V. Singh, \\Generalized coherent states and\nthe uncertainty principle,\" Phys. Rev. D 25, 3413{3416\n(1982).\n7F. A. M. de Oliveira, M. S. Kim, P. L. Knight, and V.\nBu\u0015 zek, \\Properties of displaced number states,\" Phys.\nRev. A 41, 2645{2652 (1990).\n8M. M. Nieto, \\Displaced and squeezed number states,\"\nPhys. Lett. A 229, 135{143 (1997).\n9I. Fujiwara and K. Miyoshi, \\Pulsating states for quan-\ntal harmonic oscillator,\" Prog. Theor. Phys. 64, 715{718\n(1980).\n10M. E. Marhic and P. Kumar, \\Squeezed states with a ther-\nmal photon distribution,\" Opt. Comm. 76, 143{146 (1990).\n11M. J. Engle\feld, \\A solution of a Fokker-Planck equation,\"\nPhysica A 167, 877{886 (1990).\n12C. W. Wong, \\Nonspreading wave packets,\" Am. J. Phys.\n64, 792{799 (1996).\n13S. I. Kryuchkov, S. K. Suslov, and J. M. Vega-Guzm\u0013 an,\n\\The minimum-uncertainty squeezed states for atoms and\nphotons in a cavity,\" J. Phys. B: At. Mol. Opt. Phys. 46,\n104007 (2013).\n14A. Mahalov and S. K. Suslov, \\Wigner function ap-\nproach to oscillating solutions of the 1D-quintic nonlinear\nSchr odinger equation,\" J. Nonlinear Opt. Phys. Mater. 22,\n1350013 (2013).\n15A. Mahalov, E. Suazo, and S. K. Suslov, \\Spiral Laser\nbeams in inhomogeneous media,\" Opt. Lett. 38, 2763{2766\n(2013).\n16C. C. Yan, \\Soliton like solutions of the Schr odinger equa-\ntion for a simple harmonic oscillator,\" Am. J. Phys. 62,\n147{151 (1994).\n17R. M. L\u0013 opez, S. K. Suslov, and J. M. Vega-Guzm\u0013 an, \\On\na hidden symmetry of quantum harmonic oscillators,\" J.Di\u000b. Eqns. Appl. 19, 543{554 (2013).\n18It may be of interest to note how the present author stum-\nbled upon the generalized coherent states (equation (2)\nabove). The Bohmian trajectories of the quantum particle\nin a coherent state are described in P. R. Holland, The\nQuantum Theory of Motion (Cambridge University Press,\n1993), Sec. 4.9. One can mathematically set up the fol-\nlowing problem: are there other quantum states in which\nthe Bohmian trajectories are the same as in the coherent\nstate? The solution to this problem gives the generalized\ncoherent states (2).\n19J. Plebanski, \\Wave functions of a harmonic oscillator,\"\nPhys. Rev. 101, 1825{1826 (1956).\n20M. V. Satyanarayana, \\Generalized coherent states and\ngeneralized squeezed coherent states,\" Phys. Rev. D 32,\n400{404 (1985).\n21K. B. M\u001cller, T. G. J\u001crgehsen, and J. P. Dahl, \\Displaced\nsqueezed number states: Position space representation, in-\nner product, and some applications,\" Phys. Rev. A 54,\n5378{5385 (1996).\n22M. D. Crisp, \\Application of the displaced oscillator basis\nin quantum optics,\" Phys. Rev. A 46, 4138{4149 (1992).\n23E. K. Irish, J. Gea-Banacloche, I. Martin, and K. C.\nSchwab, \\Dynamics of a two-level system coupled to a\nhigh-frequency quantum oscillator,\" Phys. Rev. B 72,\n195410 (2005).\n24E. K. Irish, \\Generalized rotating-wave approximation for\narbitrarily large coupling,\" Phys. Rev. Lett. 99, 173601\n(2007).\n25D. P. S. McCutcheon and A. Nazir, \\Quantum dot Rabi ro-\ntations beyond the weak exciton-phonon coupling regime,\"\nNew J. Phys. 12, 113042 (2010).\n26Contrary to the impression given to many students by\nvague references to a \\correspondence principle\", most\nquantum states never look classical, even in the limit of\nin\fnite energy and in\fnite quantum numbers.\n27L. Mandel and E. Wolf, Optical Coherence and Quantum\nOptics (Cambridge University Press, 1995).\n28E. Merzbacher, Quantum Mechanics 3rd. ed. (Wiley,\n1998), Secs. 14.6 and 23.4.\n29F. Ziesel, T. Ruster, A. Walther, H. Kaufmann, S.\nDawkins, K. Singer, F. Schmidt-Kaler, and U. G.\nPoschinger, \\Experimental creation and analysis of dis-\nplaced number states,\" J. Phys. B: At. Mol. Opt. Phys.\n46, 104008 (2013)."    },    {        "title": "1311.3910v1.Equation_of_state_in_the_generalized_density_scaling_regime_studied_from_ambient_to_ultra_high_pressure_conditions.pdf",        "content": "1 \n Equation of state in the generalized density scalin g regime studied \nfrom ambient to ultra-high pressure conditions \nA. Grzybowski,* K. Koperwas, and M. Paluch \nInstitute of Physics, University of Silesia, Uniwer sytecka 4, 40-007 Katowice, Poland \n \n In this paper, based on the effective intermolecul ar potential with well separated density and \nconfiguration contributions and the definition of t he isothermal bulk modulus, we derive two similar e quations of \nstate dedicated to describe volumetric data of supe rcooled liquids studied in the extremely wide press ure range \nrelated to the extremely wide density range. Both t he equations comply with the generalized density sc aling law \nof molecular dynamics versus ()T h /ρ  at different densities ρ and temperatures T , where the scaling exponent \ncan be in general only a density function () ρ ργ ln /ln dhd=  as recently argued by the theory of isomorphs. \nWe successfully verify these equations of state by using data obtained from molecular dynamics simulat ions of \nthe Kob-Andersen binary Lennard-Jones liquid. As a very important result, we find that the one-paramet er \ndensity function h(ρ) analytically formulated in the case of this proto typical model of supercooled liquid, which \nimplies the one-parameter density function γ(ρ), is able to scale the structural relaxation times  with the value of \nthis function parameter determined by fitting the v olumetric simulation data to the equations of state . We also \nshow that these equations of state properly describ e the pressure dependences of the isothermal bulk m odulus \nand the configurational isothermal bulk modulus in the extremely wide pressure range investigated by t he \ncomputer simulations. Moreover, we discuss the poss ible forms of the density functions h(ρ) and γ(ρ) for real \nglass formers, which are suggested to be different from those valid for the model of supercooled liqui d based on \nthe Lennard-Jones intermolecular potential.  \n* Corresponding author’s email: andrzej.grzybowski@ us.edu.pl \n \nI.  INTRODUCTION \nIn the last decade, our understanding of the glass transition and related phenomena has been \nconsiderably enhanced by making the observation tha t values of dynamic quantities such as structural r elaxation \ntime τ and viscosity η collected for many systems near the glass transiti on in various conditions of temperature T \nand density ()pT,ρ  at ambient and elevated pressure p can be plotted on one master curve according to a \npower law density scaling function ( )T f /γρ  with only one parameter γ, which is characteristic of a given \nmaterial independent of thermodynamic conditions.1,2,3,4,5,6,7,8,9,10,11,12,13,14,15  This scaling rule established \nphenomenologically has been also subjected to inten sive theoretical and simulation studies,16,17,18,19,20,21,22,23  2 \n which have led to a rather general conclusion that an effective short-range potential, () ()t IPL eff Ar Ur U − =  \n(where its repulsive part is given by an inverse po wer law (IPL) term, () ()IPL m\nIPL r r U / 4σε= with \nγ3≈IPL m , and At is some small constant or linear attractive backgr ound), underlies the T/γρ -scaling of the \nmolecular dynamics near the glass transition. As a reference for the systems that complies with the po wer law \ndensity scaling rule at least to a good approximati on, Dyre’s group introduced a concept of strongly correlating \nliquids , which are defined by a strong linear correlation between isochoric equilibrium fluctuations of virial  W \nand potential energy U with the correlation slope that corresponds to the  scaling exponent γ, assuming that the \nWU  correlation is strong if the correlation coefficie nt 9 . 0>R .18,19,20      \nThe simple and tempting theoretical grounds for the  power law density scaling caused that this pattern  \nof scaling has become a prominent trend in the inve stigations of glass formers. Nevertheless, already in the early \nreports on this matter,4,17,24,25  a general form of the density scaling function has  been considered, according to \nwhich, for instance, the structural relaxation time  can obey the following scaling equation \n ( )()=Thf Tρρτ,  (1) \nwhere the only density dependent function ()ρh  is not limited to the power function γρ. Thorough analyses of \nmolecular dynamics (MD) simulation data confirmed 19,22,23,26  that the scaling exponent γis not constant and can \ndepend at least on density even in simple models ba sed on the Lennard-Jones (LJ) potential such as the  Kob-\nAndersen binary Lennard-Jones (KABLJ) mixtures.27  It has been realized that the pure power law densi ty scaling is \nvalid only for potentials with a single IPL term, h owever, a hidden scale invariance of molecular dyna mics \ncharacterized by more complex potentials can be rev ealed by using reduced units 23,28,29,30  (e.g. the structural \nrelaxation time in the reduced units, 2/ 1 3 / 1 ~T τρ τ= , for τ, ρ, T in the usual LJ units). For instance, in the KABLJ  \nliquid which is a double IPL model, the reduced uni ts enable to scale the structural relaxation times versus T/γρ\n \nwith const =γ but only in the sufficiently narrow density range ( e.g., the T/γρ -scaling of ~τ in the KABLJ \nmodel occurs if the particle number density ρ ( VN/≡ , where N is the particle number and V is the system \nvolume) ranges from 1.2 to 1.6, but it is not possi ble if ρ varies from 1.2 to 2.0).31   \nThe general pattern of the density scaling (Eq. (1) ) can be better understood within the theory of iso morphs \nrecently formulated by Dyre’s group.29,32,33  According to this theory (Appendix A in Ref. 29), a system is strongly 3 \n correlating  if and only if it has isomorphs to a good approxim ation in its phase diagram, which are curves of iso morphic \nstate points in the following sense: two state poin ts ( )1 1,ρT  and ( )2 2,ρT  are isomorphic if all pairs of their physically \nrelevant microconfigurations ( )) 1 ( ) 1 (\n1 ,,Nr rK  and ( )) 2 ( ) 2 (\n1 ,,Nr rK  characterized by identical reduced coordinates \n) 2 (~ ) 1 (~\ni i r r =  (where i i r r3 / 1 ~ρ= ) have proportional configurational NVT Boltzmann fa ctors, \n( ) [ ] ( ) [ ]2) 2 ( ) 2 (\n1 12 1) 1 ( ) 1 (\n1 / ,, exp / ,, exp Tk U C Tk UB N B N r r r r K K − = − , where the constant 12 C depends only on the \nstate points ( )1 1,ρT  and ( )2 2,ρT , not on the microscopic configurations. An undoubte d achievement of this theory is \nan incontrovertible evidence 34  for the only density dependent scaling exponent ()ργ , which can be derived in the case \nof strongly correlating systems  from the function ()ρh  in Eq. (1) by logarithmic differentiating with res pect to ρln  \n   ργln ln \ndhd=  (2) \nThus, the general density scaling idea given by Eq.  (1) can be considered as a consequence of the gene ralization \nabout the power density function ()γρ ρ= h with const =γ , which assumes that the scaling exponent γ can \ndepend on density ρ based on Eq. (2). It should be noted that the scal ing exponent argued to be dependent on \ndensity well corresponds to the density dependent s lope of the WU correlation established from MD simu lations \nin the KABLJ model. This result of simulation exper iments can be well grounded and generalized within the \ntheory of isomorphs, which enables to prove 34  that strongly correlating systems  in general obey the \nconfigurational Grüneisen equation of state 35,36,37   \n () ()ρ ργ Φ+ = U W  (3) \nAlthough a general discussion on thermodynamics of condensed matter with strong pressure-energy correl ations \nhas been already done, involving Eq. (3) 34  and its version 32  for the double IPL potential model, until recently , no \nequation of state (EOS) has been formulated in the form convenient to describe PVT data of the strongly \ncorrelating systems  considered in the general density scaling case cha racterized by the density scaling \ncriterion 17,31,34  \n ()const Th=ρ (4) \nwhich should be validated at const =~τ  if we analyze the density scaling of structural re laxation times in \nterms of Eq. (1) where τ should be also replaced with τ~.31,34   4 \n  In this paper, we derive an EOS, which can be cons idered as an approximate PVT representation of the \nconfigurational Grüneisen EOS (Eq. (3)) for systems  that meet the general density scaling criterion (E q. (4)). We \ntest a version of the EOS for the LJ potential usin g PVT data from our MD simulations in the KABLJ mod el and \ndiscuss a possible form of the EOS for real glass f ormers measured in the extremely wide pressure rang e that \ncorresponds to the wide density range, in which the  density scaling law given by Eq. (1) with the dens ity \nfunction h approximated by ()γρ ρ= h with const =γ is not sufficient to scale molecular dynamics of th e \nmaterials, but it is possible to find another densi ty dependent scaling function ()ρh  that enables the scaling in \nterms of Eq. (1) and implies the density dependent scaling exponent ()ργ  based on Eq. (2).  \n \nII. EQUATION OF STATE FOR PVT DATA OF STRONGLY CORRELATING LIQUIDS  \n A few years ago, we suggested an equation of state  derived in the power law density scaling regime fi rst \nin its isothermal form,38,39  which has been later generalized 40  to describe PVT in a convenient way not limited \nonly to any constant temperature condition,  \n \n\n\n\n−\n\n\n+ = 1) (\n00\n0EOS \nEOS conf conf \nT conf conf p Bp pγ\nρρ\nγ   (5) \nwhere 2\n0 2 0 1 0 01\n0 ) ( ) ( ) ,( TTA TTA A pTconf − + − + = =−υ ρ  and the configurational isothermal bulk modulus \n)) ( exp( ) ( ) (0 2 0 00TT b p B p Bconf conf conf \nTconf conf \nT − − =  at a reference configurational pressure, \n( )MpT RT p pT p pconf conf /, ),(0 0 0 0 0 0 0 ρ − = = , which is established at the reference temperature  T0 and \npressure p0 using the system molar mass  M and the gas constant R. We determined 26,38,39,40  the physical meaning \nof the exponent EOS γ  by arguing the relation 3 /IPL EOS m≅ γ , where IPL m  is the exponent of the repulsive \nIPL term of the effective short-range intermolecula r potential, () ()tm\neff A r r UIPL − = / 4σε , which is \nsuggested to be responsible for the power law densi ty scaling of molecular dynamics near the glass tra nsition.  \nBased on the definition of the isothermal bulk modu lus, ( )T T p B ∂ ∂= ρln , we also arrived at \nanother EOS, finding its isothermal 39  and generalized 40  versions which comply with the linear pressure \ndependence, ( )0 0)( )( pp pB pBEOS T T − + = γ , where 0p is a reference pressure. We showed that the fittin g \nof PVT data to this EOS 5 \n  \n\n\n\n−\n\n\n+ = 1)(\n00\n0EOS \nEOS TpBppγ\nρρ\nγ   (6) \nwhere 2\n0 2 0 1 0 01\n0 ) ( ) ( ),( TTA TTA A pT − + − + = =−υ ρ  and the isothermal bulk modulus \n)) ( exp( )( )(0 2 0 00TTb pB pBT T − − =  at a reference pressure 0p, yields a similar value of the exponent \nEOS γ  to that found by using Eq. (5) in case of a tested  system. Such results have been obtained for both t he \nsimulation and experimental data.26,39,40  Moreover, our analyses of the PVT data collected f rom MD simulations \nin the KABLJ model and its version limited to the r epulsive IPL term confirmed the relation 3 /IPL EOS m≅ γ  \nfor both Eqs. (5) and (6) in the case of the simple  models based on the Lennard-Jones intermolecular p otential.26  \nIt is worth noting that very recently we have also derived 41  a similar equation of state for the activation vol ume \nin an analogous way to that employed in the derivat ion of Eq. (6), but using the definition of the iso thermal bulk \nmodulus for the activation volume.  \n The already mentioned recent investigations 31,34  of the generalized density scaling within the fram ework \nof the theory of isomorphs have induced us to find the counterparts of Eqs. (5) and (6) not limited to  the power \nlaw density scaling. We have pointed out that one c an do that by an appropriate generalization about t he method \nexploited 38,42  to derive Eq. (5) from the effective short-range i ntermolecular potential, \n() ()t IPL eff A U U − = R R , which involves all particle coordinates denoted b y R at a given state ( T,ρ) and \nimplies a simple relation, ( ) ( )( ) ()t IPL m\neff A U UIPL − =−R R3\n031\n0 / / ρρ ρρ , based on the Euler theorem on \nhomogeneous functions. One can note 43  that the generalized density scaling requires the following modification \nof this relation \n ( ) ( )()()()ρ ρ ρρ g Uh Ueff + =−R R31\n0/  (7) \nwhich is assumed to well separate density and confi guration contributions ( ()ρh , ()ρg , and ()RU ) to the \neffective potential eff U . Then, the configurational pressure approximately determined by the average system \nvirial per the system volume, eff conf UV VWp ∇⋅ −= ≅ R31, and  expressed as 6 \n ( )\n\n\n\n=kT MRT pIPL m\nconf 3\n0/ρρϕρ to derive Eq. (5) in the case of the power law den sity scaling, can be \ngeneralized as follows \n =kT h\nMRT pconf )(ρϕρ (8) \nBy analogy with the derivation of Eq. (5),26,38,39,42  we perform the first order Taylor series expansion  of the \nfunction φ(x) with ()()kT hx /ρ=\n in Eq. (8) about ()()kT h x /0 0 ρ= , i.e., about 0ρρ= , which \nyields ( ) ( )( )[ ]0 0 0 xxx xMRT pconf − ′+ = ϕ ϕρ. Introducing the reference configurational pressur e \n( )0 0 xMRT pconf ϕρ=\n and assuming the only temperature dependent paramet er ( ) ( )0xkM RTB ϕρ′ = , we arrive \nat the following EOS \n ()()() ( )0 0 ρ ρh hTB p pconf conf − + =  (9) \nThe physical meaning of the temperature dependent p arameter ()TB  can be found by using the generalized \ndensity scaling exponent given by Eq. (2). In the i sothermal conditions, density is an only pressure f unction \n()pρρ= . Then, we can transform Eq. (2), exploiting the co nfigurational isothermal bulk modulus, \n() () ()conf \nT\nTconf \nTconf \nconf Bph p\nph\ndhd\n∂∂=∂∂\n∂∂=ρ\nρρ\nρρ ln \nln ln \nln ln , to the following differential equation \n  () ()\n( )conf conf \nTconf p B ph ργ ρ=∂∂ln  (10) \nA general solution of Eq. (10) can be expressed as follows ( )()\n( )\n\n\n=∫conf \nconf conf \nTdp p BA hργρ exp . Assuming \nthe initial condition, ( )0 0, ρ ρ =conf pT , we find the integration \nconstant, ( )()\n( )\n0exp 0\nρρργρ\n=\n\n\n− =∫conf \nconf conf \nTdp p BhA , and the related particular solution of Eq. (10), 7 \n ( ) ( )()\n( )()\n( )\n\n\n\n\n\n\n− = ∫ ∫\n=conf \nconf conf \nTconf \nconf conf \nTdp p Bdp p Bh hργ ργρ ρ\nρρexp exp \n00, which can be approximated \nby the following density scaling function \n ( ) ( )()\n( )( ) \n− + =conf conf \nconf conf \nTp pp Bh h0\n00\n01ργρ ρ  (11)  \nusing the first order Taylor series expansion about  conf conf p p0= ,   \n()\n( )()\n( )()\n( )( )\n\n− +\n\n\n=\n\n\n\n=∫ ∫conf conf \nconf conf \nTppconf \nconf conf \nTconf \nconf conf \nTp pp Bdp p Bdp p B0\n001 exp exp \n0ργ ργ ργ. Since \nEq. (9) implies the density scaling function, ()() ()( )conf conf p pTB h h01\n0 − + =−ρ ρ  we confirm by \ncomparison with Eq. (11) that the parameter ()TB  can depend only on temperature \n ( )()\n( ) ( )0 00\nρ ργ hp BTBconf conf \nT=  (12) \nThis is because ()()()0 0 0 , , ργρh p Bconf conf \nT are determined at a constant reference pressure 0p. The relation given \nby Eq. (12) applied to Eq. (9) results in the EOS t hat possesses all parameters with the well-defined physical meaning,    \n ()\n( )()\n( )\n\n\n− + = 1\n0 00\n0ρρ\nργ hh p Bp pconf conf \nT conf conf  (13) \nIt should be noted that Eqs. (10) and (11) are also  valid after replacing its configurational quantiti es with their \nnonconfigurational counterparts. In this way, we ca n formulate another EOS, which is morphologically v ery \nsimilar to Eq. (13), \n ()\n( )()\n( )\n\n\n− + = 1\n0 00\n0ρρ\nργ hh pBppT (14) \nIf Eqs. (13) and (14) are considered in the case of  the power law density scaling, i.e., if ()EOS hγρ ρ= , then the \nscaling exponent is a material constant independent  of thermodynamic conditions and ()EOS γ ργ =  based on 8 \n Eq. (2). Consequently, in this case, Eqs. (13) and (14) can be reduced to Eqs. (5) and (6), respective ly. We have \nreported 39,40  that the important prediction made by Eqs. (5) and  (6) is the linear scaling of PVT data, \n( )EOS γρρ0 vs. ( )()0 0/ pBppT −   or  ( ) ()conf conf \nTconf conf p B p p0 0/ −  with the slope equal to EOS γ . One \ncan note that Eqs. (13) and (14) also lead to some kind of the linear scaling of PVT data that general izes \n( )EOS γρρ0to ()()0 /ρ ρh h  and EOS γ  to ()0ργ , but only if const =0ρ  is chosen at each considered \ntemperature at a reference pressure 0p. It means that the slope of the linear dependences  ()()0 /ρ ρh h  vs. \n( )()0 0/ pBppT −  or ( ) ()conf conf \nTconf conf p B p p0 0/ −  possesses a particular value, which depends on the  \nchoice of the reference state and does not always c onstitute any representative value of the scaling e xponent γ if \n()ργ  changes with density.      \n In addition, it is interesting to determine the pr essure dependences of the configurational isotherma l \nbulk modulus ()p Bconf \nT and the isothermal bulk modulus ()pBT that follow from Eqs. (13) and (14) to \ncompare them with the mentioned linear pressure fun ctions ()p Bconf \nT and ()pBT implied by Eqs. (5) and (6), \nrespectively. Differentiating the equations of stat e Eqs. (13) and (14) with respect to ρln  and exploiting Eq. \n(2), we find the following relations \n ( ) ()()()\n( ) ( )0 00ρ ργρργ\nhhp B p Bconf conf \nTconf \nT =   (15) \n ( ) ( )()()\n( ) ( )0 00ρ ργρργ\nhhpB pBT T =  (16)  \nwhich are reasonable results because the values of the fitting parameters involved in the functions  ()ρh  and \n()ργ  can be slightly different in Eqs. (13) and (14). I f ()EOS hγρ ρ= , then ()()EOS γ ργ ργ = =0 and \n()()( )EOS h hγρρ ρ ρ0 0 / / = . If we replace the right side of the latter equati on with its counterparts found from \nEqs. (5) and (6), respectively, we indeed arrive at  the mentioned linear pressure dependences ()p Bconf \nT  and \n()pBT. However, in general, Eqs. (15) and (16) can resul t in nonlinear pressure functions in isothermal \nconditions. Exploiting Eq. (11) and its counterpart  for nonconfigurational quantities, we find that th e pressure 9 \n dependences of conf \nTB  and TB can deviate from the linear character in isotherma l conditions if the scaling \nexponent γ varies with density,     \n ( )()\n( )()( )( ) [ ]conf conf conf conf \nTconf \nT p p p B p B0 0 0\n0− + = ργργργ (17) \n ( )()\n( )( ) ( )( )[ ]0 0 0\n0pp pB pBT T − + = ργργργ (18) \n \nIII. TEST OF THE EOS BY USING A PROTOTYPICAL MODEL OF SUPERCOOLED LIQUID \nIn Section II, we have derived two morphologically similar equations of state, which are based on the \nassumption that the scaling exponent can be a funct ion ()ργ , dependent only on density and related by Eq. (2) \nand the theory of isomorphs to a density scaling fu nction ()ρh  not limited to a power density function. Based \non the theory of isomorphs, one can also show 31,34  that the only density dependent function h in Eq. (1) is given \nby the polynomial ()∑=\njm\njjc h3 /ρ ρ in the case of a strongly correlating system , the molecular dynamics of \nwhich is described by the multiple IPL potential de fined by a sum of the IPL terms ∑−\njm\njjru . Therefore, Eqs. \n(13) and (14) with the density scaling function, ()∑=\njm\njjc h3 /ρ ρ , can be applied to describe PVT data of \nsystems characterized by such an IPL potential. For  instance, the following density scaling function   \n ()2 4)1 ( ρ ρ ρ c c h −+ =   (19) \nhas been argued 31,34  for the KABLJ model based on the LJ potential with  the exponents of the repulsive and attractive \nterms equal to 12 and 6, respectively. Then, Eq. (2 ) implies the related density function for the scal ing exponent  \n ( )( )ρρ ρργhc c2 4)1 ( 2 4 − +=  (20) \nThus, Eqs. (13) and (14) with the functions ()ρh  and ()ργ  given respectively by Eqs. (19) and (20) should be  \nvalid in the case of the KABLJ model.  \n It is worth noting that a known function ()ρh  with values of its parameters found from fitting P VT \ndata to the EOS given by Eqs. (13) or (14) is expec ted to enable to scale the structural relaxation ti mes according 10 \n to Eq. (1). The suggested property is of great impo rtance to the proper linkage between dynamics and \nthermodynamics near the glass transition, which is still under investigation. Therefore, besides the s tandard test \nfor a good approximation of the PVT data by Eqs. (1 3) and (14), we also verify whether these EOS meet this \nimportant criterion or not. \n  To perform the test we exploit our MD simulation data collected from the equilibrium simulation of \n1000 particles in the KABLJ model in the NVT ensemb le. Some of the used isotherms of structural relaxa tion \ntimes and PVT data (at T=0.5, 0.75, 1.0, 1.5, 2.0, 2.5, 3.0 in the LJ units ) have been earlier reported.26  However, \nthe related particle number density range, 6 . 1 2 . 1 ≤≤ρ  in LJ units, is then too narrow to be representati ve for \nthe general density scaling regime not limited to t he power law density scaling. As already mentioned in \nIntroduction, the power law density scaling of stru ctural relaxation times τ in the KABLJ model can be \nachieved 30,31  when the particle number density ranges from 1.2 t o 1.6 if the reduced units suggested by Dyre’s \ngroup are employed (e.g., 2/ 1 3 / 1 ~T τρ τ= , for τ, ρ, T in the usual LJ units) to cause the NVT molecular \ndynamics to be isomorph invariant in the sense post ulated by the theory of isomorphs.29  Nevertheless, the power \nlaw density scaling of structural relaxation times is impossible in the considerably wider range of pa rticle \nnumber densities, i.e., 0 . 2 2 . 1 ≤≤ρ , even if the structural relaxation times are expre ssed in the reduced \nunits.31  Therefore, we have performed additional MD simulat ions in the KABLJ model at temperatures T=2.0, \n3.0, 3.5, 4.0, 4.5, 5.0 using the RUMD package 44  to supplement the earlier collected simulation dat a with the \nparticle number density range, 0 . 2 6 . 1 ≤<ρ . Other details of the simulations are the same as those described \nin Ref. 26 for the narrower particle number density  range, 6 . 1 2 . 1 ≤≤ρ .    \n The structural relaxation times have been determin ed in the usual manner 30  from incoherent intermediate self-\nscattering functions 27  ( t=τ  if 1),(−=et FSq ) at the wave vector q of the first peak of the AA structure factor \nparticles at each simulation state ( T,ρ) separately, where A denotes the particle specie t hat constitutes 80% of the \nparticle content of the binary mixture. As can be s een in Fig. 1, there are only small differences in τ in the LJ units (Fig. \n1(a)) and ~τin the reduced units (Fig. 1(b)) suggested by the t heory of isomorphs. However, we perform the further  \nanalysis for the KABLJ model in the reduced units t o meet the  requirements of the theory of isomorphs . In this \ncontext, it should be noted that the quotient chara cter of Eqs. (13) and (14) allows us to apply these  EOS to describe \nPVT data in the usual LJ units.  11 \n \n \nFIG. 1. Plot of isothermal structural relaxation ti mes in the KABLJ model at different temperatures ve rsus the particle \nnumber density: (a) τ in the usual LJ umits and (b) τ~  in the reduced units, 2/ 1 3 / 1 ~T τρ τ= , where the particle number \ndensity ρ and temperature T  are in the usual LJ units. \n \n It is worth noting that the density dependent func tions ()ρh  and ()ργ  given by Eqs. (19) and (20) \nhave only one shared parameter c, which means that the number of fitting parameters  in the generalized density \nscaling isothermal EOS given by Eqs. (13) and (14) does not increase in comparison with their power la w \ndensity scaling counterparts represented by the iso thermal versions of Eqs. (5) and (6) that are chara cterized by \nthe power density function ()EOS hγρ ρ=  and the constant scaling exponent ()EOS γ ργ = . Using the \nsimulation data in the KABLJ model in the particle number density range, 6 . 1 2 . 1 ≤≤ρ  in the usual LJ units, \nwe have previously shown 26  that the scaling exponent γ that enables the power law density scaling of stru ctural \nrelaxation times of the KABLJ liquid is approximate ly the same as those found from fitting PVT data of  the \nKABLJ model to Eqs. (5) or (6). Now, we check wheth er it is possible to find the value of the paramete r c from \nfitting PVT data collected for MD simulations in th e KABLJ model in the considerably larger particle n umber \ndensity range, 0 . 2 2 . 1 ≤≤ρ  in the usual LJ units, which also enables the func tion ()ρh  given by Eq. (19) to \nscale the structural relaxation times according to Eq. (1).    12 \n  The polynomial character of the functions ()ρh  given by Eq. (19) that also affects the form of th e \nfunction ()ργ  given by Eq. (20)  causes that it is convenient to  use reduced densities ref reduc ρρ ρ /=  to \nensure that the values of the parameters c and (1- c) have consistent units. This fact and an intrinsic  feature of the \ntheory of isomorphs, which  predicts (see Eq. (2) i n Ref. 31) a micro-configuration (all particle coord inates) \n( )R3 / 1/−\nref ρρ  at a state ( T,ρ) isomorphic with the chosen reference state ( Tref ,ρref ) described by the micro-\nconfiguration  R, cause that we exploit reduced densities to verify  the new equations of state (Eqs. (13) and (14)) \nwith the density functions Eqs. (19) and (20) as we ll as to scale structural relaxation times accordin g to Eq. (1). \nNevertheless, the plots of PVT data in the KABLJ mo del with their fitting curves to Eqs. (13) and (14)  are \nshown (see Fig. 2) as pressure functions of non-red uced density for convenience of reading the figures .  \n \n \nFIG. 2. Plot of isothermal volumetric data in the K ABLJ model at different temperatures: (a) the confi gurational pressure \nversus the particle number density and (b) pressure  versus the particle number density. Values of all quantities are in the \nusuall LJ units. The solid lines in panels (a) and (b) denote fitting curves of the PVT data respectiv ely to Eq. (13) and Eq. \n(14) with the density dependent functions ()ρh  and ()ργ  given by Eqs. (19) and (20), where the normalized densities \nhigh norm ρρ ρ =  with 0 . 2=high ρ  in the usual LJ unit are assumed instead of ρ. 13 \n  It is reasonable to assume 2 . 10=ρ  to fit the PVT data to Eqs. (13) and (14), that is  the lowest density \nat which the values of the reference pressure are k nown for each isotherm of tested PVT simulation dat a, and \nconsequently 0ρ ρ=ref  to calculate the reduced densities ref reduc ρρ ρ /=  as 0/ρρ ρ =reduc . Then, based \non Eqs. (19) and (20), ( )()110 = =h hreduc ρ  and ( )() 2 210 + = = creduc γ ργ . The latter linear equation is a \nspecial case of Eq. (20), which in general results in a rational nonlinear function of the parameter c at a given \ndensity.  To make the fitting procedure more reliab le, it is better  to avoid this non-representative linear case, \nespecially that the fitting value of the parameter c tends then to infinity for numerical reasons. In o rder to do that \nwe normalize the explored density domain 0 . 2 2 . 1 ≤≤ρ  in the usual LJ units to the range, \n0 . 1 6 . 0 ≤ ≤norm ρ , where high norm ρρ ρ = with the highest considered density 0 . 2=high ρ  in the usual LJ \nunits. Using the normalized density domain, we are able to find reliable values of the fitting paramet er c as well \nas ()conf conf \nTp B0 and ()0pBT respectively for Eqs. (13) and (14) with the funct ions ()ρh  and ()ργ  given by \nEqs. (19) and (20). It should be stressed that the values of configurational reference pressures conf p0 and \nreference pressures 0p have not been fitted, because they have been fixed  in the case of each isotherm based on \nthe simulation PVT data at 2 . 10=ρ  in the usual LJ units. The fitting curves of the P VT simulation data to Eqs. \n(13) and (14) with their fitting values of the para meters are presented in Figs. 2(a) and 2(b), respec tively.  As a \nresult, we obtain a good quality of the fits and ve ry close values of the parameter 03 . 0 74 . 1± =c  and \n03 . 0 70 . 1± =c  determined from Eqs. (13) and (14), respectively, using the functions ()ρh  and ()ργ  given \nby Eqs. (19) and (20). Then, we apply the function ()ρh  defined by Eq. (19) with both the established valu es of \nthe parameter c to scale the structural relaxation times of the KA BLJ model in the reduced units (i.e., \n2/ 1 3 / 1 ~T τρ τ= ) according to Eq. (1). To do that we exploit the a ssumed reduced density, 0/ρρ ρ =reduc , \nthat implies the scaling function ( )( ) ( )2\n04\n0 0 /)1 ( / / ρρ ρρ ρρ c c h −+ = , which successfully leads (see \nFigs. 3(a) and 3(b)) to the generalized density sca ling of the structural relaxation times ~τ in terms of Eq. (1) for \nboth the values of the parameter c. It means that the generalized density scaling sho uld be also possible in the \nnormalized density range, because the function ( )0/ρρh  results in \n( )( )( ) ( )( )2 2\n04 4\n0 0 / / )1 ( / / /high high high high norm norm c c h ρρ ρ ρ ρρ ρ ρ ρ ρ −+ = , but to scale the structural 14 \n relaxation times by means of the function, ( )( )high norm norm h h ρρ ρ ρ / /0=  , we need to rescale the polynomial \ncoefficients c and (1- c) to ( )4\n0/ρ ρhigh c  and ( )2\n0/ )1 ( ρ ρhigh c− , respectively.  \n \nFIG. 3. The density scaling of all the isotherms of  the structural relaxation times in the reduced uni ts in the KABLJ model \nfrom Fig. 1(b) according to Eq. (1) with the densit y scaling functions given by Eq. (19) with the valu e of the parameter c \nfound from fitting PVT data to the equation of stat e given respectively by (a)  Eq. (13) and (b) Eq. ( 14). \n \nAs already mentioned and discussed in the context o f Eq. (2), the scaling exponent γ in the case of the \ngeneralized density scaling depends on density. It is interesting to see how varies the density depend ent exponent \n()ργ  given by Eq. (20) in the KABLJ model examined in t he density range 0 . 2 2 . 1 ≤≤ρ in the usual LJ \nunits and how the changes in ()ργ  affect the pressure dependences ()p Bconf \nT and ()pBT, which are \npredicted respectively by Eqs. (17) and (18) to dev iate from the linear character valid at low densiti es or \npressures. It is worth noting that the PVT data are  considered herein in two times larger density rang e \n( 0 . 2 2 . 1 ≤≤ρ  in the usual LJ units) that corresponds to three a nd a half times larger pressure range \n( 350 0 ≤<p  in the usual LJ units) than those ( 100 0 , 6 . 1 2 . 1 ≤< ≤≤ p ρ  in the usual LJ units) explored \nby us previously 26  to test Eqs. (5) and (6) by means of the KABLJ mod el. We found that the increase in density \nresults in a decrease in the value of the scaling e xponent from 48 . 5=γ  at 2 . 1=ρ  in the usual LJ units to 15 \n 44 . 4=γ  at 0 . 2=ρ  in the usual LJ units if we calculate the values ( )0/ρργ  from Eq. (20) with 70 . 1=c  \nobtained from fitting the PVT data to Eq. (13), and  in a very slightly different decrease in these val ue from \n40 . 5=γ  at 2 . 1=ρ  in the usual LJ units to 35 . 4=γ  at 0 . 2=ρ  in the usual LJ units if we calculate the \nvalues ( )0/ρργ  from Eq. (20) with 70 . 1=c  obtained from fitting the PVT data to Eq. (14). Th ese changes \nin γ with varying density should influence the pressure  dependences of the configurational isothermal bulk  \nmodulus and the isothermal bulk modulus according t o Eqs. (17) and (18), respectively. To examine the high \npressure behavior of conf \nTB  and TB, we choose the PVT isotherms at high temperatures,  because the PVT data \nsimulated for these isotherms cover the entire cons idered density range 0 . 2 2 . 1 ≤≤ρ  in the usual LJ units. As \nan example, we show the dependences  ()p Bconf \nT and ()pBT at T=3.5 in Fig. 4.  \n \nFIG. 4. Plots of the configurational isothermal bul k modulus versus the configurational pressure (a) a nd (b) the isothermal \nbulk modulus versus pressure calculated respectivel y from the definitions  ( )Tconf conf \nT p B ρln /∂ ∂=   and \n( )T T p B ρln /∂∂=  by using the PVT simulation data in the KABLJ mode l at T=3.5 in the usual LJ units. The dashed \nlines denote the low pressure linear fits of these dependences and their high pressure extrapolations.  The solid lines are \ngenerated in panels (a) and (b) respectively from E q. (17) and Eq. (18) with the values of their param aters found by fitting the \nPVT data (see Fig. 2) respectively to Eqs. (13) and  (14) with the density dependent functions ()ρh  and ()ργ  given by \nEqs. (19) and (20), exploiting the normalized densi ties high norm ρρ ρ = with 0 . 2=high ρ  in the usual LJ unit instead \nof ρ as it was assumed to fit the PVT data to these equ ations of state. 16 \n The expected deviation of these dependences from th e linear character is indeed observed at pressures higher \nthan 100 in the usual LJ units. For instance, the v alue of TB at the highest simulated pressure along the isothe rm \nT=3.5 is 12% smaller than its counterpart calculated from the high pressure extrapolation of the linear pressure \ndependence of TB determined at low pressures with its slope 5.36, w hich is very close to the mentioned low \ndensity limit ( γ=5.40 at 2 . 1=ρ ) of the density function of the scaling exponent γ.   \n \nIV. THE SUGGESTED FORM OF THE EOS FOR PVT DATA OF R EAL GLASS FORMERS  \n Very recently, another density scaling function ()ρh  established phenomenologically  \n () ( )ρ ρ ρ2\n2 1 ln ln exp C C h + =  (21) \nhas been suggested 31  to scale structural relaxation times of real glass  formers in terms of Eq. (1). For the \nconvenience of calculations, the polynomial functio n of ρln  is used here in Eq. (21) instead of the polynomial  \nfunction of ρ10 log , which means that the fitting parameters 1 1A C=  and 10 ln /2 2A C=  if A1 and A2 are \nthe fitting parameters of the function ()ρh  reported for real glass formers in Ref. 31. It sho uld be noted that Eq. \n(21) via Eq. (2) implies the following simple densi ty dependent function for the scaling exponent  \n () ρ ργ ln 22 1C C+ =  (22) \nwhich can be reduced to the constant scaling expone nt if the parameter 02=C . However, the parameter 1C \nindicates in general the value of the scaling expon ent ()ργ  if density tends to unity. Taking into considerati on \nthe successful description of the deviation from th e power law density scaling in the case of supercoo led van der \nWaals liquids measured in a wide pressure range (i. e., in a wide density range), which is possible by using Eq. \n(21),31  we postulate that Eqs. (21) and (22) cause the equ ations of state (Eqs. (13) and (14)) to properly de scribe \nPVT data of real glass formers, the molecular dynam ics of which obeys the generalized density scaling law (Eq. \n(1)), because these EOS are derived herein also in the density scaling regime not limited to the power  law \ndensity scaling.   \n Unfortunately, in contrast to the KABLJ model succ essfully used to verify the derived EOS with the \nfunctions ()ρh  and ()ργ  given by Eqs. (19) and (20), a reliable test of Eq s. (13) and (14) with the density functions \n()ρh  and ()ργ  given by Eqs. (21) and (22)  can be only done indi rectly due to the high pressure limit of PVT \nmeasurements, which usually do not exceed 200MPa. A lthough an exception is made by the PVT experimenta l data of 17 \n a few glass formers measured up to 700MPa,8 the very recent ultra-high pressure dielectric and  ultrasonic \nmeasurements of propylene carbonate (PC) up to 4.2G Pa and 1.7GPa, respectively, show 45  that the pressure \ndependence of the isothermal bulk modulus determine d from the ultrasonic measurements within the press ure range \nfrom 0.1MPa to about 1GPa at T=295K, which is above  the glass transition temperature of PC, cannot be satisfactorily \ndescribed in the low pressure limit by using the eq uation of state based on the Tait equation, which h as been \nparameterized for PC in Ref. 8. On the other hand, the Tait equation earlier parameterized by using PV T data of PC 6 \nmeasured in the typical pressure range of PVT exper iments, i.e., up to 200MPa, properly predicts the l ow pressure \nlimit of the dependence ()pBT found from the ultrasonic data, but these dependen ces begin to diverge with \nincreasing pressure. In this situation, Eqs. (13) a nd (14) with the functions ()ρh  and ()ργ  given by Eqs. (21) and \n(22) seem to be a good alternative to properly desc ribe PVT data in the ultra-high pressure limit. A p reliminary test for \nthe validity of these EOS can be based on the descr iption of the pressure dependence of the isothermal  bulk modulus \nof PC at T=295K, which has been  reported in Fig. 8 (a) in Ref. 45 for different ultrasonic experimenta l methods. We \nconsider herein (Fig. 5) only the data from autocor relation measurement assessed by the authors of Ref . 45 as the most \naccurate one.  \n It should be noted that the simple density scaling  function given by Eq. (21) enables us to find pres sure \nfunctions of the configurational isothermal bulk mo dulus and the isothermal bulk modulus based on the \nequations of state Eqs. (13) and (14), respectively . Since we test only the pressure dependence of the  isothermal \nbulk modulus based on the data taken from Ref. 45, we discuss in detail only the derivation of the \nfunction ()pBT. From Eq. (14), one can easily find, ()() ()( )() [ ]0 0 0 0 / 1 pBpp h hT − + = ργ ρ ρ . If we  \nsubstitute the function ()ρh  for Eq. (21) in the latter equation, we can formul ate a quadratic equation for ρln  \nwith the following coefficients C2, C1, and ()( )() [ ] ( )0 0 0 0 1 02\n2 / 1ln ln ln pB pp C CT − + + + − ργ ρ ρ . \nThen, we can find density as a function of pressure  at a given temperature \n ( )( ) ( )[ ] ( )\n22 / 1\n0 0 0 2 0 12 02 2\n22\n1 1\n2/ 1ln 4 ln 4 ln 4ln CpBpp C CC C C CT − + + + + ± −=ργ ρ ρρ  (23) \nwhich can be simplified by using the reduced densit y 0/ρρ ρ =reduc  and assuming 1≥reduc ρ  in the \nfollowing way ( )( ) ( )[ ] ( )\n22 / 1\n0 0 22\n1 1\n2/ 1 1ln 4ln CpB pp C C CT reduc − + + + −=γρ  , where ()1 1C=γ . It is \nworth noting that Eq. (23) applied to Eq. (22) also  enables us to establish the scaling exponent γ as a function of 18 \n pressure, ( ) ( ) ( ) [ ] ( )2 / 1\n0 0 1 22\n1 / 1ln 4 pB ppC C C pT − + + = γ , where the reduced density is employed. \nFinally, Eq. (18) with the scaling exponent γ given by Eq. (22), which is expressed by the  pres sure function \n()pγ  that assumes 0/ρρ ρ =reduc , yields the following pressure function for the is othermal bulk modulus \n ( )( ) ( ) [ ] ( ) ( ) ( )[ ]\n10 1 02 / 1\n0 0 1 22\n1 / 1ln 4\nCppC pB pB ppC C CpBT T\nT− + − + +=  (24) \n The authors of Ref. 45 noted that the dependence ()pBT obtained from the autocorrelation ultrasonic \nexperiment at T=295K is almost linear in the considered pressure ra nge up to 1GPa, although they observed that \nthe isothermal compression of PC cause the dependen ce ()pBT to slightly deviate from the linear behavior. An \ninteresting question arises as to how strong is the  effect of pressure on the isothermal compressibili ty of the \nmaterial squeezed above 1GPa. Assuming a gradual pr essure influence on volumetric properties of superc ooled \nliquids, one can expect that the dependence ()pBT at p > 1GPa also gradually deviates from the high press ure \nextrapolation of the linear dependence ()pBT determined at low pressures to pressures larger th an 1GPa. \nTaking into account this supposition, we can discri minate between two predictions about the isothermal  \ndependence ()pBT at pressures above 1GPa, which are based on the ty pical quadratic pressure parametrization \n() ()( )( ) ( )( )2\n02 2\n0 00 0/ ) 2 / 1 ( / pp p B pp p B pBpBpp T pp T T T − ∂ ∂ + − ∂ ∂+ == = as well as on Eq. (24) that \nfollows from Eq. (14) with the  functions  ()ρh  and ()ργ  given by Eqs. (21) and (22). Since \n() MPa 7 . 2716 0=pBT can be taken from the experimental autocorrelation  ultrasonic data at ambient \npressure p0, both the quadratic parameterization of ()pBT and Eq. (24) have two fitting parameters \n( )\n0/1 pp Tp B D=∂ ∂= and ( )\n02 2\n2 / ) 2 / 1 (pp Tp B D= ∂ ∂ =  as well as C1 and C2, respectively. We determine the \nvalues of the parameters ( 1 . 0 8 . 81 ± =D  and -1 \n2 GPa 10 . 0 83 . 0± −=D  as well as 02 . 0 65 . 11 ± =C  and \n03 . 0 15 . 72 ± =C ) by fitting the experimental dependence ()pBT for PC at T=295 (see Fig. 5). Then, we \nextrapolate the quadratic parameterization of ()pBT and Eq. (24) with the values of their parameters f itted within the \npressure range below 1GPa up to p=4GPa. In this way, we investigate the extremely wi de pressure range that becomes \nachievable experimentally by means of modern high p ressure techniques, e.g. exploiting diamond anvil p ressure cells.  \n 19 \n \n \nFIG. 5. A \n comparison of three ultra-high pressure predictions  of the pressure dependence of the isothermal bulk modulus \n(open squares) established 45  from the ultrasonic autocorrelation measurements o f PC at T=295K (above the glass transition \ntemperature of PC) and pressures lower than 1GPa. T he dotted line denotes the linear pressure function  fitted by using the \n experimental data ()pBT. The solid line is the fitting curve of the experi mental data ()pBT to Eq. (24), i.e, to Eq. (18) \nwith the function γ from Eq. (22), which is based on the density scali ng function h from Eq. (21). The dot dashed line is the \nfitting curve of the experimental data ()pBT to Eq. (18) with the function γ from Eq. (20), which is based on the density \nscaling function h from Eq. (19). The dotted and dashed lines result respectively from the linear and quadratic pressure  \nparameterizations of the experimental dependence ()pBT. Each fitting curve is extrapolated up to p=4GPa. \n \nAs a result, we find that the high pressure extrapo lation based on the quadratic parameterization of  ()pBT , \ndenoted by the dashed line in Fig. 5, diverges from  the linear behavior (the dotted line in Fig. 5) in  a significantly \nlarger degree than that established for the high pr essure extrapolation made by means of Eq. (24), dep icted by the \nsolid line in Fig. 5. For instance, the values of ()pBT  extrapolated at p=4GPa by using the quadratic pressure \nparameterization and Eq. (24) are respectively 45% and 6% less than that predicted by the linear press ure \ndependence ()pBT  found at low pressures. This result indirectly giv es evidence of the good ability of Eq. (14)  \nwith the  functions  ()ρh  and ()ργ  assumed by Eqs. (21) and (22) to describe the ultr a-high pressure PVT data. \nNevertheless, this preliminary test requires verify ing in the future by using the ultra-high experimen tal PVT data or \nat least the pressure dependences of the isothermal  bulk modulus measured above 1GPa at different temp eratures. A 20 \n similar stipulation can be made in the case of Eq. (23). As already mentioned, Eq. (23) provides a pre ssure function \nfor density, which could be used to evaluate densit y in the ultra-high pressure range if the values of  the parameters \nC1 and C2 are known. To do that one could employ the tempera ture parameterizations of the density \n( )0 0 ,pTρ ρ=  and the isothermal bulk modulus ()0pBT at the reference pressure 0p exploited by us in Eqs. \n(5) and (6). Then, one could establish the temperat ure-density dependence of structural relaxation tim es and make \nan attempt at scaling them according to Eq. (1). Ho wever, to make such an analysis more reliable, the found values \nof the parameters C1 and C2 should be prior verified by using the ultra-high p ressure experimental PVT data or at least \na larger dataset of the pressure dependences of the  isothermal bulk modulus measured from ambient pres sure to the \nhighest pressure as possible (preferably p>1GPa) at different temperatures within the explored  temperature range.     \n Finally, it is interesting to discuss  the ultra-h igh pressure prediction of the dependence ()pBT for PC \nat T=295, which results from Eq. (18) with the density f unction ()ργ  given by Eq. (20) that is a consequence \n(via Eq. (2) with Eq. (19) ) of the LJ potential wi th the exponents 12 and 6 of the repulsive and attr active terms, \nrespectively. In this case, using Eq. (14) with the  functions ()ρh  and ()ργ  given by Eqs. (19) and (20), we \ndetermine, ( ) ( ) ( ) ( )( ) ( ) [ ] ( )\ncpB pp ch c cT\n2/ 1 4 1 12/ 1\n0 0 0 02\n2 − + + −±−−=ργ ρρ , which is next \nappropriately employed in Eq. (18)  to find the pre ssure function of TB. Assuming the reference density  \n() MPa 7 . 2716 0=pBT and 3\n0 g/cm 19 . 1=ρ  at ambient pressure,45  the value of only one parameter c \nrequires estimating. By fitting the ultrasonic auto correlation data for the isothermal bulk modulus of  PC at \np<1GPa to Eq. (18) with the density function ()ργ  given by Eq. (20), we obtain 01 . 0 23 . 2± =c , where the \nnormalized density domain defined by the transforma tion, high norm ρρ ρ = , with the arbitrarily chosen \nsufficiently large density, 3g/cm 0 . 2=high ρ ,  has turned out to be better to use for the same reasons as those \npointed out in the case of the fitting of the PVT s imulation data in the KABLJ model to Eqs. (13) and (14) with \nthe functions ()ρh  and ()ργ  given by Eqs. (19) and (20). Then, we extrapolate this fitting curve of the \ndependence ()pBT to higher pressures up to p=4GPa. This prediction (depicted by the dot dashed line in Fig. \n5) is considerably closer to that established by us ing the quadratic pressure parameterization of TB (the dashed \nline in Fig. 5) than it is relative to that determi ned from Eq. (24), which is denoted by the solid li ne in Fig. 5. It 21 \n means that we obtain the significantly different pr edictions of the dependence ()pBT above 1GPa from Eq. \n(18) with the scaling exponent ()ργ  calculated from the density scaling functions ()ρh  given by Eqs. (21) and \n(19), for which real glass formers have been previo usly found 31  to meet and not to meet the generalized density \nscaling criterion (Eq. (4)), respectively. Thus, it  is reasonable to claim that the high pressure beha vior of the \nisothermal bulk modulus can reflect  the ability of  a given function ()ρh  to scale the molecular dynamics near the \nglass transition according to the generalized densi ty scaling law given by Eq. (1) and vice versa  the scaling function \n()ρh  can be used to predict at least approximately the pressure dependence of the isothermal bulk modulus.    \n \nV. SUMMARY AND CONCLUSIONS   \n We have generalized two equations of state (Eqs. ( 5) and (6)) earlier discussed 26,38,39,40  in the power law \ndensity scaling regime to describe PVT data of supe rcooled liquids, the molecular dynamics of which ob eys the \ndensity scaling law not limited to the power law de nsity scaling. The generalization about Eq. (5) is Eq. (13) \nderived by using the effective intermolecular poten tial (Eq. (7)) with well-separated density and conf iguration \ncontributions as well as the general relation (Eq. (2)) argued by the theory of isomorphs between the density \ndependent scaling exponent )(ργ  and the density scaling function )(ρh , which has been combined with the \ndefinition of the configurational isothermal bulk m odulus. However, the generalization about Eq. (6) i s Eq. (14) \nformulated from Eq. (2) combined with the definitio n of the isothermal bulk modulus.  \n We have very successfully verified both these EOS by using the PVT data obtained from MD NVT \nsimulations in the KABLJ model in the extremely wid e density range ( 0 . 2 2 . 1 ≤≤ρ  in the usual LJ units), \nwhich corresponds to the extremely wide pressure ra nge ( 350 0 ≤<p  in the usual LJ units). To perform this \ntest the functions ()ρh  and ()ργ   in Eqs. (13) and (14) have been given by Eqs. (19 ) and (20), which can be \nanalytically derived 31,34  for the intermolecular potential consisted of two IPL terms. By fitting the PVT data from \nMD simulations in the KABLJ model to each equation of state with the functions ()ρh  and ()ργ  with only \none shared parameter, we have found the value of th is parameter, and using this value in Eq. (1) with the \nfunctions ()ρh  given by Eq. (19), we have scaled the structural r elaxation times evaluated in the extremely \nwide density range ( 0 . 2 2 . 1 ≤≤ρ  in the usual LJ units) in the KABLJ model. This re sult obtained for each \nEOS is meaningful, because it confirms the validity  of the derived equations of state in the generaliz ed density 22 \n scaling regime. This scaling has also revealed a pe culiar behavior of the longest structural relaxatio n times, \nwhich seem to increase considerably slower than the  shorter relaxation times do. However, this observa tion \nrequires verifying by extremely time-consuming MD s imulations, the results of which would be able to e xceed \nas much as possible the structural relaxation times  currently reached.   \nBased on the generalized equations of state given b y Eqs. (13) and (14), we have arrived at the pressu re \ndependences of the configurational isothermal bulk modulus (Eq. (17)) and the isothermal bulk modulus (Eq. \n(18)), which can be in general nonlinear. The very good quality of the fits of the PVT data in the KAB LJ model \nto Eqs. (13) and (14) with the functions ()ρh  and ()ργ  given by Eqs. (19) and (20) has enabled us to very  \nsatisfactorily reproduce the dependences ()p Bconf \nT and ()pBT in the case of the KABLJ model, which have \nturned out to deviate gradually with increasing pre ssure from the high pressure extrapolations of thei r linear \npressure dependences valid at low pressures.   \nWe have suggested to employ Eqs. (21) and (22) resp ectively as the functions ()ρh  and ()ργ  in Eqs. \n(13) and (14), because Eq. (4) with the function ()ρh  given by Eq. (21)  has been previously used 31  as the \ncriterion for the generalized density scaling that has been met by real van der Waals liquids. Since t here is no \nPVT data measured in the sufficiently wide pressure  range, we have performed a preliminary indirect te st of this \nassumption by using the high pressure dependence of  the isothermal bulk modulus found from the ultraso nic \nautocorrelation measurements carried out 45  up to ca. p =1GPa. We have shown that Eq. (18) with the density  \nfunction ()ργ  given by Eq. (22) yields a more reasonable ultra-h igh pressure prediction of the dependence \n()pBT up to p=4GPa  than that obtained from the typical quadrati c pressure parameterization of ()pBT.   \n In the case of real glass formers, the derived equ ations of state require further studying by using \nexperimental volumetric data in the sufficiently wi de pressure range, which are expected to be measure d in the \nfuture. Then, if the structural relaxation or visco sity data are also accessible in the extremely wide  pressure \nrange, the functions ()ρh  and ()ργ   suggested by Eqs. (22) and (23) can be even chang ed. Nevertheless, these \nEOS given by Eqs. (13) and (14) are proper and conv enient templates for further ultra-high pressure \ninvestigations of volumetric properties of the mate rials approaching the glass transition, especially if their \nmolecular dynamics obeys the generalized density sc aling law given by Eq. 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Gromnitskaya, A. A. Pronin, A.  G. Lyapin, V. V. Brazhkin, and A. A. Volkov, J. Ch em. \nPhys. 137 , 084502 (2012). \n \n  "    },    {        "title": "1312.5223v1.Density_of_states_of_the__XY__model__an_energy_landscape_approach.pdf",        "content": "arXiv:1312.5223v1  [cond-mat.stat-mech]  18 Dec 2013Density of states of the XYmodel: an energy landscape approach\nCesare Nardini∗\nLaboratoire de Physique, ´Ecole Normale Sup´ erieure de Lyon and CNRS, 46 all´ ee d’Ital ie, F-69007 Lyon, France\nRachele Nerattini†and Lapo Casetti‡\nDipartimento di Fisica e Astronomia and Centro per lo Studio delle Dinamiche Complesse (CSDC),\nUniversit` a di Firenze, and Istituto Nazionale di Fisica Nu cleare (INFN),\nSezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorenti no (FI), Italy\n(Dated: December 17, 2013)\nAmong the stationary configurations of the Hamiltonian of a c lassical O( n) lattice spin model,\na class can be identified which is in one-to-one corresponden ce with all the the configurations of\nan Ising model defined on the same lattice and with the same int eractions. Starting from this\nobservation it has been recently proposed that the microcan onical density of states of an O( n)\nmodel could be written in terms of the density of states of the corresponding Ising model. Later, it\nhas been shown that a relation of this kind holds exactly for t wo solvable models, the mean-field and\nthe one-dimensional XYmodel, respectively. We apply the same strategy to derive ex plicit, albeit\napproximate, expressions for the density of states of the tw o-dimensional XYmodel with nearest-\nneighbor interactions on a square lattice. The caloric curv e and the specific heat as a function of the\nenergy density are calculated and compared against simulat ion data, yielding a very good agreement\nover the entire energy density range. The concepts and metho ds involved in the approximations\npresented here are valid in principle for any O( n) model.\nI. INTRODUCTION\nA quite recent point of view on the study of the thermodynamic prop erties of a classical N−body Hamiltonian\nsystem, commonly referred to as the “energy landscape” approa ch [1], implies the study of the properties of the\ngraph of the energy function H: ΓN→R, where Γ Nis phase space of the system. Examples of applications include\ndisordered systems and glasses [ 2,3], clusters [ 1], biomolecules and protein folding [ 4–6]. In this context a special\nrˆ ole is played by the stationary points of the energy function. The latter are the points pcin phase space such that\n∇H(pc) = 0. Based on knowledge about the stationary points of the energ y function, landscape methods allow to\nestimate both dynamical and equilibrium properties of a system. In t he present paper the attention will be focused\nonly on equilibrium properties.\nThe natural setting to understand the connection between the s tationary points of the Hamiltonian and equilibrium\nstatistical properties is the microcanonical one [ 7]. This is the statistical description in which the system is considered\nas isolated; the control parameter is the energy density ε=E/N, and the relevant thermodynamic potential is the\nentropy density sNgiven by\nsN(ε) =1\nNlogωN(ε), (1)\nwhereωN(ε) is the density of states of the system. The density of states is giv en by\nωN(ε) =/integraldisplay\nΓNdΓNδ(H−Nε) =/integraldisplay\nΓN∩ΣεdΣε\n|∇H|, (2)\nwhere Σ εis the hypersurface of constant energy NεanddΣ is the Hausdorff measure; the rightmost integral stems\nfrom a co-areaformula [ 8]. When a stationary configuration pcis considered, the integrand in ( 2) diverges. In finite- N\nsystems this divergence is compensated by the measure dΣ that shrinks in such a way that the integral in Eq. ( 2)\nremains finite. This notwithstanding, the density of states is non-a nalytic at the corresponding value of the energy\ndensityεc=H(pc)/N. Although these nonanalyticities typically disappear in the thermody namic limit N→ ∞, in\nsome special cases they may survive and give rise to equilibrium phase transitions [ 9–11]. Stationary points of the\n∗cesare.nardini@gmail.com\n†rachele.nerattini@fi.infn.it\n‡lapo.casetti@unifi.it2\nenergy correspond to topology changes in the accessible phase sp ace, and this has suggested a relation between these\ntopology changes and equilibrium phase transitions (see Refs. [ 12–14] and references therein).\nIn [15] an approximate form for the density of states of a paradigmatic c lass of classical spin models, the O( n)\nmodels, was conjectured on the basis of energy landscape conside rations. According to the conjecture, the density of\nstatesω(n)of a classical O( n) spin model on a lattice can be approximated in terms of the density o f statesω(1)of\nthe corresponding Ising model, i.e., an Ising model defined on the sam e lattice and with the same interactions. The\ncrucial observation is that all the configurations of the correspo nding Ising model are stationary configurations of a\nO(n) model for any n. The density of states would then be given by\nω(n)(ε)≈ω(1)(ε)g(n)(ε), (3)\nwhereg(n)(ε) is anunknownfunction representingthe volumeofaneighborhood ofthe Ising configurationin the phase\nspace of the O( n) model. Since the function g(n)comes from the evaluation of local integrals over a neighborhood\nof the phase space, one expects that it is regular in all the energy d ensity range. Equation ( 3) will be discussed in\ndetail in Sec. II. The assumptions made in [ 15] to derive Eq. ( 3) are rather crude and uncontrolled. However, were\nthis relation exact, there would be an interesting consequence: th e critical energy densities of the phase transitions of\nall the O( n) models on a given lattice would be the same and equal to that of the c orresponding Ising model.\nDespite the fact that Eq. ( 3) is approximate[ 16] the critical energy densities are indeed, if not equal, very close to\neach other, whenever a phase transitions is known to take place, a t least for ferromagnetic models defined on regular\nd−dimensional hypercubic lattices. In particular, the critical energy densities are the same and equal to the Ising one\nfor all the O( n) models with long-range interactions, as shown by the exact solutio n [17], and the same happens for\nall the O( n) models on an one-dimensional lattice with nearest-neighbor intera ctions. For what concerns O( n) models\nwith nearest-neighbor interactions defined on (hyper)cubic lattic es withd >1, only numerical results are available\nfor the critical energy densities. In d= 2 a Bereˇ zinskij-Kosterlitz-Thouless (BKT) phase transition is pr esent in the\nn= 2 case, occurring at a value of the energy density that differs of a bout 2% from the Ising case (see [ 15,18] for\na deeper discussion[ 19]). Ind= 3, the difference between the critical energy densities of the O( n) models and those\nof the Ising model is smaller than 3% for any n, including the n=∞case, as recently shown in [ 20]. In the most\nfrequently considered cases, that is for n= 2 (the XYmodel),n= 3 (the Heisenberg model) and for the O(4) model,\nit becomes less than 1%.\nIn two particular cases, where the prediction on the critical energ y densities is exact, the derivation of Eq. ( 3) can be\nfollowed rigorously, that is for the mean-field and for the 1 dferromagnetic XYmodels. In these cases, an expression\nvery similar to Eq. ( 3), and which reduces to the latter when ε→εc, can be derived exactly in the thermodynamic\nlimit [18]. The technical aspects of the derivation strongly rely on the pecu liarities of the two models and especially\non the fact that they are exactly solvable in the microcanonical ens emble. This feature is crucial for the analysis\npresented in [ 18] and its generalization to O( n) models with n >2 ind >1 is unlikely to be feasible.\nThe aim of the present paper is to show that an approximate expres sion for the density of states ω(n)of nearest-\nneighbor ferromagnetic O( n) models in d >1 can be derived by applying the same energy landscape consideratio ns\nas discussed above. More precisely, two approximation techniques are presented here: the first one can be seen\nas the natural generalization of the techniques applied in [ 18] for the mean-field and for the 1- dXY models and\nwill be referred to as “first-principle” approximation in the following; the second one will be named “ansatz-based”\napproximation, its starting point being the ansatz on the form of th e density of states given by Eq. ( 3) supposed\nto be valid in all the energy density range. These techniques can be a pplied to estimate the density of states of in\nprinciple any O( n) model in d >1. However we have explicitly performed the calculations only for n= 2, i.e., for the\nXYmodel in d= 2; the technical aspects of the generalization to other members of the O( n) class in d >2 will be\ndiscussed underway.\nThepaperisorganizedasfollows. InSec. IIthestationary-pointapproachandtheapproximationsintroduce din[15]\nleading to Eq. ( 3) will be recalled and discussed. In Sec. IIIthe “first-principle” and the “ansatz-based” approaches\nwill be introduced. In both approaches a crucial point is the estimat ion ofg(n). A possible way in which this can be\ndone will be presented in Sec. IIIA. In Secs. IVandVthe “first-principle” and the “ansatz-based” approximations\nwill be, respectively, discussed in details through their application to theXYmodel in d= 2. Some remarks on the\ntechnical limits of the “first-principle” approach will be included in Sec .IVA. The paper ends with some concluding\nremarks in Sec. VI.\nII. STATIONARY POINTS AND DENSITY OF STATES\nIn our analysis we are going to consider some special cases of classic al O(n) spin models defined on a regular square\nlattice in d= 2 and with periodic boundary conditions. In these models, to each la ttice site iann-component classical3\nspin vector Si= (S1\ni,...,Sn\ni) of unit length is assigned. The energy of the model is given by the Ha miltonian\nH(n)=−J/summationdisplay\n/angbracketlefti,j/angbracketrightSi·Sj=−J/summationdisplay\n/angbracketlefti,j/angbracketrightn/summationdisplay\na=1Sa\niSa\nj, (4)\nwhere the angular brackets denote a sum over all pairs of nearest -neighbor lattice sites. The exchange coupling Jwill\nbe assumed positive, resulting in ferromagnetic interactions, and w ithout loss of generality we shall set J= 1 in the\nfollowing. The Hamiltonian ( 4) is globally invariant under the O(n) group. In the case n= 1, the symmetry group\nO(1)≡Z2is a discrete one and the Hamiltonian ( 4) becomes the Ising Hamiltonian\nH(1)=−/summationdisplay\n/angbracketlefti,j/angbracketrightσiσj, (5)\nwhereσi=±1∀i. In all the other cases n≥2, theO(n) group is a continuous one. For n= 2 we obtain the XY\nmodel that will be the subject of our study. In this model the spins are constrained on the unit circle S(1)and the\ncomponents of the ith spin can be parametrized by a single angular variable ϑi∈[0,2π) such that\n/braceleftBigg\nS1\ni= cosϑi,\nS2\ni= sinϑi.(6)\nThe Hamiltonian of the XYmodel can thus be conveniently written as\nH(2)=−1\n2N/summationdisplay\ni=1/summationdisplay\nj∈N(i)cos(ϑi−ϑj), (7)\nwhereN(i) denotes the set of nearest neighbors of lattice site i. The energy density ε=H(2)/Nlies in the energy\nrange [−d,d] wheredis the lattice dimension.\nLet us now recall the derivation of Eq. ( 3) made in [ 15]. The stationary configurations of H(n)forn≥2 are given\nby the solutions ¯S= (¯S1,...,¯SN) of theNvector equations\n∇H(n)= 0, (8)\nsuchthat the constraint/summationtext\na=1n(Sa\ni)2= 1issatisfied ∀i= 1,...,N. To find explicitly allthe stationaryconfigurations\nis an essentially impossible task but in some special cases, see e.g. [ 21], or for very small systems, see e.g. [ 22] and\nreferences therein. However, as shown in Ref. [ 15], a particular class of solutions can be found, given by all the\nconfigurations in which the spins are parallel or anti-parallel to a fixe d direction (say the nth direction): S1\ni=...=\nSn−1\n0= 0∀i. Indeed, in this case, the constraint/summationtext\na=1n(Sa\ni)2= 1 can be fulfilled by choosing Sn\ni=σi∀iand\nthe stationary points equations ( 8) are satisfied by any of the 2Npossible choices of the σ’s. The Hamiltonian ( 4)\nbecomes the Ising Hamiltonian ( 5) on this class of stationary configurations. Therefore a one-to- one correspondence\nbetween a class of stationary configurations of the Hamiltonian ( 4) andallthe configurations of the Ising model[ 23];\nthe corresponding stationary values are the energy levels of the I sing Hamiltonian. We shall refer to the class of\nstationary configurations ¯S= (0,...,0,σi)∀i= 1,...,Nas “Ising stationary configurations”. Although this class\ndoes not include all the stationary points of H(n), see e.g. [ 24], the 2NIsing ones are a non-negligible fraction of\nthe whole, especially for large N. The total number of stationary configurations is expected to be O(eN) [21,25].\nWhen ferromagnetic O( n) models defined on regular d−dimensional lattices are considered, as in our case, in the\nthermodynamic limit N→ ∞the energy density levels of the Ising Hamiltonian ( 5) become dense and cover the\nwhole energy density range of all the O( n) models. This latter observation, together with the above mention ed\nproperties of the Ising points, suggests that Ising stationary co nfigurations are the most important ones, so that the\ndensity of states ω(n)of an O(n) model may be approximated in terms of these configurations.\nLet us then rewrite the density of states ω(n)as a sum of integrals over a partition of the phase space,\nω(n)(ε) =/summationdisplay\np/integraldisplay\nUpδ(H(n)−Nε)dΓ, (9)\nwherepruns over the 2NIsing stationary configurations, Upis a neighborhood of the p-th Ising configuration such\nthat{Up}2N\np=1is a proper partition of the configuration space Γ N, that coincides with phase space for spin models\ndefined by the Hamiltonian H(n)in Eq. (4).4\nTwo approximations were introduced in [ 15] to derive Eq. ( 3) from Eq. ( 9): (i) it was assumed that the integrals\nin Eq. (9) depend only on ε, i.e., the neighborhoods Upcan be chosen, or deformed, such as\ng(n)(ε,p) =/integraldisplay\nUpδ(H(n)−Nε)dΓ =g(n)(ε,q) =/integraldisplay\nUqδ(H(n)−Nε)dΓ =g(n)(ε), (10)\nfor anyp,qsuch that H(n)(p) =H(n)(q) =Nε; (ii) Since non-Ising stationary configurations have been neglected in\nthis analysis, only neighborhoods centered around stationary con figurations at energy density εhave been retained\nin the sum ( 9). These two assumptions immediately lead to Eq. ( 3), that we rewrite for convenience:\nω(n)(ε)≈ω(1)(ε)g(n)(ε). (11)\nBothassumptionsareneededtoderiveEq.( 11), andarestrictlyrelatedtoeachother. However,these twoass umptions\nmight well play a very different rˆ ole. In [ 18] it has been shown that in the two analytically tractable special case s,\nthe mean-field and the one-dimensional XYmodels, assumption ( ii) does not hold in general: it holds only when\nε→ε(n)\nc. As a consequence, one should include also stationary configuratio ns with energy ε′/ne}ationslash=εin the sum. By\nintroducing the continuous function\nG(n)(ε,ε′) =/integraldisplay\nUp|H(n)(p)\nN=ε′δ/parenleftBig\nH(n)−Nε/parenrightBig\ndΓ (12)\nsuch that\nG(n)(ε,ε) =g(n)(ε), (13)\nthe density of states can be written as\nω(n)(ε) =ω(1)(˜ε)G(n)(ε,˜ε). (14)\nIn the above expression ˜ εis a suitable function of ε. If ˜ε=ε, then using Eq. ( 13) one recovers Eq. ( 11). This precisely\nhappens in the mean-field and 1- d XYmodels when ε→ε(n)\nc, that is at the phase transition.\nTo compute G(n)(ε,˜ε) the microcanonical solutions of the models are needed, as shown in [18]. For this reason Eq.\n(14) becomes of little use when O( n) models in d >1 are considered, as in our case. Therefore in the “ansatz-based ”\napproximation we shall assume that Eq. ( 11) is valid in the whole energy density range and not only for ε=ε(n)\nc, as\nwe are going to discuss in detail in the following Sections.\nIII. FIRST-PRINCIPLE AND “ANSATZ-BASED” APPROXIMATIONS: AN OVERVIEW\nLet us now discuss how the results recalled in Sec. IIcan be generalized to O( n) spin models in d >1. Our results\ncan be grouped into two different categories according to the appr oach involved in their derivation.\nThe first attempt in the generalizationis to set up an approximationp rocedure that allows to estimate each integral\nappearing in the sum in Eq. ( 9), that yields the density of states ω(n)(ε). This approach is the direct generalization of\nthe techniques already applied to the mean-field and to the one-dime nsionalXYmodels to generic short-range O( n)\nmodels except that, for short range systems, only an approximat e evaluation of each of these integrals is possible.\nThis approximation protocol will be denoted as “first-principles” ap proximation, its starting point being simply the\ndensity of states ω(n)expressed in terms of a suitable partition of the phase space Γ of th e system. The procedure\nwill be presented in Sec. IVand the two-dimensional XYmodel will be considered as a test model of our analysis.\nThe second approach starts from the ansatz on the form of the d ensity of states given by Eq. ( 11) withg(n)(ε) as in\nEq. (10). In this approach it is assumed that the integral in Eq. ( 10) does not depend on the specific point considered\nbut only on its energy density εand that only Ising points with energy density ε′=εcontribute to the density of\nstatesω(n)(ε) in Eq. ( 11). We will denote this kind of analysis as “ansatz-based” approximat ion. The general aspects\nof the method will be presented in Sec. Vand its application to the XYmodel in two dimensions will be discussed\nin detail.\nBoth in the “ansatz-based” and “first-principle” approximations, the main point is to estimate the quantity\ng(n)(ε,p) =/integraldisplay\nUpdΓδ/parenleftBig\nH(n)−Nε/parenrightBig\n=/integraldisplay\nUpdΓδ\n−1\n2N/summationdisplay\ni=1/summationdisplay\nj∈N(i)n/summationdisplay\na=1Sa\niSa\nj−Nε\n, (15)5\nFIG. 1. Index convention for the nearest-neigbors spins ϑ(1)\niandϑ(2)\njofϑifor theXYmodel on a square lattice with edge\nL=√\nNand periodic boundary conditions. The spins in the lattice s itesi+1 and i+Lare said to be second neighbors.\nwhere the expression of H(n)given in ( 4) has been explicitly introduced. Among the O( n) class of models, Eq. ( 15)\ncan be analytically evaluated only in the cases of the mean-field and on e-dimensional XYmodels; its computation\nin the general case being as difficult as to find an exact solution of the models. However some computational\nprocedures can be set up to carry on the calculations, albeit appro ximate. In [ 26] an approximation scheme has been\nintroduced, named the Local-Mean-Field (LMF) approximation. This is a model-dependent procedure that allows to\nreduce the N−dimensional integral in Eq. ( 15) over the configuration space Γ to None-dimensional integrals over\nuncoupled variables. The uncoupling procedure becomes possible on ce suitable model-dependent collective variables\nare defined[ 27]. The LMF procedure will be presented in Sec. IIIA. We shall restrict the presentationto the case n= 2\nand to two-dimensional square lattices with periodic boundary cond itions. The generalization to higher-dimensional\nlattices and different values of nshould be possible along similar lines, but we did not work this out in detail although\nsomething similar (but in the canonical ensemble) has been done in [ 26] for the case n= 2 and d= 3.\nOther schemes than LMF for approximating the g(n)’s may be used. However, their implementation may be\nquite complicated in the case of the “first-principle” approximation, while feasible in the “ansatz-based” one. The\nreason for this will become clear in Sec. IVA. We will give an example of an alternative scheme, corresponding to a\nharmonic approximation of the Hamiltonian around each Ising station ary point, while discussing the “ansatz-based”\napproximation in Sec. V.\nA. The Local Mean-Field (LMF) approximation\nIn the case n= 2 and d= 2, i.e. for the XYmodel in two spatial dimensions, the parametrization ( 6) can be\nchosen such that the integral in Eq. ( 15) becomes\n/integraldisplay\nUpdϑ1...dϑNδ\n−1\n2N/summationdisplay\n/angbracketlefti,j/angbracketrightcos(ϑi−ϑj)−Nε\n=\n=/integraldisplay\nUpdϑ1...dϑNδ\n−N/summationdisplay\ni=1\ncosϑi2/summationdisplay\nj=1cosϑ(j)\ni+sinϑi2/summationdisplay\nj=1sinϑ(j)\ni\n−Nε\n=\n=/integraldisplay∞\n−∞dk\n2πe−ikNε/integraldisplay\nUpdϑ1...dϑNe−ik/summationtextN\ni=1/bracketleftBig\ncosϑi/summationtext2\nj=1cosϑ(j)\ni+sinϑi/summationtext2\nj=1sinϑ(j)\ni/bracketrightBig(16)\nwhere we have introduced the integral representation of the δ−function, δ(x) =1\n2π/integraltext∞\n−∞e−ikxdk;ϑ(j)\nidenotes the\nnearest-neighbor spin jof the lattice site iconsidered according to the convention reported in Fig. 1. The two spins\nidentified by ϑ(1)\niandϑ(2)\niare said to be second-neighbor spins. A generalization of Eq. ( 16) ton >2 andd >2\nshould be straightforward once a proper parametrization for the angular variables as in Eq. ( 6) is chosen and the\nspins are labeled accordingly. In Eq. ( 16),pdenotes any Ising stationary point; the LMF approximation can the n be\nintroduced as follows.\nLet us consider a particular Ising point, i.e., ¯ p=/parenleftbig¯ϑ1,...,¯ϑN/parenrightbig\nwith¯ϑi∈ {0,π} ∀i= 1,...,N. For each angular\nvariable ϑi, in Eq. ( 16) we replace the N−1 variables ϑjwithj/ne}ationslash=iwith their values ¯ϑjat point ¯ p. The variable ϑi6\nis left free to vary in all the range specified by U¯p(ϑi). In this way the angular variables in Eq. ( 16) become uncoupled\nand\n/integraldisplay∞\n−∞dk\n2πe−ikNε/integraldisplay\nUpdϑ1...dϑNe−ik/summationtextN\ni=1/bracketleftBig\ncosϑi/summationtext2\nj=1cosϑ(j)\ni+sinϑi/summationtext2\nj=1sinϑ(j)\ni/bracketrightBig\n=\n=/integraldisplay∞\n−∞dk\n2πe−ikNεN/productdisplay\ni=1/integraldisplay\nU¯p(ϑi)dϑie−ikcosϑi/parenleftBig\ncos¯ϑ(1)\ni+cos¯ϑ(2)\ni/parenrightBig (17)\nwithH(¯p) =Nε′. Eq. (17) clarifies the choice of “Local Mean-Field” as the name for the appr oximation. Indeed, after\nLMF approximation is applied, the angular variables ϑi’s in Eq. ( 17) become independent variables in U¯p(ϑi); each\ndegree of freedom interacts only with a sort of local mean-field gen erated by the spins located in the nearest-neighbor\nlattice sites. Remarkably, with this approximation the contribution o f the sines in the exponent of Eq. ( 17) vanishes,\nsince sin ϑi= 0 when ϑi∈ {0,π}, and the expression in Eq. ( 17) simplifies.\nSince the analysis is restricted to Ising stationary configurations, in Eq. (17) only two cases are possible:\n(a) The second-neighbor spins are equal. In this case cos ¯ϑ(1)\ni= cos¯ϑ(2)\ni=±1, and so\n/integraldisplay\nU¯p(ϑi)dϑie−ikcosϑi/parenleftBig\ncos¯ϑ(1)\ni+cos¯ϑ(2)\ni/parenrightBig\n=/integraldisplay\nU¯p(ϑi)dϑie∓2ikcosϑi; (18)\n(b) The second-neighbor spins are opposite. In this case cos ¯ϑ(1)\ni=−cos¯ϑ(2)\ni, and so\n/integraldisplay\nU¯p(ϑi)dϑie−ikcosϑi/parenleftBig\ncos¯ϑ(1)\ni+cos¯ϑ(2)\ni/parenrightBig\n=/integraldisplay\nU¯p(ϑi)dϑi. (19)\nTo evaluate Eqs. ( 18) and (19), the neighborhoods U¯p(ϑi)have to be defined. A priori there are no particular clues on\nthe best choice of U¯p(ϑi), the only request being that {Up(ϑ1),...,U p(ϑN)}2N\np=1has to be a partition of the phase space\nof the system. In the following, two different choices of Up(ϑi)will be considered:\nUp(ϑi)=/braceleftBigg\n[−π,π],if¯ϑi= 0,\n[0,2π],if¯ϑi=π.(20)\nUp(ϑi)=/braceleftBigg/bracketleftbig\n−π\n2,π\n2/bracketrightbig\n,if¯ϑi= 0,/bracketleftbigπ\n2,3π\n2/bracketrightbig\n,if¯ϑi=π.(21)\nIn principle the choice in Eq. ( 20) should be avoided, the neighborhoods Up(ϑi)’s being not disjoint. We will anyhow\nconsider it in the following, since it is the easiest choice that can be don e a priori for these systems. For the “first-\nprinciple” approximation only the first choice for the Up(ϑi)will be considered. The technical reasons for this choice\nwill be discussed in Sec. IVA. For the “ansatz-based” approximations, instead, both ( 20) and (21) will be considered\nand the effect of neighborhood superposition will be explicitly discuss ed.\nIV. “FIRST-PRINCIPLE” APPROXIMATION\nLet us consider the form of the density of states ω(n)(ε) given by Eq. ( 9). In the following we are going to apply our\nprocedure to derive an approximate form of the density of states ω(2)(ε) for the XYmodel in d= 2. A generalization\nof these techniques to O( n) models with n >2 ind >2 is thought to be possible on the same lines and the key points\nwill be highlighted in the following discussion.\nOnce the XYmodel in d= 2 is considered, Eq. ( 9) can be written as\nω(2)\nN(ε)≃/summationdisplay\np∈Γ1\n2π/integraldisplay∞\n−∞dke−iNkε/integraldisplay\nUpdϑ1...dϑNe−ik/summationtextN\ni=1/bracketleftBig\ncosϑi/summationtext2\nj=1cosϑ(j)\ni+sinϑi/summationtext2\nj=1sinϑ(j)\ni/bracketrightBig\n, (22)\nwherepis any Ising stationary configuration. The integral over the angula r variables ϑican be evaluated by applying\nthe LMF approximation presented in Sec. IIIA. In this case we choose a definition of the integration neighborhood s7\nas in Eq. ( 20). Similar results are supposed to hold also for a choice of Up(ϑi)as in Eq. ( 21) but the calculations\nhave not been carried out in this case. The generalization of Eq. ( 22) to other O( n) models would depend on\nthe parametrization chosen to describe the spin variables and on th e number of nearest neighbors (that is, on the\ndimensionality of the lattice).\nFrom Eqs. ( 18) and (19) we have\n/integraldisplayπ\n−πdϑ e∓2ikcosϑ= 2πJ0(2|k|) (23)\nwhenever a couple of equal second-neighbors spins is present in th e system, and\n/integraldisplay2π\n0dϑ1 = 2π (24)\nwhenever a couple of opposite second-neighbors spins is present in the system; J0(x) is the zero-order Bessel function\nofthe first kind[ 28]. We will denoteby Ncthe numberofcouplesofequalsecond-neighborspins[ 29] andbync=Nc/N\nits number density.\nFor any given value of Nc, a particular family of Ising stationary configurations is selected an d the integral in Eq.\n(22) becomes the product of two different contributions: (2 πJ0(2|k|))Ncdue to couples of equal second neighbors\nspins, and (2 π)N−Ncdue to the opposite couples. In this way Eq. ( 22) becomes\nω(2)\nN(ε)≃N/summationdisplay\nNc=0ν(Nc)(2π)N(1−nc)\n2π/integraldisplay∞\n−∞dk eN[−ik ε+nclog(2πJ0(2√\nk2))], (25)\nwhereν(Nc) is a degeneracy factor counting the number of Ising configuratio ns with a given value Ncof the collective\nvariable. Due to periodic boundary conditions, determining ν(Nc) in Eq. ( 25) reduces to a combinatorial problem\nanalogous to disposing Ncdistinct elements over Npossible empty spaces; we then have ν(Nc) =/parenleftbigN\nNc/parenrightbig\n=N!\nNc!(N−Nc)!.\nThe evaluation of the degeneracy factor ν(Nc) is crucial for this kind of analyses and we will come back on this point\nin Sec.IVA.\nSince we are interested in the large- Nbehavior of the system, the integration over kin Eq. (25) can be computed\nusing the saddle-point method; the saddle-point equation is given by k=iτwithτsatisfying the self-consistency\nequation\nI1(2τ)\nI0(2τ)=ε\n2nc. (26)\nEq. (25) can then be written as\nω(2)\nN(ε)≃N/summationdisplay\nNnc=0N!\nNc!(N−Nc)!eN[−τ ε+log2π+nclog(I0(2τ))]\n≃N/integraldisplay1\n0dnceN[−τ ε−nclognc−(1−nc)log(1−nc)+log2π+nclog(I0(2τ))],(27)\nwhere we have replaced the sum over Ncwith an integration over nc, we have neglected the term −1\nNlog2πin\nthe exponent since its contribution vanishes for N→ ∞, and we have introduced the Stirling approximation of the\nfactorial terms in the binomial coefficient to get\nN!\nNc!(N−Nc)!≃eN[−nclognc−(1−nc)log(1−nc)]. (28)\nFor each value of the energy density ε, we can approximate Eq. ( 27) as\nω(2)\nN≃N eNmaxnc∈[0,1][−τ ε−nclognc−(1−nc)log(1−nc)+log2π+nclog(I0(2τ))]. (29)\nThe entropy density s(2)(ε) in the thermodynamic limit is finally given by:\ns(2)(ε)≃max\nnc∈[0,1]f(nc,τ), (30)8\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle /SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle /SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/Minus2.0/Minus1.5/Minus1.0/Minus0.5 0.00.00.51.01.52.02.53.03.54.0\nΕT\nFIG. 2. Temperature Tas a function of the energy density εas derived from Eq. ( 30) for the XYmodel in d= 2. Our\nresults (red circles) are plotted together with the data obt ained by a Monte Carlo simulation (blue triangles). Errorba rs of the\nsimulation data are smaller than symbol sizes. The dashed bl ue line and the solid red line connecting the two sets of data a re\nmeant as guide to the eyes.\nwith\nf(nc,τ) =−τ ε−nclognc−(1−nc)log(1−nc)+ +log2 π+nclog(I0(2τ)) (31)\nandτnumerically determined from Eq. ( 26). The maximization procedure in Eq. ( 30) can be performed numerically.\nThe temperature Tand the specific heat cas a function of the energy density can be computed from Eq. ( 30) by\nnumerical differentiation and are shown as red circles in Figs. 2and3, respectively. As a comparison, data for Tand\ncas functions of the energy density εhave been computed with a Monte Carlo simulation of the XYmodel with edge\nL= 32 ind= 2, performed with the optimized cluster algorithm spinmcprovided by the ALPS project [ 30]. The\nnumerical data are plotted in Figs. 2and3as blue squares together with the results obtained from our appro ximation.\nFig.2shows that Eq. ( 30) correctly reproduces the asymptotic behavior of the function T(ε) both in the low and\nin the high energy regime, at a semiquantitative level. In particular, f orε≃ −2 our results and the simulation data\nare almost coincident. For ε≥ −1.8 the difference between the numerical and the approximate result s increases;\nthe approximate value of the temperature remains lower than the r esults obtained from the simulations although\nessentially at a constant distance. The difference between the calc ulated and the simulated temperatures never\nexceeds 15%.\nOur results for the specific heat are reported in Fig. 3(red circles). They show a peak for εp,1≃ −1.495 marked\nby the vertical red dot-dashed line, at a slightly lower energy densit y value than εp≃ −1.24 where the peak occurs\nin the simulation data (vertical dashed blue line). The overall shape o f the specific heat sketched by our results is in\nqualitative agreement with the numerical results for ε∈[εp,1,−0.6], although shifted to lower energies. For ε≥ −0.6\nthe agreement increases also quantitatively and two sets of points become essentially indistinguishable. On the other\nside, forε < εp,1the agreement becomes worse. Both from theoretical and numer ical results, we know that c(ε)→0.5\nwhenε→0, see i.e. [ 31,32]. Our results show an abrupt increase for ε≃ −1.85. This is a shortcoming of our\napproximation that is still under investigation.\nA. The degeneracy factor νand a different approximation for ω(n)\nThe analysis presented in Sec. IVshows that the “first-principle” procedure provides a practical m ethod to derive\nan approximate form of the density of states ω(2)in two dimensions. The thermodynamic functions derived from Eq.\n(30) are in reasonably good agreement with the simulations, as shown in F igs.2and3. This suggests that our idea\nof considering only Ising stationary points in the derivation of ω(2)(ε) is trustworthy and provides a good strategy to\napproximate the thermodynamic properties of continuous O( n) models, in principle for any value of nandd.\nAn important feature of the “first-principle” approximation is the n atural emergence of collective variables, like\nNc, in terms of which the stationary configurations can be parametriz ed and the density of states re-written as in9\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/Minus2.0/Minus1.5/Minus1.0/Minus0.5 0.00.00.51.01.52.0\nΕc\nFIG. 3. Specific heat cas a function of the energy density εas derived from Eq. ( 30) for the XYmodel in d= 2. Our\nresults (red circles) are plotted together with the data obt ained by a Monte Carlo simulation (blue triangles). Errorba rs of the\nsimulation data are smaller than symbol sizes. The dashed bl ue line and the solid red line connecting the two sets of data a re\nmeant as guide to the eyes.\nEq. (25). The number and the type of the collective variables depend on sev eral aspects: the model considered,\nthe dimensionality of the lattice, the definition of the integration neig hborhoods Up(ϑi), the specific Ising point p\nconsidered, and the approximation strategy applied to evaluate th e integral in Eq. ( 15). Indeed, as we are going to\nshow in Sec. VC, instead of applying the LMF approximation other strategies could h ave been adopted to compute\nthe integral in Eq. ( 15), like e.g. a harmonic expansion of H(n)around the Ising stationary points. In this case other\nquantities, as the determinant D(p) of the Hessian matrix of H(n)or the density of index ι(p) =I\nN(p) would have\nemerged in the analysis[ 33].\nLet us denote by z(n,d,p) the vector of collective variables needed in the evaluation. In the “ first-principle”\napproach the density of states ω(n)\nN(ε) can always be reduced to a form of the type\nω(n)\nN(ε) =N/summationdisplay\nz(n,d,p)ν(z(n,d,p),N)f(z(n,d,p),N), (32)\nas shown in Eq. ( 25). In the above expression ν(z(n,d,p),N) is the degeneracy factor associated to the collective\nvector of parameters zcounting the number of Ising configurations for given values z(n,d,p) andNof the collective\nvariables and of the number of degrees of freedom of the system, respectively.\nThe degeneracy factor νcan not be evaluated analytically but in some specific cases like the one discussed in Sec.\nIVand those discussed in [ 18]. In the analysis presented in Sec. IVonly integration neighborhoods as in Eq. ( 20)\nhave been considered. Indeed, as will be shown in Sec. VB, a choice of the integration neighborhoods as in ( 21)\nwould have produced the emergence of two different collective varia bles,NcandN3withN3denoting the number of\ntriplets of equal Ising spins. To compute ω(2), it would then be necessary to estimate the degeneracy factor ν(Nc,N3)\ncounting the number of Ising configurations with given values of N,NcandN3. Up to our knowledge, this quantity\nis not analytically known. To carry on the calculation one may then est imateν(z(n,d,p)) numerically. This can be\ndone for instance by performing a Monte Carlo simulation in which the I sing configurations are grouped and counted\naccording to their value of z. This method, although possibly correct, would require a strong co mputational effort\nand we preferred to leave it to future work.10\nOn the other hand the ansatz on the form of the density of states ω(n)given by Eq. ( 11) is able to reproduce with\nunexpected accuracy both the emergence of the phase transitio ns in the O( n) models and even the critical energy\ndensity values at which the transitions are located [ 15]. Then, one may consider Eq. ( 11) as the new starting point\nto approximate the thermodynamic functions of the O( n) system in the whole energy density range [ −d,d]. In this\nkind of approach, called “ansatz-based” approach, the main point remains the estimation of g(n)(ε); this implies the\nemergence of collective variables z(n,p,d), as before. This notwithstanding, it is now reasonable to assume t hat,\ngiven a particular Ising point pwith energy density ε′=H(n)(p)/N, the possible values of the collective variables\nz(n,d,p) would narrow around a typical value ˜z(n,d,p) whenN→ ∞(see e.g. Ref. [ 24]). In this limit the typical\nvalue˜z(n,d,p) would not depend on panymore but only on its energy density ε′. We then have ˜z(n,d,p)≃˜z(n,d,ε′).\nSince the ansatz in Eq. ( 11) imposes that only stationary points with energy density ε′=εhave to be considered in\nthe evaluation of ω(n)(ε), we have that g(n)(z(d,p))→g(n)(z(d,ε)) =g(n)(ε) inddimensions, with z(d,ε) suitable\nfunctions that can be easily estimated by fits of numerical simulation data.\nAll these concepts will be clarified in the next Sections where the “an satz-based” approximation will be applied to\ntheXYmodel in d= 2.\nV. “ANSATZ-BASED” APPROXIMATION\nLet us consider the density of states ω(n)as given by Eq. ( 11); then, our purpose is to estimate g(n)(ε). This can\nbe done for instance by applying the LMF approximation introduced in Sec.IIIA. Let us consider the XYmodel in\nd= 2 as test model of our procedure so that all the technical tools p resented in Sec. IIIAcan be immediately applied\nto this case; the generalization to other O( n) models should be straightforward.\nA. LMF approximation for g(2)(ε)andUp(ϑi)given by Eq. ( 20).\nWe start considering the LMF approximation with the first choice of t he neighborhoods Up(ϑi)given by Eq. ( 20).\nIn this case the calculations proceed on the same lines as in the case o f the “first-principles” approximation, the only\ndifferencebeingthat onlyIsingpointswith energydensity εareconsideredandthe collectivevariable z(2,2,ε) =Nc(ε)\ndoes not depend on the specific Ising point panymore but only on its energy density ε.\nThe density of states can then be written as\nω(2)\nN(ε) =ω(1)\nN(ε)(2π)N(1−nc(ε))\n2π/integraldisplay∞\n−∞eN[−ikε+nc(ε)log(2πJ0(2√\nk2))](33)\nwhereω(1)\nN(ε) plays the rˆ ole of ν(Nc,N) in Eq. ( 25) and is analytically known thanks to the exact solution of the 2- d\nIsing model. On the other hand, nc=nc(ε) is an unknown function that has to be determined. This has been do ne\ninterpolating the numerical data obtained by a Monte Carlo simulation of the two-dimensional XYmodel with edge\nL= 32. The result is shown as a solid blue line in Fig. 4, together with the simulation data.\nThe integral in Eq. ( 33) can be computed with the saddle-point method and the saddle-poin t equation is given by\nk=iλ;λsatisfies a self-consistency equation analogous to Eq. ( 26) given by\nI1(2λ)\nI0(2λ)=ε\n2nc(ε). (34)\nEq. (33) can then be written as\nω(2)\nN(ε)≃ω(1)\nN(ε)eN[−λε+log2π+nc(ε)log(I0(2λ))]=eN/bracketleftBig\ns(1)\nN(ε)−λε+log2π+nc(ε) log(I0(2λ))/bracketrightBig\n(35)\nvalid for N≫1;s(1)\nN(ε) represents is the entropy density of the two-dimensional Ising m odel. Dividing by Nthe\nlogarithm of the above expression, letting N→ ∞and neglecting the sub-leading terms in N, we finally get the\nfollowing expression for the entropy density of the XYmodel in d= 2\ns(2)(ε)≃s(1)(ε)+log2π−λε+nc(ε)log[I0(2λ)], (36)\nwhereλsatisfies the self-consistency equation ( 34). In Fig. 5we plot the temperature Tas a function of the energy\ndensityεas obtained from numerical differentiation of Eq. ( 36) (red points). As in Fig. 2, the values obtained with\na Monte Carlo simulation are shown for comparison (blue squares).11\nFIG. 4. Data for ncobtained by a Monte Carlo simulation of the two-dimensional XYmodel with edge L= 32 (blue solid\ncircles). Errorbars of the simulation data are smaller than the symbol sizes. The solid blue line is the interpolating fu nction\nnc(ε).\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle /SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle /SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/Minus2.0/Minus1.5/Minus1.0/Minus0.5 0.00.00.51.01.52.02.53.03.54.0\nΕT\nFIG. 5. Temperature Tas a function of the energy density εas derived from Eq. ( 36) for the XYmodel in d= 2. Our\nresults (red circles) are plotted together with the data obt ained by a Monte Carlo simulation (blue triangles). Errorba rs of the\nsimulation data are smaller than the symbol sizes. The dashe d blue line and the solid red line connecting the two sets of da ta\nare meant as guide to the eyes.\nFig.5shows that the asymptotic behavior of the function T(ε) is well reproduced at a semiquantitative level by our\napproximation in the harmonic regime (very low-energy), in the low-e nergy regime and in the high-energy limit; the\nagreement is extremely good for low energies. For ε/greaterorsimilar−1.9 the approximate results move away from the numerical\nones and the approximate value of the temperature remains lower t han the results obtained from the simulation[ 34].\nThe largest difference between theoretical and numerical values o fTis about 50%.\nIn Fig.6the values of the specific heat cobtained with a numerical differentiation of Eq. ( 36) are plotted as a\nfunction of the energy density ε. As in Fig. 5the theoretical results are displayed as red circles and are plotted\ntogether with the values of ccomputed by Monte Carlo simulation (blue squares). The specific hea t shows a peak for\nεp,2≃ −1.258 marked by the vertical red dot-dashed line. This value of the en ergy density is very close to εp≃ −1.24\nthat is the energy density value at which the peak occurs in the simula tion data. Our approximation is able to\nreproduce the correct behavior of the specific heat in the high-en ergy regime, while the agreement becomes slightly\nworse in the low-energy case, although qualitatively correct. The d ifference between calculated and simulated values\nofcis about 20% for ε < εpand smaller for ε > εp. The trend of the theoretical results up to ε≃ −1.9 seems to\nsuggests a value for c(−2)∈[0.5,0.6], a bit higher than expected. For ε≃ −2 an abrupt increase of our results is12\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/Minus2.0/Minus1.5/Minus1.0/Minus0.5 0.00.00.51.01.52.0\nΕc\nFIG. 6. Specific heat cas a function of the energy density εas derived from Eq. ( 36) for the XYmodel in d= 2. Our\nresults (red circles) are plotted together with the data obt ained by a Monte Carlo simulation (blue triangles). Errorba rs of the\nsimulation data are smaller than the symbol sizes. The dashe d blue line and the solid red line connecting the two sets of da ta\nare meant as guide to the eyes.\nobserved as in Fig. 3. Again, this is due to a shortcoming of the approximation.\nThe calculations presented in this Section have been repeated using the expression of Up(ϑi)given in Eq. ( 21). The\nresults are presented below.\nB. LMF approximation for g(2)(ε)andUp(ϑi)given by Eq. ( 21).\nWe now consider the LMF approximation with the neighborhoods Up(ϑi)given by Eq. ( 21). Given an Ising config-\nurationp, from Eq. ( 18) we have two possibilities.\n(i) if¯ϑ(1)\ni=¯ϑ(2)\ni= 1 (resp. −1) and¯ϑi= 1 (resp. −1), i.e., the configuration locally looks like\n· ↑ · · ↓ ·\n· ↑ ↑ or respectively · ↓ ↓,\n· · · · · ·(37)\nthen\n/integraldisplayπ\n2\n−π\n2e−ikcosϑi(cos¯ϑ(1)\ni+cos¯ϑ(2)\ni)dϑi=/integraldisplay3π\n2\nπ\n2e−ikcosϑi(cos¯ϑ(1)\ni+cos¯ϑ(2)\ni)dϑi=π(J0(2k)−iH0(2k)) (38)\nregardless of whether ¯ϑi= 0 orπ;H0(x) denotes the zero-order Struve function.\n(ii) If¯ϑ(1)\ni=¯ϑ(2)\ni= 1 (resp. −1) and¯ϑi=−1 (resp. 1), i.e., the configuration locally looks like\n· ↑ · · ↓ ·\n· ↓ ↑ or respectively · ↑ ↓,\n· · · · · ·(39)13\nthen\n/integraldisplayπ\n2\n−π\n2e−ikcosϑi(cos¯ϑ(1)\ni+cos¯ϑ(2)\ni)dϑi=/integraldisplay3π\n2\nπ\n2e−ikcosϑi(cos¯ϑ(1)\ni+cos¯ϑ(2)\ni)dϑi=π(J0(2k)+iH0(2k)) (40)\nregardless of whether ¯ϑi= 0 orπ.\nOn the other hand, Eq. ( 19) is simply given by\n/integraldisplayπ\n2\n−π\n2dϑi1 =/integraldisplay3π\n2\nπ\n2dϑi1 =π. (41)\nWe will denote by n3=N3/Nthe density of triplets of equal spins forming a local configuration a s in Eq. ( 37).\nCombining Eqs. ( 38) and (40) with Eq. ( 17) we get\ng(2)(ε) =πN\n2π/integraldisplay∞\n−∞dk eikNεeN[n3(ε)log(J0(2k)−iH0(2k))]×eN[(nc(ε)−n3(ε))log(J0(2k)+iH0(2k))].(42)\nIn this case, the vector of parameters z(2,2,ε) is given by z(2,2,ε) =nc(ε)∪n3(ε).\nThe integral in Eq. ( 42) can be computed with the saddle point method in the large- Nlimit. The saddle point\nequation is given by k=−iζ; making use of the properties of the Bessel and of the Struve fun ctions and performing\nsome algebra, Eq. ( 42) can be written as\ng(2)(ε)≃eN[−ζε+logπ+n3(ε)log(I0(2ζ)−L0(2ζ))]eN[(nc(ε)−n3(ε))log(I0(2k)+L0(2ζ))](43)\nwithζsatisfying the self-consistency equation\n−ε+n3(ε)\nI0(2ζ)−L0(2ζ)/parenleftbigg\n2I1(2ζ)−/parenleftbigg2\nπ+L−1(2ζ)+L1(2ζ)/parenrightbigg/parenrightbigg\n+\n+nc(ε)−n3(ε)\nI0(2ζ)+L0(2ζ)/parenleftbigg\n2I1(2ζ)+/parenleftbigg2\nπ+L−1(2ζ)+L1(2ζ)/parenrightbigg/parenrightbigg\n= 0.(44)\nAs in the case of nc(ε), the function n3(ε) can be obtained interpolating the numerical data arising from a Mon te\nCarlo simulation of the system; the simulation data for n3are not shown here.\nWe can now insert Eq. ( 43) in Eq. ( 11). By taking the logarithm of the resulting expression and neglecting the\nsub-leading term in N, we finally arrive to the following expression for the entropy density\ns(2)(ε)≃s(1)(ε)−ζε+ logπ+n3(ε) log[I0(2ζ)−L0(2ζ)] +(nc(ε)−n3(ε)) log[I0(2ζ)+L0(2ζ)] (45)\nvalid in the N→ ∞limit;ζhas to be determined numerically from Eq. ( 44).\nFig.7showsTas function of the energy density εobtained by numerical differentiation from Eq. ( 45) (red points).\nAs in Fig. 5the theoretical values are plotted together with the data obtaine d by a Monte Carlo simulation of the\nsystem (blue squares).\nFor Fig. 7the same comments can be done as for Fig. 5. The theoretical results are in good agreement with the\nnumerics in the entire energy density range. In comparison with the theoretical caloric curve in Fig. 5, the caloric\ncurve resulting from this approximation and shown in Fig. 7is closer to the numerical results: the largest difference\nbetween theory and simulation is here around 20%. This fact is the eff ect of the different choice of the integration\nneighborhoods. In particular, if the integration neighborhoods ar e superposed, as in Eq. ( 20),g(2)(ε) is overestimated\nand the difference between the theoretical and the numerical res ults is larger than in the case of a choice of the\nintegration neighborhoods as in Eq. ( 21).\nThis fact is even more evident for the specific heat. The theoretica l results are plotted in red in Fig. 8. With a\nchoice of the integration neighborhoods of Eq. ( 42) as in Eq. ( 21), the energy value of the peak of the specific heat\nderived with our approximation is εp,3≃ −1.3≃εp≃ −1.24; moreover, the entire high energy regime for ε > εp\nis in good quantitative agreement with the numerics. On the other ha nd, forε < εpthe two sets of data separate\nthemselves and in the low energy regime the same considerations as f or Figs.3and6can be done.\nC. Harmonic approximation for g(2)(ε)\nIn this last Section we present an alternative way to estimate the fu nctiong(2)appearing in Eq. ( 11), using a\nharmonic expansion of the Hamiltonian around each Ising stationary point. This approach can be seen as the natural14\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle /SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle /SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/Minus2.0/Minus1.5/Minus1.0/Minus0.5 0.00.00.51.01.52.02.53.03.54.0\nΕT\nFIG. 7. Temperature Tas a function of the energy density εas derived from Eq. ( 45) for the XYmodel in d= 2. Our\nresults (red circles) are plotted together with the data obt ained by a Monte Carlo simulation (blue triangles). Errorba rs of the\nsimulation data are smaller than the symbol sizes. The dashe d blue line and the solid red line connecting the two sets of da ta\nare meant as guide to the eyes.\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/Minus2.0/Minus1.5/Minus1.0/Minus0.5 0.00.00.51.01.52.0\nΕc\nFIG. 8. Specific heat cas a function of the energy density εas derived from Eq. ( 36) for the XYmodel in d= 2. Our\nresults (red circles) are plotted together with the data obt ained by a Monte Carlo simulation (blue triangles). Errorba rs of the\nsimulation data are smaller than the symbol sizes. The dashe d blue line and the solid red line connecting the two sets of da ta\nare meant as guide to the eyes.15\ngeneralization of the classical low temperature harmonic expansion , see e.g. the Stillinger and Weber theory [ 35].\nHowever, in the present case, Ising stationary points are not only minima but saddles of every index so that the\ngeneralization is far from being straightforward. For this reason, we only discuss the “ansatz-based” approach here.\nLet us consider for the moment only the factor g(2)(ε,¯p) in Eq. ( 15). The approximation consists in expanding the\nHamiltonian up to the harmonic order around the Ising configuration ¯p= (¯ϑ1,...¯ϑN):\ng(2)(ε,¯p)≃g(2)\nhar(ε,¯p) =/integraldisplay\nU¯pdϑ1...dϑNδ/parenleftbigg1\n2(ϑ−¯p)H(¯p)(ϑ−¯p)T−N(ε−ε¯p)/parenrightbigg\n, (46)\nwhereH(¯p) is the Hessian matrix of H(2)evaluated on the Ising stationary point ¯ pwith energy H(2)(¯p) =Nε¯p. We\nnow shift the coordinates to move the stationary point ¯ pto the origin of the coordinate system, a change of variables\nto diagonalize the Hessian matrix and a rescaling of the integration va riables according to the eigenvalues of the\nHessian. With this procedure, we have\ng(2)(ε,¯p)≃g(2)\nhar(ε,¯p) =1/radicalbig\n2ND(¯p)/integraldisplay\nU¯pdy1...dyN+dx1...dxN−δ\nN+(¯p)/summationdisplay\ni=1y2\ni−N−(¯p)/summationdisplay\ni=1x2\ni−N(ε−ε¯p)\n(47)\nwhereD(¯p) is the determinant of H(¯p),N−(¯p) is the index of ¯ p,N+(¯p) =N−N−(¯p). Observe that in the previous\nexpression, yiwith 1≤i≤N+(¯p) are the variables associated with the directions in phase space cor responding to\nthe positive eigenvalues and xiwith 1≤i≤N−(¯p) are the variables associated with the directions in the phase space\ncorresponding to the negative eigenvalues.\nTo go further, we have to specify in a suitable way the integration ne ighborhoods U¯p. As in the LMF case, this\nchoice determines the feasibility of the calculation. In the following, w e consider U¯pas the union of two balls [ 36]\ncentered on the stationary point. The first one is associated with t he directions yiand has radius α, and the second\none to the directions xiand has radius β. With this choice, we have\ng(2)\nhar(ε,¯p) =1/radicalbig\n2ND(¯p)/integraldisplay∞\n−∞dy1...dyN+dx1...dxN−δ/parenleftbig\nY2−X2−N(ε−ε¯p)/parenrightbig\nΘ/parenleftbig\n−Y2+N+α/parenrightbig\nΘ/parenleftbig\n−X2+N−β/parenrightbig\n.\n(48)\nwhere we have introduced the following variables:\nY2=N+(p)/summationdisplay\ni=1y2\niandX2=N−(p)/summationdisplay\ni=1x2\ni. (49)\nIn principle, αandβshould be considered as functions of the specific Ising configuratio n ¯p. This would however\nintroduce in the theory a huge number of free parameters. Thus, we will assume that αandβare fixed real numbers.\nMoreover, g(2)\nhardepends only very weakly on α: this can be easily understood by recalling that, in the canonical\nensemble, large values in the positive directions are damped in a Gauss ian way. For this reason, we can also assume\nthatα≫β.\nAs explained in Section IVA, in the “ansatz-based” approach the continuous weight does not really depend on the\nspecific Ising stationary point ¯ pbut only on its energy density, which corresponds to assume that\n/braceleftbigg\nD(¯p)1\nN,N+(¯p)\nN,N−(¯p)\nN/bracerightbigg\n≃/braceleftbigg\nD(ε)1\nN,N+(ε)\nN,N−(¯ε)\nN/bracerightbigg\n. (50)\nThese functions can be computed numerically as in the case of nc(ε) andn3(ε) discussed in Secs. IVandV, respec-\ntively (data not shown). We recall that in the “ansatz-based” app roach, only continuous weights corresponding to\nIsing stationary points with energy ε¯p=εcontribute to ω(2). We refer the reader to Appendix Afor the detailed\ncomputation of g(2). Under these conditions, the approximate density of states beco mes\nω(2)(ε)≃exp/braceleftBig\nN\n2/bracketleftBig\n2s(1)(ε)+1−log2−LD(ε)+logπ−n+(ε)logn+(ε)\n2−n−(ε)logn−(ε)\n2+log(n−(ε))+log(β)/bracketrightBig/bracerightBig\n(51)\nwhereLD= limN→∞1\nNlogD,n−=N−/N,n+= 1−n−. Two observations are mandatory at this level. First, we\nnote that Eq. ( 51) does not depend anymore on α; second, even if the entropy does depend on β, thermodynamic\nfunctions such as temperature or specific heat do not depend on t he free parameter β. Thus, thermodynamic functions\ndo not depend on any free parameters.\nThe behavior of the temperature and of the specific heat as a func tion of the energy density εderived from Eq.\n(51) are reported in Fig. 9and10, respectively. For the latter figures, similar considerations as in th e case of Fig. 5\nand6can be done.16\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle /SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle /SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/Minus2.0/Minus1.5/Minus1.0/Minus0.5 0.00.00.51.01.52.02.53.03.54.0\nΕT\nFIG. 9. Temperature Tas a function of the energy density εas derived from Eq. ( 51) for the XYmodel in d= 2. Our\nresults (red circles) are plotted together with the data obt ained by a Monte Carlo simulation (blue triangles). Errorba rs of the\nsimulation data are smaller than the symbol sizes. The dashe d blue line and the solid red line connecting the two sets of da ta\nare meant as guide to the eyes.\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/Minus2.0/Minus1.5/Minus1.0/Minus0.5 0.00.00.51.01.52.02.5\nΕc\nFIG. 10. Specific heat cas a function of the energy density εas derived from Eq. ( 51) for the XYmodel in d= 2. Our\nresults (red circles) are plotted together with the data obt ained by a Monte Carlo simulation (blue triangles). Errorba rs of the\nsimulation data are smaller than the symbol sizes. The dashe d blue line and the solid red line connecting the two sets of da ta\nare meant as guide to the eyes.17\nVI. CONCLUDING REMARKS\nIn this paper, we have presented two different semi-analytic ways t o approximate the density of states, hence\nthermodynamic functions, of short-range O( n) models on a regular lattice. We have performed explicit calculations in\nthe case of the XYmodel in d= 2, but the generalization of our results to higher dimensions and/o r to O(n) models\nwithn >2 should be possible along the same lines. We used an energy landscape approach inspired by our previous\nworks reported in Refs. [ 15,18].\nWe have developed two different schemes of approximation. The firs t one, called “first-principle” approximation,\nis an attempt to follow the same kind of derivation used for the 1- dand the mean-field XYmodels in [ 18]. The\nsecond one, called “ansatz-based” approximation, assumes Eq. ( 3) as an ansatz on the form of the density of states\nof O(n) models with n≥2. In both cases, the delicate point is to approximate the weights ggiven by Eq. ( 15). Two\ndifferent approximations have been used, the local mean field (LMF) and a harmonic approximation around each\nIsing configuration.\nTheXYmodel in d= 2 is not exactly solvable and any approximation scheme has to be com pared with the results\ncoming from numerical simulations. Calculated thermodynamic funct ions always show a qualitative agreement with\nthe numerical data and in some cases also a quantitative agreement . Particularly interesting is the presence of the\npeak in the specific heat reported in Figs. 3,6and8. Despite the approximations involved in its derivation, the\nspecific heat correctly shows a peak and not a divergence as it happ ens, instead, for the Ising model in d= 2 at\nε(1)\nc=√\n2. For this particular value of the energy density our numerical pro cedure correctly produces a finite value\nofcalthough the numerical convergence is more delicate as highlighted b y the scattered data present in Figs. 6and8\nforε≃ε(1)\nc. In case of Fig. 8the agreement is also quantitative as far as the location of the peak of the specific heat\nis concerned.\nAs stressed throughout the paper, the concepts presented he re are valid in principle for any O( n) model in any\nspatial dimensions. Hence the generalization of the calculations car ried out here for the XYmodel on a square lattice\nto other O( n) models in ddimensions should be possible, with possibly some technical complicatio ns, and may give a\nhint towards the development of approximation techniques that ma y be valid for estimating the density of states of\nlarge classes of Hamiltonian systems.\nAppendix A: “Ansatz-based” approximation with harmonic ex pansion\nIn this Appendix, we compute exactly the integrals in Eq. ( 48) under the assumption ε¯p=ε. We observe that the\ncomputation in this appendix can be extended also to the case εp/ne}ationslash=ε; however, we do not report the details of this\nmore general case here, as we do not need it for the development o f the “ansatz-based” approximation.\nInserting the change of measure in Eq. ( 48) due to the change of variables in Eq. ( 49), and setting εp=ε, we have\ng(2)\nhar(ε,¯p) =C(N+)C(N−)/radicalbig\n2ND(¯p)/integraldisplay∞\n0dY dX YN+−1XN−−1δ/parenleftbig\nY2−X2/parenrightbig\nΘ/parenleftbig\n−Y2+N+α/parenrightbig\nΘ/parenleftbig\n−X2+N−β/parenrightbig\n,(A1)\nwhereC(k) is the surface of the unitary ( k−1)-dimensional sphere Sk−1:\nC(k) =kπk\n2\nΓ/parenleftbigk\n2+1/parenrightbig. (A2)\nUsing the delta functional to express Xas a function of Yand with simple computations, we get\ng(2)\nhar(ε,¯p) =C(N+)C(N−)/radicalbig\n2ND(¯p)/integraldisplay∞\n0dY YN−2Θ/parenleftbig\n−Y2+N+α/parenrightbig\nΘ/parenleftbig\n−Y2+N−β/parenrightbig\n=\n=C(N+)C(N−)/radicalbig\n2ND(¯p)/integraldisplay√\nN n−β\n0dY YN−2= (A3)\n=1\nN−1C(N+)C(N−)/radicalbig\n2ND(¯p)(N n−β)N−1\n2.\nTopassfromthefirsttothesecondlineweassumedthat α≫β, asdiscussedin Section VC,sothat min {n+α,n−β}=\nn−β. The above expression for g(2)\nharcan be simplified in the large Nlimit by retaining only those factors that are\nexponential in N. We thus obtain\ng(2)\nhar(ε,¯p)∼N≫1exp/braceleftbiggN\n2/bracketleftbigg\n1−LD(¯p)+logπ\n2−n+(¯p)logn+(¯p)\n2−n−(¯p)logn−(¯p)\n2+log(n−(¯p))+log( β)/bracketrightbigg/bracerightbigg\n,(A4)18\nwhere we have used the classical asymptotic expansion of the Gamm a function. Inserting the last expression in the\nansatz (11), making use of Eq. ( 50), we obtain the approximate density of states given in Eq. ( 51).\n[1] D. J. Wales, Energy Landscapes (Cambridge University Press, Cambridge, 2004).\n[2] P. G. Debenedetti and F. H. Stillinger, Nature, 410, 259 (2001).\n[3] F. Sciortino, Journal of Statistical Mechanics: Theory and Experiment, 2005, P05015 (2005).\n[4] J. N. Onuchic, Z. Luthey-Schulten, and P. G. Wolynes, Ann ual Review of Physical Chemistry, 48, 545 (1997).\n[5] L. N. Mazzoni and L. Casetti, Phys. Rev. Lett., 97, 218104 (2006).\n[6] L. N. Mazzoni and L. Casetti, Phys. Rev. E, 77, 051917 (2008).\n[7] M. Kastner, Physica A: Statistical Mechanics and its App lications, 359, 447 (2006).\n[8] H. Federer, Geometric Measure Theory (Springer, New York, 1969).\n[9] L. Casetti and M. Kastner, Phys. Rev. Lett., 97, 100602 (2006).\n[10] M. Kastner, O. Schnetz, and S. Schreiber, Journal of Sta tistical Mechanics: Theory and Experiment, 2008, P04025 (2008).\n[11] L. Casetti, M. Kastner, and R. Nerattini, Journal of Sta tistical Mechanics: Theory and Experiment, 2009, P07036 (2009).\n[12] L. Casetti, M. Pettini, and E. G. D. Cohen, Physics Repor ts,337, 237 (2000).\n[13] M. Pettini, Geometry and topology in Hamiltonian dynamics and statisti cal mechanics (Springer, New York, 2007).\n[14] M. Kastner, Rev. Mod. Phys., 80, 167 (2008).\n[15] L. Casetti, C. Nardini, and R. Nerattini, Phys. Rev. Let t.,106, 057208 (2011).\n[16] In this form, Eq. ( 3) cannot be exact neither for finite systems nor in the thermod ynamic limit [ 18] because it would\nimply wrong values of the specific heat critical exponent α(n), that would be given by α(n)=−α(1)for anyn. However,\nit correctly reproduces the sign of the critical exponent th at entails a cusp-like behavior of the specific heat at critic ality\nrather than a divergence, as happens in the Ising case.\n[17] A. Campa, A. Giansanti, and D. Moroni, Journal of Physic s A: Mathematical and General, 36, 6897 (2003).\n[18] C. Nardini, R. Nerattini, and L. Casetti, Journal of Sta tistical Mechanics: Theory and Experiment, 2012, P02007 (2012).\n[19] As to a possible relation between the BKT transition and the 2-dIsing transition, it is interesting to notice that they do\nshare some “weak” universality as defined by Suzuki [ 37], despite being very different transitions. The critical ex ponent\nratioβ/νand the exponent δ, for instance, are equal in the two universality classes [ 38].\n[20] R. Nerattini and L. Casetti, in preparation (2013).\n[21] D. Mehta and M. Kastner, Annals of Physics, 326, 1425 (2011).\n[22] D. Mehta, Phys. Rev. E, 84, 025702 (2011).\n[23] These considerations are valid also for O( n) models defined on generic graphs and with a generic interact ion matrix Jijin\nEq. (4).\n[24] R. Nerattini, M. Kastner, D. Mehta, and L. Casetti, Phys . Rev. E, 87, 032140 (2013).\n[25] R. Schilling, Physica D: Nonlinear Phenomena, 216, 157 (2006).\n[26] R. Nerattini, Paesaggio energetico dei modelli XY, Master’s thesis, Universit` a di Firenze (2010).\n[27] Their rˆ ole reminds theoneplayedby NπandNdin[18] for themean-fieldand theone-dimensional XYmodels, respectively.\n[28] M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).\n[29] The number of couples of opposite spins will be simply gi ven byND=N−Nc.\n[30] Web page: http://alps.comp-phys.org/ .\n[31] J. M. Kosterlitz and D. J. Thouless, Journal of Physics C : Solid State Physics, 6, 1181 (1973).\n[32] V. L. Bereˇ zinskij, Sov. Phys. JETP, 32, 493 (1971).\n[33] More precisely, the quantity that would appear is D(p)1\nNand will be referred to as the “reduced determinant” of the\nHessian matrix of H(n)inp; the index Iof a stationary point pis the number of negative eigenvalues of the Hessian matrix\nof the Hamiltonian evaluated at p.\n[34] This behavior is analogous to the one obtained in [ 26] with a similar procedure applied to the partition function of the\nsystem.\n[35] F. H. Stillinger and T. A. Weber, Science, 225, 983 (1984).\n[36] A different choice of the U¯pgiven by a single ball, may look more natural at first sight. Th is choice, however, lead to the\ninconsistent result that if N−= 0, the minimum energy density value available to the system in the set U¯pis lower than\nε¯p.\n[37] M. Suzuki, Progress of Theoretical Physics, 51, 1992 (1974).\n[38] P. Archambault, S. T. Bramwell, and P. C. W. Holdsworth, Journal of Physics A: Mathematical and General, 30, 8363\n(1997)."    },    {        "title": "1402.0202v2.Free_fall_energy_density_and_flux_in_the_Schwarzschild_black_hole.pdf",        "content": "arXiv:1402.0202v2  [gr-qc]  2 May 2015Free-fall energy density and flux in the Schwarzschild black hole\nWontae Kim1∗andBibhas Ranjan Majhi2†\n1Department of Physics, Sogang University, Seoul 121-742, R epublic of Korea\n2Racah Institute of Physics, Hebrew University of Jerusalem ,\nGivat Ram, Jerusalem 91904, Israel\nOctober 8, 2018\nAbstract\nInthe four-dimensionalbackgroundofSchwarzschildblackhole, w einvestigatethe energy\ndensities and fluxes in the freely falling frames for the Boulware, Unr uh, and Israel-Hartle-\nHawking states. In particular, we study their behaviors near the h orizon and asymptotic\nspatial infinity by using the trace anomaly of a conformally invariant s calar field. In the\nBoulware state, both the energy density and flux are negative dive rgent when the observer is\ndropped at the horizon, and asymptotically vanish. In the Unruh st ate, the energy density\nis also negative divergent at the horizon while it is positive finite asympt otically. The flux in\nthe Unruh state is alwayspositive and divergentat the horizon. In t he Israel-Hartle-Hawking\nstate, the energy density depends on the angular motion of free f all, and fluxes vanish at the\nhorizon and the spatial infinity. Finally, we discuss the role of the neg ative energy density\nnear the horizon in the evaporating black hole.\n∗E-mail: wtkim@sogang.ac.kr\n†E-mail: bibhas.majhi@mail.huji.ac.il\n11 Introduction\nBlack holes are still mysterious objects in that they can be e xcited thermally like a black body;\nhowever, they can evaporate eventually through Hawking rad iation [1, 2], which yields infor-\nmation loss problem and related intriguing issues [3, 4, 5]. The original particle explanation of\nHawking radiation is based on quantum-mechanical pair crea tions consisting of the positive and\nnegative energy states with appropriate boundary conditio ns. If Hawking radiation plays a role\nof information carrier, then the full information can be rec overed by accumulating the whole\nradiation. Thus it is expected that the quantum-mechanical unitary evolution of black hole will\nbe accommodated. Of course, there is another way to look into some aspects of Hawking radi-\nation, which can be realized near the horizon along the line o f Unruh [6]. At the horizon, it has\nbeen believed that the Kruskal coordinates play a role of fre e-fall coordinate and the space-time\nis locally flat. In that region, there are no out-going modes a nd the space-time is regarded as a\nfree-fall vacuum where there is no radiation, whereas the fix ed observer in the Schwarzschild co-\nordinates which is equivalent to the accelerated observer i n the Rindler coordinates can perceive\nradiation following the Unruh effect [6].\nOn the other hand, there is an interesting observation which is of relevance to a constant\nnegative energy density that was measured by an observer orb iting close to a black hole [7].\nNow, one can extend this to a time-like radial geodesic curve as a trajectory of freely falling\nobserver and study the free-fall energy density around a bla ck hole. In the collapsing two-\ndimension dilatonic black hole [8], the negative energy den sity was found near the horizon which\nis divergent at the horizon when the free fall begins just at t he horizon [9]. It has also been\nshown that the positive and negative energy regions are sepa rated by a time like curve and the\nnegative energy region is reduced as time goes on. This calcu lation can also be done in the\ntwo-dimensional soluble Schwartzschild black hole for thr ee black hole states of the Boulware\n[10], Unruh [6], and Israel-Hartle-Hawking states [11, 12] by employing the trace anomaly of\nscalar fields [13]. It turns out that the behaviors of the ener gy-density rely on the initial free-fall\nstatus [14] in the sense that the negative energy density is d ominant near the horizon and is\ndivergent just at the horizon but it can be surprisingly posi tive finite even at the horizon in the\nUnruh state depending on the starting position of the free fa ll. The stretched horizon observed\nby the freely falling observer was also discussed in connect ion with the negative energy density\nrelated to instability of black hole [15]. And, the detector response has been studied for geodesic\ntrajectories in the context of Hawking phenomenon [16] whic h predicts higher temperature near\nthe horizon. This implies that the energy flux near the horizo n measured by these observers\nmust diverge. Recently, the negative energy region was defin ed as “the zone” [17] and its role\nwas intensively discussed in connection with the firewall ar gument [18] (for a similar prediction\nfrom different assumptions, see [19]). On the other hand, it ha s also been suggested that there\nis no apparent need for firewalls since the unitary evolution of black hole entangles a late mode\nlocated outside the event horizon with a combination of earl y radiation and black hole states\n[20], and claimed that the remaining set of nonsingular real istic states do not have firewalls but\nyet preserve information [21].\nIn this work, we would like to study the behaviors of the energ y density and flux measured\nby the freely falling observer on the Schwarzschild black ho le background. The advantage of\nlower dimensional models was to be able to determine the ener gy-momentum tensors exactly\nby solving trace anomaly equation and covariant conservati on relation. However, in the four\ndimensional black hole it is non-trivial task to obtain the e xact form of the energy-momentum\ntensors from the conformal anomaly analytically [13, 22]. F ortunately, the essential ingredient in\nblack holes mostlycomes fromthenearhorizon andtheasympt otic region wherethecomplicated\nequations are relatively simplified [23]. So we are going to c alculate the energy density and the\nflux for three different black hole states in the free-fall fram e, and study what state is relevant\n2to the angular free-fall motion at the event horizon and exam ine whether the similar divergent\nbehavior of energy density at the horizon appears or not in th e four-dimensional case. This\nis very important because this negative divergent behavior of the energy density is not the\ntwo-dimensional characteristics but more or less generic f eature.\nTheorganization of thepaperisas follows. Insection 2, the propervelocity fromthegeodesic\nequation is obtained and the radial velocity is chosen as zer o for convenience when the free-fall\nbegins toward the black hole. In the next section, the approx imate energy-momentum tensors\nnear the horizon and the spatial infinity are introduced for t hree black hole states in the self-\ncontained manner [23]. In section 4, it will be shown that in t he Boulware state the energy\ndensity is negative divergent at the horizon, and asymptoti cally vanishes. In the Unruh state, it\nis negative at the horizon whereas it is positive finite asymp totically. For the Boulware and the\nUnruh states, the angular motion of free fall is not significa nt at the horizon. However, in the\nIsrael-Hartle-Hawking state, for 2 M > L, the energy density is negative at the horizon whereas\nfor 2M < Lit can be positive. The behaviour of the flux, near the horizon and at the spatial\ninfinity, will be discussed in section 5. The final section is d evoted to conclusion and discussion.\n2 The setup: Freely falling observer in black hole spacetime\nInthissection, thecomponentsofthevelocity vectorforaf reelyfallingobserverintheSchwarzschild\nspacetime will be given. These are found out by solving geode sic equations. The four dimen-\nsional Schwarzschild black hole space time is given by the fo llowing metric,\nds2=−f(r)dt2+dr2\nf(r)+r2/parenleftig\ndθ2+sin2θdφ2/parenrightig\n, (1)\nwhere the metric function is f(r) = 1−2M/rand the event horizon is located at rH= 2M. The\nmetric coefficients are independent of time tand the angular φcoordinates. They correspond to\ntwoconserved quantities, namely theenergy andangular mom entum. Moreover, sincethemetric\n(1) is spherically symmetric, the motion of the observer is c onfined on the plane. Therefore,\nthere is no motion along the normal to the plane and hence θ=π/2 is chosen. Using this fact,\nit is easy to solve geodesic equations. Remember that in the s ubsequent analysis, without any\nloss of generality, we will consider θ=π/2 (For more details, see section 7 .4 of [24]). Solution of\ngeodesic equation leads to the following form of the proper v elocity of the freely falling observer,\nua=/parenleftigdt\ndτ,dr\ndτ,dθ\ndτ,dφ\ndτ/parenrightig\n=/parenleftigE\nf(r),±[E2−V2\neff(r)]1/2,0,L\nr2/parenrightig\n(2)\nwhere\nV2\neff(r) =f(r)/parenleftig\n1+L2\nr2/parenrightig\n. (3)\nHere,EandLare two constants which are conserved quantities correspon ding to time trans-\nlational and rotational symmetry of the spacetime metric (1 ), respectively. They are identified\nwith the energy and angular momentum per unit mass. The posit ive sign refers to the outgoing\nobserver away fromthe horizon and the negative sign corresp ondsto the ingoing observer toward\nthe horizon. In our calculations, we shall consider only the ingoing observer.\nNow, it can be shown that the energy and angular momentum are r elated by the initial\nposition of the observer, provided the initial velocity is z ero. It will be done in the following\nway. The radial velocity of the observer with respect to the S chwarzschild time is\n/parenleftigdr\ndt/parenrightig2\n=/parenleftigf(r)\nE/parenrightig2/bracketleftig\nE2−V2\neff(r)/bracketrightig\n. (4)\n3Note that the radial velocity always vanishes at r= 2M, which is independent of the energy. If\nthe initial velocity of the frame is zero at rs, then we obtain E2−V2\neff(rs) = 0, where rsis the\ninitial position of the observer. This helps us to express th e conserved quantity Eas\nE=Veff(rs) =/radicaligg\nf(rs)/parenleftig\n1+L2\nr2s/parenrightig\n, (5)\nwhich can be reduced to the two-dimensional case for L= 0 [14]. This will be used later to\nexpress the energy density and flux in terms of the initial pos ition of the observer.\n3 Components of renormalized energy-momentum tensor\nIn four dimensions, breaking of the conformal symmetry lead s to the trace anomaly, and the\nform of the anomalous energy-momentum tensor can be obtaine d by solving two equations:\nconservation of the stress-tensor and trace anomaly expres sion [13]. Unfortunately, the exact\nform of the components can not be evaluated. But still, it is p ossible to evaluate only the\nasymptotic forms near the horizon and at infinity which are ne cessary for our analysis. The\nvalue of the integration constants appearing in the solutio ns, depends on the nature of the\nvacuum. In the below, we give the components in ( t,r,θ,φ) coordinates for three vacua. Near\nthe horizon ( r→2M) these are given by [13, 22, 23]\n/angbracketleftig\nTa\nb/angbracketrightig\nB→C(0)\nB\n−1 0 0 0\n0 1/3 0 0\n0 0 1 /3 0\n0 0 0 1 /3\n, (6)\n/angbracketleftig\nTa\nb/angbracketrightig\nU→C(0)\nU\n1/f−1 0 0\n1/f2−1/f0 0\n0 0 0 0\n0 0 0 0\n, (7)\n/angbracketleftig\nTa\nb/angbracketrightig\nH→C(0)\nH\n1 0 0 0\n0 1 0 0\n0 0 2 0\n0 0 0 2\n, (8)\nwhere in the above B,UandHstand for the Boulware state [10], Unruh state [6], and Israe l-\nHartle-Hawking state [11, 12], respectively. The expressi ons forC(0)’s are\nC(0)\nB=−1\n30 212π2M4f2, C(0)\nU=L\n4π(2M)2, C(0)\nH=1\n96M4, (9)\nwhereLis called the Luminosity which is always positive and finite q uantity [23]. On the other\nhand, the energy-momentum tensor component in r→ ∞region are\n/angbracketleftig\nTa\nb/angbracketrightig\nB→ O(r−6), (10)\n/angbracketleftig\nTa\nb/angbracketrightig\nU→C(∞)\nU\n−1−1 0 0\n1 1 0 0\n0 0 0 0\n0 0 0 0\n, (11)\n4/angbracketleftig\nTa\nb/angbracketrightig\nH→C(∞)\nH\n−1 0 0 0\n0 1/3 0 0\n0 0 1 /3 0\n0 0 0 1 /3\n, (12)\nwhere\nC(∞)\nU=L\n4πr2, C(∞)\nH=1\n30 212π2M4. (13)\nIt must be noted that the above expressions are valid only in t he two regions and reflects the\nasymptotic nature of them, not the exact form. Hence any phys ical quantity obtained from\nthese components is also approximate. But interesting thin g is that we can still say about the\nnature of the quantities which are our interest of study. In t he below, we shall discuss the energy\ndensity and flux measured by a freely falling observer.\n4 Energy densities in three different states\nThe energy density for a freely falling observer in different s tates can be calculated using the\nfollowing expression\nǫ=/angbracketleftig\nTa\nb/angbracketrightig\nuaub. (14)\nIn this calculation, the results are valid up to the leading o rder in 1 /f(r) for the case of near\nhorizon limit or the inverse of radial distance rfor the case of the asymptotic infinite limit.\nThe other terms are irrelevant because we are only intereste d in the asymptotic behavior of the\nenergy density.\nLet us first start with the Boulware state. In the region of r→2Malong with θ=π/2, it\ncan be given as\nǫ(0)\nB(r|rs) =C(0)\nB/bracketleftig/angbracketleftig\nTt\nt/angbracketrightig\nutut+/angbracketleftig\nTr\nr/angbracketrightig\nurur+/angbracketleftig\nTφ\nφ/angbracketrightig\nuφuφ/bracketrightig\n=C(0)\nB/bracketleftigE2\nf(r)+1\n3f(r)(E2−V2\neff(r))+L2\n3r2/bracketrightig\n. (15)\nUsing Eqs. (3) and (5), the expression (15) can be simplified a s\nǫ(0)\nB(r|rs) =−1\n30 212π2M4f2(r)/bracketleftig4f(rs)\n3f(r)/parenleftig\n1+L2\nr2s/parenrightig\n−1\n3/bracketrightig\n, (16)\nwhich is always negative divergent at the horizon. Now if the measurement is done at the\nstarting point of free fall, we can obtain\nǫ(0)\nB(rs|rs) =−1\n30 212π2M4f2(rs)/parenleftig\n1+4L2\n3r2s/parenrightig\n. (17)\nThis tells that the freely falling observer, which begins it s journey very close to the horizon,\nsees very large negative energy density. For other limit r→ ∞, the energy density falls as\nǫ(∞)\nB→ O(1/r6).\nFor the case of the Unruh state, the energy density near the ho rizon is given by\nǫ(0)\nU(r|rs) =C(0)\nU/bracketleftig/angbracketleftig\nTt\nt/angbracketrightig\nutut+/angbracketleftig\nTt\nr/angbracketrightig\nutur+/angbracketleftig\nTr\nt/angbracketrightig\nurut+/angbracketleftig\nTr\nr/angbracketrightig\nurur/bracketrightig\n=C(0)\nU/bracketleftig\n−E2\nf2(r)+2E\nf2(r)/parenleftig\nE2−V2\neff(r)/parenrightig1/2\n−1\nf2(r)/parenleftig\nE2−V2\neff(r)/parenrightig/bracketrightig\n.(18)\n5Using Eqs. (3) and (5), we can obtain\nǫ(0)\nU(r|rs) =C(0)\nU/bracketleftig\n−2f(rs)\nf2(r)/parenleftig\n1+L2\nr2s/parenrightig\n+2/radicalbigg\nf(rs)/parenleftig\n1+L2\nr2s/parenrightig\nf2(r)/braceleftig\nf(rs)/parenleftig\n1+L2\nr2s/parenrightig\n−f(r)/parenleftig\n1+L2\nr2/parenrightig/bracerightig1/2\n+1+L2\nr2\nf(r)/bracketrightig\n. (19)\nThe energy density measured at the moment when the free fall b egins is reduced to\nǫ(0)\nU(rs|rs) =−L\n4π(2M)2/parenleftig\n1+L2\nr2s/parenrightig\nf(rs). (20)\nSince luminosity is positive whereits numerical value is L= 2.337×10−4/(πM2) [25], the energy\ndensity is always negative divergent for rs→2M. In the asymptotically infinity of r→ ∞, it\nturns out to be\nǫ(∞)\nU(r|rs) =C(∞)\nU/bracketleftig/angbracketleftig\nTt\nt/angbracketrightig\nutut+/angbracketleftig\nTt\nr/angbracketrightig\nutur+/angbracketleftig\nTr\nt/angbracketrightig\nurut+/angbracketleftig\nTr\nr/angbracketrightig\nurur/bracketrightig\n=C(∞)\nU/bracketleftigE2\nf(r)+E/parenleftig\nE2−V2\neff(r)/parenrightig1/2\n+E\nf2(r)/parenleftig\nE2−V2\neff(r)/parenrightig1/2\n+1\nf(r)/parenleftig\nE2−V2\neff(r)/parenrightig/bracketrightig\n. (21)\nNow, if we impose the condition that the observer’s initial s tarting point and measurement point\nare the same, then it leads to\nǫ(∞)\nU(rs|rs) =C(∞)\nU/parenleftig\n1+L2\nr2s/parenrightig\n. (22)\nNote that it is positive and vanishes in the limit rs→ ∞. But remember that this is the energy\ndensity. To obtain the energy, we need to integrate Eq. (21) o ver the three space volume and\nthen take the limit rs→ ∞. This will lead to a finite quantity which is proportional to s quire of\nthe Hawking temperature i.e.,T2\nH= (1/8πM)2, since luminosity is proportional to the inverse\nsquire of the mass Mof the black hole. In the two dimensional case, the energy den sity at\ninfinity is itself finite [14]. This is because there is no angu lar part contribution in the stress-\ntensor, whereas in the four-dimensional case the angular pa rt provides the 1 /r2dependence in\nC(∞)\nU, as seen from Eq. (13), which spoils the finite nature in the in finity limit. In other words,\nin the two-dimensional case there is no angular integration to obtain the energy, which means\nthe energy density itself must have the finiteness. So there i s no discrepancy between them.\nFinally, we discuss the nature of the energy density in the Is rael-Hartle-Hawking state. For\nr→2M, it is given as\nǫ(0)\nH(r|rs) =C(0)\nH/bracketleftig\n−V2\neff(r)\nf(r)+2L2\nr2/bracketrightig\n(23)\nand using Eqs. (3) and (5) in Eq. (23) leads to\nǫ(0)\nH(r|rs) =−1\n96M4/parenleftig\n1−L2\nr2/parenrightig\n. (24)\nNote that Eq. (24) is insensitive to the initial position of t he observer in the near horizon limit\nso that the initial free-fall position in this limit does not alter the essential behavior of the energy\ndensity. This is also compatible with the two-dimensional c alculations in the near horizon limit\n6ofL= 0 where it is always negative [14]. The crucial difference fro m the two-dimensional case\ncomes from the fact that for agiven non-vanishing Lthere exists additional positive contribution\nto the energy density. For 2 M > L, the energy density (24) can be still negative even at the\nhorizon whereas for 2 M < Lit can be positive at the horizon. For the latter case, there i s also\na critical free-fall position of rc=L > rHwhere the energy density vanishes. It means that the\nhighly circular free-fall motion near the horizon gives ris e to transition from the negative energy\ndensity to the positive energy density during free fall. On t he other hand, for r→ ∞, in the\nsimilar manner we can obtain\nǫ(∞)\nH(r|rs) =C(∞)\nH/bracketleftigE2\nf(r)+1\n3f(r)/parenleftig\nE2−V2\neff(r)/parenrightig\n+L2\n3r2/bracketrightig\n, (25)\nwhereC(∞)\nHis finite so that the energy density fall as O(1/r2). Forr=rsthe above equation\nis simply reduced to\nǫ(∞)\nH(rs|rs) =C(∞)\nH/bracketleftig\n1+4L2\n3r2s/bracketrightig\n, (26)\nwhich is definitely positive.\n5 Fluxes in different states\nIn this section, we shall study the asymptotic behavior of th e flux for a freely falling observer\nin three states. The flux for a freely falling observer is defin ed as [7]\nF=−/angbracketleftig\nTa\nb/angbracketrightig\nuanb(27)\nwherenais a spacelike unit normal satisfying gabnanb= +1 and gabnaub= 0. In order to\ndetermine na, let us suppose that the normal vector has the form of na= (A,B,C,D ) where\nthe unknown parameters have to be fixed by using two condition snana= +1 and naua= 0.\nSince the unknown parameters are four while the number of equ ations relating them are two, we\nimpose the condition C= 0 =Dfrom the beginning for convenience to eliminate the redunda nt\ndegrees of freedom. Then, these two conditions yield\nf(r)A2−B2\nf(r)= 1, (28)\nAE+B\nf(r)(E2−V2\neff(r))1/2= 0. (29)\nSolving the above two equations, we obtain\nA=∓/parenleftig\nE2−V2\neff(r)/parenrightig1/2\nVeff(r)/radicalbig\nf(r), B=±E/radicalbig\nf(r)\nVeff(r). (30)\nAs we mentioned earlier, the negative sign in Eq. (2) has been chosen. Now considering the\npositive sign of B, we find the components of the unit normal in the case of the pre sent metric\n(1) as\nna=/parenleftig\n−/parenleftig\nE2−V2\neff(r)/parenrightig1/2\nVeff(r)/radicalbig\nf(r),E/radicalbig\nf(r)\nVeff(r),0,0/parenrightig\n. (31)\nThe expressions for flux measured by a freely falling observe r will be given for three different\nstates for the regions r→2Mandr→ ∞in what follows.\n7For the Boulware state, the value of the flux for the limit r→2Mturns out to be\nF(0)\nB(r|rs) =−1\n30 212π2M44E/parenleftig\nE2−V2\neff(r)/parenrightig1/2\n3Veff(r)f5/2(r)\n=−1\n30 212π2M44f(rs)/parenleftig\n1+L2\nr2s/parenrightig/bracketleftig\n1−f(r)/parenleftig\n1+L2\nr2/parenrightig\nf(rs)/parenleftig\n1+L2\nr2s/parenrightig/bracketrightig1/2\n3f3(r)/radicalbigg/parenleftig\n1+L2\nr2/parenrightig(32)\nwhich diverges near the horizon. Moreover, the above vanish es forr=rs. On the other hand,\nthis falls as F(∞)\nB(r|rs)→ O(r−6) for the limit of r→ ∞.\nIn the case of the Unruh state, the flux for the near horizon lim it is given by\nF(0)\nU(r|rs) =−L\n4π(2M)2/bracketleftig2E/parenleftig\nE2−V2\neff(r)/parenrightig1/2\nf3/2(r)Veff(r)−E2\nf3/2(r)Veff(r)−E2−V2\neff(r)\nf3/2(r)Veff(r)/bracketrightig\n=L\n4π(2M)2/bracketleftig\n−2f(rs)/parenleftig\n1+L2\nr2s/parenrightig/parenleftig\n1−f(r)/parenleftig\n1+L2\nr2/parenrightig\nf(rs)/parenleftig\n1+L2\nr2s/parenrightig/parenrightig1/2\nf2(r)/parenleftig\n1+L2\nr2/parenrightig1/2+f(rs)/parenleftig\n1+L2\nr2s/parenrightig\nf2(r)/parenleftig\n1+L2\nr2/parenrightig1/2\n+f(rs)/parenleftig\n1+L2\nr2s/parenrightig/parenleftig\n1−f(r)/parenleftig\n1+L2\nr2/parenrightig\nf(rs)/parenleftig\n1+L2\nr2s/parenrightig/parenrightig\nf2(r)/parenleftig\n1+L2\nr2/parenrightig1/2/bracketrightig\n(33)\nand in particular at r=rsit can be simplified as\nF(0)\nU(rs|rs) =L\n4π(2M)21\nf(rs)/parenleftig\n1+L2\nr2s/parenrightig1/2\n. (34)\nSince luminosity is positive finite quantity, it is always po sitive and finite except at rs= 2M\nwhere it diverges. In the asymptotic infinity region, the flux for the Unruh vacuum is calculated\nas\nF(∞)\nU(r|rs) =L\n4πr2/bracketleftig2E/parenleftig\nE2−V2\neff(r)/parenrightig1/2\n/radicalbig\nf(r)Veff(r)+E2/radicalbig\nf(r)\nVeff(r)+E2−V2\neff(r)\nf3/2(r)Veff(r)/bracketrightig\n=L\n4πr2/bracketleftig2f(rs)/parenleftig\n1+L2\nr2s/parenrightig/parenleftig\n1−f(r)/parenleftig\n1+L2\nr2/parenrightig\nf(rs)/parenleftig\n1+L2\nr2s/parenrightig/parenrightig1/2\nf(r)/parenleftig\n1+L2\nr2/parenrightig1/2+f(rs)/parenleftig\n1+L2\nr2s/parenrightig\n/parenleftig\n1+L2\nr2/parenrightig1/2\n+f(rs)/parenleftig\n1+L2\nr2s/parenrightig/parenleftig\n1−f(r)/parenleftig\n1+L2\nr2/parenrightig\nf(rs)/parenleftig\n1+L2\nr2s/parenrightig/parenrightig\nf2(r)/parenleftig\n1+L2\nr2/parenrightig1/2/bracketrightig\n(35)\n8and forr=rsit is reduced to\nF(∞)\nU(rs|rs) =L\n4πr2sf(rs)/parenleftig\n1+L2\nr2s/parenrightig1/2\n, (36)\nwhere it also yields finite quantity when one integrates it ov er the three space volume.\nFinally, the flux in the Israel-Hartle-Hawking state near th e horizon vanishes as\nF(0)\nH(r|rs) = 0. (37)\nIn this calculation, the matrix we have used is Eq. (8) which i s constant near the horizon. That\nmeans the leading order of Eq. (8) gives us vanishing flux. On t he other hand, in the asymptotic\ninfinite region, it is given by\nF(∞)\nH(r|rs) =1\n30 212π2M44E/parenleftig\nE2−V2\neff(r)/parenrightig1/2\n3Veff(r)/radicalbig\nf(r)\n=1\n30 212π2M44f(rs)/parenleftig\n1+L2\nr2s/parenrightig/bracketleftig\n1−f(r)/parenleftig\n1+L2\nr2/parenrightig\nf(rs)/parenleftig\n1+L2\nr2s/parenrightig/bracketrightig1/2\n3f3/2(r)/radicalbigg/parenleftig\n1+L2\nr2/parenrightig(38)\nwhich vanishes as F(∞)\nH(rs|rs) = 0 at r=rs, so that it is zero at the spatial infinity.\n6 Conclusion and discussion\nIn the four-dimensional Schwarzschild black hole backgrou nd, we have studied the energy den-\nsities and fluxes near the horizon and asymptotic spatial infi nity by using the trace anomaly of\na conformally invariant scalar field in the three vacua. In th e Boulware state, both the energy\ndensity and flux are negative divergent at the horizon, and as ymptotically vanish. In the Un-\nruh state, the energy density is negative divergent at the ho rizon whereas it is positive finite\nasymptotically. The flux in the Unruh state is always positiv e. For the Boulware and the Unruh\nstates, the angular momentum due to the free-fall motion doe s not affect the energy density at\nthe horizon. However, in the Israel-Hartle-Hawking state, for 2M > L, the energy density is\nnegative at the horizon whereas for 2 M < Lit can be positive, and the fluxes are zero at the\nhorizon and the spatial infinity.\nIn particular, as for the flux in the Unruh state, which is alwa y positive. As was discussed,\nthe negative energy density near the horizon should be influx to reduce the black hole mass so\nthat at first sight the flux sign may be assumed to be negative. H owever, note that the energy\ndensity has already negative sign so that the flux sign should be totally positive, which is similar\nto the one-dimensional linear momentum case that the negati ve mass and negative velocity gives\nthe positive sign. Therefore, the ingoing negative energy d ensity near the horizon toward the\nblack hole and the out-going positive energy density at the s patial infinity have the same sign.\nUnfortunately, we have just analytically discussed the lim iting cases of the energy density\nand flux instead of full region including the intermediate re gion without resort to numerical\nsimulations whichmight disclose thefullbehaviorsof them . However, weobtained theimportant\nproperties of the energy density and the flux at the horizon. F or instance, in the Unruh state\ndescribing the collapsing black hole, the energy density is negative near the horizon and the\npositive asymptotically. It means that there exists at leas t one free-fall position for the energy\ndensity to vanish. For the evaporating black hole, we can nat urally imagine that the black hole\n9is surrounded by the negative energy density and the positiv e energy density is located far away\nfrom the horizon to the infinite extent. The negative energy d ensity should be compensated\nwith the positive black hole mass. The positive energy is alr eady separated from the black hole\noutside the horizon. If this positive energy carries inform ation of black hole, then the negative\nenergy carries negative information, which will result in t he unitary evolution of black hole as\nwas discussed in Ref. [17].\nNow, onemight askwhatthenovel aspectofourcalculation is . Sowewouldliketoemphasize\nsome differences between the well-established works and the p resent calculation, and also discuss\nphysical significances of the result. Hawking phenomenon [1 ] predicts that an observer at future\ninfinity perceives an outgoing thermal flux of massless parti cles with a temperature. This effect\nis certainly observer dependent and one can not immediately say anything as to what different\nobservers would measure. In this context, the important obs ervation is due to Unruh [6]. The\nmain result is that a static detector at a constant radial pat h detects thermal emission with\na temperature which satisfies the Tolman equilibrium condit ion [26] just like a usual thermal\nbath. Next question arises: what will be for the detectors wh ich are moving along the infalling\ngeodesics? Since the Killing vectors are not tangent to thes e paths, one can not use the Tolman\nrelation. Instead, since the acceleration for these frames is zero, it happens that there is no\noutgoingfluxnearthehorizonfollowingtheUnruheffect. Howe ver, theaboveheuristicargument\nwas challenged by “firewall” [18], wheretheprincipleof inf ormation theory suggests thepresence\nof it very near the horizon. Unfortunately, till now there is no conclusive discussion in literatures\nto find the fate of the freely falling observers crossing the h orizon.\nIn our calculation, we clarified whether the freely falling o bservers perceive any radiation or\nnot. Inparticular, it wasfoundthatthenegatively ”diverg ent” energyflowstothehorizonrather\nthan the high frequency outgoing modes at the horizon called the firewall [18], which has not yet\nbeen discussed in other literatures. In connection with bla ck hole complementarity, the free-fall\nobserver when crossing the horizon sees nothing special unt il he/she arrives at the coordinate\ninvariant singularity. Therefore, the present calculatio n can modify the conventional black hole\ncomplementarity [3,4,5] since the huge negative energy den sity can be found when the observer\nis dropped very near the horizon. Next, what needs to be answe red is that why such a huge\namount of energy density appears, which seems to be contradi ctory to the previous result giving\nvanishing result near the horizon in Ref. [6]. Actually, thi s is not inconsistent with the Unruh\neffect since the energy density largely consists of three comp onents such as ingoing, outgoing,\ncross components of the energy-momentum tensors. Near the h orizon, the outgoing and cross\ncomponents of energy-momentum tensors vanish, so that ther eis nooutgoing radiation following\nthe Unruh effect. However, the non-vanishing ingoing energy- momentum tensor contributes to\nthe infalling energy density, which implies that the ingoin g negative energy can be found in the\nfreely falling frame. Moreover, its magnitude is divergent at the horizon because the free-fall\nfour velocity ua, which plays a role of the vierbein ea\nτto transform the fiducial coordinates to\nthe locally flat coordinates where τis the proper time in the locally flat spacetime, is divergent\nat the horizon as seen from Eq. (2). Note that the reason for th is divergence is related to\nthe gravitational time dilation. For instance, when the fra me is dropped at rest very near\nthe horizon, then the zeroth component of the four velocity o fu0=dt/dτbecomes very large\nsince the coordinate time dtis very large while the proper time dτis finite in freely falling\nframe. In these regards, we could see the nontrivial behavio rs of the free-fall energy density near\nthe horizon which are of relevance to modification of black ho le complementarity including the\nfirewall problem even in spite of the simplest context. This i ssue deserves further attention in\nthis direction.\nAcknowledgements\nWK was indebted to Yongwan Gim for helpful comments and would like to thank In Yong\n10Park for exciting discussions. This work was supported by th e National Research Foundation of\nKorea(NRF) grant funded by the Korea government(MSIP) (201 4R1A2A1A11049571). BRM\nwas supported by a Lady Davis Fellowship at Hebrew Universit y, by the I-CORE Program of\nthe Planning and Budgeting Committee and the Israel Science Foundation (grant No. 1937/12),\nas well as by the Israel Science Foundation personal grant No . 24/12.\nReferences\n[1] S. W. Hawking, “Particle Creation by Black Holes,” Commu n. Math. Phys. 43, 199 (1975)\n[Erratum-ibid. 46, 206 (1976)].\n[2] S. W. Hawking, “Breakdown of predictability in gravitat ional collapse,” Phys. Rev. D 14,\n2460 (1976).\n[3] L. Susskind, L. Thorlacius and J. Uglum, “The stretched h orizon and black hole comple-\nmentarity,” Phys. Rev. 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Israel, “Thermo-field dynamics of black holes,” Phys . Lett. A 57, 107 (1976).\n[12] J. B. Hartle and S. W. Hawking, “Path Integral Derivatio n of Black Hole Radiance,” Phys.\nRev. D13, 2188 (1976).\n[13] S. M. Christensen and S. A. Fulling, “Trace Anomalies an d the Hawking Effect,” Phys.\nRev. D15, 2088 (1977).\n[14] M. Eune, Y. Gim and W. Kim, “Something special at the even t horizon,” Mod. Phys. Lett.\nA29, 1450215 (2014) [arXiv:1401.3501 [hep-th]].\n[15] I. Y. Park, “Indication for unsmooth horizon induced by quantum gravity interaction,”\narXiv:1401.1492 [hep-th].\n[16] M. Smerlak and S. Singh, “New perspectives on Hawking ra diation,” Phys. Rev. D 88,\n104023 (2013) [arXiv:1304.2858 [gr-qc]].\n[17] B. Freivogel, “Energy and Information Near Black Hole H orizons,” arXiv:1401.5340 [hep-\nth].\n11[18] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, “Bla ck Holes: Complementarity or\nFirewalls?,” JHEP 1302, 062 (2013) [arXiv:1207.3123 [hep-th]].\n[19] S. L. Braunstein, S. Pirandola and K. Zyczkowski, “Bett er Late than Never: Information\nRetrieval from Black Holes,” Phys. Rev. Lett. 110, no. 10, 101301 (2013).\n[20] J. Hutchinson and D. Stojkovic, “Icezones instead of fir ewalls: extended entanglement\nbeyond the event horizon and unitary evaporation of a black h ole,” arXiv:1307.5861 [hep-\nth].\n[21] D. N. Page, “Excluding Black Hole Firewalls with Extrem e Cosmic Censorship,” JCAP\n1406, 051 (2014) [arXiv:1306.0562 [hep-th]].\n[22] P. Candelas, “Vacuum polarization in Schwarzschild sp acetime,” Phys. Rev. D 21, 2185\n(1980).\n[23] R. Balbinot, A. Fabbri andI. L. Shapiro, “Vacuum polari zation in Schwarzschild space-time\nby anomaly induced effective actions,” Nucl. Phys. B 559, 301 (1999) [hep-th/9904162].\n[24] T. Padmanabhan, “Gravitation: Foundations and fronti ers,” Cambridge, UK: Cambridge\nUniv. Pr. (2010) 700 p\n[25] T. Elster, “Vacuum polarization near a black hole creat ing particles,” Phys. Lett. A 94, 205\n(1983).\n[26] R. C. Tolman, “On the Weight of Heat and Thermal Equilibr ium in General Relativity,”\nPhys. Rev. 35, 904 (1930).\n12"    },    {        "title": "1402.2556v1.New_approach_for_solving_master_equations_of_density_operator_for_the_Jaynes_Cummings_Model_with_Cavity_Damping.pdf",        "content": "New approach for solving master equations of densit y operator \nfor the Jaynes Cummings Model with Cavity Damping \n \nSeyed Mahmoud Ashrafi 1 and Mohammad Reza Bazrafkan \n \nPhysics Department, Faculty of Science, I. K. I. Un iversity, Qazvin, Iran \n \n  By introducing thermo- entangled state representati on \u0001, which can map master equations of \ndensity operator in quantum statistics as state-vec tor evolution equations, and using ’’dissipative \ninteraction picture \u0001\u0001 we solve the master equation of Jaynes- Cummings m odel with cavity damping. \nIn addition we derive the Wigner function for densi ty operator when the atom is initially in the up \nstate \u0001 and the cavity mode is in coherent state.  \n \nKeywords: Jaynes - Cummings model, entangled state representa tion, master equations. \n \nPACS:   42.50.–p, 03.65.–w  \n \n1. Introduction \n \n As is well known, decoherence dynamics and the var ious methods for controlling this process is one \nof the important topics in quantum optics and the t heory of quantum open systems can help for better \nunderstanding of how quantum decoherence is process ed. In nature most quantum systems are \nimmersed in reservoir that this systems introduce o pen system [1, 2] the time evolution of an open \nquantum system interacting with a memory-less envir onment can be expressed as a Lindblad  master \nequation as follows[3, 4]  \n\u0001 \u0001 \u0001 \u0002\u0003\u0004 \u0005 \u0005\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002\u0002 \u0002 \u0002 \u0002\u0006 \u0003\u0004 \u0003\u0004 \u0003\u0004 \u0007\b \b\u0001 \u0001 \u0001 \u0001 \u0001 \u0001 \n\u0001\u0002 \u0003 \u0004\u0005 \u0006 \u0003\u0006 \u0006\u0006 \u0003 \u0003\u0006\u0006 \u0002\u0003\u0002\u0002 \u0002 \u0002 \u0002 \u0002 \u0003\u0002 \u0003 \u0004 \u0005 \u0006 \u0007 \b \u0007 \u0007 \t \n \u0004 \u0005\t \n\u000b\u0001                                        (1)     \nWhere \u0002\u0003\u0004\u0003\u0002is, the density of the system, \u0002\u0005 is the Hamiltonian of the system and \u0002\n\u0001\u0006is the so-called \nLindblad operators that are supposed to model the e ffect of the environment on the system. Equation \n(1) describes the general case of a quantum open sy stem. Using some quantum –mechanical models \nfor the reservoir, one can find a differential equa tion for the time evolution of reduced density \noperator of the system. \n Let us consider a two-level system interact with a  resonant mode of the electromagnetic field, Jaynes - \nCoumming model [5, 6], this model is described with out the rotating wave approximation by the \nHamiltonian ( 1 )=\u0001 \n\u0001 \u0001\u0002\u0002\u0002 \u0002 \u0002 \u0002 \u0002 \u0003 \u0004\u0003 \u0004\u0007\u0005 \u0007\u0007 \u0007 \u0007 \u0003 \u0004 \u0005 \u0005 \b \u0007 \u0006 \b \b \b                                                         (2)  \n\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001 If damping is contained in J-C model, one has to so lve \u0002 \n\u0002\u0003\u0004\u0002\u0002 \u0002 \u0006 \u0006\u0004\b\u0002 \u0003\u0004 \u0005 \u0006 \n\u0002\u0003\u0002\u0002 \u0006 \u0002 \u0002 \u0003 \u0002 \u0003 \u0006 \u0007 \b \t \n \t \n                                                           (3) \nWhere \u0004\b\u0006\u0002\u0002 \u0003\t \n is for  T = 0 \n\u0001 \u0001 \u0001\u0002 \u0002\u0002\u0002 \u0002\u0002\u0002 \u0002\u0002\u0002 \u0003\b \u0004\u0007\u0004\b\u0006 \u0007 \u0007 \u0007\u0007 \u0007\u0007 \u0002 \u0006 \u0002 \u0002 \u0002 \u0002 \u0003\u0006 \u0007 \u0007 \t \n                                                         (4)  \nThere is a newly developed method, introduced by Fa n and Fan, [7] from which we can map the \ndensity operator by a vector of two-mode Fock space  whose first mode is the system mode and the \n\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\u0001\n\u0002\u0002\u0001\u0003\u0004\u0005\u0005\u0006\u0007\b\u0004\t\n\u000b\t\f\u0001\r\u000e\u000f\u0010\u0004\u0005\u0011\u0001\u0001 \r\u0007\u0010\u0005\r\u0012\u000b\u0013\u000b\u0014\u000b\u000e\u0015\r\u0016\u0015\u000b\u0005 \u0001\u0001second mode is a fictitious one, and consequently t he master equation appears as a Schr \u0003odinger- like \ntime evolution equation. \n In this present paper, instead of using phase spac e representation of density operator [8,9] we adopt ed \nthermo- entangled state (TES) representation to tre at  time evolution of density operator in the \n J-C model with cavity damping where Xu and Yuan de rived density operator for a Raman- coupled \nmodel in cavity damping [10]. This paper is organiz ed as follows  \nIn sect.2, we review the (TES) representation and i n sect.3, the equation of motion of the density \noperator is decoupled. In sec.4, we solve the maste r equation of J-C model by using of \"dissipative \ninteraction picture\" in entangled representation. I n Sec. 5 Wigner function is derived for a special \ninitial condition. Section 6 is devoted to calculat ion. In the Appendix, we derive a used relation in the \ntext. \n \n2. Brief Review of the Thermo Entangled State Repre sentation \n \nIn this section, we first briefly review the TES an d construct the TES representation in the doubled  \nFock space in the following form [11-14]\u0002\u0002\n     \b\u0001 \u0001 \u0001 \u0001\u0005\u0002 \u0002\u0002 \u0002 \t\n\u000b \f\u0006\f\u0006\n\b\u0007 \t \u0007\t \u0001 \u0001 \u0001 \u0001 \u0001\f\r \u000e \u000f \u0010\u0006 \u0007 \b \u0007 \b \u0011 \u0012 \u000f \u0010 \u000f \u0010 \u000f \u0013 \u0014 \u0002\u0003                           (5) \nWhere \u0001\u0002\u0002 \u0003\u0006 \u0004\u0007\u0007 and \u0001\u0002\u0002 \u0003\u0006 \u0004\t\t are the pairs of canonical annihilation and creatio n operators which the first \n(physical) and the second (fictitious) mode respect ively. Defining \u0001\u0002\u0002 \u0002 \u0007 \t\u0001\u0015 \u0007 and \u0001\u0002 \u0002 \u0002\u0007 \t\u0007\u0006 \b , one \ncan recognize that Eq. (5) is eigen-vector of. \n      \u0001\u0002\u0006\n\u0002\u0007\u0001 \u0001 \u0001 \u0001\n\u0001 \u0001 \u0001 \u0001\f\u0006\n\u0006                                                                    (6) \nObviously, When \f\u0001\u0006 \n\b\n\f\u0005\f \t\n\u000b \f\u0006\f \u0006 \u0007\n\b\u0001\u0001\u0001 \u0001 \u0001\u0016\n\u0006\r \u000e \u000f\u0010 \u0006 \u0006 \u0007 \u0006 \u000f\u0010\u000f\u0010\u000f\u0013 \u0014 \u000b                                               (7) \n And it is obtain that \n\u0001\n\u0001\u0002\u0002\f \f \u0006\n\u0002\u0002\f \f \u0007\u0007 \t\n\u0007 \t\u0001 \u0001\n\u0001 \u0001\u0006 \u0006 \u0006 \n\u0006 \u0006 \u0006                                                                 (8) \nBy means of technique (IWOP), one can prove that ma kes up a complete and orthonormal set \n\b\n\u0003\b\u0004\u0005\u0006 \u0003 \u0004\u0007\u0002\u0001\u0001 \u0001 \u0001 \u0001 \b\t \u0001 \u0001\b\u0017 \u0017 \u0006 \u0006 \u0007 \u0018                                              (9) \nAnd, it is convenient to introduce the state \u0003\u0004\u0003\u0002 [15] for density operator \n\u0002\u0003\u0004 \u0003\u0004 \f \u0007\u0003 \u0003\u0002 \u0002 \u0001\u0015 \u0006                                                                     (10)  \n  This means that we can uniquely represent any den sity operator \u0002\u0003\u0004\u0003\u0002of physical mode by vector \n\u0003\u0004\u0003\u0002 from the two-mode Hilbert space. This vector contai ns all information on the quantum state as \nthe density operator.  \n3. Time Evolution of Density Operator in the J-C Mo del \n \nIn other to describe Eq. (3) in the interaction pic ture, which may be called rotational interaction \npicture, we use of the unitary operator \n\u0001\u0002\u0002 \u0003\u0004 \t\n\u000b\u0003 \u0004\u0007\n\u0003 \u0004 \u0003\u0007\u0007 \u0003 \u0006 \u0007                                                        (11) \nTherefore we have \u0005 \u0005 \u0002 \u0002 \u0002 \u0002 \u0006 \u0006\u000b\n \n \n \u0005\n \u0005 \u0002 \u0002 \u0007 \u0007 \u0019 \u0019                                     (12) \nInserting Eq. (12) into Eq. (3) we derive the maste r equation in the rotational interaction picture as  \nfollows \n\u001a\u001b\u0001\u0002\n\u0002 \u0002 \u0002 \u0002 \u0002 \u0002\u0003 \u0004\u0003 \u0004\u0006\u0004 \u0003 \u0004 \u0003\n\u0004\b\u0002 \u0003\n\u0004 \u0007\f \u0007\f \u0006 \n\u0002\u0003\u0003 \u0003 \u0002\n\u0004 \u0005 \u0005 \u0002 \u0006 \u0002 \u0007\n\b \u0007 \u0002 \u0003 \u0002 \u0003 \u0006 \u0007 \b \b \b \u0004 \u0005 \t \n \t \n                           (13) \nWe write the density operator \n\r\n\u0005\u0005 \u0005\b \f \r \u0005 \b\n\b\u0005 \b\b \u0005 \b \f \r\f\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0005 \u0005\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \b \b\u0004 \u0004\n\u0004\u0004\n\u0004\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002\u0005 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0006\r \u000e \r \u000e \b \u0007 \u000f \u000f \u0010 \u0010 \u000f \u000f \u0010 \u0010 \u0006 \u0006 \u0006\u000f \u000f \u0010 \u0010 \u000f \u000f \b \u0007 \u000f \u000f \u0010 \u0010 \u0013 \u0014 \u0013 \u0014 \u000b                                 (14) \n \nUsing the Pauli spin matrices \u0005 \b \r \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0006 \u0006\r \u000e\u0005 \u0005 \u0005 \u0005 \u0005 \u0005 \u0005 \b \u0007 \b \u0006 \u0006 \u0006  and the unit matrix \f\u0002\u0002\u0005\u0005\u0006, Note \nthat \n\u0005\u0005 \u0005\b \b\u0005 \b\b \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0006 \u0006 \u0006 \u0007\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0006 \u0001 \u0001 \u0006 \u0001 \u001c \u0006 \u001c \u0001 \u0006 \u001c \u001c                      (15) \nAnd \n\f \u0005\u0005 \b\b \u0005 \u0005\b \b\u0005\n\b \u0005\b \b\u0005 \r \u0005\u0005 \b\b\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0006 \u0006\n\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0003 \u0004\u0006 \u0007\u0004\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \n\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0006 \b \u0006 \b \n\u0006 \u0007 \u0006 \u0007                                   (16) \nAccording to the above equations, Eq. (13) can be s plit into the following four equations \n\u0001\n\f \u0005 \f\n\u0001\n\u0005 \f \u0005\n\u0001 \u0001\n\b \r \r \b\n\u0001 \u0001\n\r \b \b \u0002 \u0002 \u0002 \u0002 \u0002 \u0003 \u0004\u0006 \u0006\n\u0002 \u0002 \u0002 \u0002 \u0002 \u0003 \u0004\u0006 \u0006\n\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u000e\u0003 \u0004 \u0003 \u0004\u000f \u0006\n\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u000e\u0003 \u0004 \u0003\u0004 \u0003 \u0004 \u0003\n\u0004\b\n\u0004 \u0003 \u0004 \u0003\n\u0004\b\n\u0004 \u0003 \u0004 \u0003 \u0004 \u0003 \u0004 \u0003\n\u0004\b\n\u0004 \u0003 \u0004 \u0003 \u0004 \u0003\u0004 \u0007\f \u0007\f \u0006 \n\u0004 \u0007\f \u0007\f \u0006 \n\u0007\f \u0007\f \u0007\f \u0007\f \u0006 \n\u0007\f \u0007\f \u0007\f\u0003 \u0003 \n\u0003 \u0003 \n\u0003 \u0003 \u0003 \u0003 \n\u0003 \u0003 \u0003 \u0002 \u0004 \u0002 \u0006 \u0002 \n\u0002 \u0004 \u0002 \u0006 \u0002 \n\u0002 \u0004 \u0002 \u0002 \u0006 \u0002 \n\u0002 \u0004 \u0002 \u0002 \u0007\n\u0007\n\u0007 \u0007 \n\u0007\u0002 \u0003 \u0002 \u0003 \u0006 \u0007 \b \b \u0004 \u0005 \t \n \t \n\u0002 \u0003 \u0002 \u0003 \u0006 \u0007 \b \b \u0004 \u0005 \t \n \t \n\u0002 \u0003 \u0006 \u0007 \b \b \b \b \t \n\u0006 \b \b \b \u0004\n\u0004\n\u0004\n\u0004\n\r\u0002\u0004\u000f \u0007\u0004 \u0003\n\u0004\b\u0007\f \u0006 \u0003\u0006 \u0002 \u0007 \u0002 \u0003\b\t \n               (17) \nOne can find that the first two and the last equati on are coupled, respectively. The first two equatio ns \ncan be decoupled as \n\u0001\u0002\u0002 \u0002 \u0002 \u0002 \u0006\u0004 \u0003 \u0004 \u0003\n\u0004\b\u0004 \u0007\f \u0007\f \u0006 \u0003 \u0003 \u0002 \u0004 \u0002 \u0006 \u0002 \u0007\n\u001d \u001d \u001d \u0002 \u0003 \u0002 \u0003 \u0006 \b \b \u0004 \u0005 \t \n \t \n\u0004\u0005                                       (18) \nWhere, we introduced \n\f \u0005 \u0002 \u0002 \u0002 \u0007\u0002 \u0002 \u0002 \u001d\u0006 \u001d                                                                (19) \nThe other two equations can be decoupled. By taking  \n\r \b\u0002 \u0002 \u0002 \u0007\u000f\u0004\u0002 \u0002 \u0002 \u0006 \b                                                              (20) \nObviously \n\u0001 \u0001\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u000e\u0003 \u0004 \u0003 \u0004\u000f \u0007\u0004 \u0003 \u0004 \u0003 \u0004 \u0003 \u0004 \u0003\n\u000f \u000f \u000f \u0004\b \u000f\u0004 \u0007\f \u0007\f \u0007\f \u0007\f \u0006 \u0003 \u0003 \u0003 \u0003 \u0002 \u0004 \u0002 \u0002 \u0006 \u0002 \u0007 \u0007 \u0002 \u0003 \u0006 \u0007 \b \b \b \b \t \n\u0004                     (21) \nIt is seen that solving the equation of motion in E q. (13) is equivalent to solving Eq. (18) and Eq. ( 21). \n \n4. Unraveling Master Equations through Entangled St ate Representation \n \nIn this section, we unravel the master equations Eq . (18) and Eq. (19) through (TES) representation. \nTherefore acting both hand some Eq. (18) in \f\u0001\u0006 and using Eq. (8), we can find a Schrodinger-\nlike time evolution equation \n\u001a \u001b\u0001 \u0001 \u0001 \u0001\n\u0001\u0003\u0004\u0002 \u0002 \u0002 \u0002\u0002 \u0002 \u0002 \u0002 \u0002\u0002 \b \u0003\u0004\b\n\u0002 \u0002 \u0002 \u0003 \u0004\n\b\n\u0002\u0002\u0003\u0004 \u0003\u0004\u0006\n\b\u0004 \u0003 \u0004 \u0003 \u0004 \u0003 \u0004 \u0003\n\u0004 \u0003 \u0004 \u0003\u0002 \u0003\n\u0004 \u0007\f \u0007\f \t\f \t\f \u0007\t \u0007\u0007 \t\t \u0003\u0002\u0003\n\u0004 \f \f \u0006 \n\u0010 \u0003 \u0006 \u0003\u0003 \u0003 \u0003 \u0003 \n\u0003 \u0003 \u0002 \n\u0004 \u0002 \n\n\u0004\u0001 \u0001\n\n\u0002\u001d \u0007 \u0007 \n\u001d\n\u0007\n\u001d \u001d \u001e \u001f     \u0002 \u0003 \u0006 \b \u0007 \u0007 \b \u0007 \u0007 ! \" \u0004 \u0005\t \n    # $ \n\u001e \u001f     \u0006 \b \b ! \"     # $ \n\u001e \u001f     \u0006 \b ! \"     # $ \u0005\n\u0005      (22) \nWhere  \n\u0001 \u0001 \u0001\u0002 \u0002 \u0002\u0002 \u0002 \u0002 \u0002 \u0002 \u0002\u0002 \u0003\u0004 \u0003 \u0004\u0006 \u0003\b \u0004\u0007\u0004 \u0003 \u0004 \u0003\u0010 \u0003 \u0004 \f \f \u0006 \u0007\t \u0007\u0007 \t\t\u0003 \u0003 \u0004\u0001 \u0001\u0007\n\u001d\u0006 \b \u0006 \u0007 \u0007 \u0005                               (23) By using of commutation relations \u0002\u0002 \u0002 \u0006\u0006\u0001 \u0001\u0002 \u0003\u0006 \u0007 \t \n and \u0001 \u0001\u0002\u0002 \u0002 \u0006\u0006\u0001 \u0001\u0002 \u0003\u0006 \u0007 \u0004 \u0005\t \none can find that \n\u0002 \u0002 \u0002 \u0002 \u0003\u0004 \u0006 \u0003 \u0004 \f \u0007\b \b \u0010 \u0003 \u0006\u0010 \u0003 \u0006 \n \n \n\u001d \u001d \u0002 \u0003\u0004 \u0005 \u0017 \b \b % \u0004 \u0005\t \n                                                 (2 4) \n It is difficult to solve Eq. (22) therefore we def ine ''dissipative interaction picture'' in entangled \nrepresentation [15, 16] for \u0003\u0004\u0003\u0002\u001d as follows \n\u0002\n\b\u0003\u0004 \u0003\u0004\u0007\u0003\u0006\n\u0011\u0003 \f \u0003\n\u0002 \u0002 \u0007\n\u001d \u001d \u0006                                                        (25) \nFor new variable \u0003\u0004\u0011\u0003\u0002\u001dwe have \n\u0002 \u0002 \n\b \b \u0003\u0004\n\u0002 \u0002 \u0002\u0003\u0004 \u0003\u0004 \u0006 \u0003\u0004 \u0003\u0004 \u0007\u0003 \u0003\u0006 \u0006 \u0011 \u0011\u0011\n\u0011\u0002 \u0003\n\u0010 \u0003 \u0003 \u0010 \u0003 \f \u0010 \u0003\f\n\u0002\u0003\n \n \u0002\n\u0002\u0007\u001d\n\u001d \u001d \u001d \u001d \u0006 \u0006                            (26) \nWhere  \n& '\u0002 \u0002 \n\b \b \n\u0001 \b\u0002 \u0002 \u0003\u0004 \u0003\u0004\n\u0002 \u0002 \t \u0007\u0003 \u0003\u0006 \u0006 \u0011\n\u0003\n\u0004 \u0003 \u0004 \u0003\u0010 \u0003 \f \u0010 \u0003\f\n\u0004 \f \f\n \n \n\n\u0003 \u0003 \u0004 \u0001 \u0001\u0007\n\u001d \u001d \n\u0007\u0006\n\u0006 \b \u0005                                               (27)  \nNow we have \n\u0002 \u0002 \u0003\u0004\u0006 \u0003 \u0004 \f\u0007\u0011 \u0011\u0010 \u0003 \u0010 \u0003\u001d \u001d \u0002 \u0003 \u0017\u0006\u0004 \u0005\t \n                                                             (28) \n \nTherefore we can integrate Eq. (26) and find \n\f\u0002 \u0003\u0004 \t\n\u000b\u000e \u0003 \u0004\u000f \u0003\f\u0004 \u0006\u0003\n\u0011\n\u0011 \u000b\u0003 \u0002\u0003\u0010 \u0003\u0002 \u0002 \u001d \u001d \u001d \u0017 \u0017 \u0006\u0018                                           (29) \n To find more simplified form of (29) we write \n\u0001\n\f\u0002 \u0002 \u0002 \u000e \u0003 \u0004\u000f \u000e \u0003\u0004 \u0003\u0004 \u000f\u0006\u0003\n\u0011\u0002\u0003\u0010 \u0003 \u0003 \u0003\u0004 \u0001 \u0004 \u0001\f\n\u001d \u001d \u0017 \u0017\u0006 \u0007 \u0018\n                                                   (30) \nWhere the time dependent parameter \u0003\u0004\u0003\u0004\u001dis defined as \n\b \b\n\f\b\u0003\u0004 \t \u000e\t \u0005\u000f \u0007\n\b\u0003\u0003 \u0003\u0004 \u0003 \u0004 \u0003\u0004\u0003 \u0004 \u0002\u0003\n\u0004\n \n \u0003 \u0003 \u0004\u0004 \u0004 \n\u0003 \n \u0017\u0017\b \b \n\u001d\u0017 \u0006 \u0007 \n\b\u0018\u0006\u0005 \u0005                                      (31) \nThen substituting Eq. (30) into Eq. (29) yields \n\u0001\u0002 \u0002 \u0003\u0004 \t\n\u000b\u000e \u0003\u0004 \u0003\u0004\u000f \u0003\f\u0004 \u0006\u0011 \u000b\u0003 \u0003 \u0003\u0002 \u0004 \u0001 \u0004 \u0001 \u0002 \f\n\u001d \u001d \u001d \u001d \u0006 \u0007 \n                                        (32)             \n \nUsing Eq. (8) and Eq. (10) one can decompose Eq. (3 2) as follows  \n\u0001 \u0001\n\u0001 \u0001\u0001\n\u0002 \u0002 \u0002 \u0002 \n\u0002 \u0002 \u0002 \u0002 \u0003 \u0004\n\u0001\u0002 \u0002 \u0003\u0004 \t\n\u000b\u000e \u0003\u0004 \u0003\u0004\u000f \u0003\f\u0004 \f \u0006\n\u0003\f\u0004 \f\u0006\n\u0003\f\u0004 \f \u0006\n\u0002 \u0002 \u0003 \u0004 \u0003\f\u0004 \u0003 \u0004 \f\u0007\u0011\n\u0007 \u0007 \t \t\n\u0007 \u0007 \u0007 \u0007 \u0003 \u0003 \u0003\n\f \f\n\f \f\n\u0012 \u0012 \u0004 \u0004 \u0004 \u0004 \n\u0004 \u0004 \u0004 \u0004 \u0002 \u0004 \u0001 \u0004 \u0001\u0002 \u0001\n\u0002 \u0001\n\u0002 \u0001\n\u0004 \u0002 \u0004 \u0001\f\n\f \f \n\u001d \u001d \u001d \u001d \n\f \f \n\u001d \u001d \u001d \u001d \u001d \u001d \u001d \u001d \n\u0007 \u0007 \b \n\u001d\n\u0007 \u0007 \u0007 \n\u001d\n\u001d \u001d \u001d \u0006 \u0007 \u0006 \n\u0006 \u0006 \n\u0006 \u0006 \n\u0006 \u0006                                     (33) \n Where \u0001\u0002 \u0002 \u0002\u0003 \u0004\u0007 \u0007 \u0012 \f\u0004 \u0004 \u0004\f\n\u001d \u001d \u0007\u0015  is defined displaced operator of first mode, then to find density operator \n\u0003\u0004\u0003\u0002\u001d in Schrodinger picture we should return rotational  interaction picture  \n\u0001 \u0001\n\u0001 \u0001\u0002\n\b\n\u0002 \u0002\u0002 \u0002 \u0002\u0002 \u000e\b \u000f\b\n\u0002 \u0002\u0002 \u0002 \u0002\u0002 \u000e\b \u000f\u0001 \b\u0003\u0004 \u0003\u0004\n\t \u0003\u0004\n\u0002 \u0002 \t \u0003 \u0004 \u0003\f\u0004 \u0003 \u0004 \f \u0007\u0003\u0006\n\u0011\n\u0003\u0007\t \u0007\u0007 \t\t\n\u0011\n\u0003\u0007\t \u0007\u0007 \t\t\u0003 \f \u0003\n\u0003\n\u0012 \u0012 \n\n\n\u0002 \u0002 \n\u0002\n\u0004 \u0002 \u0004 \u0001\u001d \u001d \n\u0007 \u0007 \n\u001d\n\u0007 \u0007 \n\u001d \u001d \u001d \u0006\n\u0006\n\u0006 \u0006                                     (34) \nTo solve above equation one can immediately underst and that this equation is similar in form to the \nmaster equation of lossy channel (or cavity at zero  temperature) [17, 18], therefore we have \n\u0001 \u0001\u0002\u0002 \u0002\u0002\u0001 \u0001 \b \b\n\f\u0002 \u0002 \u0002 \u0002 \u0003\u0004 \u0003 \u0004 \u0003\f \u0004 \u0003 \u0004 \u0007\n\u0010\u0003 \u0003 \u0001\u0007\u0007 \u0007\u0007 \u0001 \u0001 \n\u0001\u0013\u0003 \f \u0007 \u0012 \u0012 \u0007 \f\n\u0001\n \n \n\u0002 \u0004 \u0002 \u0004 \u0016\u0007 \u0007 \n\u001d \u001d \u001d \u001d \n\u0006\u001e \u001f       \u0006 ! \"       # $ \u000b                                    (35) \nConsequently density operator in Schrodinger pictur e reads as follows \n\u0001 \u0001\u0002\u0002 \u0002\u0002 \u0003 \u0004 \u0003 \u0004\u0001 \u0001 \b \b\n\f\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0003\u0004 \u0003 \u0004 \u0003\f \u0004 \u0003 \u0004 \u0007\u0010\u0001\u0004 \u0003\u0007\u0007 \u0004 \u0003\u0007\u0007 \u000b \u0001 \u000b \u0001 \n\u0001\u0013\u0003 \f \u0007 \u0012 \u0012 \u0007 \f\u0001\n \n \u0003 \u0003 \u0002 \u0004 \u0002 \u0004 \u0016\u0007 \b \u0007 \n\u001d \u001d \u001d \u001d \n\u0006\u0006\u000b                                 (36) \nSimilarly, the explicit form of evolution of densit y operator \u0002\u000f\u0002in Eq. (21) can be derived. Acting on \nboth sides of Eq. (21) with \f\u0001\u0006 we have \n\u0001 \u0001 \u0001 \u0001\u0003\u0004\u0002 \u0002\u0002 \u0002 \u0002 \u0002 \u0002\u0002 \u0002 \u0002 \u0003\b\u0004 \u0003 \u0004 \u0003\u0004\b\n\u0002\u0002\u0003\u0004 \u0007\n\b\u000f \u0004 \u0003 \u0004 \u0003 \u0004 \u0003 \u0004 \u0003\n\u000f\n\u000f \u000f\u0002 \u0003\n\u0007\t \u0007\u0007 \t\t \u0004 \u0007\f \u0007\f \t\f \t\f \u0003\u0002\u0003\n\u0006 \u0010 \u0003\u0003 \u0003 \u0003 \u0003 \u0002\n\u0004 \u0002 \n\n\u0002\u0007 \u0007 \u001e \u001f     \u0006 \u0007 \u0007 \u0007 \b \b \b ! \"     # $ \n\u001e \u001f     \u0006 \b ! \"     # $            (37) \nWhere  \n\u0001 \u0001\n\u0001\u0002 \u0002 \u0002 \u0002 \u0002 \u0003\u0004 \u000e\u0003 \u0004 \u0003 \u0004 \u000f \n\u0002 \u0002 \u000e \u000f\u0007\u0004 \u0003 \u0004 \u0003\n\u000f\n\u0004 \u0003 \u0004 \u0003\u0010 \u0003 \u0004 \u0007 \t \f \u0007 \t\f\n\u0004 \f \f\u0003 \u0003 \n\u0003 \u0003 \u0004\n\u0004\u0007 \u0007\u0007\n\u0007\u0006 \u0007 \b \b \b \n\u0006 \u0007 \b                                      (38) \nIt is a simple task to check following commutation relations \n\u0001\n\u0001 \u0001 \u0001 \u0001\u0002\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0006 \r \b \u0006 \u0006 \u0006\n\u0002\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0006 \r \b \u0006 \u0006 \u0007\u0006 \u0007 \t \u0007 \u0014 \u0006\u0014 \n\u0006 \t \u0007 \t \u0015 \u0006\u0015 \u0007 \u0001 \u0007\n\u0007 \u0001 \u0007\u0002 \u0003 \u0002 \u0003\u0006 \u0007 \u0006 \b \u0006 \u0006 \t \n \t \n\u0002 \u0003 \u0002 \u0003 \u0006 \u0007 \u0006 \u0007 \u0006 \u0006 \u0004 \u0005 \t \n \t \n                      (39) \nTo solve Eq. (37) we defined ''dissipative interacti on picture'' for \u0003\u0004\u000f\u0003\u0002 as follows \n\u0002\n\b\u0003\u0004 \u0003\u0004\u0007\u0003\u0006\n\u000f \u000f\u0011\u0003 \f \u0003\n\u0002 \u0002 \u0007\u0006                                                   ( 40) \n \nTherefore, we have  \n\u0002 \u0002 \n\b \b \u0003\u0004\n\u0002 \u0002 \u0002 \u0003\u0004 \u0003\u0004 \u0006 \u0003\u0004 \u0003\u0004 \u0007\u0003 \u0003\u0006 \u0006 \u000f\u0011 \u0011\u0011\n\u000f \u000f \u000f \u000f\u0011\u0002 \u0003\n\u0010 \u0003 \u0003 \u0010 \u0003 \f \u0010 \u0003\f\u0002\u0003\n \n \u0002\n\u0002\u0007\u0006 \u0006                         (41) \nMaking use of commutation relations Eq. (39) one ca n find \u0002\u0006\u0007\u0007\u0007\u0001 \b\n\u0001\u0002 \u0002 \u0002\u0003\u0004 \t \u000e \u000f \n\u0002 \u0002 \u0002 \u0002 \u0011\u0012\u0013\u0014 \u0013\u0015\u0016\u0014 \u0011\u0012\u0013\u0014 \u0013\u0015\u0016\u0014 \u0007\b \b \b \b \u0003\u0006\u0011 \u0004 \u0003 \u0004 \u0003\n\u000f\n\u0004 \u0003 \u0004 \u0003\u0010 \u0003 \u0004 \f \f\n\u0003 \u0003 \u0003 \u0003\u0004 \f \u0014 \f \u0015 \n\u0003 \u0003 \n\u0003 \u0003 \u0004 \u0007 \u0007\n\n \n \n \n \u0004 \u0007 \u0007\u0002 \u0003\u0007\u0004 \u0005\t \n \u0007\n\u0007\u0006 \u0007 \b \n\u001e \r \u000e \r \u000e\u001f     \u000f \u000f \u0010 \u0010 \u0006 \u0007 \u0007 \b \u0007 \u000f \u000f ! \" \u0010 \u0010 \u000f \u000f \u000f \u000f \u0010 \u0010   \u0013 \u0014 \u0013 \u0014   # $          (42) \nBecause of \u0002 \u0002 \u0003\u0004\u0006 \u0003 \u0004\f\u0011 \u0011\n\u000f \u000f\u0010 \u0003 \u0010 \u0003\u0002 \u0003 \u0017%\u0004 \u0005\t \n we can not integrate Eq. (40) thus we defined new operators \n\u0002\u0002\u0003\u0004\u0006 \u0003\u0004\u0016\u0003 \u0017\u0003 as follows \n\u0001 \u0002 \u0002 \u0002 \u0003\u0004 \u0011\u0012\u0013\u0014 \u0003 \u0004\u0006\n\b\n\u0002 \u0002 \u0002 \u0003\u0004 \u0013\u0015\u0016\u0014 \u0003 \u0004\u0007\n\b\u0004 \u0003 \u0004 \u0003\n\u0004 \u0003 \u0004 \u0003\u0003\u0016\u0003 \u0004 \f \f\n\u0003\u0017\u0003 \u0004 \u0014\f \u0015\f\u0003 \u0003 \n\u0003 \u0003 \n\u0004 \u0007 \u0007\n\n\u0004\u0007\n\u0007\u0006 \u0007 \b \n\u0006 \b                                                  (4 3) \n Now by using of appendix we can decompose Eq. (41)  and find \n\f \f \f \f \u0002\u0002\u0003\u0004 \t\n\u000b \u0003\u0004 \t\n\u000b \u0003\u0004 \t\n\u000b \u0003\u0006 \u0004 \u0003\f \u0004 \u0007\u0003 \u0003 \u0003 \u000b\n\u000f \u000f \u0011 \u0011\u0003 \u0017\u000b\u0002\u000b \u0016\u000b\u0002\u000b \u0002\u000b \u0002\u000b\u0018\u000b\u000b\u0002 \u0002 \u001e \u001f \u001e \u001f \u001e \u001f                   \u0017 \u0017 \u0006! \" ! \" ! \"                   # $ # $ # $ \u0018 \u0018 \u0018 \u0018                   (44) \nWhere  \n\b\u0002\u0002\u0003\u0006 \u0004 \u0003\u0004\u0006 \u0003 \u0004\n\u0017 \u0011\u0012\u0013\u0014\u0003 \u0004\u0013\u0015\u0016\u0014\u0003 \u0004\u0011\u0012\u0013\u0003 \u0003 \u0004\u0004 \u0006\n\b \b \u0018\u000b\u000b \u0016\u000b \u0017\u000b\n\u000b \u000b \u000b \u000b\n \n \u0004 \u0003 \u0002 \u0003\u0017 \u0017\u0006\u0004 \u0005\t \n\r \u000e \u000f \u0010 \u0017 \u0017 \u0006 \u0007 \u0007 \u000f \u0010 \u000f \u0010 \u000f \u0013 \u0014                                           (45) \nAnd  \n\f \f \u0003\u0006 \u0004 \u0003\u0006 \u0004\u0007\u0003 \u000b\n\u0002\u000b \u0002\u000b\u0018\u000b\u000b \u0019\u000b\u000b\u0017 \u0017 \u0017\u0006 \u0018 \u0018                                                        (46) \nTherefore we have \n\u0001\n\b \b \u0005 \u0005 \u0003\u0006 \u0004 \u0001\n\f \f \n\u0002 \u0002 \u0002 \u0002 \u0003\u0004 \u0003\u0004 \u0003\u0004 \u0003\u0004\u0003\u0006 \u0004\u0002 \u0002 \u0002 \u0002 \u0003\u0004 \t\n\u000b \u0013\u0015\u0016\u0014\u0003 \u0004\u0003 \u0004 \t\n\u000b \u0011\u0012\u0013\u0014\u0003 \u0004\u0003 \u0004 \u0003\f \u0004\b \b\n\u0003\f\u0004\u0003 \u0003\n\u0019\u000b\u000b \u0004 \u000b \u0004 \u000b \u0004 \u000b \u0004 \u000b\n\u000f \u000f \u0011 \u0011\n\u0003\u0014 \u0003\u0015 \u0003 \u0003\u0019\u000b\u000b\n\u000f\u0011\u000b \u000b\u0003 \f \u0004 \u0014\f \u0015\f \u0002\u000b \u0004 \f \f \u0002\u000b\n\f \f \f\u0003 \u0003 \u0003 \u0003 \n\u000b \u000b \u000b \u0007 \u000b \u0007\n \n \u0002 \u0004 \u0004 \u0007 \u0007 \u0002 \n\u0002\f \f \u0017 \u0007 \u0007 \n\u0017 \u0007 \u0007 \u001e \u001f \u001e \u001f             \u0006 \b \u0007 \b ! \" ! \"             # $ # $ \n\u0006\u0018 \u0018 \n(47) \nWhere \n\u0005 \b \n\f \f\u0003\u0004 \u0011\u0012\u0013\u0014\u0003 \u0004 \u0006 \u0003\u0004 \u0013\u0015\u0016\u0014\u0003 \u0004 \u0007\n\b \b \u0003 \u0003\n\u0004 \u000b \u0004 \u000b \u000b \u000b\u0003 \u0004 \f \u0002\u000b \u0003 \u0004 \f \u0002\u000b\u0003 \u0003 \n \n \u000b \u0004 \u000b \u0004 \u0015 \u0007 \u0015 \u0007 \u0018 \u0018                            (48) \n  To find the density operator \u0003\u0004\u000f\u0003\u0002  we decompose any of the operators \u0001\n\u0005 \u0005 \u0002 \u0002\u0003\u0004 \u0003\u0004\u0003 \u0003\f\u000b \u0007 \u000b \u0007\f\u0007 and \n\b \b \u0002 \u0002 \u0003\u0004 \u0003\u0004\u0003\u0014 \u0003\u0015 \f\u000b \u000b \f\u0007as a product of operators, one related to the physi cal mode and the other to the fictitious \nmode, therefore we have \n\u0001 \u0001 \u0001\n\u0005 \u0005 \u0005 \u0005 \n\u0001 \u0001\n\u0005 \u0005 \u0005 \u0005 \n\u0001 \u0001\n\u0005 \u0005 \u0005 \u0005 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0003\u0004 \u0003\u0004 \u0003 \u0004 \u0003 \u0004\n\u0002 \u0002\u0002 \u0002 \n\u0002 \u0002 \u0002 \u0002\n\u0001\n\u0005 \u0005\u0002 \u0003\f\u0004 \u0003\f\u0004 \f\n\u0002\u0003\f\u0004\f\n\u0002\u0003\f\u0004\f\n\u0002 \u0002 \u0002\u0003 \u0004 \u0003\f\u0004 \u0003 \u0004 \f\u0003 \u0003 \u0007 \t \u0007 \t\n\u000f \u000f\u000b\n\u0007 \u0007 \t \t\n\u000f\n\u0007 \u0007 \u0007 \u0007 \n\u000f\n\u000f\f \f\n\f \f\n\f \f\n\u0012 \u0012 \u000b \u0007 \u000b \u0007 \u000b \u000b \n\u000b \u000b \u000b \u000b \n\u000b \u000b \u000b \u000b \u0002 \u0002 \u0001\n\u0002 \u0001\n\u0002 \u0001\n\u000b \u0002 \u000b \u0001\f \f \n\f \f \n\f \f \u0007 \b \u0007 \b \n\u0007 \u0007 \n\u0007 \u0007 \u0006 \u0006 \n\u0006 \u0006 \n\u0006 \u0006 \n\u0006 \u0007 \u0006                                         (49) \nAnd  \u001a\u001b\u001a\u001b\u0001\u0001\b\b \b \b \u0005 \u0005 \n\u0001 \u0001\n\b \b \b \b\n\u0001 \u0001 \u0001\n\b \b \b \b \b \b\n\b\u0002 \u0002 \u0002 \u0002 \u0002 \u0002\b\u0002 \u0002 \b \u0003\u0004 \u0003\u0004 \u0001\n\u0005 \u0005 \n\u0002 \u0002 \u0002 \u0002 \b \b \u0001\n\u0005 \u0005 \n\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \b \b \u0001\n\u0005 \u0005 \n\u0002\b\u0002 \u0002 \u0003\f\u0004 \u0003 \u0004 \u0003\f\u0004 \u0003 \u0004 \f\n\u0002 \u0002 \u0003 \u0004 \u0003\f\u0004 \u0003 \u0004 \f\n\u0002 \u0002 \u0003 \u0004 \u0003\f\u0004 \u0003 \u0004 \f\t\u0007 \u0014 \u0015 \u0003 \u0003\n\u000f \u000f \u0011\n\u0007 \u0007 \n\u000f\n\u0007 \u0007 \u0007 \t \t \u0007 \n\u000f\n\u0007\f \f \f \f \u0012 \u0012 \n\f \f \f \u0012 \u0012 \f\n\f \f \f \u0012 \u0012 \f\n\f \f\u000b \u0001\u000b \u0001 \u000b \u000b \u000b \u0007 \u000b \u0007\n\u000b \u000b\u0001 \u000b\u0001 \u000b \n\u000b \u000b \u000b \u000b \u000b \u000b \n\u000b\u0002 \u000b \u0002 \u000b \u0001\n\u000b \u0002 \u000b \u0001\n\u000b \u0002 \u000b \u0001\f \f \f \n\f \f \n\f \f \f \n\f\u0007 \b \u0007 \u0007 \n\u0007\n\b \u0007 \u0007 \u0007 \u0006 \u0007 \u0006 \n\u0006 \u0007 \u0006 \n\u0006 \u0007 \u0006 \n\u0006\u0001 \u0001 \u0001\n\b \b \b \b \b\u0002 \u0002 \u0002 \u0002 \u0002 \b\u0001\n\u0005 \u0005 \u0002 \u0002 \u0003 \u0004 \u0003\f\u0004 \u0003 \u0004 \f \u0007\u0007 \u0007 \u0007 \u0007 \u0007 \n\u000f\u0012 \u0012 \f \f\u000b \u000b \u000b \u000b \u000b \u000b \u0002 \u000b \u0001\f \f \b \u0007 \u0007 \u0007 \u0007 \u0006       (50) \n By means of \u0005\u0002\u0002\u0006 \u0002 \u0002 \u0002 \u0002 \b\u0016\u0017 \u0016 \u0017 \u0016 \u0017 \f \f\f \f\u0002 \u0003\u0007\u0004 \u0005\t \n \b\u0006 , one can recast Eq. (50) into \n\u0001 \u0001 \u0001\n\b \b \b \b \b \b \n\b\u0001 \u0001 \u0001\n\b\b \b \b \b \b \b \n\b\u0001\n\b\b \b \b \b \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \b \b \u0003\u0006 \u0004 \u0001\n\u0005 \u0005 \n\u0003\u0006 \u0004 \b \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \b \b \u0001\n\u0005 \u0005 \n\u0003\u0006 \u0004 \b \u0002 \u0002 \u0002 \u0002 \b \b \n\u0005\u0002 \u0002 \u0002 \u0003\u0004 \u0003 \u0004 \u0003\f\u0004 \u0003 \u0004 \f\n\u0002 \u0002 \u0002\u0003 \u0004 \u0003\f\u0004 \u0003 \u0004 \f\n\u0002\u0003\u0007 \u0007 \u0007 \u0007 \u0007 \u0007 \u0019\u000b\u000b\n\u000f \u000f \u0011\n\u0019\u000b\u000b \u0007 \u0007 \u0007 \u0007 \u0007 \u0007 \n\u000f\n\u0019\u000b\u000b \u0007 \u0007 \u0007 \u0007 \u0003 \f \f \f \u0012 \u0012 \f \f\n\f \f \f \u0012 \u0012 \f \f\n\f \f \f \u0012 \u000b \u000b \u000b \u000b \u000b \u000b \n\u000b\u000b \u000b \u000b \u000b \u000b \u000b \n\u000b\u000b \u000b \u000b \u000b \u0002 \u000b \u0002 \u000b \u0001\n\u000b \u0002 \u000b \u0001\n\u000b\f \f \f \n\f \f \f \n\f \f \f \u0017 \b \u0007 \u0007 \u0007 \n\u0017\u0007 \b \u0007 \u0007 \u0007 \n\u0017\u0007 \u0007 \b \u0006 \u0007 \u0006 \n\u0006 \u0007 \u0006 \n\u0006\u0001 \u0001 \u0001\n\b \b \b \b \n\b\u0001\n\b\b \b \n\b\u0001\n\b \u0005 \b \u0005 \b \b \b \u0002 \u0002 \u0002 \u0002 \b \b \u0001\n\u0005\n\u0003\u0006 \u0004 \u0018 \u0002 \u0002 \u0018 \u0018 \u0001 \u0001\n\b \u0005 \u0005 \b \n\u0003\u0006 \u0004 \u0018 \u0002 \u0002 \u0018 \u0018 \u0001\n\u0005 \b \u0005 \b \u0002\u0002\u0004 \u0003\f\u0004 \u0003 \u0004 \f\n\u0002 \u0002 \u0002 \u0002 \u0002\u0003 \u0004 \u0003 \u0004 \u0003\f\u0004 \u0003 \u0004 \u0003 \u0004 \f\n\u0002 \u0002 \u0002\u0003 \u0004 \u0003\f\u0004 \u0003 \u0004 \f\u0006\u0007 \u0007 \u0007 \u0007 \n\u000f\n\u0019\u000b\u000b \u0007 \u0007 \n\u000f\n\u0019\u000b\u000b \u0007 \u0007 \n\u000f\u0012 \f \f\n\f \f \u0012 \u0012 \u0012 \u0012 \f\n\f \f \u0012 \u0012 \f\u000b \u000b \u000b \u000b \n\u000b\u000b \u000b \n\u000b \u000b\u000b \u000b\u000b \u000b \u000b \u0002 \u000b \u0001\n\u000b \u000b \u0002 \u000b \u000b \u0001\n\u000b \u000b \u0002 \u000b \u000b \u0001\f\n\f\n\f \f \f\u0007 \b \u0007 \u0007 \n\u0017\u0007 \u0007\n\u0017\u0007 \b \b \u0007\u0007 \u0006 \n\u0006 \u0007 \u0007 \u0006 \n\u0006 \b \u0007 \u0007 \u0006      (51) \nWhere we have employed the following relations \n\u0001 \u0001 \u0001\u0005\u0002 \u0002 \u0002\u0003 \u0004 \u0003 \u0004 \t\n\u000b \u0003 \u0004 \u0003 \u0004\u0006\b\n\u0005\u0002 \u0002 \u0002\u0003 \u0004 \u0003 \u0004 \t\n\u000b \u0003 \u0004 \u0003 \u0004\u0007\b\u0012 \u0012 \u0012 \n\u0012 \u0012 \u0012 \f \r \f\r \f\r \f \r \n\f \r \f\r \f\r \f \r \f \f \n\f \f \u001e \u001f     \u0006 \u0007 \b ! \"     # $ \n\u001e \u001f     \u0006 \u0007 \b ! \"     # $                                    (52) \n Now using the result of the master equation lossy channel [18, 19], we can find the density \noperator \u0002\u0003\u0004\u000f\u0003\u0002  in the Schrodinger picture           \n\u0001 \u0001 \b\u0001\n\u0005 \b \u0005 \b \b \b \b \u0002\u0002 \u0002\u0002 \u0003 \u0004 \u0003 \u0004\u0003\u0006 \u0004 \u0018 \u0002 \u0002 \u0018 \u0018 \u0001 \u0001 \b \b\u0005 \b \u0005 \b\n\f\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0003\u0004 \u0003 \u0004 \u0003\f\u0004 \u0003 \u0004 \u0007\n\u0010\u0001\u0004 \u0003\u0007\u0007 \u0004 \u0003\u0007\u0007 \u0019\u000b\u000b \u0007 \u0007 \u000b \u0001 \u0001 \n\u000f \u000f\n\u0001\u0013\u0003 \f \f \u0007\f \u0012 \u0012 \f \u0007 \f\n\u0001\n \n \u0003 \u0003 \u000b\u000b \u000b\u000b \u000b \u000b \u000b \u0002 \u000b \u000b \u0002 \u000b \u000b \f \f \f\u0016\u0007 \b \u0007 \u0007 \u0017\b \b \u0007 \u0007\n\u0006\u0006 \b \u0007 \u0007 \u000b\n (53) \n5. Wigner Function for Density Operator \u0002\u0002\u001d \n \nIn this section we find the Wigner functions of den sity operators Eq. (36) for the given initial state . \nSuppose the atom is initially in the up state \u0001 and the cavity mode is in a coherent state \f\f, \ntherefore the initial condition for density operato r is \n\f \f \f \u0007 \u0002 \f \f \u0006 ( \u0001 \u0001                                                             (54) \n Then substituting Eq. (54) into Eq. (19) and Eq. ( 20) yields \n \f \f \u0003\f\u0004 \u0003\f\u0004 \u0007\u000f\u0002 \u0002 \f \f \u001d\u0006 \u0006                                                         (55) \nNoticing \n\u0001 \u0001 \b\u0002\u0002 \u0002\u0002 \u0003 \u0004\n\f \f \f \f \f \t\n\u000b\u000e \u0003 \u0005\u0004\u000f \u0007\u0007\u0007 \u0007\u0007 \f \f \f \f \f\u000b \u000b \u000b \u000b \u000b \u000b \f \f \f \f \f \f \f \u0007 \u0007 \u0007 \b \u0007 \u0007 \u0006 \u0007           (56)                             \nThe equation Eq. (36) becomes  \n\u0003 \u0004 \u0003 \u0004\b \b \f \f \u0002\u0003\u0004 \u0003 \u0004 \u0003 \u0004\u0007\u0004 \u0003 \u0004 \u0003\u000b\u0003 \f \f\n \n \u0003 \u0003 \u0002 \f \u0004 \f \u0004 \u0007 \b \u0007 \b \n\u001d \u001d \u001d \u0006 \b \b                               (57) \nWe note that Eq. (57) shows that, if the initial co ndition for density operator \u0003\f\u0004\u0002\u001d is the coherent \nstate \f\f, then at later times it continues to remain the co herent state with a time dependent complex \nparameter \n \u0003 \u0004\b\f \f \u0003\u0004 \u0003 \u0004\u0007\u0004 \u0003\u0003 \f\n\u0003\f \f \u0004 \u0007 \b \n\u001d\u0006 \b                                                    (58) Now, by inserting Eq. (31) into Eq. (58), we arrive  at \n\u0003 \u0004\b\f \f \b \b \u0003\u0004 \u0007\n\b \b \u0004 \u0003\u0004 \u0004\u0003 \f\n\u0004 \u0004\n\u0003\u0004 \u0004 \f \f \n\u0003 \n \u0003 \n \u0007 \b \r \u000e \u000f \u0010 \u0006 \b \u001d \u000f \u0010 \u000f \u0010 \u000f \u0013 \u0014 \b \b \u0005                                     (59) \nThe Wigner function for density operator is defined  as \n\f\u0002\u0002 \u0003 \u0006\u0004 \u000e \u0003 \u0004\u0003\u0004\u000f\u0006\u001a \u0003 \u0013\b\n \u0003\f \f\u0002 \u0006                                                          (60) \nWhere the Wigner operator \f\u0002\u0003 \u0004\n\fhas the form \n\u0001\n\f\u0002 \u0002 \u0002 \u0003 \u0004 \b\u0019\t\n\u000b\u000e \b\u0003 \u0004\u0003 \u0004\u000f \u0019\u0007 \n \u0007 \u0007 \f \f \f \f\u0006 \u0007 \u0007 \u0007                                           (61) \nInserting Eq. (57) and Eq. (60) into Eq. (59) yield s \n\b\n\f\u0003 \u0006\u0004 \b\t\n\u000b\u000e \b \u0003\u0004 \u000f \u0007 \u001a \u0003 \u0003\f \f \f \u001d\u0006 \u0007 \u0007                                                   ( 62) \nObviously, for \u0005\u0003\n\u0007 one can find \f\b\u0003\u0004\n\b\u0004\u0003\n\u0004\u0004\f\n\u0003 \n \u0006\n\b\u0005 , being independent of the initial state, this \nmeans that the system loses its memory. In the same  method one can find \u0003 \u0006\u0004\u000f\u001a \u0003\ffor \u0002\u0003\u0004\u000b\n\u000f\u0003\u0002 . \n \n6. Conclusions  \n   \n In summery, for the first time we introduced a new  type of the interaction picture related to \ndissipative processes, dissipative interaction pict ure in entangled state representation, and employed  \nthis picture to treat the time evolution of density  operators in the J-C model with cavity damping. Th is \ntechnical method provides us with a fresh view of s olving master equation in another problem as well. \nMoreover we have derived the Wigner function \u0003 \u0006\u0004\u001a \u0003\f\u001d  of density operator for special initial \ncondition and we have shown that the system loses i ts memory when \u0005\u0003\n\u0007 . \n \nAppendix \n \nHere, we reproduce the proof of Eq. (44) given in [ 1]. Suppose that  \n\u0002\u0002 \u0002 \u0002 \u0002 \u0003\u0004 \u000e \u0003\u0004 \u0003\u0004\u000f \u0003\u0004\u0006 \u0003\f\u0004 \u0005 \u0007\u0002\u001b\u0003 \u0016\u0003 \u0017\u0003 \u001b\u0003 \u001b \u0002\u0003\u0006 \b \u0006                                     (63) \nWhere \n\u0002 \u0002 \u0002 \u0002 \u0003\u0004\u0006 \u0003 \u0004 \u0003\u0004\u0006 \u0003 \u0004 \f\n\u0002\u0002\u0003\u0004\u0006 \u0003 \u0004 \u0003\u0006 \u0004\u0016\u0003 \u0016\u0003 \u0017\u0003 \u0017\u0003\n\u0016\u0003 \u0017\u0003 \u0018\u0003\u0003\u0002 \u0003 \u0002 \u0003\u0017 \u0017\u0006 \u0006 \u0004 \u0005 \t \n \t \n\u0002 \u0003 \u0017 \u0017\u0006\u0004 \u0005\t \n                                              (64) \nIt is difficult to solve Eq. (63) because of commut ation relation \u0002 \u0002\u0002 \u0002\u0003\u0004 \u0003\u0004\u0006 \u0003 \u0004 \u0003 \u0004 \f \u0016\u0003 \u0017\u0003 \u0016\u0003 \u0017\u0003\u0002 \u0003 \u0017 \u0017 \b \b % \u0004 \u0005\t \n, \nin order to we defined new operator \u0002\u0003\u0004\n\u0003 as follows \n\f\n\f\u0002 \u0002 \u0002 \u0003\u0004 \t\n\u000b\u000e \u0003 \u0004 \u000f \u0003\u0004\u0006\n\u0002 \u0002 \u0002 \u0003\u0004 \t\n\u000b\u000e \u0003 \u0004 \u000f \u0003\u0004\u0007\u0003\n\u0003\n\u0003 \u0017\u0003 \u0002\u0003 \u001b\u0003\n\u001b\u0003 \u0017\u0003 \u0002\u0003 \n\u0003\u0017 \u0017 \u0006 \u0007 \n\u0017 \u0017 \u0006\u0018\n\u0018                                                         (65) \nTherefore we have \n\f \f \n\f \f \n\f\u0002 \u0002 \u0002 \u0002 \u0002 \u0003\u0004 \u0003\t\n\u000b\u000e \u0003 \u0004 \u000f\u0004 \u0003\u0004 \t\n\u000b\u000e \u0003 \u0004 \u000f \u0003\u0004\n\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0003\u0004\t\n\u000b\u000e \u0003 \u0004 \u000f \u0003\u0004 \t\n\u000b\u000e \u0003 \u0004 \u000f\u000e \u0003\u0004 \u0003\u0004\u000f \u0003\u0004\n\u0002\u0002 \u0002 \t\n\u000b\u000e \u0003 \u0004 \u000f \u0003\u0004 \u0003\u0004\u0007\u0003 \u0003\n\u0003 \u0003\n\u0003\u0002 \u0002 \u0002 \n\u0003 \u0017\u0003 \u0002\u0003 \u001b\u0003 \u0017\u0003 \u0002\u0003 \u001b\u0003\n\u0002\u0003 \u0002\u0003 \u0002\u0003\n\u0017\u0003 \u0017\u0003 \u0002\u0003 \u001b\u0003 \u0017\u0003 \u0002\u0003 \u0016\u0003 \u0017\u0003 \u001b\u0003\n\u0017\u0003 \u0002\u0003 \u0016\u0003\u001b\u0003\u0017 \u0017 \u0017 \u0017 \u0006 \u0007 \b \u0007 \n\u0017 \u0017 \u0017 \u0017 \u0006 \u0007 \u0007 \b \u0007 \b \n\u0017 \u0017 \u0006 \u0007 \u0018 \u0018 \n\u0018 \u0018 \n\u0018            (66) \nBy substituting Eq. (65) into Eq. (66) yields \f \f \n\f \f \n\f\u0002 \u0002 \u0002 \u0002 \u0002 \u0003\u0004 \t\n\u000b\u000e \u0003 \u0004 \u000f \u0003\u0004\t\n\u000b\u000e \u0003 \u0004 \u000f \u0003\u0004\n\u0005\u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u0002 \u000e \u0003\u0004 \u0003 \u0004\u0006 \u0003\u0004 \u0003 \u0004\u0006 \u0003 \u0004\u0006 \u0003\u0004 \u0007\u0007\u0007\u000f \u0003\u0004\n\b\n\u0002 \u0002 \u000e \u0003\u0004 \u0003\u0006 \u0004 \u000f \u0003\u0004\u0007\u0003 \u0003\n\u0003 \u0003\n\u0003\u0002\n\u0003 \u0017\u0003 \u0002\u0003 \u0016\u0003 \u0017\u0003 \u0002\u0003 \n\u0003\n\u0002\u0003\n\u0016\u0003 \u0017\u0003 \u0016\u0003 \u0002\u0003 \u0017\u0003 \u0017\u0003 \u0016\u0003 \n\u0003\n\u0016\u0003 \u0018\u0003\u0003 \u0002\u0003 \n\u0003\u0017 \u0017 \u0017 \u0017 \u0006 \u0007 \n\u0002 \u0003 \u0002 \u0002 \u0003\u0003\u0017 \u0017 \u0017 \u0017 \u0006 \u0007 \b \b \u0004 \u0005 \u0004 \u0004 \u0005\u0005\t \n \t \t \n\n\u0017 \u0017 \u0006 \b \u0018 \u0018 \n\u0018 \u0018 \n\u0018                  (67) \nObviously, because of commutation relation  \n\f \f \u0002 \u0002 \u0003\u0004 \u0003\u0006 \u0004 \u0006 \u0003 \u0004 \u0003 \u0006 \u0004 \f \u0006\u0003 \u0003\n\u0016\u0003 \u0018\u0003\u000b\u0002\u000b\u0016\u0003 \u0018\u0003 \u000b \u0002\u000b\u0017 \u0002 \u0003\u0017 \u0017 \u0017 \u0017 \b \b \u0006 \u0004 \u0005\u0004 \u0005 \t \n\u0018 \u0018                                     (68) \nWe can easily integrate Eq. (67) and find \n\f \f \f \f \u0002 \u0002 \u0002 \u0003\u0004 \t\n\u000b\u000e \u0003\u0004\u000f\t\n\u000b\u000e \u0003\u0004\u000f\t\n\u000b\u000e \u0003\u0006 \u0004 \u000f\u0007\u0003 \u0003 \u0003 \u000b\n\u001b\u0003 \u0002\u000b\u0017\u000b \u0002\u000b\u0016\u000b \u0002\u000b \u0018\u000b\u000b \u0002\u000b \u0017 \u0017 \u0006\u0018 \u0018 \u0018 \u0018                        (69) \n \nReferences \n \n1.  Gardiner C and  Zoller P 2000   Quantum Noise (Ber lin: Springer) \n2. Breuer H P and Petruccione F 2002 the Theory of Ope n Quantum System. (Oxford: Oxford University \nPress) \n3. Lindblad G 1976 Commun Math. Phys. 48  119 \n4. Gorini V and Kossakowski  Sudarshan A  1976  J. Mat h.Phys. 17   821 \n5. Schleich W P 2001 Quantum optics in Phase Space   ( Berlin: Wiley/VCH)  \n6. Gerry C and  Knight P  2005 Introductory Quantum Op tics  (New York: Cambridge University Press) \n7. Fan H Y and Fan Y 1998 Phys. Lett. A  246   242  \n8. Walls D F and  Milburn G j 1994 Quantum optics (New  York: Springer)  \n9. Cahill K and Glauber R  1969  Phys. Rev. 177  1857 \n10. Xu Y, Yuan J, Song J and Liu Q 2010 Int. J.  Theor,  Phys. 49  2180  \n11. Fan  H Y and  Klauder  J  R  1994  Phys. Rev. A 49  704  \n12. Umezawa  H , Matsumoto  H and Tachiki  M  1982 Ther mo Field Dynamics and Condensed State \n(Amesterdam :  North Holland) \n13. Takahashi Y and Umezeawa H 1975 Collect. Phenom.  2 55  \n14. Fan H Y and Lu H L 2007 Mod. Phys. Lett. B  21  183 \n15. Bazrafkan M R and Ashrafi M 2013 J. Russia Laser.  34  1  \n16. Ashrafi S M and Bazrafkan M R 2013 chin. Phys. Let.   30  11 \n17. Fan H Y and   Hu L U 2009 Commun. Theor. Phys. 51  729 \n18. Fan H Y and   Hu   L U 2008 Opt. Commun.  281   5571  \n \n \n \n \n "    },    {        "title": "1402.6348v1.Thermal_properties_of_supernova_matter__The_bulk_homogeneous_phase.pdf",        "content": "APS/123-QED\nThermal properties of supernova matter: The bulk homogeneous phase\nConstantinos Constantinou,1, 2,\u0003Brian Muccioli,1,yMadappa Prakash,1,zand James M. Lattimer2,x\n1Department of Physics and Astronomy, Ohio University, Athens, OH 45701\n2Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800\n(Dated: November 11, 2018)\nWe investigate the thermal properties of the potential model equation of state of Akmal,\nPandharipande and Ravenhall. This equation of state approximates the microscopic model\ncalculations of Akmal and Pandharipande, which feature a neutral pion condensate. We treat\nthe bulk homogeneous phase for isospin asymmetries ranging from symmetric nuclear matter to\npure neutron matter and for temperatures and densities relevant for simulations of core-collapse\nsupernovae, proto-neutron stars, and neutron star mergers. Numerical results of the state variables\nare compared with those of a typical Skyrme energy density functional with similar properties at\nnuclear densities, but which di\u000bers substantially at supra-nuclear densities. Analytical formulas,\nwhich are applicable to non-relativistic potential models such as the equations of state we are\nconsidering, are derived for all state variables and their thermodynamic derivatives. A highlight of\nour work is its focus on thermal response functions in the degenerate and non-degenerate situations,\nwhich allow checks of the numerical calculations for arbitrary degeneracy. These functions are\nsensitive to the density dependent e\u000bective masses of neutrons and protons, which determine the\nthermal properties in all regimes of degeneracy. We develop the \\thermal asymmetry free energy\"\nand establish its relation to the more commonly used nuclear symmetry energy. We also explore\nthe role of the pion condensate at supra-nuclear densities and temperatures. Tables of matter\nproperties as functions of baryon density, composition (i.e., proton fraction) and temperature are be-\ning produced which are suitable for use in astrophysical simulations of supernovae and neutron stars.\nKeywords: Supernova matter, potential models, thermal e\u000bects.\nPACS numbers: 21.65.Mn,26.50.+x,51.30.+i,97.60.Bw\nI. INTRODUCTION\nThe equation of state (EOS) of dense, hot matter is\nan essential ingredient in modeling neutron stars and\nhydrodynamical simulations of astrophysical phenomena\nsuch as core-collapse supernova explosions, proto-neutron\nstars, and compact object mergers. In broad terms, two\nmajor regions for the EOS can be identi\fed at relatively\nlow temperatures or entropies. At sub-nuclear densities\n(nof 10\u00007to\u00180:1 fm\u00003), matter is in an inhomogeneous\nmixture of nucleons (neutrons and protons), light nuclear\nclusters (alpha particles, deuterons, tritons etc.), and\nheavy nuclei. Leptons, mainly electrons, are also present\nto balance the nuclear charges. Uniform matter, and\nheavy nuclei become progressively more neutron-rich as\nthe density rises. Above about 0.01 fm\u00003, nuclei deform\nin resonse to competition between surface and Coulomb\nenergies, which may also lead to pasta-like geometrical\ncon\fgurations. By the density 0.1 fm\u00003, the inhomo-\ngeneous phase gives way to a uniform phase of nucle-\nons and electrons. Above the nuclear saturation density,\nn0'0:16 fm\u00003, the uniform phase may become popu-\nlated with more exotic matter, including Bose (pion or\n\u0003cconstan@helios.phy.ohiou.edu\nybm956810@ohio.edu\nzprakash@phy.ohiou.edu\nxjames.lattimer@stonybrook.edukaon) condensates, hyperons and decon\fned quark mat-\nter. The appearance of Bose condensates and decon\fned\nquark matter may be through \frst-order or continuous\nphase transitions.\nAt large-enough temperatures below n0, the inhomo-\ngeneous phase disappears and is again replaced by a uni-\nform phase of nucleons and electrons. At su\u000eciently high\ntemperatures at every density, thermal populations of\nhadrons and pions should appear.\nThe composition and thermodynamic properties of\nmatter at a given density n, temperature T, and over-\nall charge fraction (parametrized by the electron con-\ncentrationYe=ne=n) is determined by minimizing the\nfree energy density. In all realistic situations, matter is\ncharge-neutral, but the net baryonic charge is non-zero\nand equalized by the net leptonic charge. It can generally\nbe assumed that baryonic species are in strong interac-\ntion equilibrium, but equilibrium does not always exist\nfor leptonic species which are subject to weak interac-\ntions. In circumstances in which dynamical timescales\nare long compared to weak interaction timescales, the\nfree energy minimization is also made with respect to\nYe. Such matter is said to be in beta equilibrium and\nits properties are a function of only density and temper-\nature, and, if neutrinos are trapped in matter, the total\nnumber of leptons per baryon. Below n0, where generally\nthe only baryons are neutrons and protons and the only\nleptons are electrons and possibly neutrinos, charge neu-\ntrality dictates that the number of electrons per baryon\nYeequals the proton fraction x=np=n, but at higherarXiv:1402.6348v1  [astro-ph.SR]  25 Feb 20142\ndensities, the charge fractions of muons, hyperons, Bose\ncondensates and quarks, if present, have to be included.\nBeta equilibrium may not occur during gravitational col-\nlapse or dynamical expansion, such as occurs in Type II\nsupernovae and neutron star mergers.\nThe free energy can be calculated using a variety of\nmethods, but it is generally a complicated functional of\nthe main physical variables n,TandYeand cannot be\nexpressed analytically. In order to e\u000eciently describe\nthe EOS, it is customary to build three-dimensional ta-\nbles of its properties. An essential criterion is that full\nEOS tables be thermodynamically consistent so as not\nto generate spurious and unphysical entropy during hy-\ndrodynamical simulations. Beginning with the work of\nLattimer and Swesty (hereafter referred to as LS)[1], ex-\namples of such tables include the works of Shen et al [2],\nShen et al [3, 4], and others [5, 6]. We refer the reader\nto Refs. [5, 7{9] in which comparisons of outcomes in su-\npernova simulations, for pre-bounce evolution and black\nhole formation, respectively, have been made using di\u000ber-\nent EOSs. A parallel study [10] of neutron star mergers\nwith di\u000berent EOSs has also been undertaken.\nThe EOS, in addition to controlling the global hydro-\ndynamical evolution, also determines weak interaction\nrates including those of electron capture and beta decay\nreactions and neutrino-matter interactions. These reac-\ntion rates depend sensitively on the properties of mat-\nter, including the magnitudes of the neutron and proton\nchemical potentials and e\u000bective nucleon masses, among\nother aspects. Also of considerable importance are the\nspeci\fc heats and susceptabilities of the constituents,\nwhich determine, respectively, the thermal and transport\nproperties of matter. Thermal properties, especially, may\nbe easier to diagnose from neutrino observations of super-\nnovae: the timescale for black hole formation, in cases\nwhere that happens, appears to be an important exam-\nple [10].\nOne of the most realistic descriptions of the properties\nof interacting nucleons is the potential model Hamilto-\nnian density of Akmal, Pandharipande, and Ravenhall\n(APR hereafter) [11], which reproduces the microscopic\npotential model calculations of Akmal and Pandhari-\npande (AP) [12]. An interesting feature of the AP model\nis the occurrence of a neutral pion condensate at supra-\nnuclear densities for all proton fractions. The AP model\nis especially relevant because it satis\fes several impor-\ntant global criteria that have been gleaned from nuclear\nphysics experiments and astrophysical studies of neutron\nstars, especially those concerning the neutron star max-\nimum mass and their typical radii.\nBoth isospin-symmetric and isospin-asymmetric prop-\nerties of cold baryonic (neutron-proton) matter in the\nvicinity of n0are of considerable importance, as they\ngovern the masses of nuclei, nucleon-pairing phenomena,\ncollective motions of nucleons within nuclei, the transi-\ntion density from inhomogeneous to homogeneous bulk\nmatter, the radii of neutron stars, and many observables\nin medium-energy heavy-ion collisions [13]. One of themost important isospin-symmetric properties at n0is the\ndensity derivative of the pressure P, or, the incompress-\nibilityK0of matter which is now rather well-determined:\nK0= 9(dP=dn )n0;x=1=2;T=0'230\u000630 MeV from Refs.\n[14, 15] and 240\u000620 MeV from Ref. [16]\nAnother isospin-symmetric nuclear constraint stems\nfrom the thermal properties of nuclei and bulk matter.\nFermi liquid theory holds that the thermal properties of\nthe equation of state are largely controlled by the nu-\ncleon e\u000bective masses. In short, experiments indicate\nthat nucleon e\u000bective masses are reduced from their bare\nvalues (m) atn0for symmetric matter to approximately\nm\u0003\n0=m'0:8\u00060:1 [17, 18] and microscopic theory sug-\ngests they further decrease at higher densities. The ex-\ntraction ofm\u0003\n0from nuclear level densities is complicated\nby uncertain contributions from the surface energy as\nwell as possible energy dependences in m\u0003.\nAdditionally, of great signi\fcance is the in\ruence of\nisospin-asymmetry on the properties of nucleonic mat-\nter, not only on the e\u000bective masses, but also on its\nenergyE(n;x;T ), particularly the symmetry energy pa-\nrameterSv= 1=8(@2E=@x2)n0;x=1=2;T=0and its sti\u000bness\nparameter L= 3=8(@2E=@n@x2)n0;x=1=2;T=0. Starting\nfrom the Bethe-Weizacker mass formula [19, 20] and its\nmodernization [21, 22] for nuclei containing a fraction x\nof protons, most mass formulas characterize the symme-\ntry energy of nucleonic matter by these two parameters.\nFrom a variety of experiments, including measurements\nof nuclear binding energies, neutron skin thicknesses of\nheavy nuclei, dipole polarizabilities, and giant dipole res-\nonance energies [23, 24] Svlies in the range 30-35 MeV\nandLlies in the range 40-60 MeV. Recent developments\nin the prediction of the properties of pure neutron mat-\nter by Gandol\f, Carlson and Reddy [25] and by Hebeler\nand Schwenk [26] suggest very similar values for Svand\nLcompared to those derived from nuclear experiments.\nIt is worth noting that there exists a phenomenological\nrelation [27] between neutron star radii and zero temper-\nature neutron star matter pressures near n0, which is\nnearly that of pure neutron matter and largely a func-\ntion of the Lparameter [23]. Astrophysical observations\nof photospheric radius expansion in X-ray bursts [28] and\nquiescent low-mass X-ray binaries [29] have been used\n[30, 31] to conclude that the radii of neutron stars with\nmasses in the range 1.2-1.8 M \fare between 11.5 km and\n13 km, and therefore predict that L'45\u000610 MeV, al-\nthough the astrophysical model dependence of this result\nmay signi\fcantly enlarge its uncertainty. Nevertheless,\nthis range overlaps that from nuclear experiments and\nalso that from neutron matter theory, suggesting that\nsystematic dependencies are not playing a major role in\nthe astrophysical determinations.\nA potentially more important astrophysical constraint\noriginates from mass measurements of neutron stars. A\nconsequence of general relativity is the existence of a\nmaximum neutron star mass for every equation of state.\nCausality arguments, together with current radius es-\ntimates, indicate this is in the range of 2-2.8 M \f[23].3\nThe largest precisely measured neutron star masses are\n1:97\u00060:04 M\f[32] and 2:01\u00060:04 M\f[33]. It is likely\nthat the true maximum mass is at least a few tenths of\na solar mass larger than these measurements.\nAn important issue concerns the quality and relevance\nof experimental information that could constrain the\nthermal properties of dense matter. Calibrating the ther-\nmal properties of bulk matter from experimental results\ninvolves disentangling the e\u000bects of several overlapping\nenergy scales (associated with shell and pairing e\u000bects,\ncollective motion, etc.) that determine the properties of\n\fnite sized nuclei. The level densities of nuclei (inferred\nthrough data on, for example, neutron evaporation spec-\ntra and the disposition of single particle levels in the va-\nlence shells of nuclei [34, 35] depend on the Landau e\u000bec-\ntive masses, m\u0003\nn;p, of neutrons and protons. These masses\nare sensitive to both the momentum and energy depen-\ndence of the nucleon self-energy leading to the so-called\nthek-mass and the !-mass, emphasized, for example, in\nRefs. [36{39]. For bulk matter, in which the predomi-\nnant e\u000bect is from the k-mass,m\u0003\nn;p=m= 0:7\u00060:1 has\nbeen generally preferred. The speci\fc heat and entropy\nof nuclei receive substantial contributions from low-lying\ncollective excitations, as shown in Refs. [40, 41], a sub-\nject that needs further exploration to pin down the role\nof thermal e\u000bects in bulk matter.\nFollowing the suggestions in Refs. [42, 43], the liquid-\ngas phase transition has received much attention with the\n\fnding that the transition temperature for nearly isospin\nsymmetric matter lies in the range 15-20 MeV [44]. Al-\nthough the critical temperature depends on the incom-\npressibility parameter K0, it is also sensitive to the spe-\nci\fc heat of bulk matter in the vicinity of n0which de-\npends on the e\u000bective masses. Further information about\nthe e\u000bective masses can be ascertained from \fts of the\noptical model potential to data [45], albeit it at low mo-\nmenta. The density and the saturating aspect of the high\nmomentum dependence of the real part of the optical\nmodel potential has been crucial in explaining the \row\nof momentum and energy observed in intermediate en-\nergy (<1 GeV) collisions of heavy-ions, preserving at the\nsame time the now well-established value of the incom-\npressibility K0= 230\u000630 MeV, as demonstrated in Refs.\n[46{48]. Notwithstanding these activities, further e\u000borts\nare needed to calibrate the \fnite temperature properties\nof nucleonic matter to reach at least the level of accu-\nracy to which the zero temperature properties have been\nassessed.\nRelatively few EOSs have been constructed from un-\nderlying interactions satisfying all these important con-\nstraints [5]. However, AP and APR satisfy nearly all\nof them. For APR, K0= 266 MeV, S2'32:6 MeV,\nL'58:5 MeV, and m\u0003\n0=m= 0:7, within two standard\ndeviations of the experimental ranges. The maximum\nneutron star mass supported by the APR model is in ex-\ncess of 2M\f, and the radius of a 1 :4M\fstar is about 12\nkm. Despite its obvious positive characteristics, no three-\ndimensional tabular EOS has been constructed with theAPR equation of state. Furthermore, its \fnite tempera-\nture properties for arbitrary degeneracy and proton frac-\ntions, including the e\u000bects of its pion condensate, have\nnot been studied to date.\nThe chief motivation for the present study is to perform\na detailed analysis of the EOS of AP through a study\nof the properties predicted by its APR parametrization.\nParticular attention is paid to the density dependence of\nnucleon e\u000bective masses which govern both the qualita-\ntive and quantitative behaviors of its thermal properties.\nAnother objective of the present work is to document the\nanalytic relations describing the thermodynamic prop-\nerties of potential models. These are essential ingredi-\nents in the generation of EOS tables based upon mod-\nern energy density functionals that employ Skyrme-like\nenergy density functionals. Importantly, the analytic ex-\npressions developed here can be utilized to update LS-\ntype liquid droplet EOS models that take the presence\nof nuclei at subnuclear densities and subcritical temper-\natures into account. This would represent a signi\fcant\nimprovement to existing EOS tables in that they could\nbe replaced with ones including realistic e\u000bective masses.\nSome aspects of the thermal properties of hot, dense\nmatter have been explored in Ref. [49] for isospin sym-\nmetric matter, but the comparative thermal properties\nof di\u000berent Skyrme-like interactions remain largely unex-\nplored. In view of the lack of systematic studies contrast-\ning the predicted thermodynamic properties of the APR\nmodel with those of other Skyrme energy density func-\ntionals, we are additionally motivated to perform such\nstudies for one particular case, that of the SKa force due\nto Kohler [50]. This is one of the EOS's tabulated in the\nsuite of EOS's provided in Ref. [51] which is reproduced\nhere in detail. The methods developed here are general\nand can be advantageously used for other Skyrme-like\nenergy density functions in current use.\nFor both the APR and Ska models, we compute the\nEOS for uniform matter for temperatures ranging up\nto 50 MeV, baryon number densities in the range 10\u00007\nfm\u00003to 1 fm\u00003, and proton fractions between 0 (pure\nneutron matter) and 0.5 (isospin-symmetric nuclear mat-\nter). Ideal gas photonic and leptonic contributions (both\nelectrons and muons) are included for all models. The\nresults presented here for densities below 0.1 fm\u00003in\nthe homogeneous phase serve only to gauge di\u000berences\nfrom the more realistic situation in which supernova\nmatter contains an inhomogeneous phase. Work toward\nextending calculations to realistically describe the low\ndensity/temperature inhomogeneous phase containing \f-\nnite nuclei is in progress at various levels of sophisti-\ncation (Droplet model, Hartree, Hartree-Fock, Hartree-\nFock-Boguliobov, etc.) beginning with an LS-type liquid\ndroplet model approach and will be reported separately.\nIn addition, hyperons and a possible phase transition to\ndecon\fned quark matter are not considered in this work.\n. The organization of this paper is as follows. In Sec.\nII, we brie\ry discuss some of the features of the APR\nand Ska Hamiltonians, and the ingredients involved in4\ntheir construction. We then present their single-particle\nenergy spectra and potentials using a variational proce-\ndure in Sec III. In Sec. IV, properties of cold, isospin-\nsymmetric matter and consequences for small deviations\nfrom zero isospin asymmetry are examined. Analyses\nof results for the two models include those of energies,\npressures, neutron and proton chemical potentials, and\ninverse susceptibilities. Section V contains our study of\nthe behavior of all the relevant state variables for the\nAPR and SKa models at \fnite temperature. The numer-\nical results, valid for all regimes of degeneracy, are juxta-\nposed with approximate ones in the degenerate and non-\ndegenerate limits for which analytical expressions have\nbeen derived. Contributions from leptons and photons\nare also summarized in this section. In Sec. VI, we ad-\ndress the transition from a low-density to a high-density\nphase in which a neutral pion condensate is present us-\ning a Maxwell construction. The numerical results of this\nsection constitute the equation of state of supernova mat-\nter for the APR model in the bulk homogeneous phase.\nOur summary and conclusions are given in Sec. VII. The\nappendices contain ancillary material employed in this\nwork. In Appendix A, we provide a detailed derivation of\nthe single-particle energy spectra for the potential models\nused. General expressions for all the state variables of the\nAPR model valid for all neutron-proton asymmetries are\ncollected in Appendix B. The formalism to include contri-\nbutions from leptons (electrons and positrons) and pho-\ntons is presented in Appendix C, wherein both the exact\nand analytical representations are summarized. Numeri-\ncal methods used in our calculations of the Fermi-Dirac\nintegrals for arbitrary degeneracy are summarized in Ap-\npendix D. Appendix E contains thermodynamically con-\nsistent prescriptions to render EOS's causal when they\nbecome acausal at some high density for both zero and\n\fnite temperature cases.\nII. POTENTIAL MODELS\nIn this work, we study the thermal properties of uni-\nform matter predicted by potential models. We focus\non an interaction derived from the work of Akmal and\nPandharipande (hereafter AP) [12], using an approxima-\ntion developed by Akmal, Pandharipande and Ravenhall\n(hereafter APR) [11], and a Skyrme [52] force developed\nby K ohler (Ska henceforth) [50]. We pay special attention\nto the \fnite temperature properties of these two models\nfor the physical conditions expected in supernovae and\nneutron star mergers, which has heretofore not received\nmuch attention.\nThe Hamiltonian density of Ska [50] is a typical exam-\nple of the approach based on e\u000bective zero-range forces\npioneered by Skyrme [52], which are typically called\nSkyrme forces. These were further developed to describe\nproperties of bulk matter and nuclei in Ref. [53]. Skyrme\nforces are easier to use in this context than \fnite-range\nforces (see, e.g., Ref [54]). To date, a vast number ofvariants of this approach exist in the literature [55] which\nhave varying success in accounting for properties of nu-\nclei and neutron stars. The strength parameters of the\nSkyrme-like energy density functionals are calibrated at\nnuclear and sub-nuclear densities to reproduce the prop-\nerties of many nuclei, their behavior at high densities\nbeing constrained largely by neutron-star data.\nThe Hamiltonian density of APR is a parametric \ft\nto the AP microscopic model calculations in which the\nnucleon-nucleon interaction is modeled by the Argonne\nv18 2-body potential [56], the Urbana UIX 3-body po-\ntential [57], and a relativistic boost potential \u000ev[58]\nwhich is a kinematic correction when the interaction is\nobserved in a frame other than the rest-frame of the nu-\ncleons. These microscopic potentials accurately \ft scat-\ntering data in vacuum and thus incorporate the long\nscattering lengths of nucleons at low energy. Addition-\nally, they have also been successful in accounting for the\nbinding energies and spectra of light nuclei. An inter-\nesting feature of AP, incorporated in the Skyrme-like\nparametrization of the APR model, is that at supra-\nnuclear densities a neutral pion condensate appears. De-\nspite the softness induced by the pion condensate in the\nhigh density equation of state, the APR model is capa-\nble of supporting a neutron star of 2.19 M \f, in excess of\nthe recent accurate measurements of the masses of PSR\nJ1614-2230 (1 :97\u00060:04 M\f) [32] and PSR J0348+0432\n(2:01\u00060:04 M\f) [33].\nBeing non-relativistic potential models, both the APR\nand Ska models have the potential to become acausal\n(that is, the speed of sound exceeds the speed of light)\nat high density. A practical \fx to keep their behaviors\ncausal which is thermodynamically consistent is possible\nand is adopted in this work (see Appendix E).\nOur choice of these two models was motivated by sev-\neral considerations, including the facts that (i) both mod-\nels yield similar results for the equilibrium density, bind-\ning energy, symmetry energy, and compression modulus\nof symmetric matter, as well as for the maximum mass\nof neutron stars and (ii) the two models di\u000ber signi\f-\ncantly in other properties such as their Landau e\u000bective\nmasses (important for thermal properties), derivatives of\ntheir symmetry energy at nuclear density (important for\nthe high density behavior of isospin asymmetry energies),\nskewness (i.e., the derivative of the compression modulus)\nat nuclear density, and their predicted radii correspond-\ning to the maximum mass con\fguration. The impact of\nthe di\u000berent features of these two models for their ther-\nmal properties is one of the main foci of our work here.\nThe methods used to explore their thermal e\u000bects are ap-\nplicable and easily adapted to other Skyrme-like energy\ndensity functionals.5\nA. Hamiltonian density of APR\nExplicitly, the APR Hamiltonian density is given by\n[11]\nHAPR =\u0014~2\n2m+ (p3+ (1\u0000x)p5)ne\u0000p4n\u0015\n\u001cn\n+\u0014~2\n2m+ (p3+xp5)ne\u0000p4n\u0015\n\u001cp\n+g1(n)[1\u0000(1\u00002x)2)] +g2(n)(1\u00002x)2;(1)\nwheren=nn+npis the baryon density, x=np=nis the\nproton fraction, and\nni=1\n\u00192Z\ndkik2\ni\n1 +e(\u000fki\u0000\u0016i)=T(2)\n\u001ci=1\n\u00192Z\ndkik4\ni\n1 +e(\u000fki\u0000\u0016i)=T(3)\nare the number densities and kinetic energy densities of\nnucleon species i=n;p, respectively. The quantities \u000fki,\n\u0016iandTare the single-particle spectra, chemical poten-\ntials and temperature (with Boltzmann's constant kBset\nto unity), respectively. The \frst two terms on the right-\nhand side of this expression are due to kinetic energy and\nmomentum-dependent interactions while the last terms\nare due to density-dependent interactions. Compared to\na classical Skyrme interaction, such as Ska (described be-\nlow), this model has a more complex density dependence\nin the single-particle potentials and e\u000bective masses. Due\nto the occurrence of a neutral pion condensation at supra-\nnuclear densities, the potential energy density functions\ng1andg2take di\u000berent forms on either side of the tran-\nsition density. In the low density phase (LDP)\ng1L=\u0000n2h\np1+p2n+p6n2+ (p10+p11n)e\u0000p2\n9n2i\n(4)\ng2L=\u0000n2\u0010p12\nn+p7+p8n+p13e\u0000p2\n9n2\u0011\n; (5)\nwhereas, in the high density phase (HDP)\ng1H=g1L\u0000n2\u0002\np17(n\u0000p19) +p21(n\u0000p19)2\u0003\nep18(n\u0000p19)\n(6)\ng2H=g2L\u0000n2\u0002\np15(n\u0000p20) +p14(n\u0000p20)2\u0003\nep16(n\u0000p20):\n(7)\nThe values of the parameters p1throughp21, as well\nas their dimensions which ensure that HAPR has units\nof MeV fm\u00003, are presented in Table I. Alternate choices\nfor the underlying microscopic physics lead to di\u000berent\n\fts to the above generic form, so even though p13;p14and\np21are all 0 in our case, we carry the terms containing\nthese coe\u000ecients in the algebra of Appendix B.\nThe trajectory in the n\u0000xplane, for any temperature,\nalong which the transition from the LDP to the HDP\noccurs is obtained by solving\ng1L[1\u0000(1\u00002x)2] +g2L(1\u00002x)2\n=g1H[1\u0000(1\u00002x)2] +g2H(1\u00002x)2:(8)p1337:2 MeV fm3p14 0\np2\u0000382:0 MeV fm6p15287:0 MeV fm6\np389:8 MeV fm5p16\u00001:54 fm3\np4 0:457 fm3p17175:0 MeV fm6\np5\u000059:0 MeV fm5p18\u00001:45 fm3\np6\u000019:1 MeV fm9p19 0:32 fm\u00003\np7214:6 MeV fm3p20 0:195 fm\u00003\np8\u0000384:0 MeV fm6p21 0\np9 6:4 fm6\np1069:0 MeV fm3\np11\u000033:0 MeV fm6\np12 0:35 MeV\np13 0\nTABLE I. Parameter values for the Hamiltonian density of\nAkmal, Pandharipande, and Ravenhall [11]. Values in the\nlast column are speci\fc to the high density phase (HDP).\nThe dimensions are such that the Hamiltonian density is in\nMeV fm\u00003.\nThe solution gives a transition density nt= 0:32 fm\u00003for\nsymmetric nuclear matter ( x= 1=2) andnt= 0:195 fm\u00003\nfor pure neutron matter ( x= 0). For intermediate val-\nues ofx, the transition density is approximated to high\naccuracy by the polynomial \ft\nnt(x) = 0:1956+0:3389x+0:2918x2\u00001:2614x3+0:6307x4:\n(9)\nIn calculations of subsequent sections, the transition from\nthe LDP to the HDP at zero and \fnite temperatures will\nbe made through the use of the above polynomial \ft. The\nmixed phase region is determined via a Maxwell construc-\ntion for the numerical purposes of which ntis used as an\ninput. We show in Sec. VI that while the transition is in-\ndependent of Tfor anyx, the two densities which de\fne\nthe boundary of the phase-coexistence region do exhibit\na weak dependence on temperature.\nB. Hamiltonian density of Ska\nThe Hamiltonian density of Ska [50] based on the\nSkyrme energy density functional approach is expressed\nas\nHSka=~2\n2mn\u001cn+~2\n2mp\u001cp\n+n(\u001cn+\u001cp)\u0014t1\n4\u0010\n1 +x1\n2\u0011\n+t2\n4\u0010\n1 +x2\n2\u0011\u0015\n+(\u001cnnn+\u001cpnp)\u0014t2\n4\u00121\n2+x2\u0013\n\u0000t1\n4\u00121\n2+x1\u0013\u0015\n+to\n2\u0010\n1 +xo\n2\u0011\nn2\u0000to\n2\u00121\n2+xo\u0013\n(n2\nn+n2\np)\n\u0014t3\n12\u0010\n1 +x3\n2\u0011\nn2\u0000t3\n12\u00121\n2+x3\u0013\n(n2\nn+n2\np)\u0015\nn\u000f\n(10)6\nTerms involving \u001ciwithi=n;pare purely kinetic in ori-\ngin whereas terms involving n\u001ciandni\u001ciarise from the\nexchange part of the nucleon-nucleon interaction. The\nlatter determine the density dependence of the e\u000bective\nmasses (see below). The remaining terms, dependent on\npowers of the individual and total densities give the po-\ntential part of the energy density. The various strength\nparameters are calibrated to desired properties of bulk\nmatter and of nuclei chie\ry close to the empirical nu-\nclear equilibrium density. Many other parametrizations\nof the Skyrme-like energy density functional also exist\n[55] and are characterized by di\u000berent values of observ-\nable physical quantities (see below). The parameters to\nthrought3,xothroughx3, and\u000ffor the Ska model [50]\nare listed in Table II.\niti xi\u000f\n0\u00001602:78 MeVfm60.02 1/3\n1 570:88 fm30\n2\u000067:7 fm30\n38000:0 MeVfm7-0.286\nTABLE II. Parameter values for the Ska Hamiltonian den-\nsity [50].The dimensions are such that the Hamiltonian den-\nsity is in MeV fm\u00003.\nIII. SINGLE-PARTICLE ENERGY SPECTRA\nThe single-particle energy spectra \u000fki;(i=n;p) that\nappear in the Fermi-Dirac (FD) distribution functions\nnki=\u0002\n1 +e(\u000fki\u0000\u0016i)=T\u0003\u00001are obtained from functional\nderivatives of the Hamiltonian density (see appendix A\nfor derivation):\n\u000fki=k2\ni@H\n@\u001ci+@H\n@ni: (11)\nThe ensuing results can be expressed as\n\u000fkn=~2k2\n2m+Un(n;k)\n\u000fkp=~2k2\n2m+Up(n;k); (12)\nwheremis the nucleon mass in vacuum, and UnandUp\nare the neutron and proton single-particle momentum-\ndependent potentials, respectively. Utilizing these spec-\ntra, the Landau e\u000bective masses m\u0003\niare\nm\u0003\ni\u0011kFi\"\f\f\f\f@\u000fki\n@k\f\f\f\f\nkFi#\u00001\n; (13)\nwherekFiare the Fermi-momenta of species i. Physi-\ncal quantities such as the thermal energy, thermal pres-\nsure, susceptibilities, speci\fc heats at constant volume\nand pressure, and entropy all depend sensitively on these\ne\u000bective masses as highlighted in later sections.APR single-particle potentials\nFrom Eq. (1) and Eq. (12), the explicit forms of the\nsingle-particle potentials for the LDP Hamiltonian den-\nsity of APR are\nUnL(n;k) = (p3+Ynp5)ne\u0000p4nk2\n+f[p3+p5\u0000p4n(p3+Ynp5)]\u001cn\n+ [p3\u0000p4n(p3+Ypp5)]\u001cpgep4n\n+ 4Ypg1L\nn+ 2(Yn\u0000Yp)g2L\nn\n+ 4YnYpf1L+ (Yn\u0000Yp)2f2L\nUpL(n;k) = (p3+Ypp5)ne\u0000p4nk2\n+f[p3+p5\u0000p4n(p3+Ypp5)]\u001cp\n+ [p3\u0000p4n(p3+Ynp5)]\u001cngep4n\n+ 4Yng1L\nn+ 2(Yp\u0000Yn)g2L\nn\n+ 4YnYpf1L+ (Yn\u0000Yp)2f2L; (14)\nwithYp=xandYn= 1\u0000x, and where\nf1L=dg1L\ndn\u00002g1L\nnandf2L=dg2L\ndn\u00002g2L\nn:(15)\nIn the HDP,\nUnH(n;k) =UnL(n;k)\u00004Yp(Yn\u0000Yp)\nn(\u000eg1\u0000\u000eg2)\n+ 4YnYp\u000ef1+ (Yn\u0000Yp)2\u000ef2 (16)\nUpH(n;k) =UpL(n;k) +4Yn(Yn\u0000Yp)\nn(\u000eg1\u0000\u000eg2)\n+ 4YnYp\u000ef1+ (Yn\u0000Yp)2\u000ef2: (17)\nThe functions \u000eg1; \u000eg 2; \u000ef 1, and\u000ef2are de\fned in Ap-\npendix B. The corresponding e\u000bective masses from Eq.\n(13) are\nm\u0003\ni\nm=\u0014\n1 +2m\n~2(p3+Yip5)ne\u0000p4n\u0015\u00001\n; (18)\nwhereYi= (1\u0000x) for neutrons ( i=n) andYi=xfor\nprotons (i=p). Subsuming the k2-dependent parts of\nUi(n;k) in Eq. (14) into the kinetic energy terms in Eqs.\n(12), the single-particle energies may be expressed as\n\u000fki=~2k2\n2m\u0003\ni+Vi(n); (19)\nwhere the functional forms of Vi(n) are readily ascer-\ntained from the relations in Eq. (14). The quadratic\nmomentum-dependence of the single particle spectra, al-\nbeit density and concentration dependent through the\ne\u000bective masses, is akin to that of free Fermi gases. Con-\nsequently, the thermal state variables can be calculated\nas for free Fermi gases, but with attendant modi\fcations\narising from the density-dependent e\u000bective masses as\nwill be discussed later.7\nSkyrme single-particle potentials\nExplicit forms of the single-particle potentials for the\nSka Hamiltonian are given by\nUn(n;k) = (X1+YnX2)nk2+ (X1+X2)\u001cn+X1\u001cp\n+ 2n(X3+YpX4) +n1+\u000f[(2 +\u000f)X5\n+ 2Yn+\u000f(Yn2+Yp2)]\nUp(n;k) = (X1+YpX2)nk2+ (X1+X2)\u001cp+X1\u001cn\n+ 2n(X3+YnX5) +n1+\u000f[(2 +\u000f)X6\n+ 2Yp+\u000f(Yn2+Yp2)]; (20)\nwhere\nX1=1\n4h\nt1\u0010\n1 +x1\n2\u0011\n+t2\u0010\n1 +x2\n2\u0011i\nX2=1\n4\u0014\nt2\u00121\n2+x2\u0013\n\u0000t1\u00121\n2+x1\u0013\u0015\nX3=t0\n2\u0010\n1 +x0\n2\u0011\n;X4=\u0000t0\n2\u00121\n2+x0\u0013\nX5=t3\n12\u0010\n1 +x3\n2\u0011\n;X6=\u0000t3\n12\u00121\n2+x3\u0013\n:(21)\nFrom Eq. (13), the density-dependent Landau e\u000bective\nmasses are\nm\u0003\ni\nm=\u0014\n1 +2m\n~2(X1+YiX2)n\u0015\u00001\n: (22)\nThe single-particle spectra have therefore the same struc-\nture as in Eq. (19), but with the potential terms Vi(n)\ninferred from Eq. (20).\nIV. ZERO TEMPERATURE PROPERTIES\nAt temperature T=0, nucleons are restricted to their\nlowest available quantum states. Therefore, the Fermi-\nDirac distribution functions that appear in the integrals\nof the number density and the kinetic energy density be-\ncome step-functions:\nnki=\u0012(\u000fki\u0000\u000fFi); (23)\nwhere\u000fFiis the energy at the Fermi surface for species\ni. Consequently,\nni=1\n\u00192ZkFi\n0k2\nidki=k3\nFi\n3\u00192(24)\n\u001ci=1\n\u00192ZkFi\n0k4\nidki=k5\nFi\n5\u00192=3\n5nik2\nFi: (25)\nThus, the kinetic energy densities can be written as sim-\nple functions of the number density nand the proton\nfractionx:\n\u001cp=1\n5\u00192(3\u00192np)5=3=1\n5\u00192(3\u00192nx)5=3(26)\n\u001cn=1\n5\u00192(3\u00192nn)5=3=1\n5\u00192(3\u00192n(1\u0000x))5=3:(27)We can therefore write\nH(np;nn;\u001cp;\u001cn;T= 0) =H(n;x);\nand use standard thermodynamic relations to get the var-\nious quantities of interest, some examples of which are\nlisted below beginning with x= 1=2 for isospin symmet-\nric nuclear matter. General expressions for arbitrary x\nare provided in Appendix B.\nA. Isospin symmetric nuclear matter\nThe APR Hamiltonian\nIt is convenient to write HAPR as the sum of a kinetic\npartHk, a part consisting of momentum-dependent in-\nteractionsHm, and a density-dependent interactions part\nHd. The energy per particle of symmetric nuclear matter\nEcan then be similarly decomposed as\nE\u0011HAPR\nn=Ek+Em+Ed; (28)\nwhere\nEk=3\n5~2k2\nF\n2m;kF= (3\u00192n=2)1=3\nEm=3\n5nk2\nFe\u0000p4n(p3+p5=2)\nEdL=g1L\nn; EdH=g1H\nn=EdL+\u000eg1\nn:(29)\nThe corresponding pressure is\nP=n2@E\n@n=Pk+Pm+Pd\nPk=2\n3nEk;Pm=\u00125\n3\u0000p4n\u0013\nnEm\nPdL=n(Ed+f1L)\nPdH=PdL\u0000\u000eg1+n\u000ef1: (30)\nThe nucleon chemical potential takes the form\n\u0016=@H\n@n=\u0016k+\u0016m+\u0016d\n\u0016k=5\n3Ek=~2k2\nF\n2m\n\u0016m=nk2\nFe\u0000p4n\u001a\np5\u00124\n5\u0000p4n\n2\u0013\n+p3\u00128\n3\u0000p4n\u0013\u001b\n\u0016dL=dg1L\ndn; \u0016dH=\u0016dL+\u000ef1: (31)8\nThe inverse susceptibility is given by\n\u001f\u00001=@\u0016\n@n=\u001f\u00001\nk+\u001f\u00001\nm+\u001f\u00001\nd\n\u001f\u00001\nk=2\n3\u0016k\nn\n\u001f\u00001\nm=\u0000p4\u0016m+3\n5k2\nFe\u0000p4n\n\u0003\u001a4\n3p5\u001210\n3\u0000p4n\u0013\n+2\n3p3\u001225\n3\u00004p4n\u0013\u001b\n\u001f\u00001\ndL= 8f1L\nn+ 4h1L\n\u001f\u00001\ndH=\u001f\u00001\ndL\u00002\nn2(\u000eg1\u0000\u000eg2) +\u000eh1; (32)\nwhere\nh1L=df1L\ndn\u00002f1L\nn(33)\nand\u000eh1can be found in App.B. The nuclear matter in-\ncompressibility is given by\nK= 9dP\ndn=Kk+Km+Kd\nKk= 10Ek= 6~2k2\nF\n2m\nKm=\u0000\n40\u000048p4n+ 9p2\n4n2\u0001\nEm\nKdL= 18Ed+ 9 [4f1L+nh1L]\nKdH=KdL+ 9n\u000eh 1: (34)\nThe speed of sound can be written in terms of \u0016andK\nor\u001f\u00001as\n\u0010cs\nc\u00112\n=K\n9(\u0016+m)=n\u001f\u00001\n\u0016+m(35)\nFrom this relation, it can be shown that the APR model\nbecomes acausal ( cs=c= 1) atn= 0:841 fm\u00003in the\ncase of symmetric matter.\nThe speed of sound cs, and the response functions K\nand\u001fare generated by density \ructuations. Evidently,\nthey are not independent of each other (relationships be-\ntween them in the case of general asymmetry are given\nin Appendix B). Each quantity, however, is useful in its\nown right for a number of applications. For example, cs\nis necessary in implementing causality (see Appendix E),\nKis essential to the calculation of the liquid-gas phase\ntransition (Sec.V), and \u001fis required in the numerical\nscheme by which the mixed-phase region, at the onset\nof pion condensation, is constructed (Sec.VI). At \fnite\ntemperature, this group also includes the speci\fc heats\nat constant volume and pressure, CVandCP. The latter\ncan be used to identify phase transitions, address causal-\nity at \fnite Tand, furthermore, are related to hydrody-\nnamic time-scales as in the collapse to black holes.\nThe Skyrme Hamiltonian\nSimilarly to the APR Hamiltonian we write HSkaas\nthe sum of a kinetic part Hk, momentum-dependent in-teractionsHm, and a density-dependent interactions Hd.\nThe energy per particle is then given by\nE\u0011HSka\nn=Ek+Em+Ed; (36)\nwhere\nEk=3\n5~2k2\nF\n2m; Em=3\n5nk2\nF\u0012\nX1+1\n2X2\u0013\nEd=n\u0014\nX3+1\n2X4+n\u000f\u0012\nX5+1\n2X6\u0013\u0015\n: (37)\nContributions to the pressure arise from\nPk=2\n3nEk;Pm=5\n3nEm\nPd=n\u0012\nEd+\u000fn\u000f+1\u0012\nX5+1\n2X6\u0013\u0013\n: (38)\nThe nucleon chemical potential receives contributions\nfrom\n\u0016k=5\n3Ek; \u0016m=8\n3Em\n\u0016d= 2Ed+\u000f\u0012\nX5+1\n2X6\u0013\nn\u000f+1: (39)\nThe inverse susceptibility is composed of terms involving\n\u001f\u00001\nk=2\n3\u0016k\nn; \u001f\u00001\nm=25\n12\u0016m\nn+4m\n~2X2\u0016k\n\u001f\u00001\nd=\u0016d\nn+n\u000f\u0014\u0012\nX5+1\n2X6\u0013\n\u000f+X6\u0015\n+X4:(40)\nThe nuclear matter incompressibility is determined by\nthe terms\nKk= 10Ek; Km= 40Em\nKd= 18Ed+ 9\u000f(\u000f+ 3)n1+\u000f\u0014\nX5+1\n2X6\u0015\n:(41)\nCombining the above results with Eq.(35) we \fnd that\nSka violates causality for baryon densities above n=\n1:028 fm\u00003.\nB. Isospin asymmetric matter\nHere, we focus on the energetics of matter with neu-\ntron excess beginning with some general considerations\nthat are model independent. The neutron-proton asym-\nmetry is commonly characterized by the parameter \u000b=\n(nn\u0000np)=nwhich is connected to the proton fraction x\nthrough the simple relation \u000b= 1\u00002x.\nThe expansion of the energy per particle E(n;\u000b) =\nH=nof isospin asymmetric matter in powers \u000b, is given\nby:\nE(n;\u000b) =E(n;0) +X\nl=2;4;:::Sl(n)\u000bl(42)9\nwhere\nSl=1\nl!@lE(n;\u000b)\n@\u000bl\f\f\f\f\n\u000b=0;l= 2;4;::: (43)\nSimilarly, the pressure of isospin-asymmetric matter can\nbe written as\nP(n;\u000b) =n2@E(n;\u000b)\n@n(44)\n=P(n;0) +n\n3X\nl=2;4;:::Ll(n)\u000bl(45)\nwhere\nLl= 3ndSl(n)\ndn(46)\nEvaluating Eqs. (42)-(45) for pure neutron matter at the\nsaturation density n0of symmetric matter to O(\u000b2) gives\nE(n0;1)'E0+Sv (47)\nP(n0;1)'Ln0\n3(48)\nwhereE0=E(n0;0) is the saturation energy of nu-\nclear matter, Sv=S2(n0) is its symmetry energy pa-\nrameter that characterizes the energy cost involved in\nrestoring isospin symmetry from small deviations, and\nL=L2(n0) is its sti\u000bness parameter. By the de\fnition\nofn0,P(n0;0) = 0.\nOnly even powers of \u000bsurvive in the two series in\nEqs. 42 and 45 above because the two nucleon species\nare treated symmetrically in the Hamiltonian. Further-\nmore, due to the near complete isospin invariance of the\nnucleon-nucleon interaction, the density dependent po-\ntential terms are generally carried only up to O(\u000b2); that\nis,Sl(n) andLl(n) forl >2 receive contributions just\nfrom the kinetic energy and the momentum-dependent\ninteractions. Finally, as demonstrated in Refs. [59{\n62],S2(n)\u001dS4(n);S6(n);:::and hence coe\u000ecients with\nl= 2 su\u000ece in describing bulk matter even when \u000b\u00181.\nWhile the full calculations are rather involved, the\ndominance of S2(n) can be illustrated in a simple man-\nner by turning to the isospin-asymmetric free gas whose\nkinetic energy can be expressed as\nEkin=1\n3EF\u00141\n2n\n(1 +\u000b)5=3+ (1\u0000\u000b)5=3o\n\u00001\u0015\n;(49)\nwhere\nEF=~2k2\nF\n2m=~2\n2m\u00123\u00192n\n2\u00132=3\n(50)\nis the Fermi energy of non-interacting nucleons in sym-\nmetric nuclear matter. Through a Taylor expansion of\nterms involving \u000b(terms in odd powers of \u000bcanceling),\nthe various contributions from kinetic energy are\nSkin\n2(n) =1\n3EF; Skin\n4(n) =1\n81EF; Skin\n6(n) =7\n2187EF:::\n(51)the series converging rapidly to the exact result of\n(EF=3) (22=3\u00001). At the empirical nuclear equilib-\nrium density of n0= 0:16 fm\u00003,Skin\n2(n0)'12:28\nMeV, whereas its associated sti\u000bness parameter is Lkin=\n(2=3)EF0'24:56 MeV.\nAs mentioned earlier, in the presence of interactions,\nS4(n);S6(n);:::are modi\fed solely by the momentum-\ndependent terms which, predominantly, give rise to the\ne\u000bective mass while preserving the relative sizes of the\nSl's and their derivatives (For APR, at n0,S2=S4'35\nandL2=L4'18 whereas for Ska, S2=S4'29 and\nL2=L4'17.). Thus, we can write\nP(n;\u000b)'n2\u0002\nE0(n;0) +\u000b2S0\n2(n)\u0003\n(52)\nwhere the primes denote derivatives with respect to the\ndensityn.\nBy expanding E0(n;0) andS0\n2(n) about the saturation\ndensityn0of symmetric matter (noting that E0(n0;0) =\n0), we obtain\nE0(n;0)'K0\n9n0\u000e+Q0\n54n0\u000e2(53)\nS0\n2(n)'L\n3n0+KS2\n9n0\u000e+QS2\n54n0\u000e2(54)\nwhere\u000e= (n=n 0)\u00001, and\nK0= 9n2\n0d2E(n;0)\ndn2\f\f\f\f\nn0; Q 0= 27n3\n0d3E(n;0)\ndn3\f\f\f\f\nn0(55)\nL= 3n0dS2(n)\ndn\f\f\f\f\nn0; KS2= 9n2\n0d2S2(n)\ndn2\f\f\f\f\nn0(56)\nQS2= 27n3\n0d3S2(n)\ndn3\f\f\f\f\nn0(57)\nThe skewnessSis related to K0andQ0via\nS=k3\nFd3E\ndk3\nF\f\f\f\f\n\u000b=0;n0= 6K0+Q0 (58)\nand the symmetry term K\u001cof the liquid drop formula for\nthe isospin asymmetric incompressibility [63] is related\ntoSv,L,K0, andKS2via\nK\u001c=KS2\u0000LSv\nK0: (59)\nAt the equilibrium density n0\u000bof isospin asymmetric\nmatter,\nP(n0\u000b;\u000b) = 0 =E0(no\u000b;0) +S0\n2(n0\u000b)\u000b2: (60)\nThe insertion of Eqs. (53)-(54) into Eq. (60), while re-\ntaining terms up to O(\u000e), leads to [64, 65]\n\u000e\u000b\u0011n0\u000b\nn0\u00001 =\u00003L\nK0\u000b2\u0011\u0000C\u000b2(61)\nto lowest order in \u000b2. This relation allows us to trace the\nloci of the minima of the energy per particle for changing10\nasymmetries. Further improvement to cover higher val-\nues of\u000brequires keeping terms to O(\u000e2) in Eqs.(53)-(54):\n\u000e\u000b=3K0\nQ0\u0010\n1 +KS2\nK0\u000b2\u0011\n\u0010\n1 +QS2\nQ0\u000b2\u0011\n\u00038\n>><\n>>:\u00001 +2\n641\u00002LQ0\u000b2\u0010\n1 +QS2\nQ0\u000b2\u0011\nK2\n0\u0010\n1 +KS2\nK0\u000b2\u001123\n751=29\n>>=\n>>;:(62)\nIn this expression, we have discarded terms involving L4\nbecause, as we mentioned earlier, these are very small\nand make no signi\fcant contributions. Additionally, for\nAPR,KS2=K0\u00180:4 andQS2=Q0\u0018\u00001:2. The large ( >\n1) magnitude ofjQS2=Q0jmeans that for \u000b\u00150:7 (which\nwas the reason for going beyond \u000b2in the \frst place),\nwe incur signi\fcant error upon expanding Eq. (62) in a\nTaylor series in \u000b. This problem does not arise for Ska\nwhereKS2=K0\u00180:3 andQS2=Q0\u0018\u00000:6. In the latter\ncase, Eq. (62) can be reduced to the simple form\n\u000e=\u00003L\nK0\u000b2\u0014\n1 +\u0012Q0L\n2K2\n0\u0000KS2\nK0\u0013\n\u000b2\u0015\n: (63)\nWe stress that Eq. (63) is applicable only in situations\nwherejKS2=K0jandjQS2=Q0jare much smaller than 1.\nIf this condition does not hold (such as in APR), the\nmore general expression (62) must be used.\nFinally, we calculate the incompressibility at the sat-\nuration density n0\u000bof asymmetric matter in terms of\nsymmetric matter equilibrium properties, to O(\u000b2) (see\nalso, Refs. [64, 65]). Using Eq. (52) we get, for general\nn,\nK(n;\u000b) = 9@P(n;\u000b)\n@n(64)\n=K(n;0)\u0000\n1 +A(n)\u000b2\u0001\n(65)\nwhere\nK(n;0) = 9\u0002\n2nE0(n;0) +n2E00(n;0)\u0003\n(66)\nA(n) =9\nK(n;0)\u0002\n2nS0\n2(n) +n2S00\n2(n)\u0003\n(67)\nAtn=n0\u000b,\nK(n0\u000b)'K(n0;0) +dK(n;0)\ndn\f\f\f\f\nn0(n0\u000b\u0000n0) (68)\n=K0+\u0012\n4K0+Q0\n3\u0013\n\u000e\u000b (69)\n'K0\u0014\n1\u000012L\nK0\u0012\n1 +Q0\n12K0\u0013\n\u000b2\u0015\n(70)\n\u0011K0(1 +B\u000b2) (71)and\nA(n0\u000b)'9\nK0 \n2n0dS2(n)\ndn\f\f\f\f\nn0+n2\n0d2S2(n)\ndn2\f\f\f\f\nn0!\n(72)\n=9\nK0\u0012\n2n0L\n3n0+n2\n0KS2\n9n2\n0\u0013\n(73)\n=6L\nK0\u0012\n1 +KS2\n6L\u0013\n\u0011A: (74)\nHence, toO(\u000b2),\nK(n0\u000b;\u000b)'K0[1 + (A+B)\u000b2] (75)\n\u0011K0(1 + ~A\u000b2); (76)\nwhere the coe\u000ecient Arepresents modi\fcations to the\ncompressibility evaluated at n0due to changing asym-\nmetry, whereas the coe\u000ecient Bencodes alterations due\nto the shift of the saturation point of matter as the asym-\nmetry varies.\nC. Results and analysis\nIn this section, the zero temperature results obtained\nfrom the APR and Ska Hamiltonians are presented.\nColumns 2 and 3 in Table III contain the key symmetric\nnuclear matter properties for both models at their\nrespective equilibrium densities (nearly the same). Note\nthat while the energy per particle E(n0)\u0011E0and the\ncompression modulus K0for both models are similar,\nthe e\u000bective masses m\u0003\n0=mare somewhat di\u000berent near\nnuclear densities. Signi\fcant di\u000berences are seen in\nthe skewness parameters S, the Ska model being more\nasymmetric than the APR model at its equilibrium\ndensity.\nAPR Ska Experiment Reference\nn0(fm\u00003)0.160 0.155 0:17\u00060:02[45, 66{68]\nE0(MeV) -16.00 -15.99\u000016\u00061 [45, 68]\nK0(MeV) 266.0 263.2 230\u000630 [14, 15]\n240\u000620 [16]\nQ0(MeV) -1054.2 -300.2\u0000700\u0006500 [69]\nSv(MeV) 32.59 32.91 30-35 [23, 24]\nL(MeV) 58.46 74.62 40-70 [23, 24]\nKS2(MeV) -102.6 -78.46\u0000100\u0006200 This work\nQS2(MeV) 1217.0 174.5 ?\nS(MeV) 541.8 1278.9 680\u0006530 This work\nm\u0003\n0=m 0.70 0.61 0:8\u00060:1 [17, 18]\nTABLE III. Entries in this table are at the equilibrium den-\nsityn0of symmetric nuclear matter for the APR and Ska\nmodels.E0is the energy per particle, K0is the compression\nmodulus,Q0is related to the third derivative of E,Sis the\nskewness,m\u0003\n0=mis the ratio of the Landau e\u000bective mass to\nmass in vacuum, Svis the nuclear symmetry energy parame-\nter, andL,KS2, andQS2are related to the \frst, second, and\nthird derivative of the symmetry energy, respectively.11\n 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0  0.1  0.2  0.3  0.4  0.5m*/m\nn (fm-3)APR\npnx = 0.1\n0.3\n0.5\n 0  0.1  0.2  0.3  0.4  0.5  0.6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\nn (fm-3)Ska\npnx = 0.1\n0.3\n0.5\nFIG. 1. Left panel: Ratios of the neutron (solid) and proton (dotted) Landau e\u000bective masses to the vacuum mass versus\nbaryon density nfor the APR model from Eq. (18). Right panel: Same as the left panel but for the Ska model from Eq. (22).\nValues of the proton fraction xare as indicated in the \fgure.\nAmong the most important quantities to be discussed\nare the nucleon Landau e\u000bective masses as they are criti-\ncal to the thermal properties of the equation of state. We\nshow ratios of the neutron and proton Landau e\u000bective\nmasses to the vacuum mass versus baryon density nfor\nvalues ofx= 0:5;0:3 and 0.1, respectively, in Fig. 1.\nThe left panel is for the APR model from Eq. (18) and\nthe right panel contains similar results for the Ska model\nfrom Eq. (22). At the equilibrium density n0of symmet-\nric nuclear matter, m\u0003\n0for Ska is smaller than for APR,\nand sincejX2j<2X1andjp5j<2p3, this means that m\u0003\nis also smaller for Ska at every xatn0. Therefore, de\fn-\ningaSka=X1+YiX2andaAPR =p3+Yip5, we must\nhaveaSka>aAPRe\u0000bn0for anyYi2[0;1] from Eqs. (18)\nand (22). It then follows from p4>0 thatm\u0003\niis smaller\nfor Ska at all densities for every value of x2[0;1] and for\nboth neutrons and protons. Furthermore, since p5<0\nandX2<0, we have that m\u0003\nn(n;x)>m\u0003\n0>m\u0003\np(n;x) for\nn>0 andx<1=2.\nFigure 2 shows the energy particle Eas a function of\nbaryon density nfor values of x= 0:5;0:3 and 0:1 for\nthe two models. Our calculated results of APR (solid\ncurves) agree well with those tabulated in Table VI and\nVII of Ref. [11] (shown by crosses for x= 0:5 in this\n\fgure). We also contrast the microscopic AP results for\npure neutron matter and symmetric nuclear matter with\nthose obtained from the APR \ft in Table IV. (As noted in\nthe introduction, results below n'0:1 fm\u00003can be used\nto establish di\u000berences from the inhomogeneous phase of\nsupernova matter containing nuclei, light nuclear clus-\nters, etc.) The asterisks in Fig. 2 show the densities at\nwhich the transition from the low density phase (LDP)\nto the high density phase (HDP) occurs due to pion con-\ndensation. While there is good agreement between the\nresults of the two models up to and slightly beyond the\nequilibrium density, the Ska model is seen to have bothn (fm\u00003)AP(SNM) APR(SNM) AP(PNM) APR(PNM)\n0.04 -6.48 -5.63 6.45 6.42\n0.08 -12.13 -11.56 9.65 9.58\n0.12 -15.04 -14.98 13.29 13.28\n0.16 -16.00 -16.00 17.94 17.99\n0.20 -15.09 -15.16 22.92 23.57\n0.24 -12.88 -12.96 27.49 28.04\n0.32 -5.03 -5.14 38.82 39.41\n0.40 2.13 2.62 54.95 54.72\n0.48 15.46 15.14 75.13 74.59\n0.56 34.39 32.92 99.74 99.45\n0.64 58.35 56.22 127.58 129.57\n0.80 121.25 119.97 205.34 206.22\n0.96 204.02 207.14 305.87 305.06\nTABLE IV. AP vs APR energies in MeV for symmetric nu-\nclear matter (SNM) and pure neutron matter (PNM) ex-\ntracted from Ref. [11].\nhigher energies and pressures (slopes of the energy) than\nthe APR model at high densities for all values of x. This\nfeature essentially stems from the emergence of the pion\ncondensate in the HDP of APR which softens the corre-\nsponding EOS. Both equations of state become acausal\nat high densities; a scheme to retain causality will be\noutlined later.\nRows 5 and 6 in Table III list the symmetry energy\nSvand its slope parameter Lfor the two models.\nAlthoughSvfor both the models are similar, values\nofLdi\u000ber signi\fcantly. The higher value of Lfor the\nSka model leads to a greater energy and pressure of\nisospin asymmetric matter than for the APR model near\nnuclear saturation densities, a feature that persists to\nhigher densities.\nThe density dependent symmetry energy S2(n) can in12\n-20 0 20 40 60 80 100 120 140\n 0  0.1  0.2  0.3  0.4  0.5  0.6E (MeV)\nn (fm-3)T = 0 MeV\nAPRSka\nx = 0.1\n0.3\n0.5\nFIG. 2. Zero temperature energy per particle Eversus baryon\nnumber density for the APR (solid curves) using Eqs. (B25)-\n(B28) and Ska (dashed curves) models at the indicated values\nof the proton fraction x. The crosses on the APR curve for\nx= 1=2 show values from column 6 of Table VI in Ref. [11].\nAlthough not shown here, we have veri\fed that similar agree-\nment is obtained with the APR results in column 5 of Table\nVII in Ref. [11] for pure neutron matter (x=0). The cusps in\nthe APR curves are due to the onset of neutral pion conden-\nsation.\ngeneral be written as S2=S2k+S2m+S2dwithS2kas in\nEq. (51). Contributions from the momentum-dependent\nand density-dependent parts, S2mandS2d, depend on\nthe model used. For the APR model,\nS2m=1\n3k2\nFne\u0000p4n(p3+ 2p5);\nS2d=1\nn(\u0000g1+g2); (77)\nwhereas for the Ska model\nS2m=1\n3k2\nFn(X1+ 2X2) andS2d=n\n2(X4+X6n\u000f):(78)\nNote that the terms S4(n) andS6(n) receive contribu-\ntions from the momentum-dependent interaction part as\nwell because of terms involving ni\u001ciin theH's of Eqs.\n(1) and (10). Explicitly,\nS4m=1\n34k2\nFne\u0000p4n(p3\u0000p5);\nS6m=7\n37k2\nFne\u0000p4n\u0012\np3\u00001\n5p5\u0013\n(79)\nfor the APR model, and for the Ska model\nS4m=1\n34k2\nFn(X1\u0000X2);\nS6m=7\n37k2\nFn\u0012\nX1\u00002\n5X2\u0013\n: (80)\nIn Fig. 3, the extent to which the functions S2(n)\n(which we call the symmetry energy), S4(n) andS6(n)from Eqs. (51), (79), and (80) contribute to the dif-\nference between pure neutron matter and nuclear mat-\nter energy, \u0001 E(n) =E(n;\u000b = 1)\u0000E(n;\u000b = 0) (for\nwhich we reserve the term \"asymmetry energy\") is ex-\namined. The left (right) panel shows results for the APR\n(Ska) model. The symmetry energy S2(n) adequately\naccounts for the total \u0001 E(n) up to twice n0. How-\never, for densities well in excess of n0, contributions from\nS4(n)S6(n)\u0001\u0001\u0001become important although S2(n) re-\nmains dominant. The jumps in the symmetry energies for\nAPR atn=p19= 0:32 fm\u00003(at which transition from\nthe LDP to HDP occurs for x= 0:5) are due to the de\f-\nnitions ofS2(n); S4(n); S6(n)\u0001\u0001\u0001which involve deriva-\ntives taken at x= 0:5. As the transition to the HDP\noccurs at lower values of nasxdecreases toward x= 0,\nthe conventional de\fnitions of S2(n); S4(n);S6(n)\u0001\u0001\u0001\nfail to capture the true behavior of \u0001 E(n) in the pres-\nence of a phase transition. That is to say,\nS(n)\u0011X\nl=2;4;:::Sl(n)6= \u0001E(n) (81)\nin the vicinity of a phase transition driven by density and\ncomposition, regardless of the order to which the sum is\ncarried out.\nResults for the coe\u000ecients A; B; C; and ~Athat de-\nscribe the isospin asymmetry dependence to O(\u000e\u000b) of the\nequilibrium density and compression moduli for the APR\nand Ska models are displayed in Table V. Since asymme-\ntry lowers the equilibrium density, transitions occurring\nat supra-nuclear densities do not a\u000bect these results. One\nobserves that even though HAPR andHSkaare calibrated\nto very similar values of the symmetry energy and the\ncompression modulus, these asymmetry coe\u000ecients vary\nsigni\fcantly.\nModelABC ~A=A+B\nAPR 0.933 -1.766 0.659 -0.833\nSka 1.403 -3.079 0.851 -1.676\nTABLE V. Results for the coe\u000ecients that describe the\nisospin asymmetry dependence to O(\u000e\u000b) of the equilibrium\ndensity and compression moduli.\nThe extent to which Eq. (61), inserted into Eq. (42)\nexpanded toO(\u000b2), adequately describes the loci of en-\nergy minima in the energy per particle of subnuclear mat-\nter for arbitrary \u000bis demonstrated in Fig. 4 for the two\nmodels. The dark circles show locations of the minima\nresulting from the exact calculations using Eqs. (1) and\n(10) as the proton fraction xis varied toward that of pure\nneutron matter. The leading order results shown by the\ndotted curves accurately trace the loci of minima down to\nx= 0:2. Considering the O(\u000e2\n\u000b) contribution in Eq. (62)\nimproves agreement with the exact results even down to\nx= 0:1.\nIn Fig. 5, we show the pressure as a function of nfor\nrepresentative values of x. For allx, including for neutron13\n 0 10 20 30 40 50 60 70 80 90\n 0  0.1  0.2  0.3  0.4  0.5Symmetry Energies (MeV)\nn (fm-3)S2+S4+S6∆EAPR\nS2+S4\nS2\n 0  0.1  0.2  0.3  0.4  0.5 0 10 20 30 40 50 60 70 80 90\nn (fm-3)S2+S4+S6∆E\nSka\nS2+S4\nS2\nFIG. 3. Left panel: Symmetry energies for APR (from Eqs. (51),(77) and (79)) vs baryon density n. Right panel: Same as in\nthe left panel but for Ska (Eqs. (51),(78) and (80)).\n-20-15-10-5 0 5 10\n 0  0.03  0.06  0.09  0.12  0.15E (MeV)\nn (fm-3)x = 0.10\n0.15\n0.2\n0.28\n0.4Ο(δα)\nΟ(δα2)\nAPR\n 0  0.03  0.06  0.09  0.12  0.15-20-15-10-5 0 5 10\nn (fm-3)x = 0.10\n0.15\n0.2\n0.28\n0.4O(δα)\nO(δα2)\nSka\nFIG. 4. Left panel: Loci of minima in the energy per particle versus baryon density for the APR (left panel) and Ska (right\npanel) models for di\u000berent proton fractions. The dark circles are exact results from Eqs. (1) and (10). The dotted curves show\nO(\u000e\u000b) results from Eq. (61), whereas the O(\u000e2\n\u000b) (Eq. (62)) contributions are shown as dashed lines.\nmatter (not shown), the Ska model has higher pressure\nthan that for the APR model. As with the energy per\nparticle shown in Fig. 2, the larger sti\u000bness of the Ska\nmodel relative to the APR model is caused by appear-\nance of a pion condensate in the HDP of the latter. The\ndistinctive jumps in pressure for the APR model are due\nto the phase transition to a pion condensate, i.e., from\nthe LDP to the HDP which occurs at lower densities for\nincreasingly asymmetric matter.\nThe neutron and proton chemical potentials, \u0016nand\n\u0016p, versus baryon density for the two models are shown\nin the \frst two panels of Fig. 6. Due to its relative sti\u000b-\nness, results for the Ska model are systematically larger\nthan those for the APR model for all values of the pro-\nton fraction x. It is worthwhile to mention here that\n^\u0016=\u0016n\u0000\u0016p(with modi\fcations from e\u000bects of tempera-ture to be discussed in subsequent sections), shown in the\nrightmost panel of Fig. 6, controls the reaction rates as-\nsociated with electron captures and neutrino interactions\nin supernova matter.\nThe inverse susceptibilities are shown in Fig. 7 for\nthe APR and Ska models at representative proton frac-\ntions. The largest qualitative and quantitative di\u000berences\nbetween the two models occur at supra-nuclear densi-\nties ford\u0016n=dnnandd\u0016p=dnp. The cross derivatives\nd\u0016n=dnp=d\u0016p=dnnare qualitatively similar for the two\nEOSs, but relatively small quantitative di\u000berences be-\ntween the two models exist. In the case of the APR\nmodel, in which a pion condensate appears, these deriva-\ntives are required ingredients in the Maxwell construction\nwhich determines the phase boundary densities at which\nthe pressure and an average chemical potential are equal14\n 0 20 40 60 80 100 120 140 160\n 0  0.1  0.2  0.3  0.4  0.5  0.6P (MeV fm-3)\nn (fm-3)T = 0 MeV\nAPR\nSka\nx = 0.10.5\nFIG. 5. Pressure versus baryon density for the APR (Eqs.\n(B29)-(B33)) and Ska models at di\u000berent proton fractions.\nThe jumps in the APR results are due to phase transition to\na pion condensate at the values of xindicated.\n(this ensures mechanical and chemical equilibria). These\nderivatives are also utilized in constructing the full dense\nmatter tabular EOS as will be discussed later.\nV. FINITE TEMPERATURE PROPERTIES\nIn this section, properties of the APR and Ska models\nat \fnite temperature Tare calculated. At \fnite T, the\nHamiltonian density is a function of four independent\nvariables; namely, the number densities niand the kinetic\nenergy densities \u001ciof the two nucleon species. These are,\nin turn, proportional to the F1=2andF3=2Fermi-Dirac\n(FD) integrals [70], respectively:\nni=1\n2\u00192\u00122m\u0003\niT\n~2\u00133=2\nF1=2i (82)\n\u001ci=1\n2\u00192\u00122m\u0003\niT\n~2\u00135=2\nF3=2i (83)\nwhereF\u000bi=Z1\n0x\u000b\ni\ne\u0000 iexi+ 1dxi (84)\nxi=1\nT\u0012\nk2\ni@H\n@\u001ci\u0013\n=1\nT~2k2\ni\n2m\u0003\ni\u0011\"ki\nT(85)\n i=1\nT\u0012\n\u0016i\u0000@H\n@ni\u0013\n=\u0016i\u0000Vi\nT\u0011\u0017i\nT:(86)\nThe quantity  i, generally termed as the degeneracy pa-\nrameter, is related to the fugacity de\fned by zi=e i. In\nthe above equations, one must keep in mind that m\u0003\niis a\nfunction of the number densities of both nucleon species\ni=n;p. Consequently, derivatives of the FD integralswith respect to the densities take the forms\n@F1=2i\n@ni=F1=2i\nni\u0012\n1\u00003\n2ni\nm\u0003\ni@m\u0003\ni\n@ni\u0013\n(87)\nand@F1=2i\n@nj=\u00003\n2ni\nm\u0003\nj@m\u0003\ni\n@njF1=2i: (88)\nFD integrals of di\u000berent order are connected through\ntheir derivatives with respect to  i:\n@F\u000bi\n@ i=\u000bF(\u000b\u00001)i: (89)\nTherefore,\n@F\u000bi\n@ni=@F\u000bi\n@F1=2i@F1=2i\n@ni\n=@F\u000bi\n@ i\u0012@F1=2i\n@ i\u0013\u00001@F1=2i\n@ni\n= 2\u000bF(\u000b\u00001)i\nF\u00001=2i@F1=2i\n@ni: (90)\nSimilarly, cross derivatives with respect to density of\nFermi integrals are given by\n@F\u000bi\n@nj= 2\u000bF(\u000b\u00001)i\nF\u00001=2i@F1=2i\n@nj: (91)\nUtilizing the relations\n@\n@n=@\n@nn@nn\n@n\f\f\f\f\nx+@\n@np@np\n@n\f\f\f\f\nx= (1\u0000x)@\n@nn+x@\n@np\n@\n@x=@\n@nn@nn\n@x\f\f\f\f\nn+@\n@np@np\n@x\f\f\f\f\nn=\u0000n@\n@nn+n@\n@np;\nthe derivatives of F\u000biwith respect to nandxare ob-\ntained as\n@F\u000bi\n@n= 2\u000bF(\u000b\u00001)i\nF\u00001=2i\u0014\n(1\u0000x)@F\u000bi\n@nn+x@F\u000bi\n@np\u0015\n(92)\n@F\u000bi\n@x= 2\u000bF(\u000b\u00001)i\nF\u00001=2in\u0014@F\u000bi\n@np\u0000@F\u000bi\n@nn\u0015\n: (93)\nUsing Eqs. (90)-(93), we arrive at the following ex-\npressions for the density derivatives of the degeneracy\nparameter and the kinetic energy density:\n@ i\n@ni=2\nF\u00001=2i@F1=2i\n@ni;@ i\n@nj=2\nF\u00001=2i@F1=2i\n@nj(94)\n@ i\n@n=2\nF\u00001=2i@F1=2i\n@n;@ i\n@x=2\nF\u00001=2i@F1=2i\n@x(95)15\n-50 0 50 100 150 200 250 300 350 400\n 0  0.1  0.2  0.3  0.4  0.5  0.6µn (MeV)\nn (fm-3)T = 0 MeV\nAPRSka\nx = 0.10.3\n0.5\n 0  0.1  0.2  0.3  0.4  0.5  0.6-100-50 0 50 100 150 200 250 300\nµp (MeV)\nn (fm-3)T = 0 MeV\nAPR\nSka\nx = 0.10.30.5\n 0 50 100 150 200 250 0  0.1  0.2  0.3  0.4  0.5  0.6\nT = 0 MeV\nAPR Exact\nO(S2)O(S4) x = 0.1\n0.3\nn (fm-3)µn - µp (MeV)\n 50 100 150 200 250\n 0  0.1  0.2  0.3  0.4  0.5  0.6 0  0.1  0.2  0.3  0.4  0.5  0.6\nT = 0 MeV\nSkaExact\nO(S2)O(S4)\nx = 0.1\n0.3\nFIG. 6. The \frst (second) panel shows the neutron (proton) chemical potential versus baryon density nfor the APR (Eqs.\n(B58)-(B62)) and Ska models for di\u000berent values of x. The rightmost panel shows ^ \u0016=\u0016n\u0000\u0016p. The jumps in the APR results\nare due to phase transitions to a pion condensate.\n 0 500 1000 1500 2000\n 0.01  0.1dµn / dnn (MeV fm3)\nn (fm-3)T = 0 MeV\nAPRSka\nx = 0.10.30.5\n 0.01  0.1-100-50 0 50 100 150 200 250 300\ndµp / dnp (MeV fm3)\nn (fm-3)T = 0 MeV\nAPR Skax = 0.1\n0.3\n0.5\n-1000-500 0 500 1000\n 0.01  0.1dµn / dnp (MeV fm3)\nn (fm-3)T = 0 MeV\nAPRSka\nFIG. 7. Neutron and proton inverse susceptibilities versus baryon density for the APR (Eqs. (B63)-(B72)) and Ska models\nat the indicated proton fractions x. Recall that d\u0016n=dnp=d\u0016p=dnn. Note that the cross derivatives have a very weak\nx-dependence. The jumps in the APR results are due to phase transitions to a pion condensate.\n@\u001ci\n@ni=\u001ci\nni\"\n3F2\n1=2i\nF3=2iF\u00001=2i\n+5\n2ni\nm\u0003\ni@m\u0003\ni\n@ni \n1\u00009\n5F2\n1=2i\nF3=2iF\u00001=2i!#\n(96)\n@\u001ci\n@nj=5\n2\u001ci\nm\u0003\ni@m\u0003\ni\n@nj \n1\u00009\n5F2\n1=2i\nF3=2iF\u00001=2i!\n(97)\n@\u001ci\n@n=\u001ci\u00145\n21\nm\u0003\ni@m\u0003\ni\n@n\n+3F1=2i\nF3=2iF\u00001=2i\u0012\n(1\u0000x)@F1=2i\n@nn+x@F1=2i\n@np\u0013\u0015\n(98)\n@\u001ci\n@x=\u001ci\u00145\n21\nm\u0003\ni@m\u0003\ni\n@x\n+3F1=2i\nF3=2iF\u00001=2in\u0012@F1=2i\n@np\u0000@F1=2i\n@nn\u0013\u0015\n:(99)\nThese relations will be used in subsequent discussions of\nthe \fnite-temperature properties. For a rapid evaluation\nof the FD integrals, two numerical techniques that giveaccurate results for varying degrees of degeneracy are de-\nscribed in Appendix D.\nA. Thermal e\u000bects\nTo infer the e\u000bects of \fnite temperature we focus on\nthe thermal part of the various state variables; that is, the\ndi\u000berence between the T= 0 and the \fnite- Texpressions\nfor a given thermodynamic function X:\nXth=X(n;x;T )\u0000X(n;x;0) (100)\nThis subtraction scheme discards terms that do not de-\npend on the kinetic energy density. The thermal energy\nis given by\nEth=E(T)\u0000E(0)\n=1\nnX\ni\u0014~2\n2m\u0003\ni\u001ci\u00003\n5TFini\u0015\n(101)\nwhere\nTFi=~2k2\nFi\n2m\u0003\ni: (102)16\nThe thermal pressure takes the form\nPth=P(T)\u0000P(0)\n=2\n3X\niQi\u0014~2\n2m\u0003\ni\u001ci\u00003\n5TFini\u0015\n;(103)\nwhereQi= 1\u00003\n2n\nm\u0003\ni@m\u0003\ni\n@n: (104)\nThe quantities Qiare the consequence of the momentum-\ndependent interactions in the Hamiltonian which lead to\nthe Landau e\u000bective mass. For a free gas, Qi= 1 and\nPth= 2Eth=3 as usual. The entropy per particle can be\nwritten as\nS=1\nnTX\ni\u00145\n3~2\n2m\u0003\ni\u001ci+ni(Vi\u0000\u0016i)\u0015\n=1\nnX\nini\u00145\n3F3=2i\nF1=2i\u0000lnzi\u0015\n: (105)\nThe thermal free energy density can be expressed as\nFth=F(T)\u0000H(0) =H(T)\u0000nTS\u0000H(0)\n=X\ni\u0014~2\n2m\u0003\ni\u001ci\u00003\n5TFini\u0000Tni\u00125\n3F3=2i\nF1=2i\u0000lnzi\u0013\u0015\n(106)\nin terms of which the thermal contribution to the chem-\nical potentials are\n\u0016ith=\u0016i(T)\u0000\u0016i(0) =@Fth\n@ni\f\f\f\f\nnj:(107)\nwhere\u0016i(T) =T i+Vi (108)\nThe total free energy\nF=X\ni\u0014~2\n2m\u0003\ni\u001ci\nn\u0000TYi\u00125F3=2i\n3F1=2i\u0000 i\u0013\u0015\n+Fd(109)\ncan be expressed, with the aid of\n\u001ci=2m\u0003\niT\n~2F3=2i\nF1=2ini; (110)\nas\nF=X\ni\u0014\nTYi\u0012\n\u00002F3=2i\n3F1=2i+ i\u0013\u0015\n+Fd (111)\nThe second derivative of the above with respect to the\nproton fraction xevaluated at x= 1=2 yields the sym-\nmetry energy at \fnite temperature:\nS2(T) =1\n8d2F\ndx2\f\f\f\f\nx=1=2(112)\n=\u0000T\n3F3=2\nF2\n1=2\u0014dF1=2\ndx\n+\u00121\n2F1=2\u00003F1=2\n4F3=2F\u00001=2\u0013\u0012dF1=2\ndx\u00132\n\u00001\n4d2F1=2\ndx2#\n+S2d (113)where\nF\u000b\u0011F\u000bi(x= 0:5)\ndF1=2\ndx\u0011dF1=2n\ndx\f\f\f\f\nx=1=2=\u0000dF1=2p\ndx\f\f\f\f\nx=1=2\n=\u00002F1=2\u0012\n1 +3\n4m\u0003dm\u0003\ndx\u0013\n(114)\nd2F1=2\ndx2\u0011d2F1=2n\ndx2\f\f\f\f\nx=1=2=d2F1=2p\ndx2\f\f\f\f\nx=1=2\n=6F1=2\nm\u0003dm\u0003\ndx\u0012\n1 +1\n8m\u0003dm\u0003\ndx\u0013\n(115)\nm\u0003\u0011m\u0003\nn(x= 1=2) =m\u0003\np(x= 1=2) (116)\ndm\u0003\ndx\u0011dm\u0003\nn\ndx\f\f\f\f\nx=1=2=\u0000dm\u0003\np\ndx\f\f\f\f\nx=1=2(117)\nNote that\nd2m\u0003\ndx2=2\nm\u0003dm\u0003\ndx: (118)\nThus the thermal contributions to the symmetry energy\nare\nS2;th=S2(T)\u0000S2(0) (119)\nFor the calculation of the speci\fc heat at constant vol-\nume, we begin by writing the energy per particle as\nE=1\nnX\ni~2\n2m\u0003\ni\u001ci+n-dependent terms\nThen\nCV=@E\n@T\f\f\f\f\nn=1\nnX\ni~2\n2m\u0003\ni@\u001ci\n@T\f\f\f\f\nni\nThe condition that niare constant implies\ndni\ndT= 0 =@ni\n@T\f\f\f\f\nF1=2i+@ni\n@F1=2i\f\f\f\f\nT@F1=2i\n@T\f\f\f\f\nni\n)@ni\n@T\f\f\f\f\nF1=2i=\u0000@ni\n@F1=2i\f\f\f\f\nT@F1=2i\n@T\f\f\f\f\nni(120)\nBut\n@F1=2i\n@T\f\f\f\f\nni=@ i\n@T\f\f\f\f\nni@F1=2i\n@ i=1\n2F\u00001=2i@ i\n@T\f\f\f\f\nni(121)\nwhere Eq. (89) was used in obtaining the second equality.\nSolving for@ i\n@T\f\f\f\nnigives\n@ i\n@T\f\f\f\f\nni=\u0000@ni\n@T\f\f\f\f\nF1=2i \n@ni\n@F1=2i\f\f\f\f\nT1\n2F\u00001=2i!\u00001\nUsing Eq. (82) for the derivatives of niwith respect to\nTandF1=2iwe get\n@ i\n@T\f\f\f\f\nni=\u00003\nTF1=2i\nF\u00001=2i(122)17\nTheT-derivative of Eq. (83) is\n@\u001ci\n@T\f\f\f\f\nni=\u001ci \n5\n2T+1\nF3=2i@F3=2i\n@T\f\f\f\f\nni!\n=\u001ci \n5\n2T+1\nF3=2i@ i\n@T\f\f\f\f\nni@F3=2i\n@ i!\n=\u001ci \n5\n2T\u00009\n2TF2\n1=2i\nF3=2iF\u00001=2i!\n(123)\nwhere equations (89) and (122) have been exploited for\nthe last line. Thus\nCV=5\n2nTX\ni~2\u001ci\n2m\u0003\ni \n1\u00009\n5F2\n1=2i\nF3=2iF\u00001=2i!\n(124)\nThe starting point of the calculation of the speci\fc heat\nat constant pressure is\nCP=CV+T\nn2\u0000@P\n@T\f\f\nn\u00012\n@P\n@n\f\f\nT(125)\nThe temperature derivative of the pressure at \fxed den-\nsity is given by\n@P\n@T\f\f\f\f\nn=2\n3X\ni~2\n2m\u0003\niQi@\u001ci\n@T\f\f\f\f\nn\n=5\n3TX\ni~2\n2m\u0003\niQi\u001ci \n1\u00009\n5F2\n1=2i\nF3=2iF\u00001=2i!\n(126)\nwhere Eq.(123) was used in going from the \frst line to the\nsecond. The density derivative of the pressure at \fxed\ntemperature is\n@P\n@n\f\f\f\f\nT=~2\n3d\ndn X\niQi\u001ci\nm\u0003\ni!\n+dPd\ndn\n=~2\n3X\ni\u0014Qi\nm\u0003\nid\u001ci\ndn+\u001ci\nm\u0003\nidQi\ndn\u0000\u001ciQi\nm\u00032\nidm\u0003\ni\ndn\u0015\n+dPd\ndn(127)\nThe density derivatives of the kinetic energy density are\ngiven in Eqs. (96)-(99) and those of m\u0003,Q, andPdin\nAppendix B.\nFinally, the inverse susceptibilities are given by\n\u001fij;th=\u001fij(T)\u0000\u001fij(0) =\u0012@\u0016ith\n@nj\u0013\u00001\n(128)\nwhere\n\u001fii(T) =\u0012@\u0016i\n@ni\u0013\u00001\n=\u0012\nT@ i\n@ni+@Vi\n@ni\u0013\u00001\n=\"\nT\u0012@F1=2i\n@ i\u0013\u00001@F1=2i\n@ni+@Vi\n@ni#\u00001\n=\u00142T\nniF1=2i\nF\u00001=2i\u0012\n1\u00003\n2ni\nm\u0003\ni@m\u0003\ni\n@ni\u0013\n+@Vi\n@ni\u0015\u00001\n;\n(129)\u001fij(T) =\u0014\n\u00003TF1=2i\nF\u00001=2i1\nm\u0003\ni@m\u0003\ni\n@nj+@Vi\n@ni\u0015\u00001\n;i6=j:\n(130)\nResults\nWe now present numerical results. Comparisons of\nthese results with analytical results in degenerate and\nnon-degenerate situations will be presented in the next\nsub-section.\nWe begin by examining results of the total pressure\n(from Eq. (103)) as it varies with temperature and den-\nsity in the sub-nuclear regime for isospin symmetric mat-\nter (x= 0:5). Our results for the APR and Ska models\nare shown in Fig. 8. The prominent feature in this \fgure\nis the onset of a liquid-gas phase transition, the critical\ntemperature and density for which are obtained by the\ncondition\ndP\ndn\f\f\f\f\nnc;Tc=d2P\ndn2\f\f\f\f\nnc;Tc= 0: (131)\nThe critical temperatures (densities) for the APR and\nSka models were found to be 17.91 MeV (0.057 fm\u00003)\nand 15.12 MeV (0.056 fm\u00003), respectively, so that\nPc\nncTc=\u001a\n0:347; for APR\n0:303; for Ska:(132)\nThese results provide an interesting contrast with the\nvalue 0.375 for a Van der Waals-like equation of state\nand the experimental values that lie in the range 0.27-\n0.31 for noble gases (see, e.g. Ref. [71], p.69).\nIn Fig. 9, we show how the critical temperatures and\ndensities vary as a function of proton fraction Ypin the\nleft panel. Both quantities are scaled to their respective\nvalues for symmetric nuclear matter ( Yp= 0:5). The fall-\no\u000b of the critical temperature with Ypis similar for the\nAPR and Ska models, whereas the fall-o\u000b of the critical\ndensity with Ypfor the Ska model is steeper than for\nthe APR model. The critical proton fractions beyond\nwhich the phase transition disappears are similar for both\nmodels, that for the APR model being slightly larger than\nfor the SkA model. As is evident from the right panel in\nthis \fgure, Pc=ncTcexhibits very little variation with Yp.\nThe thermal properties are dominated by the behavior\nof the e\u000bective masses. For all densities, at a given value\nofx, the APR e\u000bective masses are larger than for Ska. As\na result, thermal contributions to entropy, energy, pres-\nsure, free energy, etc. are larger in the case of APR at\nthe same density. This explains the relative behaviors in\nFigures 10, 11, 12 and 14. The reverse behavior is seen\nin the thermal part of chemical potentials in Figure 13.\nThis behavior can be understood through the limiting\ncases (145) and (162) where the e\u000bective masses enter\nwith an overall negative sign.\nThe thermal energy (from Eq. (101)) is shown in Fig.\n10 for the two models at proton fractions xof 0.5 and 0.1,18\n-1-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4\n 0  0.04  0.08  0.12  0.16P (MeV fm-3)\nn (fm-3)T = 1 MeV612Tc = 17.91 MeV20APR\nx = 0.5\n 0  0.04  0.08  0.12  0.16  0.2-1 0 1 2 3 4\nn (fm-3)T = 1 MeV610Tc = 15.12 MeV17Ska\nx = 0.5\nFIG. 8. Pressure of isospin symmetric matter vs baryon density (from Eq. (103)) for the APR (left) and Ska (right) models at\nthe indicated temperatures. The point ( P;n) on the critical temperature curve of each model at which dP=dn =d2P=dn2= 0\nis indicated by the downward arrow.\n 0 0.2 0.4 0.6 0.8 1\n 0  0.1  0.2  0.3  0.4Tc / Tc(x=0.5)\nxAPR\nSka\n 0 0.2 0.4 0.6 0.8 1\n 0  0.2  0.4nc / nc(x=0.5)\nxAPR\nSka\n 0 0.1 0.2 0.3 0.4 0.5 0  0.1  0.2  0.3  0.4\nAPR\nPc / (Tc nc)\n 0  0.1  0.2  0.3  0.4 0 0.1 0.2 0.3 0.4\nxSka\nFIG. 9. Left panel: Critical temperatures (scaled with their respective values for proton fraction x= 0:5) vsx. The inset\nshows critical densities (scaled with their respective values for x= 0:5) vsx. Right panel: Critical parameter Pc=ncTcvsx.\nand at temperatures Tof 20 and 50 MeV, respectively,\nfor the two models. Common to both models are the\nfeatures that the thermal energy (i) decreases, and (ii) is\nnearly independent of the proton fraction with increasing\ndensity. Maximal di\u000berences (with respect to x) are seen\nto be in the vicinity of n0= 0:16 fm\u00003for both mod-\nels. Di\u000berences between the two models increase with\nincreasing density, particularly for densities in excess of\nn0. These common and di\u000berent features arise due to\na combination of e\u000bects involving the dependence of the\nthermal energy on the e\u000bective masses as the degree of\ndegeneracy changes with density as will be discussed in\nthe next sub-section with analytical results in hand.\nIn Fig. 11, the di\u000berence between the pure neu-\ntron matter and nuclear matter free energies \u0001 Fth=F(n;T;x = 0)\u0000F(n;T;x = 0:5) is shown for the two\nmodels atT= 20 and 50 MeV, respectively. For both\ntemperatures shown, the APR model has a larger \u0001 Fth\nthan that of the Ska model. This feature can be under-\nstood in terms of the larger thermal energies of the APR\nmodel relative to those of the Ska model at the same den-\nsity and temperature which dominate over the opposing\ne\u000bects of entropy.\nThe thermal pressures (from Eq. (103)) for the two\nmodels are shown in Fig. 12 for x= 0:5 and 0.1, and\nT= 20 and 50 MeV as functions of density. Both mod-\nels display the same trend of rising almost linearly with\ndensity until around 1.5 n0before beginning to saturate\nat higher densities. This trend is independent of pro-\nton fraction and temperature, however; the sti\u000bness in19\n 0 5 10 15 20 25 30\n 0.1  0.2  0.3  0.4  0.5Eth (MeV)T = 20 MeV\nAPR\nSka0.1x = 0.5\nn (fm-3) 0.1  0.2  0.3  0.4  0.5 0 10 20 30 40 50 60 70 80\nT = 50 MeV\nAPR\nSka0.1x = 0.5\nFIG. 10. Thermal energy per particle (Eq. (101)) at the\nindicated proton fractions and temperatures for the APR and\nSka models.\n 0 5 10 15 20 25 30 35\n 0  0.1  0.2  0.3  0.4  0.5∆Fth (MeV)\nn (fm-3)APR\nSka\nT = 20 MeV50 MeV\nFIG. 11. Di\u000berence between the pure neutron matter and nu-\nclear matter free energies (Eq. (106)) at the indicated tem-\nperatures for the APR and Ska models.\npressure is more pronounced for the higher temperature\nand lower proton fraction. The agreement between the\nresults of the two models becomes progressively worse as\nthe density increases. As with the thermal energy in Fig.\n10, these results are a consequence of the increasing de-\ngeneracy with increasing density and the behavior of the\ne\u000bective masses in the two models as our discussion in\nthe next sub-section will reveal.\nThe neutron and proton thermal chemical potentials\n(from Eq. (107)) plotted as functions of baryon density\nare presented in the left and right panels of Fig. 13, re-\nspectively. Chemical potentials of fermions inclusive of\ntheir zero temperature parts decrease with temperature\nat a \fxed density, hence the negative values of their ther-\nmal counterparts. We observe larger neutron and proton\nthermal chemical potentials from the Ska model when\n 0 0.5 1 1.5 2 2.5 3 3.5\n 0.1  0.2  0.3  0.4  0.5Pth (MeV fm-3)\nT = 20 MeVAPR\nSka\nx = 0.10.5\nn (fm-3) 0.1  0.2  0.3  0.4  0.5 0 2 4 6 8 10 12 14 16 18\nT = 50 MeVAPR\nSka\nx = 0.10.5FIG. 12. Thermal pressure vs baryon density (Eq. (103)) at\nthe indicated proton fractions and temperatures.\ncompared with the APR model for all but the lowest\nbaryon densities and at both temperatures. The di\u000ber-\nence between the two models is greatest at intermidi-\nate densities (between n0and 2n0) and at high temper-\natures. In the case of the neutron thermal chemical po-\ntential there is little di\u000berence between isospin symmetric\n(x= 0:5) and neutron rich matter ( x= 0:1). This is not\nthe case for the proton chemical potential which displays\na much greater di\u000berence as isospin asymmetry increases.\nIn Fig. 14, we present our results for the entropy\nper baryon for the APR and Ska models. Our results\nshow that the APR model provides a larger entropy per\nbaryon than the Ska model for all baryon densities, pro-\nton fractions and temperatures. The magnitude of the\nobserved di\u000berence is independent of proton fraction x\nand increases with baryon density nand temperature T.\nFor extremely low densities, ( n\u001cn0) the di\u000berence in\nentropy per baryon between the models is negligible as\ninteractions play a minor role in a nearly ideal gas for\nthis quantity.\nIn Figs. 15 through 17 we present results from Eqs.\n(128)-(129) of the thermal inverse susceptibilities for the\nAPR and Ska models. The neutron-neutron and proton-\nproton thermal inverse susceptibilities (Figs. 15 and 16,\nrespectively) show no signi\fcant di\u000berence between the\ntwo models at all baryon densities, proton fractions and\ntemperatures. The neutron-proton thermal inverse sus-\nceptibility (Fig. 17) shows a signi\fcant di\u000berence be-\ntween the two models at densities less than n0. The\nmagnitude of this discrepency is independent of pro-\nton fraction and only mildly dependent on tempera-\nture. This di\u000berence can be attributed to the e\u000bective\nmasses as it is explicitly shown in Eqs. (166) and (167)\n(the non-degenerate limit is appropriate for small densi-\nties). The leading terms in \u001fiigo asT=nithus APR\nand Ska are similar because the e\u000bective mass enters\nonly as a correction. On the other hand, \u001fijdi\u000ber sig-\nni\fcantly since their leading terms are proportional to20\n-50-40-30-20-10 0 10\n 0 0.1  0.2  0.3  0.4  0.5µn,th (MeV)\nT = 20 MeVAPRSka\nx = 0.50.1\nn (fm-3) 0 0.1  0.2  0.3  0.4  0.5-200-150-100-50 0 50\nT = 50 MeVAPRSka\nx = 0.50.1\n-70-60-50-40-30-20-10 0 10\n 0.1  0.2  0.3  0.4  0.5µp,th (MeV)\nT = 20 MeVAPRSka\nx = 0.10.5\nn (fm-3) 0.1  0.2  0.3  0.4  0.5-250-200-150-100-50 0 50\nT = 50 MeVAPRSka\nx = 0.10.5\nFIG. 13. Thermal neutron (left) and proton (right) chemical potentials vs baryon density (Eq. (107)).\n 0 1 2 3 4 5\n 0.1  0.2  0.3  0.4  0.5S (kB)T = 20 MeV\nAPR\nSka0.1x = 0.5\nn (fm-3) 0.1  0.2  0.3  0.4  0.5 0 1 2 3 4 5 6\nT = 50 MeV\nAPR\nSka0.1x = 0.5\nFIG. 14. Entropy per baryon in units of kBvs baryon density\n(Eq. (105)).\n(T=m\u0003\ni) (dm\u0003\ni=dnj) and thererefore their behavior is pri-\nmarily in\ruenced by the e\u000bective mass.\nIn Fig. 18 results for the speci\fc heats at constant\nvolume and at constant pressure, CVandCP(from Eqs.\n(124) and (125)) are shown as functions of baryon den-\nsity for the APR and Ska models at temperatures of 20\nand 50 MeV, respectively. Beginning with the value of\n1.5 characteristic of a dilute ideal gas, CVsteadily de-\ncreases with increasing density as degeneracy begins to\nset in. As the EOS of the Ska model is sti\u000ber than that of\nthe APR model at high densities, the fall o\u000b of CVwith\ndensity is correspondingly more rapid. For both models,\nCVexhibits little dependence on proton fraction for both\ntemperatures shown. Results of CP, shown in the right\npanel of this \fgure, exhibit characteristic maxima that\nindicate the occurrence of a liquid-gas phase transition\nat low densities. At n=ncandT=Tc,dP=dn!0\n(see Fig. 8 in which Pvsnfor the two models are shown\nat various temperatures) which causes CP(which is in-\n 0 500 1000 1500 2000 2500 3000 3500\n 0.01  0.1dµn,th/dnn (MeV fm3)T = 20 MeV\nAPR & Ska\nx = 0.10.5\nn (fm-3) 0.01  0.1 0 2000 4000 6000 8000 10000\nT = 50 MeV\nAPR & Ska\nx = 0.10.5FIG. 15. Neutron-neutron inverse susceptibility vs baryon\ndensity (Eqs. (128)-(129)) for the APR and Ska models at\nthe indicated proton fractions x. The two models are visually\nindistinguishable at both temperatures and proton fractions.\nversely proportional to dP=dn ) to diverge. For isospin\nsymmetric matter at T= 20 MeV, the maximum in CP\nis greater for the APR model than that for the Ska model.\nThis feature can be understood in terms of T= 20 MeV\nbeing closer to the Tc= 17:91 MeV of the APR model\nthan to the Tc= 15:12 MeV for the Ska model. As\nforCV, there is little dependence on proton fraction for\nCP. Note that an abrupt jump in CPalso occurs for the\nAPR model at the densities for which a transition from\nthe LDP to the HDP takes place due to the onset of pion\ncondensation (see the inset in the \frst of the right panel\n\fgure for its presence also at T= 20 MeV.)21\n 0 4000 8000 12000 16000 20000\n 0.01  0.1dµp,th/dnp (MeV fm3)T = 20 MeV\nAPR & Ska\nx = 0.1\n0.5\nn (fm-3) 0.1 0 10000 20000 30000 40000 50000\nT = 50 MeV\nAPR & Ska\nx = 0.1\n0.5\nFIG. 16. Proton-proton inverse susceptibility vs baryon den-\nsity (Eqs. (128)-(129)). Just as in the case of \u001f\u00001\nnn, the two\nmodels are indistinguishable at both temperatures and proton\nfractions.\n 0 50 100 150 200 250 300 350\n 0.01  0.1dµn,th/dnp (MeV fm3)T = 20 MeV\nAPRSkax = 0.1\n0.5\nn (fm-3) 0.01  0.1 0 100 200 300 400 500 600 700 800 900\nT = 50 MeV\nAPRSkax = 0.1\n0.5\nFIG. 17. Neutron-proton susceptibility vs baryon density\n(Eqs. (128) and (130)). Because d\u0016n=dnp=d\u0016p=dnn, only\none of the cross derivatives is shown. Unlike \u001f\u00001\nnnand\u001f\u00001\npp,\n\u001f\u00001\nnpexhibits strong model dependence at low densities.\nB. Limiting cases\nIn this section, we study the limiting cases when de-\ngenerate (low T, highnsuch thatT=EFi\u001c1) and non-\ndegenerate (high T, lownsuch thatT=EFi\u001d1) condi-\ntions prevail. In these limits, compact analytical expres-\nsions for all thermodynamic variables can be obtained.\nFrom a comparison of the exact, but, numerical, results\nwith their analytical counterparts, the density and tem-\nperature ranges in which supernova matter is degener-\nate, partially degenerate or non-degenerate can be estab-\nlished. In addition, such a comparison also provides a\nconsistency check on our numerical calculations of the\nthermal variables. Because of the varying concentrationsof neutrons and protons (and leptons, considered in a\nlater section) encountered, one or the other species may\nwell lie in di\u000berent regimes of degeneracy.\nDegenerate limit\nIn this case, we make use of Landau's Fermi Liq-\nuid Theory (FLT) [72, 73], which allows for a model-\nindependent discussion of the various thermodynamical\nfunctions. The temperature dependence of these func-\ntions is governed by the nature of the single particle\nspectrum. For the APR and Skyrme Hamiltonians, this\ndependence is characterized by a density dependent ef-\nfective mass.\nIn FLT, the entropy density sand the number density\nnmaintain the same functional forms as those of a free\nFermi gas. For a single-component gas,\ns=\u0000X\nk;\u001b[nk\u001blnnk\u001b+ (1\u0000nk\u001b) ln (1\u0000nk\u001b)] (133)\nn=X\nk;\u001bnk\u001bandnk\u001b=1\ne(\u000fk\u001b\u0000\u0016)=T+ 1; (134)\nwherekis the wave number, and \u001bstands for spin degrees\nof freedom, respectively. Note that the quasiparticle en-\nergy\u000fkis itself a function of the distribution function nk.\nThe distribution of particles close to the zero tempera-\nture Fermi energy EFdetermines the general behavior\n(degenerate versus non-degenerate) of the system.\nThe low temperature expansion of sis standard and\nto orderTyields\ns=\u00192\n3N(0)T=\u00192\nkFvFnT; (135)\nwhereN(0) is the density of states at the Fermi surface:\nN(0) =X\n~k\u000e(\u000fk\u001b\u0000\u0016) =3n\nkFvF: (136)\nThe quantity vFis the Fermi velocity:\nvF=@\u000fo\nks\n@k\f\f\f\f\nk=kF=kF\nm\u0003(137)\nThe above equation serves as a de\fnition of the quasi-\nparticle e\u000bective mass m\u0003. Including the 2 spin degrees\nof freedom, n=k3\nF=(3\u00192) so thatN(0) =m\u0003kF=\u00192. The\nentropy density in Eq. (135) is often written as\ns= 2anT =\u00192\n2n\u0014T\nTF\u0015\n: (138)\nAbove, the level density parameter aand the Fermi tem-\nperatureTFare\na=\u00192N(0)\n6n=\u00192\n2kFvF=\u00192\n4TF\nTF=1\n2kFvF=k2\nF\n2m\u0003: (139)22\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6\n 0.1  0.2  0.3  0.4  0.5Cv (kB)\nT = 20 MeVAPR\nSkax = 0.10.5\nn (fm-3) 0.1  0.2  0.3  0.4  0.5 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6\nT = 50 MeVAPR\nSkax = 0.10.5\n 0 2 4 6 8 10 12\n 0.1  0.2  0.3  0.4  0.5Cp (kB)T = 20 MeV\nAPR Skax = 0.10.5\nn (fm-3) 0.2  0.3 0.5 1 1.5Cp (kB)\nn (fm-3)\n 0.1  0.2  0.3  0.4  0.5 0.5 1 1.5 2 2.5 3 3.5\nT = 50 MeV\nAPR\nSkax = 0.10.5\nFIG. 18. Left panels: Speci\fc heat constant volume, CV(from Eq. (124)) vs baryon density. Right panels: Speci\fc heat at\nconstant pressure, CP(from Eq. (125)) vs baryon density.\nIn normal circumstances, the leading correction to s\nabove is of order ( T=TF)2unless there exist soft collective\nmodes which give rise to a ( T=TF)3ln (T=TF) behavior\n[73].\nThe generalization to a multi-component gas is\nstraightforward. The sums in Eq. (133) and Eq. (134) go\nover particle species so that the end result for the entropy\ndensity reads as\ns=\u00192\n3TX\niNi(0) = 2TX\niaini(140)\nwhereai=\u00192\n2kFivFi=\u00192\n2m\u0003\ni\nk2\nFi(141)\nThe rest of the thermal variables follow from thermo-\ndynamics, particularly the Maxwell relations. The ther-\nmal energy is obtained fromZ\ndE=Z\nTdS =2\nnX\niainiZ\nTdT\n)Eth=T2\nnX\niaini (142)\nThe thermal pressure arises from\nZ\ndp=ZT\n0\u0012\ns\u0000nds\ndn\u0013\ndT\n=X\ni\u0014\naini\u0000nd(aini)\ndn\u0015\nT2:\nUsingai=\u00192\n2m\u0003\ni\n(3\u00192ni)2=3, we get\nnd(aini)\ndn=aini\u00002ain\n3\u0012\n1\u00003\n2n\nm\u0003\nidm\u0003\ni\ndn\u0013\n(143)\nThis allows us to write the thermal pressure as\nPth=2T2\n3X\niainiQi; (144)whereQiis given by Eq. (104). The thermal chemical\npotentials are obtained from\nZ\nd\u0016i=\u0000Zds\ndnidT=\u0000d\ndni0\n@X\njajnj1\nAT2\n)\u0016ith=\u0000T22\n4ai\n3+X\njnjaj\nm\u0003\njdm\u0003\nj\ndni3\n5: (145)\nThus, the susceptibilities are\nd\u0016i;th\ndni=\u0000T2\n3\u0012\n\u00002\n3ai\nni+ 2ai\nm\u0003\nidm\u0003\ni\ndni\n+3niai\nm\u0003\nid2m\u0003\ni\ndn2\ni+ 3njaj\nm\u0003\njd2m\u0003\nj\ndn2\ni!\n(146)\nd\u0016i;th\ndnj=\u0000T2\n3 \nai\nm\u0003\nidm\u0003\ni\ndnj+aj\nm\u0003\njdm\u0003\nj\ndni\n+3niai\nm\u0003\nid2m\u0003\ni\ndnidnj+ 3njaj\nm\u0003\njd2m\u0003\nj\ndnidnj!\n(147)\nThe free energy is given by\nFth=Eth\u0000TS=\u0000Eth=\u0000T2X\niaiYi (148)\nfrom which we get the symmetry energy\nS2;th=T2a\n9\"\n1 +3\n2m\u0003dm\u0003\ndx\u00009\n4m\u00032\u0012dm\u0003\ndx\u00132#\n(149)\na=\u00192\n2m\u0003\n~21\n\u00003\u00192n\n2\u00012=3(150)\nwherem\u0003anddm\u0003=dxare given by Eqs.(116) and (117)\nrespectively.23\nFrom the relation for the thermal energy, the speci\fc\nheat at constant volume is\nCV=2T\nnX\niaini=S=2Eth\nT: (151)\nIn the degenerate limit, to lowest order in temperature\nCP=CV: (152)\nNon-degenerate limit\nIn the ND limit, the degeneracy (and hence the fugac-\nity) is small, so that the FD functions can be expanded\nin a Taylor series about z= 0:\nF\u000bi'\u0000(\u000b+ 1)\u0012\nzi\u0000z2\ni\n2\u000b+1+:::\u0013\n: (153)\nThen theF1=2series is perturbatively inverted to get the\nfugacity in terms of the number density and the temper-\nature:\nzi=ni\u00153\ni\n\r+1\n23=2\u0012ni\u00153\ni\n\r\u00132\n; (154)\nwhere\u0015i=\u00122\u0019~2\nm\u0003\niT\u00131=2\n(155)\nand\r= 2 (the spin orientations) :\nSubsequently, these are used in the other FD integrals so\nthat they, too, are expressed as explicit functions of the\nnumber density and the temperature:\nF3=2i=3\u00191=2\n4ni\u00153\ni\n\r\u0014\n1 +1\n25=2ni\u00153\ni\n\r\u0015\n(156)\nF1=2i=\u00191=2\n2ni\u00153\ni\n\r(157)\nF\u00001=2i=\u00191=2ni\u00153\ni\n\r\u0014\n1\u00001\n23=2ni\u00153\ni\n\r\u0015\n(158)Finally, we insert these into equations (101)-(107) from\nwhich we get:\nEth=1\nnX\ni(\n3\n2Tni\"\n1 +ni\n4\u0012\u0019~2\nm\u0003\niT\u00133=2#\n\u00003\n5TFini)\n(159)\nPth=X\ni(\nTQini\"\n1 +ni\n4\u0012\u0019~2\nm\u0003\niT\u00133=2#\n\u00002\n5TFini)\n(160)\nS=1\nnX\nini(\n5\n2\u0000ln\"\u00122\u0019~2\nm\u0003\niT\u00133=2ni\n2#\n+ni\n8\u0012\u0019~2\nm\u0003\niT\u00133=2)\n(161)\n\u0016ith=\u0000T(\n\u0000ln\"\u00122\u0019~2\nm\u0003\niT\u00133=2ni\n2#\n\u0000ni\n2\u0012\u0019~2\nm\u0003\niT\u00133=2\n+3\n2ni\nm\u0003\nidm\u0003\ni\ndni\"\n1 +ni\n4\u0012\u0019~2\nm\u0003\niT\u00133=2#\n+3\n2nj\nm\u0003\njdm\u0003\nj\ndni2\n41 +nj\n4 \n\u0019~2\nm\u0003\njT!3=23\n59\n=\n;\n\u0000TFi\u0014\n1\u00003\n5ni\nm\u0003\nidm\u0003\ni\ndni\u0015\n+3\n5nj\nm\u0003\njdm\u0003\nj\ndniTFj:(162)\nThus\nFth=X\ni(\nTYi\"\n\u00001 + ln\"\u00122\u0019~2\nm\u0003\niT\u00133=2ni\n2#\n+ni\n4\u0012\u0019~2\nm\u0003\niT\u00133=2#\n\u00003\n5TFiYi)\n(163)\nS2;th=T\n8(\n8\u0012\n1 +3\n4m\u0003dm\u0003\ndx\u0013\"\n1 +n\n8\u0012\u0019~2\nm\u0003T\u00133=2#\n\u00004\"\n1 +3\n8m\u00032\u0012dm\u0003\ndx\u00132#\n+3n\n4m\u0003\u0012\u0019~2\nm\u0003T\u00133=2dm\u0003\ndx\u0012\n1 +1\n8m\u0003dm\u0003\ndx\u0013)\n\u0000TF\n3\u0012\n1 +3\n2m\u0003dm\u0003\ndx\u0013\n(164)\nTF=\u00123\u00192n\n2\u00132=3~2\n2m\u0003(165)24\nd\u0016i\ndni=T\nni\u0012\n1\u00003ni\nm\u0003\nidm\u0003\ni\ndni\u0013\"\n1 +ni\n2\u0012\u0019~2\nm\u0003\niT\u00133=2\n+2\n3TFi\nT#\n+O \u0012dm\u0003\ndn\u00132\n;d2m\u0003\ndn2!\n(166)\nd\u0016i\ndnj=\u0000T(\n3\n2m\u0003ndm\u0003\ni\ndnj\"\n1 +ni\n2\u0012\u0019~2\nm\u0003\niT\u00133=2\n+2\n3TFi\nT#\n+3\n2m\u0003\njdm\u0003\nj\ndni2\n41 +nj\n2 \n\u0019~2\nm\u0003\njT!3=2\n+2\n3TFj\nT3\n59\n=\n;\n+O \u0012dm\u0003\ndn\u00132\n;d2m\u0003\ndn2!\n(167)\nCV=1\nnX\ni(\n3\n2ni\"\n1\u0000ni\n8\u0012\u0019~2\nm\u0003\niT\u00133=2#)\n: (168)\nThe second derivatives and the squares of the \frst deriva-\ntives of the e\u000bective mass are neglected because they rep-\nresent higher order corrections.\nForCP, we need the temperature and density deriva-\ntives of pressure in the non-degenerate limit, for which\nwe use Eq.(125) in conjuction with\nP=X\ni\"\nTQini(\n1 +ni\n4\u0012\u0019~2\nm\u0003\niT\u00133=2)#\n+Pd(169)\nto get\n@P\n@T\f\f\f\f\nn=X\ni\"\nQini(\n1\u0000ni\n8\u0012\u0019~2\nm\u0003\niT\u00133=2)#\n(170)\n@P\n@n\f\f\f\f\nT=X\ni\"\nT(\n1 +ni\n4\u0012\u0019~2\nm\u0003\niT\u00133=2)\u0012@Qi\n@nni+QiYi\u0013#\n+X\ni\"\nTQ2\niniYi\n4\u0012\u0019~2\nm\u0003\niT\u00133=2#\n+dPd\ndn: (171)\nResults\nThis section is devoted to comparisons of results from\nthe analytical formulas obtained in the previous section\nfor the limiting cases with those from the exact calcula-\ntions presented earlier. In addition to providing us with\nphysical insights about the general trends observed, these\ncomparisons will allow us to delineate the range of den-\nsities for which matter with varying isospin asymmetry\nand temperature can be regarded as either degenerate\nor non-degenerate. We will restrict our comparisons to\nresults from the APR model only as those for the Ska\nmodel yield similar conclusions.\nIn Fig. 19, we show the thermal energies Ethas a\nfunction of baryon density nforT= 20 MeV (left panel)\nand 50 MeV (right panel) for proton fractions of x= 0:5\nand 0.1, respectively. The T2dependence implied by the\n 0 5 10 15 20 25 30\n 0.1  0.2  0.3  0.4  0.5Eth (MeV)\nT = 20 MeV APRx = 0.10.5\nDegenerate\nNon-\nDegenerate\nExact\nn (fm-3) 0.2  0.4  0.6  0.8 0 10 20 30 40 50 60 70 80\nT = 50 MeV APRx = 0.10.5\nDegenerate\nNon-Degenerate\nExactFIG. 19. Thermal energy per particle (Eq. (101)) and lim-\niting cases (Eqs. (142) and (159)) vs baryon density at the\nindicated temperatures and proton fractions.\ndegenerate approximation in Eq. (142) is borne out by\nthe the exact results at high densities. Also, the larger\nthe temperature, the larger is the density at which the\ndegenerate approximation approaches the exact result.\nThe e\u000bective masses introduce an additional density de-\npendence to the\u0018n\u00002=3behavior characteristic of a free\ngas of degenerate fermions for which Ethwould be larger\nthan that with momentum dependent interactions. Note\nthat in the degenerate limit, both the approximate and\nexact results are nearly x- independent. With increas-\ning temperature, the non-degenerate approximation in\nEq. (159)) reproduces the exact results the agreement\nextending up to nuclear density and even slightly beyond.\nAs for a free Boltzmann gas, the thermal energy is pre-\ndominantly linear in Tin the non-degenerate limit and is\nonly slightly modi\fed by the density dependence of the\ne\u000bective masses. The \u0018n\u00002=3fall o\u000b with density arises\nfrom the last term in Eq. (101) (the degeneracy energy of\nfermions at T= 0) with sub-dominant corrections from\nthe density dependence of the e\u000bective masses. E\u000bects\nof isospin asymmetry are somewhat more pronounced in\nthe non-degenerate case when compared to the degen-\nerate limit. Results for highly asymmetric matter from\nEq. (159)) begin to deviate from the exact results at\nlower densities than for symmetric matter because the\ntwo components are in di\u000berent regimes of degeneracy.\nIn Fig. 20, thermal contributions to the symmetry\nenergy,S2;thfrom Eq. (113) and its limiting cases from\nEqs. (149) and (164) for the APR model are shown as\nfunctions of baryon density at temperatures of 20 and 50\nMeV, respectively. Agreement between the degenerate\nlimit and the exact result is obtained around 3 n0forT=\n20 MeV and at much larger densities ( n > 1 fm\u00003) for\ntheT= 50 MeV. The non-degenerate limit coincides\nwith the exact result for densities less than \u00190:5n0for\nthe 20 MeV temperature. At T= 50 MeV the non-\ndegenerate limit has a greater range of baryon densities25\n-2 0 2 4 6 8 10\n 0 0.1  0.2  0.3  0.4  0.5APRT = 20 MeV\nn (fm-3)DegerateDegenerateNon-\nExactS2 th (MeV)\n 0  0.2  0.4  0.6  0.8-2 0 2 4 6 8 10\nAPRT = 50 MeV\nn (fm-3)DegerateDegenerateNon-\nExactS2 th (MeV)\nFIG. 20. Thermal contributions to the symmetry energy,\nS2;th, from Eq. (113) compared with its limiting cases (Eqs.\n(149) and (164)) at the indicated temperatures.\nfor which it agrees with the exact result, reaching up\nto 1-1.5n0. A noteworthy feature in this \fgure is that\nboth the exact and the degenerate result for S2;thbecome\nnegative after a certain baryon density. Note that for free\nfermions,S2;thin Eq. (149) is strictly positive, pointing\nto the fact that derivatives of m\u0003with respect to proton\nfractionxare at the root of driving S2;thnegative. In\nwhat follows, we examine the rate at which the identity\n\u0001Fth=P\niSi;thwithieven (odd terms cancelling) is\nful\flled.\nThe left panel of Fig. 21 shows the di\u000berence of the\nexact free energies \u0001 Fth=Fth(n;x= 0;T)\u0000Fth(n;x=\n0:5;T) atT= 20 MeV. Also shown are contributions\nfrom various Si;that the same temperature. To be spe-\nci\fc, we consider only the degenerate limit results for\nSi;thin this comparison. It turns out that only S2turns\nnegative at a \fnite baryon density, whereas S4; S6;\u0001\u0001\u0001\nwhich contain higher derivatives of m\u0003with respect to\nthe proton fraction xare all positive whose magnitudes\ndecrease very slowly. We have calculated up to thirty\nterms inSi;thand show how their sums compare with\n\u0001Fth. It is clear that the convergence to the exact result\nis relatively poor, in contrast to the rapid convergence\nof symmetry energies at zero temperature (see Fig. 3).\nThe situation is better, although by no means impres-\nsive, for \u0001Fth=Fth(n;x= 0:02;T)\u0000Fth(n;x= 0:5;T)\natT= 20 MeV. These results indicate that the happen-\nstance of rapid convergence of symmetry energies at zero\ntemperature cannot automatically be taken to hold for\ntheir thermal parts as well. It should be noted, however,\nthat the latter represent relatively small corrections to\nthe total symmetry energy where the main contribution\nis due to the zero temperature component. The asymp-\ntotic nature of the Taylor series expansion of \u0001 Fthin even\npowers of (1\u00002x) at \fnite temperature is at the origin of\nsuch poor convergence for large isospin asymmetries. Ex-\nact, albeit numerical, calculations of the Fermi integralsare necessary for high isospin asymmetry.\n 0 0.5 1 1.5 2 2.5 3 3.5 4\n 0.2  0.3  0.4  0.5Thermal Symmetry Energy (MeV)\nn (fm-3)APRT = 20 MeV\n∆Fth =\nS2,th\nS2,th+...+S12,thS2,th+...+S30,thFth(n,Yp=0,T)\n-Fth(n,Yp=0.5,T)\n 0.2  0.3  0.4  0.5  0.6 0 0.5 1 1.5\nn (fm-3)APRT = 20 MeV\n∆Fth =\nS2,th\nS2,th+...+S12,thS2,th+...+S30,th\nFth(n,Yp=0.02,T)\n-Fth(n,Yp=0.5,T)\nFIG. 21. Thermal symmetry free energies Si;thand their con-\ntributions to \u0001 Fth=P\niSi;thas de\fned in the insets.\n-50-40-30-20-10 0\n 0 0.1  0.2  0.3  0.4  0.5T = 20 MeVAPRExact\nDegenerateNon-Degenerate\nx = 0.1\n0.5Fth (MeV)\nn (fm-3) 0  0.2  0.4  0.6  0.8-140-120-100-80-60-40-20 0\nT = 50 MeVAPRExact\nDegenerateNon-Degenerate\nx = 0.1\n0.5Fth (MeV)\nn (fm-3)\nFIG. 22. Thermal free energy (Eq. (106)) and its limiting\ncases (Eqs. (148) and (163) vs baryon density at the indicated\nproton fractions and temperatures.\nIn Fig. 22, we show results for the thermal free energy\nfrom Eq. (106) and its limiting cases from Eqs. (148)\nand (163) as functions of baryon density. The degenerate\nlimit and the exact result of Fthare in agreement for\ndensities greater than 1 :5n0forT= 20 MeV and only\nfor much larger densities ( n\u00155n0) forT= 50 MeV.\nThe convergence between the degenerate limit and the\nexact result of Fthis independent of proton fraction for\nboth temperatures. The non-degenerate limit begins to\ndi\u000ber from the exact result at around n0forT= 20 MeV\nand about 2 n0forT= 50 MeV. For both temperatures\nshown, the convergence between the non-degenerate limit\nand the exact solution is nearly independent of proton\nfraction.\nResults for the exact thermal pressures Pth(from Eq.\n(103)) and those of its limiting cases (Eqs. (144) and\n(160) are presented in Fig. 23 for the APR model. For26\n 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5\n 0.1 0.2 0.3 0.4 0.5 0.6Pth (MeV fm-3)\nT = 20 MeVAPRx = 0.10.5DegenerateNon-\nDegenerate\nExact\nn (fm-3) 0.2  0.4  0.6  0.8 0 5 10 15 20 25\nT = 50 MeVAPRx = 0.10.5DegenerateNon-\nDegenerate\nExact\nFIG. 23. Thermal pressure (Eq. (103)) and limiting cases\n(Eqs. (144) and (160)) vs baryon density.\nboth temperatures considered, the initial rise of Pth(in\nthe non-degenerate regime) is linear with slope \u0018Tmod-\nulated by the factors Qihighlighting the role of density\ndependent e\u000bective masses relative to a free fermi gas for\nwhich the slope would be T. The linear rise is halted as\nmatter begins to become increasingly degenerate when ef-\nfective mass corrections begin to gain importance. Quan-\ntitative agreement of the exact results with those from\nthe limiting form of the degenerate expression is, how-\never, reached at densities much larger than shown in this\n\fgure. Note that isospin asymmetry e\u000bects are more pro-\nnounced for Pththan forEthexcept at very low and very\nhigh densities.\nThermal contributions to the neutron and proton\nchemical potentials \u0016n;thand\u0016p;thversus baryon density\nnare shown in Fig. 24 in which comparisons between\nbetween results from the exact (Eq. (107)) and limit-\ning cases (Eqs. (145) and (162)) are made. For \u0016n;th\n(left panels), good agreement is found between the non-\ndegenerate limit and the exact result for densities up to\nn0forT= 20 MeV and up to 2 n0forT= 50 MeV. Re-\nsults in the degenerate limit rapidly approach the exact\nresults, unlike in the cases of EthandPth. Note that this\nlevel of quantitative agreement, in both non-degenerate\nand degenerate cases, required derivatives of the den-\nsity dependent e\u000bective masses (Eqs. (145) and (162)).\nIsospin asymmetry e\u000bects are not very pronounced for\n\u0016n;th.\nThe thermal contribution to the proton chemical\npotential\u0016p;th(right panels) exhibits a greater di\u000ber-\nence between isospin symmetric and asymmetric matter\nwhen compared to \u0016n;th. The agreement between the\nexact results for \u0016p;thand those of the limiting cases is\nmuch the same as it was for \u0016n;th. Both the degenerate\nand non-degenerate limits agree to a greater degree for\nthe higher temperature and for isospin symmetric matter.\nIn Fig. 25, we present the exact results for the entropyper baryon (Eq. (105)) and its limiting cases (Eqs. (140)\nand (161)) as functions of baryon density n. The exact\nresults show little di\u000berence between isospin symmetric\nand asymmetric matter. A comparison of the results in\nthe two panels reveals the range of densities over which\nthe non-degenerate and degenerate approximations re-\nproduce the exact results. The agreement between the\nexact results and those of the limiting cases is almost in-\ndependent of proton fraction, although what little di\u000ber-\nence there is points to isospin symmetric matter having\na slightly better agreement.\nIn Figs. 26, 27, and 28, thermal contributions to the\ninverse susceptibilities \u001f\u00001\nnn,\u001f\u00001\npp, and\u001f\u00001\nnp(Eqs. (128),\n(129) and (130)) are shown together with their limit-\ning cases in Eqs. (146), (147) (166) and (167) (the\nnon-degenerate limits are in the insets of all three \fg-\nures). Note that where expected, the degenerate and\nnon-degenerate approximations provide an accurate de-\nscription of the exact results. It is intriguing that for\ndensities slightly above the nuclear density, neither of\nthe approximations works very well.\nIn the left panels of Fig. 29, we present our results\nof the speci\fc heat at constant volume from Eq. (124)\nand its limiting cases from Eqs. (151) and (168) for the\nAPR model. Results shown are for for isospin symmetric\n(x= 0:5) and neutron rich matter ( x= 0:1) at temper-\natures of 20 and 50 MeV, respectively. The degenerate\nlimit (151) converges with the exact result for densities\nlarger than 0 :4 fm\u00003atT= 20 MeV and for densities\nlarger than (1 fm\u00003) forT= 50 MeV with little to no\ndependence on proton fraction. As expected, the non-\ndegenerate limit holds at low densities, the agreement\nwith the exact result extending to slightly above n0at\nthe higher temperature. The extent of disagreement is\nsomewhat dependent on the proton fraction with neu-\ntron rich matter di\u000bering from the exact result at slightly\nlower baryon densities than for symmetric matter.\nThe right panels of Fig. 29 show the speci\fc heat at\nconstant pressure from Eq. (125) and its limiting cases\nfrom Eqs. (152) and (125) using Eqs. (168), (170), and\n(171) as functions of baryon density. The degenerate\nlimit ofCP(Eq. 152) provides good agreement with\nthe exact solution at densities greater than about n0at\nT= 20 MeV. At T= 50 MeV, the degenerate limit of\nCPprovides a good approximation to the exact result at\ndensities greater than 2 n0, but does not converge until\nlarge densities ( n > 1 fm\u00003). The non-degenerate limit\nofCPin Eq. (125) using Eqs. (168), (170), and (171) is\nin agreement with the exact solution up to n0forT= 20\nMeV and up to almost 2 n0forT= 50 MeV. However, the\nliquid-gas phase transition pushes the exact solution to\nlargerCPwhen compared to the e\u000bect of this transition\non the degenerate limit. Even including the e\u000bects of the\nliquid-gas phase transition, the agreement between the\nnon-degenerate limit and the exact solution is very good.\nThe rate of converence between the two limits and the\nexact solution is independent of proton fraction.27\n-50-40-30-20-10 0 10\n 0.1  0.2  0.3  0.4  0.5µn,th (MeV)\nT = 20 MeVAPRx = 0.1\n0.5Deg.Non-Deg.\nExact\nn (fm-3) 0.2  0.4  0.6  0.8-200-150-100-50 0 50\nT = 50 MeVAPRx = 0.1\n0.5Deg.Non-Deg.\nExact\n-70-60-50-40-30-20-10 0 10\n 0.1  0.2  0.3  0.4  0.5µp,th (MeV)\nT = 20 MeVAPRx = 0.10.5Deg.Non-Deg.\nExact\nn (fm-3) 0.2  0.4  0.6  0.8-250-200-150-100-50 0 50\nT = 50 MeVAPRx = 0.10.5Deg.Non-Deg.\nExact\nFIG. 24. Proton (left) and neutron (right) thermal chemical potentials (Eq. (107)) with limits (Eqs. (145) and (162)) vs baryon\ndensity.\n 0 1 2 3 4 5 6\n 0.1  0.2  0.3  0.4  0.5S (kB)T = 20 MeV\nAPRx = 0.10.5Degenerate\nNon-\nDegenerate\nExact\nn (fm-3) 0.2  0.4  0.6  0.8 0 1 2 3 4 5 6\nT = 50 MeV\nAPRx = 0.10.5Degenerate\nNon-\nDegenerate\nExact\nFIG. 25. Entropy per baryon (Eq. (105)) and its limiting\ncases (Eqs. (140) and (161)) vs baryon density at the indi-\ncated proton fractions and temperatures.\nResults for leptons\nHere we present results of our calculations for the con-\ntribution from leptons to the energy Eeper baryon and\nthe electron chemical potential \u0016eas functions of baryon\ndensityn. Other state variables follow in a straightfor-\nward manner and are summarized in Appendix C. We\npresent the exact results obtained using the scheme in\nRef. [74] (Eqs. (D9) and (D11) labelled JEL in \fgures)\nand those of the relativistic approach with mass correc-\ntions (Eqs. (C10) and (C9) labelled Rel in \fgures). Com-\nparisons are made both at T= 0 and 50 MeV, and in\nisospin symmetric and neutron rich matter.\nIn Fig. 30, we display the energy per baryon Eeof\nelectrons and positrons as a function of baryon density\nn. The two approaches (JEL and Rel) are in complete\nagreement at all nfor both temperatures and for isospin\n 0 50 100 150 200\n 0.1 0.2 0.3 0.4 0.5dµn,th/dnn (MeV fm3)\nT = 20 MeVAPRx = 0.1\n0.5\nn (fm-3) 0 200 400 600 800\n 0.01  0.1\nn (fm-3) 0 500 1000 1500\n 0.01  0.1\nn (fm-3) n (fm-3)\n 0.2  0.4  0.6  0.8 0 100 200 300 400 500 600\nT = 50 MeVAPRx = 0.1\n0.5FIG. 26. Neutron-neutron inverse susceptibility vs baryon\ndensity (Eqs. (128) and (129)) for the APR model and its\nlimiting cases at the indicated proton fractions x. The de-\ngenerate limit (Eq. (146)) and the non-degenerate limit (Eq.\n(166)), see inset, are both shown.\nsymmetric and asymmetric matter. Isospin symmetric\nmatter provides a larger contribution to the energy of\nleptons than neutron rich matter. This is expected as\nthe system is charge neutral, thus the quantity of leptons\nis dependent on the number of protons. For both tem-\nperatures considered, the contribution from positrons is\nnegligible.\nThe electron chemical potential \u0016eis shown as a func-\ntion of baryon density nin Fig. 31. As was the case\nwith the contribution to energy from leptons, the two\napproaches (JEL and Rel.) are in complete agreement\nfor all baryon densities at both temperatures, and for\nboth isospin symmetric and asymmetric matter.28\nVI. EQUATION OF STATE WITH A PION\nCONDENSATE\nWe have seen in earlier sections that the APR Hamil-\ntonian density incorporates a phase transition involving\na neutral pion condensate and that at the transition den-\nsity several of the state variables exhibited a jump. In\nthis section, we discuss how an equation of state that sat-\nis\fes the physical requirements of stability is constructed\nin the presence of this \frst-order phase transition.\nMechanical stability requires that the inequality\ndP\ndn\u00150 (172)\nis always satis\fed. However, in the case of APR model,\nthe transition from the LDP to the HDP is accompanied\nby a decrease in pressure pointing to a negative incom-\npressibility. We deal with this unphysical incompress-\nibility by means of a Maxwell construction which takes\nadvantage of the thermodynamic equilibrium conditions\nPL(nL) =PH(nH) (173)\n\u0016L(nL) =\u0016H(nH) (174)\nto establish the mixed-phase region such that\ndP\ndn= 0: (175)\nThe entropy density is discontinuous across the region\n(even though it contains none of the terms in the Hamil-\ntonian that drive the phase change) thus generating a\nlatent heat\nl=T[sH(nH)\u0000sL(nL)] (176)\n 0 500 1000 1500 2000\n 0 0.1 0.2 0.3 0.4 0.5dµp,th/dnp (MeV fm3)\nT = 20 MeVAPR\n0.1\nx = 0.5\nn (fm-3) 0 500 1000 1500 2000\n 0.01  0.1\nn (fm-3) 0 500 1000 1500 2000\n 0.01  0.1\nn (fm-3) n (fm-3)\n 0.2  0.4  0.6  0.8 0 500 1000 1500 2000\nT = 50 MeVAPR\n0.1\nx = 0.5\nFIG. 27. Proton-proton inverse susceptibility vs baryon den-\nsity (Eqs. (128) and (129)) and its limiting cases. Both the\nexact result and its degerate limit (Eq. (146)) are shown. The\ninset compares the non-degenerate limit (Eq. (166)) with the\nexact result.\n 0 50 100 150 200\n 0.1 0.2 0.3 0.4 0.5dµp,th/dnn = dµn,th/dnp (MeV fm3)\nT = 20 MeVAPR\nx = 0.1\n0.5\nn (fm-3) 0 50 100 150 200\n 0.01  0.1\nn (fm-3) 0 100 200 300 400 500 600\n 0.01  0.1\nn (fm-3) n (fm-3)\n 0.2  0.4  0.6  0.8 0 100 200 300 400 500 600\nT = 50 MeVAPR\nx = 0.1\n0.5FIG. 28. Mixed inverse susceptibilities (Eqs. (128) and (130)\nand the limiting cases (Eqs. (147) and (167)) vs baryon den-\nsities. Asd\u0016n=dnp=d\u0016p=dnn, only one of the mixed deriva-\ntives is shown.\nwhich signi\fes a \frst-order transition.\nThe numerical implementation of the coexistence con-\nditions in Eqs. (173)-(174) is accomplished as in Ref.\n[75] where the average chemical potential (as electrons\ncontribute similarly in both phases)\n\u0016=Yn\u0016n+Yp(\u0016p+\u0016e) (177)\nand the function\nQ=nt\u0016\u0000P (178)\nare expanded in a Taylor series about nt(the density at\nwhich transition from the LDP to HDP occurs) to \frst\nand second order respectively, yielding\n\u0016(n) =\u0016(nt) + (n\u0000nt)d\u0016\ndn\f\f\f\f\nnt(179)\nQ(n) =Q(nt) +(n\u0000nt)2\n2d\u0016\ndn\f\f\f\f\nnt: (180)\nThen the LDP and the HDP counterparts are set equal,\nas stipulated by equilibrium, forming a system of two\nequations the solution of which gives the densities that\nde\fne the boundary of the coexistence region\nnL=nt+\u0016H(nt)\u0000\u0016L(nt)\n\u00160\nL(nt)1=2\u0002\n\u00160\nL(nt)1=2+\u00160\nH(nt)1=2\u0003(181)\nnH=nt+\u0016L(nt)\u0000\u0016H(nt)\n\u00160\nH(nt)1=2\u0002\n\u00160\nL(nt)1=2+\u00160\nH(nt)1=2\u0003(182)\nThe primes (0) denote derivatives with respect to the\nnumber density n.\nThese results serve as initial guesses which are further\nimproved by adopting an iterative procedure. We de\fne29\n 0 0.5 1 1.5 2\n 0.1  0.2  0.3  0.4  0.5Cv (kB)T = 20 MeVAPR\nx = 0.1\n0.5Degenerate\nNon-\nDegenerateExact\nn (fm-3) 0.2  0.4  0.6  0.8 0 0.5 1 1.5 2\nT = 50 MeVAPR\nx = 0.1\n0.5Degenerate\nNon-\nDegenerateExact\n 0 2 4 6 8 10 12\n 0.1  0.2  0.3  0.4Cp (kB)T = 20 MeVAPR\nx = 0.10.5\nDeg.Non-Deg.Exact\nn (fm-3) 0.8 1 1.2 1.4\n 0.3\n 0.2  0.4  0.6  0.8 0 0.5 1 1.5 2 2.5 3 3.5\nT = 50 MeVAPR\nx = 0.5\n0.1Deg.\nNon-\nDeg.Exact\nFIG. 29. Left: Speci\fc heat at constant volume from Eq. (124) and its limiting cases from Eqs. (151) and (168). Right:\nSpeci\fc heat at constant pressure from Eq. (125) and its limiting cases (Eqs. (152) and (125) using Eqs. (168), (170), and\n(171)).\n 0 20 40 60 80 100 120 140 160 180\n 0  0.1  0.2  0.3  0.4  0.5Ee (MeV)\nn (fm-3)T = 50 MeV\nT = 50 MeVx = 0.5\nx = 0.10 MeV\n0 MeVRel.\nJEL\nFIG. 30. Contribution to energy per particle from leptons\nvs baryon density at the indicated temperatures and proton\nfractions. The solid lines are obtained using the approximate\nanalytical expression Eq. (C10) and the crosses correspond\nto a full numerical calculation using Eq. (D9).\nthe functions\nf(nL;nH) =PL(nL)\u0000PH(nH) (183)\ng(nL;nH) =\u0016L(nL)\u0000\u0016H(nH) (184)\nand expand to \frst order in Taylor series about the mth\n 0 50 100 150 200 250 300 350 400 450\n 0  0.1  0.2  0.3  0.4  0.5µe (MeV)\nn (fm-3)T = 0 MeVx = 0.5\nx = 0.150 MeVRel.\nJELFIG. 31. Electron chemical potential vs baryon density at the\nindicated temperatures and proton fractions. The solid lines\nare obtained using the approximate analytical expression Eq.\n(C9) and the crosses correspond to a full numerical calculation\nusing Eq. (D11). The positron chemical potential has the\nsame magnitude but opposite sign.\niterative solution\nf(nm+1\nL;nm+1\nH) =f(nm\nL;nm\nH) + (nm+1\nL\u0000nm\nL)@f\n@nL\f\f\f\f\nnm\nL\n+ (nm+1\nH\u0000nm\nH)@f\n@nH\f\f\f\f\nnm\nH(185)\ng(nm+1\nL;nm+1\nH) =g(nm\nL;nm\nH) + (nm+1\nL\u0000nm\nL)@g\n@nL\f\f\f\f\nnm\nL\n+ (nm+1\nH\u0000nm\nH)@g\n@nH\f\f\f\f\nnm\nH: (186)\nEquations (185) and (186) are independent of each other30\nand can thus be used to determine nLandnH. If we\nassume that nm+1\nL andnm+1\nH are the \\true\" solutions of\nthe system (i.e. f(nm+1\nL;nm+1\nH) =g(nm+1\nL;nm+1\nH) = 0),\nthen\nnm+1\nL=nm\nL+f(nm\nL;nm\nH)@g\n@nH\f\f\f\nnm\nH\u0000g(nm\nL;nm\nH)@f\n@nH\f\f\f\nnm\nH\n@f\n@nH\f\f\f\nnm\nH@g\n@nL\f\f\f\nnm\nL\u0000@f\n@nL\f\f\f\nnm\nL@g\n@nH\f\f\f\nnm\nH\n(187)\nnm+1\nH=nm\nH+f(nm\nL;nm\nH)@g\n@nL\f\f\f\nnm\nL\u0000g(nm\nL;nm\nH)@f\n@nL\f\f\f\nnm\nL\n@f\n@nL\f\f\f\nnm\nL@g\n@nH\f\f\f\nnm\nH\u0000@f\n@nH\f\f\f\nnm\nH@g\n@nL\f\f\f\nnm\nL\n(188)\nThis process is repeated until the di\u000berence nm+1\u0000nm\nis less than some prescribed value.\nResults\n 0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29 0.31 0.33 0.35\n 0  0.1  0.2  0.3  0.4  0.5n (fm-3)\nYpntnH\nnL\nT = 0 MeV50 MeV\nAPR\nFIG. 32. The curve labeled ntshows the trajectory in the\nn\u0000Ypplane along which the transition from the LDP to\nthe HDP occurs according to Eq. (8). Results for ntare a\nreproduction of those in Fig. 7 of Ref. [11]. The crosses show\nresults from our polynomial \ft in Eq. (9). Curves labeled\nnLandnHindicate the mixed-phase boundary at zero and\n50 MeV temperatures, respectively, determined by a Maxwell\nconstruction as described in the text.\nThe transition densities between the LDP and HDP\nphases from Eq. (9) are shown by the solid curve (and\ncrosses) in Fig. 32 as a function of proton fraction at zero\nand 50 MeV, respectively. In addition, results from the\ndetermination of the mixed phase region (curves labeled\nnLandnH) using a Maxwell construction are presented\nas a function of proton fraction. The range of baryon\ndensities in the mixed phase region has only slight de-\npendence on the proton fraction and temperature. As\nthe neutral pion condensate is mainly driven by densitye\u000bects in the APR model, e\u000bects of temperature in the\nrange considered are small .\nIn Fig. 33, we show the total pressure (left panels) and\nthe average chemical potential (right panels) as functions\nof baryon density using a Maxwell construction. Results\nof our calculations are shown for Yp= 0:1;0:3;and 0:5,\nand atT= 20 and 50 MeV, respectively. The mixed\nphase region exists in the horizontal portions of the pres-\nsure and chemical potential curves. For both P and \u0016,\nthe abrupt transitions into and out of the mixed phase\nregions after Maxwell construction are evident.\nA comparison between the free energies of APR and\nSka is presented in Fig. 34. The two models are in close\nagreement up to n\u00180:2 fm\u00003but for higher densities,\nAPR is softer due to pion condensation.\nIn Fig. 35, the total entropy as a fuction of the aver-\nage chemical potential is shown for representative proton\nfractions at temperatures of 20 and 50 MeV, respectively.\nThe vertical portions in these curves show the entropy\njumps across the mixed phase region after Maxwell con-\nstruction.\nIn Fig. 36, we present the individual contributions\nof nucleons and leptons to the total speci\fc heat den-\nsities at constant volume and pressure. The contribu-\ntion from leptons was obtained using the JEL scheme\n(see Appendix D) while the nucleonic contribution was\ncalculated by adapting the general results of section V\n(Eqs.(124)-(125)) to APR and Ska . The two models are\nin agreement for densities up to n0, whereas for larger\ndensities, the speci\fc heat densities of APR are higher\n(bothcVandcP). Except for the highest densities shown\nin these \fgures, the dominant contributions arise from\nnucleons.\nThe individual contributions of nucleons and leptons\nto the total entropy density for the APR and Ska models\nare displayed in Fig. 37. Note that in the degenerate\nlimits'cV'cP. As with the speci\fc heat densities,\nthe largest contributions are from nucleons for densities\nof relevance in core-collapse supernovae.\nThermal variables for constant entropy, that is isen-\ntropes, often provide valuable guidance to the hydrody-\nnamic evolution of a system, as in ideal hydrodynamics\n(meaning without viscous terms) the entropy density cur-\nrent is conserved. Ever since the observation by Bethe et\nal., [76], who pointed out that the entropy in supernova\nevolution is low, a great deal of qualitative understand-\ning has been gained by studying isentropes for the various\nthermodynamical variables. In view of this, we present\nsome isentropes in what follows.\nIsentropes of the APR model in the T-nplane are\nshown in Fig. 38. The crosses in this \fgure show re-\nsults from the degenerate limit expression\nT=S\n2[anYn+ (ap+ae)Yp](189)\nwith excellent agreement for S\u00141. The level den-\nsity parameters anandapabove are as in Eq. (141),\nwhereas that for the electrons is ae= (\u00192=2)(EFe=k2\nFe)31\n 0 10 20 30 40 50 60 70 80 90\n 0.1  0.2  0.3  0.4P (MeV fm-3)T = 20 MeV\nn (fm-3)0.1x = 0.4 APR\nSka\n 0.1  0.2  0.3  0.4 0 10 20 30 40 50 60 70 80 90\nT = 50 MeV\n0.1x = 0.4 APR\nSka\n-50 0 50 100 150 200 250 300\n 0.1  0.2  0.3  0.4µ (MeV)\nT = 20 MeV\nn (fm-3)0.1x = 0.4 APR\nSka\n 0.1  0.2  0.3  0.4-150-100-50 0 50 100 150 200 250 300\nT = 50 MeV0.1 x = 0.4 APR\nSka\nFIG. 33. Pressure (left) (Eq. (103)) and average chemical potential (right) (Eq. (177)) for the APR (solid) and Ska (dashed)\nmodels at the indicated proton fractions and temperatures. The \rat portions of the APR curves are due to the Maxwell\nconstruction for the mixed-phase region, the boundaries of which are given by Eqs. (181)-(182).\n-200-100 0 100 200 300 400\n 0  0.2  0.4  0.6  0.8F (MeV)\nT = 20 MeVYp = 0.10.4\nn (fm-3) APR\nSka\n 0.2  0.4  0.6  0.8  1-200-100 0 100 200 300 400\nT = 50 MeVYp = 0.4\n0.1 APR\nSka\nFIG. 34. Free energy (Eq. (111)) vs baryon density for the\nAPR (solid) and Ska (dashed) models. Results for Yp=\n0:1 and 0:4 atT= 20 MeV (left) and 50 MeV (right) are\npresented. The onset of pion condensation appears as a cusp\nat the appropriate densities.\nas electrons are relativistic for near nuclear and supra-\nnuclear densities. We have veri\fed that a similarly ex-\ncellent agreement is obtained for the Ska model (results\nnot shown).\nIsentropes of the APR model in the Pth-nplane are\nshown in Fig. 39 in which the exact numerical results\nare compared with those in the degenerate limit [77]:\nPth=2n\n3\u00192S2P\niYi\nTFiQi\n\u0010P\niYi\nTFi\u00112;i=n;p;e: (190)\nWe observe nearly identical results for S\u00142. For nucle-\nons,Qiare those from Eq. (104). For electrons, Qe= 1=2\nandTFe=k2\nFe=(2EFe) =\u00192=(4ae).\nIsentropes of the APR model in the \u0016th-nplane are\n 0 0.5 1 1.5 2 2.5 3\n 0  100  200  300S (kB)T = 20 MeV\nµ (MeV)x = 0.10.4 APR\nSka\n 0  100  200  300 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5\nT = 50 MeV\nµ (MeV)x = 0.10.4 APR\nSkaFIG. 35. Total entropy of baryons (Eq. (105)) and leptons\n(Eq. (C12)) vs average chemical potential (Eq. (177)) at\nthe temperatures and proton fractions shown after Maxwell\nconstruction.\nshown in Fig. 40. To compare the exact results with\nthose from the degenerate limit results, it was necessary\nto expand the expressions for the entropy and the nu-\ncleon thermal chemical potentials to O(T3) andO(T4)\nrespectively:\nS= 2TX\ni=n;p;eaiYi\u000016T3\n5\u00192X\ni=n;p;ea3\niYi (191)\n\u0016i=n;p=\u0000T2\"\nai\n3+aini\nm\u0003\nidm\u0003\ni\ndni+ajnj\nm\u0003\njdm\u0003\nj\ndni#\n+4T4\n5\u00192\"\n\u0000a3\ni+3a3\nini\nm\u0003\nidm\u0003\ni\ndni+3a3\njnj\nm\u0003\njdm\u0003\nj\ndni#\n;i6=j\n(192)32\n 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0  0.2  0.4  0.6  0.8cv (fm-3)\nT = 20 MeVx = 0.4Total\nn+p\ne-+e+\nn (fm-3) APR\nSka\n 0.2  0.4  0.6  0.8 0 0.2 0.4 0.6 0.8 1 1.2\nT = 50 MeVx = 0.4Total\nn+p\ne-+e+ APR\nSka\n 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0  0.2  0.4  0.6  0.8cp (fm-3)\nT = 20 MeVx = 0.4Total\nn+p\ne-+e+\nn (fm-3) APR\nSka\n 0.2  0.4  0.6  0.8 0 0.2 0.4 0.6 0.8 1 1.2\nTotal\nn+p\ne-+e+\nT = 50 MeVx = 0.4 APR\nSka\nFIG. 36. Contributions from nucleonic and leptonic constituents for the speci\fc heat densities at constant volume (left panels)\nand constant pressure (right panels). The nucleonic contributions are from Eqs. (124)-(125)) for the APR (solid) and Ska\n(dashed) models and leptonic contributions (dotted) are from (Eqs. (D17)-(D18)).\n 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0  0.2  0.4  0.6  0.8s (fm-3)\nT = 20 MeVx = 0.4Total\nn+p\ne-+e+\nn (fm-3) APR\nSka\n 0.2  0.4  0.6  0.8 0 0.2 0.4 0.6 0.8 1 1.2\nTotal\nn+p\ne-+e+\nT = 50 MeVx = 0.4 APR\nSka\nFIG. 37. Nucleonic (Eq. (105)) and leptonic (dotted) (Eq.\n(D13)) contributions for the total entropy density for the APR\n(solid) and Ska (dashed) models at the indicated proton frac-\ntions and temperatures.\nThen, the average thermal chemical potential is given by\n\u0016av;th =\u0000T2\u0014Ynan\n3\u0012\n1 +3n\nm\u0003ndm\u0003\nn\ndn\u0013\n+Ypap\n3\u0012\n1 +3n\nm\u0003pdm\u0003\np\ndn\u0013\n+2aeYp\n3\u0015\n+4T4\n5\u00192\u0014\n\u0000Yna3\nn\u0012\n1\u00003n\nm\u0003ndm\u0003\nn\ndn\u0013\n\u0000Ypa3\np\u0012\n1\u00003n\nm\u0003pdm\u0003\np\ndn\u0013\u0015\n: (193)\nThe temperature used in the above expression is obtained\nfrom Eq. (191) by perturbative inversion:\nT=S\n2PaiYi\u0014\n1 +2S2\n5\u00192Pa3\niYi\n(PaiYi)3\u0015\n;i=n;p;e (194)\n 1 10 100\n 0  0.1  0.2  0.3  0.4T (MeV)\ns = 0.250.51.02.0\nAPR\nx = 0.1\nn (fm-3) 0.1  0.2  0.3  0.4  0.5 1 10 100\ns = 0.250.51.02.0\nAPR\nx = 0.4\nn (fm-3)FIG. 38. Curves of constant entropy in the ( T;n)-plane for the\nAPR model. Solid curves show results from exact numerical\ncalculations and the crosses show results from the degenerate\nlimit expression in Eq. (189) at the indicated proton fractions.\nAt this level of approximation (made necessary by the\nweak density dependence of the chemical potential in the\ndegenerate limit), we get fairly good consistency between\nthe exact and the approximate results for S\u00141.\nVII. CONCLUSIONS\nOur primary objective in this work has been to build\nan equation of state of supernova matter in the bulk ho-\nmogeneous phase based on the zero-temperature APR\nHamiltonian density which has been devised to reproduce\nthe results of the microscopic potential model calcula-\ntions of Akmal and Pandharipande for nucleonic matter\nwith varying isospin asymmetry. One of the main fea-\ntures of the APR model is that it incorporates a neutral33\n 0.01 0.1 1 10 100\n 0  0.1  0.2  0.3  0.4Pth (MeV fm-3)\ns = 0.250.51.02.0APR\nx = 0.1\nn (fm-3) 0.1  0.2  0.3  0.4  0.5 0.01 0.1 1 10 100\ns = 0.250.51.02.0APR\nx = 0.4\nn (fm-3)\nFIG. 39. Isentropes in the ( Pth;n)-plane for the APR model\nat the indicared proton fractions. Solid curves are from the\nexact numerical calculations. Results (crosses) from the de-\ngenerate limit expression are from Eq. (190).\n-2-1.5-1-0.5 0\n 0  0.1  0.2  0.3  0.4µth (MeV fm-3)0.25\n0.5\ns = 1.0\nAPR\nx = 0.1\nn (fm-3) 0 0.1  0.2  0.3  0.4  0.5-2-1.5-1-0.5 0\n0.25\n0.5\ns = 1.0\nAPR\nx = 0.4\nn (fm-3)\nFIG. 40. Isentropes in the ( \u0016th;n)-plane for the APR model.\nSolid curves are results from the exact numerical calculations\nand the crosses are from expressions in the degenerate limit\nin Eqs. (193)-(194)) at the indicated proton fractions.\npion condensate at supra-nuclear densities found in the\ncalculations of AP for all values of proton fraction. Con-\nsequently, its high density behavior is somewhat soft in\nits pressure variation, yet it is able to support a neutron\nstar in excess of 2 M \frequired by recent observations.\nOur principal contribution in this work is the extension\nof the APR model to \fnite temperature for use in numer-\nical simulations of core-collapse supernovae. In order to\nprovide a contrast, we have also calculated the \fnite tem-\nperature properties of a model (termed Ska) using an en-\nergy density functional stemming from Skyrme e\u000bective\nforces. The methods developed in this work are applica-\nble and easily adapted to investigate thermal properties\nof other Skyrme-like energy density functionals.We have studied the behavior of the state variables\nenergyE, pressureP, the neutron and proton chemi-\ncal potentials \u0016nand\u0016p, entropy per baryon S, the free\nenergyF, and the response functions such as the com-\npressibility K, the inverse susceptibilities \u001fij, and spe-\nci\fc heatsCVandCPof the APR and the Ska models\nas functions of the temperature T, the baryon density\nn, and the proton-to-baryon fraction x. The two EOS's\nare quantitatively similar for densities up to \u00181.5n0, but\ndi\u000ber signi\fcantly at higher densities. The cross suscep-\ntibilities\u001fnp,\u001fpnand the ratio Pc=(ncTc) evaluated at\nthe critical density ncof the liquid-gas phase transition\nare the only exceptions to the above general observation.\nWe have also calculated several properties of isospin-\nsymmetric matter at the saturation density and com-\npared with experimental results, although the latter, in\nsome cases, are associated with large uncertainties. Con-\nsiderable attention has been paid to the symmetry en-\nergyS2as a function of the density and the temper-\nature. Our results reveal a weak dependence on the\ntemperature which leads to the conclusion that S2is\ndetermined mainly by the density dependent e\u000bective\nmass. It is also evident herein that, in the case of\nmatter with a phase transition, the quantities S2, and\nFsym=F(x= 0)\u0000F(x= 1=2) are fundamentally di\u000ber-\nent.\nWe also \fnd that the density jump across the coex-\nistence region of the LDP to the HDP transition of the\nAPR model depends weakly on the temperature, the pro-\nton fraction, and the leptonic contributions.\nThat thermal e\u000bects are, in general, less pronounced in\ndegenerate matter is expected as this is the regime where\nT=TF\u001c1; i.e., temperature e\u000bects are overwhelmed by\ndensity e\u000bects. However, when looking at the thermal\npart of any given thermodynamic quantity, the afore-\nmentioned density e\u000bects are entirely determined by the\ne\u000bective masses. As we have seen, the density depen-\ndence of the e\u000bective mass for nucleons interacting via\nSkyrme or Skyrme-like forces is responsible for several\ndegenerate limit e\u000bects not encountered in a free gas. In\nparticular, as a function of density the thermal pressure\nPth\rattens (whereas in a free gas it increases monotoni-\ncally),\u0016i;thbecome positive (strictly negative in the free\ncase), and S2;thbecomes negative (always positive for\na free gas). The results of Eqs. (144),(145), and (149)\nin which terms involving the derivatives of the e\u000bective\nmass with respect to the density encode e\u000bects of momen-\ntum dependent interactions and modify the expressions\nfrom what would have been their free forms. The role of\nthe e\u000bective mass in the non-degenerate limit, although\npresent, is minimal for most of the state variables. In-\ntriguingly, our results indicate that, for the temperatures\n(up to 30 MeV) and proton fractions (0.38-0.42) of most\nrelevance to supernova evolution, densities in the vicinity\nof the nuclear saturation density can be considered nei-\nther degenerate nor non-degenerate. The quantitative\nresults presented in this work (particularly, the neutron\nand proton chemical potentials) can be used to advan-34\ntage to determine the rates of electroweak reactions such\nas electron capture and neutrino-matter interactions in\nhot dense matter.\nBased on the APR model, work on the inhomogeneous\nphase at subnuclear densities where nuclei coexist with\nleptons, nucleons, nuclei, and light nuclear clusters as\nwell as pasta-like con\fgurations is in progress and will\nbe reported in a separate work.\nACKNOWLEDGEMENTS\nWe have bene\ftted greatly from the unpublished Ph.D.\nthesis of Matthew Carmell from Stony Brook University.\nComputational help from Kenneth Moore at Ohio Uni-\nversity during the initial stages of this work is gratefully\nacknowledged. This work was supported in part by the\nUS DOE under Grants No. DE-FG02-87ER-40317 (for\nC.C. and J.M.L) and No. DE-FG02-93ER-40756 (for\nB.M. and M.P.).\nAppendix A: SINGLE-PARTICLE SPECTRA\nHere we provide a derivation of the expression in Eq.\n(11) which is a direct consequence of the fact that the\nexpectation value of the Hamiltonian is stationary with\nrespect to variations of its eigenstates [53, 78]:\n\u000e\n\u000e\u001ek \nE\u0000X\nk\u000fkZ\nj\u001ek(~ r)j2d3r!\n= 0; (A1)\nwhere\u000fkis the eigenvalue corresponding to the eigenstate\n\u001ek,E=hHiandkis the set of all relevant quantum\nnumbers.\nFor a many-body Hamiltonian, \u001ekare the single parti-\ncle states making up the Slater determinant, and there-\nfore the set of all \u000fkis the single-particle energy spectrum\nof the Hamiltonian.\nConsider now a nucleonic Hamiltonian density\nH=H(\u001ci;ni), where\n\u001ci(~ r) =X\nk;sjr\u001ek(~ r;s;i )j2(A2)\nni(~ r) =X\nk;sj\u001ek(~ r;s;i )j2(A3)\nare the kinetic energy density and number density re-\nspectively, of the nucleon species with isospin i.\nThe variation of the number density with respect to \u001e\nis\n\u000eni=X\nk;s[\u000e\u001e\u0003(~ r;s;i )\u001e(~ r;s;i ) +\u001e\u0003(~ r;s;i )\u000e\u001e(~ r;s;i )]:\n(A4)\nImposing time-translational invariance leads to\n\u001e(~ r;s;i ) =i2s\u001e\u0003(~ r;\u0000s;i) (A5)\nand\u000e\u001e(~ r;s;i ) =i2s\u000e\u001e\u0003(~ r;\u0000s;i): (A6)Therefore,\n\u000eni=X\nk;s[\u000e\u001e\u0003\u001e+ (\u00001)\u001e(\u0000s)\u0002(\u00001)\u000e\u001e\u0003(\u0000s)]\n=X\nk;s[\u000e\u001e\u0003\u001e+\u000e\u001e\u0003(\u0000s)\u001e(\u0000s)]\n= 2X\nk;s\u000e\u001e\u0003\u001e; (A7)\nas the sum is over all spins.Similarly,\n\u000e\u001ci= 2X\nk;sr\u000e\u001e\u0003\nk\u0001r\u001ek: (A8)\nFurthermore,\nE=X\niZ\nd3rH(\u001ci;ni): (A9)\nCombining this with (A4) and (A6) implies\n\u000eE=X\niZ\nd3r\u0014@H\n@\u001ci\u000e\u001ci+@H\nni\u000eni\u0015\n=Z\nd3rX\ni2\n4@H\n@\u001ci(2X\nk;sr\u001e\u0003\nk\u0001r\u001ek) +@H\nni(2X\nk;s\u000e\u001e\u0003\nk\u001ek)3\n5\n=Z\nd3rX\nk;s\"\n2\u000e\u001e\u0003\nkX\ni\u0012\n\u0000r@H\n@\u001cir+@H\nni\u0013\n\u001ek#\n:(A10)\nThe minus sign is a consequence of the anti-hermiticity\nof theroperator:hr\u001ej=h\u001ejry=h\u001ej(\u0000r).\nFinally, by inserting (A10) into (A1) we get\n0 =Z\nd3rX\nk;s2\u000e\u001e\u0003\nk\"X\ni\u0012\n\u0000r@H\n@\u001cir+@H\n@ni\u0013#\n\u001ek\n\u0000Z\nd3rX\nk;s2\u000e\u001e\u0003\nk\u000fk\u001ek\n=Z\nd3rX\nk;s2\u000e\u001e\u0003\nk\"X\ni\u0012\n\u0000r@H\n@\u001cir+@H\n@ni\u0013\n\u0000\u000fk#\n\u001ek\n)X\ni\u0012\n\u0000r@H\n@\u001cir+@H\n@ni\u0013\n\u0000\u000fk= 0\n)\u0000r@H\n@\u001cir+@H\n@ni\u0000\u000fki= 0: (A11)\nThus in momentum space,\nk2\ni@H\n@\u001ci+@H\n@ni=\u000fki: (A12)\nAppendix B: APR STATE VARIABLES\nIn this appendix we summarize results pertaining to\nthe zero temperature state variables of APR. Combining\nthe density-dependent parts (see below) of these, with35\nthe appropriate thermal expressions from sections VI and\nVII yields the corresponding expressions at \fnite tem-\nperature. It is convenient to write HAPR as the sum of\na kinetic partHk, a part consisting of the momentum-\ndependent interactions Hm, and a density-dependent in-\nteractions partHd:\nHAPR =Hk+Hm+Hd (B1)\nwhere\nHk=~2\n2m(\u001cn+\u001cp) (B2)\nHm= (p3+ (1\u0000x)p5)ne\u0000p4n\u001cn\n+ (p3+xp5)ne\u0000p4n\u001cp (B3)\nHd=g1(n)[1\u0000(1\u00002x)2)] +g2(n)(1\u00002x)2(B4)\nFurthermore, the following quantities are necessary:\n\u000eg1=g1H\u0000g1L\n=\u0000n2\u0002\np17(n\u0000p19) +p21(n\u0000p19)2\u0003\nep18(n\u0000p19)\n(B5)\n\u000eg2=g2H\u0000g2L\n=\u0000n2\u0002\np15(n\u0000p20) +p14(n\u0000p20)2\u0003\nep16(n\u0000p20)\n(B6)\nf1L=dg1L\ndn\u00002g1L\nn\n=\u0000n2[p2+ 2p6n\n+ (p11\u00002p2\n9p10n\u00002p2\n9p11n2)e\u0000p2\n9n2i\n(B7)\nf1H=f1L+\u000ef1 (B8)\n\u000ef1= [2p19(p17\u0000p19p21)n\n+f3(2p19p21\u0000p17) +p18p19(p17\u0000p19p21)gn2\n+fp18(2p19p21\u0000p17)\u00004p21gn3\n\u0000p18p21n4\u0003\nep18(n\u0000p19)(B9)\nh1L=df1L\ndn\u00002f1L\nn\n=\u0000n2\u0002\n2p6\u00002p2\n9(p10+ 3p11n\n\u00002p2\n9p10n2\u00002p2\n9p11n3)e\u0000p2\n9n2i\n(B10)\nh1H=h1L+\u000eh1 (B11)\n\u000eh1= [2p19(p17\u0000p19p21)\n+f6(2p19p21\u0000p17) + 4p18p19(p17\u0000p19p21)gn\n+f6p18(2p19p21\u0000p17)\n+p2\n18p19(p17\u0000p19p21)\u000012p21\t\nn2\n+\b\np2\n18(2p19p21\u0000p17)\u00008p18p21\t\nn3\n\u0000p2\n18p21n4\u0003\nep18(n\u0000p19)(B12)w1L=dh1L\ndn\u00002h1L\nn\n=\u0000n2\u0000\n\u00003p11+ 6p2\n9p10n+ 12p2\n9p11n2\n\u00004p4\n9p10n3\u00004p4\n9p11n4\u0001\n2p2\n9e\u0000p2\n9n2(B13)\nw1H=w1L\u0000\u000ew1 (B14)\n\u000ew1= [6f(2p19p21\u0000p17) +p18p19(p17\u0000p19p21)g\n+f18p18(2p19p21\u0000p17)\n+6p2\n18p19(p17\u0000p19p21)\u000024p21\t\nn\n+\b\n9p2\n18(2p19p21\u0000p17)\n+p3\n18p19(p17\u0000p19p21)\u000036p18p21\t\nn2\n+\b\np3\n18(2p19p21\u0000p17)\u000012p2\n18p21\t\nn3\n\u0000p3\n18p21n4\u0003\nep18(n\u0000p19)(B15)\nf2L=dg2L\ndn\u00002g2L\nn\n=\u0000n2\u0010\n\u0000p12\nn2+p8\u00002p2\n9p13ne\u0000p2\n9n2\u0011\n(B16)\nf2H=f2L+\u000ef2 (B17)\n\u000ef2= [2p20(p15\u0000p20p14)n\n+f3(2p20p14\u0000p15) +p16p20(p15\u0000p20p14)gn2\n+fp16(2p20p14\u0000p15)\u00004p14gn3\n\u0000p16p14n4\u0003\nep16(n\u0000p20)(B18)\nh2L=dh2L\ndn\u00002h2L\nn\n=\u0000n2\u00142p12\nn3\u00002p2\n19p13(1\u00002p2\n9n)e\u0000p2\n9n2\u0015\n(B19)\nh2H=h2H+\u000eh2 (B20)\n\u000eh2= [2p20(p15\u0000p20p14)\n+f6(2p20p14\u0000p15) + 4p16p20(p15\u0000p20p14)gn\n+f6p16(2p20p14\u0000p15)\n+p2\n16p20(p15\u0000p20p14)\u000012p14\t\nn2\n+\b\np2\n16(2p20p14\u0000p15)\u00008p16p14\t\nn3\n\u0000p2\n16p14n4\u0003\nep16(n\u0000p20)(B21)\nw2L=dw2L\ndn\u00002w2L\nn\n=\u0000n2\u0014\n\u00006p12\nn4+ 4p4\n9p13(1 +n\u00002p2\n9n2)e\u0000p2\n9n2\u0015\n(B22)\nw2H=w2L+\u000ew2 (B23)\n\u000ew2= [6f(2p20p14\u0000p15) +p16p20(p15\u0000p20p14)g\n+f18p16(2p20p14\u0000p15)\n+6p2\n16p20(p15\u0000p20p14)\u000024p14\t\nn\n+\b\n9p2\n16(2p20p14\u0000p15)\n+p3\n16p20(p15\u0000p20p14)\u000036p16p14\t\nn2\n+\b\np3\n16(2p20p14\u0000p15)\u000012p2\n16p14\t\nn3\n\u0000p3\n16p14n4\u0003\nep16(n\u0000p20)(B24)36\nThe subscripts LandHimply the low density and the\nhigh density phase respectively.\nExpressions for the state variables are collected below.\nEnergy per particle\nE\nA=Ek\nA+Em\nA+Ed\nA=H\nn(B25)\nEk\nA=(3\u00192)5=3\n5\u00192~2\n2mn2=3[(1\u0000x)5=3+x5=3] (B26)\nEm\nA=(3\u00192)5=3\n5\u00192n\np3[(1\u0000x)5=3+x5=3]\n+p5[(1\u0000x)8=3+x8=3]o\nn5=3e\u0000p4n\n(B27)\nEd\nA=1\nn\b\ng1[1\u0000(1\u00002x)2)] +g2(1\u00002x)2\t\n(B28)\nPressure\nP=Pk+Pm+Pd=n2@H=n\n@n(B29)\nPk=2\n3nEk\nA(B30)\nPm=\u00125\n3\u0000p4n\u0013\nnEm\nA(B31)\nPdL=n\u001aEdL\nA+f1L[1\u0000(1\u00002x)2]\n+f2L(1\u00002x)2\t\n(B32)\nPdH=PdL+ (\u0000\u000eg1+n\u000ef1)[1\u0000(1\u00002x)2]\n+(\u0000\u000eg2+n\u000ef2)(1\u00002x)2(B33)\nIncompressibility\nK=Kk+Km+Kd= 9@P\n@n(B34)\nKk= 10Ek\nA(B35)\nKm= (40\u000048p4n+ 9p2\n4n2)Em\nA(B36)\nKdL= 18Ed\nA+ 9\b\n(4f1+nh1)[1\u0000(1\u00002x)2]\n+(4f2+nh2)(1\u00002x)2\t\n(B37)\nKdH=KdL+ 9n\u0000\n\u000eh1[1\u0000(1\u00002x)2]\n+\u000eh2(1\u00002x)2\u0001\n(B38)Second derivative of pressure with respect to density\nd2P\ndn2=d2Pk\ndn2+d2Pm\ndn2+d2Pd\ndn2(B39)\nd2Pk\ndn2=20\n271\nnEk\nA(B40)\nd2Pm\ndn2=\u0012200\n27\u000056\n3p4n+p2\n9n2\u0000p3\n4n3\u00131\nnEm\nA(B41)\nd2PdL\ndn2=2\nnEdL\nA+\u001210f1L\nn+ 7h1L+nw1L\u0013\n[1\u0000(1\u00002x)2]\n+\u001210f2L\nn+ 7h2L+nw2L\u0013\n(1\u00002x)2(B42)\nd2PdH\ndn2=d2PdL\ndn2+ (\u000eh1+n\u000ew 1)[1\u0000(1\u00002x)2]\n+(\u000eh2+n\u000ew 2)(1\u00002x)2(B43)\nSymmetry energy\nS2=S2k+S2m+S2d=1\n8@2H=n\n@x2\f\f\f\f\nx=1=2(B44)\nS2k=10\n91\n25=3(3\u00192)5=3\n5\u00192~2\n2mn2=3(B45)\nS2m=10\n91\n25=3(3\u00192)5=3\n5\u00192~2\n2mn5=3e\u0000p4n(p3+ 2p5)\n(B46)\nS2d=1\nn(\u0000g1+g2) (B47)\nFirst derivative of symmetry energy with respect to\ndensity\ndS2\ndn=dS2k\ndn+dS2m\ndn+dS2d\ndn(B48)\ndS2k\ndn=2\n3S2k\nn(B49)\ndS2m\ndn=S2m\nn\u00125\n3\u0000p4n\u0013\n(B50)\ndS2dL\ndn=S2dL\nn+1\nn(\u0000f1L+f2L) (B51)\ndS2dH\ndn=dS2dL\ndn+1\nn2(\u000eg1\u0000\u000eg2)\n\u00001\nn(\u000ef1\u0000\u000ef2) (B52)37\nSecond derivative of symmetry energy with respect\nto density\nd2S2\ndn2=d2S2k\ndn2+d2S2m\ndn2+d2S2d\ndn2(B53)\nd2S2k\ndn2=\u00002\n9S2k\nn2(B54)\nd2S2m\ndn2=S2m\nn2\u001210\n9\u000010\n3p4n+p2\n4n2\u0013\n(B55)\nd2S2dL\ndn2=1\nn2(\u00002f1L+ 2f2L\u0000nh1L+nh2L)(B56)\nd2S2dH\ndn2=d2S2dL\ndn2\u00002\nn3(\u000eg1\u0000\u000eg2)\n+2\nn2(\u000ef1\u0000\u000ef2)\u00001\nn(\u000eh1\u0000\u000eh2) (B57)\nChemical potentials\n\u0016i=\u0016ik+\u0016im+\u0016id=@H\n@ni(B58)\n\u0016ik=5\n3(3\u00192)5=3\n5\u00192~2\n2mn2=3\ni (B59)\n\u0016im=(3\u00192)5=3\n5\u00192e\u0000p4n\n\u0003\u001a\np5\u00148\n3n5=3\ni\u0000p4\u0010\nn8=3\ni+n8=3\nj\u0011\u0015\n+p3\u00148\n3n5=3\ni+5\n3n2=3\ninj+n5=3\nj\n\u0000p4\u0010\nn8=3\ni+n5=3\ninj+nin5=3\nj+n8=3\nj\u0011io\n(B60)\n\u0016idL=1\nn2[4njg1L+ 4ninjf1L\n+2(ni\u0000nj)g2L+ (ni\u0000nj)2f2L\u0003\n(B61)\n\u0016idH=\u0016idL\u00004\nn3nj(ni\u0000nj)(\u000eg1\u0000\u000eg2)\n+1\nn2[4ninj\u000ef1+ (ni\u0000nj)2\u000ef2] (B62)Inverse susceptibilities\n\u001fii=\u001fiik+\u001fiim+\u001fiid=@\u0016i\n@ni(B63)\n\u001fiik=2\n3\u0016ik\nni(B64)\n\u001fiim=\u0000p4\u0016im+(3\u00192)5=3\n5\u00192e\u0000p4n\n\u0003\u001a\np5\u001440\n9n2=3\ni\u00008\n3p4n5=3\ni\u0015\n+p3\u001440\n9n2=3\ni+10\n9n\u00001=3\ninj\n\u0000p4\u00128\n3n5=3\ni+5\n3n2=3\ninj+n5=3\nj\u0013\u0015\u001b\n(B65)\n\u001fiidL=1\nn2[8njf1L+ 4ninjh1L\n+4(ni\u0000nj)f2L+ (ni\u0000nj)2h2L\u0003\n(B66)\n\u001fiidH=\u001fiidL+8\nn4nj(ni\u00002nj)(\u000eg1\u0000\u000eg2)\n\u00008\nn3nj(ni\u0000nj)(\u000ef1\u0000\u000ef2)\n+4ninj\nn2\u000eh1+(ni\u0000nj)2\nn2\u000eh2 (B67)\n\u001fij=\u001fijk+\u001fijm+\u001fijd=@\u0016i\n@nj(B68)\n\u001fijk= 0 (B69)\n\u001fijm=\u0000p4\u0016im+(3\u00192)5=3\n5\u00192e\u0000p4n\n\u0003\u001a\n\u00008\n3p4p5n5=3\nj\n+p3\u00145\n3n2=3\ni+5\n3n2=3\nj\n\u0000p4\u0012\nn5=3\ni+5\n3n2=3\ninj+8\n3n5=3\nj\u0013\u0015\u001b\n(B70)\n\u001fijdL=1\nn2[4g1L+ 4nf1L+ 4ninjh1L\n\u00002g2L+ (ni\u0000nj)2h2L\u0003\n(B71)\n\u001fijdH =\u001fijdL\u00004\nn4[(ni\u0000nj)2\u00002ninj](\u000eg1\u0000\u000eg2)\n+4\nn3(ni\u0000nj)2(\u000ef1\u0000\u000ef2)\n+4ninj\nn2\u000eh1+(ni\u0000nj)2\nn2\u000eh2 (B72)38\nSpeed of Sound\n\u0010cs\nc\u00112\n=dP\nd\"(B73)\n=1\n(1\u0000x)\u0016n+x\u0016p+mK\n9(B74)\n=n\n(1\u0000x)\u0016n+x\u0016p+m(B75)\n\u0003\u0002\n\u001fnn(1\u0000x)2+x(1\u0000x)(\u001fnp+\u001fpn) +\u001fppx2\u0003\nHere,\"includes the nucleon rest mass.\nLandau e\u000bective mass\nm\u0003\ni=\u00141\nm+2\n~2(np3+nip5)e\u0000p4n\u0015\u00001\n(B76)\nDerivatives of m\u0003\niwith respect to n,x,ni, andnj\ndm\u0003\ni\ndn=\u0000m\u0003\ni\nn\u0012\n1\u0000m\u0003\ni\nm\u0013\n(1\u0000np4) (B77)\ndm\u0003\ni\ndx=\u0006(n)\n(p)2\n~2p5m\u00032\nine\u0000p4n(B78)\ndm\u0003\ni\ndni=\u00002\n~2m\u00032\ni[p3(1\u0000np4) +p5(1\u0000nip4)]e\u0000p4n\n(B79)\ndm\u0003\ni\ndnj=\u00002\n~2m\u00032\ni[p3(1\u0000np4)\u0000nip4p5)]e\u0000p4n(B80)\nd2m\u0003\ni\ndn2=m\u0003\ni\nn2\u0012\n1\u0000m\u0003\ni\nm\u0013\n\u00001\nndm\u0003\ni\ndn(1\u0000np4) (B81)\nd2m\u0003\ni\ndndni=m\u0003\ni\nn2\u0012\n1\u0000m\u0003\ni\nm\u0013\n\u00001\nndm\u0003\ni\ndni(1\u0000np4) (B82)\nd2m\u0003\ni\ndndnj=m\u0003\ni\nn2\u0012\n1\u0000m\u0003\ni\nm\u0013\n\u00001\nndm\u0003\ni\ndnj(1\u0000np4) (B83)Single-particle energy spectrum\n\u000fki=k2\niTi+Vi (B84)\nTi=@H\n@\u001ci=~2\n2m\u0003\ni(B85)\nVi=@H\n@ni=@Hm\n@ni+@Hd\n@ni(B86)\n@Hm\n@ni=f[p3+p5\u0000p4(np3+nip5)]\u001ci\n+ [p3\u0000p4(np3+njp5)]\u001cjge\u0000p4n(B87)\n@Hd\n@ni=\u0016id (B88)\nDerivatives of Viwith respect to niandnj\n(for use in the \fnite-T susceptibilities)\n@Vim\n@ni=\u001a\n[p3+p5\u0000p4(np3+nip5)]\u0012@\u001ci\n@ni\u0000p4\u001ci\u0013\n\u0000p4(p3+p5)\u001ci\u0000p4p3\u001cj\n+ [p3\u0000p4(np3+njp5)]\u0012@\u001cj\n@ni\u0000p4\u001cj\u0013\u001b\ne\u0000p4n\n(B89)\n@Vid\n@ni=\u001fiid (B90)\n@Vim\n@nj=\u001a\n[p3+p5\u0000p4(np3+nip5)]\u0012@\u001ci\n@nj\u0000p4\u001ci\u0013\n\u0000p4p3\u001ci\u0000p4(p3+p5)\u001cj\n+ [p3\u0000p4(np3+njp5)]\u0012@\u001cj\n@nj\u0000p4\u001cj\u0013\u001b\ne\u0000p4n\n(B91)\n@Vid\n@nj=\u001fijd (B92)39\nDerivatives of Qiwith respect to n,ni, andnj\ndQi\ndn=\u00003\n2m\u0003\ni\u0014dm\u0003\ni\ndn\n\u0000n\nm\u0003\ni\u0012dm\u0003\ni\ndn\u00132\n+nd2m\u0003\ni\ndn2#\n(B93)\ndQi\ndni=\u00003\n2m\u0003\ni\u0014dm\u0003\ni\ndn\n\u0000n\nm\u0003\nidm\u0003\ni\ndndm\u0003\ni\ndni+nd2m\u0003\ni\ndndni\u0015\n(B94)\ndQi\ndnj=\u00003\n2m\u0003\ni\u0014dm\u0003\ni\ndn\n\u0000n\nm\u0003\nidm\u0003\ni\ndndm\u0003\ni\ndnj+nd2m\u0003\ni\ndndnj\u0015\n(B95)\nAppendix C: CONTRIBUTIONS FROM LEPTONS\nAND PHOTONS\nCharge neutrality requires that the total charge of the\nprotons be exactly cancelled by that of the electrons. At\nT= 0, this can be stated in terms of the number densities\nasnp=ne\u0000, where the electron (with its 2 spin degrees\nof freedom) number density ne\u0000is given by\nne\u0000= 2ZkFe\u0000\n0d3k\n(2\u0019)3=k3\nFe\u0000\n3\u00192(C1)\nso that the electron Fermi momentum is kFe\u0000=\n(3\u00192ne\u0000)1=3. The chemical potential of the electrons is\nequal to their energy on the Fermi surface:\n\u0016e\u0000=\u000fFe\u0000= (k2\nFe\u0000+m2\ne)1=2: (C2)\nBecause electromagnetic interactions yield negligible cor-\nrections [79], electrons can be treated as a free Fermi gas\nand hence their contributions to the energy density and\nthe pressuse of the system are\n\"e\u0000= 2ZkFe\u0000\n0d3k\n(2\u0019)3(k2+m2\ne)1=2\n=1\n8\u00192\u0002\nkFe\u0000\u000fFe\u0000(2k2\nFe\u0000+m2\ne)\n+m4\neln\u0012me\nkFe\u0000+\u000fFe\u0000\u0013\u0015\n(C3)\npe\u0000=2\n3ZkFe\u0000\n0d3k\n(2\u0019)3k2\n(k2+m2e)1=2\n=1\n24\u00192\u0002\nkFe\u0000\u000fFe\u0000(2k2\nFe\u0000\u00003m2\ne)\n+3m4\neln\u0012kFe\u0000+\u000fFe\u0000\nme\u0013\u0015\n(C4)\nAt \fniteT, one must consider the net electric charge of\nelectrons and positrons because in supernovae tempera-\nture rises well above the 1 MeV threshold for e\u0000e+pairproduction. Accordingly, the charge neutrality condition\nbecomesnp=ne\u0000\u0000ne+\u0011ne, where the net lepton\ndensity is given by\nne= 2ZkFe\u0000\n0d3k\n(2\u0019)3\u00141\n1 +ek\u0000\u0016e\nT\u00001\n1 +ek+\u0016e\nT\u0015\n(C5)\nwith the chemical potentials of electrons and positrons\nbeing equal in magnitude, but opposite in sign. In the\nrange of densities and temperatures pertaining to super-\nnovae\u0016e; T\u001dmeand thus the relativistic limit applies:\n\u000fk= (k2+me)1=2'k\u0012\n1 +m2\ne\n2k2\u0013\n(C6)\n1\n1 +e\u000fk\u0006\u0016e\nT'1\n1 +ek\u0006\u0016e\nT\n\u0006@\n@\u0016e\u0012m2\ne\n2k1\n1 +ek\u0006\u0016e\nT\u0013\n(C7)\nThen, Eq. (C5) can be integrated analytically with the\nresult\nne=\u00163\ne\n3\u00192\u0014\n1 +\u0016\u00002\ne(\u00192T2\u00003\n2m2\ne)\u0015\n(C8)\nwhich can be solved for the chemical potential\n\u0016e=\u00123\u00192ne\n2\u00131=3\n\u00038\n><\n>:0\n@1\u0000\"\n1 +\u0012\u00192T2\n3\u0000m2\ne\n2\u00133\u00122\n3\u00192ne\u00132#1=21\nA1=3\n+0\n@1 +\"\n1 +\u0012\u00192T2\n3\u0000m2\ne\n2\u00133\u00122\n3\u00192ne\u00132#1=21\nA1=39\n>=\n>;\n(C9)\nThe total energy density, total pressure, and total en-\ntropy density of the leptons in the relativistic regime are\n\"e=\"e\u0000+\"e+\n=\u00164\ne\n4\u00192\u0002\n1 +\u0016\u00002\ne(2\u00192T2\u0000m2\ne)\n+\u00192T2\u0016\u00004\ne\u00127\u00192T2\n15\u0000m2\ne\n3\u0013\u0015\n(C10)\npe=pe\u0000+pe+\n=\u00164\ne\n12\u00192\u0002\n1 +\u0016\u00002\ne(2\u00192T2\u00003m2\ne)\n+\u00192T2\u0016\u00004\ne\u00127\u00192T2\n15\u0000m2\ne\u0013\u0015\n(C11)\nse=\"e+pe\u0000\u0016ene\nT\n=\u00162\neT\n3\u0014\n1 +\u0016\u00002\ne\u00127\u00192T2\n15\u0000m2\ne\n2\u0013\u0015\n(C12)40\nIn the limit me!0,pe=1\n3\"e. The speci\fc heats at\nconstant volume and constant pressure can be obtained\nby\nCVe=1\nne@\"e\n@T\f\f\f\f\nne\n=1\nne \n@\"e\n@\u0016e\f\f\f\f\nT@\u0016e\n@T\f\f\f\f\nne+@\"e\n@T\f\f\f\f\n\u0016e!\n(C13)\nCPe=@\n@T\u0012\"e+pe\nne\u0013\f\f\f\f\npe\n=1\nne \n@\"e\n@\u0016e\f\f\f\f\nT@\u0016e\n@T\f\f\f\f\npe+@\"e\n@T\f\f\f\f\n\u0016e!\n\u0000(\"e+pe)\nn2e \n@ne\n@\u0016e\f\f\f\f\nT@\u0016e\n@T\f\f\f\f\npe+@ne\n@T\f\f\f\f\n\u0016e!\n(C14)\nwhere\n@\"e\n@\u0016e\f\f\f\f\nT=\u00163\ne\n\u00192\u0014\n1 +\u0016\u00002\ne\u0012\n\u00192T2\u0000m2\ne\n2\u0013\u0015\n(C15)\n@\u0016e\n@T\f\f\f\f\nne=\u00002\u00192T\n3\u0016eh\n1 +\u00192\u0016\u00002e\u0010\nT2\n3\u0000m2e\n2\u00192\u0011i (C16)\n@\"e\n@T\f\f\f\f\n\u0016e=T\u00162\ne\u0014\n1 +\u0016\u00002\ne\u00127\u00192T2\n15\u0000m2\ne\n6\u0013\u0015\n(C17)\n@\u0016e\n@T\f\f\f\f\npe=\u0000\u00162\neT\n3\u00192ne\u0014\n1 +3\u00192\n\u00162e\u00127\u00192T2\n15\u0000m2\ne\n2\u0013\u0015\n(C18)\n@ne\n@\u0016e\f\f\f\f\nT=\u00162\ne\n\u00192\u0014\n1 +\u00192\u0016\u00002\ne\u0012T2\n3\u0000m2\ne\n2\u00192\u0013\u0015\n(C19)\n@ne\n@T\f\f\f\f\n\u0016e=2\u0016eT\n3: (C20)\nFinally, we present the derivatives of the electron chem-\nical potential with respect to the proton and neutron\nnumber densities. These are essential for our subsequent\ndiscussion of the low-to-high-density phase transition of\nHAPR and of our treatment of it by means of a Maxwell\nconstruction. At T= 0, we have\n@\u0016e\n@np=k2\nFe\u0000\n3ne\u0000\u0016eand@\u0016e\n@nn= 0; (C21)\nwhereas at \fnite temperature ( T >1 MeV)\n@\u0016e\n@np=3\u00192\n\u00192T2\u0000m2\n2+ 3\u00162eand@\u0016e\n@nn= 0:(C22)\nWhenT < 1 MeV, numerical evaluation of the relevant\nFD integrals is required. The numerical methods adopted\nin this work are outlined in Appendix D.\nThe contributions from photons are adequately given\nby the standard blackbody relations for the energy den-\nsity, the pressure, and the entropy density:\n\"\r=\u00192\n15T4\n(~c)3; p\r=\"\r\n3;ands\r=4\n3\"\r\nT;(C23)respectively. These remain very small compared to the\nbaryonic and leptonic contributions for all temperatures\nrelevant to the supernova problem and, for most practical\npurposes, can be ignored with no repercussions.\nAppendix D: NUMERICAL NOTES\nThe electronic state variables involve relativistic\nFermi-Dirac integrals, the general form of which is\nF\u0015( ;x) =Z1\n0\u000b\u0015\u0000\u000b\nx+ 1\u00011=2\n1 +e\u000b\u0000 d\u000b (D1)\nwhere\nx=me\nT(D2)\n\u000b=(k2+m2\ne)1=2\nT+x (D3)\n =\u0016e\u0000me\nT(D4)\nIn particular, the number density, the energy density, and\nthe pressure are given by\nne=p\n2\n\u00192T5=2m1=2\ne(F3=2+xF1=2) (D5)\n\"e=p\n2\n\u00192T7=2m1=2\ne(F5=2+ 2xF3=2+x2F1=2) (D6)\npe=p\n2\n3\u00192T7=2m1=2\ne(F5=2+ 2xF3=2) (D7)\nrespectively.\nWe evaluate these quantities numerically, using the JEL\nmethod [74] whereby they are expressed algebraically\nin terms of the mass, the temperature, and the chemical\npotential:\nne=m3\ne\n\u00192fg3=2(1 +g)3=2\n(1 +f)M+1=2(1 +g)N(1 +f=a)1=2\n\u0003MX\nm=0NX\nn=0pmnfmgn\u0014\n1 +m+\u00121\n4+n\n2\u0000M\u0013f\n1 +f\n+\u00123\n4\u0000N\n2\u0013fg\n(1 +f)(1 +g)\u0015\n(D8)\nUe=\"e\u0000mene\n=m4\ne\n\u00192fg5=2(1 +g)3=2\n(1 +f)M+1(1 +g)NMX\nm=0NX\nn=0pmnfmgn\n\u0002\u00143\n2+n+\u00123\n2\u0000N\u0013g\n1 +g\u0015\n(D9)\npe=m4\ne\n\u00192fg5=2(1 +g)3=2\n(1 +f)M+1(1 +g)NMX\nm=0NX\nn=0pmnfmgn(D10)41\nwhere\n =\u0016e\u0000me\nT= 2(1 +f=a)1=2ln\u0014(1 +f=a)1=2\u00001\n(1 +f=a)1=2+ 1\u0015\n(D11)\ng=T\nme(1 +f)1=2\u0011t(1 +f)1=2: (D12)\nThe coe\u000ecients pmnforM=N= 3 anda= 0:433 are\ndisplayed in table VI.\nThe entropy density and the free energy density follow\npmnn= 0n= 1n= 2n= 3\nm= 05.34689 18.0517 21.3422 8.53240\nm= 116.8441 55.7051 63.6901 24.6213\nm= 217.4708 56.3902 62.1319 23.2602\nm= 36.07364 18.9992 20.02285 7.11153\nTABLE VI. JEL coe\u000ecients pmnforM=N= 3 anda=\n0:433\nfrom standard thermodynamic relations:\nse=1\nT(\"e+pe\u0000\u0016ene) (D13)\nFe=\"e\u0000Tse (D14)\nFurthermore, by taking derivatives of ne,Ue, andpewith\nrespect to andtwe can get the susceptibilities and the\nspeci\fc heats:\n@\u0016e\n@np\f\f\f\f\nnn=T \n@ne\n@ \f\f\f\f\nt\u0000t2@ne\n@t\f\f\f\f\n !\u00001\n(D15)\n@\u0016e\n@nn\f\f\f\f\nnp= 0 (D16)\nCVe=1\nneme0\n@@Ue\n@t\f\f\f\f\n \u0000@Ue\n@ \f\f\f\f\nt@ne\n@t\f\f\n \n@ne\n@ \f\f\f\nt1\nA(D17)\nCPe=1\nneme0\nB@@Ue\n@t\f\f\f\f\n \u0000@Ue\n@ \f\f\f\f\nt@pe\n@t\f\f\f\n \n@pe\n@ \f\f\f\nt1\nCA\n\u0000Ue+pe\nn2eme0\nB@@ne\n@t\f\f\f\f\n \u0000@ne\n@ \f\f\f\f\nt@pe\n@t\f\f\f\n \n@pe\n@ \f\f\f\nt1\nCA(D18)\nwhere\n@\n@ \f\f\f\f\nt=f\n1 +f=a \n@\n@f\f\f\f\f\ng+t2\n2g@\n@g\f\f\f\f\nf!\n(D19)\n@\n@t\f\f\f\f\n =g\nt@\n@g\f\f\f\f\nf: (D20)\nThe non-relativistic Fermi-Dirac integrals\nF\u0015( ) =Z1\n0x\u0015\n1 +ex\u0000 dx (D21)\nx=1\nT~2k2\n2m\u0003;  =\u0016\u0000V(n)\nT(D22)that are relevant to the thermodynamics of the nucleons\nare treated by the method developed in [80]. There, three\ndi\u000berent approximations and corresponding intervals are\ngiven for each of F3=2,F1=2, andF\u00001=2:\nF\u0015( ) =e 2\n66664\u0000(\u0015+ 1) +e nX\ns=0pses \nnX\ns=0qses 3\n77775;\u00001< \u00141\n(D23)\nF\u0015( ) =nX\ns=0ps s\nnX\ns=0qs s;1\u0014 \u00144 (D24)\nF\u0015( ) = \u0015+12\n666641\n\u0015+ 1+1\n 2nX\ns=0ps \u0000s\nnX\ns=0qs \u00002s3\n77775;4\u0014 <1\n(D25)\nIn our code, we have used the coe\u000ecients of the n= 4\ncase as they appear in [80].\nThese integrals have also been computed using the non-\nrelativistic version of the JEL approach:\nF3=2=3f(1 +f)1=4\u0000M\n2p\n2MX\nm=0pmfm(D26)\nF1=2=f(1 +f)1=4\u0000M\np\n2(1 +f=a)\n\u0003MX\nm=0pmfm\u0014\n1 +m\u0000\u0012\nM\u00001\n4\u0013f\n1 +f\u0015\n(D27)\nF\u00001=2=\u0000f\na(1 +f=a)3=2F1=2\n+p\n2f(1 +f)1=4\u0000M\n1 +f=aMX\nm=0pmfm\u0002\n(1 +m)2\n\u0000\u0012\nM\u00001\n4\u0013f\n1 +f\u0012\n3 + 2m\u0000\u0014\nM+3\n4\u0015f\n1 +f\u0013\u0015\n(D28)\nwith\n =\u0016\u0000V(n)\nT= 2(1 +f=a)1=2+ ln\u0014(1 +f=a)1=2\u00001\n(1 +f=a)1=2+ 1\u0015\n(D29)\nThe coe\u000ecients M,a, andpmin the above equations\nare contained in Table VI under the n=0 column. The\nagreement between the two methods is excellent.42\nAppendix E: CAUSAL EQUATIONS OF STATE\nIt is not unusual for equations of state from non-\nrelativstic potential models to become acausal at some\nhigh density. Causality is preserved as long as the speed\nof soundcsis less than or equal to the speed of light c.\nIn this appendix, we present a thermodynamically consis-\ntent method by which an EOS based on a non-relativistic\npotential model can be modi\fed so that it remains causal\nat arbitrary high densities, both at zero temperature and\nat \fnite temperature.\nZERO TEMPERATURE CASE\nIn terms of the pressure Pand energy density \u000f, the\ncondition for an EOS to remain casual is\n\u0010cs\nc\u00112\n\u0011\f=dP\nd\u000f=dP\ndn\u0012d\u000f\ndn\u0013\u00001\n\u00141: (E1)\nIncluding the rest-mass energy density mn, the total en-\nergy density is\n\u000f=\"+mn; (E2)\nwhere\"is the internal (or speci\fc) energy density of mat-\nter. The pressure and its density derivative are then\nP=nd\"\ndn\u0000\"=n\u0016\u0000\" anddP\ndn=nd\u0016\ndn:(E3)\nWe can thus write (E1) as a \frst order di\u000berential equa-\ntion (DE):\nd\u0016\ndn\u0000\f\nn\u0016=\fm\nn: (E4)\nThe integrating factor of Eq. (E4) is given by\nf(n) = exp\u001a\n\u0000\fZdn\nn\u001b\n=n\u0000\f; (E5)\nand has the property\nd\ndn[n\u0000\f\u0016] =n\u0000\f\fm\nn: (E6)\nIntegration of Eq. (E6) leads to\n\u0016=d\"\ndn=\u0000m+c1n\f; (E7)\nwherec1is a constant of integration. A second integra-\ntion results in\n\"=\u0000mn+c1n\f+1\n\f+ 1+c2 (E8)\nwith another constant of integration c2, and therefore\nP=c1\f\n\f+ 1n\f+1\u0000c2: (E9)The integration constants c1andc2are determined by\nthe boundary conditions\n\"(nf) =\"f andP(nf) =Pf; (E10)\nwherenfis the causality \fxing density, about 0.9-0.95\nna(at which the EOS becomes acausal), which is chosen\nsuch that\ndP\nd\u000f\f\f\f\f\nna= 1; (E11)\nand the functional forms of \"(n) andP(n) are those ob-\ntained from the original Hamiltonian density.\nFrom Eqs. (E10), we get\nc1=\u000ff+Pf\nn\f+1\naandc2=1\n\f+ 1(\f\u000ff\u0000Pf):(E12)\nThus the energy density and the pressure are given by\n\"=\u0000mn+(\u000ff+Pf)\n\f+ 1\u0012n\nnf\u0013\f+1\n+\f\u000ff\u0000Pf\n\f+ 1(E13)\nP=\f\n\f+ 1(\u000ff+Pf)\u0012n\nnf\u0013\f+1\n\u0000\f\u000ff\u0000Pf\n\f+ 1:(E14)\nEquations (E13)-(E14) can be used for n\u0015nawith\f\u0014\n1 so that causality is never violated. Thermodynamic\nconsistency is built-in, because Eqs. (E13)-(E14) obey\nthe general identity (E3).\nFINITE TEMPERATURE CASE\nAt \fnite temperature, the causality condition becomes\n\f=dP\nd\u000f\f\f\f\f\ns=dP\ndn\f\f\f\f\ns\u0012d\u000f\ndn\f\f\f\f\ns\u0013\u00001\n\u00141: (E15)\nWe transform the \frst term to the variables nandTby\nthe use of Jacobians to get\ndP\ndn\f\f\f\f\ns=\rdP\ndn\f\f\f\f\nTwith\r=CP\nCV: (E16)\nThe second term of (E15) can be written as\nd\u000f\ndn\f\f\f\f\ns=d(\"+mn)\ndn\f\f\f\f\ns=\u0016+m (E17)\nby employing the identity\n\u0016=d\"\ndn\f\f\f\f\ns=dF\ndn\f\f\f\f\nT(E18)\nwhereFis the free energy density.\nThe pressure and its density derivative at \fnite tem-\nperature change to\nP=ndF\ndn\f\f\f\f\nT\u0000F=n\u0016\u0000F anddP\ndn\f\f\f\f\nT=nd\u0016\ndn\f\f\f\f\nT:\n(E19)43\nThus the \fnite-T equivalent of (E4) is:\nd\u0016\ndn\f\f\f\f\nT\u0000\f=\r\nn\u0016=\f=\rm\nn(E20)which leads to (by full analogy with the zero-T case)\nc1=Ff+mnf+Pf\nn\f=\r+1\na(E21)\nc2=1\n\f=\r+ 1\u0014\f\n\r(Ff+mnf)\u0000Pf\u0015\n(E22)\nF=\u0000mn+(Ff+mnf+Pf)\n\f=\r+ 1\u0012n\nnf\u0013\f=\r+1\n+\f=\r(Ff+mnf)\u0000Pf\n\f=\r+ 1(E23)\nP=\f=\r\n\f=\r+ 1(Ff+mnf+Pf)\u0012n\nnf\u0013\f=\r+1\n\u0000\f=\r(Ff+mnf)\u0000Pf\n\f=\r+ 1: (E24)\nNote that\fand\rshould be evaluated at nf.\n[1] J. 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Perdew*ǂ, Adrienn Ruzsinszky*, Jianwei Sun*, and Kieron Burke**  \n \n*Department of Physics, Temple University, Philadelphia, PA 19122  \nǂDepartment  of Chemistry, Temple University, Philadelphia, PA 19122  \n**Department of Chemistry and Department of Physics, University of California,  \n         Irvine, CA 92697  \n \nAbstract:   Approximations to the exact density functional for the exchange -\ncorrelation energy  of a many -electron ground state can be constructed by satisfying \nconstraints that are universal , i.e., valid for all electron densities. Ged anken \ndensities  are designed  for the purpose of this construction , but need not be realistic . \nThe uniform electron gas is an old gedanken  density. H ere, we  propose a spherical \ntwo-electron gedanken density in which the dimensionless density gradient can be \nan arbitrary positive  constant wherever the density is non -zero. The Lieb -Oxford \nlower b ound on the exchange  energy can be satisfied within a generalized gradient \napproximation (GGA) by bounding its enhancement factor or simplest GGA \nexchange -energy de nsity. This enhancement -factor  bound is well  known to be \nsufficient , but our gedanken  density shows that it is also necessary. T he \nconventional  exact exchange -energy density  satisfies no such local bound , but \nenergy densities are not unique, and the simplest GGA  exchange -energy density is \nnot an approximation to it . We further derive a strongly and optimally  tightened \nbound on the exchange enhancement factor of a two -electron density, which is \nsatisfied by the loc al density approximation but is violated by all  published GGA ’s \nor meta -GGA ’s. Finally, s ome consequences of the non -uniform density -scaling \nbehavior for the asymptotics of the exchange enhancement factor of a GGA or \nmeta -GGA  are given .    \n \n 2 \n      1.    Gedanken densities: What and why?   (Introduction and s ummary)  \n       Kohn -Sham density functional theory [1,2] for the ground -state energy  \nE and \ndensity  \n)(rn of a many -electron system  is widely used for atoms, molecules,  and \nsolids  because of its balance between useful accuracy and computational \nefficiency. Only the exact density functional for the exchange -correlation en ergy \nneeds to be approximated. The original local density (LDA) approximation  [1,2] is\n \n          \n )),(( )( ][3rn rrnd n Eunif\nxcLDA\nxc                                                                  (1)                                    \nwhere  \n)( )( )( n n nunif\ncunif\nxunif\nxc    is the exchange -correlation energy per electron of \njellium, an electron gas of uniform density  \nn. In the absence of a magnetic field, \nonly the total density is needed in the exact theory, but the approximations work  \nbetter under a  generalization to spin -density functional theory [2], where they \nbenefit from more input information.  Here  we will simpli fy the notation by \ndisplaying the approximations only  for spin-unpolarized systems.  For the exchange \nenergy, which is our focus here, the generalization to arbitrary spin -polarization is \ntrivial  [3].   \n        Jellium is not a real system, but it is one for which the LDA  is exact. This is a \ngedanken density, in the same sense that an imagined  experiment is a gedanke n \nexperiment.  Jellium is also a quasi -realistic paradigm for the valence -electron \ndensity in a simple metal .  The electron gas of non -uniform b ut slowly -varying \ndensity, on which the density -gradient expansion is based , is another gedanken \ndensity. Semi local approximations can greatly increase the accuracy without much \nreducing the computational efficiency in comparison to LD A, and here we \nintroduce new gedanken densities which may be useful for their further \ndevelopment.  Model densities can also be interesting in other contexts, e.g., to \nexplore exchange and correlation in syste ms of reduced dimensionality [4 ]. \n         The simplest semi local  approximation is the generalized gradient \napproximation (GGA)  [5-12]: \n        \n),,( )( ][3sn Fn rnd n EGGA\nxcunif\nxGGA\nxc                                                               (2) \nwhere \n3/1 2) 3)(4/3( )( n nunif\nx   is the exchange energy per electron of jellium  and \n0),(sn FGGA\nxc\n is the enhancement factor over local exchange due to both correlation \nand the dimensionless density gradient  3 \n             \n] )3(2/[3/43/12n n s   .                                                                      (3) \nWe can write  \n            \n),( )( ),( sn Fs Fsn FGGA\ncGGA\nxGGA\nxc   .                                                          (4) \nThe high -density limit of \n),(sn FGGA\nxc  is the exchange enhancement factor\n)(s FGGA\nx , \nand 𝐹𝑥𝐺𝐺𝐴(𝑠=0)=𝐹𝑥𝐿𝐷𝐴=1 is required to  recover LDA in the uniform -density  \nlimit.  \n         Unlike LDA, GGA is not unique. The nonempirical approach [2,6,9 -12] to \nGGA construction requires the satisfaction of known exact constraints on the \nfunctional \n][nExc  for all possible densities \nn .  One of these constraints \n(automatically satisfied by LDA) is the Lieb -Oxford lower bound [13]. The \noriginal bound was for the indirect part of the electron -electron interaction , \n][nU Vee\n where \n][nU  is the Hartree electrostatic interaction of the density with \nitself , for any wavefunction (not necessarily a ground -state) of density  \n).(rn  \nPerdew  [9] used the inequality \n][ ][ nU V nEee xc   to find  a version that is more  \nuseful  for density functional theory,  \n           \n),( ][ ][3n rndB n BEnEunif\nxLDA\nx xc                                                              (5) \nwhere of course  \nxcE and \nxE are negative.  The optimal  (smallest possibl e) constant \nis in the range 1.67  < B < 2.273. From the low-density limit for jellium, this range \nis narrowed  [9] to 1.93  < B < 2.273. Levy and Perdew [14 ] and r ecently Räsänen et \nal. [15] have conjectured that the optimum B is close to  1.93, but the only rigorous \nform of the bound has B=2.27 3 (or the slightly tighter 2.215 derived by Chan and \nHandy [1 6]). The right -hand side of Eq. (5) is clearly also a lower bound on \n][nEx  \nwithin density functional theory , since  \n               \n][ ][ ][ n BE nE nELDA\nx xc x  .                                                                (6) \nVia the spin -scaling equality for exchange  [3], spin-density GGA’s are guaranteed \nto satisfy this rigorous bound on the exchange energy if the spin -unpolarized \nenhancement factor satisfies  \n              \n)(s FGGA\nx < 2.27 3/21/3 = 1.80 4   (all s, spin -unpolarized ).                   (7)                                                      \nIn this form, the bound was used to construct the Perdew -Wang 1991 (PW91) [9-\n11] and the Perdew -Burke -Ernzerhof (PBE)  [11,12 ] GGA’s.  In section 2, we will 4 \n discuss the relevance of this and related newer exact constaints to functional \nconstruction.  \n         While Eq. (7) is clearly a sufficient condition [12,17 ] for a GGA to satisfy  the \nLieb-Oxford bound of Eq. (6) on the exchange energy for all possible densities , is \nit also a necessary condition  [17,18 ]?  In other words, are GGA enhancement \nfactors with \n)(s FGGA\nx  > 1.80 4 over some range of \ns  strictly forbidden by the  Lieb-\nOxford lower bound ?  Refs. [17,19,20 ] (and the success of many functionals \nviolating the bound) suggest  that the answer is no.  \n        Our answer to the last question is yes. To show this, in section 3 we construct \na new gedanken two -electron spherical density in which the dimensionless density \ngradient \ns  takes the same arbitrary positive value over the entire density.  The \nexistence of such densities implies that GGA’s with enhancem ent factors that go \nabove 1.804 will violate the Lieb -Oxford bound for some of those densities.  \n        By far the most important gedanken densities are the uniform or slowly -\nvarying densities: an infinite set for which the computationally -efficient semilo cal \napproximations can and should be exact, having some resemblance to real densities \nof interest (those of valence electrons in simple metals).  Our new gedanken \ndensity of section 3 is less realistic, and serves a more limited purpose: to establish \nthat bounds like Eqs . (7) or (11 ) are necessary constraints on semilocal \napproximations.  We would like to find another gedanken density, in which \n  is \neveryw here infinite, but have not found one yet.  \n        An early  [20] and recent  [19] objection to the inequalit y of Eq. (7 ) was that it  \nis contradicted by the behavior of the conventional exact exchange energy density \nin the density tail of an atom or molecule. But this objection overlooks the fact that \nthe exact exchange energy density is not unique, and that the simplest GGA  \nexchange -energy density  does not (and cannot) model the conventional choice. We \ndiscuss this i ssue further in section  4, where we also present  non-conventional \nexact exchange energy densities with exact exchange enhancement factors  that are \n(at least for a cuspless  two-electron atom) everywhere b ounded from above  by \n1.804 . \n        It is now well -known that the GGA form is too simple to be highly accurate, \neven for those densities for which semilocal approximations are well -suited. Much \nbetter dissociation energies, surface energies, and equilibrium geometries  [21] can \nbe found from the semilocal (hence still computationally efficient) meta -GGA  [21-5 \n 26], which uses three ingredients: the local density \n),(rn  its gradient \n),(rn  and the \norbital kinetic energy density  \n       \n2/)( 2)(2occup\nr r\n     (spin -unpolarized),                                               (8) \n     where \n)(r\n is a Kohn -Sham orbital.  Ref. [26] suggests that a meta -GGA                                                                                                                 \n        \n),,,( )( ][3  sn Fn nrd n EMGGA\nxcunif\nxMGGA\nxc                                                          (9) \nwith \n         \nunif W /) (  ,                                                                                  (10)  \ncan recognize and assign a different appropriate GGA description to covalent  \nsingle (\n0 ), metallic  (\n1 ), and weak (\n)1  bonds. H ere the Weizsäcker \nexpression \nn nW8/2  is exact for a two -electron density (\n)0  and \n3/53/22)3)(10/3( nunif \n is exact for a uniform density (\n,0s\n)1 . \nis the sole \ningredient of the Becke -Edgecombe [ 27] electron localization function (ELF), but \nhere we think of it as a dime nsionless deviation from single orbital shape \n(DDSOS).  \n        The high -density limit of the meta -GGA enhancement  factor \n0),,(sn FMGGA\nxc  \nis again the exchange enhancement factor \n),(s FMGGA\nx  with \n1 )1 ,0( LDA\nxMGGA\nx F s F \n, and again the sufficient and necessary condition to \nsatisfy the global Lieb -Oxford bound on the exchange energy for all possible \ndensities with \n0  is \n)0 ,(s FMGGA\nx < 1.80 4.  But, in section 5, we will derive the \nmuch tighter bound  \n           \n)0 ,(s FMGGA\nx  < 1.174       (all s, spin -unpolarized )                             (11)                  \nand conjecture that this tight bound  for meta -GGA exchange might remain true for \nall \n .  \n       Eq. (1 1) is strongly violated by existing GGA’s a nd meta -GGA’s. For the PBE \nGGA [12 ], \n)(s FGGA\nx  varies from \n... 2195.012  s at small \ns  to 1.804 at large  \ns.  For \nthe revTPSS meta -GGA  [24], 𝐹𝑥𝑀𝐺𝐺����(𝑠,𝛼=1)=1+0.1235 𝑠2+⋯ and \n𝐹𝑥𝑀𝐺𝐺𝐴(𝑠,𝛼=0)=1.15+0𝑠2+⋯  at small  \ns while  \n804.1),(s FMGGA\nx  at large  \ns\n. These functionals we re constructed to satisfy Eq. ( 7), not Eq. (1 1). 6 \n        In section  6, we discuss non -uniform density -scaling of the exchange energy to \nthe true two -dimensional limit. We argue that, to satisfy the right scaling behavior, \nwe must have  \n        \n2/1)( lim\n s s FGGA\nx s ,                                                                              (12) \nand \n        \n2/1)0 ,( lim\n  s s FMGGA\nx s  .                                                                     (13) \n \n \n2.  Relevance of these constraints to functional construction  \n         It can be argued that many approximate exchange functionals   (e.g., B88 [ 7]) \nviolate Eq. (7 ), that nearly all violate Eq. (13 ), and that so far all beyond LDA \nviolate Eq.  (11).  Many of these functionals are usefully accurate for real systems. \nSo, are these constraints merely pedantic?  We will argue that the answer  is no, \nfrom several different perspectives.  \n         It is precisely because we want to refine the functionals, and especially the \nmeta -GGA’s, that we now focus on exact constraints to which many properties of \nreal systems are not so sensitive. The satisfaction of exact constraints can subtly \nimprove a functional for a given system by controlling the way the functional \napproaches  extreme limits, even when the system is n ot close to those extremes. \nExact constraints that are s atisfied by LDA, such as Eq. (11 ), are of special \ntheoretical interest [ 28,29,30 ], and should be preserved in beyond -LDA \nfunctionals.  \n        The exact density functional and its exact constraints are universal, for all \nallowable densities, and not just for the densities of real atoms, molecules, and \nsolids. While the latter systems have g reat practical interest, their more familiar \nproperties  may not sensitively sample all the exact constraints. If the form of a \nfunctional (e.g. , GGA or meta -GGA) permits the satisfaction of an exact constraint \nfor all densities, then that constraint should  be taken seriously  in the nonempirical \nconstruction of functional s of that form .  \n       Even for the known properties of real systems, the existing functionals of any \nbeyond -LDA form are far from optimal .  The GGA form can satisfy a certain set of \nconstraints, but cannot satisfy them all simultaneously  [31,32 ]. GGA’s that work 7 \n best for the atomization energies of molecules (like B88) tend to overestimate the \nlattice constants of solids, while those that predict accurate lattice constants (like \nPBEsol  [32]) strongly overestimate atomization energies  [32]. Meta -GGA’s often \nresolve the se dilemma s [21,23,24 ], but can still fail when confronted with \nproperties for which they were not previously tested, such as the critical pressures \nfor structural phase transitions in solids  [33]. Similar considerations may apply to  \nsmall energy differences between different structures of a molecule. We doubt  that \nan empirical approach to functional construction can deal with such problems.  \n        The semilocal functionals such as meta -GGA are most appropriate to certain \nclasses of  densities: (1) uniform or slowly -varying densities, for which they can be \nexact by construction, and (2) compact densities, such as the densities of atoms, \nwhere the exact exchange and correlation holes are necessarily confined to a region \nclose to the el ectron they surround. For such densities, the meta -GGA exchange \nand meta -GGA corre lation energies can be separately  accurate. For multi -center \nbonded systems, such as molecules and solids, the exact exchange hole and the \nexact correlation hole can be separately spread out over two  or more centers. In \nmany of those cases (e.g., near the equilibrium geometries of sp-bond ed systems), \nthe exact combined exchange -correlation hole is still localized around the electron \nit surrounds, and semilocal functionals  can work via  a cancellation of errors \nbetween exchange and correlation. (This error cancellation would be lost if we \ncombined exact exchange with  semilocal correlation.) Although an error \ncancellation is expected on the basis of strong qualitative arguments, the \nquantitative degree of cancellation is not something that can be predicted or \ncontrolled.  Thus even a meta -GGA that satisfies all possible exact constraints is \nstill not guaranteed to work ; testing and benchmarking are always needed.  \n         Going beyond the meta -GGA form is necessary for certain strongly -\ncorrelated or stretched -bond situations. Some of the fully nonlocal approximations, \nsuch as local hybrid functional s [34] or self -interaction corrections  [35,36 ], require \na good meta -GGA as a starting point. The self -interaction correction in particular \nrequires applying the meta -GGA to the densities of typically localized  orbitals,  \nconstructed by unitary transformation  of the occupied Kohn -Sham orbital s. These  \none-electron  (\n0 ) densities are somewhat more challenging to a meta -GGA than \nthe ground -state density of the spin-polarized one-electron atom. We exp ect that \nsatisfaction of Eqs. (11) and (13 ), along with more familiar constraints, could \ngreatly improve the accuracy of self -interaction -corrected results.  8 \n          Finally, we note that a functional can ea sily satisfy a bound like Eq. (6 ) for \nall realistic densities, and still violate it for some allowable densities. The \ngedanken density of the next section s uggests  that, to satisfy such a bound for all \nallowable densities , requires a corresponding bound like \nB Fx on the enhancement \nfactor . \n \n3. A gedanken two -electron spherical density with constant non -zero s \n       Here we will introduce a new gedanken density t o show that the bound of Eq s. \n(7) or (11 ) is a necessary   condition for  a GGA or meta -GGA to satisfy the \ncorresponding lower bound on the integrated exchange energy for all possible \nelectron densities.       \n       Consider the density  \n        \n3/ )( rA rn     \n) (1 0 Rr R                                                                         (14) \n                 0  (otherwise).  \nHere \nA , \n0R, and \n1R >\n0R are positive constants, and t he density is normalized to two  \nelectrons:  \n                   \n,2 ln4 / 43 21\n0  yA rAr drR\nR                                                              (15) \nwhich fixes \nA as a function of \n0 1/RRy . (Other electron numbers are also possible, \nbut for N=2 it is easy to evaluate the exact exchange energy.)  The reduced gradient \nis then  \n                 \n3/1]ln})3/{2)[(2/3( y s   ,                                                                   (16) \nthe same positive but arbitrary constant over the whole range where the density is \nnon-zero. When y  varies from 1 to \n , \ns varies from 0 to \n  (Fig. 1) . Clearly , when \ns is constant where the density is non -zero, the enhancement factor \n)(s FGGA\nx  can be \nfactored out of the integral for  \nGGA\nxE . Then that  GGA will satisfy the global \nexchange -only Lieb -Oxford bound for this family of  densities  only when  \n)(s FGGA\nx  \n<  1.804 for all s.  \n        For this gedanken density, we can evaluate \nLDA\nxE  and the exact\nxE (which for \ntwo-electron ground -state densities equals \n])[)2/1( nU , finding  \n          \n3/4\n03/4 3/12] /[ln]/11}][ )2/{()3(3[ y y R ELDA\nx     ,                                      (17) 9 \n            \n2\n0 ] /[ln]/)ln1(1)[/2( y yy R Ex  ,                                                       (18) \nwhere \n0 1/RRy . The ratio \nLDA\nx xEE/  (Fig. 2 ) maximizes at 1.0875 for\n3.5y  (\n)06.1s\n, and tends slowly to zero in the \n1y  (\n)0s  and \ny  \n) (s  limits.  \nThis result is consistent with Eq . (11). The gedanken density  of Eq.  (14) in the \nlimit \n1y becomes a thin spherical shell of high uniform two-electron density, for \nwhich no general -purpose semilocal approximation could be expected to work.  \n        A possible objection is that this density is too s harply cut off  to be allowabl e, \ni.e, to arise from a potential . However, we can make allowable densities by \nintroducing regions \n2/ 2/0 0  Rr R  and \n2/ 2/1 1  Rr R over which the \ndensity is rounded to approach zero with zero slope at \n2/0R and \n.2/1R  \n can \nbe any positive length less than \n02R . If a GGA violates Eq. ( 7), there will always \nbe a range of s or y for which this GGA violates the global Lieb -Oxford bound for \nallowable  rounded densities with small -enough \n . This is true regardless of what \nhappens in the rounding regions, from which the contributions to the GGA \nexchange energy must be negative.  \n        As \n  approaches zero from above, t he reduced gradient \ns  becomes very large \nover the small region where the rounding occurs.  But, for a GGA which satisfies \nEq. (7 ), the contribution to \nGGA\nxE from the rounding regions m ust vanish as fast as or \nfaster  than \n .  For an even smoother rounding leading to the same conclusion s, see \nAppendix A.  \n        The orbital kinetic energy density \nW  diverges like \n2s  as \ns , and \n/1s\nin the rounding region as \n0 . Thus the kinetic energy diverges like \n /1 )/1(2\n when \n.0  Eq. ( 14) is not a Lieb allowable [ 37] density.  \n        In summary, our gedanken density of Eq. ( 14) is not an allowable density, but \nit is the limit of a sequence of allowable densities associated with a convergent \nsequence of approximate exchange energies  and bounds . Our approximations \nshould obey the  Lieb-Oxford bound on the exchange energy for every al lowable \ndensity in the sequence.   \n       The Becke 1988 [7 ] GGA has an enhancement factor \n)(88s FB\nx  that grows \nwithout bound as s increases. It will violate the global Lieb -Oxford bound on \n][nEx\nfor all \n19 3\n0 0 1 102)9/ 4exp( / x s RRy     where  \n804.1)17.3 (088s FB\nx . While  our \ngedanken density of Eq. ( 14) does not so far present a serious practical challenge 10 \n even to Becke 1988 exchange, it will present a much more serious cha llenge after \nthe bound of Eq. ( 7) is tightened to that of Eq. (1 1) in section 5.  \n        Here we briefly mention an earlier gedanken density [14] that gave us some \nlimited guidance for the 1996 construction of the PBE GGA: Consider a one -\nelectron density that is lattice -periodic over a large crystal. As the volume \n  of the \ncrystal tends to infinity, the reduced gradient \ns  tends to infinity everywhere  like \n3/1\n.  A GGA whose enhancement factor \n)(s FGGA\nx  exceeds 1.804 in the limit \ns  \nwill violate the general Lieb -Oxford bound for this density.  \n \n4. Energy densities and GGA enhancement factors              \n         In Eq. (2), \n),( )( sn Fn nGGA\nxcunif\nx  is a function of position \nr  that can be interpreted \nas a GGA exchange -correlation energy density, and \n)( )( s Fn nGGA\nxunif\nx plays the same \nrole for exchange. The best -known exact exchange energy density is the \nconventional one  [38] \n                \nrr rr rd r rnconv\nx  '/)',(' )4/1()( )(2 3    (spin -unpolarized),               (19) \nwhere  \n)'()( 2)',(*r r rroccup \n\n   is the Kohn -Sham one -particle density matrix. \nThen it may be  tempting to equate  \n                  \nunif\nxconv\nxGGA\nxF  / ,                                                                                (20) \nas in Ref. [ 19] or in the earlier Ref. [ 20].  In these references , the right -hand side of \nEq. ( 20) was evaluated  numerically for atoms and molecules, and found to diverge \nin the tail of the electron density. That unbounded result, wh ich contradicts Eq. (7), \ncan also be found analytically, since in the tail \n)2/(1rconv\nx  [7] while \nunif\nx  decays \nexponentially with   r.  The right -hand side of Eq. ( 20), plotted vs. s in molecules, \npresents a band of values [19] rather than a single value for each s. \n         Energy densities are not observables, and are not uniquely defined in density \nfunctional theory  (with the electron gas of uniform density as the sole exception) . \nOne can add any function that integrates to zero to any choice of energy density, \nand the integ ral will not change; thus this produces another equally valid energy \ndensity. This fact is very well known in studies of the non -interacting kinetic 11 \n energy \nsT . There are two natural choices of kinetic energy density, which both \nintegrate to \nsT : The first is the positive \n  of Eq. ( 8), and the second is  \n            \n  occup\nr r\n    ).( )2/1)(( 2~ 2 *                                                                     (21) \nEach can be useful in its own context, but they differ substantially from each other. \nWeighted sums like \n~)1(c c are also possible choices. It is precisely for this \nreason that the choice of kinetic energy density must be specified when meta -\nGGA’s are being constructed.  \n          The same reasoning applies to exchange or exchange -correlation energy \ndensities. They are not uniquely defined. For example, \nconv\nxn  and \n3/22nc nconv\nx  \nintegrate to the same exchange energy. There is no reason to equate the simplest \nGGA exchange energy density to  \nconv\nxn , as in Eq . (20). Indeed, they should not be \nequated, because the second -order gradient expansion of the conventional \nexchange energy density involves ill-behaved  terms that cannot be expressed in \nterms of n and s alone  [39,40 ]. An integration by parts, which changes the  \nconventional exchange energy density to an unconventional one, must be \nperformed to express the gradient expansion in terms of the  latter  two variables \nalone  [41]. Several years ago, exchange energy densities were defined [ 42] in \nterms of the exchange potentia l. Such  energy densities are unambiguously defined \nfor any exchange energy functional, exact or approximate , but their  interpretation \nand use is too demanding at present.  \n         This situation is familiar in other areas of physics. The scalar and  vector \npotentials of electromagnetic theory [ 43] are not measurable, and neither is the \nwavefunction of quantum mechanics [ 44], so these objects are gauge or unitarily \nvariant, while measurable properties are gauge or unitarily invariant.  \n         We have  never asserted the existence of exact exchange energy densities \nsatisf ying a bound like that of Eq. ( 7), but we close this section by presenting \nsuggestive but in conclusive evidence that such exact exchange energy densities \nmight exist : \n         A family of exact exchange energy densities can be generated by coordinate \ntransformation  [38,45 -47]: \n          \nu u ru rud r rnx /) ,]1[( )4/1()()(2 3       .                                      (22) 12 \n When the parameter \n1 , the conve ntional Eq. ( 19) is recover ed, with \n)2/(11rx\n in the density tail for an atom or molecule. Any \n  between 0 and 1 \nmakes \nx  decay exponentially for an atom or molecule, with the fastest decay for \n2/1\n[38,45].  Note that any member of this family can be as easily constructed \nfrom a knowledge of the  orbitals as can the conventional choice (\n1 ), and is an \nequally valid choice for an exact exchange energy density . Now let us define an \neffective, \n -dependent exchange enhancement factor  \n           \nunif\nx x xF/  .                                                                                     (23) \nWe have evalua ted Eq. ( 23) and plotted it vs. \nr  for a simple cuspless two -electron \natom of density  \n            \n)2 exp()21()2/1()(3r r rn      ,                                                         (24) \nfor which s is a function of \nr , increasing from zero at \n0r  to \n  as \nr . \n(Our numerical tests confirm that the integrated exact exchange energy \n492.0  is \nindependent of \n .)  Our results (independent of the scale parameter \n ) are plotted \nin Fig. 3 . While \n\nxF diverges in the density tail for \n1 , it everywhere satisfies the \nbound of  Eq. (7 ) for 0.75  >\n > 0.50.  At \n75.0 , it tends to a positive constant as \nr\n. \n         Finally, note that it is the integrated  exchange and correlation energies that \nare to be approximated to satisfy exact constraints.  The associated energy densities \nare not relevant  to experiment , and not relevant to theory except for example in the \nconstruction of local hybrid functionals  [34].  While there are many possible \nexchange energy densities for a given GGA, some more bounded than others, there \nis only one of the simple form \n)( )( n ns Funif\nxGGA\nx , and that one should  satisfy the \nbound of Eq. (6).  \n \n5. Tight bound  on the exchange -enhancement factor for a two -electron density  \n        The Lieb -Oxford bound of Eq. ( 6) is valid for any density, but Lieb and \nOxford [13 ] also derived tighter bounds for one - and two -electron d ensities . Their \nderivation of  Eq. (25 ) below was presented as a more rigorous version of an \nearlier one by Gadre, Barto lotti, and Handy [48 ]. Because the GGA form cannot  \ndistinguish between a two -electron density and one with mo re electrons, \nenforcement of a  tight Lieb -Oxford b ound within GGA  would lead to little 13 \n improvement over LDA  for most  systems. Thus this is an exact constraint that \ncannot be usefully imposed on GGA construction, but can be very useful for meta -\nGGA’s.  \n        For an arbitrary  spin-polarized one-electron density  \n)(1rn , where \nx xc ee E EnU V  ][\n is a pure self -interaction correction, the optim al bound is \nknown [13 ,48]: \n               \n3/4\n13\n1 092.1][ rnd nEx .                                                                       (25) \nFor a spin -unpolarized t wo-electron ground state of density  \n2n, we can take \n2/2 1n n\n and \n][2][1 2 nE nEx x . Then  Eq. (2 5) implies  \n                \n][ 174.1][ }])3(3/{4)[2/092.1(][2 23/12 3/1\n2 n E n E nELDA\nxLDA\nx x    .                    (26) \nTwo-electron ground states have \n0 . Thus our two-electron gedanken density of \nsection 3  tells us that a sufficient and necessary condition f or a meta -GGA to \nsatisfy Eq. ( 26) for all two -electron densities is Eq. ( 11), \n             \n174.1)0 ,(s FMGGA\nx   (all s, spin -unpolarized).                                    (27) \nEq. ( 27) remains  optimally tight, as it would not be if it were  derived from the \nLieb-Oxford bound  on \n][. nU Vee  for a two -electron density. We have no proof \nof  the analog of Eq. ( 27) for \n0 , but we suspect that it may be  true, because we \nsuspect that \n)0 ,( ),(     s F s FMGGA\nxMGGA\nx  as in the M eta-GGA  Made Simple  of Ref. \n[25] (which works rather well with PBE GGA correlation and satisfies \n29.1),(s FMGGA\nx\n). We know of no two-electron  spin-unpolarized density with \n174.1 /LDA\nx xEE\n. \n         Capelle and Odashima [ 49] have suggested the possibility of a tightened \nLieb-Oxford bound for the exchange -correlation energy , and we suspect that they \nwere right to do so.  However, the possibilities for tightening this bound are limited, \nreducing B of Eq. ( 5) from 2.273 to  a value 1.93  or higher (and thus  the bound on \nthe spin -unpolarized \nxF  from 1.804 to 1.53 ), as mentio ned after Eq. ( 5). In \ncontrast, our suspected bound \n174.1),(s FMGGA\nx is dramatically tightened  over Eq. \n(7). We will explore the consequences of this assumption in future work.    \n         While LDA sati sfies Eq. (27), published GGA’s and meta -GGA’s violate it. \nSo why do GGA’s work even for the exchange energy of the  He atom and similar 14 \n two-electron densities? The answer must be that , for these two -electron densities,  \nGGA’s make an error cancellation between regions o f small s (where their \nexchange enh ancement is too low, around 1 , as it must be to recover the uniform -\ndensity limit ) and larger s (where their exchange enhancemen t is too high,  \nviolating Eq. ( 27)).  The Meta -GGA Made Simple [25 ], and to a lesser extent other \nmeta -GGA’s, have \n)0 ,0( s FMGGA\nx  considerably higher than 1  but less than 1.174  \n(see Fig. 1 of Ref. [25 ]).  The bound of Eq. (27) cannot be applied at the GGA \nlevel, even f or the He atom, because it would destroy this  error cancellation, but it \ncan hopefully be applied at the meta -GGA level.  Many -electron densities  on the \nother hand  are energetically dominated by  regions with  \n1s , where standard \nfunctionals are not seriously challenged even by our tightened lower bound.  \n         A meta -GGA for exchange that satisfies the conjectured general bound \n][ 174.1][ n E n ELDA\nxMGGA\nx\n might work wi th semilocal (sl) functionals  for correlation, \nwhich typically satisfy \n][ 94.0][ n E nELDA\nxsl\nc , making \n][ 14.2][ n E nELDA\nxsl\nxc . \n \n6. Non-uniform density scaling : Implications for the asymptotics of the \nexchange enhancement factor   \n         In this section, we will explore the implications of  non-uniform density \nscaling [50 ] for the large -s and large -\n behaviors of the exchange enhancement \nfactor \nxF . We start from a density \n),,( zyxn  having a finite \n][nEx , then define the \none-dimensionally scaled density \n).,,( ),,()1(zyxn zyxn   The scaled  density has the \nsame electron number as the unscaled one, but is more compressed (\n1 ) or \nexpanded (\n1 ) along the x direction. When \n , the density collapses from \nthree dimensions to two, in which the exch ange energy is still finite and negative. \nLevy [50 ] proved that  \n                  \n. ][ lim)1(   nEx                                                                          (28) \nThe LDA and most existing GGA’ s and meta -GGA’s violate Eq. ( 28) [51-53]. The \nPW91 GGA [9-11] and the VT(8,4) GGA [54 ] and its related meta -GGA [55 ] \nsatisfy Eq. ( 28), but they incorrectly [52,53 ] make the left -hand side vanish. A \nfinite limit is achieve d by the GGA of Chiodo et al. [56 ]. \n         Starting from the definitions of s (Eq. ( 3)) and \n (Eq. ( 10), we easily find \nthat, under non -uniform scaling to the true two-dimensional limit (\n) , 15 \n                   \n),,( ),,(3/2zyxf zyxs  ,                                                                    (29) \n                 \n0),,(zyx .                                                                                 (30) \nIf the unscaled s is nonzero over some region in which the unscaled density is non -\nzero, then the meta -GGA exc hange energy (Eq. ( 9)) has a finite non -zero limit for \nEq. (28) when Eq. ( 13) is satisfied. This determines how the exchange \nenhancement factor vanishes as \ns . \n            Levy [50 ] also defined a two -dimensional scaling of \n),,( zyxn  to \n               \n),,( ),,(2 )2(zyxn zyx n   .                                                                     (31) \nClearly this is the product of two one -dimensional scalings along different axes \nwith the same scale parameter. Applying a third yields the three -dimensional or \nuniform scaling  \n               \n),,( ),,(3 )3(zyxn zyxn   .                                                                   (32) \nOur exchange functi onals and essentially all other sensible exchange functionals  \nare designed to behave correctly [57 ] under uniform scaling: \n  \n              \n][ ][)3( 1nE nEx x\n .                                                                               (33) \nMoreover, the exchange component of a sensible approximate exchange -\ncorrelation energy function al is found  when the density is scaled uniformly by \n , \nthe functional is divided by \n , and \n  is taken to \n , because this is a known \nproperty of the exact exchange -correlation functional  [58]. \n           Funct ionals  that satisfy both Eq. (28) and Eq. (33 ) will of course also scale \ncorrectly [ 50] under two -dimensional scaling to the one -dimensional limit : \n              \n\n ][ lim)2( 1\n  nEx .                                                                       (34) \n          In the low -density limit under non -uniform three -dimensional scaling, \ncorrelation scales like exchange.  We can define  \n             \n][ lim][)3( 1\n0   nE nBxc xc\n ,                                                                   (35) \nwhich itself  has one -dimensional and two -dimensional scaling limits  that will be \nsatisfied by a GGA (or in generalization by a meta -GGA) if [ 14] \n              \n ),( lim2/1sn FsGGA\nxc s .                                                                     (3 6) 16 \n When the exchange part of \nGGA\nxcF  satisfies Eq. (1 3), it is very easy to satisfy Eq. \n(36).   \n \nNotes added in proof:      Mirtschink, Seidl, and Gori -Giorgi [59] have discussed \nthe “local Lieb -Oxford bound”. Peverati and Truhlar  [60] have written that further \nimprovement of approximate density functionals “may involve breaking some of \nthe constraints that we are following right now or – less likely as it seems to us – \nadding new constraints”. We hope that the new constraints of o ur paper will lead to \npractical improvements.  \n \nAcknowledgments:   Our work on exact constraints has been inspired by that of \nMel Levy. The work of JPP and JS was supported in part by the National Science \nFoundation under grant DMR -1305135.  That of KB was supporte d by DOE under \ngrant DE -FG02 -08ER46496 . We thank G.I. Csonka for  assistance with the \nmanuscript, and S.B. Trickey as well as our referees for comments and corrections.  \n \nAppendix:  Well -behaved densities tending  to the gedanken density of Eq. ( 14)               \n         In this appendix, we c heck that the density of Eq. ( 14) can be achieved as a \nlimit of densities that satisfy physically reasonable physical conditions:  \n                                                      ∞>𝑛(𝐫)>0                                                (A1)  \neverywhere in real -space,  𝑛(𝐫) is normalizable, and 𝑛(𝐫) has finite no n-interacting \nkinetic energy [37 ]. We will also require that its second derivative be finite \neverywhere, so as to be able to find the  corresponding Kohn -Sham potential (i.e., \nthe density is non-interacting  𝑣 -representable.)  \n        To construct suc h a density from that of Eq. (14 ), we begin by extending the \ndensity to be finite in all regions of space. We write  \n𝑛𝑒(𝐫)=𝐴\n𝑅03[10−15𝑟\n𝑅0+6(𝑟\n𝑅0)2\n],𝑟≤𝑅0 \n                                                        =𝐴/𝑟3,𝑟≥𝑅0,                                              (A2)  17 \n where the form for  𝑟<𝑅 is a quadratic chosen to match the gedanken density and \nits first two derivatives at  𝑅0. Note that 𝑛𝑒(𝐫) coincides with the gedanken density \nexactly between  𝑅0 and  𝑅1, but remains finite everywhere.  \n        Next, we multiply by a damping factor to ensure that the density drops rapidly \noutside the shell. We define the damping function  \n𝑓𝑚(x)=(∑𝑥𝑗\n𝑗!𝑚\n𝑗=0)exp(−𝑥)\n√1+(𝑥𝑚\n𝑚!)2,    𝑥≥0 \n                                                                      =1,𝑥≤0    ,                                          (A3)  \nwhich switches f rom a constant (1)  to a decaying exponential at 𝑥=0. Here 𝑚 \ndetermine s the number of derivatives that remain continuous at 𝑥=0, while the \ndenominator ensures that a simple exponential decay is recovered at large 𝑥. We \nchoose 𝑚=2 to ensure that second -derivatives are well -behaved at the transition.  \n       Our well -behaved  density can now be defined as  \n                                               𝑛𝛾(𝐫)=𝑛𝑒(𝐫)𝑓2(𝑅0−𝑟\n𝛾+1\n2)𝑓2(𝑟−𝑅1\n𝛾+1\n2)                 (A4)  \nand as 𝛾→0, it approaches the gedanken density of the text. To ensure \nnormalization, 𝐴 must become a function of 𝛾 whose 𝛾→0 limit is that of Eq . (15) \nof the text.  For any finite 𝛾, this is a Lieb -allowed density, and for sufficiently \nsmall 𝛾, its exchange energy can be made arbitrarily close to the gedanken density \nof the main text.  \n       In Fig 4 , we plot dens ities for  𝑅0=1, and  𝑅1=2 with 𝛾 =0.1, as well as the \ngedanken density. 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Perdew, Phys. Rev. A 32, 2010 ( 1985).  21 \n [58] M. Levy  and J.P. Perdew, Int. J. Quantum Chem. 49, 539 (1994).  \n[59] A. Mirtschink, M. Seidl, and P. Gori -Giorgi, J. Chem. Theory Comput. 8, \n3097 (2012).  \n[60] A. Peverati and D.G. Truhlar, Phil. Trans. Roy. Soc. A 372, UNSP20120476 \n(2014).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 22 \n Fig. 1: Plot of the reduced density gradient s (Eq. ( 16)) vs. \n0 1/RRy for the two -\nelectron gedanken density of Eq. ( 14). \n \n \n \n \n \n \n \n \n \n \n \n \n23 \n Fig. 2: Plot of the exchange -energy ratio \nLDA\nx xEE/ (Eqs. (17)  and (18)) vs. \n0 1/RRy\nfor the two -electron gedanken  density of Eq. ( 14). \n \n \n \n \n \n \n \n \n \n \n \n \n \n24 \n Fig. 3 : Plot of the effective \n -dependent exchange enhancement factor of Eq. ( 23) \nvs. \nr  for the cuspless two -electron (\n)0  density of Eq. ( 24). Different values of \n\n correspond to different choices for the exact exchange energy density  under a \ncoordinate transformation . The curve for \n75.0 tends to a nonzero  constant as\nr\n. All curves in 0.75 >\n  > 0.5 satisfy the bound of Eq. ( 7). Note that s = \n0.00, 0.54, 1.06, 1.98, and 3.67 at \nr  = 0, 1, 2, 3, and 4, respectively.  \n \n \n \n \n \n \n 00.511.522.5\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0\nr\nFx =0.5=0.75=0.85=1.025 \n  \nFig. 4:  Plot of the smoothed two -electron gedanken density \nn  of Eq. (A4) vs. \ndistance \nr  from the origin.  The 𝛾=0.001 curve already converges t o the \ngedanken  density of Eq . (14). \n \n \n \n \n \n \n \n \n \n \n \n 26 \n Fig. 5:  Plot of the reduced density gradient s for the smoothed two -electron \ngedanken density of Eq. (A4) vs. distance r from the origin.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n27 \n Fig. 6: Plot of the Kohn -Sham potential \n2/1 2/12/ )2/1( n n vs   for the smoothed \ntwo-electron gedanken density of Eq. (A4) vs. distance r from the origin.  For the \ngedanken density of  Eq, (1 4), this reduces to \n)8/(32r vs in the range \n1 0 Rr R\n.  Here \n)8/(12 is evaluated only for \n1.0  , to make the two curves plottable \non the same scale.  \n \n \n \n"    },    {        "title": "1403.7612v1.Nonlinear_channels_of_Werner_states.pdf",        "content": "NONLINEAR CHANNELS OF WERNER STATES.\nV.I. Man'ko1\u0003and R.S. Puzko2\n1P.N. Lebedev Physical Institute, Russian Academy of Science\nMoscow, Russia 119991\n2Moscow Institute of Physics and Technology (State University)\nDolgoprudny, Russia 117303\n\u0003Corresponding author e-mail: manko @ sci.lebedev.ru\nAbstract\nThe nonlinear positive map of density matrix of two-qubit Werner state called nonlinear channel is\nstudied. The map \u001a!\b(\u001a) is realized by rational function \b. The in\ruence of the map onto\nthe entanglement properties of the transformed density matrix is discussed. The violation of Bell\ninequality (CHSH inequality) for the two-qubit state \b( \u001a) is investigated. The nonlinear channels\nunder discussion create the entangled state from separable Werner state. The quantum spin-tomograms\nof the states are studied.\nKeywords: Werner state, nonlinear quantum channels, quantum tomogram, Bell inequality,quantum\nentanglement, separable states.\n1 Introduction\nThe entanglement [1] of composite quantum system states is related to purely quantum correlations\nof the subsystems, e.g. for two-qubit states this phenomenon is studied for Werner states [2]. The Peres-\nHorodecki criterion [3{5] of the entanglement is known as necessary and su\u000ecient condition for two-qubit\nstate and for qubit-qutrit state. The density matrix of any system state can be transformed into another\ndensity matrix. Linear transforms of density matrix are known as positive maps [6] (see also [7]). Some\nspeci\fc positive maps are known as completely positive maps [6, 8] called quantum channels. Recently\nsome nonlinear maps of density matrices were discussed [9, 10]. In [15, 16] the spin-tomograms of qudit\nstates were introduced. The tomograms are fair probability distributions which determine the density\nmatrix of the spin-system state.\nThe aim of our work is to study how the nonlinear positive map (nonlinear channel) in\ruences the\nentanglement properties, if it is applied to the Werner state. We will discuss the properties of the\nspin-tomogram of Werner state under the action of nonlinear channels. The standard linear quantum\nchannels are intensively discussed in the literature (see e.g. [12]).We consider the speci\fc nonlinear map.\nThis map provides the new density matrix which is obtained from the initial one by taking n-th power\nof this initial matrix with adding the corresponding factor responsible for normalization condition of the\ndensity operator. The important property of the entangled qudit-system state is the possibility to violate\nBell inequality (CHSH). We are going to study in this work the in\ruence of nonlinear channels on the\nviolation of CHSH inequality.\n1arXiv:1403.7612v1  [quant-ph]  29 Mar 2014The paper is organized as follows.\nIn Sec. 2 we review the properties of Werner state and its density matrix \u001aw;1which depends on the\nreal parameter p. The entanglement phenomenon is studied for density matrix \u001an\nw;1=Tr[\u001an\nw;1] is Sec. 3.\nThe tomograms and violation of Bell inequalities for two series of transformed (by nonlinear channels)\nstates are discussed in Sec. 4. The conclusion and prospective are presented in Sec. 5.\n2 Properties of non-linear channels \u001an\nw;1=Tr[\u001an\nw;1]for casesn= 1;2;3\nDensity matrix for Werner states with parameter pcan be written as\n\u001aw;1=0\nBBBB@A0 0C\n0B0 0\n0 0B0\nC0 0A1\nCCCCA; (1)\nwhereA(p) =1+p\n4,B(p) =1\u0000p\n4,C(p) =p\n2. Eigenvalues for this matrix are\n\u00151=1 + 3p\n4;\u00152;3;4=1\u0000p\n4: (2)\nNon-negativity of density matrix restricts parameter pto domain\u00001\n3\u0014p\u00141. To examine the entan-\nglement of Werner state we used Peres-Horodecki criterion. The ppt-matrix for \u001aw;1takes form\n\u001appt\nw;1=0\nBBBB@A0 0 0\n0B C 0\n0C B 0\n0 0 0 A1\nCCCCA; (3)\nand has eigenvalues \u00151=1\u00003p\n4,\u00152;3;4=1+p\n4, which must be non-negative for separable state. Therefore\n\u001aw;1describes entangled states in the domain (1 =3;1] of the parameter p.\nDensity matrix \u001aw;2=1\nTr[\u001a2\nw;1]\u001a2\nw;1is obtained from matrix \u001aw;1by the nonlinear transform. The\nnormalized density matrix has the form similar to matrix \u001aw;1, but with rede\fned numbers A(p) =\n(1+p)2+4p2\n4(1+3p2),B(p) =(1\u0000p)2\n4(1+3p2),C(p) =p(1+p)\n1+3p2. Its eigenvalues are \u00151=(1+3p)2\n4(1+3p2),\u00152;3;4=(1\u0000p)2\n4(1+3p2). Unlike\n\u001aw;1the density matrix is non-negative for all real values of parameter p. The ppt-matrix for \u001aw;2\nhas eigenvalues \u00151=1\u00006p\u00003p2\n4(1+3p2),\u00152;3;4=1+2p+5p2\n4(1+3p2), they are non-negative if ( \u00001\u00002p\n3\n3\u0014p\u0014\u00001 +2p\n3\n3).\nAccording to Peres-Horodecki criterion \u001aw;2describes entangled states if ( p<\u00001\u00002p\n3\n3)[(\u00001+2p\n3\n3<p).\nMatrix\u001aw;3=1\nTr[\u001a3\nw;1]\u001a3\nw;1is also similar to \u001aw;1withA(p) =(1+p)3+12p2(1+p)\n4(6p3+9p2+1),B(p) =(1\u0000p)3\n4(6p3+9p2+1),\nC(p) =6p(1+p)2+8p3\n4(6p3+9p2+1). For positivity of matrix \u001aw;3it is necessary for its eigenvalues to be non-negative.\nFirst of them \u00151=(1+3p)3\n4(6p3+9p2+1)changes sign in points \u00001=3,\u00001\n2(1 +1\n3p\n3+3p\n3). Others three eigenvalues\nare equal to \u00152;3;4=(1\u0000p)3\n4(6p3+9p2+1)and they are positive if(1\u0000p)3\n4(6p3+9p2+1)< p < 1. Thus\u001aw;3is density\nmatrix for\u00001=3\u0014p\u00141. Transposed matrix has following eigenvalues: \u00151=1\u00009p\u00009p2\u000015p3\n4(6p3+9p2+1),\u00152;3;4=\n2Figure 1: Negativity (the sum of absolute values of eigenvalues of density matrix) for cases \u001appt\nw;1(upper-left\nplot),\u001appt\nw;2(upper-right plot) and \u001appt\nw;3(lower plot).\n1+3p+15p2+13p3\n4(6p3+9p2+1), which are non-negative for \u00001\u0014p\u0014\u00001\n5(1 + 21\n3p\n3\u000023p\n3). The entanglement appears for\nthe parameter of Werner state in the domain \u00001\n5(1 + 21\n3p\n3\u000023p\n3))<p\u00141.\n3 Case of arbitrary integer n\nFor general integer nproviding matrix \u001an\nw;1we can obtain the factorized form of matrix \u001aw;n. We have\nthe decomposition \u001aw;1=SDS\u00001, whereD- diagonal matrix with eigenvalues of \u001aw;1on the diagonal,\nandS- unitary matrix with columns taken as eigenvectors of \u001aw;1. Using this decomposition it is easy\nto calculate the matrix \u001an\nw;1:\n\u001an\nw;1= (SDS\u00001)n=SDS\u00001\u0001SDS\u00001\u0001:::\u0001SDS\u00001\u0001SDS\u00001=SDnS\u00001: (4)\n3Eigenvectors of \u001aw;1read\n\u00151!\u0000 !a1= (1p\n20 01p\n2)T;\n\u00152;3;4!\u0000 !a2= (0 1 0 0 )T;\n\u0000 !a3= (0 0 1 0 )T;\n\u0000 !a4= (1p\n20 0\u00001p\n2)T;(5)\nwhere\u0015i- eigenvalues (2). Thus \u001aw;1has the following decomposition:\n\u001aw;1=SDS\u00001=0\nBBBB@1p\n20 01p\n2\n0 1 0 0\n0 0 1 0\n1p\n20 0\u00001p\n21\nCCCCA0\nBBBB@1+3p\n40 0 0\n01\u0000p\n40 0\n0 01\u0000p\n40\n0 0 01\u0000p\n41\nCCCCA0\nBBBB@1p\n20 01p\n2\n0 1 0 0\n0 0 1 0\n1p\n20 0\u00001p\n21\nCCCCA: (6)\nApplying this decomposition, we obtain for \u001aw;nthe expression:\n\u001aw;n=1\n3(1\u0000p)n+ (1 + 3p)n0\n@1\n2((1+3p)n+(1\u0000p)n) 0 01\n2((1+3p)n\u0000(1\u0000p)n)\n0 (1\u0000p)n0 0\n0 0 (1 \u0000p)n0\n1\n2((1+3p)n\u0000(1\u0000p)n) 0 01\n2((1+3p)n+(1\u0000p)n)1\nA: (7)\nEigenvalues of this matrix read\n\u00151=(1 + 3p)n\n3(1\u0000p)n+ (1 + 3p)n; (8)\n\u00152;3;4=(1\u0000p)n\n3(1\u0000p)n+ (1 + 3p)n: (9)\nIfngoes to in\fnity, all the eigenvalues become non-negative numbers for all the values of real parameter\np:\np<0!\u00151= 0;\u00152;3;4=1\n3;\np= 0!\u00151;2;3;4=1\n4;\np>0!\u00151= 1;\u00152;3;4= 0:(10)\nThus, in this limit the matrix \u001aw;nsatis\fes the conditions for density matrix for all real values of parameter\np. The ppt-matrix \u001ap\nw;npthas the following eigenvalues:\n\u0015ppt\n1=1\n23(1\u0000p)n\u0000(1 + 3p)n\n3(1\u0000p)n+ (1 + 3p)n; (11)\n\u0015ppt\n2;3;4=1\n2(1\u0000p)n+ (1 + 3p)n\n3(1\u0000p)n+ (1 + 3p)n: (12)\nBefore utilize criterion for separable state it is worth to separate cases of odd and even n.\nFor odd integers n, there is singularity in eigenvalues of \u001aw;nand\u001appt\nw;nin pointp= 1 +4\nnp\n3\u00003. The\neigenvalue\u00151goes to zero if p=\u00001=3,\u00152;3;4go to zero in point p= 1. Therefore \u001aw;nis non-negative\nfor all odd nforp2[\u00001=3; 1]. The eigenvalues \u0015ppt\n1and\u0015ppt\n2;3;4goes to zero if p= 1\u00004\nnp\n3+3andp=\u00001,\nrespectively. Finally, \u001appt\nw;nis positive for p2[\u00001; 1\u00004\nnp\n3+3]. So the matrix \u001aw;ndescribes separable state\n4forp2[\u00001=3; 1\u00004\nnp\n3+3] and entangled state in domain (1 \u00004\nnp\n3+3; 1]. In the limit n!1 , where integers\nnare odd numbers, the domain of parameter pproviding entangled state becomes (+0; 1].\nFor evenn, the matrix \u001aw;nis non-negative for all real p. As for\u001appt\nw;n, its eigenvalues \u0015ppt\n2;3;4are always\npositive and \u0015ppt\n1changes sign in points p= 1\u00004\nnp\n3+3andp= 1 +4\nnp\n3\u00003. Thus,\u001aw;ndescribes entangled\nstate if (p<1 +4\nnp\n3\u00003)[(p>1\u00004\nnp\n3+3). In the limit n!1 , where integers nare even numbers, the\ndomain becomes ( p<\u00001)[(p>0).\nUtilizing obtained eigenvalues one can calculate negativity for the case of arbitrary n. In the domain\nof parameter p, where\u001aw;ndescribes entangled state, negativity takes value:\nN=\f\f\f\f2(1 + 3p)n\n3(1\u0000p)n+ (1 + 3p)n\f\f\f\f: (13)\nFrom this formula it can be easily found that negativity reaches its maximum value at the point p= 1\nfor all integers n. In Figure 1 the dependence of negativity on parameter pis illustrated for cases of\nn= 1;2;3. For the separable states negativity for density matrix \u001appt\nw;nis equal to 1. As for the entangled\nstates, the negativity re\rects the strength on entanglement which is maximum at p= 1.\n4 Quantum tomography for nonlinear channels transforming density\nmatrix of Werner state\nTo describe the spin states the tomographic probability distributions identi\fed with the system states\ncan be used [15, 16]. For direction\u0000 !n(\u0012; ), which is determined by the parameters of unitary matrix u\nthe tomogram for spin with density operator ^ \u001acan be obtained from formula:\nW(m;\u0000 !n) =hmju^\u001au+jmi; (14)\nwheremis spin projection on the direction\u0000 !n. Quantum tomogram for system with density operator ^ \u001a\nfor two spins with j= 1=2 can be given by formula:\nW(m1;\u0000 !a;m 2;\u0000 !b) =hj1;m1;j2;m2jU^\u001aU+jj1;m1;j2;m2i; (15)\nwhereU=u1\nu2, the product being tensor product of unitary matrices\nui=0\n@cos\u0010\n\u0012i\n2\u0011\nei\n2('i+ i)sin\u0010\n\u0012i\n2\u0011\nei\n2('i\u0000 i)\n\u0000sin\u0010\n\u0012i\n2\u0011\ne\u0000i\n2('i\u0000 i)cos\u0010\n\u0012i\n2\u0011\ne\u0000i\n2('i+ i)1\nA;i= (1;2): (16)\nThe matrix elements of the matrices depend on Eulers angles \u0012i,'i, i. For the Werner state tomographic\nprobability for system of two qubits, depending on directions\u0000 !n1and\u0000 !n2, read\nW(\";\u0000 !n1;\";\u0000 !n2) =A(p)\u0012\ncos2\u0012\u00121\n2\u0013\ncos2\u0012\u00122\n2\u0013\n+ sin2\u0012\u00121\n2\u0013\nsin2\u0012\u00122\n2\u0013\u0013\n+\n+B(p)\u0012\nsin2\u0012\u00121\n2\u0013\ncos2\u0012\u00122\n2\u0013\n+ cos2\u0012\u00121\n2\u0013\nsin2\u0012\u00122\n2\u0013\u0013\n+C(p)\n2sin (\u00121) sin (\u00122) cos ( 1+ 2);(17)\n5W(\";\u0000 !n1;#;\u0000 !n2) =A(p)\u0012\nsin2\u0012\u00121\n2\u0013\ncos2\u0012\u00122\n2\u0013\n+ cos2\u0012\u00121\n2\u0013\nsin2\u0012\u00122\n2\u0013\u0013\n+\n+B(p)\u0012\ncos2\u0012\u00121\n2\u0013\ncos2\u0012\u00122\n2\u0013\n+ sin2\u0012\u00121\n2\u0013\nsin2\u0012\u00122\n2\u0013\u0013\n\u0000C(p)\n2sin (\u00121) sin (\u00122) cos ( 1+ 2);(18)\nW(#;\u0000 !n1;\";\u0000 !n2) =A(p)\u0012\nsin2\u0012\u00121\n2\u0013\ncos2\u0012\u00122\n2\u0013\n+ cos2\u0012\u00121\n2\u0013\nsin2\u0012\u00122\n2\u0013\u0013\n+\n+B(p)\u0012\ncos2\u0012\u00121\n2\u0013\ncos2\u0012\u00122\n2\u0013\n+ sin2\u0012\u00121\n2\u0013\nsin2\u0012\u00122\n2\u0013\u0013\n\u0000C(p)\n2sin (\u00121) sin (\u00122) cos ( 1+ 2);(19)\nW(#;\u0000 !n1;#;\u0000 !n2) =A(p)\u0012\ncos2\u0012\u00121\n2\u0013\ncos2\u0012\u00122\n2\u0013\n+ sin2\u0012\u00121\n2\u0013\nsin2\u0012\u00122\n2\u0013\u0013\n+\n+B(p)\u0012\nsin2\u0012\u00121\n2\u0013\ncos2\u0012\u00122\n2\u0013\n+ cos2\u0012\u00121\n2\u0013\nsin2\u0012\u00122\n2\u0013\u0013\n+C(p)\n2sin (\u00121) sin (\u00122) cos ( 1+ 2):(20)\nHereA(p),B(p),C(p) are elements of density matrix \u001aw;1. The density matrices \u001aw;nhave form similar\nto the form of matrix \u001aw;1. The only di\u000berence is connected with the di\u000berence of numbers A(p),B(p),\nC(p), determining the matrix elements of the matrix \u001aw;n. It means that the tomogram of state with\ndensity matrix \u001aw;nremains the same, but with following numbers:\nA(p) =1\n2(1\u0000p)n+ (1 + 3p)n\n3(1\u0000p)n+ (1 + 3p)n; (21)\nB(p) =(1\u0000p)n\n3(1\u0000p)n+ (1 + 3p)n; (22)\nC(p) =1\n2(1 + 3p)n\u0000(1\u0000p)n\n3(1\u0000p)n+ (1 + 3p)n: (23)\nHaving the tomogram we can examine Bell inequality [13,14,17] connected with quantum correlations in\nthe system. For directions\u0000 !n\u000band\u0000 !n\f, which we mark by indices \u000band\fthe functionhM\u000b\ficorresponding\nto matrices \u001aw;nreads\nhM\u000b\fi=W(\";\u0000 !n\u000b;\";\u0000 !n\f)\u0000W(\";\u0000 !n\u000b;#;\u0000 !n\f)\u0000W(#;\u0000 !n\u000b;\";\u0000 !n\f) +W(#;\u0000 !n\u000b;#;\u0000 !n\f) = (24)\n= 2C(p)(cos\u0012\u000bcos\u0012\f+ sin\u0012\u000bsin\u0012\fcos ( \u000b+ \f); (25)\nwhere we used di\u000berence between elements of matrix \u001aw;n:C(p) =A(p)\u0000B(p). We obtain the expression\nfor correlation B:\nB=hMabi+hMaci+hMdbi\u0000hMdci= 2C(p)(cos\u0012acos\u0012b+ sin\u0012asin\u0012bcos( a+ b) +\n+ cos\u0012acos\u0012c+ sin\u0012asin\u0012ccos( a+ c) +\n+ cos\u0012dcos\u0012b+ sin\u0012dsin\u0012bcos( d+ b)\u0000\n\u0000cos\u0012dcos\u0012c\u0000sin\u0012dsin\u0012ccos( d+ c)): (26)\nAs it can be seen from (25) the expression hMabi+hMaci+hMdbi\u0000hMdciis the product of two functions:\nf(p) = 2C(p),depending only on parameter p, and the function of angle arguments \u0010(\u0000 !a;\u0000 !b;\u0000 !c;\u0000 !d). Thus\n6Figure 2: Dependence maximum value of hMabi+hMaci+hMdbi\u0000hMdcion parameter p. Graphics\ncorrespond to cases \u001aw;1(upper-left plot), \u001aw;2(upper-right plot) and \u001aw;3(lower plot). The horisontal\nline on all plots shows the Bell limit for separable states.\nthe task of \fnding maximum of hMabi+hMaci+hMdbi\u0000hMdcihas two aspects. This function reaches\nits maximum when both f(p) and function \u0010(\u0000 !a;\u0000 !b;\u0000 !c;\u0000 !d) reach their maxima. As it can be seen from\n(25), the function of angles\n\u0010(\u0000 !a;\u0000 !b;\u0000 !c;\u0000 !d) = cos\u0012acos\u0012b+ sin\u0012asin\u0012bcos( a+ b) + cos\u0012acos\u0012c+ sin\u0012asin\u0012ccos( a+ c) +\n+ cos\u0012dcos\u0012b+ sin\u0012dsin\u0012bcos( d+ b)\u0000cos\u0012dcos\u0012c\u0000sin\u0012dsin\u0012ccos( d+ c)(27)\nremains the same for all powers nand all parameters p. Maximum of function f(p) can be easily\ncalculated. In the domain of values pwhere\u001aw;nis non-negative, max[f(p)] =f(p)jp=1= 1. So for all\npowersnone has inequality:\nmax(hMabi+hMaci+hMdbi\u0000hMdci)\u0014max(\u0010(\u0000 !a;\u0000 !b;\u0000 !c;\u0000 !d)): (28)\nFigure 2 displays the dependence of correlation Bon parameter pfor di\u000berent powers of power\nn= 1;2;3. The correlation Bwas calculated numerically. Taking into account the domain of pwhere\n7the states are separable, one can see that for these states the Bell inequality B\u00142 is satis\fed. Also the\ncorrelation for entangled states meets the bound B\u00142p\n2.\n5 Conclusion\nTo resume we point out the main results of our work. We constructed and studied the nonlinear\npositive maps of Werner state density matrix \u001aw;1!\u001an\nw;1=Tr[\u001an\nw;1] for arbitrary integer n. It was found\nthat for di\u000berent domains of Werner state parameter p, the entanglement in case of both odd and even\nn depends on this domain. We obtained the property of violation of Bell inequality for the nonlinearly\ntransformed states. The Cirelson bound 2p\n2 [11] was checked to be reached for transformed Werner\nstate. The correlation between Bell inequality violation and appearing the entanglement was discussed\nfor all integer n. We have shown that in the Werner state the nonlinear channel (for n= 2) creates the\nentangled state from separable one in correspondence with remark of [14]. For example, the Werner state\n\u001aw;1withp=1\n6is separable, but the state \u001aw;2for this value of phas negativity of entangled state.\nReferences\n[1] E. Schr odinger, Discussion of probability relations between separated states , Proc. Cambridge Philos.\nSoc., 31, 555 (1935).\n[2] R. F. Werner, Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable\nmodel , Phys. Rev. A, 40, 4277 (1989).\n[3] A. Peres, Separability Criterion for Density Matrices , Phys. Rev. Lett., 77, 1413 (1996).\n[4] M. Horodecki, P. Horodecki and R. Horodecki, Separability of Mixed States: Necessary and Su\u000ecient\nConditions , Phys. Lett. A, 223, 1 (1996).\n[5] M. Horodecki, P. Horodecki and R. Horodecki, Separability of mixed states: necessary and su\u000ecient\nconditions , arXiv: quant-ph/9605038 (1996).\n[6] E. S. G. Sudarchan, P. M. Mathews and Jayaseetha Rau, Stochastic Dynamics of Quantum-\nMechanical Systems , Phys. Rev., 121, 920 (1961).\n[7] W. F. Stinespring, Positive functions on C\u0003-algebras , Proc. Amer. Math. Soc., 6, 211-216 (1955).\n[8] K. Kraus, States, E\u000bects and Operations: Fundamental Notions of Quantum Theory , Springer-\nVerlag, Berlin (1983).\n[9] S. A. Ibort, V. I. Man'ko, G. Marmo, A. Simoni and F. Ventiglia, An introduction to tomographic\npicture of quantum mechanics , Phys. Scr., 79(2009).\n[10] V. N. Chernega and O.V. Man'ko, Tomographic and improved subadditivity conditions for two qubit\nand qudit with j= 3=2, arXiv: 1401.6539 [quant-ph] (2014).\n[11] B. S. Cirel'son, Quantum generalizations of Bell's inequality , Lett. Math. Phys., 4, 93 (1980).\n8[12] A. S. Holevo, Statistical Structure of Quantum Theory , Lecture Notes in Physics, Monographs,\nSpringer (2001).\n[13] J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Proposed experiment to test local hidden-\nvariable theories ,Phys. Rev. Lett., 23, 880 (1969).\n[14] V. N. Chernega, O. V. Man'ko, V. I. Man'ko, Subadditivity condition for spin-tomograms and density\nmatrices of arbitrary composite and noncomposite qudit systems , arXiv: 1403.2233 [quant-ph] (2014).\n[15] V. V. Dodonov, V. I. Man'ko, Positive distribution description for spin states , Phys. Lett. A, 229,\n335 (1997).\n[16] V. I. Man'ko and O. V. Man'ko, Spin state tomography , J. Exp. Theor. Phys., 85, 430 1997.\n[17] M. A. Man'ko and V. I. Man'ko, Quantum correlations expressed as information and entropic in-\nequalitues for composite and noncomposite systems , arXiv: 1403.1490 [quant-ph] (2014).\n9"    },    {        "title": "1404.0825v1.Density_Functionals_in_the_Presence_of_Magnetic_Field.pdf",        "content": "arXiv:1404.0825v1  [math-ph]  3 Apr 2014Density Functionals in the Presence of Magnetic Field\nAndre Laestadius∗\nDepartment of Mathematics, KTH, 100 44 Stockholm, Sweden\n(Dated: April 29, 2018)\nAbstract\nIn this paper density functionals for Coulomb systems subje cted to electric and magnetic fields\nare developed. The density functionals depend on the partic le density,ρ, and paramagnetic cur-\nrent density, jp. This approach is motivated by an adapted version of the Vign ale and Rasolt\nformulation of Current Density Functional Theory (CDFT), w hich establishes a one-to-one cor-\nrespondence between the non-degenerate ground-state and t he particle and paramagnetic current\ndensity. Definition of N-representable density pairs ( ρ,jp) is given and it is proven that the set of\nv-representable densities constitutes a proper subset of th e set ofN-representable densities. For\na Levy-Lieb type functional Q(ρ,jp), it is demonstrated that (i) it is a proper extension of the\nuniversal Hohenberg-Kohn functional, FHK(ρ,jp), toN-representable densities, (ii) there exists a\nwavefunction ψ0such thatQ(ρ,jp) = (ψ0,H0ψ0)L2, whereH0is the Hamiltonian without external\npotential terms, and (iii) it is not convex. Furthermore, a c onvex and universal functional F(ρ,jp)\nis studied and proven to be equal the convex envelope of Q(ρ,jp). For both QandF, we give\nupper and lower bounds.\n1I. INTRODUCTION\nThe theoretical foundation of Density Functional Theory (DFT) is the Hohenberg-Kohn\ntheorem [1] that states that the particle density of a quantum mec hanical system determines\nthe scalar potential up to a constant. Arguments have been put f orward that this theorem\ncould be generalized to include systems with magnetic fields [2–4]. The se arguments rely on\neither the paramagneticcurrent density or the totalcurrent de nsity being used together with\nthe particle density to determine the scalar potential and vector p otential of the system.\nNonetheless, for the formulation with paramagnetic current dens ity, counterexamples have\nbeen constructed that exclude the existence of a Hohenberg-Ko hn theorem for such a formu-\nlation (see for instance [5] where this was first demonstrated, or [ 6] for more mathematical\ndetails in the one-electron case). Moreover, the existence of a Ho henberg-Kohn theorem for\nthe formulation with the total current density is still an open quest ion [6, 7], since the proofs\nof [3] and [4] do not hold. (The error in [3] was highlighted in [6] and the e rror in [4] was\npointed out in [7].)\nHowever, in an adapted version of the Vignale and Rasolt formulation of Current Den-\nsity Functional Theory (CDFT), the particle density and the param agnetic current density\ndetermine the non-degenerate ground-state [2, 6]. This allows a Ho henberg-Kohn functional\nto be defined, from which other density functionals can be develope d. Following Lieb’s pro-\ngramme for DFT [8], the issue of establishing a mathematically rigorous CDFT formulated\nwith the paramagnetic current density will here be addressed.\nThe aims of this article are the following:\n(i)Define the set of N-representable particle and paramagnetic current densiti es.The\ndefinition is motivated by Proposition 3, and Proposition 4 shows that this set is convex.\n(ii)ForN-representable particle and paramagnetic current densiti es, study a Levy-Lieb\ntype functional Q(ρ,jp).In Theorem 5, Q(ρ,jp) is proven to be a proper extension of\nthe universal Hohenberg-Kohn functional and, moreover, it is pr oven that there exists a\nminimizer such that Q(ρ,jp) = (ψ0,H0ψ0)L2. Proposition 8 demonstrates that Q(ρ,jp) is\nnot a convex functional, which motivates\n(iii)Investigate a convex and universal particle and paramagnet ic current density func-\ntional,F(ρ,jp).In Theorem 11, it is proven that F(ρ,jp) equals the convex envelope of\n2Q(ρ,jp). Furthermore, in Theorem 13, the minimization of\nF(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)\nis connected with a set of Euler-Lagrange equations.\n(iv)Give upper and lower bounds for particle and paramagnetic cu rrent density function-\nals.Bounds for both Q(ρ,jp) andF(ρ,jp) are found in Theorem 14, Proposition 16 and\nCorollary 17. Proposition 16 and Corollary 17 require that the vortic ity is zero.\nII. PRELIMINARIES\nWe will in this paper consider a system of Ninteracting electrons. The Hamiltonian of\nthe system is given by (in suitable units)\nH(v,A) =N/summationdisplay\nk=1/parenleftbig\n(i∇k−A(xk))2+v(xk)/parenrightbig\n+/summationdisplay\n1≤k<l≤N|xk−xl|−1, (1)\nwherev(x) is the scalar potential and A(x) the vector potential, with components Ak(x)\nk= 1,2,3, such that B(x) =∇×A(x), whereB(x) is the magnetic field. The following will\nbe assumed: (i) there is a lowest eigenvalue e0ofH(v,A) with dimker( e0−H) = 1, (ii) the\nsolution of H(v,A)ψ=e0ψfulfilsψ/ne}ationslash= 0 almost everywhere (a.e.), and (iii) the magnetic\nfield vanishes outside some large sphere ( Bhas compact support) and we may take Ak(x)\nto be bounded. See [6] for further discussion about assumptions ( i) and (ii).\nSome different function spaces will be used in the forthcoming discus sion. A function\nfthat satisfies/integraltext\nRn|f|p<∞, for somep∈[1,∞), is said to belong to the normed space\nLp(Rn) with norm ||f||Lp(Rn)=/parenleftbig/integraltext\nRn|f|p/parenrightbig1/p. ForR>0, letBR={x∈Rn||x| ≤R}. Then\nf∈Lp\nloc(Rn) if for any BRwe have that ||f||Lp(BR)=/parenleftBig/integraltext\nBR|f|p/parenrightBig1/p\n<∞. The normed space\nL∞(Rn) consists of those functions fthat satisfy ||f||L∞(Rn)= esssup {|f||x∈Rn}<∞.\nA function f∈L2(Rn) that satisfies/integraltext\nRn|∇f|2<∞belongs to H1(Rn). Furthermore,\nf∈L2(Rn) belongs to H1\nA(Rn) if/integraltext\nRn|(i∇ −A)f|2<∞for some non-zero A(x). Both\nH1andH1\nAare Hilbert spaces with norms ||f||2\nH1=/integraltext\nRn|f|2+/integraltext\nRn|∇f|2and||f||2\nH1\nA=\n/integraltext\nRn|f|2+/integraltext\nRn|(i∇−A)f|2respectively. Foravector u, ifeachcomponentof u, (u)l,l= 1,2,3,\nbelongs toLpfor somep∈[1,∞], we write u∈(Lp)3.\nFor the proofs set forth in this article, some different notions of co nvergence will be used.\nA sequence {ψk} ⊂Lp(Rn) is said to converge (in Lp-norm) toψ∈Lp(Rn) if and only if\n3/integraltext\nRn|ψk−ψ|p→0, and we write ψk→ψ. Moreover, denote the inner product of a Hilbert\nspaceHby (·,·)H. A sequence {ψk} ⊂His then said to converge weakly to ψ∈Hif and\nonly if (ψk,φ)H→(ψ,φ)Hfor allφ∈H, and we write ψk⇀ ψ. The inner product of\nH1(Rn) is given by ( ψ,φ)H1(Rn)=/integraltext\nRnψφ+/integraltext\nRn∇ψ· ∇φ. In particular, weak convergence\nonH1(Rn) implies weak convergence in the L2(Rn) sense, i.e., ( ψk,φ)L2(Rn)=/integraltext\nRnψkφ→\n/integraltext\nRnψφ= (ψ,φ)L2(Rn).\nAlso note that a function (or functional) f:D→Ris convex on Dif forx1,x2∈Dand\n0≤λ≤1, we have f(λx1+(1−λ)x2)≤λf(x1)+(1−λ)f(x2).\nNow, letψdenote the wavefunction describing the system. For simplicity, spin will not\nbe treated. Henceforth, assume that ψ(x1,...,xN) is antisymmetric in its coordinates xi\nand belongs to\nWN={ψ∈H1(R3N)|||ψ||L2(R3N)= 1}. (2)\nAssumeAk∈L∞(R3),k= 1,2,3, andv∈L3/2(R3)+L∞(R3), and define the ground-state\nenergy\ne0(v,A) = inf{Ev,A(ψ)|ψ∈WN}, (3)\nwhereEv,A(ψ) is a functional on WNgiven by\nEv,A(ψ) =/summationdisplay\nk/parenleftbigg/integraldisplay\nR3N|(i∇k−A(xk))ψ|2+/integraldisplay\nR3N|ψ|2v(xk)/parenrightbigg\n+/summationdisplay\nk<l/integraldisplay\nR3N|ψ|2|xk−xl|−1.(4)\nWe shall interpret the inner-product ( ψ,H(v,A)ψ)L2as the number Ev,A(ψ), which is well-\ndefined for ψ∈WN.\nForψ∈WN, define the particle density and the paramagnetic current density to be,\nrespectively,\nρψ(x) =N/integraldisplay\nR3(N−1)|ψ(x,x2,...,xN)|2dx2...dxN,\njp\nψ(x) =NIm/integraldisplay\nR3(N−1)ψ(x,x2,...,xN)∇xψ(x,x2,...,xN)dx2...dxN. (5)\nLetH(v,A) for the special case v= 0 andA= 0 be denoted H0, that is,\nH0=−N/summationdisplay\nk=1∆k+/summationdisplay\n1≤k<l≤N|xk−xl|−1,\n4and set\n(ψ,H0ψ)L2=/summationdisplay\nk/integraldisplay\nR3N|∇kψ|2dx1...dxN+/summationdisplay\nk<l/integraldisplay\nR3N|ψ|2|xk−xl|−1dx1...dxN,(6)\nforψ∈WN, even though H0ψ /∈L2. The kinetic energy of ψ, denotedT(ψ), and the\nexchange-correlation energy, denoted Exc(ψ), are given by, respectively,\nT(ψ) =N/summationdisplay\nk=1/integraldisplay\nR3N|∇kψ|2dx1...dxN,\nExc(ψ) = (ψ,/summationdisplay\n1≤k<l≤N|xk−xl|−1ψ)L2−1\n2/integraldisplay\nR3/integraldisplay\nR3ρψ(x)ρψ(y)\n|x−y|dxdy.\nNote that (6) can be written as ( ψ,H0ψ)L2=T(ψ)+Exc(ψ)+1\n2/integraltext\nR3/integraltext\nR3ρψ(x)ρψ(y)\n|x−y|dxdy.\nTo put this work into context, the case A= 0 will first be discussed. A particle density ρ\nis said to be v-representable if there exists a Hamiltonian H(v), with ground-state ψ0, such\nthatρ=ρψ0. The Hohenberg-Kohn theorem then states that a v-representable particle\ndensityρdetermines the scalar potential v(x) up to a constant [1]. For such densities, we\ncan define\nFHK(ρ) = (ψρ,H(vρ)ψρ)L2−/integraldisplay\nR3ρvρ= (ψρ,H0ψρ)L2,\nwhereψρis the ground-state of H(vρ) and where vρis determined by ρ(according to the\nHohenberg-Kohn theorem). This scheme, however, suffers from the fact that the functional\nFHK(ρ) is not explicitly computable, and that the set of v-representable particle densities\nis unknown. To remedy this situation, Lieb [8] extended the Hohenbe rg-Kohn functional\nFHK(ρ) toFLL(ρ) forρ∈IN, where\nFLL(ρ) = inf{(ψ,H0ψ)L2|ψ∈WN,ρψ=ρ}, (7)\nand\nIN=/braceleftBig\nρ/vextendsingle/vextendsingle/vextendsingleρ≥0,ρ1/2∈H1(R3),/integraldisplay\nR3ρ=N/bracerightBig\n.\nThe ground-state energy, e0(v), can then be obtained from\ne0(v) = inf/braceleftBig\nFLL(ρ)+/integraldisplay\nR3ρv/vextendsingle/vextendsingle/vextendsingleρ∈IN/bracerightBig\n,\n5which is the so-called Levy-Lieb constrained search formalism [8, 9]. Moreover, Lieb [8] has\nproved that the functional FLL(ρ) is not convex and that the functional\nFc(ρ) = sup/braceleftBig\ne0(v)−/integraldisplay\nR3ρv/vextendsingle/vextendsingle/vextendsinglev∈L3/2(R3)+L∞(R3)/bracerightBig\n,\nwhich is convex, equals the convex envelope of FLL(ρ) onL1(R3)∩L3(R3). Furthermore,\ne0(v) = inf/braceleftBig\nFc(ρ)+/integraldisplay\nR3ρv/vextendsingle/vextendsingle/vextendsingleρ∈L1(R3)∩L3(R3)/bracerightBig\n.\nThis is the programme we now wish to undertake for paramagnetic cu rrent density func-\ntionals.\nIn the remainder of this paper the system Hamiltonian, H(v,A), will also account for a\nmagnetic field B(x)/ne}ationslash= 0. For such a system, both the particle density and the paramagn etic\ncurrent density are needed to describe the system.\nIII. PARAMAGNETIC CURRENT DENSITY FUNCTIONALS\nWe will here begin the pursuit of describing a system of Ninteracting electrons in terms\nof density functionals with both ρandjpas variables. The Vignale and Rasolt formulation\nof CDFT uses the paramagnetic current density jptogether with the particle density ρ. The\nstatement in [2] that the potentials vandAare determined by the density pair ( ρ,jp) can\nbe reformulated to correctly state that ( ρ,jp) determines the non-degenerate ground-state\nwavefunction ψ. This ground-state ψmay be the solution to many different Schr¨ odinger\nequations of the form H(v,A)ψ=e0ψ, wheree0is the lowest eigenvalue and ψassumed\nto be non-degenerate. Hence, the density pair ( ρ,jp) does not necessarily determine the\npotentialsvandA.\nWith this correspondence between a density pair ( ρ,jp) and a non-degenerate ground-\nstateψ, the aim is now to generalize some previous results for particle densit y function-\nals, i.e., functionals that only depend on ρ. This generalization will follow Lieb’s pro-\ngramme for DFT [8], and will constitute of the following: (i) give mathem atical criteria for\nN-representable density pairs ( ρ,jp), (ii) extend a universal Hohenberg-Kohn functional,\ndenotedFHK(ρ,jp), to a Levy-Lieb-type functional, denoted Q(ρ,jp), which has the N-\nrepresentable densities as domain, (iii) study the convex envelope o fQ(ρ,jp), and (iv) give\nupper and lower bounds for both Q(ρ,jp) andF(ρ,jp).\n6A. The Hohenberg-Kohn functional and N-representable densities\nThe starting point is the following theorem:\nTheorem 1 Assume that H(v1,A1)andH(v2,A2)have non-degenerateground-states ψand\nφrespectively. Then ρψ=ρφandjp\nψ=jp\nφimplyψ=const.φ.\nRemarks. (i) For a proof we refer either to [2] or Theorem 9 in [6].\n(ii) Note that Theorem 1 differs from the Hohenberg-Kohn theorem [1] since no claim is\nmade that the densities determine the potentials.\nTheorem 1 will now beapplied. The first issue to address is a Hohenber g-Kohn functional\nthat depends on both the density ρand the paramagnetic current density jp.\nDefinition. A density pair ( ρ,jp) is said to be v-representable if there exists a Hamil-\ntonianH(v,A), with ground-state ψ0, such that ρ=ρψ0andjp=jp\nψ0. This set of densities\nwill be denoted AN, that is,\nAN={(ρ,jp)|ρ=ρψ,jp=jp\nψ,ψis a non-degenerate ground-state of some H(v,A)}.\nFor(ρ,jp)∈AN, letψρ,jpdenote the non-degenerate ground-stateofsome H(v,A), which\nis determined by ( ρ,jp) according to Theorem 1. The Hohenberg-Kohn functional, given b y\nFHK(ρ,jp) = (ψρ,jp,H0ψρ,jp)L2,\nis then well-defined for ( ρ,jp)∈AN. Let the set of those potentials vandAsuch that\nH(v,A) has a non-degenerate ground-state be denoted VN, i.e.,\nVN={(v,A)|H(v,A) has a non-degenerate ground-state }.\nFor (v,A)∈VN, one has\ne0(v,A) = min/braceleftBig\nFHK(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈AN/bracerightBig\n.\nThis is the so-called variational principle of CDFT.\n7Theorem 2 For a given potential pair (v,A)∈VN, the ground state energy functional\nassumes its minimum value for the true ground state densitie s if the admissible densities are\ninAN, i.e.,\ne0(v,A) = min/braceleftBig\nFHK(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈AN/bracerightBig\n.\nProof.Fix (v,A)∈VN. For any ( ρ,jp)∈ANthere exist potentials ˜ vand˜Asuch that\nH(˜v,˜A) has a non-degenerate ground-state ψρ,jp, and one can define\nGv,A(ρ,jp) = (ψρ,jp,H(v,A)ψρ,jp)L2.\nSinceψρ,jpneed not be the ground state of H(v,A), by the variational principle for wave-\nfunctions\nGv,A(ρ,jp)≥e0(v,A).\nFurthermore, by the fact that ( v,A)∈VN, there exists a non-degenerate ground state ψ0\nofH(v,A). Letρ0=ρψ0andjp\n0=jp\nψ0, that is, the corresponding ground state particle and\nparamagnetic current density, which clearly belong to AN. For (ρ0,jp\n0)∈ANthere exists\nψρ0,jp\n0that satisfies ψρ0,jp\n0= const.ψ0, by Theorem 1. Hence\nGv,A(ρ0,jp\n0) = (ψρ0,jp\n0,H(v,A)ψρ0,jp\n0)L2=e0(v,A).\nOne may then conclude that\ne0(v,A) = min{Gv,A(ρ,jp)|(ρ,jp)∈AN}\n= min/braceleftBig\nFHK(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈AN/bracerightBig\n./squaresolid\nThe next step is to define a Levy-Lieb-type functional. This functio nal will be denoted\nQ(ρ,jp) and will depend on density pairs ( ρ,jp) that are said to be N-representable. To\nthat end, first note\nProposition 3 (i) Ifψ∈WN, thenρψ∈IN,jp\nψ∈(L1(R3))3and/integraltext\nR3|jp\nψ|2ρ−1\nψ≤T(ψ), and\n(ii) the functional (ρ,jp)/ma√sto→/integraltext\nR3|jp|2ρ−1<∞is convex.\n8Proof.(i) Letψ∈WN, then by Theorem 1.1 of [8], ρψ∈IN. Furthermore, one has\n/integraldisplay\nR3|jp\nψ|2ρ−1\nψ≤3/summationdisplay\nk=1/integraldisplay\nR3/parenleftbigg\nN2/integraldisplay\nR3(N−1)|ψ|2/integraldisplay\nR3(N−1)|∂kψ|2/parenrightbigg\nρ−1\nψ=N/integraldisplay\nR3N|∇1ψ|2=T(ψ).\nTo see that each component of jp\nψis inL1(R3), note that\n/integraldisplay\nR3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nR3(N−1)Im(ψ∂kψ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay\nR3N|ψ∂kψ| ≤/parenleftbigg/integraldisplay\nR3N|ψ|2/parenrightbigg1/2/parenleftbigg/integraldisplay\nR3N|∂kψ|2/parenrightbigg1/2\n<∞.\nTo prove (ii), set ρ=λρ1+(1−λ)ρ2andjp=λjp\n1+(1−λ)jp\n2, where 0<λ<1. Since\nρ2\nρ1|jp\n1|2+ρ1\nρ2|jp\n2|2≥2jp\n1·jp\n2,\nit follows that\nρ/parenleftbigg\nλ|jp\n1|2\nρ1+(1−λ)|jp\n2|2\nρ2/parenrightbigg\n=λ2|jp\n1|2+λ(1−λ)/parenleftbiggρ2\nρ1|jp\n1|2+ρ1\nρ2|jp\n2|2/parenrightbigg\n+(1−λ)2|jp\n2|2\n≥λ2|jp\n1|2+2λ(1−λ)jp\n1·jp\n2+(1−λ)2|jp\n2|2=|jp|2.\nOne may then conclude\n/integraldisplay\nR3|jp|2ρ−1≤λ/integraldisplay\nR3|jp\n1|2ρ−1\n1+(1−λ)/integraldisplay\nR3|jp\n2|2ρ−1\n2,\nwhich shows the convexity. /squaresolid\nMotivated by Proposition 3, the set of N-representable density pairs ( ρ,jp) is now\ndefined as follows.\nDefinition. A density pair ( ρ,jp) is said to be N-representable if ( ρ,jp)∈YN, where\nYN=/braceleftBig\n(ρ,jp)/vextendsingle/vextendsingle/vextendsingleρ∈IN,jp∈(L1(R3))3,/integraltext\nR3|jp|2ρ−1<∞/bracerightBig\n.\nA convex combination of N-representable densities is also N-representable, but a v-\nrepresentable density need not be N-representable. To summarize\nProposition 4 (i) The set YNis convex, and\n(ii)AN/subsetnoteqlYN.\nRemark. The proof of part (ii) will be given after Proposition 8.\n9Proof of (i). Recall that the set INconsists of those non-negative densities ρthat satisfy\nρ1/2∈H1(R3) and/integraltext\nR3ρ=N. Note that the functional ρ/ma√sto→/integraltext\nR3(∇ρ1/2)2is convex and that\nINis a convex set [8]. Since by Proposition 3 (ii), ( ρ,jp)/ma√sto→/integraltext\nR3|jp|2ρ−1<∞is a convex\nfunctional, it follows that YNis a convex set. /squaresolid\nNote that for ( ρ,jp)∈YN,/integraltext\nR3ρvand/integraltext\nR3jp·Aare finite since v∈L3/2(R3) +L∞(R3)\nandAk∈L∞(R3). However, if a given AhasAk/∈L∞(R3) for some k,/integraltext\nR3jp·Ais still\nfinite if (ρ,jp)∈YA, whereYA={(ρ,jp)∈YN|ρ∈L1(R3,|A|2)}. Note that if\nψ∈˜WN,A={ψ∈ ⊗N\nk=1H1\nA(R3)|ψ∈WN,/integraldisplay\nR3ρψ|A|2<∞},\nthen (ρ,jp)∈YA, and\n/integraldisplay\nR3|(jp)kAk| ≤/parenleftbigg/integraldisplay\nR3|jp|2ρ−1/parenrightbigg1/2/parenleftbigg/integraldisplay\nR3ρ|A|2/parenrightbigg1/2\n<∞. (8)\nThe proof of (8) follows directly from/integraltext\nR3|(jp)kAk|=/integraltext\nR3|(jp)kρ−1/2||ρ1/2Ak|and using\nSchwarz’s inequality.\nB. The Levy-Lieb-type functional Q(ρ,jp)\nWe now turn to finding an extension of the functional FHK(ρ,jp). A Levy-Lieb-type\nfunctional, denoted Q(ρ,jp), will be introduced (cf. [8] and [9]) and proven to satisfy\nQ(ρ,jp) =FHK(ρ,jp) for (ρ,jp)∈AN. The domain of Q(ρ,jp) will consist of those ρand\njpthat are elements of YN.\nDefinition. For (ρ,jp)∈YN, we define a Levy-Lieb-type functional\nQ(ρ,jp) = inf{(ψ,H0ψ)L2|ψ∈WN,ψ/ma√sto→(ρ,jp)},\nwhereψ/ma√sto→(ρ,jp) means that ρψ=ρ,jp\nψ=jp.\nNote that Q(ρ,jp) is the generalization of FLL(ρ), see (7), when also describing the\nsystem with the paramagnetic current density. Theorem 3.3 of [8] states that there exists a\nψ0∈H1(R3N) such that FLL(ρ) = (ψ0,H0ψ0)L2andψ0/ma√sto→ρforρ∈IN. A similar result is\nalso true for the functional Q(ρ,jp). Furthermore, on YN,Q(ρ,jp)≥/integraltext\nR3|jp|2ρ−1. The next\ntheorem summarizes the claims made so far about Q(ρ,jp).\n10Theorem 5 (i) There exists a ψ0such thatQ(ρ,jp) = (ψ0,H0ψ0)L2andψ0/ma√sto→(ρ,jp),\n(ii)Q(ρ,jp)is the proper extension of FHK(ρ,jp)fromANtoYNin the sense that for\n(ρ,jp)∈AN,Q(ρ,jp) =FHK(ρ,jp), and\n(iii)/integraltext\nR3|jp|2ρ−1≤Q(ρ,jp)onYN.\nProof.(i) Let{ψk}∞\nk=1be a minimizing sequence, that is lim k(ψk,H0ψk)L2=Q(ρ,jp) and\nψk/ma√sto→(ρ,jp) for allk. From [8] (Theorem 3.3), ψk⇀ ψ0inH1(R3N) andψk→ψ0in\nL2(R3N) for some H1-functionψ0(after passing to a subsequence, which we for simplicity\ncontinue to denote ψk). Then by Theorem 1.3 and Theorem 3.3 in [8], ρψ0=ρ. Since taking\nweak limits, one has lim k(ψk,H0ψk)L2≥(ψ0,H0ψ0)L2. It remains to show that ψ0/ma√sto→jpa.e.\nLetg(x) =χM(x) be the characteristic function of any (measurable) set M⊂R3and let\n(u)ldenote thel:th component of the vector u. Now, using the weak convergence of {ψk}∞\nk=1\ninH1(R3N) and the norm-convergence in L2(R3N), we have for l= 1,2,3,\nlim\nk→∞/integraldisplay\nR3(jp\nψk)lg= lim\nk→∞NIm/integraldisplay\nR3/integraldisplay\nR3(N−1)ψk(∂lψk)g\n= lim\nk→∞NIm/parenleftbigg/integraldisplay\nR3N(ψk−ψ0)(∂lψk)g+/integraldisplay\nR3Nψ0(∂lψk)g/parenrightbigg\n=/integraldisplay\nR3(jp\nψ0)lg.\nThis givesjp\nψ0(x) =jp\nψk(x) =jp(x) a.e.\n(ii) Fix (ρ,jp)∈ANand letψ0be as in part (i). The claim in (ii) will be shown\nby demonstrating that ( ψ0,H0ψ0)L2= (ψρ,jp,H0ψρ,jp)L2, whereψρ,jpsatisfiesFHK(ρ,jp) =\n(ψρ,jp,H0ψρ,jp)L2. Now, since ψρ,jp∈WNandψρ,jp/ma√sto→(ρ,jp),\n(ψ0,H0ψ0)L2=Q(ρ,jp)≤(ψρ,jp,H0ψρ,jp)L2.\nOn the other hand, since ψρ,jpis the ground state of some Hamiltonian H(v,A),\ne0(v,A) = (ψρ,jp,H(v,A)ψρ,jp)L2= (ψρ,jp,H0ψρ,jp)L2+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)\n≤(ψ0,H(v,A)ψ0)L2= (ψ0,H0ψ0)L2+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2),\nand (ψ0,H0ψ0)L2≥(ψρ,jp,H0ψρ,jp)L2.\n(iii) Fix (ρ,jp)∈YN. Letψ∈WNsuch thatρψ=ρandjp\nψ=jp. By Proposition 3 (i),\n/integraltext\nR3|jp|2ρ−1≤T(ψ). We then have\n/integraldisplay\nR3|jp|2ρ−1≤inf/braceleftBig\nT(ψ)/vextendsingle/vextendsingle/vextendsingleψ∈WN,ψ/ma√sto→(ρ,jp)/bracerightBig\n≤inf{(ψ,H0ψ)L2|ψ∈WN,ψ/ma√sto→(ρ,jp)}=Q(ρ,jp)./squaresolid\n11The situation is now as follows. The set of v-representable density pairs, AN, is a proper\nsubset of the N-representable density pairs, YN. The Hohenberg-Kohn functional FHK,\ndefined onAN, hasbeenextended totheLevy-Lieb-type functional Q(ρ,jp), which isdefined\nonYN. Combining the variational principle of CDFT with Theorem 5, one has f or (v,A)∈\nVN,\ne0(v,A) = min/braceleftBig\nQ(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈AN/bracerightBig\n.\nThe admissible set ANover which the minimization is performed can be exchanged by YN\nif the minimum is replaced by infimum. Note that Q(ρ,jp) + 2/integraltext\nR3jp·A+/integraltext\nR3ρ(v+|A|2)\nremains finite since we require v∈L3/2(R3)+L∞(R3) andAk∈L∞(R3).\nTheorem 6 Forv∈L3/2(R3)+L∞(R3)andAk∈L∞(R3),\ne0(v,A) = inf/braceleftBig\nQ(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈YN/bracerightBig\n.\nProof.Fix (ρ,jp)∈YN. Then\ne0(v,A) = inf/braceleftBig\n(ψ,H(v,A)ψ)L2/vextendsingle/vextendsingle/vextendsingleψ∈WN/bracerightBig\n≤inf/braceleftBig\n(ψ,H(v,A)ψ)L2/vextendsingle/vextendsingle/vextendsingleψ∈WN,ρψ=ρ,jp\nψ=jp/bracerightBig\n=Q(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2).\nSince (ρ,jp)∈YNwas arbitrary, we have\ne0(v,A)≤inf/braceleftBig\nQ(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈YN/bracerightBig\n.\nFor the reverse inequality, let {ψk}∞\nk=1⊂WNbe a minimizing sequence for e0(v,A), i.e.,\ne0(v,A)+1\nk>(ψk,H(v,A)ψk)L2. Putρk=ρψkandjp\nk=jp\nψk, then\ne0(v,A)+1\nk>(ψk,H0ψk)L2+2/integraldisplay\nR3jp\nk·A+/integraldisplay\nR3ρk(v+|A|2)\n≥Q(ρk,jp\nk)+2/integraldisplay\nR3jp\nk·A+/integraldisplay\nR3ρk(v+|A|2)\n≥inf/braceleftBig\nQ(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈YN/bracerightBig\n./squaresolid\n12The admissible set over which the minimization is performed can be exte nded even fur-\nther. First a definition.\nDefinition. X={(ρ,jp)|ρ∈L1(R3)∩L3(R3),jp∈(L1(R3))3}.\nNext, for (ρ,jp)∈X, define a functional ˜Q(ρ,jp) given by\n˜Q(ρ,jp) =Q(ρ,jp) if (ρ,jp)∈YN,\n=∞,otherwise.\nThe energy e0(v,A) can be computed using ˜QonX. This is implied by the following\nargument. Let\n˜e(v,A) = inf/braceleftbigg\n˜Q(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈X/bracerightbigg\n.\nOne directly has ˜ e(v,A)≤e0(v,A), since˜Q(ρ,jp) =Q(ρ,jp) if (ρ,jp)∈YN. On the other\nhand, since ˜Q(ρ,jp) =∞if (ρ,jp)/∈YN, we have ˜e(v,A) =e0(v,A). Thus\nTheorem 7 Forv∈L3/2(R3)+L∞(R3)andAk∈L∞(R3),\ne0(v,A) = inf/braceleftbigg\n˜Q(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈X/bracerightbigg\n.\nC. Convex envelope of Q(ρ,jp)\nSo far the variational principle of CDFT has been replaced by the follo wing optimization\nproblem\ne0(v,A) = inf/braceleftbigg\nQ(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ/parenleftbig\nv+|A|2/parenrightbig/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈YN/bracerightbigg\n.\nNote thatQ(ρ,jp) could be exchanged by ˜Q(ρ,jp) and the admissible set YNextended to X.\nHowever, just as Lieb has demonstrated that FLL(ρ) is not convex (Theorem 3.4 in [8]), the\nsame is also true about Q(ρ,jp). A proof of this fact as well as a proof of (ii) in Proposition\n4 now follows.\n13Proposition 8 Q(ρ,jp)is not a convex functional.\nProof of Proposition 8 and Proposition 4 (ii). Choosev(x) as in the proof of Theorem\n3.4 in [8] such that it has M= 2L+ 1 ground-states ψk. Setρk=ρψkandjp\nk=jp\nψkfor\nk= 1,2,...,Mand note that for all k,\ne0(v,0) =Q(ρk,jp\nk)+/integraldisplay\nR3ρkv. (9)\nLet ˜ρ=1\nM/summationtextM\nk=1ρkand˜jp=1\nM/summationtextM\nk=1jp\nk. By definition, FLL(˜ρ)≤Q(˜ρ,˜jp). One has\ne0(v,0)<FLL(˜ρ)+/integraldisplay\nR3˜ρv≤Q(˜ρ,˜jp)+/integraldisplay\nR3˜ρv,\nwhere the first strict inequality follows by Theorem 3.4 of [8] (˜ ρcannot be a ground-state\ndensity of this v(x)). Using (9), we obtain\n1\nMM/summationdisplay\nk=1Q(ρk,jp\nk)<Q(˜ρ,˜jp),\nwhich shows that Q(ρ,jp) is not convex.\nFor the proof of part (ii) in Proposition 4, assume that ˜ ρand˜jpare the ground-state\ndensities of some other potential pair (˜ v,˜A), then\ne0(˜v,˜A) =Q(˜ρ,˜jp)+2/integraldisplay\nR3˜jp·˜A+/integraldisplay\nR3˜ρ(˜v+|˜A|2)\n>1\nMM/summationdisplay\nk=1/parenleftbigg\nQ(ρk,jp\nk)+2/integraldisplay\nR3jp\nk·˜A+/integraldisplay\nR3ρk(˜v+|˜A|2)/parenrightbigg\n.\nThis gives that for at least one k,\ne0(˜v,˜A)>Q(ρk,jp\nk)+2/integraldisplay\nR3jp\nk·˜A+/integraldisplay\nR3ρk(˜v+|˜A|2).\nBut this is a contradiction and hence AN/subsetnoteqlYN./squaresolid\nThe next step will be to obtain a convex and universal density funct ional, denoted\nF(ρ,jp). For that purpose the Legendre transform will be used. The fun ctionalF(ρ,jp)\nwill be defined on the whole space X. (Recall that Xis the space of those ( ρ,jp) such that\nρ∈L1(R3)∩L3(R3) andjp∈(L1(R3))3.)\nDefinition. The convex functional F(ρ,jp), defined on X, is given by\nF(ρ,jp) = sup/braceleftBig\ne0(v,A)−2/integraldisplay\nR3jp·A−/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsinglev∈L3/2+L∞,Ak∈L∞/bracerightBig\n.\n14Remarks. (i) SinceFis the supremum over vandAof linear functionals in ρandjp, it is\nconvex.\n(ii) Furthermore, from the fact that e0(v,A)−2/integraltext\nR3jp·A−/integraltext\nR3ρ(v+|A|2)≤Q(ρ,jp) for\n(ρ,jp)∈YN, it follows that F(ρ,jp)≤Q(ρ,jp) for all (ρ,jp)∈YN.\nThe functional F(ρ,jp) can be used to compute the ground-state energy, which follows\nfrom a direct generalization of Lieb’s proof for the functional\nsup/braceleftBig\ne0(v,0)−/integraldisplay\nR3ρv/vextendsingle/vextendsingle/vextendsinglev∈L3/2+L∞/bracerightBig\n.\nOne may minimize F(ρ,jp)+2/integraltext\nR3jp·A+/integraltext\nR3ρ(v+|A|2) on either YNorX.\nTheorem 9\ne0(v,A) = inf/braceleftBig\nF(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈X/bracerightBig\n= inf/braceleftBig\nF(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈YN/bracerightBig\n.\nProof.Denote the first expression of e0(v,A) asM−(v,A) and the second one as M+(v,A).\nNote thatM−(v,A)≤M+(v,A). Now, fix v0∈L3/2+L∞andAk\n0∈L∞. By the definition\nofF(ρ,jp), we have for ( ρ,jp)∈X,\nF(ρ,jp)≥e0(v0,A0)−2/integraldisplay\nR3jp·A0−/integraldisplay\nR3ρ(v0+|A0|2) =F0(ρ,jp),\nwhere the last equality is a definition. Thus\nM−(v0,A0)≥inf/braceleftBig\nF0(ρ,jp)+2/integraldisplay\nR3jp·A0+/integraldisplay\nR3ρ(v0+|A0|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈X/bracerightBig\n=e0(v0,A0).\nBut sincev0∈L3/2+L∞andAk\n0∈L∞was arbitrary, we obtain M−(v,A)≥e0(v,A).\nOn the other hand, for ( ρ,jp)∈YNwe have that F(ρ,jp)≤Q(ρ,jp), and consequently\nM+(v,A)≤inf/braceleftBig\nQ(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ/parenleftbig\nv+|A|2/parenrightbig/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈YN/bracerightBig\n=e0(v,A).\nThuse0(v,A)≤M−(v,A)≤M+(v,A)≤e0(v,A)./squaresolid\nAs the reader may recall, Q(ρ,jp) =FHK(ρ,jp) onAN. In fact,F(ρ,jp) =Q(ρ,jp) =\nFHK(ρ,jp) onAN, since\n15Proposition 10 If(ρ,jp)∈AN,F(ρ,jp) =Q(ρ,jp).\nProof.Assume (ρ,jp)∈AN, thenFHK(ρ,jp) =Q(ρ,jp). For some vandA,\ne0(v,A) =Q(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2).\nConversely, using Theorem 9,\ne0(v,A)≤F(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2).\nThusF(ρ,jp)≥Q(ρ,jp). However, since the reverse inequality also holds, F(ρ,jp) =\nQ(ρ,jp)./squaresolid\nThe main result of this section is the following:\nTheorem 11 Fis the convex envelope of Q.\nBefore proving Theorem 11, some preparation is required. First de fine\nDefinitions. (i) A functional, f, is weakly lower semi continuous (weakly l.s.c.) if\nf(φ)≤liminf\nk→∞f(φk)\nwhen{φk}converges weakly to φ.\n(ii) LetZbe a normed space and let f:D→R, whereD⊂Z, and set\nΛf,D={g|gis weakly l.s.c. and convex, and g(φ)≤f(φ) for allφ∈D}.\nThe convex envelope on Zof the functional fis then defined to be\nCEf(φ) = sup{g(φ)|g∈Λf,D}.\n(iii) IfZis a normed space we let Z∗denote the dual space of Z, which is the space of all\nbounded linear functionals on Z. Moreover, the dual pairing between an element z∈Zand\nz∗∈Z∗will be denoted /an}b∇acketle{tz,z∗/an}b∇acket∇i}htZ,Z∗. IfZ=Lp(R3), thenZ∗=Lq(R3) with 1/p+1/q= 1,\nand/an}b∇acketle{tz,z∗/an}b∇acket∇i}htZ,Z∗=/integraltext\nR3z(x)z∗(x)dx. In particular,\nX∗={(v′,A′)|v′∈L3/2(R3)+L∞(R3),A′∈(L∞(R3))3}.\n16Proposition 12 Fis weakly lower semi continuous.\nProof.The proof of this fact is standard, but is included for the sake of co mpleteness. Since\nFis convex, it suffices to show that Fis lower semi continuous in norm. For any λ∈R,\ndefine the set\nKλ={(ρ,jp)|F(ρ,jp)≤λ}\n=/braceleftBig\n(ρ,jp)/vextendsingle/vextendsingle/vextendsinglee0(v,A)−2/integraldisplay\nR3jp·A−/integraldisplay\nR3ρ/parenleftbig\nv+|A|2/parenrightbig\n≤λ,∀(v,A)∈X∗/bracerightBig\n.\nNow, assume that {ρn,jp\nn} ⊂Kλand thatρn→ρinL1- andL3-norm and that each\ncomponent of jp\nnconverges to the respective component of jpinL1-norm. Then for each\nv∈L3/2+L∞andAk∈L∞,\nλ≥lim\nn/parenleftbigg\ne0(v,A)−2/integraldisplay\nR3jp\nn·A−/integraldisplay\nR3ρn/parenleftbig\nv+|A|2/parenrightbig/parenrightbigg\n=e0(v,A)−2/integraldisplay\nR3jp·A−/integraldisplay\nR3ρ/parenleftbig\nv+|A|2/parenrightbig\n,\nwhich follows from the fact that norm convergence implies weak conv ergence. This shows\nthatKλis norm closed and that Fis norm l.s.c. /squaresolid\nNow, letfbe a convex functional defined on a convex subset D⊂Zof a normed space\nZ. Then the Legendre transform of f, denotedf∗, defined on the set\nD∗={z∗∈Z∗|sup\nz∈D{/an}b∇acketle{tz,z∗/an}b∇acket∇i}htZ,Z∗−f(z)}<∞},\nis given by\nf∗(z∗) = sup{/an}b∇acketle{tz,z∗/an}b∇acket∇i}htZ,Z∗−f(z)|z∈D}.\nProof of Theorem 11. Letf= CEQandD=YN. Note that −CEQ≤0, since 0 ∈ΛQ,YN.\nThen, for (v′,A′)∈X∗,\nf∗(v′,A′) = sup/braceleftbigg/integraldisplay\nR3ρv′+/integraldisplay\nR3jp·A′−CEQ(ρ,jp)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈YN/bracerightbigg\n.\nBy the definition of the convex envelope, it follows that CE Q(ρ,jp)≤Q(ρ,jp) onYN. This\ngives\nf∗(v′,A′) = sup/braceleftbigg\n−CEQ(ρ,jp)+/integraldisplay\nR3ρv′+/integraldisplay\nR3jp·A′/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈YN/bracerightbigg\n≥ −inf/braceleftbigg\nQ(ρ,jp)−/integraldisplay\nR3ρv′−/integraldisplay\nR3jp·A′/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈YN/bracerightbigg\n=−e0(−v′−|A′/2|2,−A′/2).\n17By taking the Legendre transform one more time, one obtains for ( ρ.jp)∈X\n(f∗)∗(ρ,jp) = sup/braceleftbigg/integraldisplay\nR3ρv′+/integraldisplay\nR3jp·A′−f∗(v′,A′)/vextendsingle/vextendsingle/vextendsingle(v′,A′)∈X∗/bracerightbigg\n≤sup/braceleftbigg\ne0(−v′−|A′/2|2,−A′/2)+/integraldisplay\nR3ρv′+/integraldisplay\nR3jp·A′/vextendsingle/vextendsingle/vextendsingle(v′,A′)∈X∗/bracerightbigg\n= sup/braceleftbigg\ne0(v,A)−2/integraldisplay\nR3jp·A−/integraldisplay\nR3ρ/parenleftbig\nv+|A|2/parenrightbig/vextendsingle/vextendsingle/vextendsingle(v,A)∈X∗/bracerightbigg\n=F(ρ,jp),\nwherev=−v′−|A′/2|2∈L3/2(R3)+L∞(R3) andA=−A′/2∈(L∞(R3))3. We may then\nconclude that, for ( ρ,jp)∈X,\n(f∗)∗(ρ,jp)≤F(ρ,jp).\nNow, from an infinite dimensional extension of Fenchel’s theorem it fo llows that if the\noriginal functional is convex and weakly lower semi continuous, the n the double Legendre\ntransform of the functional equals the functional itself [8]. Thus forf= CEQwe obtain\nCEQ(ρ,jp) =f(ρ,jp) = (f∗)∗(ρ,jp)≤F(ρ,jp).\nConversely, since Fis convex, weakly lower semi continuous and is bounded above by Q,\ni.e.F∈ΛQ,YN, we have that\nF(ρ,jp)≤sup{f(ρ,jp)|f∈ΛQ,YN}= CEQ(ρ,jp).\nIt then follows that for all ( ρ,jp)∈X, CEQ(ρ,jp) =F(ρ,jp)./squaresolid\nSinceF(ρ,jp) is convex, one may seek to obtain a connection between a set of Eu ler-\nLagrange equations and the minimization of F(ρ,j) +2/integraltext\nR3jp·A+/integraltext\nR3ρ(v+|A|2) onYN.\nLetZbe a normed space and fa real-valued functional on Z,f:Z→R. Iffis convex on\nZ, givenz0∈Z, there exists z∗∈Z∗, not necessarily unique, such that\nf(z)≥f(z0)+/an}b∇acketle{tz−z0,z∗/an}b∇acket∇i}htZ,Z∗\nholds for all z∈Z. We now introduce the concept of Fr´ echet differentiability and Fr´ echet\nderivative. Let f:Z→Rbe defined on an open domain Df⊂Z. If, for a fixed z∈Z\nand for each h∈Z, there exists δf(z;h)∈Rthat is linear and continuous with respect to\nhsuch that\nlim\n||h||Z→0|f(z+h)−f(z)−δf(z;h)|\n||h||Z= 0,\n18thenfissaidtobeFr´ echetdifferentiableat zandδf(z;h)issaidtobetheFr´ echetdifferential\noffatzwith increment h. The Fr´ echet differential is unique, and if it exists then\nlim\nα→0f(z+αh)−f(z)\nα\nexists and equals the Fr´ echet differential. We write δf(z;h) =/an}b∇acketle{th,f′(z)/an}b∇acket∇i}htZ,Z∗and callf′the\nFr´ echet derivative of f. Note that if f′(z0) exists, we have for all z\nf(z)≥f(z0)+/an}b∇acketle{tz−z0,z∗/an}b∇acket∇i}htZ,Z∗,\nwherez∗=f′(z0) is unique. For a functional f(z1,z2),f′\nzkwill be used to denote partial\nderivative. We are now ready to formulate and prove\nTheorem 13 Assume that F′\nρ(ρ0,jp\n0)andF′\njp(ρ0,jp\n0)exist and/integraltext\nR3ρ0=Nand that\nF′\nρ(ρ0,jp\n0)+v+|A|2+µ0= 0,\nF′\njp(ρ0,jp\n0)+2A= 0,\na.e. for some µ0∈R,v∈L3/2(R3)+L∞(R3)andA∈(L∞(R3))3. Then(ρ0,jp\n0)minimizes\ninf/braceleftBig\nF(ρ,jp)+2/integraldisplay\nR3jp·A+/integraldisplay\nR3ρ(v+|A|2)/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈YN/bracerightBig\n.\nIf in addition, F(ρ0,jp\n0) =Q(ρ0,jp\n0), then(ρ0,jp\n0)∈AN.\nProof.Setw1=−F′\nρ(ρ0,jp\n0)∈L3/2(R3) +L∞(R3) andw2=−F′\njp(ρ0,jp\n0)∈(L∞(R3))3.\nSinceF′exists at (ρ0,jp\n0) andFis convex, we have for ( ρ,jp)∈YN,\nF(ρ,jp)≥F(ρ0,jp\n0)+/integraldisplay\nR3(jp\n0−jp)·w2+/integraldisplay\nR3(ρ0−ρ)w1.\nBy assumption, w1=v+|A|2+µ0andw2= 2Aa.e. Since/integraltext\nR3ρ0=N,\nF(ρ,jp)≥F(ρ0,jp\n0)+2/integraldisplay\nR3(jp\n0−jp)·A+/integraldisplay\nR3(ρ0−ρ)(v+|A|2)+µ0(N−/integraldisplay\nR3ρ).\nHowever, for any ( ρ,jp)∈YN,/integraltext\nR3ρ=N, and hence the conclusion follows.\nFor the second part, assume F(ρ0,jp\n0) =Q(ρ0,jp\n0). UsingQ(ρ,jp)≥F(ρ,jp), we obtain\nQ(ρ,jp)≥F(ρ,jp)≥F(ρ0,jp\n0)−/integraldisplay\nR3(ρ−ρ0)w1−/integraldisplay\nR3(jp−jp\n0)·w2\n=Q(ρ0,jp\n0)−/integraldisplay\nR3(ρ−ρ0)w1−/integraldisplay\nR3(jp−jp\n0)·w2.\n19Ifwedefine ˜Q(ρ,jp) =Q(ρ0,jp\n0)−/integraltext\nR3(ρ−ρ0)w1−/integraltext\nR3(jp−jp\n0)·w2, wehave ˜Q(ρ,jp)≤Q(ρ,jp).\nNow,\nQ(ρ0,jp\n0)+/integraldisplay\nR3jp\n0·w2+/integraldisplay\nR3ρ0w1= inf/braceleftBig\n˜Q(ρ,jp)+/integraldisplay\nR3jp·w2+/integraldisplay\nR3ρw1/vextendsingle/vextendsingle/vextendsingle(ρ,jp)∈YN/bracerightBig\n≤e0/parenleftBig\nw1−|w2|\n42\n,w2\n2/parenrightBig\n≤Q(ρ0,jp\n0)+/integraldisplay\nR3jp\n0·w2+/integraldisplay\nR3ρ0w1.\nBy settingw1=v+|A|2∈L3/2(R3)+L∞(R3) andw2= 2A∈(L∞(R3))3, it follows that\ne0(v,A) =Q(ρ0,jp\n0)+2/integraldisplay\njp\n0·A+/integraldisplay\nρ0(v+|A|2).\nFromTheorem5(i), weknowthatthereexistsa ψ0∈WNsuchthatQ(ρ0,jp\n0) = (ψ0,H0ψ0)L2\nandψ0/ma√sto→(ρ0,jp\n0). But then\ne0(v,A) = (ψ0,H0ψ0)L2−2/integraldisplay\nR3jp\n0·+/integraldisplay\nR3ρ0(v+|A|2) = (ψ0,H(v,A)ψ0)L2,\nwhich shows that ( ρ0,jp\n0)∈AN./squaresolid\nThe last order of business in this section will be to obtain a lower bound forFonX.\nThe motivation is the following. From Theorem 3.8 in [8], we have\nFc(ρ) = CEFLL(ρ)≥/integraldisplay\nR3(∇ρ1/2)2,ifρ∈IN,\n≥ ∞,otherwise.\nWe shall now take convex combinations of the two convex functiona lsρ/ma√sto→/integraltext\nR3(∇ρ1/2)2and\n(ρ,jp)/ma√sto→N2/integraltext\nR3|jp|2ρ−1and use Theorem 11 to obtain\nTheorem 14 Define for (ρ,jp)∈Xand0≤λ≤1\nJλ(ρ,jp) =λ/integraldisplay\nR3(∇ρ(x)1/2)2+(1−λ)/integraldisplay\nR3|jp(x)|2ρ(x)−1,if(ρ,jp)∈YN,\n=∞,otherwise.\nThenJλ(ρ,jp)≤F(ρ,jp)for(ρ,jp)∈Xand0≤λ≤1.\nProof.From [8] we have that ρ/ma√sto→/integraltext\nR3(∇ρ1/2)2is convex and bounded above by T(ψ) for\nρψ=ρ,ψ∈WN. From Proposition 3 and Theorem 5, we can then conclude that Jλis\nconvex and Jλ≤QonYN. We now want to show that Jλis weakly l.s.c. (since Jλis convex\nwe will show that it is norm-l.s.c) so we can conclude that Jλ≤CEQ=F.\n20Letρn→ρinL1(R3)∩L3(R3)-norm and jp\nn→jpin (L1(R3))3-norm. We want to show\nthatCλ= liminf n→∞Jλ(ρn,jp\nn)≥Jλ(ρ,jp). IfCλ=∞we are done, so we will assume that\nCλ<∞.\nNote that the case λ= 1 follows from Theorem 3.8 in [8]. It then suffices to show the\nresult forλ= 0. By the same argument as in the proof of Theorem 3.8 in [8], assum eρ∈IN.\n(Ifρ<0 on a set of positive measure, then ρn<0 andJ0(ρn,jp\nn) =∞for sufficiently large\nn. Similarly we have that/integraltext\nR3ρ/ne}ationslash=NgivesJ0(ρn,jp\nn) =∞for sufficiently large n.) Since\nC0<∞, (ρn,jp\nn)∈YN. Setgn=jp\nn/ρ1/2\nn. Then{gn}(or at least a subsequence of {gn}) is\nbounded in L2(R3)3, and by the Banach-Alaoglu theorem there exists a g∈L2(R3)3and a\nsubsequence {gnk}such thatgnk⇀ginL2(R3)3.\nThe next step is to show that g=jp/ρ1/2a.e. First note since ρnk→ρ≥0 inL1(R3)-\nnorm, there exists a subsequence, which we continue to denote {ρnk}, and a non-negative\nF∈L1(R3) such that ρnk(x)≤F(x) andρnk(x)→ρ(x) a.e. From\n|ρnk(x)1/2−ρ(x)1/2|2≤2(ρnk(x)+ρ(x))≤2(F(x)+ρ(x)),\nwe have by dominated convergence, ρ1/2\nnk→ρ1/2inL2(R3). Let (u)ldenote the l:th compo-\nnent of the vector u. It then follows that gnkρ1/2→jpinL1(R3)3, since forl= 1,2,3,\n/integraldisplay\nR3|(gnk)lρ1/2−(jp)l| ≤/integraldisplay\nR3|(gnk)lρ1/2−(gnk)lρ1/2\nnk|+/integraldisplay\nR3|(gnk)lρ1/2\nnk−(jp)l|\n≤/parenleftbigg/integraldisplay\nR3|(gnk)l|2/parenrightbigg1/2/parenleftbigg/integraldisplay\nR3|ρ1/2\nnk−ρ1/2|2/parenrightbigg1/2\n+/integraldisplay\nR3|(jp\nnk)l−(jp)l| →0,\nask→0, where we used that ρ1/2\nnk→ρ1/2inL2(R3) and (jp\nnk)l→(jp)linL1(R3) for\nl= 1,2,3.\nNow, letM⊂R3be an arbitrary measurable set. Since ρ1/2χM∈L2(R3) and by the\nweak convergence of gnktoginL2(R3)3, one obtains\nlim\nk→∞/integraldisplay\nR3(gnk)lρ1/2χM=/integraldisplay\nM(g)lρ1/2.\nOn the other hand, since χM∈L∞(R3) and norm-convergence implies weak-convergence,\nlimk→∞/integraltext\nR3(gnk)lρ1/2χM=/integraltext\nM(jp)l. Then/integraltext\nM(g)lρ1/2=/integraltext\nM(jp)l, which gives g=jp/ρ1/2a.e.\nLastly, by the w.l.s.c. of the L2(R3)-norm,\nC0= liminf\nk→∞/integraldisplay\nR3|jp\nnk(x)|2ρnk(x)−1= liminf\nk→∞||gnk||2\nL2(R3)\n≥ ||g||2\nL2(R3)=/integraldisplay\nR3|jp(x)|2ρ(x)−1=J0(ρ,jp)./squaresolid\n21D. Upper and lower bounds for densities with vanishing vorticity\nThelastissuetobeaddressediswhenthedensity pair( ρ,jp)isrestrictedtotheconstraint\n∇×(jp/ρ) = 0. Thequantity ∇×(jp/ρ)iscalledthevorticity. Weshallbeginbyconstructing\na determinantal wavefunction that yields a prescribed density pair (ρ,jp), i.e., finding a\nfunction in WNthat is a determinant and that reproduces a given density pair ( ρ,jp)∈YN.\nThis can be achieved by a straightforward generalization of Theore m 1.2 of [8] (see also [10]\nwhere a determinantal construction is considered without the con straint∇×(jp/ρ) = 0 for\nN/ne}ationslash= 3 but without an explicit upper bound for the kinetic energy). Defin e, as in ref. [8], a\nfunction on the real line given by\nf(x1) =2π\nN/integraldisplayx1\n−∞/integraldisplay∞\n−∞/integraldisplay∞\n−∞ρ(s,x2,x3)dsdx2dx3.\nNote thatf(−∞) = 0,f(∞) = 2πand\ndf\ndx1=2π\nN/integraldisplay\nR2ρ(x1,x2,x3)dx2dx3.\nTo obtain a determinant ψDthat yields a given density pair ( ρ,jp)∈YN, put\nψD(x1,...,xN) = (N!)−1/2det[φk(xl)]k,l,\nwhere, fork= 0,1,...,N−1,\nφk(x) =/parenleftbiggρ(x)\nN/parenrightbigg1/2\nei(kf(x1)−M(x1)+S(x)). (10)\nNote that ( φk,φl)L2=δkl. It is immediate that ρψD=/summationtextN−1\nk=0|φk(x)|2=ρ. Moreover, from\nthe calculation\nIm(φk∇φk) =ρ\nN/parenleftbigg/parenleftbigg\nkdf\ndx1−dM\ndx1/parenrightbigg\nˆex+∇S/parenrightbigg\n,\nit follows that\njp\nψD=N−1/summationdisplay\nk=0Im(φk∇φk) =ρ∇S+/parenleftBigg\nρ\nNdf\ndx1N−1/summationdisplay\nk=0k−ρdM\ndx1/parenrightBigg\nˆex\n=ρ∇S+ρ/parenleftbigg1\n2(N−1)df\ndx1−dM\ndx1/parenrightbigg\nˆex. (11)\n22Proposition 15 Given(ρ,jp)∈YNthat fulfils ∇×(jp/ρ) = 0, there exists a determinant\nψD∈L2(R3N)such that ||ψD||L2= 1and\nT(ψD)≤/parenleftbigg\n1+(4π)2(N2−1)\n12/parenrightbigg/integraldisplay\nR3(∇ρ1/2)2+/integraldisplay\nR3|jp|2ρ−1<∞. (12)\nProof.TakeψD= (N!)−1/2det[φk(xl)]k,l, withφkas in (10) for k= 0,1,...,N−1. From\n(11),jp\nψD=jpifSandMare chosen such that ∇S=jp/ρand\nM(x1) =f(x1)\nNN−1/summationdisplay\nk=0k=1\n2(N−1)f(x1).\nWe are done if we can show (12). To that end, note that\n|∇φk|2=1\nN/parenleftBigg\n(∇ρ1/2)2+ρ/parenleftbigg/parenleftbigg\nkdf\ndx1−dM\ndx1/parenrightbigg\nˆex+∇S/parenrightbigg2/parenrightBigg\n.\nThe kinetic energy of ψDsatisfies\nT(ψD) =N−1/summationdisplay\nk=0/integraldisplay\nR3|∇φk|2\n=/integraldisplay\nR3(∇ρ1/2)2+/parenleftBigg\n1\nNN−1/summationdisplay\nk=0k2−(N−1)\n42/parenrightBigg/integraldisplay\nR3ρ/parenleftbiggdf\ndx1/parenrightbigg2\n+/integraldisplay\nR3ρ|∇S|2\n=/integraldisplay\nR3(∇ρ1/2)2+(N2−1)\n12/integraldisplay\nR3ρ/parenleftbiggdf\ndx1/parenrightbigg2\n+/integraldisplay\nR3|jp|2ρ−1. (13)\nFor the second term in the r.h.s. of (13), note that\n/integraldisplay\nR3ρ(x)/parenleftbiggdf\ndx1/parenrightbigg2\ndx=/parenleftbigg2π\nN/parenrightbigg2/integraldisplay\nRg(x1)6dx1,\nwhere\ng(x1)2=/integraldisplay\nR2ρ(x1,x2,x3)dx1dx2.\nFrom [8],g∈H1(R) and moreover\ng(x1)4≤4/integraldisplay\nRg(x1)2dx1/integraldisplay\nR/vextendsingle/vextendsingle/vextendsingle/vextendsingledg\ndx1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx1≤4N/integraldisplay\nR3/parenleftbig\n∇ρ1/2/parenrightbig2dx.\nThus, (13) now gives\nT(ψD)≤/parenleftbigg\n1+(4π)2(N2−1)\n12/parenrightbigg/integraldisplay\nR3(∇ρ1/2)2+/integraldisplay\nR3|jp|2ρ−1<∞,\n23where all terms are finite since ( ρ,jp)∈YN./squaresolid\nRemark. Note that for ψDchosen as in Proposition 15, the exchange-correlation en-\nergy,Exc(ψD), does not depend on the paramagnetic current density. This can be seen\nfrom\nExc(ψD) =−1\n2N2/integraldisplay\nR3/integraldisplay\nR3ρ(x)ρ(y)\n|x−y|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN−1/summationdisplay\nk=0eik(f(x1)−f(y1))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndxdy\n=−1\n2N/integraldisplay\nR3/integraldisplay\nR3ρ(x)ρ(y)\n|x−y|FN(f(x1)−f(y1))dxdy,\nwhereFN(t) is the Fej´ er kernel, given by FN(t) = sin2(Nt/2)/(Nsin2(t/2)).\nProposition 16 For(ρ,jp)∈YNfulfilling ∇×(jp/ρ) = 0, we have\nQ(ρ,jp)≤/parenleftbigg\n1+(4π)2(N2−1)\n12/parenrightbigg/integraldisplay\nR3(∇ρ1/2)2+/integraldisplay\nR3|jp|2ρ−1+1\n2/integraldisplay\nR3/integraldisplay\nR3ρ(x)ρ(y)\n|x−y|dxdy.\nProof.First note that\n(ψ,H0ψ) =T(ψ)+Exc(ψ)+1\n2/integraldisplay\nR3/integraldisplay\nR3ρψ(x)ρψ(y)\n|x−y|dxdy.\nNow, given ( ρ,jp)∈YNfulfilling ∇×(jp/ρ) = 0, there exits, by Proposition 15, a determi-\nnantal wavefunction ψDsuch thatρψD=ρandjp\nψD=jp. We then have\nQ(ρ,jp)≤/parenleftbigg\n1+(4π)2(N2−1)\n12/parenrightbigg/integraldisplay\nR3(∇ρ1/2)2+/integraldisplay\nR3|jp|2ρ−1+(ψD,/summationdisplay\n1≤k<l≤N|xk−xl|−1ψD)\n≤/parenleftbigg\n1+(4π)2(N2−1)\n12/parenrightbigg/integraldisplay\nR3(∇ρ1/2)2+/integraldisplay\nR3|jp|2ρ−1+1\n2/integraldisplay\nR3/integraldisplay\nR3ρ(x)ρ(y)\n|x−y|dxdy,\nwhere the last inequality follows from the fact that Exc(ψD)≤0, sinceψDis a determinant.\n/squaresolid\nWe conclude this section by applying Proposition 16 and Theorem 14. T he following\ncorollarygives bothanupper andlower boundfor QandFinterms ofJ0(ρ,jp) =/integraltext\nR3|jp|ρ−1\nandJ1(ρ,jp) =/integraltext\nR3(∇ρ1/2)2.\n24Corollary 17 Let(ρ,jp)∈YNbe such that ∇×(jp/ρ) = 0. Then for 0≤λ≤1,\nλJ0(ρ,jp)+(1−λ)J1(ρ,jp) =Jλ(ρ,jp)≤F(ρ,jp)≤Q(ρ,jp)\n≤aN+(b+cN2)J1(ρ,jp)+J0(ρ,jp),\nwherea= 4/(3√\n3π),b= 1−(4π)2/12andc= (4π)2/12+4/(3√\n3π).\nProof.The statement follows directly from Proposition 16 and Theorem 14 a nd the fact\nthat\n1\n2/integraldisplay\nR3/integraldisplay\nR3ρ(x)ρ(y)\n|x−y|dxdy≤C1||ρ||2\nL6/5(R3)≤C1N3/2||ρ||1/2\nL3(R3)\n≤C1C2N3/2J1(ρ,jp)1/2≤1\nπ4\n3√\n3(N+N2J1(ρ,jp)),\nwhere the Hardy-Littlewood-Sobolev inequality ( C1= 2(4/π1/2)2/3/3) and Sobolev’s in-\nequality for gradients ( C2= 2/(31/221/3π2/3)) have been used [11]. /squaresolid\nIV. SUMMARY\nThis paper has aimed at giving CDFT formulated with the paramagnetic current den-\nsity a mathematically rigorous foundation. It has focused on definin g and investigating\ndensity functionals that depend on the particle density and the par amagnetic current den-\nsity.N-representable density pairs ( ρ,jp) have been defined. A Hohenberg-Kohn functional,\nFHK(ρ,jp), has been extended to a Levy-Lieb-type functional, denoted Q(ρ,jp), with the set\nofN-representable densities as domain. It has been proven that ther e exists a wavefunction\nψ0such thatQ(ρ,jp) = (ψ0,H0ψ0)L2andρψ=ρ,jp\nψ=jp. Moreover, a universal and convex\ndensity functional F(ρ,jp) has been proven to exist such that it equals the convex envelope\nofQ(ρ,jp). On the set of v-representable densities, the functionals FHK,QandFall agree.\nFurthermore, a connection between the minimization of F(ρ,jp) and a set of Euler-Lagrange\nequations has been established.\nForN-representable density pairs ( ρ,jp) fulfilling ∇×(jp/ρ) = 0, both upper and lower\nbounds of FandQin terms of convex functionals that are given explicitly have been\nobtained.\n25ACKNOWLEDGMENTS\nThe author is very thankful to Michael Benedicks and Anders Szep essy for useful com-\nments and discussions.\n∗andrela@math.kth.se; Department of Mathematics, KTH, 100 44 Stockholm, Sweden\n[1] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (Nov 1964),\nhttp://link.aps.org/doi/10.1103/PhysRev.136.B864\n[2] G. Vignale and M. Rasolt, Phys. Rev. Lett. 59, 2360 (Nov 1987),\nhttp://link.aps.org/doi/10.1103/PhysRevLett.59.2360\n[3] G. Diener and M. Rasolt, J. Phys.: Condens. Matter 3, 9417 (1991)\n[4] X.-Y. Pan and V. Sahni, Int. Jour. Quant. Chem. 110, 2833 (2010),\nhttp://dx.doi.org/10.1002/qua.22862\n[5] K. Capelle and G. Vignale, Phys. Rev. B 65, 113106 (Feb 2002),\nhttp://link.aps.org/doi/10.1103/PhysRevB.65.113106\n[6] A. Laestadius and M. Benedicks, To appear in Int. Jour. Qu ant. Chem.\n[7] E.I.Tellgren, S.Kvaal, E.Sagvolden, U.Ekstr¨ om, A.M. Teale, andT.Helgaker, Phys. Rev. A\n86, 062506 (Dec 2012), http://link.aps.org/doi/10.1103/PhysRevA.86.062506\n[8] E.H.Lieb,Int. Jour. Quant. Chem. 24,243(1983), http://dx.doi.org/10.1002/qua.560240302\n[9] M. Levy, Proc. Natl. Acad. Sci. USA 76, 6062 (1979)\n[10] E. H. Lieb and R. Schrader, Phys. Rev. A 88, 032516 (Sep 2013),\nhttp://link.aps.org/doi/10.1103/PhysRevA.88.032516\n[11] E. H. Lieb and M. Loss, Analysis (American Mathematical Society, Providence, Rhode Island ,\n2001)\n26"    },    {        "title": "1404.7089v1.Density_matrix_form_of_Gross_Pitaevskii_equation.pdf",        "content": "Density matrix form of Gross-Pitaevskii equation\nV. N. Chernega, O. V. Man'ko\u0003, V. I. Man'ko\nP. N. Lebedev Physical Institute, Russian Academy of Sciences\nLeninskii Prospect 53, Moscow 119991, Russia\n\u0003Corresponding author e-mail: omanko@sci.lebedev.ru\nAbstract\nWe consider the generalized pure state density matrix which depends on di\u000berent time moments. The\nevolution equation for this density matrix is obtained in case where the density matrix corresponds to\nthe solutions of Gross-Pitaevskii equation.\nKeywords: wave function, density matrix, von Neumann equation, nonlinear Schr odinger equation.\n1 Introduction\nThe wave function  (x;t) satis\fes the linear Schrodinger equation [1]. The density matrix introduced\nin [2{4] satis\fes the von Neumann equation. There exists nonlinear Schrodinger equation for which\npotential energy term provides qubic nonlinearity of the equation. The equation was used by Gross [5] and\nPitaevskii [6]. The von Neumann equation for density matrix can be derived from the linear Schrodinger\nequation. Recently [7] the generalized density matrix of pure and mixed states was introduced and\nthis matrix depends on two time moments. For equal times the generalized density matrix coincides\nwith the standard density matrix . The equation for this generalized density matrix was discussed\nin [7]. The analog of nonlinear von Neumann equation written for Wigner function [8] in case of Gross-\nPitaevskii equation was discussed in [9]. The aim of our article is to obtain nonlinear analog of Gross-\nPitaevskii equation written for generalized density matrix introduced in [7]. The paper is organized\nas follows. In Sec.2 the derivation of von Neumann equation for density matrix is reviewed. In Sec.3\nthe derivation of nonlinear equation for generalized density matrix corresponding to Gross-Pitaevskii\nequation is demonstrated. The conclusions and perspectives are given in Sec.4.\n2 Schr odinger and von Neumann equations\nThe wave function  (x;t) of the quantum system satis\fes the Schr odinger evolution equation\ni@ (x;t)\n@t=\u00001\n2@2 (x;t)\n@x2+U(x) (x;t): (1)\n1arXiv:1404.7089v1  [quant-ph]  28 Apr 2014We assume Planck constant ~= 1 and mass m= 1. The term U(x) is the real potential energy. To\nderive von Neumann equation we write the equation for the function  \u0003(x0;t) which reads:\n\u0000i@ \u0003(x0;t)\n@t=\u00001\n2@2 \u0003(x0;t)\n@x02+U(x0) \u0003(x0;t): (2)\nFrom the system of equations (1,2) it follows the equation for density matrix\n\u001a (x;x0;t) = (x;t) \u0003(x0;t) (3)\nin position representation. The equation reads\ni@\u001a (x;x0;t)\n@t=\u00001\n2(@2\u001a (x;x0;t)\n@x2\u0000@2\u001a (x;x0;t)\n@x02) + (U(x)\u0000U(x0))\u001a (x;x0;t): (4)\nThe same equation (4) is valid for convex sum of pure state density matrices\n\u001a(x;x0;t) =X\nnpn\u001a n(x;x0;t); (5)\nwhere 1\u0015pn\u00150;P\nnpn= 1:For stationary states with wave functions of the factorized form\n n(x;t) =\u001en(x) exp(\u0000iEnt) (6)\nthe density matrix \u001a n(x;x0;t) does not depend on time.\nIn [7] the generalized density matrix of pure state was introduced\nR (x;x0;t;t0) = (x;t) \u0003(x0;t0): (7)\nFor equal times t=t0one has equality\nR (x;x0;t;t) =\u001a(x;;x0;t): (8)\nUsing the same method of derivation which is used to obtain von Neumann equation from Schr\u0013 odinger\nequation one can get the equation for the generalized density matrix which reads [7]\ni@R (x;x0;t;t0)\n@t+i@R (x;x0;t;t0)\n@t0=\n\u00001\n2[@2R (x;x0;t;t0)\n@x2\u0000@2R (x;x0;t;t0)\n@x02] + [U(x)\u0000U(x0)]R (x;x0;t;t0): (9)\nIf the times coincide t=t0the above equation provides the von Neumann equation for density matrix\n\u001a(x;;x0;t). In view of linearity of Eq.(9) the same equation is valid for the matrix\nR (x;x0;t;t0) =X\nnpnR n(x;x0;t;t0): (10)\nHere the generalized density matrix R(x;x0;t;t0) is convex sum of pure state matrices R n(x;x0;t;t0).\nThe generalized density matrix of pure stationary state contains dependence on time, i.e.\nR n(x;x0;t;t0) = exp[\u0000iE(t\u0000t0)]\u001en(x)\u001e\u0003\nn(x0): (11)\nThus for stationary states the matrix R n(x;x0;t;t0) contains the information corresponding to the both\npast and future time moments.\n23 Nonlinear Gross-Pitaevskii equation\nThe nonlinear Schr odinger equation with qubic nonlinearity has the form\ni@ (x;t)\n@t=\u00001\n2@2 (x;t)\n@x2+U(x) (x;t) +g (x;t)j (x;t)j2: (12)\nOur purpose is to use this equation where gis nonlinearity constant and to derive the corresponding\nequation for the generalized density matrix R (x;x0;t;t0).\nFirst, we write the equation for complex conjugate function  \u0003(x0;t) analogously to the case of\nnonlinear Schr odinger equation\n\u0000i@ \u0003(x0;t0)\n@t0=\u00001\n2@2 \u0003(x0;t0)\n@x02+U(x0) \u0003(x0;t0) +g \u0003(x0;t0)j (x0;t0)j2: (13)\nThen we multiply all the terms in Eq.(12) by the function  \u0003(x0;t0) and all the terms in Eq.(13) by the\nfunction (x;t). We get two equations\ni@ (x;t) \u0003(x0;t0)\n@t=\u00001\n2@2 (x;t) \u0003(x0;t0)\n@x2+U(x) (x;t) \u0003(x0;t0) +g (x;t) \u0003(x0;t0)j (x;t)j2:(14)\nand\n\u0000i@ (x;t) \u0003(x0;t0)\n@t0=\u00001\n2@2 (x;t) \u0003(x0;t0)\n@x02+U(x0) (x;t) \u0003(x0;t0) +g (x;t) \u0003(x0;t0)j (x0;t0)j2:(15)\nTaking the di\u000berence of equations (12) and (15) we get the equation\ni@R (x;x0;t;t0)\n@t+i@R (x;x0;t;t0)\n@t0=\u00001\n2[@2R (x;x0;t;t0)\n@x2\u0000@2R (x;x0;t;t0)\n@x02]\n+[U(x)\u0000U(x0)]R (x;x0;t;t0) +gR (x;x0;t;t0)[R (x;x;t;t )\u0000R (x0;x0;t0;t0)]: (16)\nThe obtained equation is nonlinear and nonlocal equation for the generalized density matrix corre-\nsponding to the Gross-Pitaevskii equation.\nConclusion\nTo resume we point out the main results of our work. We reviewed the derivation of von Neumann\nequation for density matrix from the Schr odinger equation. Using the method to provide this derivation\nwe applied it to derive the equation for generalized density matrix which contains time-dependence even\nfor stationary states. We extended the derivation to the case of nonlinear cubic Schr odinger equation.\nThe obtained equation gives also the equation for standard density matrix in the case of taking into\naccount the nonlinearity. The forms of the obtained equation in terms of Wigner-Weyl function, Husimi\nfunction and tomographic probability distribution will be given in future publications.\n3References\n[1] E. Schr odinger, Naturwissenchaften ,Bd. 14 , s. 664 (1926)\n[2] L. D. Landau, Z. Physik ,45, 430 (1927).\n[3] J. von Neumann, \"Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik\", Nach. Ges.\nWiss. G ottingen ,11, 245 (1927).\n[4] J. von Neumann, Mathematische Grundlagen der Quantenmechanik , Springer, Berlin (1932).\n[5] E. P. Gross, Nuovo Cim. ,20, 454 (1961)\n[6] L. P. Pitaevskii, Sov. Phys. JETP ,13, 451 (1961)\n[7] M. A. Man'ko, V. I. Man'ko, \"Tomographic entropic inequalities in the probability representation\nof quantum mechanics,\" arXiv:1208.5695 [quant-ph], in: R. Bijker (Ed.), Beauty in Physics: Theory\nand Experiment ,AIP Conference Proceedings, New York, 1488 , 110 (2012).\n[8] E. Wigner, Phys. Rev. ,40, p. 749 (1932)\n[9] S. De Nicola, R. Fedele, M. A. Manko, and V. I. Manko, Eur. Phys. J. B ,36, 385 (2003)\n4"    },    {        "title": "1405.4815v1.Zero_energy_Majorana_states_in_a_one_dimensional_quantum_wire_with_charge_density_wave_instability.pdf",        "content": "arXiv:1405.4815v1  [cond-mat.mes-hall]  19 May 2014Zero-energy Majorana states in a one-dimensional quantum w ire with charge density\nwave instability\nE. Nakhmedov1,2and O. Alekperov2\n1Institut f¨ ur Theoretische Physik, Universit¨ at W¨ urzbur g, D-97074 W¨ urzburg, Germany\n2Institute of Physics, Azerbaijan National Academy of Scien ces, H. Cavid ave. 33, AZ1143 Baku, Azerbaijan\n(Dated: March 9, 2022)\nOne-dimensional lattice with strong spin-orbit interacti ons (SOI) and Zeeman magnetic field is\nshown to lead to the formation of a helical charge-density wa ve (CDW) state near half-filling.\nInterplay of the magnetic field, SOI constants and the CDW gap seems to support Majorana bound\nstates underappropriate value of the external parameters. Explicit calculation of the quasi-particles’\nwave functions supports a formation of the localized zero-e nergy state, bounded to the sample end-\npoints. Symmetry classification of the system is provided. R elative value of the density of states\nshows a precise zero-energy peak at the center of the band in t he non-trivial topological regime.\nPACS numbers: 75.70.Tj,71.70.Ej, 85.75.-d, 72.15.Nj\nRecently, new exotic topological states of con-\ndensed matter, capable of supporting non-Abelian quasi-\nparticle1, have been suggested2–5, which can be used\nas a fault-tolerant platform for topological quantum\ncomputation6,7. These topological phases reveal chiral\nMajorana edge particles, being their own antiparticles,\nwhich are represented by the non-Abelian statistics with\nnon-commutative fermionic exchange operators.\nSuggestion by Read and Green3, that Majorana states\ncan be realized at the vortex cores of a two-dimensional\n(2D)px+ipysuperconductor,hasprovokednewadvances\ninengineeringasemiconductionnanostructurewith zero-\nenergy state. Kitaev showed2a possible realization of a\nsingle Majorana fermion at each end of a p-wave spinless\nsuperconductingwire. Theeffective p-wavesuperconduc-\ntors were shown8–11to be realized in a semiconductor\nfilm, in which s-wave pairing is induced by the proximity\neffect in the presence of spin-orbit interactions (SOI) and\nZeeman magnetic field.\nFormation of zero-energy Majorana bound states in\none-dimensional(1 D)quantumwireintheproximitytoa\ns-wavesuperconductorandinthepresenceofSOIandthe\nmagnetic field has been argued recently in Refs. [12,13].\nIn this Letter we predict a new realization mechanism\nof Majorana quasi-particles in a 1 Dcrystal with charge\ndensity wave (CDW) instability. We consider the model\nof a 1Dcrystal around half-filling with strong spin-orbit\ninteractions and in the presence of Zeeman magnetic\nfield. There is an instability against formation of CDW\nand spin-density wave (SDW) in a such model. The key\nto the quantum topological order is the coexistence of\nSOI with CDW or SDW state and an externally induced\nZeeman coupling of spins. We show that for the Zee-\nman coupling below a critical value, the system is a non-\ntopological CDW semiconductor. However, above the\ncritical value of Zeeman field, the lowest energy excited\nstateisazero-energyMajoranafermionstatefortopolog-\nical CDW crystal. Thus, the system is transmuted into\na non-Abelian CDW state with increasing the external\nmagnetic field.\nSDW and CDW arebroken-symmetryground states ofhighly anisotropic, so called quasi-1 Dmetals which are\nthoughtto ariseasthe consequenceofelectron-phononor\nelectron-electroninteractions14,15. Thesestateshavetyp-\nical1Dcharacter,and they can be convenientlydiscussed\nwithin the framework of various 1D models16. CDW\nand SDW states are successfully realized in quasi-1D\nstructures such as organic molecules of ( TMTSF )2PF6,\n(MDTTF )2Au(CN)2, (DMET)2Au(CN)2, andAu,\nIn,Geatomic wires grown by self-assembly on vicinal\nSi(553),Si(557) orGe(001) surfaces17,18.\nThe model considered here is essentially 1D Hubbard\nmodel with on-site Coulomb interactions in the presence\nof both Rashba and Dresselhaus SOI and Zeeman mag-\nnetic field. Non-interacting part ˆH0of the Hamiltonian\nˆH=ˆH0+ˆHintin momentum space reads\nˆH0=/summationdisplay\n0<k<G/ 2/summationdisplay\nσ,σ′/braceleftbigg\nξkc†\nk,σck,σ′δσ,σ′+ωZc†\nk,σ(σx)σσ′ck,σ′+\nαRsin(kd)c†\nk,σ(σz)σσ′ck,σ′+αDsin(kd)c†\nk,σ(σy)σσ′ck,σ′+\n(k↔k−G/2)/bracerightbigg\n, (1)\nwhereαRandαDare constants of Rashba and Dressel-\nhaus SOI19, correspondingly, ωZ=g/planckover2pi1µBB/2 is Zeeman\nenergy of a magnetic field B,ξk=ǫk−µwithǫk=\n−2tcos(kd), andµis the Fermi energy. At half-filling\nµ= 0, and the electron-hole symmetry ξk−G/2=−ξkfor\none-particle states is realized. G= 2π/dis the reciprocal\nlattice vector with dbeing the unit cell size. Interaction\ntermHintin the Hamiltonian is written\nˆHint=1\n2N/summationdisplay\n0<q<G/summationdisplay\nσ/braceleftbigg/summationdisplay\n−G/2<k<G/ 2−q\nq−G/2<k′<G/2U(k,k′;q)c†\nk+q,σck,σc†\nk′−q,−σck′,−σ+\n/summationdisplay\nG/2−q<k<G/ 2\n−G/2<k′<q−G/2U(k,k′;q)c†\nk,σck+q−G,σc†\nk′,−σck′−q+G,−σ/bracerightbigg\n,(2)2\nwhereUis a strength of the Hubbard interaction and N\nis the number of lattice sites. Note that the momentum\nsummation in Eq. (1) is taken over positive part of the\nBrillouinzone,andthefirstfourtermsinHamiltoniande-\nscribetherightmoving( k>0)particles. Theleftmoving\n(k−G/2<0) particles are taken into account by adding\nthe terms with k↔k−G/2. The electron-hole order pa-\nrameter at the density-wave instability is introduced as\n∆σ=V\nN/summationtext\n0<k<G/ 2/an}bracketle{tc†\nk−G/2,σck,σ/an}bracketri}htunderassumptionthat\nU(k,k′,q) =Vδ(q−G/2). The complex-conjugate order\nparameterisobtainedbysummingtheelectron-holepair-\ning over negative momentum part of the Brillouin zone,\n∆∗\nσ=V\nN/summationtext\n−G/2<k<0/an}bracketle{tc†\nk+G/2,σck,σ/an}bracketri}ht. CDW and SDW or-\nder parameters are defined as ∆ CDW= (∆↑+∆↓)/2\nand ∆ SDW= (∆ ↑−∆↓)/2, respectively. Assuming\n∆↑= ∆↓for CDW, thereby we eliminate SDW order-\ning, and ∆ CDW= ∆↑= ∆↓. For SDW we assume\n∆↑=−∆↓, at the same time CDW formation is elim-\ninated, and ∆ SDW= ∆↑=−∆↓. Further we use a com-\nmon notation ∆ for both CDW and SDW ordering, and\nreplaceˆHintin the mean field approximation by ˆHMF\nint\nˆHMF\nint=/summationdisplay\n0<k<G/ 2,σ¯σ{∆c†\nk,σck−G/2,σ+∆∗c†\nk−G/2,σck,σ},\n(3)\nwhere ¯σ= 1 for CDW ordering, and ¯ σ=−σ=∓1 for\nSDW state. Hamiltonian ˆHMF=ˆH0+ˆHMF\nintis written\nin the basis Ψ†=/parenleftBig\nc†\nk,↑c†\nk,↓c†\nk−G/2,↓−c†\nk−G/2,↑/parenrightBig\nas\nˆHMF=/summationdisplay\n0<k<G/ 2{Ψ†ˆHΨ+ξk+ξ−k+G/2}+2\nV|∆|2,(4)\nwith\nˆH=ξkτz⊗σ0+αRsink τ0⊗σz+αDsink τz⊗σy+\nωZτz⊗σx+τj(∆)⊗σj; (5)\nwhere the Pauli matrices σandτoperate in spin and\nparticle-hole spaces, ⊗is the Kronecker product of ma-\ntrices. In the last term, j=yfor CDW and j=xfor\nSDW pairing\nτy(∆) =/parenleftbigg\n0−i∆\ni∆∗0/parenrightbigg\n,andτx(∆) =/parenleftbigg\n0 ∆\n∆∗0/parenrightbigg\n,(6)\nThe first term of Eq. (5) in the linearized form, −/planckover2pi1∂yτz,\nwith the third Zeeman term, ωZσxconstitutes the mas-\nsive Dirac equation. The charge density ordering, how-\never, with the last term τj(∆)σjtransforms the model to\nthe four-band model.\nThe pole of the single particle Green’s function\nG−1(E,k) =E−ˆHdetermines the quasiparticle energy\nE2\nCDW=ξ2\nk+α2sin2k+|∆|2+ω2\nZ±\n±2/radicalBig\nξ2\nkα2sin2k+ω2\nZ|∆|2+ξ2\nkω2\nZ; (7)\nE2\nSDW=/parenleftbigg\n|ξk|±/radicalBig\nα2sin2k+ω2\nZ/parenrightbigg2\n+|∆|2,(8)a\n/Minus3/Minus2/Minus10123/Minus1.0/Minus0.50.00.51.0\nkE\nb\n/Minus3/Minus2/Minus10123/Minus1.5/Minus1.0/Minus0.50.00.51.01.5\nkE\nc\n/Minus3/Minus2/Minus10123/Minus2/Minus1012\nkE\nd\n/Minus3/Minus2/Minus10123/Minus3/Minus2/Minus10123\nkE\nFIG. 1: Energy spectrum is plotted according to Eq. (7) for\nthe fixed values of t= 0.5, ˜α= 0.8, and for the following\nvalues of the dimensionless parameters, (a) ˜∆ = 0.0, ˜ωZ=\n0.05, ˜µ= 0.0; (b)˜∆ = 0.5, ˜ωZ= 0.7, ˜µ= 0; (c) ˜∆ = 0.7,\n˜ωZ=√\n1.3, ˜µ=−0.1; (d)˜∆ = 0.7, ˜ωZ=√\n2.18, ˜µ=−0.3\n.\nfor CDW and SDW states, correspondingly. SO cou-\npling constant αin the expressions for the energy spec-\ntrum is a renormalized constant α=/radicalbig\nα2\nR+α2\nD. Equa-\ntion (8) does not allow a zero-energy mode due to a fi-\nnite gap ∆ at the origin. However, experimental evi-\ndences in many quasi-1D materials, e.g. in Bechgaard\nsalt (TMTSF )2PF6suggest a realization of an uncon-\nventionalSDWwithanorderparameter ∼∆1sinkyield-\ning a zero-energy state. The dispersive CDW or SDW\ngap can be derived from the extended Hubbard model\nwith nonlocal interaction20. Further, we will discuss only\nthe topological CDW state.\nA small deviation from half-filling at T= 0 was shown\nby Brazovskii et al.21to create a band of kink states\nwithin the Peierls gap. This picture is changed at finite\ntemperatures. According to the phase diagrams in the\ntemperature-chemical potential ( T,µ) and temperature-\ndensity (T,n) planes, callculated in Ref.[ 22] on the base\nof Brazovskii et al. theory21, for fixed electron density\n1<n<n L, wherenLis Leung’s density23at the triple\npoint of the normal (N), commensurate (C) and incom-\nmensurate (IC) phases, the kink band shrinks with in-\ncreasing temperature until it vanishes at the IC-C tran-\nsition. For fixed temperature 0 < T < T Lthe kink\nband arises at some electron density n >1 and broad-\nens with increasing density until the kinks become soft.\nAt finite temperatures ( T <T 0) and for small deviation\nof the chemical potential from half-filling |µ|< T0=\n1.056Tc(0) = (2/π)∆, whereTc(0) = (4Weγ/π)e−1/λis\nthe transitiontemperatureat µ= 022, the systemisinC-\nphasewith vanishingmismatchingbetween the electronic\nstateskandG/2−k.\nSolution of the energy spectrum for different values\nof ˜α, ˜µ,˜∆, and ˜ωZis plotted in Fig. 1, where the\ndimensionless parameters with tilde are given in the\nunit of the halved band width 2 t. Solution of Eq. (7)3\nαR=αD= ∆ =ωZyields a usual cosine-band in the\nreduced Brillouin zone. SOI results in two shifted cosine-\nbands along k-axes, whereas Zeeman splitting doubles\nthe band along the energy axes, opening a gap at the an-\nticrossing point (see, Fig. 1a). Formation of the density\nwave opens a gap at the boundary of the Brillouin zone.\nTheenergyspectrumatthecenteroftheBrillouinzone\nfor the topological CDW with gapped “bulk” states and\nzero-energy end-states can be written as\nE(0)\nCDW=E(0) =/vextendsingle/vextendsingle/vextendsingleωZ−/radicalBig\nµ2\nt+|∆|2/vextendsingle/vextendsingle/vextendsingle,(9)\nwhereµt=−2t−µ. A magnetic-field dominated gap\nat the center of the band for ω2\nZ>|∆|2+µ2\ntturns to\nthe pairing-dominated one for ω2\nZ<|∆|2+µ2\nt, (Figs.\n1d and b, correspondingly). A quantum phase transi-\ntion from topological non-trivial to trivial phase occurs\natω2\nZ=|∆|2+µ2\nt. The gap at k= 0 vanishes under\nthis condition emerging Majorana fermion states at the\nends of the wire, which is plotted in Fig. 1 for the di-\nmensionless parameters ˜ α= 0.8,˜∆ = 0.7, ˜ωZ=√\n1.3,\nand ˜µ=−0.1.\nIt is possible to check that the Hamiltonian ˆHre-\nspects time-reversal symmetry (TRS) UTˆH∗(k)U−1\nT=\nˆH(−k) with TRS operator T=UTKin the absence\nof the magnetic field, and particle-hole symmetry (PHS)\nUPˆH∗(k)U−1\nP=−ˆH(−k) with PHS operator P=UPK.\nHere,Kisthecomplexconjugateoperator, UT=σ0⊗iσy\nandUP=σx⊗σ0satisfyingT2=−1 andP2= 1. The\nTRS operator transforms k→ −kas well asck↑⇔c†\nk↓\nandck↓⇔ −c†\nk↑, resulting in ∆ ↔∆∗for the order pa-\nrameter and keeping the excitation spectrum unchanged\nξ−k=ξk. Instead, the PHS operator transforms\nck↑⇔c†\nk−G/2↓andck↓⇔ −c†\nk−G/2↑,(10)\nkeeping unchanged the order parameter ∆. PHS entails\nan energy spectrum symmetric about the Fermi level.\nAccording to symmetry classification the system belongs\ntoDIIIclasswhichcanbetopologicallynontrivial24pro-\nvided that both TRS and PHS are satisfied. An external\nmagnetic field breaks TRS and drives the system from\nDIIItoDclass, which possessesa single Majoranazero-\nenergy mode at each end of the wire.\nThe main feature of Majorana fermion is that it is\nown ’anti-particle’. This property can be proved for a\n1Dunconventional CDW model14,20with dispersive and\ncomplex order parameter ∆ k= ∆0sin(kd) by mapping\nit to the Kitaev’s model2for the p-wave superconductor.\nHamiltonian of a 1 Dunconventional CDW model be-\ncomes invariant under the particle-hole transformations\ncv\nk≡ck↔cc†\nk≡c†\nk−G/2andcv†\nk↔cc\nkin momentum\nspace ordv\nn↔dc†\nnanddv†\nn↔dc\nnin site-representation,\nwhere the spin index is neglected due to the spin degen-\neration. The PHS transforms it to the Kitaev’s one\nˆH0=/summationdisplay\nn{−2t(dv†\nndv\nn+1+dv†\nn+1dv\nn)+2i∆0dv†\nndv†\nn+1+\n2i∆∗\n0dv\nn+1dv\nn}, (11)which revealsthe Majoranaend states. It is easy to show\nthat the PHS conditions (10) transform our Hamiltonian\n(1) and (3) to the form, describing the s-wave type su-\nperconductor with misaligned spins but with the same\nmomentakof Cooper pairs, which should reveal again\nthe Majorana quasi-particles.\nMajorana bound states arise at the interface of triv-\nial and topological regions under certain condition by\nvarying the parameters of 1 Dwire. In order to under-\nstand the localized character of the zero energy state we\nrewrite Hamiltonian in the real coordinatespace. We lin-\nearize the cosine energy spectrum around the Fermi level\nkF=G/4 asξk=ǫk−µ= 4tsin(k+kF)d\n2sin(k−kF)d\n2≈\nvF/planckover2pi1(k−kF)→vF/planckover2pi1(−i∂\n∂y−kF) for right-mover, and\nξk−G/2≈ −vF/planckover2pi1(k+kF)→ −vF/planckover2pi1(i∂\n∂y−kF) for left-\nmover, and the SO coupling term sin( dk)→ −id∂\n∂y.\nOne can see that µz=vFkF/planckover2pi1; at half-filling µ= 0 and\nµt=vFkF/planckover2pi1= 2t. Schr¨ odinger equation, corresponding\nto zero energy, reads\n/bracketleftbigg\n−µt−i(vF/planckover2pi1+νσαR)∂\n∂y/bracketrightbigg\nψR\nσ+/parenleftbigg\nωZ−νσαD∂\n∂y/parenrightbigg\nψR\n−σ+\n∆ψL\nσ= 0/bracketleftbigg\nµt+i(vF/planckover2pi1+νσαR)∂\n∂y/bracketrightbigg\nψL\nσ+/parenleftbigg\nωZ+νσαD∂\n∂y/parenrightbigg\nψL\n−σ+\n∆∗ψR\nσ= 0, (12)\nwhere−σ=↓,↑,andνσ=±forσ=↑,↓,correspondingly.\nFor long enough wire L≫1, we choose the magnetic\nfieldω2\nZ< µ2\nt+|∆|2fory∈[0,L] andω2\nZ> µ2\nt+|∆|2\noutside this interval. By choosing the wave functions\nΨT(y) = exp{iky}(bR\n↑, bR\n↓, bL\n↓,−bL\n↑)T, one gets the\ndeterminant equation det|¯H|= 0 to find k, where\nH=vF/planckover2pi1(k−kF)τz⊗σ0+αRk τ0⊗σz+\nωZτz⊗σx+αDk τz⊗σy+∆σy⊗τy.(13)\nThe allowed values of kare obtained from the equation,\n(v2\nF/planckover2pi12−α2)k2−2k(µtvF/planckover2pi1±i|∆|α)−L= 0, where L=\nω2\nZ−|∆|2−µ2\nt. ForL= 0 this equation has a real root\nk= 0, corresponding to a single allowed state in the gap.\nSince there is no other state for a quasiparticle to move,\nthis stateis localizedand it seemsto be protectedagainst\nlocal perturbations. For L /ne}ationslash= 0,ktakes complex values,\nsignaling on realization of a gapped state. In this case\nthe wave function decays exponentially in both sides of\ny= 0 but with different localization lengths. General\nsolution for kreads\nkν=kF±i|¯∆|¯α+ν/radicalBig\n(kF¯α±i|¯∆|)2+ ¯ω2\nZ(1−¯α2)\n1−¯α2,\n(14)\nwhere ¯α=α\nvF/planckover2pi1,¯∆ =∆\nvF/planckover2pi1, ¯ωZ=ωZ\nvF/planckover2pi1, andν=±. The\nwave function decays exponentially if, generally speaking\n|∆|,α/ne}ationslash= 0. Forα= 0,k±=µt±√\nω2\nZ−|∆|2\nvF/planckover2pi1and the trivial\nCDW state is gapped for ωZ<|∆|, which is destroyed\nforωZ>|∆|.4\na\n/Minus2/Minus101205101520253035Ρ\nb\n/Minus2/Minus1012010203040506070Ρ\nc/Minus0.6/Minus0.4/Minus0.20.00.20.40.60.00.20.40.60.81.0\n/Minus6/Minus4/Minus20246050100150Ρ\nd/Minus0.6/Minus0.4/Minus0.20.00.20.40.60.00.10.20.30.40.5\n/Minus4/Minus2024050100150200Ρ\nFIG. 2: The relative change in the DOS δρ(ǫ,V)/ρ(0)(ǫ) for\n(a) ˜α= 0.8,˜∆ = 0.0 and ˜ωZ= 0.3, (b) ˜αR= 0.6,˜∆ = 0.7\nand ˜ωZ= 0.5, (c) ˜αR= 0.3,˜∆ = 1.0 and ˜ωZ=√\n2.0, and (d)\n˜αR= 0.3,˜∆ = 1.0 and ˜ωZ= 1.6. The inelastic scattering rate\nischosentobe ˜ η= 0.05. Insetin(c)showsazero-energypeak,\ncorresponding to Majorana quasi particle, which disappear s\nin inset (d) by destroying the condition.\nMajorana bound state is formed by varying the pa-\nrameters ∆, ωZ, andµ. We consider a linearized Hamil-\ntonian, Eq. (13), for the relevant momenta near k= 0\nandµt= 0, assuming a spatial variation of the mag-\nnetic field ωZ= ∆ +byneary= 0, which crosses a\nconstant gap ∆ >0. For simplicity, Dresselhaus SOI\nis neglected, αD= 0, and ∆ is chosen to be real. Fol-\nlowing Oreg et al.13, the squared, due to the particle-\nhole symmetry, Hamiltonian (13), H2, is reduced to the\ndiagonal form by mean of the unitary operator U=\n1\n2(τz+iτy+iσxτz+σxτy),\n˜H=UH2U†= [ω2\nZ+∆2+(αRk)2]−αR/planckover2pi1bσzτz+2ωZσzτ0,\n(15)\nwith spectrum E2= (ωZ±∆)2±αR/planckover2pi1b. The term, pro-\nportionalto bσz, appearsin Hamiltoniandue tothe topo-\nlogical defect at the ends of the wire, which bridges two\nedges of the conduction and valence bands. The bound\nstate may form if ∆ varies in space and crosses ωz.\nZero-energy Majorana state in the Peierls gap can be\nexperimentally detected from the tunneling experiments,\nwhere the conductivity of the tunneling contact is ex-\npressed through the one-particledensity of states (DOS),\nρ(ǫ,T), as\nδG(V,T)\nG(0)=/integraldisplay+∞\n−∞dǫ\n4Tδρ(ǫ)\nρ(0)/bracketleftbigg1\ncosh2ǫ−eV\n2T+1\ncosh2ǫ+eV\n2T/bracketrightbigg\n.\n(16)\nAtT= 0 this expression is written δG(ǫ)/G(0)=\n[ρ(ǫ,0)−ρ(0)]/ρ(0)=δρ(ǫ)/ρ(0), whereρ(0)is the DOS\nof a pure system. The DOS is found from the conven-\ntional expression ρ(ǫ) =/integraltextπ\n−πdk\n2π/summationtext\nnδ(ǫ−En(k)), where\nEn(k) is the energy spectrum for n= 1,2,3,4 given\nby Eq. (7). The delta-function is regularized for nu-merical calculations, replacing it by Lorenzian function\nδ(ǫ−En(k)) =η/{(ǫ−En(k))2+η2}, whereηis the\nrate of inelastic processes. A formation of the Majorana\nquasi-particle in the center of the band is clearly seen\nin the relative value of the DOS δρ(ǫ)/ρ(0). Evolution\nof the central peak in δρ(ǫ)/ρ(0)is depicted in Fig. 2,\nwhere the central peak emerges only for special values\nof the external parameters satisfying the critical condi-\ntionω2\nZ=|∆|2+µ2\nt. Note that midgap states have been\nobserved recently in a topological superconducting phase\nby Mourik et al.25and by Das et al.26in zero-bias mea-\nsurements on InSbandInAsnanowires, contacted with\none normal (gold) and one superconducting electrode.\nAn artificial string of Au,In,Ge,Pbatoms on vicinal\nSi(557),Si(553) andGe(001) surfaces seems to be suit-\nable for experimental realizations. These structures with\na large lateral chain spacing ( ∼1.6nm) can be built27by\nplacing metallic atoms side-by-side on a non-conducting\ntemplate by using e.g. a scanning tunneling microscope.\nAngle-resolved photoemission data indicates a 1 Delec-\ntron pocket with very weaktransversedispersion in these\nstructures. The ratio of the parallel and transverse hop-\nping integrals t/bardbl/t⊥was determined from a tight-binding\nfit to the Fermi contour to be larger than 6018. There-\nfore, the structures are three-dimensional with practi-\ncallyin-wiremotionofparticles. Thesestructuresexhibit\na Peierls instability below ∼150−200K. Recently, a\nspin polarized CDW has been observed28inPb/Si(557),\nwhere the Fermi surface nested charge density instability\noccurs by appropriate choice of band- filling, spin-orbit\ncoupling and external parameters. The Rashba param-\neter in this structure was found to be 1.9 eV ˚Afor the\nvalue of the Rashba splitting 0 .2˚A−1. High values of the\nband gap and the SOI constants may allow to realize a\ntopological CDW phase at higher temperatures, making\na significant step comparedto previous mechanism to de-\ntect Majorana state in topological superconductors.\nWe showed in this paper a possible realization of zero-\nenergy Majorana state in the CDW phase of a 1D crys-\ntal. CDW state in 1D crystal is realized due to nesting of\nthe Fermi level. The wave function of this state “mixes”\nan electron state ψk,σwith a momentum k >0 above\nthe Fermi level with an hole state ψk−G/2,σwith a mo-\nmentumk−G/2<0 below the Fermi level, which resem-\nblestheBogolyubov-deGenneswavefunctionwithmixed\nelectron and hole states too. A quasiparticle excitation\nin the topologicalCDWstate emergesas a localized zero-\nenergy state in the middle of the Brillouin zone. Since\nCDW phase is realized at higher temperatures, this new\nmechanism facilitates an observation of Majorana parti-\ncles and their implementation for the quantum computa-\ntions.\nE.P. would like to acknowledge B. Trauzettel and J.\nC. Budich for valuable discussions. This work was sup-\nported by the Scientific Development Foundation of the\nAzerbaijan Republic under Grant Nr. EIF-2012-2(6)-\n39/01/1.5\n1F. Wilczek, Fractional Statistics and Anyon Superconduc-\ntivity, (World Scientific, Singapore, 1990).\n2A. Yu. Kitaev, Sov.- Phys. Usp. 44, 131 (2001).\n3N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).\n4D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001).\n5L. FuandC. L. Kane, Phys. Rev.Lett. 100, 096407 (2008).\n6A. Yu. Kitaev, Ann. Phys. 303, 2(2003).\n7C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das\nSarma, Rev. Mod. Phys. 80, 1083 (2008).\n8M. Sato, Y. Takahashi, and S. Fujimoto, Phys. Rev. Lett.\n103, 020401 (2009).\n9J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,\nPhys. Rev. Lett. 104, 040502 (2010).\n10J. Alicea, Phys. Rev. B 81, 125318 (2010).\n11A. C. Potter and P. A. Lee, Phys. Rev. Lett. 105, 227003\n(2010).\n12R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.\nLett.105, 077001 (2010).\n13Y. Oreg, G. Rafael, and F. von Oppen, Phys. Rev. Lett.\nbf 105, 177002 (2010).\n14A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. -P. Su,\nRev. Mod. Phys. 60, 781 (1988).\n15G. Gr¨ uner, Rev. Mod. Phys. 60, 1129 (1988); ibid, 66, 1\n(1994).\n16J. Solyom, Adv. Phys. 28, 201 (1979).\n17C. Blumenstein, J. Sch¨ afer, S. Mietke, S. Meyer, A.\nDollinger, M. Lochner, X. Y. Cui, L. Patthey, R. Matz-\ndorf, and R. Claessen, Nat. Phys. 7, 776 (2011).\n18P. C. Snijders and H. H. Weitering, Rev. Mod. Phys. 82,207 (2010).\n19Rashba and Dresselhaus SOI, resulting from the bulk in-\nversion asymmetry and structural inversion asymmetry,\ncorrespondingly, are expressed in a 2 D{yz}plane as\nHR=αR(pyσz−pzσy) andHD=αD(pyσy−pzσz), which\nare reduced to the forms of HR=αRpyσzandHD=pyσy\nfor a 1Dwire lying along y-direction.\n20K. Maki, B. D´ ora, and A. Virosztek, in book The Physics\nof Organic Superconductors and Conductors , Springer Se-\nries in materials Science Vol. 110, p. 569-587 (2008).\n21S. A. Brazovskii, S. A. Gordyunin, and N. N. Kirova,\nPis’ma Zh. Eksp. Teor. Fiz. 31, 486 (1980)[Sov. JETP.\nLett.31, 456 (1980)].\n22J. Mertsching and H. J. Fischbeck, Phys. Status Solidi (b),\n103, 783 (1981).\n23M. C. Leung, Phys. Rev. B 11, 4272 (1975).\n24A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud-\nwig, Phys. Rev. B 78, 195125 (2008).\n25V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A.\nM. Bakkers, and L. P. Kouwenhoven, Science 336, 1003\n(2012).\n26A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H.\nShtrikman, Nature Phys. 8,887 (2012).\n27N. Nilius, T. M. Wallis, and W. Ho, Science 297, 1853\n(2002).\n28C. Tegenkamp, D. L¨ ukermann, H. Pfn¨ ur, B. Slomski, G.\nLandolt, and J. H. Dil, Phys. Rev. Lett. 109, 266401\n(2012)."    },    {        "title": "1406.3383v1.Absorbing_Phase_Transitions_and_Dynamic_Freezing_in_Running_Active_Matter_Systems.pdf",        "content": "arXiv:1406.3383v1  [cond-mat.soft]  12 Jun 2014\nAbsorbing Phase Transitions and Dynamic Freezing in Runnin g Ac-\ntiveMatterSystems\nCharlesReichhardt andCynthia J.OlsonReichhardt,∗\nReceivedXthXXXXXXXXXX20XX, AcceptedXthXXXXXXXXX20XX\nFirstpublishedon theweb XthXXXXXXXXXX200X\nDOI:10.1039/b000000x\nWe examine a two-dimensional system of sterically repulsiv e interacting disks where each particle runs in a random dire ction.\nThis system is equivalent to a run-and-tumbledynamicssyst em in the limit where the run time is infinite. At low densities , we\nfind a strongly fluctuating state composed of transient clust ers. Above a critical density that is well below the density a t which\nnon-active particles would crystallize, the system can org anize into a drifting quiescent or frozen state where the fluc tuations\nare lost and large crystallites form surrounded by a small de nsity of individual particles. Although all the particles a re still\nmoving, their paths form closed orbits. The average transie nt time to organize into the quiescent state diverges as a pow er law\nuponapproachingthe critical density from above. We compar eour results to the randomorganizationobservedfor period ically\nsheared systems that can undergo an absorbing transition fr om a fluctuating state to a dynamical non-fluctuating state. I n the\nrandomorganizationstudies,thesystemorganizestoastat einwhichtheparticlesnolongerinteract;incontrast,wefi ndthatthe\nrandomlyrunningactivematterorganizestoa stronglyinte ractingdynamicallyjammedstate. We showthat the transiti onto the\nfrozenstate is robust againsta certainrangeof stochastic fluctuations. We also examinethe effectsof addinga small nu mberof\npinned particles to the system and find that the transition to the frozen state shifts to significantly lower densities and arises via\nthenucleationoffacetedcrystalscenteredat theobstacle s.\n1 Introduction\nNonequilibrium collections of interacting particles ofte n\nform strongly fluctuating or turbulent states and can exhibi t\nnonequilibrium phase transitions between distinct dynami cal\nregimes1–3. Examples of such systems include sheared col-\nloids4–7, turbulentliquidcrystals8, granularmatter9,10, driven\nvortex matter in type-II superconductors11,12, and active mat-\nter or self-driven particle assemblies13–16. In certain cases a\nnonequilibriumtransitioncanbeidentifiedtooccurbetwee na\nfluctuatingstateandaquiescentstateinwhichfluctuations are\nalmost completely absent but the system is still out of equi-\nlibrium or even in motion; here, the system can become dy-\nnamicallytrappedinto an absorbingphase1,2. The absence of\nfluctuations in the absorbing or quiescent state makes it im-\npossibleforthesystem toescapefromthisstate.\nRecently, periodically sheared dilute colloidal suspensi ons\nwere shown to be an outstanding example of a system ex-\nhibiting the hallmark features of a transition into an absor b-\ning phase4–7. Thermal fluctuations are negligible in this sys-\ntem, so the dynamics is governed by collisions between the\nparticles. Both numericaland experimentalstudies compar ed\nthe position of the colloidal particles at the beginning and\nTheoretical Division, Los Alamos National Laboratory, Los Alamos, New\nMexico 87545, USA. Fax: 1 505 606 0917; Tel: 1 505 665 1134; E-m ail:\ncjrx@lanl.govend of each shear cycle. This protocol permits the identi-\nfication of two distinct dynamical regimes. In the fluctuat-\ning or irreversible state, the particles are at different lo ca-\ntions at the beginning and end of each cycle, and perform an\nanisotropic random walk over the course of many cycles. In\nthe reversible state, particles return to the same position af-\nter each cycle and fluctuations vanish when the particles or-\nganize into a configuration that prevents particle-particl e col-\nlisions from occurring4,5. For fixed particle density, the re-\nversiblestate appearsbelowa critical oscillatoryshear a mpli-\ntudeγc, while at high shear amplitudes the system remains\nin the fluctuating state. When the periodic shear is first ap-\nplied, the system always starts in an irreversible fluctuati ng\nstate, but after a number of cycles it either organizes into\na steady irreversible state or reaches a reversible quiesce nt\nstate. As γcis approached, the number of cycles or the to-\ntal timeτrequired for the system to reach one of these states\ndiverges as a power law τ∝ |γ−γc|−ν||, withν||≈1.35\nin two-dimensional (2D) simulations and ν||≈1.5in three-\ndimensional (3D) experiments5. Major classes of absorbing\nphase transitions include directed percolation1,17,18and con-\nserved directed percolation2,18, which have predicted expo-\nnentsin 2Dof ν||= 1.295andν||= 1.225,respectively.\nIn the colloidal system the absorbing state is characterize d\nby the organization of the particles into a random pattern in\nwhich they do not interact with each other,leading the trans i-\n1–10|1tion to be termed “random organization”5,18. A similar peri-\nodic drive protocol has been used to analyze other nonequi-\nlibrium systems, such as periodically driven superconduct -\ning vortices11,12and granular media18–20, by identifying the\ntransient time required to transition from irreversible to re-\nversibleflow. Other studieshave shownthat a transitionfro m\nan irreversible to a reversible state can occur even when the\nreversible state is strongly interacting, such as in jammed\nsolids21–24,andinsomecasesthistransitionisassociatedwith\napowerlaw divergenceofatimescale, indicativeofacritic al\npoint21,22,24.\nAnother type of nonequilibriumsystems is self-driven par-\nticles or active matter13,16,25, which include swimming bacte-\nria undergoingrun-and-tumbledynamics26–30, flocking parti-\ncles31,32, pedestrian motion33, and self-driven colloidal parti-\ncles34–40. One class of active matter is run-and-tumble self-\ndriven particles, which move or run in a randomly chosen\ndirection for a period of time before undergoing a tumbling\nevent and then moving in a freshly chosen random direction.\nRun-and-tumble dynamics have been used to model systems\nsuchasswimmingbacteria26–29,38. Anotheractivematterclass\nisactiveBrownianparticlessuchasJanuscolloids. Hereea ch\nparticle is self-driven but its direction of motion slowly d if-\nfuses34,36–39. When the active particles have additional steric\ninteractions,atransitioncanoccurfromauniformliquids tate\nto a phase-separated state consisting of high density clust ers\nseparated by a low density dilute gas29,37–48. The high den-\nsity state can be characterized as having glass43,46or crys-\ntalline37,39,42,44,45order. The cluster structures are only tran-\nsient and exhibit fluctuations in which the clusters can brea k\nup and reform over time, as observed in recent experiments\nwhere the clusters were described as “living crystals”37,39.\nWhen the particle density is increased, the system eventual ly\nentersa dense jammed state in which slow rearrangementsof\ntheparticlescanstill occurduetotheiractivity41,49.\nHere we investigate whether active matter systems can ex-\nhibit absorbing transitions similar to the random organiza -\ntion from a strongly fluctuating state to a dynamically froze n\nsteady state. Active matter systems such as run-and-tumble\nand active Brownian particles are usually modeled by includ -\ning a stochastic term either in the randomization of the run-\nning direction during a tumble or in the slow diffusion of the\nrunning direction, respectively. Deterministic dynamics can\noccur in certain limits, such as infinite run lengths in the ru n-\nand-tumble systems, or in the absence of a diffusive term in\nthe active Brownian particles. In these deterministic limi ts,\neach particle moves in a fixed direction but can still interac t\nwith other particles to create a fluctuating state. When the\ndynamicsisfreeofstochasticfluctuations,asinrecentnum er-\nical studies of binary rotating cross-shaped particles, it was\nshownthataftersometimeparticleswithdifferentrotatio ndi-\nrections could organize into a phase-separated state50. Thissystem is similar to the random organization studies in that\nthe particles are initially in a fluctuating state and after s ome\ntimeeithersettleintoadynamicallyfrozenstateorremain ina\ncontinuouslyfluctuatingstate5. Otherstudiesofgrowingactin\nfilaments found evidence of organizationinto a quiescent ab -\nsorbingstate consistingofspiral patterns14,15.\nWeconsiderrun-and-tumblestericallyrepulsivedisksint he\nlimit where the particles have an infinite run time. The initi al\npositions of the particles and the run directions are select ed\nrandomly. Atlowdensitythesystemformsastronglyfluctuat -\ningstatecontainingshort-livedclusters;however,above acrit-\nical density φcthe system can organizeinto a non-fluctuating\nor quiescent state comprised of a single drifting crystalli ne\ncluster surrounded by a number of individual particles that\nare traveling in closed orbits. By starting from the high den -\nsity limit and decreasing the density to φc, we find that the\ntime required for the system to organize to a frozen state in-\ncreases with decreasing φand diverges as a power law at φc\nwithν||= 1.21±0.3, consistentwith the exponentsfoundin\nthe random organization system and conserved directed per-\ncolation. We show that the transition is robust against the a d-\ndition of some thermal noise; however, at high enough noise\nit is no longer possible for a completely dynamically frozen\nstate to form. When we add a small number of pinned par-\nticles, we find that the system can organize to a dynamically\nfrozen pinned state at densities well below the obstacle-fr ee\nφc. The pinned frozen states contain faceted hexagonal crys-\ntalsthat arecenteredatthe obstaclesites.\n2 SimulationandSystem\nWeconsidera2Dsystemofsize L×Lwithperiodicboundary\nconditions in the xandy-directions. The sample contains N\nactive disks with radius Rdthat interact with each other via a\nstericharmonicrepulsion. Thediskdensity φ=πNR2\nd/L2is\nthe total fraction of the sample area covered by the disks. A\nsample filled with inactive disks crystallizes into a triang ular\nlattice at a density of φ≈0.9. The motion of an individual\ndiskilocated at riis obtained by integratingthe overdamped\nequationsofmotion,\nηdri\ndt=Fm\ni+Fs\ni. (1)\nHereηis the damping constant, Fm\ni=Fdˆ miis the motor\nforce, and Fsdescribes the particle-particleinteractions. The\nmotor force consists of a fixed force Fdapplied in a fixed di-\nrectionˆ mithatisrandomlyselectedforeachdiskatthebegin-\nning of the simulation and then never changed. For run-and-\ntumble dynamics this corresponds to an infinite run length,\nwhile for active Brownian particles it would correspond to\nzero rotational diffusion of the particle motion. The steri c\ndisk-disk repulsion is Fs\ni=/summationtextN\ni/negationslash=jk(2Rd− |rij|)Θ(2Rd−\n2| 1–10\n01×1072×1073×1074×1075×107\ntime01234 V\nφ = 0.697φ = 0.545φ = 0.436\nφ = 0.473\nFig. 1The velocityfluctuations V(t)of a single probe particle\ndriveninthe xdirectionfor diskdensities φ= 0.436,0.473, 0.545,\nand0.697, from toptobottom. The curves have been vertically\nshiftedforclarity. For φ= 0.473and above, there isa transition\nfroma strongly fluctuatingstate toa non-fluctuating statea fter an\naverage timethat decreases withincreasing φ. Below a critical φc,\nthe system remains inthe fluctuatingstate.\n|rij|)ˆrij, whererij=ri−rjandˆ rij=rij/|rij|. We set\nRd= 0.5and select parameters k= 20andFd= 0.5such\nthat disks cannot squeeze past a jammed configurationdue to\nthedrivingforce. Pinnedparticlesorobstaclesare introd uced\nintheformof Npimmobiledisksthathavethesamestericre-\npulsiveinteractionsasthemobiledisks. Theirdensityisg iven\nbyφp=NpπR2\nd/L2.\n3 Results\nTo characterizethe activityin the system, we measurethe ve -\nlocity fluctuations V(t)at varied φof a single probe parti-\nclepmovingin the positive x-direction, where V(t) = ˙xp(t)\nandˆ mp=ˆ x.Figure 1 shows a series of V(t)curves taken\natφ= 0.436, 0.473, 0.545, and 0.697. Atφ= 0.697, the\nsystem starts in a fluctuating state with transient clusters or\nliving crystals, but after 1.5×106simulation time steps the\nfluctuations almost completely vanish and V≈0. The other\nparticlesin the system exhibit the same behavioras the prob e\nparticle, and have the same transition to a nonfluctuating ve -\nlocity occurring at the same time. The time required to reach\nthe non-fluctuatingstate is labeled τ. Forφ= 0.545we find\nthe same transition to a non-fluctuating state but τis larger,\nandasφisfurtherdecreased τcontinuestoincreaseasshown\nforφ= 0.473. Forφ <0.46,thesystemneverreachesa non-\nfluctuatingstate evenforextremelylongsimulationtimes.\nIn any individual simulation at a given value of φthere is0.4 0.5 0.6 0.7 0.8\nφ050100150200 τ\n10-210-1100\n(φ - φc)/φc10-210-1100101102103τ\nFig. 2The transient time τtoreacha dynamically frozenstate vs φ.\nThe solidline isa fitto τ= (φ−φc)−ν||, withφc= 0.462and\nν||= 1.21. Theτaxis isdivided by 106simulationtime steps.\nInset: The same data ona log-log plotshowing thepower lawfit .\nsome variation in the value of τwhen different random ini-\ntial conditions are chosen; however, the average value of τ\nrobustly increases with decreasing φand we find that τdi-\nverges at a critical density φc. In Fig. 2 we plot τversusφ,\nwherethesolidlineisafittotheform τ= (φ−φc)−ν||using\nφc= 0.462andν||= 1.21. The inset of Fig. 2 shows a log-\nlog plot of τversus(φ−φc)/φcfit usingν||= 1.21±0.06.\nThe diverging time scale for the transition from a fluctuatin g\nstate to a dynamically frozen state is very similar to what is\nobservedintheperiodicallyshearedcolloidalsystem,whe rea\ndivergingtime scaleappearsata criticalshearamplitudew ith\nexponent ν||= 1.33measured in 2D colloidal simulations5.\nOther simulations of a diverging time scale at the transitio n\nto a dynamically frozen state for a periodically sheared dis k\nsystem give ν||= 1.318. These exponents are close to the\nvaluesν||= 1.295andν||= 1.225expected for 2D directed\npercolation (DP)17,18and 2D conserved directed percolation\n(CDP)2,18, respectively. Our regressionfit givesa value of ν||\nclosertotheexpectedvaluefor2DCDP; however,ourresults\nare not accurate enoughto positively distinguish between D P\nand CDP. We note that in our system the particle number is\nconserved.\nThe non-fluctuating state in the random organization sys-\ntem consists of particles that are moving but that have orga-\nnized into a dynamic configuration in which they no longer\ncollide with one another5. In contrast, we find that the active\nmatter system organizes into a non-fluctuating cluster wher e\nmost of the particles are in contact with each other. We note\nthattheclusterstateisstilldynamicalsinceitcontinues todrift\nthroughthesysteminadirectiondeterminedbythenetsumof\n1–10|3x (a)y\nx (b)y\nx (c)y\nx (d)y\nFig. 3Snapshots of diskpositions where the arrows indicate the\ndirectionofthe motorforce foreachdisk. (a)Thefluctuatin gstateat\nφ= 0.242where only small transient clusters form. (b)The\nfluctuatingstate at φ= 0.436where a livingcrystal cluster forms.\n(c)Thetransientfluctuatingstateat φ= 0.545. (d)Thedynamically\nfrozensteady stateat φ= 0.545, where asingle largecrystal forms\nanddrifts through the system.themotorforcesofalltheparticlescontainedinthecluste r. In\nFig. 3 we illustrate the disk positionsalongwith the direct ion\nofthemotorforceforeachdisk. Figure3(a)showsa fluctuat-\ning state for φ < φcatφ= 0.242. At this density the system\nforms transient clusters that change rapidly. At φ= 0.436\nin Fig. 3(b), below φc, the system phase separates into living\ncrystalclumpswithtriangularorderingwhichbreakaparto ver\ntime,asobservedinotheractivematterstudies37,39,42–46.\nInFig.3(c)weillustratethetransientactivefluctuatings tate\natφ= 0.545which is above φc, while in Fig. 3(d) we show\nthe dynamically frozen steady state that forms at later time s.\nInthefrozenstate,thedisksformasinglecrystallinestru cture\nthatdoesnotchange,andtheparticleswithintheclusterma in-\ntain the same neighbors over time. The entire cluster slowly\ndrifts across the sample as a function of time, while the disk s\nthatarenotpartoftheclusterfollowclosedorbitswhichre peat\nover time due to the periodic boundary conditions. The sys-\ntem can be regardedasorganizingnot into a randomstate but\nratherintoajammednon-fluctuatingstate. Duetotheabsenc e\nof fluctuations, the particles in the cluster remain trapped in\nthe cluster. In the fluctuating state the disks can explore va ri-\nousconfigurationsandwhentheyencounteraconfigurationin\nwhich all fluctuationsare suppressed,they becometrappedi n\nthisconfigurationeventhoughalloftheparticlesarestill sub-\njecttoamotorforceandtheentireclusterassemblyismovin g\nas a rigid object. The number of configurations that produce\nfrozen states increases with increasing φ, so the average time\nτrequired for the particles to find a dynamically frozen state\ndecreaseswithincreasing τasshownin Figs.1,2.\nIn order to better understand the stability of the frozen\nstates, we color the particles according to the net componen t\nof the motor force along the ydirection. Particles that have a\nnet motion in the positive ydirection are plotted with a dark\ncolor and a light arrow,while particles that have a net motio n\nin the negative ydirection are drawn with a light color and a\ndark arrow. Figure 4(a)shows the configurationat φ= 0.545\ninthefluctuatingstatewherethesystemformstransientliv ing\ncrystals. A stable cluster can form when groups of particles\ncometogetherthathaveonaverageoppositeswimmingdirec-\ntions. In Fig. 4(b)we illustrate the dynamicallyfrozenste ady\nstate atφ= 0.545. Particles in the lower half of the cluster\nare generally moving in the positive ydirection, while parti-\ncles in the upper half of the cluster are generally moving in\nthe negative y-direction, so that the cluster can be regarded\nas a blocked or jammed state in which the particles can no\nlonger move past one another. Figure 4(c) shows the dynam-\nically frozen steady state at φ= 0.73, where again we find a\nsimilar phenomenon in which the lower half of the cluster is\nmoving against the upper half. As the density increases, the\nsize of the cluster grows until eventually the system forms a\ntriangular lattice with embedded voids, as shown in Fig. 4(d )\nforthedynamicallyfrozensteadystateat φ= 0.848. Atthese\n4| 1–10\nx (a)y\nx (b)y\nx (c)y\nx (d)y\nFig. 4Snapshots of the diskpositions where the arrows indicate th e\ndirectionof the motor force for eachdisk. Darkparticles wi thlight\narrows have a net motioninthe positive ydirection; light particles\nwithdark arrows have a net motioninthe negative ydirection. (a)\nThetransient fluctuating stateat φ= 0.545. (b) The dynamically\nfrozensteady stateat φ= 0.545where a single large clusterforms.\nParticlesinthe lower half of the cluster are generallymovi ng inthe\npositiveydirection, while particles inthe upper half of the cluster\naregenerally moving inthe negative ydirection, creating a jammed\nstate. (c) The dynamically frozensteady state at φ= 0.73. (d) The\ndynamically frozensteady stateat φ= 0.848, where asingle large\ncrystallinestructure forms thatdrifts through the system .\n100 10000ω10-810-710-610-5S(ω)\n00.5 11.5\nφ/φc01×10-62×10-63×10-64×10-6S0(b)\n(a)\nFig. 5(a) Power spectrum S(ω)of the time series V(t)of the\nvelocityfluctuations for φ/φc= 0.0732(lower curve) and\nφ/φc= 0.988(upper curve). Thedashed lineis a fitto 1/fαwith\nα= 1.0. (b) The noise power S0vsφ/φcobtained byintegrating\nS(ω)over a fixedinterval of ω.S0peaks just below φc.high densities the frozen states no longer have the same clea r\ntop/bottomasymmetryfeaturesandthecrystallinestateca nbe\nstabilized with more random up/down particle arrangements .\nEventually at φ= 0.9the system would naturally crystallize\nonitsownevenintheabsenceofmotorforces. Sincethenum-\nber of random configurations that lead to a stable crystallin e\nstate increasesas φ= 0.9isapproachedfrombelow,the time\nτrequiredto reachan absorbingstate decreaseswith increas -\ningφaboveφc.\nThe transient times τgenerallyincrease forincreasingsys-\ntem sizeL; however, the value of φcdoes not shift. The dy-\nnamically frozen steady states are likely affected by the pe -\nriodic boundary conditions, which limit the total number of\npossible configurations. In any real experiments performed\nwith active matter, the system will also be bounded and have\nalimitonthenumberofpossibleconfigurations,soweexpect\nthatthetransitiontoadynamicallyfrozensteadystatetha twe\nfindcanbereadilyobservedinexperiment.\n4 NoiseFluctuations\nIn the sheared colloidal system of Ref.5, two transient times\nwere observed. The first is the time τrequiredfor the system\nto settle into the non-fluctuating state on the low shear am-\nplitude side of transition, and the second is the time requir ed\nfor the system to settle into a steady fluctuating state as the\ncritical shear amplitude is approached from above5. In our\nsystem, for φ < φ cwe do not find a clear transient signa-\ntureforthesettlingofthefluctuationsintoasteadyfluctua ting\nstate; however, we can measure the changes in the velocity\nfluctuationsuponapproaching φcfrombelow. InFig.5(a)we\nplot powerspectra obtainedfromthe time series of the veloc -\nity fluctuations, S(ω) =|/integraltext\nV(t)e−iωtdt|2atφ/φc= 0.0732\nandφ/φc= 0.988. Here we find that upon approaching φc\nfrombelow,thenoisecharacteristicchangesfromwhiteatl ow\nφ/φcto a1/fαform close to the transition, with a large in-\ncreaseinthenoisepowerat lowfrequencies. Thesolidlinei n\nFig. 5(a)isa fit with α= 1.0.\nWe can further characterize the noise by measuring the\nnoise power S0, obtained by integrating S(ω)over a range\nof low frequencies51–53. In Fig. 5(b) we plot S0versusφ/φc\nshowing that the noise power peaks just below φc. The noise\nmeasurements taken in the frozen steady state for φ > φ c\nshow very low noise power due to the lack of fluctuations.\nA peak in the noise power has previously been proposed to\nindicate the presence of a critical point in both equilibriu m54\nand nonequilibrium systems55. The increase in the low fre-\nquency noise power indicates that the fluctuations are occur -\nring on larger length scales and thus take longer time scales\nto occur. This is correlated with the living crystals becomi ng\nlongerlivedas φcisapproached. Oncethelivingcrystalstruc-\nturebecomeslargeenough,thesystemcanfindadynamically\n1–10|5-0.4-0.200.20.4V(t)\n05×1071×108\ntime0.50.60.70.80.91P6\n05×1071×108\ntime05×1071×108\ntime05×1071×1082×108\ntime(a) (b) (c) (d)\n(e) (f) (g) (h)\nFig. 6(a,b,c,d)V(t)for asystem at φ/φc= 1.43where stochastic noise isadded inthe form of thermal kicks o f magnitude FT= 0(a), 2.0\n(b),6.0(c),and 9.0(d). (e,f,g,h)The corresponding fract ionof particles withsix-foldneighbors P6(t)for a system at φ/φc= 1.43and\nFT= 0(e), 2.0(f),6.0(g),and 9.0(h).\nfrozenconfigurationandthenremainstrappedinthatstate.\n5 Thermal Noise\nWe next consider the robustness of the transition into the dy -\nnamically frozen state in the presence of stochastic noise,\nintroduced by adding Langevin kicks FTto the equations\nof motion with the following properties: ∝angbracketleftFT(t)∝angbracketright= 0and\n∝angbracketleftFT\ni(t)FT\nj(t′)∝angbracketright= 2ηkBTδijδ(t−t′). Intheabsenceofother\nparticles or a motor force, these kicks induce a random mo-\ntion of an individualparticle. We find that close-packedcry s-\ntals atφ= 0.9with no motor force thermally disorder when\nFT>8.0. In general we find that for φ > φ c, a transition\nto the dynamically frozen state still occurs provided that t he\nthermalfluctuationsaresmallenough;however,evenathigh er\nvaluesof FTweobserveadistinctdifferenceinthedynamics\nofthe systemfor φ > φccomparedto φ < φc.\nIn Fig. 6(a-d) we plot V(t)for a system at φ/φc= 1.43\nwith different thermal noise levels FT= 0.0, 2.0, 6.0, and\n9.0. In Fig. 6(e-h)we show the correspondingfractionofsix-\nfold coordinated particles P6(t)vs time. To compute P6, we\nfindzi, the number of neighborsof particle i, from a Voronoi\nconstruction,andthenobtain P6=N−1/summationtextN\niδ(zi−6). Inthe\nfrozenclusterstate,mostoftheparticleshavesixneighbo rsso\nP6≈0.9asshowninFig.6(e)for FT= 0,wherethevelocity\nfluctuationsare also small as indicated in Fig. 6(a). The tra n-\nsition to the quasistatic cluster state remains robust even for\nfiniteFT, as shownin Fig. 6(b,f)at FT= 2.0where the sys-\ntemrapidlypassesfromthefluctuatingstatetothefrozencl us-ter state and remains trapped for further times with only mi-\nnor rearrangements. The frozen cluster state can be regarde d\nas a solid that only breaks or melts once a critical level of\nthermal fluctuations is added. For increasing FT, the behav-\nior becomesincreasinglyintermittent. A frozencluster fo rms\nfor a period of time but can undergo an instability that sends\nthe system back into a fluctuating state for another period of\ntime before a new frozen cluster appears. This process is il-\nlustrated in Fig. 6(c,g) for FT= 6.0, where the regions of\nstronglyfluctuating V(t)arecorrelatedwithstronglyfluctuat-\ning values of P6(t). We do not observe this type of two-step\nintermittent behavior in the fluctuating regime at φ < φ cfor\nFT= 0. The higher temperature systems no longer become\npermanently absorbed into the frozen state; instead, we ob-\nserve temperature-induced transitions for φ > φ cfrom the\nfrozen state at low temperature to the intermittent state at in-\ntermediatetemperatures. Whenthe stochasticfluctuations are\nlarge enough the system always remains in the fluctuating\nstate, as shown in Fig. 6(d,h) for FT= 9.0. The value of\nFTat which the system remainspermanentlyin a fluctuating\nstate decreaseswithdecreasing φ/φc. Theseresultsshowthat\nthe transition to the dynamically frozen state can still be r e-\nalized in the presence of fluctuations provided the addition al\nstochasticfluctuationsaresufficientlysmall.\n6| 1–10\n-0.6-0.4-0.200.20.4V\n02×1064×1066×1068×106\ntime0.30.40.50.60.70.8P6(a)\n(b)\nFig. 7(a)V(t)for a system at φ= 0.363containing Np= 2\nobstacles or pinned disks. (b)The corresponding P6(t). Even\nthoughφ < φ c,the system shows a transitiontoa dynamically\nfrozenstate consistingof a pinned cluster.\n6 Dynamic Freezing in the Presence of Obsta-\ncles\nWe next consider the effects of adding a small numberof ob-\nstacles to the system in form of stationary or pinned disks.\nIn this case we find that even a very low obstacle density can\ncausethesystemtoorganizeintoadynamicfrozenstateatva l-\nues ofφwell below the obstacle-free critical value φc; how-\never,thenatureofthefrozenstatesinthepresenceandabse nce\nof obstacles are quite different. Figure 7(a) shows V(t)for a\nsystem at φ= 0.363containing Np= 2obstacles. There\nis a clear transition from a fluctuating state to a dynamicall y\nfrozen state even though φ < φ c. The corresponding P6(t)\ninFig.7(b)showslargevariationsinthefluctuatingstatew ith\nvalues as large as P6= 0.81indicating transient clustering,\nwhile at the transition to the frozen state we find P6≈0.84\nasthesystemformsapinnedhexagonal-facetedcrystalthat is\ncenterednearoneoftheobstacles.\nThe obstacles serve as cluster nucleation sites. Figure 8\nshowssnapshotsatdifferenttimesofthediskandobstaclep o-\nsitions for the system in Fig. 7. In the initial fluctuating st ate\nin Fig. 8(a), there are small transient clusters. Figure 8(b )\nshows the system at a later time but still in the fluctuating\nstate where a cluster nucleates around one of the obstacles.\nThis cluster eventually breaks apart as illustrated in Fig. 8(c)\nwhich shows an even later time in the fluctuating state. Fig-\nure 8(d) shows the configuration in the dynamically frozenx (a)y\nx (b)y\nx (c)y\nx (d)y\nFig. 8Snapshots of diskand obstacle positions ina system with\nφ= 0.363andNp= 2where the arrows indicate the directionof\nthe motor force foreach disk. Moving disks are colored asinF ig.4.\nReddisks are immobile obstacles. (a) The initialfluctuatin gstate\ncontaining small transient clusters. (b) Ata later time,tr ansient\nclusters are nucleated by the obstacles. (c) Ata stilllater time, the\ntransient cluster shown in(b) has broken apart. (d)The dyna mically\nfrozen steadystate where a faceted crystal forms around the\nobstacles.\n1–10|7x (a)y\nx (b)y\nx (c)y\nx (d)y\nFig. 9Snapshots of diskand obstacle positions indynamically\nfrozenstates of systems containing obstacles where the arr ows\nindicate the directionof the motor force for each disk. Movi ng disks\narecolored as inFig.4. Reddisks areimmobile obstacles. (a ) At\nφ= 0.363andNp= 20, the cluster nolonger has facets. (b) At\nφ= 0.363andNp= 100, the cluster starts tobecome much more\ndisordered. (c)At φ= 0.363andNp= 200, multipleclusters start\ntoform. (d) At φ= 0.1212andNp= 60cluster formationstill\noccurs.\nsteady state, where the particles form a single cluster that has\nthe shape of a faceted hexagonal crystal. Unlike the frozen\nsteadystateintheobstacle-freesystem,theclusterinFig .8(d)\nis not drifting but is pinned by the obstacles. The cluster ex -\nhibitstheadditionalfeaturethatthe particlesinthe uppe rhalf\noftheclusteraremovinginthenegative ydirectiononaverage\nwhile the particles in the lower half of the cluster are movin g\nin the positive ydirection on average. In particular, the parti-\ncles on the outer edges of the crystal have their motor forces\noriented toward the center of the cluster. There are a small\nnumber of particles outside of the cluster that are moving in\nclosedorbits.\nIngeneralfor φ < φc,whenasmallnumberofobstaclesare\npresent the system eventually forms a faceted pinned crysta l\ncenteredaroundthe defects. When more obstacles are added,\nthe transient time to reach the pinned state decreases and th e\nsystemmayformmultiplepinnedclusters. InFig.9(a),asys -\ntem with φ= 0.363andNp= 20forms a single cluster that\nno longer has the faceted features found for lower obstacle0 0.01 0.02 0.03\nφp0100200 τ\n0.1 1\n|φ −pφ  |p\nc01101001000τ\n0.1 0.2 0.3 0.4\nφ00.010.02\nφp\nFluctuatingDynamically\nFrozen(a) (b) (c)\nFig. 10(a)Thetransienttime τtoreachapinnedcluster statevsthe\nobstacle density φpforφ= 0.363(circles), 0.303(squares), 0.2424\n(diamonds), and 0.182(triangles). (b) A log-logplot of τvs\n|φp−φp\nc|, where the solidline isa fitto τ∝ |φp−φp\nc|−βwith\nβ= 1.9. (c)φpvsφshowing the separationbetween the region\nwhere thesystem remains fluctuatingand the dynamically fro zen\nregion where the system eventually forms a pinned cluster st ate.\ndensities. Addingmoreobstaclestothesamesystemproduce s\nthefrozensteadystateshowninFig.9(b),wherethecluster ing\nis much more disorderedwhen Np= 100. Figure 9(c) shows\nthe same systemwith Np= 200,wheremultiplepinnedclus-\nters form. Thecluster states canalso occurat lowerdensiti es,\nas illustrated in Fig. 9(d) for a system with φ= 0.1212and\nNp= 60.\nForφ < φ cwe find that there is a critical obstacle den-\nsityφp\ncbelow which the system never reaches a dynamically\nfrozen steady state. In Fig. 10(a) we plot the transient time τ\nrequired to reach a pinned cluster versus the obstacle densi ty\nφpforφ= 0.363,0.303,0.2424, and0.182, showing that τ\ndivergesat differing critical pinning densities φp\nc. The curves\ncan befit tothe form τ∝ |(φp−φp\nc)|−βwithβ≈1.9±0.4,\nas shown in the log-log plot of Fig. 10(b). The fact that the\ntransition from the fluctuating to the dynamical state shows a\ndifferent scaling than in the obstacle-free case suggests t hat\ntheadditionofobstacleschangestheclassofthetransitio n. In\nFig. 10(c) we show the line dividing the dynamically frozen\nand fluctuating steady states on a plot of obstacle density φp\nversusdiskdensity φ. Asφdecreases,thedensityofobstacles\nrequiredtoinducetheformationofadynamicallyfrozensta te\nincreases.\nThere have been several studies examining the effect of\nquenched disorder on active matter systems. For flocking\nmodels it was found that the coherence of the system passes\nthrough a maximum when random noise is added56,57. For\nrun and tumble dynamics it was shown that the effective mo-\nbility decreases with increasing run length58. In both classes\nof system, a fluctuatingstate is always present,so thereis n ot\na sharp transition to a completely pinned state as we observe\nin thiswork.\n8| 1–10\n7 Summary\nWe examine a system of run-and-tumbledisks in the limit of\ninfiniteruntimeandfindthataboveacriticaldensity,atran si-\ntion occurs from a fluctuating state to an absorbing quiescen t\nstate. In the dynamicallyfrozenstate, a driftingcluster f orms\nand the fluctuations in the system almost completely vanish.\nThetransienttimerequiredforthesystemtoorganizeintot he\ndynamically frozen state divergesas a power law as the criti -\ncaldensityisapproachedwithanexponentof ν||≈1.21. This\nbehaviorisverysimilartothatobservedintherandomorgan i-\nzationtransitiontoanon-fluctuatingstateinshearedcoll oidal\nsystemswherethe transienttimesalso show powerlawdiver-\ngences with comparable critical exponents. A key differenc e\nbetweentherandomorganizationandactivemattersystemsi s\nthat in the quiescent state, the colloidal particles cease t o in-\nteract with each other, while the active matter particles fo rm\na strongly correlated or jammed configuration in which most\nof the particles are in constant contact. We also find that the\nmagnitudeofthefluctuationsdivergeas φcisapproachedfrom\nbelow. Thedynamictransitionisrobustagainsttheadditio nof\na certain range of stochastic or thermal fluctuations. When a\nsmall numberof obstaclesare addedto thesystem, the transi -\ntion to a frozen state can occur at densities much lower than\nthe obstacle-free critical density, and in this case the fro zen\nstate ischaracterizedbytheformationofpinnedfacetedcr ys-\ntals.\n8 Acknowledgements\nThis work was carried out under the auspices of the NNSA\nof the U.S. DoE at LANL under Contract No. DE-AC52-\n06NA25396.\nReferences\n1 H. Hinrichsen, Adv. Phys. ,2000,49,815.\n2 S. L¨ ubeck and P.C.Heger, Phys. 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Reichhardt and C.J.Olson Reichhardt, arXiv:1402.326 0.\n10| 1–10\n"    },    {        "title": "1406.4902v1.Quench_between_a_Mott_insulator_and_a_Lieb_Liniger_liquid.pdf",        "content": "Quench between a Mott insulator and a Lieb-Liniger liquid\nGarry Goldstein and Natan Andrei\nDepartment of Physics, Rutgers University and\nPiscataway, New Jersey 08854\nIn this work we study a quench between a Mott insulator and a repulsive Lieb-Liniger liquid.\nWe find explicitly the stationary state when a long time has passed after the quench. It is given\nby a GGE density matrix which we completely characterize, calculating the quasiparticle density\ndescribing the system after the quench. In the long time limit we find an explicit form for the local\nthree body density density density correlation function and the asymptotic long distance limit of the\ndensity density correlation function. The latter is shown to have gaussian decay at large distances.\nI. INTRODUCTION\nThe question of whether an isolated quantum system\nequilibrates is of fundamental interest to our understand-\ning of nonequilibrium dynamics. This question is ex-\ntremely difficult to study in the condensed matter con-\ntext as there are few truly isolated systems. However\nrecent advances in the study of ultracold atoms have\nbrought this question to the forefront. Thanks to an\nunprecedented amount of tunability and isolation in cold\natom systems it has become possibly to study complex\nnonequilibrium dynamics of many body systems. Trig-\ngered by spectacular experimental advances [1–9] there\nhas been great effort recently in the theoretical study\nof thermalization of isolated systems [10–18]. The most\nimportant questions asked are whether a system equili-\nbrates, what are the dynamics of the equilibration pro-\ncess and what ensemble if any describes the long time\nlimit of an isolated system?\nOne of the most surprising recent experimental and\ntheoretical results is that at long times the quenched\nLieb-Liniger gas [19] retains memory of its initial state\n[1, 18, 20] and does not appear to relax to thermody-\nnamic equilibrium. This is due to the fact that the Lieb\nLiniger hamiltonian:\nHLL=1\u0002\n\u00001dxn\n@xby(x)@xb(x) +c\u0000\nby(x)b(x)\u00012o\n;(1)\nhas an infinite number of conserved charges Ii. Here\nby(x)is the bosonic creation operator at the point xand\ncis the coupling constant. These conserved quantities in\nturn imply that there is a complete system of eigenstates\nfor the Lieb Liniger gas which may be parametrized by\nsets of rapiditiesfkig. To understand the equilibration of\nthis gas it was recently proposed that it is insufficient to\nconsider only thermal ensembles but it is also necessary\nto include these nontrivial conserved quantities. It was\nshown [18, 20] that the gas relaxes to a state given by the\ngeneralized Gibbs ensemble GGE with its density matrix\nbeing given by\n\u001aGGE =1\nZexp\u0010\n\u0000X\n\u000biIi\u0011\n(2)\nWhere the Iiare the conserved quantities given by\nt!\"Trap\t\r Released\t\r \nFigure 1: The Lieb Liniger gas is initialized in a nonequilib-\nrium state - the Mott insulator (say by applying an external\nlattice). The lattice is removed and the system is allowed\nto relax for a long time. Its correlation functions are then\nmeasured.\nIijfkgi=Pkijfkgiand the\u000biare the generalized in-\nverse temperatures and Zis a normalization constant\ninsuringTr[\u001aGGE] = 1. It was shown that correla-\ntion functions of the Lieb-Liniger gas at long times may\nbe computed by taking their expectation value with re-\nspect to the GGE density matrix, e.g. h\u0002 (t!1 )i=\nTr[\u001aGGE\u0002]. It was also later shown [21] that the the\nGGE ensemble is equivalent to a pure state \u001aGGE\u0018= \f\f\f~k0ED\n~k0\f\f\ffor an appropriately chosen\f\f\f~k0E\n.\nIn this work we consider the repulsive Lieb-Liniger\nmodel. We consider the case when the system is ini-\ntialized in a deep lattice. So the state of the system is\nwell described by a Mott insulator:\nj\b0i=1Y\nj=\u00001\u00021\n\u00001'(x+jl)by(x)j0i(3)\nwith'(x) =e\u0000x2=\u001b\n(\u0019\u001b= 2)1=4, withp\u001b\u001cl. The lattice is then\nreleased so that the system evolves under the Hamilto-\nnian given in Eq. (1). The Lieb-Liniger gas is allowed\nto relax for a long time, j\ti=eiHLLtj\b0i, thereby es-arXiv:1406.4902v1  [cond-mat.quant-gas]  18 Jun 20142\ntablishing a GGE with \u001aGGE\u0018=j\tih\tj, see figure (1).\nWe note that using a saddle point approach the authors\nin [21] were able to identify the limit value j\tih\tjwith\f\f\f~k0ED\n~k0\f\f\f, where the quasiparticle distribution character-\nizing~k0satisfies the appropriate Bethe-Ansatz equation\n(see below).\nIn this work we establish the exact quasiparticle den-\nsity of this GGE. We use it to calculate various correla-\ntion functions for the final state. In particular we find a\nGaussian decay of the density density correlation func-\ntion at large distances for the final state. Overall we ob-\ntain a complete characterization of the final state. Our\nresults are applicable for any coupling constant c > 0.\nWe would like to note that a quench between a BEC and\na Lieb-Liniger gas has been previously considered in [22].\nThe rest of the paper is organized as follows. In Sec-\ntion II we calculate the quasiparticle density of the final\nstate; in Section III we compute the two and three body\nlocal correlation functions for the final state; in Section\nIV we compute the long distance density density corre-\nlation functions of the final state, here we also generalize\nour results (we see that the final correlation functions are\nheavily independent of certain types of disorder) and in\nSection V we conclude.\nII. QUASIPARTICLE DENSITY\nWe would like to characterize the final state obtained\nin the quench between a Mott insulator and the Lieb-\nLiniger gas, it is given by a GGE. For the Lieb-Liniger\ngas it is possible to reduce the GGE density matrix to a\npure state [21]. That is for any local correlation function\n\u0002:\ntr[\u0002\u001aGGE] =D\n~k0\f\f\f\u0002\f\f\f~k0E\n(4)\nfor an appropriately chosen eigenstate of the Lieb Liniger\nHamiltonian\f\f\f~k0E\n. Furthermore following the authors of\n[21] we are able to specify this pure state\f\f\f~k0E\n. To do so\nlet us denote by L\u001ap(k)dkas the number of particles in\nthe interval [k;k+dk],L\u001ah(k)dkas the number of holes\nin the interval [k;k+dk]andL\u001at(k)dkas the number of\nstates in the interval [k;k+dk]so that\u001at(k) =\u001ap(k) +\n\u001ah(k). HereLis the length of the system. Then the\ndensity of particles in k-space corresponding to\f\f\f~k0E\nis\ngiven by the following equations [21]:\nL\u0002\ndk\u001ap(k)kn=In(t= 0): (5)\nTherefore to characterize the density of quasiparticles\nof the final sate one needs to compute the various con-\nserved quantities In(t= 0). To do so note that the con-\nserved charges Inare given by integrals of local densities:\nIn=\u0002\ndxJn(x) (6)with the local densities being given by\nJ0(x) =by(x)b(x),J1(x) =iby(x)@xb(x),\nJ2(x) = @xby(x)@xb(x) +c\u0000\nby(x)b(x)\u00012,\nJ3(x) =by(x)@3\nx3b(x)\u00003c\n2\u0000\nby(x)\u00012@x(b(x))2etc.\nSince these changes are local and the wave function in\nEquation (3) is given by a sum of local non-overlapping\nsingle particle terms we have that:\nIn(t= 0) =L\nlh0j\u0002\ndx'(x)b(x)In\u0002\ndy'(y)by(y)j0i\n(7)\nHowever for a single particle state it is straightforward\nto calculate the value of a local conserved quantity; in-\ndeedInjki=knjki. From this we obtain that the con-\nserved charges are zero for odd nand for even nthey are\ngiven by:\nIn(t= 0) =L\nl\u00122\n\u001b\u0013n\n2n!\n2n\n2\u0000n\n2!\u0001 (8)\nIn particular these are finite [22]. Therefore accord-\ning to Equation (5) the quasiparticle density \u001ap(k)is a\ndistribution whose moments are given by Equation (8).\nHowever it is straightforward to obtain such a distribu-\ntion it is given by:\n\u001ap(k) =\u001b1\n2\n\u00191\n2lexp\u0012\n\u0000k2\u001b\n2\u0013\n(9)\nWenotethatthisexpressioniscompletelyindependent\nof the coupling constant cand works for all c>0. This\ndistribution completely characterizes all the properties\nof the final state and completely determines \u001aGGE. For\nfuture use we would like to compute the total density of\navailable states \u001at(k)as well as the ratio\u001ap(k)\n\u001at(k). The total\nquasiparticle density is given by the equation [23]:\n\u001at(k) =1\n2\u0019+1\n2\u0019\u0002\ndqK (k;q)\u001ap(q);(10)\nhereK(k;q) =2c\nc2+(k\u0000q)2. We note that for l\u001dp\u001b\n\u001at(k)\u0018=1\n2\u0019. Furthermore the occupation probability is\ngivenbyf(k)\u0011\u001ap(k)\n\u001at(k)\u0018=2p\u0019\u001b\nlexp\u0010\n\u0000k2\u001b\n2\u0011\n. Wenotethat\nthe charges In(t= 0)grow very rapidly so are high order\nchargesareimportantassuchtheresultspresentedin[24]\nabout exponential decay of density density correlations\nwill not apply as we will see in Eq. (20).\nIII. LOCAL CORRELATION FUNCTIONS\nWe would like to calculate the local correla-\ntion functionshby(0)b(0)i,hby(0)by(0)b(0)b(0)iand\nhby(0)by(0)by(0)b(0)b(0)b(0)i. The first of these\nquantities measures the density of particles in the system\nthe second measures the effect of interactions while the\nthird measure the the three particle recombination rate.3\nFrom conservation of particle number the density of par-\nticles is given in the long time limit by \u001a=hby(0)b(0)i=\n1\nl. The two body correlation function is given by [25]:\n\nby(0)by(0)b(0)b(0)\u000b\u0018=\n\u0018=2\u0001dk1\n2\u0019\u0001dk2\n2\u0019f(k1)f(k2)(k2\u0000k1)2\n(k2\u0000k1)2+c2+:::::(11)\nThis integral may be done explicitly:\n\nby(0)by(0)b(0)b(0)\u000b\n=2\nl2\u00002p\n\u0019c2\u001b\nl2\u0002\n\u0002\"\nexp\u0012\u001bc2\n4\u0013\nErfc r\n\u001bc2\n4!#\n(12)\nThe three body integrals may be done similarly [25].\nThey are given by:\n\nby(0)by(0)by(0)b(0)b(0)b(0)\u000b\u0018=\n\u0018=6\u0001\ndk1dk2dk3f(k1)f(k2)f(k3)\u0002\n\u0002(k2\u0000k1)2\n(k2\u0000k1)2+c2(k3\u0000k1)2\n(k3\u0000k1)2+c2(k3\u0000k2)2\n(k3\u0000k2)2+c2+::::(13)\nrSimplifying we can evaluate this expression in various\nlimits. In the case that \u001bc2\u001c1we obtain that\n\nby(0)by(0)by(0)b(0)b(0)b(0)\u000b\u0018=6\nl3(14)\nIn the case that \u001bc2\u001d1we obtain that:\n\nby(0)by(0)by(0)b(0)b(0)b(0)\u000b\u0018=9\u000229\n2\nl3c6\u001b3(15)\nAs such we see a strong suppression of the three body\ndecay rates for large coupling constants c.\nIV. DENSITY DENSITY CORRELATIONS\nThe density density correlation function for generic\nstates has been previously calculated. We can use this\nto find the density density correlation function for our\nGGE. It is given by [26]:\nh\u001a(x)\u001a(0)i=\u001a2+1X\nk=2\u0000k(x) (16)\nwith the dominant term being given by [26]:\n\u00002(x) =\u00001\n4\u00192\u00021\n\u00001dk1!(k1)f(k1)\u00021\n\u00001dk2!(k2)f(k2)\u0002\n\u0002\u0012k1\u0000k2+ic\nk1\u0000k2\u0000ic\u0013\u0014p(k1;k2)\nk1\u0000k2\u00152\nexp (xp(k1;k2))\n(17)\nHere!(k) = exp\u0010\n\u00001\n2\u0019\u00011\n\u00001K(k;q)f(q)dq\u0011\nand the\nfunctionp(k1;k2)is given by:\np(k1;k2) =\u0000i(k1\u0000k2)+\u00021\n\u00001dtf(t)P(t;k1;k2)(18)The function P(t;k1;k2)is defined through the following\nintegral equation:\n1 + 2\u0019P(t;k1;k2) =\u0012k1\u0000t+ic\nk1\u0000t\u0000ic\u0013\u0012k2\u0000t\u0000ic\nk2\u0000t+ic\u0013\n\u0002\n\u0002exp\u0012\u0002\nK(t;s)f(s)P(s;k 1;k2)\u0013\n:\n(19)\nIn the case when l\u001dp\u001bthese equations greatly sim-\nplify. Indeed in that case !(k)\u0018=e\u00001andp(k1;k2) =\n\u0000i(k1\u0000k2). With these simplifications the integral in\nEq. (17) becomes,\nh\u001a(x)\u001a(0)i\u0018=\u001a2+1\n4\u00192e2l2exp\u0012\n\u0000x2\n\u001b\u0013\n(20)\nWe see a gaussian decay of correlation functions at large\ndistances.\nWe would like to note that the final result e.g. the final\nquasiparticle density see Eq. (9) is independent of posi-\ntional disorder in the Mott insulator. Indeed if the cen-\nters of the particles in the Mott insulator do not form a\nuniform lattice but instead have random locations (which\nwould happen for example for a partial filling of the first\nMott band) none of the conserved quantities, see Equa-\ntion (8), change. In particular if the particles have an\naverage interparticle separation \u0016lthen the final state has\nexactly the same correlations as those given in the main\ntext except with l!\u0016lsee Equations (12), (14), (15) and\n(20). As such we can characterize these states as well.\nV. CONCLUSIONS\nWe have studied a quench between a Mott insulator\nand a Lieb-Liniger liquid. We have found that at long\ntime the system equilibrates to a GGE. We have charac-\nterized the GGE quasiparticle density exactly and found\nit to be gaussian. We were able to compute local two\nand three body correlation functions as well as density\ndensity correlations. We have found the density density\ncorrelation to decay in a Gaussian manner with distance.\nWe argued our results to be robust to certain kinds of\ndisorder. The methods used in this paper open the pos-\nsibilitytocomputethefinalstateafteraquenchwhenthe\ninitial state is given by a collection of isolated few body\nstates (such as a collection of dimer molecules on a lat-\ntice). The conserved quantities can be calculated in that\ncase as well allowing us to obtain the final quasiparticle\ndensity as well as various correlation functions.\nThe results presented here have direct experimental\napplicability. Indeed it is not too difficult to prepare\na gas of bosons in the Mott insulator state [27]. The\nthree body density density correlations may be measured\nthrough trap loss rates [25] or through the third moment\nof the number of particles [28]. The density density cor-\nrelators studied may be measured through time of flight\ninterferometry [27], all directly experimentally relevant.4\nAcknowledgments : This research was supported by NSF grant DMR 1006684 and Rutgers CMT fellowship.\n[1] T. Kinoshita, T. Wenger, and D. S. Weiss, Nature (Lon-\ndon)440, 900 (2006)\n[2] M. Greiner, O. Mandel, T. W. Hansch and I. Bloch, Na-\nture419, 51, (2002).\n[3] S.Hofferberth, I.Lesanovsky, B.Fischer, T.Schummand\nJ. Schniedmayer, Nature 449, 324 (2007).\n[4] E. Haller, M. Gusatvsson, M. J. Mark, J. G. Danzl, R.\nHart, G. Pupillo and H.-C. Nagerl, Science 325, 1224\n(2009).\n[5] S. Trotsky, Y.-A. Chen, A. Flesch, I. P. McCulloch, U.\nSchollwock, J. Eisert and I. Bloch, Nature Phys. 8, 325\n(2012).\n[6] M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P.\nSchauss, T. Fukuhara, C. 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Bou-\nchoule, Phys. Rev. Lett. 105, 230402 (2010)."    },    {        "title": "1406.5838v1.New_inequality_for_density_matrices_of_single_qudit_states.pdf",        "content": "arXiv:1406.5838v1  [quant-ph]  23 Jun 2014New inequality for density matrices of single qudit states\nV. N. Chernega, O. V. Man’ko, V. I. Man’ko\nP.N. Lebedev Physical Institute, Russian Academy of Sciences\nLeninskii Prospect, 53, Moscow 119991, Russia\nEmail: omanko@sci.lebedev.ru\nAbstract\nUsing the monotonity of relative entropy of composite quant um systems we obtain new\nentropic inequalities for arbitrary density matrices of si ngle qudit states. Example of qutrit\nstate inequalities and the ”qubit portrait” bound for the di stance between the qutrit states\nare considered in explicit form.\nKey words: Hermitian matrix, composite quantum system, noncomposite quant um system,\nentropic and information inequalities\nPACS: 42.50.-p,03.65 Bz\n1 Introduction\nThere exist different entropic inequalities for composite quantum sy stems [1, 2, 3, 4, 5, 6, 7,\n8]. Recently it was formulated [9, 10, 11, 12, 13] that the inequalities known for composite\nsystems uncluding subadditivity condition and strong subadditivity c ondition can be generalized\nfor noncomposite systems like single qudit. The aim of this work is to us e the developed approach\n[9, 10, 11, 12, 13] and to get the new entropic inequalities for single q udit states on the base of\nknown properties of relative entropy of composite systems like the monotonicity of the relative\nentropy. We obtain a generic new matrix inequality for Hermitian matr ices. In case where the\nmatrices coincide with the two density matrices of bipartite quantum system states the new\ninequality coincides with the monotonicity property of the relative en tropy. But it is valid for\narbitrary pair of density matrices also. The bound for distance bet ween two arbitrary density\nmatrices is obtained and expressed in terms of matrix elements of th ese matrices. The paper is\norganized as follows. In next Sec. 2 the property of relative entro py are considered. The example\nof qutrit and 4 ×4 - density matrices are presented in Sec. 3. Conclusion and perspe ctives are\ngiven in Sec. 4.\n2 Relative entropy\nThe relative entropy between density matrices ρandσis defined as\nS(ρ||σ) = Tr(ρlnρ−ρlnσ), (1)\n1whereρandσare density matrices of quantum states. Also if the density matrice s are the density\nmatrices of bipartite system states, i.e. ρ/mapsto→ρ(1,2) andσ/mapsto→σ(1,2), one has inequality\nTr(ρ(1,2)lnρ(1,2)−ρ(1,2)lnσ(1,2))≥Tr(ρ(1)lnρ(1)−ρ(1)lnσ(1)), (2)\nwhere\nρ(1) = Tr 2ρ(1,2), σ(1) = Tr 2σ(1,2).\nThis inequality reflects the monotonicity property of relative entro py.\nAnother analogous inequality holds\nTr(ρ(1,2)lnρ(1,2)−ρ(1,2)lnσ(1,2))≥Tr(ρ(2)lnρ(2)−ρ(2)lnσ(2)), (3)\nwhere\nρ(2) = Tr 1ρ(1,2), σ(2) = Tr 1σ(1,2).\nNow we extend the inequalities for arbitrary matrices. For example, given a density N×Nmatrix\nρin block form\nρ=/parenleftigg\nA B\nC D/parenrightigg\nand two N×N-matrices\nρA=/parenleftigg\nA0\n0 0/parenrightigg\n, ρD=/parenleftigg\nD0\n0 0/parenrightigg\n.\nLet us construct positive map ρ/mapsto→ρ1where\nρ1=ρA+ρD≡/parenleftigg\nA+D0\n0 0/parenrightigg\n.\nThe block Dism×m-matrix, i.e. block Ais (N−m)×(N−m)-matrix and m≤N−m.\nThe following matrix inequality holds for the density matrix/parenleftigg\nA B\nC D/parenrightigg\ngiven in block form\nTr/bracketleftigg/parenleftigg\nA+D0\n0 0/parenrightigg\nln/parenleftigg\nA+D0\n0 0/parenrightigg\n−/parenleftigg\nA+D0\n0 0/parenrightigg\nln/parenleftigg\nA B\nC D/parenrightigg/bracketrightigg\n≥0.\nAnother inequality for this matrix reads\nTr\n\nTrATrB0\nTrCTrD0\n0 0 0\nln\nTrATrB0\nTrCTrD0\n0 0 0\n−\nTrATrB0\nTrCTrD0\n0 0 0\nln/parenleftigg\nA B\nC D/parenrightigg\n≥0.\nThis inequality means the nonnegativity condition for relative entrop y of the initial density matrix/parenleftigg\nA B\nC D/parenrightigg\nand its ”portraits” obtained by applying two positive maps\n/parenleftigg\nA B\nC D/parenrightigg\n/mapsto→/parenleftigg\nA+D0\n0 0/parenrightigg\n,/parenleftigg\nA B\nC D/parenrightigg\n/mapsto→\nTrATrB0\nTrCTrD0\n0 0 0\n.\n2Thenewmatrixinequality connectedwithmonotonicitypropertyofr elativeentropyoftwo density\nmatrices\nρ=/parenleftigg\nA B\nC D/parenrightigg\n, σ=/parenleftigg\na b\nc d/parenrightigg\nis given by the relation\nTr/bracketleftigg/parenleftigg\nA B\nC D/parenrightigg\nln/parenleftigg\nA B\nC D/parenrightigg\n−/parenleftigg\nA B\nC D/parenrightigg\nln/parenleftigg\na b\nc d/parenrightigg/bracketrightigg\n≥Tr/bracketleftigg/parenleftigg\nA+D0\n0 0/parenrightigg\nln/parenleftigg\nA+D0\n0 0/parenrightigg\n−/parenleftigg\nA+D0\n0 0/parenrightigg\nln/parenleftigg\na+d0\n0 0/parenrightigg/bracketrightigg\n.\nThis inequality holds for two arbitrary nonnegative matrices with Tr ρ= Trσ= 1.\n3 Qutrit and qudit examples\nThe example of qutrit density matrices ρandσprovides the new inequality\nTr\n\nρ11ρ12ρ13\nρ21ρ22ρ23\nρ31ρ32ρ33\n\nln\nρ11ρ12ρ13\nρ21ρ22ρ23\nρ31ρ32ρ33\n−ln\nσ11σ12σ13\nσ21σ22σ23\nσ31σ32σ33\n\n\n≥\nTr/bracketleftigg/parenleftigg\nρ11+ρ33ρ12\nρ21ρ22/parenrightigg/bracketleftigg\nln/parenleftigg\nρ11+ρ33ρ12\nρ21ρ22/parenrightigg\n−ln/parenleftigg\nσ11+σ33σ12\nσ21σ22/parenrightigg/bracketrightigg/bracketrightigg\n.\nThe relative entropy of two states is known to be nonnegative char acteristic of a distance between\nthe states. The new above inequality gives the bound for the distan ce which equals to relative\nentropy of two ”qubits”. The bound equals to the maximal ”qubits” distance obtained by all\nthe permutations of indices 1,2,3, in the qutrit states. Thus the pro blem of distance bound of\nqudits expressed in terms of the relative entropy is reduced to the problem of distance of ”qubit\nportrait” [14, 15] of the qudit states. This property is analogous t o the property of entanglement\nof composite qudit system states which can be characterized by th e entanglement of the qubit\nportrait of these states. One can write the chain of the inequalities for arbitrary density matrices\nρandσ. We demonstrate such chain for density 4 ×4-matrices ρandσwhich can be associated\neither with two-qubit state or with the states of qudit with j= 3/2. One has the chain of\ninequalities\nTr\n\n\nρ11ρ12ρ13ρ14\nρ21ρ22ρ23ρ24\nρ31ρ32ρ33ρ34\nρ41ρ42ρ43ρ44\n\nln\nρ11ρ12ρ13ρ14\nρ21ρ22ρ23ρ24\nρ31ρ32ρ33ρ34\nρ41ρ42ρ43ρ44\n−ln\nσ11σ12σ13σ14\nσ21σ22σ23σ24\nσ31σ32σ33σ34\nσ41σ42σ43σ44\n\n\n\n≥\nTr\n\n\nρ11+ρ44ρ12ρ13\nρ21ρ22ρ23\nρ31ρ32ρ33\n\nln\nρ11+ρ44ρ12ρ13\nρ21ρ22ρ23\nρ31ρ32ρ33\n−ln\nσ11+σ44σ12σ13\nσ21σ22σ23\nσ31σ32σ33\n\n\n\n≥\n3Tr/braceleftigg/parenleftigg\nρ11+ρ33+ρ44ρ12\nρ21 ρ22/parenrightigg/bracketleftigg\nln/parenleftigg\nρ11+ρ33+ρ44ρ12\nρ21 ρ22/parenrightigg\n−ln/parenleftigg\nσ11+σ33+σ44σ12\nσ21 σ22/parenrightigg/bracketrightigg/bracerightigg\n≥0.\nThus the distance of two-qubits or j= 3/2 qudit states has the bound determined by relative\nentropy of ”qutrit” states which also has bound determined by rela tive entropy of ”qubit” states.\nThe obtained inequalities are valid for arbitrary density matrices inclu ding the noncomposite\nsystem states of a single qudit.\nForgivenfinitedensitymatrix ρandarbitraryHermitian N×N-matrixBonecangetinequality\n0≤exp[−Tr(ρlnρ)]≤/bracketleftig\n(Tre−B)(TreB)/bracketrightig1/2.\nForB=ρlnρone has:\nexp[−Tr(ρlnρ)]≤ {[Trexp(−ρlnρ)][Trexp( ρlnρ)]}1/2.\nIfbk(k= 1,2,...,N) are eigenvalues of the matrix Bthe inequality means that there exists the\nbound for the sum\nN/summationdisplay\nk=1N/summationdisplay\nj=1exp(bk−bj)≥N2.\nForN= 2 the inequality provides obvious relation\n2[1cosh(b1−b2)]≥4\nwhich is saturated for b1=b2. One can also check that for any set of reals bk,k= 1,2,...,Nand\nnonnegative numbers wksuch that/summationtext\nkwk= 1 the following inequality holds\nlnN/summationdisplay\nk=1[exp(−bk)+N/summationdisplay\nk=1(wklnwk+wkbk)≥0.\nThus for Nrealbkand probability vector /vector w={wk}one has the generic inequality. If the\nqudit state is associated with spin tomogram [16, 17, 18] or probabilit y vector /vector w(u) on the unitary\ngrouputheobtainedinequality for bk=−wk(u)providessomeuncertainty relationformeasurable\nprobabilities wk(u).\nConclusion\nTo conclude we formulate the main result of our work. Using the ”por trait” approach [14, 15]\nto the density matrices of single qudit states the analog of monoton icity property of relative\nentropy valid for bipartite quantum system was constructed for s ystems without subsystems.\nNewmatrixinequalities forqutrit density matriceswere obtained. Th echain ofmatrixinequalities\nfor single qudit states generalizing the chain of the analogous inequa lities for multipartite qudit\nstates was demonstrated on example of qudit j= 3/2 systems and two-qubit system. The generic\ninequality for Nreal numbers and a probability N-vector was obtained. Also an ”uncertainty\nrelation” for spin tomogram was obtained. The possible applications o f the obtained inequalities\nto analyze quantum correlations in the states of noncomposite sys tems will be considered in future\npublication.\n4References\n[1] M. A. Nielsen, and I. L. Chuang, ”Quantum computation and quan tum information”, Cam-\nbridge University Press, Cambridge (2000).\n[2] E. H. Lieb, and M. B. Ruskai, ”Proof of the Strong Subadditivity o f Quantum Mechanichal\nEntropy”, J. Math. Phys. ,14, 1938-1948 (1973).\n[3] M. A. Nielsen, D. Petz, ”A simple proof of the strong subadditivity inequality”,\narXiv:quant-ph/0408130 (2005).\n[4] Eric A. Carlen, Elliott H. Lieb, ”A Minkowski Type Trace Inequality a nd Strong Subaddi-\ntivity of Quantum Entropy II: Convexity and Concavity”, Lett. Math. Phys. ,83, 107 (2008);\narXiv:0710.4167 (2007).\nEric A. Carlen, Elliott H. Lieb, ”A Minkowski Type Trace Inequality and Strong Subaddi-\ntivity of Quantum Entropy”, arXiv:math/0701352 (2007).\n[5] H. Araki, E. H. Lieb, ”Entropy Inequalities”, Commun. Math. Phys. ,18, 160170 (1970).\nEric A. Carlen, and Elliott H. Lieb, ”Bounds for Entanglement via an Ex tension of Strong\nSubadditivity of Entropy”, Letters in Mathematical Physics ,101, 1-11, (2012).\nE. H. Lieb, ”Convex Trace Function and Proof of Wigner-Yanase-D yson Conjecture”, Adv.\nMath.,11, 267288 (1973).\nE. H. Lieb, ”Some Convexity and Subadditivity Properties of Entrop y”,Bull. AMS ,81, 113\n(1975).\nM. B. Ruskai, ”Inequalities for Quantum Entropy: A Review with Cond itions for Equality”,\nJ. Math. Phys. ,43, 43584375 (2002); erratum 46, 019901 (2005).\nM. B Ruskai, ”Inequalities for traces on Von Neumann algebras”, Commun. Math. Phys. ,26,\n280289 (1972).\n[6] Eric A. Carlen, and Elliott H. Lieb, ”Remainder Terms for Some Quan tum Entropy Inequal-\nities”, arXiv:1402.3840 [quant-ph] (2014).\n[7] D. W. Robinson, and D. Ruelle, ”Mean Entropy of States in Classica l Statistical Mechanis”,\nCommunications in Mathematical Physics ,5, 288 (1967).\nO. Lanford III, and D. W. Robinson, Jour. Mathematical Physics ,9, 1120 (1968).\n[8] M. B. Ruskai, “Lieb’s simple proof of concavity ( A,B)→TrApK†B1−pKand remarks on\nrelated inequalities,” arXiv:quant-ph/0404126v1 (2004).\n[9] V. N. Chernega, and O. V. Man’ko, ”Generalized qubit portrait of the qutrit state density\nmatrix”, J. Russ. Laser Res. ,34, 383 (2013).\n[10] V. N. Chernega, and O. V. Man’ko, ”Tomographic and improved s ubadditivety conditions\nfor two qubits and qudit with j= 3/2”,J. Russ. Laser Res. ,35, 27 (2014).\n5[11] M. A. Man’ko, and V. I. Man’ko, “Quantum strong subadditivity c ondition for systems\nwithout subsystems,” arXiv:1312.6988 (2013); Phys. Scr. ,T160, 014030 (2014).\n[12] Margarita A. Man’ko, and Vladimir I. Man’ko, ”The maps of matrice s and portrait maps\nof density operators of composite and noncomposite systems”, a rXiv:1404.3650[quant-ph]\n(2014).\n[13] V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, ”Minkovskii-typ e inequality for arbi-\ntrary density matrix of composite and noncomposite systems”, ar Xiv:1405-4956v1[quant-ph]\n(2014).\n[14] V. N. Chernega, and V. I. Man’ko, J. Russ. Laser Res. ,28, 103 (2007).\n[15] C. Lupo, V. I. Man’ko, and G. Marmo, J. Phys. A: Math. Theor. ,40, 13091 (2007).\n[16] V. V. Dodonov, and V. I. Man’ko, Phys. Lett. A ,229, 335 (1997).\n[17] V. I. Man’ko, and O. V. Man’ko, ”Tomography of spin states”, J. Exp. Theor. Phys. ,85, 430\n(1997).\n[18] O. V. Man’ko, ”Tomography of spin states and classical formula tion of quantum mechanics”,\nProc. of the International Symposium on Symmetries in Scien ce X, Bregenz, Austria, 1997\n(Symmetries in Science X) , p. 207, Plenum Publ. Corpr. N.Y., B. Gruber and H. Ramek eds.\n(1998).\n6"    },    {        "title": "1406.7216v1.Calculating_and_visualizing_the_density_of_states_for_simple_quantum_mechanical_systems.pdf",        "content": "arXiv:1406.7216v1  [quant-ph]  27 Jun 2014Calculating and visualizing the density of states\nfor simple quantum mechanical systems\nDeclan Mulhall∗\nDepartment of Physics/Engineering\nUniversity of Scranton\nScranton, Pennsylvania 18510-4642, USA\nMatthew J. Moelter†\nDepartment of Physics\nCalifornia Polytechnic State University\nSan Luis Obispo, California 93407, USA\nAbstract\nWe present a graphical approach to understanding the degene racy, density of states, and cumu-\nlative state number for some simple quantum systems. By taki ng advantage of basic computing\noperations we define a straightforward procedure for determ ining the relationship between discrete\nquantum energy levels and the corresponding density of stat es and cumulative level number. The\ndensity of states for a particle in a rigid box of various shap es and dimensions is examined and\ngraphed. It is seen that the dimension of the box, rather than its shape, is the most important\nfeature. In addition, we look at the density of states for a mu lti-particle system of identical bosons\nbuilt on the single-particle spectra of those boxes. A simpl e model is used to explain how the\nN-particle density of states arises from the single particle system it is based on.\nPACS numbers: 24.60.-k,24.60.Lz,25.70.Ef,28.20.Fc\n1I. INTRODUCTION\nThe concept of the density of states (DOS) is used in many areas of physics. For example,\nitisimportantforreactionratesinnuclear physics, thecalculationo fspecific heatcapacities,\nblack-body radiation, phonon spectra, and so on.1–6The DOS arises naturally and early in\nstatistical physics. To calculate average quantities in statistical p hysics, one could do an\nintegral over phase space, but this is typically very complex. The alt ernative is to express\nthe variable of interest in terms of the energy of the system. The v olume element in phase\nspace is replaced by a weighting factor in an energy integral, which is o ften much easier to\nwork with. This weighting factor is the density of states and is the su bject of this paper.\nConsider the problem of calculating /an}bracketle{tf/an}bracketri}ht, the expected outcome of a measurement of some\nphysical quantity f. If you know the allowed quantum states of the system and can calc ulate\nfi, the value of fin theith state, then /an}bracketle{tf/an}bracketri}ht=/summationtext\nifiPi, wherePiis the probability of the\nsystem being in the ith state. This is sometimes referred to as a sum over microstates. It is\noften easier to write fas a function of energy and perform a sum over the allowed energies .\nWe then have /an}bracketle{tf/an}bracketri}ht=/summationtext\nif(ǫi)P(ǫi), where the probability of the system being on the energy\nlevelǫis given by P(ǫ) =dǫe−βǫ/Z. HereZ=/summationtext\nje−βǫjis the partition function, β= 1/kT,\nanddǫis the degeneracy of the level or number of states with energy ǫ. If the system has\nan energy ǫ, then each of these dǫstates is equally likely. Now, for our expected value of f\nwe have\n/summationdisplay\nifiPi→/summationdisplay\nǫdǫf(ǫ)P(ǫ). (1)\nIn a system where ǫis continuous (or effectively so), the sum becomes an integral and dǫ\nis replaced by g(ǫ)—the DOS—a measure of how many states there are in a small range dǫ\naroundǫ. The DOS g(ǫ) is no longer an absolute number of states; it is now a weighting\nfactor in an integral over energy. We must be careful here becau seg(ǫ) is often confused\nwith the level density; indeed, the terms are often used interchan geably. We use the term\n“level” to mean an allowed value of energy and the DOS is the level dens ity multiplied by\na degeneracy factor.\nIn this paper we present a procedural approach to obtain the DOS for systems that\narise in modern physics and statistical mechanics courses. In the n ext section we apply the\nprocedure toa singleparticle inarigidbox. InSec. IIItheprocedur e forvisualizing theDOS\nis summarized and the effect of the shape of the box on the DOS is exa mined. Section IV\n2addresses systems of Nnoninteracting bosons, where N-particle spectra are calculated and\nthe dependance of their DOS on Nis compared with a simple model.\nII. VISUALIZING THE DENSITY OF STATES FOR A PARTICLE IN A BOX\nThe particle in a box is one of the first examples students encounter in quantum me-\nchanics. It is simple enough to solve from scratch by hand and exhibit s much of the salient\nnonclassical behavior. Furthermore, it serves as a basic template for a host of interesting\ntopics: scattering, double-well potentials, and perturbation the ory to name a few. We will\ndiscuss the DOS in the context of a particle in a box of various dimensio ns. First we intro-\nduce the concept of degeneracy via numerical results for the spe ctrum. This leads naturally\nto the idea of a cumulative state number from which the DOS naturally follows. Then an\nanalytic approach to obtaining the DOS is presented.\nWe start with a (nonrelativistic) particle of mass Minside a box. We assume the box is\nrigid, by which we mean that the potential energy is infinite outside th e box and zero inside\nthe box. In a one-dimensional box of length Lthe energy levels are given by7ǫn=ǫ0n2,\nwhereǫ0=π2¯h2/(2ML2) and the quantum number nis an integer. In a two-dimensional\nsquare box with sides of length L, the energy levels are given by ǫn=ǫ0(n2\nx+n2\ny), where\nnxandnyare the quantum numbers corresponding to the two spatial dimens ions. In the\nthree-dimensional case of a cubical box of side length L, we have ǫn=ǫ0(n2\nx+n2\ny+n2\nz).\nIt is natural to associate each quantum number with a number line an d each integer\nvalue with a point along this line. In two dimensions the pair of quantum n umbers defines a\nplane or two-dimensional space called n-space.8This space could also be called momentum\nspace by using the identity px= ¯hkx=nxπ¯h/L. The energy depends on the sum of the\nsquares of nxandny. One way of thinking about the distribution of these energies is to\nlocate them in n-space. Using the horizontal axis for nxand the vertical axis for ny, we can\nwrite all energy values at their corresponding grid points. This is don e in Fig. 1, where at\neach pair of quantum numbers ( nx,ny) the energy is written (in units of ǫ0); for example, at\ngrid points ( nx,ny) = (1,3) and (3 ,1) we see a “10.” Suppose nis the radius of a circle in\nn-space such that n2=n2\nx+n2\ny. Rewriting the energy as ǫn=ǫ0n2we see that in n-space\na given energy (in units of ǫ0) corresponds to a circle of radius/radicalBig\nǫ/ǫ0. (The radius nis\nrelated to the magnitude of the momentum vector by |/vector p|=nπ¯h/L). In Fig. 1 the circles\n3corresponding to energies “36” and “65” are shown as dotted cur ves with radii√\n36 = 6 and\n√\n65≈8, respectively.\nWhen energies correspond to more than one independent state we say they are “degener-\nate.” If we make a list of all energies corresponding to the various qu antum numbers ( nx,ny)\nand order them by energy, we can make a plot of the “number of sta tes with energy ǫ” vsǫ.\nThis quantity is dǫ, the degeneracy of the energy ǫ, and for discrete spectra it is an integer.\nA plot of dǫis a series of spikes, of height dǫ, at each allowed ǫ.\nTo further illustrate this idea, imagine a hypothetical single-particle spectrum where the\nlowest nine energies are {ǫ}={2,2,3,3,3,3,5,5,5}. We have plotted dǫvsǫfor this system\nin Fig. 2 (upper left panel). At this point it is helpful to introduce the c umulative state\nnumberN(ǫ), defined as the number of states with energy ≤ǫ; its graph is a staircase\nwhere each step has a height dǫand a width determined by the gap to the next energy\n(lower left panel of Fig. 2). In other words, given an ordered list of energies{ǫi}we have\nN(ǫ) =iforǫi< ǫ < ǫ i+1.\nA plot of the spikes dǫgives a visual measure of the degeneracies of the energies. These\ndegeneracies are quite delicate in the sense that most perturbatio ns to the potential will\nbreak them and the picture for dǫwill change dramatically. Each dǫ-high spike will turn\ninto a cluster of dǫseparate spikes, each one unit high. The spacing of the spikes will be\ndetermined by the strength of the perturbation. The correspon ding change in N(ǫ) is that\neach step in the unperturbed system that was dǫhigh will now become a series of short\nsteps, each one unit high (upper and lower right panels of Fig. 2). It is hoped that Figs. 1\nand 2 will be a useful starting point for student discussion.\nA smooth DOS function g(ǫ) is useful because at higher energies one is interested in the\nnumber of states in an interval, or the density of spikes along the ǫ-axis. The DOS and\ncumulative state number are related by g(ǫ)dǫ=dN(ǫ), so the DOS is recognized as the\nslope ofN(ǫ). We must be careful here because the slope of a staircase is infinit e at each\nstep, so we mean slope in an averaging sense.\nTo getg(ǫ) fromN(ǫ) we can take a numerical derivative using a finite difference scheme,\ng(ǫ) =dN(ǫ)\ndǫ=N(ǫ+δǫ/2)−N(ǫ−δǫ/2)\nδǫ. (2)\nIt is worth saying explicitly that if the energies are put into bins of widt hδǫ, with center ǫ,\ntheng(ǫ)=(number of states in bin)/(width of bin).\n4Now that we have a numerical representation for g(ǫ), we would like to get an analytic\nexpression for g(ǫ) that is valid for the statistical region where the DOS is large and well\napproximatedbyasmoothfunction. Wewill dothisfortherigidboxpo tentialswhere ǫ=n2\n(in units of ǫ0). Inn-space the states with energy in an interval dǫcentered on ǫcorrespond\nto a set of points in a spherical shell of thickness dnwith all-positive coordinates. In Fig. 1\nthe number of states between the quarter circles of radius nandn+dnis proportional to the\narea of the curved band. This result is anapproximation, because nxandnyare discrete and\nthere are fluctuations, but we will see that this is a good approximat ion nevertheless. The\nnumber of states is the “volume” of this shell but by definition it is also g(ǫ)dǫ, so in 2-D we\nhaveg(ǫ)dǫ= (1/2)πndn. Ifn(ǫ)isasmoothfunctionwecanwrite g(ǫ) = (1/2)πn(ǫ)dn/dǫ.\nUsingn(ǫ) =√ǫanddn/dǫ= 1/(2√ǫ) we see that the DOS is constant: g(ǫ) =π/4. Since\nthe DOS is the slope of N(ǫ), the function N(ǫ) should be well approximated by the straight\nlineN(ǫ) = (π/4)ǫ, as shown in Fig. 3. The numerical results are convincing, particular ly\nwheretheenergyrangeincreasesandthefluctuationsaboutthe smoothfunctionsgetsmaller\n(see Fig. 4).\nFor a particle in a 3-D rigid box with sides Lx,Ly, andLz, we will be working with a 3-D\nn-space and again each point ( nx,ny,nz) corresponds to an allowed value of energy where\nǫ= (π2¯h2)/(2M)[(nx/Lx)2+ (ny/Ly)2+ (nz/Lz)2]. For a cube of side length Lwe have\nǫ= (π2¯h2)/(2ML2)n2=ǫ0n2withn2= (n2\nx+n2\ny+n2\nz). Now the appropriate construction\nto get the DOS is a shell of thickness dnand radius nin the positive octant of a sphere,\nwhich leads to g(ǫ) = (1/2)πn2dn/dǫ. Again, using n(ǫ) =√ǫanddn/dǫ= 1/(2√ǫ), we\nsee that the DOS now has the form g(ǫ) = (π/4)√ǫandN(ǫ) = (π/6)ǫ3/2. In Figs. 5 and 6\nwe compare the numerical results with these smooth functions for different energy ranges.\nAgain as the energy increases, the numerical fluctuations get sma ller.\nWe remark here that if we had Nnon-interacting independent particles in a rigid cube,\nthen the total energy would be the sum of the individual energies, a nd this sum could be\nrelated to the surface [xx do you mean volume? xx] of a D= 3N-dimensional hypersphere.\nThisproblemissimilartothecaseofasingleparticleinarigidboxina D-dimensionalspace.\nWe would need to discuss the volume of a sphere of radius RinD-dimensions, denoted by\nVDand given by9\nVD=πD/2\nΓ(D/2+1)RD=CDRD. (3)\n5Here, Γ(D+ 1) =D!, giving C2=πandC3=4\n3πas expected. The surface area of this\nD-sphere is given by S D=DCDRD−1. Where before we were concerned with the length\nof a quarter circle of radius n(D= 2) and the area of an octant of a sphere of radius n\n(D= 3), we now have the area of the positive portion of the D-dimensional sphere, given by\n(1/2D)SD(theneedfor nx,ny...tobepositivegives afactorof1 /2foreachdimension). The\ncumulative state number is given by the volume of phase space enclos ed by the boundary\nn=√ǫ,\nN(ǫ) =1\n2DπD/2\nΓ(D/2+1)ǫD/2, (4)\nand differentiation gives\ng(ǫ) =1\n2D+1DπD/2\nΓ(D/2+1)ǫ(D/2)−1. (5)\nThe forms of N(ǫ) andg(ǫ) for some representative Dare given in Table I. Later we will see\nthat the DOS for one particle in Ddimensions is proportional to the DOS of Dparticles in\none dimension, so long as the particles do not interact.\nIII. PROCEDURE FOR CALCULATING AND VISUALIZING THE DENSITY\nOF STATES\nFor any quantum-mechanical system we can determine the cumulat ive state number and\nthe corresponding density of states using the following procedure :\n1. Solve the relevant quantum-mechanical problem to get a list of th e energies of the sys-\ntem. (This can be an analytical expression or a list of energies obtain ed numerically.)\n2. Sort the list of energies to get a set {ǫ1, ǫ2, ǫ3, ..., ǫ n}up to some maximum energy\nǫmax.\n3. Create the cumulative state number N(ǫ) by making the set\n{ǫ1,1},{ǫ2,2},{ǫ3,3}, ...{ǫn,n}.\n4. Find the DOS by taking the numerical derivative of the cumulative s tate number, as\nin Eq. (2). Choose a window size and locate the window so that its cent er is atǫ. The\nvalue of g(ǫ) is the number of energies that are within that window divided by the\nwindow size. Evaluate this for all locations of the window, and you now have a list of\nordered pairs {ǫ,g(ǫ)}.\n6As supplementary materials10we include a Matlab11program used to obtain Figs. 5\nand 6; with appropriate modifications the program can be used to pr oduce Figs. 3 and 4.\nWe will apply this procedure to calculate and visualize the DOS for part icles in rigid\nboxes with various geometries.\nA. Cubical box\nWe first will consider a cube of side length L.\nStep 1: The energies are ǫ= (π2¯h2)(n2\nx+n2\ny+n2\nz)/(2MV2/3), where we have used\nL2=V2/3withVthe volume of the box. If we set ( π2¯h2)/(2M) = 1 and V= 1, then the\nenergies are all integers.\nStep 2: Using Mathematica12we determined the lowest 15,954 energy levels. To\ncalculate the energies we need all the triplets {nx, ny, nz}within the sphere of radius nmax\ninn-space. We made a list of n2\nx+n2\ny+n2\nzwith 1≤nx≤nmax, 1≤ny≤/radicalBig\nn2\nmax−n2\nx, and\n1≤nz≤/radicalBig\nn2max−n2x−n2y; thislist hadalltheenergies ǫ≤n2\nmax(wechose nmax= 32). This\nisavery degenerate system—of the15,954energy levels only818ar edistinct andtheaverage\ndegeneracyis19.50. AsseeninFig.6, for ǫ∼1000wehave dǫ∼60. Specifically, theenergies\nfrom 933–954, along with their degeneracies, are given by: {ǫ, g(ǫ)}={933,24},{934,39},\n{936,24},{937,27},{938,24},{},{940,6},{941,66},{942,18},{944,9},{945,48},{946,\n24},{947,15},{948,18},{949,12},{950,63},{952, 12},{953,45},{954,42}.\nStep 3: PlotN(ǫ). It is instructive to construct the set of points for the plot explic itly\nusing the list of energies. There are only 818 steps, one for each va lue ofǫ. The height of\nthe step at some energy ǫisdǫ, the degeneracy, so this staircase has steps that get higher\nwith energy (e.g., the step at ǫ= 950 has a height of 63).\nStep 4: While calculating the DOS, the window size needs to be adjusted to giv e a\nreasonable-looking graph. If the window is too small, the graph will loo k noisy; if it is\ntoo big, the graph may appear too coarse. We use trial and error t o select an appropriate\nwindow size. We have included code in the online supplement10in which the reader can\nselect the maximum energy and window size and generate plots like Figs . 5 and 6.\nFigures 5 and 6 display d(ǫ),N(ǫ), andg(ǫ) for this system. Again, it is evident that the\nagreement between the analytic result and the explicit counting of e nergies improves as the\nenergy interval increases.\n7B. Rectangular box\nThe rigid rectangular box is a straightforward adjustment to the c ubical case and gives us\na look at a non-degenerate spectrum. In order to remove the deg eneracies from the single-\nparticle spectrum that exist in the cubical case, we will make the side s of the box have\nincommensurate lengths. The choice Lx/ne}ationslash=Ly/ne}ationslash=LzwithLx= 1,Ly= 2/e, andLz=e/2\ngives us a box of unit volume and a non-degenerate spectrum.\nStep 1: The energies are ǫ= (π2¯h2)[(nx/Lx)2+(ny/Ly)2+(nz/Lz)2]/(2M). Again, the\nobvious choice for our energy scale is to set π2¯h2/(2M) = 1.\nStep 2: This time we use a different method than in the cubical case. The sph ere in\nn-space corresponds to an ellipse in phase space. This is a nice illustrat ion of the difference\nbetween these abstract spaces. The method used was an exhaus tive calculation of all the\nenergies corresponding to the points 1 ≤nx≤nmax, 1≤ny≤nmax, and 1≤nz≤nmax,\nafter which we selected ǫ≤(2nmax/e)2. The fact that nyandnzare multiplied by a factor\nof 0.74 and 1.36 respectively ensured that the sphere radius of/radicalBig\n2/enmaxwas enclosed in\nthe ellipse in phase space, and we had a complete set of energies. Alth ough the method is\ninefficient, it has the advantage of transparency.\nAll the energies in this system are unique. In order to have degener acies we would need\ntwo sets of integers, {nx,ny,nz}and{mx,my,mz}, such that nx2+e2n2\ny/4 + 4n2\nz/e2=\nmx2+e2my2/4 + 4mz2/e2, which is impossible as eis transcendental. The lowest 62,440\nenergies of the system were calculated.\nStep 3: The cumulative state number N(ǫ) consists of steps of unit height, though the\nwidths of the steps (level spacings) vary. For our rectangular bo x there are 41,435 steps,\none for each energy. A plot of dǫvsǫis a series of horizontal spikes of height 1.\nStep 4: In Fig. 7 we show g(ǫ) andN(ǫ) for this system. [xx Fig. 7 has a label of ρ\nwhich has not been defined. xx]\n[xx LEFT OFF HERE xx]\nC. Spherical box\nThe energies for a rigid sphere of radius Rareǫln= ¯h2k2\nln/(2MR2), where klnis\nthenth zero of jl(r), thelth spherical Bessel function.13Eachǫlnhas a degeneracy of\n82l+ 1. It is easy to confuse jl(r) withJl(r), the ordinary Bessel function, also known as\nthe Bessel function of the first kind; the two are related by13jl(r) =/radicalBig\nπ/(2r)Jl+1/2(r).\nNote that the nth zero of Jl(r) is written Kln.Mathematica has a useful add-on called\nNumericalMath‘BesselZeros’ thatliststhezerosofthevariousBessel functions. InSec. VI\nwe define the problem of the rigid cylindrical box, where the energies are proportional to\nK2\nln, so to avoid confusion we provide the following check: the first thre e zeros of J0(r)\nare{2.4048,5.5201,8.6537}, and the first three zeros of j0(r) are{3.9374,7.8748,11.8122}.\nActually, the distinction between Klnandklnis not important as far as g(ǫ) goes for the\nsphere and the cylinder, as the two sets of zeros are intersperse d.14\nStep 1: The role of πin the energies of the rectangular wells is taken over by the kln.\nIn the rectangular box the zeros of the sine function are multiples o fπ. This allowed us to\nfactor it out and choose the energy scale to be in units of ¯ h2π2/(2M) for the rectangular\nboxes. Now we have ǫ= ¯h2(kln/R)2/(2M). To compare with the square wells we use an\nenergy scale with ¯ h2π2/(2M) = 1; in these units, ǫln= [kln/(πR)]2. Also, if the volume of\nthe box is to be unity, then R= [3/(4π)]1/3, so finally we have ǫln=k2\nln/(3π2/4)2/3. This is\nthe spectrum we will compare to those of the rectangular and cubic al boxes.\nStep 2: We calculated all energies with kln<95. To ensure a complete set, we used the\nbruteforcemethodandcalculatedallthetriplets {kln, l, n}with0≤l≤95and1 ≤n≤95,\nsorted these 9,120 numbers, and selected the first 95. Making the triplets as opposed to just\nthe list of klnmade it easy to accommodate the 2 l+ 1 degeneracy when making the final\nspectrum.\nSteps 3 & 4 : See Fig. 7.\nD. Weyl’s theorem\nIt is clear from Sec. II and Fig. 7 that the general trend for the th ree boxes we examined\nisN(ǫ) =αǫβ. The energy exponent does not vary much with the shape of the bo x. A plot\nof lnN(ǫ) vs lnǫillustrates this nicely, as shown in Fig. 8. The deviation from the straig ht\nline occurs only at low energies, where the shape of the box matters . UsingMathematica\nwe performed linear fits to the log-log plots and obtained the results shown in Table II.\nIftheboxissufficientlylarge, aparticleinarigidboxshouldnotbeawar eoftheparticular\nshape of the box, so long as λD≪V, whereVis the volume of the box. Here, λ= 2π/|/vectork|is\n9the deBroglie wavelength ( |/vector p|= ¯h|/vectork|) andDis the dimension of the space. A slow particle\nwill know about the edge, as it has a long wavelength, and a fast part icle or high-energy\nstate will not be sensitive to the shape because its wavelength is sma ll. This is the content\nof Weyl’s theorem,15which can be paraphrased as “high energy eigenvalues of the wave\nequation are insensitive to the shape of the boundary.” A nice accou nt of the origin of the\ntheorem is given by Kac,16and an explicit proof at the level of this paper for the case of the\ncylinder and sphere is given by Lambert.17\nIV.NPARTICLES: STATISTICAL MECHANICS\nStatisticalmechanicscanbeintroducedbyanalyzingamodelsyste mofNnon-interacting\nparticles on a single-particle spectrum, often chosen as a set of eq ually-spaced levels.7This is\na valuable pedagogical tool for exploring the statistical distributio ns for classical particles,\nidentical bosons, and identical fermions. This type of model is rich w ith subtleties: the\nparticles not only don’t see each other beyond obeying the Pauli exc lusion principle if they\nare identical fermions, but their presence has no influence on the e nergies of the levels they\nfill.18Real particles, on the other hand, have interactions, so how does this model give\nuseful results? One can think of the effect of all the other particle s as being a smooth mean\nfield that can be modeled as a potential for that single particle. Anot her subtlety is the\nrobustness of the spectra of these N-body systems: they are very insensitive to the details\nof the single-particle spectrum upon which they are built, depending only on its DOS.\nWe made a simple model of the the N-particle system to serve as a starting point for\nstudents to explore what happens to the DOS as Nincreases, and to illustrate that the DOS\nof many-body systems like this are insensitive to the details of the sin gle-particle spectra\nthey are based on. In this section Eis used for the total energy of the system, reserving ǫ\nfor the single-particle energy. Subscripts will be used on gandNto indicate the number of\nparticles in the system, so g7(E) is the DOS for a system of seven particles. We compared\nN-particle systems built on four distinct single-particle spectra: the square, rectangular,\nand spherical 3-D boxes, and a spectrum with a specified DOS but ra ndom levels. A simple\nanalytical model for the DOS is presented and compared with the nu merical results. The\nfocus is on the relationship between the original single-particle ener gies that the Nparticles\npopulate and the resulting gN(E). The model demonstrates that gN(E) depends on the\n10densityof the single-particle spectrum, as opposed to the details of the en ergies themselves.\nThis is why we can replace the exact single-particle energies (zeros o f Bessel functions, etc.)\nwith any set of numbers as long as they have the same range and den sity.\nUsing a simple algorithm, the spectrum of Nidentical bosons on a set of single-particle\nenergies{ǫi}was calculated. The range of energies was 0 →Emax, withEmaxlimited only\nby computational power and user patience. The choice of bosons a voided the subtleties of\nFermi statistics. The algorithm is best illustrated with an example. Le t’s look at identical\nbosons on the single-particle energies of the rigid cube in three dimen sions. First we make a\nlist ofthesingle-particle-state energies: {ǫi}={3,6,6,6,9,9,9,11,11,11,12,14,14,...}. We\nchooseEmax= 53 so the list includes the first 140 states. This means that we are g uaranteed\ntohave acompletelist ofenergies upto E= 53. Forourpurposes an N-particlestateisalist\nofoccupied single-particle states, so(4 ,5,5,27)isafour-particlestatewithaparticleinstate\n4, two particles in state 5, and a particle in state 27. Now list the indice s of all possible one-\nparticle states: {(i)}={(1),(2),(3),...,(140)}; the energy of state ( i) isǫi. Next list the\nindex pairs of all possible two-particle states: {(i,j)}={(1,1),(1,2),(1,3),...,(140,140)},\nsort them and drop duplicates; the energy of the state ( i,j) isǫi+ǫj. Drop all states with\nE >53 as you are only guaranteed a complete set of energies for E≤53 . In general,\nlist all possible N-particle states, sort them, drop duplicates, and sum the corres ponding\nsingle-particle energies. Once you have these spectra you can use the procedure in Sec. III\nto getgN(E) andNN(E).\nSystems based on a random single-particle spectrum were included f or comparison with\nthose based on the rigid boxes because this removed the role of geo metry. The distribution\nof the random numbers was such that the DOS was g(ǫ) = (π/4)ǫ1/2; this was accomplished\nby taking a set {xi}of 500 random numbers uniformly distributed on the interval [0 ,1],\nsorting them, raising them to the power of 2 /3, and multiplying them by [(6 /π)500]2/3. The\nlogic behind this is worth mentioning. Given a set of numbers {x}with a distribution f(x),\nwe can make a function y(x) such that the set {y(x)}has a distribution αyβby noting\nthatf(x)dx=αyβdy. In our case f(x) = 1, and we have dy/dx= (1/α)y−βso that\ny(x) = [(β+1)/α]1/(β+1)x1/(β+1).\nAll of these systems had a cumulative state number of the form NN(E) =αEβ, with both\nβand lnαbeing linear in N. In Fig. 9 we show NN(E) vsE[xx Fig. 9 actually plots vs ǫ\ninstead of E. Please correct wording as necessary. xx] for Nparticles on the single-particle\n11energies of a spherical well, while Figs. 10 and 11 show graphs of βand lnαas functions of\nNfor the various systems we examined. The values of αandβfor the random spectra were\nbased on 100 separate single-particle spectra.\nTo compare with the theory, we used g1(ǫ) = 0.40ǫ1/2. [xx I don’t understand why this\nsentence is here. Should it be part of the next paragraph? xx]\nGiven a single-particle spectrum with a one-particle DOS g1(E), we can obtain a naive\nexpression for the N-body density of states gN(E) iteratively. The density of N-body states\nwithenergy Eisaproductofthedensity of( N−1)-bodystateswithenergy E′andone-body\nstates with energy E−E′, or\ngN(E) =/integraldisplayE\n0gN−1(E′)g1(E−E′)dE′. (6)\nThis expression double-counts some states. Suppose we add to th e three-particle state\n{n1,n2,n3}={3,17,17}a fourth particle in state 24; then E=ǫ3+2ǫ17+ǫ24, and there\nis a contribution to the integral in Eq. (6) from g1(ǫ24)g3(ǫ3+ 2ǫ17). However, we can get\nto the same state by adding a particle in state 17 to the three-part icle state {n1,n2,n3}=\n{3,17,24},andthisgivesacontributiontotheintegralinEq. (6)from g1(ǫ17)g3(ǫ3+ǫ17+ǫ24).\nThis is the origin of the double counting and its effect is to increase the number of states\nby a number close to Nat high energies, where the occupancies of the single-particle stat es\nare low. We did not fix this problem. When iterating Eq. (6) we need the following result\nfor integer nandm:\n/integraldisplayE\n0(E′)n(E−E′)mdE′=n!m!\n(n+m+1)!En+m+1. (7)\nTable III gives several examples of the DOS for Nbosons based on single-particle spectra\nwith level density g1(E) =Eb, forb=−1/2,0,1/2,1, corresponding to a box in 1, 2, 3,\nand 4 dimensions.\nImmediately we see some trends. The DOS for one boson in a p-dimensional rigid box,\nEq. (5), is the same as that of pbosons in a one-dimensional rigid box [xx You see this\nby comparing what to what? xx]. This result is a by-product of separ ation of variables\nin the Cartesian case. However, if we put the particles into a spheric al box, we also have\nseparation of variables, but they [xx What is “they” referring to? x x] are not equivalent.\nTheaing1(E) =aEbchanges, but the powers of gN(E) do not [xx It is not clear where this\nresult comes from. xx]. In general, upon iterating Eq. (6), for Nbosons on a single-particle\n12spectrum with g1(E) =aEb, we get\ngN(E) =b!N\n(Nb+N−1)!aNENb+N−1, (8)\nNN(E) =b!N\n(Nb+N)!aNENb+N=αEβ. (9)\nThis simple treatment exhibits the properties of the systems we hav e examined [xx Exhibits\nwhat properties? This sentence seems incomplete. xx]. The results in Table III and Figs. 9–\n11 illustrate this [xx Illustrate what perfectly? xx] perfectly. In Fig s. 10 and 11 we used the\nvaluesa= 0.4 andb= 1/2 in Eq. (9) for comparison. In Eq. (9) we see that ln( α) is the\nsum ofNln(a) and a term that depends on b, so it is not very sensitive to the value of a.\nThe theoretical value of αis higher than for any of the systems examined. This is partially\ndue to the double counting discussed above, which makes NN(E) larger, but that may not\nbe the whole story. In the case of ln( α) andβthe random system was closest to the theory,\nbut we will not speculate as to why.\n[xxIfoundthis last section, andinparticular the last fewparagraph stobevery confusing\noverall. I really lost sight of what you are trying to show. Please cons ider revising some of\nthis section to try to clarify things as much as possible. xx]\nV. CONCLUSIONS\nWe have provided a graphical approach to introduce students to t he density of states\n(DOS) for simple quantum systems. In Sec. I we introduced the con cepts of degeneracy,\ncumulative state number N(ǫ), and DOS g(ǫ) for these systems. In Sec. III we gave a\nprocedure for visualizing the DOS and used it to get the DOS for a par ticle in rigid boxes\nof various shapes. We found that the shape of the box had little effe ct on the DOS, a\ndemonstration of Weyl’s theorem. We then built N-particle systems on the single-particle\nspectra of Sec. III and found that the DOS was insensitive to the s pecific energies of the\nsingle-particle spectra, the single-particle DOS ( g1(ǫ)) being the important factor. Finally,\nwe compared our numerical results for NN(E) with an expression based on iterating the\nsingle-particle density of states and saw how the DOS for N-particle systems was simply\nrelated to g1(ǫ).\n13VI. ADDITIONAL PROBLEMS\nThefollowing problemsprovideadditionalpracticewiththeprocedur eusedinSectionIII.\nThe programs available in Ref. 10 may be a helpful starting point.\n1. Examine changes to the density of states if we continuously chan ge one of the dimen-\nsions of the rigid rectangular box. We expect the density of states to exhibit different\nbehavior as we change continuously from a “2-D” system ( Lz≪Lx,Ly), to a “3-D”\nsystem (Lz≈Lx,Ly) to a “1-D” system ( Lz≫Lx,Ly). LetLx=Ly=L0= 1, and\nthen letLzvary over the range 0 .01L0≤Lz≤100L0.\n2. The energy levels for a particle in a rigid cylindrical box of height Hand radius R\nare given by13ǫqln= ¯h2(q2π2/H2+K2\nln/R2)/(2M), where the Klnare zeroes of the\n(regular) Bessel functions. Vary the height-to-radius ratio for the cylindrical box. As\nH/Rgoes from near 0 to 1, you should expect a transition from “2-D” be havior to\n“3-D” behavior. What does this look like in terms of g(ǫ)?\n3. Redo the treatment of Sec. II for a relativistic particle in the 2-D rigid square box.\nThe important difference is that relativistically ǫ=pc= ¯hkc, wherekis the wave\nnumber and cis the speed of light.\nAcknowledgments\nWe thank T. Bensky for helpful discussions. For their hospitality du ring sabbatical leaves\nM.J.M.thanksthefacultyoftheSchoolofPhysics attheDublinIn stituteofTechnologyand\nD. M. is grateful to the physics department at Temple University. T he reviewers provided\ndetailed comments that significantly clarified our presentation.\n∗Electronic address: mulhalld2@scranton.edu\n†Electronic address: mmoelter@calpoly.edu\n1Daniel Schroeder, An Introduction to Thermal Physics (Addison-Wesley, San Francisco, 2000).\n2Ralph Baierlein, Thermal Physics (Cambridge, New York, 1999).\n3P. C. Riedi, Thermal Physics , 2nd edition (Oxford, New York, 1988).\n144Frederick Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York,\n1965).\n5Franz Mandl, Statistical Physics , 2nd edition (Wiley, New York, 1988).\n6Charles Kittel and Herbert Kroemer, Thermal Physics , 2nd edition (Freeman, New York, 1980).\n7Raymond A. Serway, Clement J. Moses, and Curt A. Moyer, Modern Physics , 3rd edition\n(Thompson Brooks/Cole, 2005).\n8This visual approach is discussed in Ref. 2 on pp. 75–79. A rel ated treatment from the momen-\ntum viewpoint is in Appendix IV of Ref. 3, pp. 297–304.\n9Claude Garrod, Statistical Mechanics and Thermodynamics (Oxford, New York, 1995).\n10Sample programs demonstrating how our calculations are car ried out are available as supple-\nmentary material at <AIP.to.insert.URL> .\n11Matlab software from The Mathworks, <www.mathworks.com> .\n12Mathematica software from Wolfram Research, <www.wolfram.com> .\n13Richard L. Liboff, Introductory Quantum Mechanics (Holden-Day, 1980).\n14Richard L. Liboff, “Density of states and other quantum proper ties of a spherical cavity,” Phys.\nRev. A,43(11), 5765–5769 (1991).\n15H. Weyl, “Das asymptotische Verteilungsgesetz der Eigenwe rte linearer partieller Differential-\ngleichungen,” Math. Ann. 71, 441–479 (1912).\n16Mark Kac, “Can One Hear the Shape of a Drum?,” Amer. Math. Mon. ,73(4), 1–23 (1966).\n17R. H. Lambert, “Density of States in a Sphere and Cylinder,” A m. J. Phys. 36(5), 417–420\n(1968).\n18Anecdotally, one of the authors (D. M.) has been surprised by the persistent student miscon-\nception that “fermions don’t interact.” This false notion c an be traced to the commonly used\n“system of noninteracting fermions” that introduces stati stical mechanics in the intermediate\ncourse. The equidistant single-particle spectrum is a comm on starting point when looking at\nthe spectra of many-body systems. It is confusing that after all the work invested in solving the\nsingle-particle Schr¨ odinger equation, wejustthrow away ourhardwonzeros-of-Bessel-functions-\nbased energies, and replace them with the integers.\n19Njema Frazier, B. Alex Brown, and Vladimir Zelevinsky, “Str ength functions and spreading\nwidths of simple shell model configurations,” Phys. Rev. C 54(4), 1665–1674 (1996).\n15Tables\nTABLE I: The DOS for a particle in a rigid box of Ddimensions. The energy ǫis in units of the\nlowest single particle energy ǫ0.\nDN(ǫ)g(ǫ) =dN/dǫ\n1ǫ1/2 1\n2ǫ−1/2\n2π\n4ǫπ\n4\n3π\n6ǫ3/2 π\n4ǫ1/2\n4π2\n32ǫ2 π2\n16ǫ\n5π2\n60ǫ5/2π2\n24ǫ3/2\n6π3\n384ǫ3 π3\n128ǫ2\nTABLE II: Parameters for N(ǫ) =αǫβfor the three different 3-D “boxes.” The theory gives\nα=π/6≈0.524 and β= 3/2; see Table I.\nα β\nCube (theory) π/6≈0.524 3/2 = 1.5\nCube (numerical) 0.29 1.58\nRectangular box 0.32 1.56\nSpherical box 0.46 1.51\n16TABLE III: The DOS for N-particle systems built on single-particle spectra with a D OSg1(E) =\naEb. We set a= 1 for simplicity but gN(E) includes a factor of aN.\nb=−1/2b= 0 b= 1/2 b= 1\ng1(E)E−1/21 E1/2E\ng2(E) π Eπ\n8E2 1\n3!E3\ng3(E) 2πE1/2 1\n2!E2 2π\n105E7/2 1\n5!E5\ng4(E)π2E1\n3!E3 π2\n1920E5 1\n7!E7\ng5(E)4\n3π2E3/2 1\n4!E4 4π2\n135135E13/2 1\n9!E9\ng6(E)1\n2π3E2 1\n5!E5 π3\n2580480E8 1\n11!E11\ng7(E)8\n15π3E5/2 1\n6!E6 8π3\n654729075E19/2 1\n13!E13\ng8(E)1\n6π4E3 1\n7!E7 π4\n10218700800E11 1\n15!E15\n...\ngN(E)πN/2\n(N/2−1)!EN/2−11\n(N−1)!EN−1πN/2\n2N(3N/2−1)!E3N/2−11\n(2N−1)!E2N−1\n17Figure captions\n02468100246810\n2\n2510172637506582101\n5813202940536885104\n101318253445587390109\n172025324152658097116\n2629344150617489106125\n37404552617285100117136\n50535865748598113130149\n6568738089100113128145164\n82859097106117130145162181\n101104109116125136149164181200ny\nnx\nFIG. 1: The energies for different combinations of nxandnyfor a particle in a 2-D square box.\nThe numbers at the grid sites are n2\nx+n2\ny. The quarter circles represent constant n=/radicalBig\nn2x+n2y;\nhere we show n=√\n36 andn=√\n65. Certain values of ngive circles that intersect with the grid\npoints. These correspond to the allowed energies ǫ=n2ǫ0. Notice that n= 6 does not intersect\nany grid point, while n=√\n65≈8 (dashed line) intersects four grid points, corresponding to the\ndegeneracy of that energy level.\n180246024 dε\nε0 5024 dε\nε\n024602468 N(ε)\nε024602468 N(ε)\nε\nFIG. 2: The degeneracy dǫ(upper panels) and corresponding cumulative state number N(ǫ) (lower\npanels) for the lowest nine levels of a simple, discrete syst em. On the left, the degeneracy of the\nsystem is intact while on the right, the degeneracy has been l ifted. The effect on dǫis dramatic;\nless so for N(ǫ).\n0 20 40 60 80 100024dεa)\n0 20 40 60 80 100050b)N(ε)\n0 20 40 60 80 10000.51c)g(ε)\nε  ∝  nx2+ ny2\nFIG. 3: Top: Degeneracy dǫof the allowed energies ǫ(in units of ǫ0) of a single particle in a 2-D\nsquare box, up to nx=ny= 10. Middle: The corresponding N(ǫ). Theory (smooth line) gives\nN(ǫ) = (π/4)ǫ. Bottom: A plot of g(ǫ) obtained using the derivative scheme in Eq. (2) with a\nwindow width of 10. Theory (smooth line) gives g(ǫ) =π/4.\n190 500 1000 150005dεa)\n0 500 1000 150005001000b)N(ε)\n0 500 1000 150000.5c)g(ε)\nε  ∝ nx2+ny2\nFIG. 4: Top: Degeneracy dǫof the allowed energies ǫ(in units of ǫ0) of a single particle in a 2-D\nsquare box, up to nx=ny= 40. Middle: The corresponding N(ǫ). Theory (smooth line) gives\nN(ǫ) = (π/4)ǫ. Bottom: A plot of g(ǫ) obtained using the derivative scheme in Eq. (2) with a\nwindow width of 60. Theory (smooth line) gives g(ǫ) =π/4.\n0 20 40 60 80 100010dεa)\n0 20 40 60 80 1000500\nb)N(ε)\n0 20 40 60 80 10005c)g(ε)\nε  ∝ nx2+ny2+nz2\nFIG. 5: Top: Degeneracy dǫof the allowed energies ǫ(in units of ǫ0) of a single particle in a\n3-D rigid cubic box, up to nx=ny=nz= 10. Middle: The corresponding N(ǫ) is compared\nto the theoretical result (smooth line) N(ǫ) = (π/6)ǫ3/2. Bottom: A plot of g(ǫ) obtained using\nthe derivative scheme in Eq. (2) with a window width of 10. The ory (smooth line) gives g(ǫ) =\n(π/4)ǫ1/2.\n200500 1000 1500050dεa)\n0500 1000 150002x 104\nb)N(ε)\n0500 1000 1500020c)g(ε)\nε  ∝ nx2+ny2+nz2\nFIG. 6: Top: Degeneracy dǫof the allowed energies ǫ(in units of ǫ0) of a single particle in a 3-D\ncubical box, up to nx=ny=nz= 40. Middle: The corresponding N(ǫ) (lower curve) is compared\nto the theoretical result (smooth line, upper) N(ǫ) = (π/6)ǫ3/2. Bottom: A plot of g(ǫ) obtained\nusing the derivative scheme in Eq. (2) with a window width of 5 0. Theory (smooth line) gives\ng(ǫ) = (π/4)ǫ1/2.\n210200 400 600 80005001000\nερ(ε)0200 400 600 800012x 104\nεN(ε)\nFIG. 7: The upper panel shows the cumulative state number for the spectra for a single particle in\ncubical, rectangular, andsphericalboxes. Thespectrahav ebeennormalizedsothat¯ h2π2/(2M) = 1\nand the volume of each box is 1. The three curves are essential ly indistinguishable. The lower\npanel is a histogram of the energies for each box. The bins are 40 units wide.\n024680246810\nln εln  N(ε)Cube\nRectangle\nSphere\nFIG. 8: A log-log plot of the cumulative state number for the s ingle-particle spectra of cubical,\nrectangular, and spherical boxes. The energies are in units of ¯h2π2/(2M) and the volume of each\nbox is 1.\n221 2 3 4 50246810\nln εln  N(ε)\nFIG. 9: A log-log plot of the cumulative state number for N-particle systems ( N= 1,2,...,5) on\nthe single-particle spectrum of the sphere. The steeper lin es correspond to higher values of N, and\ndifferent line types have been used for clarity. The slopes fol lowN(β+1).\n012345612345678910\nN (number of particles)βRectangle\nCube\nSphere\nRandom\nTheory\nFIG. 10: Theexponentof EinN(E) =αEβforsystems ofvarious particle number. Each systemis\nmade by putting non-interacting particles on the correspon ding single-particle spectrum described\nin the text. The dashed line is a theoretical value derived us ing the values a= 0.4 andb= 1/2 in\nEq. (8). These results were obtained using a linear fit to the l ogN(E) data. (The line is a guide\nfor the eye.) The error bars represent the standard deviatio n of the average values from the 100\nrandom spectra.\n230 2 4 6−30−25−20−15−10−505\nN (number of particles) ln(α)\nCube\nRectangle\nSphere\nRandom\nTheory\nFIG. 11: A graph of ln αvsNfor the same systems as in Fig. 10. Again, these results were\nobtained using a linear fit to the ln N(E) data. The error bars represent the standard deviations\nof the average values from the 100 random spectra.\n24"    },    {        "title": "1407.1959v3.Excited_State_Density_Functional_Theory_Revisited__on_the_Uniqueness__Existence__and_Construction_of_the_Density_to_Potential_Mapping.pdf",        "content": "arXiv:1407.1959v3  [cond-mat.mtrl-sci]  7 Jun 2015Excited-State Density-Functional Theory Revisited: on th e Uniqueness, Existence,\nand Construction of the Density-to-Potential Mapping\nPrasanjit Samal,1,∗Subrata Jana,1and Sourabh S. Chauhan1\n1School of Physical Sciences, National Institute of Science Education & Research, Bhubaneswar 751005, INDIA.\n(Dated: May 31, 2021)\nThe generalized constrained search formalism is used to add ress the issues concerning density-to-\npotential mapping for excited states in time-independent d ensity-functional theory. The multiplicity\nof potentials for any given density and the uniqueness in den sity-to-potential mapping are explained\nwithin the framework of unified constrained search formalis m for excited-states due to G¨ orling,\nLevy-Nagy, Samal-Harbola and Ayers-Levy. The extensions o f Samal-Harbola criteria and it’s link\nto the generalized constrained search formalism are reveal ed in the context of existence and unique\nconstruction of the density-to-potential mapping. The clo se connections between the proposed cri-\nteria and the generalized adiabatic connection are further elaborated so as to keep the desired\nmapping intact at the strictly correlated regime. Exemplifi cation of the unified constrained search\nformalism is done through model systems in order to demonstr ate that the seemingly contradic-\ntory results reported so far are neither the true confirmatio n of lack of Hohenberg-Kohn theorem\nnor valid representation of violation of Gunnarsson-Lundq vist theorem for excited states. Hence\nthe misleading interpretation of subtle differences betwee n the ground and excited state density\nfunctional formalism are exemplified.\nI. INTRODUCTION\nSince its advent, density-functional theory (DFT) [ 1–\n10] is routinely applied for calculating the electronic,\nmagnetic, spectroscopic and thermodynamic properties\nof atoms, molecules and materials in ground and excited\nstates. In the last couple of decades, studying excited-\nstates employing DFT has become the main research in-\nterest [8–48]. Thus one of the most natural approach to\ndo excited-state DFT is to adopt the time-independent\ndensity functional formalism [ 23,33,41,49] in which\nthe individual excited-state energies are determined from\nthe stationary states of the energy density functional.\nHowever, the question is whether there exists any such\nfunctional(s) for excited states analogous to the ground-\nstate. Not only energy functionals but also the most\nfundamental and essential requirement for excited-state\ndensity functional theory ( eDFT) is to establish the one-\nto-one mapping similar to the Hohenberg-Kohn theorem\nwhich is the main intent of the present work. Although\nthe issue of density ρ(/vector r) to potential ˆ v(/vector r) mapping for\nexcited states has been addressed in the past [ 50–55],\nbut the question still remains unanswered. So the cur-\nrentworkwill answerthe critiquesofdensity-to-potential\nmapping based on the generalized/unified constrained\nsearch(CS) due to Perdew-Levy(PL) [ 21], G¨ orling [ 24–\n26], Levy-Nagy(LN) [ 27–29], Samal-Harbola(SH) [ 56–59]\n∗E-mail: psamal@niser.ac.inand Ayers-Levy-Nagy [ 60–62].\nInthepresentwork,wewillcriticallyanalyseandmake\nfurtherance to the eDFT ideas proposed by Samal and\nHarbola[ 41,58]. Accordingtoit, (i)theCSapproachcan\nbe extended to excited-state in the light of the stationary\nstate formalism of G¨ orling [ 24–26] and variational eDFT\nformalism by Levy-Nagy [ 27–29]; (ii) within the varia-\ntionaleDFT formalism, the construction of the Kohn-\nSham(KS) system by comparing only the ground-state\ndensity is insufficient and can’t explain the existence of\nmultiple potentials; (iii) the density-to-potential map-\npingineDFTcanbeachievedthroughthefollowingcrite-\nria: comparethe groundstatesofthe trueandKSsystem\nenergetically such that it can account for the most close\nresemblance of the densities in a least square sense. SH\nshowedit bycomparingthe expectation value ofthe orig-\ninal ground-state KS Hamiltonian (obtained using the\nHarbola-Sahni [ 63] exact exchange potential) with that\nof the alternativeKS systems. Finally, the kinetic energy\nof true and KS system need to be kept closest. This is\nalso another way of comparing the ground states based\non the differential virial theorem(DVT) [ 64]; (iv) the CS\napproach is capable of generating all the potentials for a\ngivenexcited state densityand at the same time uniquely\nestablishes the density-to-potential mapping.\nThe work is organized as follows. In Sec.II, the gen-\neralized/unified CS eDFT will be briefly discussed from\nthe prospective of density-to-potential mapping. It will\nbe shown that there exist multitude of potentials for a\ngiven density. In Sec.III, furtherance of SH eDFT will be2\npresented in order to show the existence and unique con-\nstruction of the desired density-to-potential mapping. In\nthis, we will show, how the proposed eDFT is also con-\nsistent with the generalized adiabatic connection(GAC)\nKS formalism [ 11,24–26,69–77] and in principle appli-\ncable to (non-)coulombic densities. In Sec.IV, we will\nshowthe existenceofmultiple potentialsforgivenground\nor lowest excited states can never be ruled out even\nwithin Li. et al.[ 50] demonstration of Gunnarsson and\nLundqvist(GL) theorem [ 11,12]. However, based on the\ntheories presented in Sec.II & III, these seemingly con-\ntradictory results will be explained in order to justify the\nnon-violation of Hohenberg-Kohn(HK) [ 1] and GL the-\norems. Thus the density-to-potential mapping will be\ndemonstrated within [ 50] approach by making use of the\nunifiedeDFT for the two model systems (i.e. 1 Dquan-\ntum harmonic oscillator(QHO) with finite boundary and\ninfinite well external potentials). For completeness, in\nSec. V, same set of model systems will be used to ex-\nemplify density-to-potential mapping based on the CS\nformalism [ 78]. Finally, we will provide firm footing to\ndensity-to-potential mapping based on the proposed cri-\nteria ofeDFT.\nII. UNIFIED CONSTRAINED-SEARCH\nFORMULATION OF eDFT\nAlthough in principle the ground-state CS formalism\n[3–7] has all the information about the excited-states,\nthe desired density-to-potential mapping for individual\nexcited-states are not so trivial and straightforward. To\ndoso, seriesofattempts beingmadebasedonthe original\nCS approach [ 21–29,48,56–62]. In the recent past, the\nform of functional for ground state (both for degenerate\nand non-degenerate) has been extended [ 57,58,65–68] to\nstudytheexcitedstates. Nowwewillbrieflydescribehow\nthe generalized CS formalism explains the existence of\nmultiple potentials for any given fermionic density with-\nout hindering the density-to-potential mapping.\nLet’s consider Nfermions trapped in a local external\npotential ˆ vext(/vector r), described by the Hamiltonian\nˆH[ˆv;N] =ˆT+ˆVee+N/summationdisplay\ni=1ˆvext(/vector ri), (1)\nwhereˆTandˆVeeare the kinetic and electron-electron\ninteraction operators with the corresponding stationarystates are given by\nˆH[ˆv(/vector r),N]Ψk(/vector r) =Ek[ˆv(/vector r),N]Ψk(/vector r),(2)\nwhere ˆvext(/vector r)≡ˆv(/vector r). In Eq.( 2), Ψk(/vector r)≡Ψk[ˆv(/vector r),N]\nare the pure state v−representable stationary quan-\ntum states i.e. it is coming from the solution of the\nSchr¨ odinger equation. But for N−representable densi-\nties (i.e./integraltext\nρ(/vector r)d/vector r=N) and therefore wavefunctions (i.e./integraltext\nΨ[N]2d/vector r=N), similar to the HK universal functional\nthere exists an analogous functional which is stationary\nw.r.t all the variations that do not change the density\n(i.e.δΨ→ρ) and is given by\nQS[ρ;N] =δΨ[N]→ρ(/vector r)/an}b∇acketle{tΨ|ˆT+ˆVee|Ψ/an}b∇acket∇i}ht.(3)\nNow according to the Perdew and Levy extremum prin-\nciple [21] and generalized CS formalism [ 25,58,60], the\nenergy of the kthexcited state is given by\nEk=E[ρk;N] =QS[ρk;N]+/integraldisplay\nρk(/vector r)vext(/vector r)d/vector r.(4)\nIn Eq.(4), the minimization occurs only over G¨ orling’s\nstationary-statefunctional QS[ρk] and the corresponding\nwavefunctions are given by\nΨS\nk= ΨS[ρk,N] = arg min\nΨ[N]→ρk/an}b∇acketle{tΨ[N]|ˆT+ˆVee|Ψ[N]/an}b∇acket∇i}ht.(5)\nOn the other hand, in the LN [ 27–29] variational con-\nstrained minimization approach for excited-states leads\nto thekthstationary state energy\nEk[ρ,ρ0] = min\nρ[ˆv]→N/braceleftBig/integraldisplay\nρ(/vector r)vext(/vector r)d/vector r+F[ρ,ρ0]/bracerightBig\n=/integraldisplay\nρk(/vector r)vext(/vector r)d/vector r+F[ρk,ρ0], (6)\nwhereρ0is the ground state density of the system un-\nder consideration. The LN energy density functional dif-\nfers from the HKS ground-state and the stationary state\neDFT functional due to the bifunctional F[ρ,ρ0], which\nis defined by\nFk[ρ,ρ0] = min\nΨ[N]→ρ,/angbracketleftΨ[N]|Ψj[ˆv;N]/angbracketright=0,j<k/an}b∇acketle{tΨ|ˆT+ˆVee|Ψ/an}b∇acket∇i}ht\n=F[ρk,ρ0], (7)\nfor thekthexcited state. So the energy of the kthexcited\nstate can be re-expressed as\nEk[ρ,ρ0] = min\nρ[ˆv]→N/braceleftBig/integraldisplay\nρk(/vector r)vext(/vector r)d/vector r+\nmin\nΨ[N]→ρ,/angbracketleftΨ[N]|Ψj[ˆv;N]/angbracketright=0,j<k/an}b∇acketle{tΨ|ˆT+ˆVee|Ψ/an}b∇acket∇i}ht/bracerightBig\n(8)3\nwith the minimizing wavefunction denoted by\nΨLN\nk[ˆv;N] = arg min\nΨ[N]→ρ,/angbracketleftΨ[N]|Ψj[ˆv;N]/angbracketright=0,j<k/an}b∇acketle{tΨ|ˆT+ˆVee|Ψ/an}b∇acket∇i}ht.\n(9)\nIn the LN bifunctional F[ρ,ρ0], ifρ=ρ0then the func-\ntional reduces to HK universal functional and the same\nholds true for QS[ρ]. Also F[ρ,ρ0] is the generaliza-\ntion of the eDFT stationary state functional QS[ρ] as\ndescribed in the Theorems4 ,5&6of [60]. These the-\norems are in fact an artifact of the orthogonality con-\nstraint. Since all the lower states Ψ j[ˆv;N](j < k) are\ndetermined from the external potential ˆ vext(which is a\nunique functional of ground state density ρ0according to\nHK [1] theorem), implies that the ground state density\nplaysan important rolein LN-formalism. So, in principle\none can also write the excited-state density bifunctional\nasFk[ρ,ˆvext] instead of Fk[ρ,ρ0]. If the electronic densi-\nties arev−representable then Eq.( 7) modifies to\nEk[ˆvext;N] =/integraldisplay\nρk(/vector r)vext(/vector r)d/vector r+Fk[ρ,ˆvext].(10)\nFromthegeneralizedCSenergyfunctionalsforanyeigen-\ndensitygivenbytheEq.( 4)&Eq.(10)thereexistmultiple\npotential functions [ 41,58,60]. In general, these gener-\nalized multiple local external potentials can be obtained\nthrough the Euler Lagrange equation\nδ\nδρ/bracketleftBig\nEk−µ/braceleftBig/integraldisplay\nρk(/vector r)d/vector r−N/bracerightBig/bracketrightBig\n= 0 (11)\nvext(/vector r) =µ−/parenleftBigδF[ρ,ρ0]\nδρ/parenrightBig/vextendsingle/vextendsingle/vextendsingle\nρ=ρk(12)\nOR wext(/vector r) =µ−/parenleftBigδQS[ρ(/vector r)]\nδρ/parenrightBig/vextendsingle/vextendsingle/vextendsingle\nρ=ρk,(13)\nwhereµ=/parenleftBig\nδE[ρe]\nδρ/parenrightBig\nN. The actual potential is one of\nthese which should be uniquely mapped to the given\ndensity as will be shown in the following sections. In\nparticular, the local external potentials will be identi-\ncalvext(/vector r) =wext(/vector r) iff the density ρk(/vector r) is pure state\nv-representable and Ekis the corresponding eigen en-\nergy. Now to obtain the KS like equation for the genera-\ntion ofρkand to obtain Ek, one needs to first construct\na non -interacting system with some external potential\nˆv′\nextsuch that it’s mthexcited state density ρˆv′\nextm(/vector r) (say)\nmay be the same as ρk(/vector r) of the original system ˆ vext. In\nstationary-state eDFT [25,41,58], this is done by gen-\neralized adiabatic connection (GAC) [ 11,24,26,69–77].\nWhereas,inLNvariational eDFT[27–29,60], thisisdone\nby the constrained minimization of the expectation value\n/an}b∇acketle{tΨ[ˆv′\next,ρˆv′\nextm(/vector r)]|ˆT+{ˆVee= 0}|Ψ[ˆv′\next,ρˆv′\nextm(/vector r)]/an}b∇acket∇i}ht, whereΨ[ˆv′\next,ρˆv′\nextm(/vector r)] gives the desire density of interest. Out\nof many such non-interacting Ψ[ˆ v′\next,ρˆv′\nextm(/vector r)] s (differ-\nent systems), the unique one is chosen whose ground-\nstate density ρˆv′\next\n0(/vector r)(say) resembles with the ground-\nstate density ρˆvext\n0(/vector r) of the original system “most closely\nin a least-square sense”(i.e. the LN criterion). The\nmatching of the ground-state densities actually matches\nthe external potentials ˆ v′\nextand ˆvextaccording to the HK\ntheorem [ 1]. But the difference occurs between the ki-\nnetic energies of the two systems. As matter of which,\nthe discrepancy in the ρ⇐⇒ˆvmapping arises because\nthe LN criterion strictly depends upon the behavior of\nthe bifunctional.\nIII. PROPOSED CONSTRAINED-SEARCH\nFORMULATION OF eDFT\nThe CS formulation described in the previous section\nimplies that the content of the excited state function-\nalsQS[ρe] andF[ρe,ρ0] differs from the HK universal\nfunctional F[ρ] except their stationarity with respect to\nvariation in the external potential. Actually, only in the\ncase of ground-state, all the three functionals are identi-\ncal to one another and in general there exists a close link\nbetween G¨ orling QS[ρe] and Levy-Nagy F[ρe,ρ0] [60]. So\nin the unified eDFT formalism, for a given excited-state\neigendensity ρe(/vector r), bothQS[ρe]andF[ρ,ˆvext]arestation-\nary about the corresponding ˆ vextwhich also holds for the\ndesired excited-state ΨS\nk≡ΨLN\nk[41,58,60]. Now due to\nthe presenceoforthogonalityconstraintin F[ρ,ˆvext], sev-\neral choices for the set of low lying states can be made\nto which ΨLN\nkwill be orthogonal and for each choice,\nthere may exists a generalized potential function ˆ wext.\nSo some extra deciding factors are required for setting\nup theρ⇐⇒ˆvmapping which is the intent of the cur-\nrent section.\nNow resorting back to the work of Samal-Harbola [ 58],\nwe would also like to re-emphasis that the direct or in-\ndirect comparison of ground states are not sufficient to\nestablish the ρ(/vector r)⇐⇒ˆvext(/vector r) mapping or to construct\nthe KS system for excited-states [ 56]. Given the discus-\nsionson unified CS eDFT in the previoussection, we now\npresent a consistent approach to address the density-to-\npotential mapping issues. Fundamentally rigorous and\ncrucial tenets of the proposed eDFT are: ( i) There ex-\nist waysformapping an excited-statedensity ρe(/vector r) to the\ncorrespondingmany-electronwavefunctionΨ( /vector r) which in\nturn maps to the external potential ˆ vext(/vector r) through the4\nρ-stationarywavefunctions[ 25,58,60]. In this, the wave-\nfunction depends upon the ground-state density ρ0im-\nplicitly. ( ii) The KS system is to be defined through\na comparison of the kinetic energy, ground-state den-\nsity and variation of the energy w.r.t. symmetry of the\nexcited-states.\nTheclaimis, unifiedCSapproachcanprovidethemap-\nping from an excited-state density ρe(/vector r) to many-body\nwavefunction. Stationary state formalism [ 25,58] pro-\nvides a straightforward method of mapping ρe(/vector r)⇐⇒\nˆvext(/vector r), just by making sure whether /an}b∇acketle{tΨk|ˆT+ˆVee|Ψk/an}b∇acket∇i}htis\nstationary or not, subject to the condition that Ψ kgives\nρe. But [25,41,58,60] shows that different Ψ k(/vector r)s cor-\nrespond to potentials ˆ vk\next(/vector r). The same problem also\npervades through the variational eDFT approach as pro-\nposed by LN [ 27,28,58]. Thus unified CS gives, many\ndifferent wavefunctions Ψ k(/vector r) and the corresponding ex-\nternal potential ˆ vk\next(/vector r) can be associated with a given\ndensity. Nowifinadditiontotheexcited-statedensitywe\nalso have the ground-state information ρ0, then ˆvext(/vector r)\ncan be uniquely determined out of all possible multiple\npotentials ˆ vk\next(/vector r). Hence with the knowledge of ρ0, it\nis quite trivial to select a particular Ψ that belongs to\na given [ ρe,ρ0] combination by comparing ˆ vk\next(/vector r) with\nthe actual ˆ vext(/vector r). Alternatively, one can think of it as\nfinding Ψ variationally for a [ ρe,ˆvext] combination. Its\nbecause the knowledge of ρ0and ˆvextis equivalent. Now\nwiththeaboveinformation,thebifunctional F[ρe,ρ0]can\nbe redefined as\nF[ρe,ρ0] =/an}b∇acketle{tΨ[ρe,ρ0]|ˆT+ˆVee|Ψ[ρe,ρ0]/an}b∇acket∇i}ht.(14)\nThe above theoretical formulation is similar to that of\nLN [27] but avoids the orthogonality constraint imposed\nby LN formalism. This is because, the densities for dif-\nferent excited states for a given ground-state density ρ0\n(that correspondstoaunique externalpotential ˆ vext) can\nbe found in following manner: take a density and search\nfor Ψ that makes /an}b∇acketle{tΨ|ˆT+ˆVee|Ψ/an}b∇acket∇i}htstationary and simulta-\nneously make sure whether the corresponding potential\nˆwext/parenleftBig\ni.e. wext=−δF[ρ,ρ0]\nδρ/vextendsingle/vextendsingle/vextendsingle\nρ=ρe/parenrightBig\nresembles the given\nρ0( or ˆvext); if not, search for another density and re-\npeat the procedure until the correct ρis found. Thus it\nis clear that excited state orbitals Ψ are now functional\nof [ρe,ρ0]. So the correct density ρis excited state den-\nsity of the potential and the Ψ obtained in this method is\nalsoexcited state wavefunctioncorrespondingto that po-\ntential and density. After finding the correct density ρe,\nmake a variation over it so that ( ρe→ρe+δρ) and againperform the CS to find Ψ[ ρe+δρ;ρ0]. In this case, choose\nthat (ˆwext+δˆwext) which converges to ˆ vextasδρ→0.\nThe above propositions for the excited-states in terms\nof their densities are quite reasonable, particularly be-\ncause it’s development is parallel to that for the ground-\nstate DFT. On the other hand, to construct a Kohn-\nSham [2] system for a given density is not so trivial; and\nto carry out accurate calculations for excited-states, it is\nof prime importance to construct a KS system. Further,\na KS system will be meaningful if the orbitals involve in\nan excitation match with the corresponding excitations\nin the true system. Samal-Harbola [ 58] have shown that\nthe KSsystem constructedusingthe Levy-Nagycriterion\nfails in this regard. But using the form of the functional\naboveaKSsystemcanbedefinedforexcitedstate. Actu-\nally, the state dependence of the excited-state exchange-\ncorrelation functional [ 57,65–68] leads to the discrepan-\ncies while one compares the ground-states either direct\nor indirect manner. But in principle, obtaining a KS\nsystem is plausible. Now by defining the non-interacting\nkinetic energy Ts[ρe,ρ0] and using it to further define the\nexchange-correlation functional as\nExc[ρe,ρ0] =F[ρe,ρ0]−EH[ρe]−Ts[ρe,ρ0],(15)\nsolvesthe purpose. So the Eulerequation forthe excited-\nstate densities becomes\nvext=µ−/braceleftBigδTs[ρe,ρ0]\nδρ(/vector r)+VH[ρe]+δExc[ρe,ρ0]\nδρ(/vector r)/bracerightBig\n,(16)\nwhich is equivalent to solving the KS equation\n/braceleftbigg\n−1\n2∇2+ ˆvs(/vector r)/bracerightbigg\nΨi(/vector r) =εiΨi(/vector r),(17)\nwhere\nvs(/vector r) =vext(/vector r)+δ/braceleftBig\nF[ρ,ρ0]−Ts[ρ,ρ0]/bracerightBig\nδρ(/vector r)/vextendsingle/vextendsingle/vextendsingle\nρ(/vector r)=ρe[vext(/vector r)].\n(18)\nIn ground state DFT, one can easily find the Ts[ρ0] by\nminimizing the kinetic energy for a given density; here\nTs[ρ0] for a given density is obtained by occupying the\nlowest energy orbitals for a non-interacting system. But\nineDFT, to define Ts[ρe,ρ0] is not easy, as for the\nexcited-states it is not clear which orbitals to occupy for\na given density. Particularly because a density can be\ngenerated by many different configurations of the non-\ninteracting systems. Levy-Nagy select one of these sys-\ntems by comparing the ground-state density correspond-\ning to the excited-state non-interacting system with the5\ntrue ground-state density. However, LN criterion is not\nsatisfactory as pointed out by Samal and Harbola [ 56].\nThe reason of this discrepancy is due to the inconsis-\ntency of the ground-state density of an excited state KS\nsystem with the true ground-state density. The ground\n-state density corresponding to the excited-state KS sys-\ntem is not same as the ground-state density of the true\nsystem. This means the desired state is not associated\nwith ˆvext(/vector r), rather it comes from a different local poten-\ntial ˆv′\next(/vector r). To settle this inconsistency, KS system must\nbe so chosen that it is energetically very close to the orig-\ninal system and it can be ensured through the following\ncriterion. Criterion I: the non-interacting kinetic energy\nTs[ρe,ρ0]obtained through the CS need to be very close\nto the actual T[ρe,ρ0], where Ts[ρe,ρ0] andT[ρe,ρ0] are\ndefined as\nTs[ρe,ρ0] = min\nΦ→ρe/an}b∇acketle{tΦ|ˆT+ˆVee= 0/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright|Φ/an}b∇acket∇i}ht\nT[ρe,ρ0] = min\nΨ→ρe/an}b∇acketle{tΨ|ˆT+ˆVee|Ψ/an}b∇acket∇i}ht. (19)\nSo defining ∆ T=T−Tssmallest not only ensures that\nDFT exchange-correlation energy remains closer to the\nconventional quantum mechanical exchange-correlation\nenergy but also keeps the structure of the KS potential\nappropriate for the desired excited-state which is shown\nbelow. Based on the DVT [ 64], it can be argued how\nfor a given density ρeone can have different exchange -\ncorrelation ˆ vxcand external ˆ vextpotentials. According to\nDVT, the exact expression for the gradient of the exter-\nnal potential (for interacting system) for a given excited-\nstate density ρeis\n−∇ˆvext=−1\n4ρe(/vector r)∇∇2ρe(/vector r)+1\nρe(/vector r)/vectorZ(/vector r;Γ1(/vector r;/vector r′))\n+2\nρe(/vector r)/integraldisplay\n[∇ˆu(/vector r,/vector r′)]Γ2(/vector r,/vector r′)d/vector r′,(20)\nwhere ˆu=1\n|/vector r−/vector r′|. This equation represents an exact rela-\ntion between the gradient of the external potential ˆ vext,\nthee−einteraction potential ˆ u(/vector r,/vector r′) and the density\nmatrices ρ(/vector r),Γ1(/vector r;/vector r′) andΓ2(/vector r,/vector r′). The vector field /vectorZ\nin Eq.(20)is related to the kinetic-energy density tensor\nvia\nZα[/vector r;Γ1(/vector r;/vector r′)] =/bracketleftBig1\n4/parenleftBig∂2\n∂r′α∂r′′\nβ+∂2\n∂r′\nβ∂r′′α/parenrightBig\nΓ1(/vector r′;/vector r′′)/bracketrightBig\n/vector r′=/vector r′′=/vector r\n(21)\nSo,/vectorZcan be called a ”local” functional of Γ1. Similarly,\nfor KS potential Eq.( 20) reduces to\n∇ˆvKS=−1\n4ρe(/vector r)∇∇2ρe(/vector r)+1\nρe(/vector r)/vectorZKS(/vector r;Γ1(/vector r;/vector r′)).\n(22)As a given ground-state density ρ0fixes the external po-\ntential uniquely via HK theorem, which implies that ρ,\nΓ1andΓ2are also fixed from Eq.( 20). Since the den-\nsity matrices generated by some eigenfunction Ψ of the\nHamiltonian ˆH. So the fixed pair of excited-state and\nground-state density i.e. [ ρe,ρ0] may be arising from dif-\nferent configurations −different configurations can be\nthought of as arising from different external potential or\ndifferent exchange-correlationpotential and this is due to\nthe different Γ1andΓ2for a fixed ρe. Suppose a given\ndensityρeis generated through an ithKS system, then\n∇ˆvi\nKS=−1\n4ρe(/vector r)∇∇2ρe(/vector r)+1\nρe(/vector r)/vectorZi\nKS(/vector r;Γi\n1(KS)(/vector r;/vector r′)).\n(23)\nIf the density is generated through a jthexternal poten-\ntial then\n−∇ˆvj\next=−1\n4ρe(/vector r)∇∇2ρ(/vector r)+1\nρ(/vector r)/vectorZj(/vector r;Γj\n1(/vector r;/vector r′))\n+2\nρe(/vector r)/integraldisplay\n[∇u(/vector r,/vector r′)]Γj\n2(/vector r,/vector r′)d/vector r′.(24)\nAs a matter of which\n−∇ˆvxc=/vectorZKS(/vector r;Γ1(/vector r;/vector r′))−/vectorZ(/vector r;Γ1(/vector r;/vector r′))\nρe(/vector r)+\n/integraltext\n[∇ˆu(/vector r,/vector r′)][ρe(/vector r)ρe(/vector r′)−Γ2(/vector r,/vector r′)]d/vector r′\nρe(/vector r)(25)\nbecomes\n−∇ˆvij\nxc=/vectorZi\nKS−/vectorZj\nρ(/vector r)+/vector εj\nxc, (26)\nwhere/vector εj\nxcis the field due to the Fermi-Coulomb hole of\nthejthsystem [Γj\n2] . So the kinetic energy difference\nbetween the true system and KS system is given by\n∆T=1\n2/integraldisplay\n/vector r./braceleftBig\n/vectorZKS/parenleftBig\n/vector r;[Γ1(KS]/parenrightBig\n−/vectorZ/parenleftBig\n/vector r;[Γ1]/parenrightBig/bracerightBig\nd/vector r.(27)\nThis difference should be kept the smallest for the true\nKS system so that it gives the KS system consistent with\nthe original system. As a matter of which, we conclude\nthat one way to establish the ρe⇐⇒ˆvextmapping via\nthe LN formalism [ 27–29] is: if among the several poten-\ntials−which have the same excited -state density, one\ncan choose the correct KS potential by comparing the\nground-state density i.e. keep that KS-potential whose\nground-state density resembles with the true ground-\nstate density. Keeping the ground-state density close we\nactually keep the external potential fixed via HK theo-\nrem. Thus LN criterion is exact for non-interacting sys-\ntemasthereisnointeraction, sothe ground-statedensity\nmatch perfectly.6\nThis proposal of LN for ρe⇐⇒ˆvextmapping was car-\nried by Samal and Harbola [ 58] but they argued in a\nslightly different way. They proposed that both for in-\nteracting and non-interacting case among all the mul-\ntiple potentials, choose the correct KS potential whose\nground-state density differ from the exact ground-state\n“most closely by least-square sense” which is done in the\nfollowing manner. If ρ0(/vector r) is the exact ground state den-\nsity and ˜ρ0(/vector r) is that of the KS system (OR the alternate\npotentials ˆ wext) then SH proposition can be further im-\nprovedintuitively. Criterion II: the mean square distance\nbetweenρ0(/vector r)and˜ρ0(/vector r)should remain very close to zero .\nThus\n∆[ρ0(/vector r),˜ρ0(/vector r)] = min\nv[ ˜ρ0,ρe]/braceleftbigg/integraldisplay\n∞|[ρ0(/vector r)−˜ρ0(/vector r)]|2d/vector r/bracerightbigg1\n2\n≥0,\n(28)\nwhere the integration is carried out in the Sobolev space.\nThis criterion is more appropriate in the context of\nρe⇐⇒ˆvextthan the one proposed by [ 58,61]. The\ncriterion as given in Eq.( 28) will be fully satisfied if one\nmakes use of the excited state functionals [ 41,57,66–68].\nOtherwise it may fail in certain situations as pointed out\nby SH [58].\nInsteadofstickingtothe Criterion I & II ,onecaneven\ngo beyond the same through Criterion III: compare the\nground states of the true and alternate systems energet-\nically. It can be done in the following manner in order\nto select the KS system for a given density. The alterna-\ntive approachis to compare the ground-stateexpectation\nvalue of the KS system and the true system, instead of\ncomparing their ground-state densities and kinetic ener-\ngies. The procedure for comparing ground-state energy\nlevel is the following. First solve the exact DFT equa-\ntion (say Harbola-Sahni [ 63] etc) for ground-state of the\ntrue system and obtain the ground-state of KS Hamil-\ntonianH0. If the expectation value of the ground state\nHamiltonianofthetruesystemis /an}b∇acketle{tH0/an}b∇acket∇i}httrueandthatofthe\nKS system is /an}b∇acketle{tH0/an}b∇acket∇i}htKS, then one need to choose that KS\nsystem whose /an}b∇acketle{tH0/an}b∇acket∇i}htKS≃ /an}b∇acketle{tH0/an}b∇acket∇i}httrue. These criteria are well\nconnected to the GAC-KS [ 11,24–26,69–77] as discussed\nbelow.\nSince GAC-KS in principle helps for the self-consistent\ntreatment of excited states and could be considered as a\nplausible extension of HK theorem to the same. So now\nthe furtherance of the propositions made by SH [ 58] as\ndiscussedpreviouslywill be justified within the GAC-KS.\nIndeed, relying on the principles of GAC-KS, unified CS\nformalism along with the SH criteria can also establishthe density-to-potential mapping at the strictly corre-\nlated regime which will be shown below. In GAC, the\nλdependent Hamiltonian which is also used in the PL\nextremum principle is given by\nˆHλ[ˆv,N] =ˆT+λˆVee+N/summationdisplay\ni=1ˆv(/vector ri),(29)\nwith the corresponding equation of state\nˆHλ[ˆv,N]Ψλ[ˆv,N] =Eλ[ˆv,N]Ψλ[ˆv,N],(30)\nwhereλis the coupling constant with 0 ≤λ≤1 allowing\nthe electron-electron interaction to be triggered. Unlike\nthe adiabatic connection (AC)-DFT, the external poten-\ntial ˆv(/vector r), is independent of λ. Analogous to the Levy-\nLieb CS functionals, the GAC for the conjugate density\nfunctionals Fλ[ρ] (density fixed AC) and Eλ[ˆv] (potential\nfixed AC) are given by\nFλ=1[ρ] =Fλ=0[ρ]+/integraldisplay1\n0dFλ[ρ]\ndλdλ , (31)\nEλ=1[ˆv] =Eλ=0[ˆv]+/integraldisplay1\n0dEλ[ˆv]\ndλdλ . (32)\nSimilar to Eq.( 31) and (32), the excited-state function-\nalsTλ[ρ,ρ0],QS\nλ[ρ],Fλ[ρ,ρ0] andEλ[ρ,ρ0] can be de-\nfined. Upon finding these eDFT functionals, one can\ndefine the GAC by starting at a ρstationary wavefunc-\ntion forλ= 1 and then by gradually turning off ( λ= 0)\nthe electron-electron interaction. Thus the ρ-stationary\nwavefunctions for 0 ≤λ≤1 will form the GAC in eDFT.\nSince the ρ-stationary wave functions for a given ρare\nnumerable and the adiabatic connections do not overlap\nwith each other, states Φ iof non-interacting model sys-\ntems equals to the ρ-stationary wave functions at λ= 0\n(i.e.Φi= ΨS\nλ=0[ρ]) and can be assigned to real electronic\nstates Ψ j= Ψ[ρ,ν,α= 1] [25]. These assigned model\nstates are the eigenstates of the GAC-KS formalism. As\ndiscussed above, they are eigenstates of a Hamiltonian\noperator with local multiplicative potential. In this way,\nthe GAC will define the path of going from a non- inter-\nactingsystemtoaninteractingsystemviaa ρ−stationary\npath. Although for each of the interacting system, one\ncan still end up with multiple non-interactingKS system.\nBut with the criteria discussed previously it’s possible to\nselect the appropriate ones. So once the ρ⇐⇒ˆvextfor\nthe interacting system is fixed, it do carries over to the\nKS system via GAC and vice versa. This shows how7\nthe proposed unified CS formalism not only establishes\nthedensity-to-potentialmappingconcretelybutalsocon-\nstructs the KS system successfully. In the following sec-\ntions we will exemplify what we have proposed so far\nthrough two model systems. This will be done in order\naddress the critiques about density-to-potential mapping\nineDFT.\nIV. eDFT BEYOND THE HK AND GL\nTHEOREM\nThe issue of non-uniqueness in the density-to-potential\nmapping is alsopersuaded [ 50] in the context ofGL theo-\nrem [11,12]. In [50], it has been demonstrated for higher\nexcited states of the considered 1 Dmodel system there\nis no equivalence of the GL/HK theorem. But the crit-\nical analysis of [ 50] presented in this section will out-\nline how the multiplicity of potentials still can’t be ruled\nout even in the case of ground as well as lowest excited\nstates. So one need to go beyond [ 50] approach in order\nto address the validity of HK & GL theorem for such\nstate. In fact, relying on the principles of unified eDFT\napproach [ 25,41,58,60] as discussed in Sec. II along\nwith the proposed criteria of Sec. III, it will be shown\nhere why the claim made in [ 50] lacks merit to address\nthe excited-state density-to-potential mapping. To vali-\ndate the density-to-potential mapping (i.e. the analogue\nof HK/GL theorem) in [ 50] proposed approach, we will\nconsider as test cases: the examples of the 1 DQHO with\nfinite boundary and then the infinite potential well.\nFor clarity in understanding let’s first briefly discuss\nthe theoretical formulation of [ 50]. The Schr¨ odinger\nequation of two non-interacting fermions subjected to lo-\ncal one dimensional potentials v(x) andw(x) s.t.v(x)/ne}ationslash=\nw(x)+C, whereCis a constant are given by\n/bracketleftBig\n−1\n2d2\ndx2+v(x)/bracketrightBig\nΦi(x) =εiΦi(x),(33)\n/bracketleftBig\n−1\n2d2\ndx2+w(x)/bracketrightBig\nΨi(x) =λiΨi(x).(34)\nSuppose that the eigenfunctions of the local potential\nw(x) generates the ground/excited-state eigendensity of\nv(x) as one of it’s eigendensity but with some arbitrary\nconfiguration which is either same or different from the\noriginal one. Then one possible way of achieving this\nis: the wavefunctions Ψ( x) of the potential w(x) can be\nassociatedtothewavefunctionsΦ( x)ofthepotential v(x)via the following unitary transformation i.e./parenleftbigg\nΨk(x)\nΨl(x)/parenrightbigg\n=/parenleftbigg\ncosθ(x) sinθ(x)\n−sinθ(x) cosθ(x)/parenrightbigg/parenleftbigg\nΦi(x)\nΦj(x)/parenrightbigg\n=/parenleftbigg\nΦi(x)cosθ(x)+Φj(x)sinθ(x)\n−Φi(x)sinθ(x)+Φj(x)cosθ(x)/parenrightbigg\n,(35)\nAs a matter of which the density preserving constraint\nwill be satisfied and the ground/excited state density of\ntwo potentials remain invariant i.e.\nρ(x) =|Φi(x)|2+|Φj(x)|2=|Ψk(x)|2+|Ψl(x)|2.(36)\nNow the potentials can be obtained from the Eqs.( 33)\nand (34) by inverting the same\nv(x) =εi+¨Φi(x)\n2Φi(x)=εj+¨Φj(x)\n2Φj(x)(37)\nw(x) =λk+¨Ψk(x)\n2Ψk(x)=λl+¨Ψl(x)\n2Ψl(x).(38)\nAlso from Eqs.( 33) and (34), the difference between any\ntwoeigenvalues∆ and∆′correspondingto the potentials\nv(x) andw(x) are given by\n∆ =εj−εi=1\n2Φi(x)Φj(x)d\ndx[Φj(x)˙Φi(x)−Φi(x)˙Φj(x)],\n(39)\n∆′=λk−λl=1\n2Ψk(x)Ψl(x)d\ndx[Ψl(x)˙Ψk(x)−Ψk(x)˙Ψl(x)].\n(40)\nNow by plugging the values Ψ k(x) and Ψ l(x) from\nEq.(35) back in Eq.( 40), the rotation θ(x) can be ob-\ntained from the following\nd\ndx[˙θ(x){Φ2\ni(x)+Φ2\nj(x)}+{Φj(x)˙Φi(x)−Φi(x)˙Φj(x)}]\n= ∆′[2Φi(x)Φj(xcos2θ(x)+{Φ2\nj(x)−Φ2\ni(x)}sin2θ(x)]\n(41)\nor\nρ(x)¨θ(x)+˙ρ(x)˙θ(x)+f(Φi(x),Φj(x),∆,∆′,θ) = 0,(42)\nwhere\nf=2∆Φ i(x)Φj(x)−∆′[Φi(x)Φj(x)cos2θ(x)\n+{Φ2\nj(x)−Φ2\ni(x)}sin2θ(x)].(43)\nThe Eq.( 42) is the central equation of [ 50] theoretical\nframework which need to be solved numerically with\nproper initial conditions in order to obtain the alternate\npotential w(x) for any given density and eigenvalue dif-\nferences. In this work, the adopted numerical procedure\nto solve the above mentioned differential equation is very\nmuch accurate even at the boundary where obtaining ap-\npropriate structure and behavior of the multiple poten-\ntials and the corresponding wavefunctions are important\nand crucial.8\nA. Results: 1DQuantum Harmonic Oscillator\nThe first model system for demonstrating the density-\nto-potential mapping is the 1 DQHO defined by\nv(x) =1\n2ω2x2,where −l≤x≤l .(44)\nSo the wavefunctions and energy eigenvalues of the nth\neigenstate are given by\nΦn(x) =/parenleftBigω\nπ/parenrightBig1\n41√\n2nn!Hn(√ωx)exp(−ωx2\n2),(45)\nεn= (n+1\n2)ω , (46)\nwheren= 0,1,2.....\n(atomic units are adopted i.e. /planckover2pi1= 1 and me= 1)\n-1-0.500.51ρ[V],Ψ[V] V[ρ]Δ'=10.0\nx-20020406080100\n-2-1.5 -1-0.5 00.5 11.5 2\nFIG. 1. Upper panel: Shows (red color) the ground state\ndensity of the 1D QHO and the corresponding transformed\nwavefunctions Ψ k(blue) and Ψ l(green) for ∆′= 10.0. Lower\npanel: Shows the alternate potential associated with above\nwavefunctions and density.\nB. Fermions in The Ground State\nNow consider two non-interacting fermions occupying\nthe ground-state of the QHO i.e. n= 0 =m. So the\neigenvalue difference for this state ∆ = ε0−ε0= 0 and\nthe density is given by\nρ(x) = 2/parenleftBigω\nπ/parenrightBig1\n2exp(−ωx2). (47)-1-0.500.51\n-20020406080100\n-2 -1 0 1 2ρ[V],Ψ[V]V[ρ]Δ'=46.0\nx\nFIG. 2. The figure caption is same as Fig. 1but with ∆′=\n46.0.\nThus the correspondingequation for rotation θ(x) can be\nobtained from the Eq.( 42) and is given by\nρ(x)¨θ(x)+˙ρ(x)˙θ(x)−∆′[2/parenleftBigω\nπ/parenrightBig1\n2exp(−ωx2)cos2θ(x)] = 0.\n(48)\nNow Eq.( 48) has to be solved with proper initial condi-\ntions. The initial conditions can be fixed by taking into\nconsideration the symmetry of the differential Eq.( 48)\nand the normalization condition of the wavefunction.\nFrom Eq.( 48) it is clear thatdθ\ndx|(x=0)= 0 as both Φ( x)\nandρ(x) are symmetric about x= 0. Now another con-\ndition is that Ψ k(x) and Ψ l(x) must also be normalized.\nSo if we plot the renormalization R\n/integraldisplayl\n−l|Ψk,l(x)|2dx−1 =R= 0 (49)\nas a function of θ(x= 0), then the points where R= 0\ncorrespondsto the normalization of Ψ k(x) and Ψ l(x) [50]\nand it will provide the initial condition on θ(x= 0). Af-\nter finding θ(x), the transformed set of normalized wave-\nfunctions Ψ k(x) and Ψ l(x) is being obtained. Again us-\ning these wavefunctions the potential w(x) can be deter-\nmined from the Eq.( 38). In Fig. 1and Fig. 2, we have\nshown two different potentials which are obtained for\nthe eigenvalue differences ∆′= 10.0 and ∆′= 46.0 re-\nspectively along with the corresponding wavefunctions.\nThe important point of observation here is that the\nground-state density ρQHO=ρ0[v(x) =vQHO(x),N=9\n2] now corresponds to some arbitrary excite-state hav-\ning density ρe[w(x)/ne}ationslash=vQHO(x),N= 2]. As a mat-\nter of which, for the fixed ρQHO\n0and ∆′, the system\ngets transformed to some other system w(x) for which\nQS[ρe[w(x)] =ρQHO\n0] and/or F[ρe[w(x)] =ρQHO\n0,˜ρ0]\nwill be stationary. The corresponding stationary states\nare basically the transformed wavefunctions which are\ngiven by Eq.( 34) ΨS\nl[w(x)] = Ψ l[ρe,˜ρ0]. In this ˜ ρ0is\nthe ground state density of the newly generated poten-\ntialw(x) and ˜ρ0/ne}ationslash=ρQHO\n0. Now from the proposed cri-\nteria it follows that ∆ T/ne}ationslash= 0 , ∆[ ρ0(x),˜ρ0(x)]>0 and\nnew system is energetically far off from the original one.\nHence the given ground-state density should be uniquely\nmapped to the original QHO potential v(x) although\nthere exist several multiple potentials w(x). This result\nis consistent with the generalized/unified CS formalism\n[25,41,58,60].\n-0.4-0.200.20.40.6\n-50510152025303540\n-3 -2 -1 0 1 2 3xρ[V],Ψ[V] V[ρ]Δ ' \u0000 \u0001 \u0002 \u0003 \u0004\nFIG. 3. The figure caption is same as Fig. 1but for the lowest\nexcited state density being produced with ∆′= 15.0.\nC. Fermions in The Lowest Excited State\nAs the second example, we consider the lowest excited-\nstateoftheQHO.Sothetwonon-interactingfermionsare\nnow occupying the n= 0 and m= 1 state. For this case,\nε0=1\n2ω,ε1=3\n2ωand ∆ = ε1−ε0=ωwith the density\nρ(x) =/parenleftBigω\nπ/parenrightBig1\n2exp(−ωx2)(1+2ωx2),(50)-0.6-0.4-0.200.20.40.6\n-50510152025303540\n-3 -2 -1 0 1 2 3Δ'=35.0ρ[V],Ψ[V] V[ρ]\nx\nFIG. 4. The figure caption is same as Fig. 3but with ∆′=\n35.0.\nand the corresponding equation for rotation θ(x) is given\nby\nρ(x)¨θ(x)+ ˙ρ(x)˙θ(x)+2ωx/parenleftbigg2ω2\nπ/parenrightbigg1\n2\nexp(−ωx2)\n−∆′[2x/parenleftbigg2ω2\nπ/parenrightbigg1\n2\nexp(−ωx2)cos2θ(x)+\nω\nπexp(−ωx2){2ωx2−1}sin2θ(x)] = 0.(51)\nSince in this case Φ 0(x) is symmetric, Φ 1(x) is antisym-\nmetric, so ρ(x) symmetric around x= 0. Thus Eq.( 51)\nimplies that θ(x) should be symmetric at x= 0. The\ninitial conditions ondθ\ndx|(x=0)is obtained from the behav-\nior of the renormalization Ras a function ofdθ\ndx|(x=0).\nFollowing the same procedure as before, in this case also\nwe have obtained different potentials for the fixed lowest\nexcited state density which are shown in the Fig. 3and\nFig.4. These two alternative potentials and the trans-\nformed wavefunctions correspond to two different eigen-\nvalue differences ∆′= 16.0 and ∆′= 35.0. As described\nin the ground state case, in this case also the structure\nof the potential is different from the original 1D QHO\nas the potential should follow the structure of the wave-\nfunctions. However, according to the unified CS eDFT,\nthe results are never due to the violation of GL theorem.\nThis is because the ground and lowest excited states of\nthe newly found potential are quite different from that of\nthe QHO. So following similar argument as in the previ-\nouscase,nowthelowestexcited-statedensityoftheQHO10\ncorresponds to some different eigendensity of the multi-\nple potentials. Thus the multitude of potentials poses no\nissues for the validity of the GL theorem.\n-0.8-0.6-0.4-0.200.20.40.6\n-5051015\n-3 -2 -1 0 1 2 3Δ'=8.0ρ[V],Ψ[V]V[ρ]\nx\nFIG. 5. The figure caption is same as Fig. 1but for one of the\nhigher excited state density being produced with ∆′= 8.0.\n-0.6-0.4-0.200.20.40.60.8\n-50510152025303540\n-3 -2 -1 0 1 2 3Δ'=30.0ρ[V],Ψ[V]V[ρ]\nx\nFIG. 6. The figure caption is same as Fig. 5but with ∆′=\n30.0.-2-1.5-1-0.500.511.52\n-300-200-1000100200300400500600\n0 0.2 0.4 0.6 0.8 100.511.522.533.54\n0 0.2 0.4 0.6 0.8 1Δ'=200.0\n\u0005[V] V[\n\u0006]\nx\n\u0007(x)\nx\nFIG. 7. Upper panel: Shows the alternate wavefunctions Ψ k\nand Ψ l(green & red) resulting the ground state density of\n1D potential well for ∆′= 200.0. Lower panel: Shows the\nalternate potentials (green & red) and the density (magenta )\nassociated with above wavefunctions.\nD. Fermions in Higher Excited States\nHere we consider one of the higher excited-state of 1D\nQHO (i.e. two non-interacting fermions are in the n= 0\nandm= 2states). Forthiscase,theeigenvaluedifference\nis ∆ =ε2−ε0= 2ωand the density corresponding to it\nis given by\nρ(x) =/parenleftBigω\nπ/parenrightBig1\n2exp(−ωx2){1+(1−2ωx2)2}.(52)\nSimilarly, the corresponding equation for rotation θ(x) is\ngiven by\nρ(x)¨θ(x)+˙θ(x)˙θ(x)+4ω/parenleftBigω\n2π/parenrightBig1\n2(2ωx2−1)exp(−ωx2)\n−∆′[/parenleftBigω\n2π/parenrightBig1\n2(2ωx2−1)exp(−ωx2)cos2θ(x)\n+/parenleftBigω\nπ/parenrightBig1\n2exp(−ωx2){1\n2(2ωx2−1)2−1}sin2θ(x)] = 0.\n(53)\nNow by solving Eq.( 53) for rotation θ(x) in analogous\nwith the ground-state of the QHO and after taking care\nof the normalization of the transformed wavefunctions,\nthe potential w(x) is obtained for ∆′= 8.0,30.0. The11\n-2-1.5-1-0.500.511.52\n-200-1000100200300400500\n0 0.2 0.4 0.6 0.8 100.511.522.533.54\n0 0.2 0.4 0.6 0.8 1ρ(x)\nx\nxψ[V] V[ρ]Δ'=600.0\nFIG. 8. The figure caption is same as Fig. 7but with ∆′=\n600.0.\npotentialsalongwith thewavefunctionsareshownin Fig.\n5& Fig.6. Similar to ground and lowest excited-state,\nhere too the given density is a different eigendensity of\nthe new potentials. If it would have the same eigenden-\nsity ofw(x) thenw(x) should have been identical to the\nvQHO(x). But it is not the case. That’s why the gener-\nated potentials are completely different from the QHO.\nE. Results: 1DInfinite Potential Well\nAs our second case study, we consider the model sys-\ntem same as that reported in [ 50] (i.e. particles are\ntrapped inside an 1 Dinfinite potential well). For an\ninfinite potential well with length varying from 0 to 1,\nthentheigenfunction Φ n(x) and the energy eigenvalue\nεnare given by\nΦn(x) =√\n2sin(nπx) ;εn=n2π2\n2,(54)\nwheren= 1,2,3..... The density ρ(x) corresponding to\nthe two potentials v(x) andw(x) is given by Eq.( 36).-2-1.5-1-0.500.511.52\n-2000200400600\n0 0.2 0.4 0.6 0.8 100.511.522.533.54\n0 0.2 0.4 0.6 0.8 1Δ'=1000.0ψ[V]\b(x)\nx\nxV[\n\t]\nFIG. 9. The figure caption is same as Fig. 7but with ∆′=\n1000.0.\nF. Fermions in The Ground State\nFortwospinlessnon-interactingparticlesin n= 1 =m\nstates, the energies of two states and the difference are\nε1=π2\n2=ε2; ∆ =ε2−ε1= 0.(55)\nThe density corresponding to these states is\nρ(x) = 4[sin2(πx)], (56)\nand the equation corresponding to Eq.( 42) for the rota-\ntionθ(x) is\nρ(x)¨θ(x)+ ˙ρ(x)˙θ(x)−∆′[4sin2πxcos2θ(x)] = 0.(57)\nSince Φ 1(x) is symmetric and ρ(x) is symmetric about\nx=1\n2. Thus Eq.( 57) indicates that θ(x) should be sym-\nmetric such that ˙θ(1\n2) = 0. With this initial condition\nand choosing any value of ∆′one can solve for θ(x) and\nsubsequently obtain the Ψ ks. Now using these Ψ ks,\nthe alternate potentials w(x) will be obtained by using\nEq. (38). Since the transformed wavefunction Ψ k(x)\nmust also be normalized. This condition will be fulfilled\nby choosing the appropriate value of θ(1\n2) at which the\nΨk(x) should be normalized. Once Ψ k(x) is normalized\nthen Ψ l(x) will also be normalized. Again by adopting12\n-2-1.5-1-0.500.511.52\n00.511.522.533.5\n0 0.2 0.4 0.6 0.8 1\n-50050100150200250300350400\n0 0.2 0.4 0.6 0.8 1Δ'=200.0\nρ(x)\nx\nxV[ρ]ψ[V]\nFIG. 10. The figure caption is same as Fig. 7butfor the lowest\nexcited state density being produced with ∆′= 200.0.\nthe same procedure as that described in the case of 1 D\nQHO, the alternative multiple potentials are obtained by\nmaking use of the following renormalization Rcondition\n/integraldisplay1\n0|Ψk(x)|2dx−1 =R= 0. (58)\nAll the wavefunctions, densities and multiple potentials\nare shown in the Figs.( 7to9). Here we have generated\nthe multiple potentials for ∆′= 200.0,600.0 & 1000 .0\nrespectively. As is expected, the wavefunctions are to-\ntally different from the ground state of the 1 Dinfinite\nwell. Although the density remains to be the same in all\nthe cases. But its not the ground state eigendensity of\nthe multiple potentials. So this poses no issue for the HK\ntheorem.\nG. Fermions in The Lowest Excited State\nNow consider two fermions occupying the n= 1, m=\n2 (i.e. the lowest excited-state) eigenstates of the infinite\npotential well. Here too we have obtained several mul-\ntiple potentials unlike [ 50]. For this excited-state, the\nenergy eigenvalues are ε1=π2\n2,ε2= 2π2with ∆ =3π2\n2.\nHence the density arising from these two states is given-1.5-1-0.500.511.52\n-50050100150200250300\n0 0.2 0.4 0.6 0.8 100.511.522.533.5\n0 0.2 0.4 0.6 0.8 1Δ'=600.0\nρ(x)\nxV[ρ] Ψ[V]\nx\nFIG. 11. The figure caption is same as Fig. 10but with ∆′=\n600.0.\nby\nρ(x) = 2[sin2(πx)+sin2(2πx)]. (59)\nSimilar to the previous examples, the equation for the\nrotation θ(x) is the following\nρ(x)¨θ(x)+ ˙ρ(x)˙θ(x)+6π2sin(πx)sin(2πx)\n−∆′[4sin(πx)sin(2πx)cos2θ(x)\n+2{sin2(2πx)−sin2(πx)}sin2θ(x)] = 0.(60)\nHere Φ 1(x) is symmetric, Φ 2(x) is antisymmetric and\nρ(x) symmetric about x=1\n2. Thus Eq.( 60) predicts that\nθ(x) is antisymmetric such that θ(1\n2) = 0. In this case\nalso normalization of both Ψ k(x) and Ψ l(x) are taken\ncare and the proper R(renormalization) values are ob-\ntained w.r.t.dθ\ndx(1\n2). Quite interestingly, in this case\nalso we have successfully generated multiple potentials\nfor ∆′= 200.0,600.0 & 1000 .0. This is where [ 50] failed\nto explain the validity of GL theorem. As expected, the\npotential follows the wavefunctions pattern. This is ob-\nvious at the boundary where the wavefunctions are per-\nfectly vanishing, the potential shoots up to a very large\npositive value. The potentials along with wavefunctions\nare shown in the Figs.( 10to12). Following the same ar-\ngument as in the case of the previous model system, the\nmultiplicity of potentials obtained here are nothing to do\nwith the GL theorem.13\n00.511.522.533.5\n0 0.2 0.4 0.6 0.8 1ρ(x)\nx\nxV[ρ] ψ[V]\n-50050100150200250300\n0 0.2 0.4 0.6 0.8 1-1-0.500.511.52\nΔ'=1000.0\nFIG. 12. The figure caption is same as Fig. 10but with ∆′=\n1000.0.\nH. Fermions in Higher Excited States\nNow to complete our exploration on 1 Dwell, we have\nconsidered here the second excited-state of it. This is\nthe only excited-state for which [ 50] reported multiple\nexternal potentials for various eigenvalue differences. We\ntoo generated multiple potentials and the corresponding\nwavefunctions for ∆′= 200.0,600.0 & 1000 .0 which are\nshown in the Figs.( 13to15). The results follow the trend\nsimilar to that of the ground and lowest excited-state. In\nall the cases, we have noticed that the potentials and\nthe corresponding rotation angles can never attain flat\nstructure at the boundary unlike [ 50].\nTo conclude this section, we would like to shed some\nlight on the structure of the generated potentials at\nthe boundary as its very important to be determined\naccurately. Since the wavefunctions die out towards\nthe boundary. Thus the potentials obtained by the\nSchr¨ odinger equation inversion (i.e. Eq. 38) for specified\neigenvaluedifferenceswill attain largepositivevalue. Ac-\ntually, in our approach we have gone way beyond [ 50] to\ngenerate the accurate structure of the potential which is\nclear from the results. The important point to be noted\nis that the singularity of the potential plays the most\ncrucial role if one directly solving the Schr¨ odinger equa--2-1.5-1-0.500.511.52\n-1000100200300400500\n0 0.2 0.4 0.6 0.8 100.511.522.533.54\n00.2 0.4 0.6 0.8 1Ψ[V]\nΔ'=200.0\nρ(x)\nxxV[ρ]\nFIG. 13. The figure caption is same as Fig. 7but for one of the\nhigher excited state density being produced with ∆′= 200.0.\ntion. But in getting the potential structure whether by\ninverting the Schr¨ odinger equation or CS method solely\ndepends on the wavefunction behaviorin a given domain.\nSo better access of the wavefunction’s behavior will by\ndefault lead to reliable potential structure.\nV. RESULTS WITHIN THE CS FORMALISM\nIn this section, we will discuss the results in connec-\ntion with the density-to-potential mapping based on the\nCS-formalism discussed earlier. According to it, there\nexist multiple potentials for a given ground or excited\nstate (eigen)density. But for the case of excited state\ndensity, when it is produced as some different excited-\nstate of these multiple potentials (except the actual one)\nthe corresponding ground-states are completely different\nfrom that of the original system. Similarly, one can pro-\nduce potentials whose ground-state density may be same\nas the excited-state density of the original system. The\nresults we have obtained for the systems of our study are\nfully consistent with the unified CS eDFT. The Zhao-\nParr [78] CS method is being used to show the multiplic-\nity of potentials for a given density.\nTobegin the CS exemplification (shownin Fig. 16), lets\nconsider fournon-interacting particles in an 1 Dpoten-14\n-2-1.5-1-0.500.511.52\n-1000100200300400500600\n0 0.2 0.4 0.6 0.8 100.511.522.533.54\n0 0.2 0.4 0.6 0.8 1Δ'=600.0Ψ[V]\nρ(x)\nx\nxV[ρ]\nFIG. 14. The figure caption is same as Fig. 13but with ∆′=\n600.0.\ntial well, where two fermions are in n= 1 state and one\nfermion each in n= 2 and n= 4 state. As a result, this\ngivessomeexcitedstatedensity ρe(x) associatedwith the\nabove configuration which is shown in the Fig. 16(a) and\nis given by\nρe(x) =ρV0\ne(x) = 2|Ψ1(x)|2+|Ψ2(x)|2+|Ψ4(x)|2,(61)\nwhere Ψ i(x) s are the wavefunctions of the 1 Dpoten-\ntial well. In all our results shown in the figures ( 16)\nto (22), we have adopted notation ρ(ni(fj)), where ni\ndenotes the quantum number of the eigenfunctions of\nthe potential VorVi(i= 1,2,3,4) andfj, the occu-\npation. Using CS [ 78] the excited state density ρe(x)\ngiven by Eq.( 61) is produced through another alterna-\ntive potential V1(say) whose n= 1 state is occupied\nwith 2 fermions (i.e. f1= 2) and n= 2,n= 3 with one\nfermion each (i.e. f2= 1 =f3). Now the ground state\ndensity of the potential V1is different from that of the V0\n(i.e. particle in an infinite potential well) which is given\nby ˜ρ(1)\n0(Fig.16a). As per our formalism, there can be\nmany such multiple potentials having the given density\nas it’s eigendensity associated with some combination of\neigenfunctions. So it is possible that one can also ob-\ntain second alternative potential V2(say) whose ground-\nstate density is same as the above excited state density\n(ρe(x)) of the original system ( V0). In this way, we have-2-1.5-1-0.500.511.52\n-2000200400600800\n0 0.2 0.4 0.6 0.8 100.511.522.533.54\n0 0.2 0.4 0.6 0.8 1V[ρ] Ψ[V]\nxρ(x)\nxΔ'=1000.0\nFIG. 15. The figure caption is same as Fig. 13but with ∆′=\n1000.0.\nstudied six such excited states of the 1 Dpotential well\n(Figs.16to21) and for each case we are able to produce\nsymmetrically different multiple potentials for fix densi-\nties. Also in each case, we have produced the alternative\npotential whose ground-state density is nothing but the\ngiven excited-state density of the original configuration\n(i.e. 1Dpotential well).\nAsourfinalcasestudy, wehaveconsideredtheexcited-\nstates of the 1 DQHO. This is also an interesting model\nsystem like the potential well. The results for this case,\nare shown in Fig. 22. Now consider the Fig. 22a, in this\ncase we have produced three symmetrically different al-\nternative potentials V1,V2andV3(shown in Fig. 22b)\nwhose ground-states densities (i.e. ρ(1)\n0(x),ρ(2)\n0(x) and\nρ(3)\n0(x))are same as the different excited-states densi-\nties (i.e. ρ(1)\ne(x),ρ(2)\ne(x) andρ(3)\ne(x)) of the QHO po-\ntentialV(x). Here ρ(1)\ne(x) corresponds to the config-\nuration [ n= 0(f0= 1),n= 3(f3= 1)]. Similarly,\nρ(2)\ne(x) andρ(3)\ne(x) are arising from the excited-state\nconfigurations [ n= 1(f1= 1),n= 2(f2= 1)] and\n[n= 2(f2= 1),n= 3(f3= 1)] respectively. In Fig. 22(d),\nwe have produced a different potential V1whose excited-\nstate density corresponding to the configuration [ n=\n0(f0= 1),n= 2(f2= 1)] is same as the excited-state\ndensityρe(x) ([n= 0(f0= 1),n= 3(f3= 1)]) of the15\n(a)\n(b)(c)\n(d)\nFIG. 16. (a) ρe[n1(2),n2(1),n4(1)] is the excited-state den-\nsity of 1 Dpotential well with ground-state ρ0. ˜ρ(1)\n0is\nthe ground-state density of potential V1whose excited-state\nconfiguration [ n1(2),n2(1),n3(1)] results the same ρe. (b)\nV2[ρe] is the potential whose ground-state configuration re-\nsults the same ρeof (a) and is shown along with V1[˜ρ(1)\n0].\n(c)ρe[n1(2),n3(1),n4(1)] is the excited-state density of 1 D\npotential well with ground-state ρ0and produced in an al-\nternative configuration [ n1(2),n2(1),n4(1)] (V1[˜ρ(1)\n0]) besides\nthe ground-state configuration leading to V2[ρe]. (d) Shows\nall the alternative potentials of (c).\noriginal 1 DQHO potential. Although we have produced\nso many potentials, but our criteria will only select the\noriginal potentials (i.e. the infinite potential well in the\nprevious and QHO in the current study) for any given\n(i.e. either ground or excited-state) density. Thus estab-\nlishes the excited-state ρ(x)⇐⇒ˆv(x) mapping uniquely.\nVI. DISCUSSIONS\nNow the conceptually basic questions of eDFT: what\nare the consequences as well as similarities and differ-\nences between the results of the CS formalism and that\nobtained in connection to the HK/GL theorem? Sec-\nondly, whether there arisen any critical scenario which is\ninconsistentwiththe HKand/orGLtheorem(s)? Thisis\nbecause several multiple potentials are obtained for non-\ninteracting fermions in the ground as well as lowest ex-\ncited state. Not only that, [ 50,55] have also claimed that\nfor higher excited-states there is no analogue of HK theo-\nrem. So the seeminglycontradictoryresults maygiverise(a)\n(a)\n(b)(c)\n(d)\nFIG. 17. (a) ρe[n1(1),n2(2),n4(1)] is theexcited-state density\nof 1Dpotential well with ground-state ρ0. ˜ρ(1)\n0and ˜ρ(2)\n0are\nthe ground -state densities of V1andV2whose excited-state\nconfigurations [ n1(2),n2(1),n3(1)] and [ n1(2),n2(1),n4(1)]\nresults the same ρe. (b)V3[ρe] is the potential whose\nground-state configuration gives the same ρeof (a) and is\nshown along with V1,V2. (c)ρe[n2(2),n3(1),n4(1)] is the\nexcited-state density produced in alternative configurati on\n[n1(2),n2(1),n3(1)] (V1[˜ρ(1)\n0]), besides the ground-state con-\nfiguration leading to V2[ρe]. (d) Shows all the alternative\npotentials in (c).\ntothewrongconclusionaboutthevalidityofHK/GLthe-\norem and non-existence of density-to- potential mapping\nfor excited-states. However, the generalized/unified CS\nformalism overrules all these claims by showing that the\nground-statedensity of a given symmetry (potential) can\nbetheexcited-statedensityofdifferingsymmetry(poten-\ntial). Now this excited-state will have a corresponding\nground state which will be obviously quite different from\nthe ground-state of the original system. As a matter of\nwhich there will exist a different potential according to\nHK theorem. This is also true for the excited -state den-\nsity of the actual system: when it becomes either the\nground-state or some arbitrary excited -state density of\nanother potential. So the unified CS formalism justifies\nthe non-violation of HK/GL theorem for such states.\nNow based on the unified/generalized CS eDFT, one\ncan very nicely interpret ours as well as [ 50] results. Ac-\ntually by keepingthe excited/groundstate density fix via\naunitary transformationneverguaranteethe symmetries\nof the states involve will remain intact. This is because\nby changing the ∆′value and keeping either ground or16\n(a)\n(b)(c)\n(d)\nFIG. 18. (a) ρe[n1(1),n3(2),n4(1)] is the excited-state den-\nsity of 1 Dinfinite potential well with ground-state ρ0.\n˜ρ(1)\n0and ˜ρ(2)\n0are the ground-state densities of V1andV2,\nwhose excited-state configurations [ n1(2),n2(1),n3(1)] and\n[n1(2),n3(1),n4(1)] results the same ρe. (b)V3is the po-\ntential whose ground-state density is same as ρeof (a) and\nis shown along with V1,V2. (c)ρe[n2(1),n3(2),n4(1)] is the\nexcited-state density produced via the alternative configu ra-\ntions [n1(2),n2(1),n3(1)] (V1[˜ρ(1)\n0]) besides the ground-state\nconfiguration leading to V2[ρe]. (d) Shows all the alternative\npotentials of (c).\nthe excited state density fix, we are forcing the system\nto change itself accordingly without hindering only the\nfixed density constraint. Since ∆′is not fixed. So in\nprinciple one can make several choices for ∆′and for\neach choice, the system will converge to different poten-\ntials (systems/configurations) which can give the desired\ndensity of ground/ excited-state of the original system\n(potential/configurations)as one of it’s eigendensity. Ac-\ntually, the converged potentials are those for which the\nG¨ orling and LN functionals are stationary and minimum\nrespectively. So everything is again automatically fits\nintorealmofgeneralizedCSformalismandnothingreally\ncontradicting or posing issues for the eDFT formulations\nprovided by [ 24–29,41,58,60,61]. Also the transformed\nquantum states leading to multitude of potential for a\ngiven density are energeticallyfar off from the actual sys-\ntem and even the ground-states are also very different.\nThus the generalized CS formalism proposed in this work\nalong with the SH criteria can be considered as the most\nessentialstepsforestablishingthe ρ(/vector r)⇐⇒ˆvext(/vector r)which\nfurther elaborated below.\n(a)\n(b)(c)\n(d)\nFIG. 19. (a) ρe[n1(1),n2(1),n4(2)] is the excited-state\ndensity of 1 Dinfinite potential well with ground-state\nρ0. ˜ρ(1)\n0, ˜ρ(2)\n0and ˜ρ(3)\n0are the ground-state densi-\nties ofV1,V2andV3whose excited-state configurations\n[n1(2),n2(1),n3(1)],[n1(2),n2(1),n4(1)] and [ n1(2),n4(2)] re-\nsults the same ρe. (b)V4is the potential whose ground-\nstate density is same as ρeof (a) and is shown along with\nV1,V2andV3. (c)ρe[n1(1),n3(1),n4(2)] is the excited-\nstate density produced in the alternative configurations\n[n1(2),n2(1),n3(1)] (V1[˜ρ(1)\n0]), [n1(2),n2(1),n4(1)] (V2[˜ρ(2)\n0])\nand [n1(2),n3(1),n4(1)] (V3[˜ρ(3)\n0]) besides the ground-state\nconfiguration leading to V4[ρe]. (d) Shows all the alternative\npotentials of (c).\nNow the question is out of these existing multiple po-\ntentials in association with a fix density and ∆′, which\npotential in principle should be picked in view of the\nρ(x)⇐⇒ˆv(x)? The criteria of selecting the exact poten-\ntial out of all possibilities have already been discussed in\nSec.III. First of all it is quite obvious from the Figs.( 1to\n6) and from Fig. 7toFig.15that the ground-state den-\nsities of the generated alternate potentials are different\nfrom that of the original potential. This is also true\neven for the results of the CS formalism as shown in the\nFigs.(16to22). So when we are fixing the excited-state\ndensity at the same time we should have taken care of\nthe ground-state of the newly found system and the old\none. Similarly, when several multiple potentials are gen-\nerated for a given ground-state density, the same is not\nproduced as the ground state eigendensity of the alter-\nnate potentials. So it’s quite obvious that there is no\nviolation of the HK theorem. The criteria of taking care\nofthe ground-statesofthe twosystemisgiveninEq.( 28).17\n(a)\n(b)(c)\n(d)\nFIG. 20. (a) ρe[n2(1),n3(1),n4(2)] is the excited-state den-\nsity of 1 Dinfinite potential well with ground-state ρ0.\n˜ρ(1)\n0and ˜ρ(2)\n0are the ground state densities of V1and\nV2, whose excited-state configurations [ n1(2),n2(1),n3(1)]\nand [n1(2),n2(1),n4(1)] results the same ρe. (b) V3\nis the potential whose ground state density is same\nasρeof (a) and is shown along with V1,V2. (c)\nρe[n1(2),n4(2)] is the excited-state density produced in al-\nternative configurations [ n1(2),n2(1),n3(1)] (V1[˜ρ(1)\n0]) and\n[n1(2),n2(1),n4(1)] (V2[˜ρ(2)\n0]) besides the ground-state con-\nfiguration leading to V3[ρe]. (d) Shows all the alternative\npotentials of (c).\nAdditionally the kinetic energies of the two systems need\nto be kept closest, which we have pointed out on the ba-\nsis of DVT. So in all the non-interacting model systems\nreportedhere, ∆ Tshould havebeen zero. But the drasti-\ncally differing structures of the transformed and original\nwavefunctions are nothing but the manifestation of non-\nvanishing difference of kinetic energies and thus leading\nto the multiple potentials. Furthermore, the most signif-\nicant differences between the symmetries of the old and\nnew systems implies that principally there exist discrep-\nancies in the expectation values of the Hamiltonian w.r.t.\nthe ground-states of various multiple potentials. This is\nwhat trivially follows from the reported results. Hence,\nthe proposed criteria uniquely maps a given density of\nthe 1DQHO/infinite well to a potential which is noth-\ning but the 1 DQHO/infinite well and discards rest of\nthe multiple potentials.\n(a)\n(b)(c)\n(d)\nFIG. 21. (a) ρe[n2(2),n3(2)] is the excited-state density\nof 1Dinfinite potential well with ground-state ρ0. ˜ρ(1)\n0\nand ˜ρ(2)\n0are the ground-state densities of V1andV2,\nwhose excited-state configurations [ n1(2),n2(1),n4(1)] and\n[n1(2),n4(2)] results the same ρe. (b)V3is the potential\nwhose ground-state density is same as ρeof (a) and is\nshown along with V1,V2. (c)ρe[n1(1),n2(1),n3(1),n4(1)]\nis the excited-state density produced in alterna-\ntive configurations [ n1(2),n2(1),n3(1)] (V1[˜ρ(1)\n0]) and\n[n1(2),n2(1),n4(1)] (V2[˜ρ(2)\n0]) besides the ground-state con-\nfiguration leading to V3[ρe]. (d) Shows all the alternative\npotentials of (c).\nVII. SUMMARY AND CONCLUDING\nREMARKS\nIn this work, we have tried to obtain a consistent the-\nory foreDFT based on the stationary state, variational\nand GAC formalism of modern DFT. We have provided\na unified and general approach for dealing with excited-\nstates which follows from previous attempts made by\nPerdew-Levy, G¨ orling, Levy-Nagy-Ayers and in particu-\nlar the work of Samal-Harbola in the recent past. In this\ncurrent attempt, we have answered the questions raised\nabout the validity of HK and GL theorems to excited-\nstates. We have settled the issues by explaining why\nthere exist multiple potentials not only for higher excited\nstates but also for the ground as well as lowest excited\nstate of given symmetry. In fact, the existing eDFT for-\nmalism allows the above possibility and at the same time\nkeeps the uniqueness of density-to-potential mapping in-\ntact. So we have established in a rigorous fundamental\nfooting the non-violation of the HK and GL theorem.18\n(a)\n(b)(c)\n(d)\nFIG. 22. (a) ρ(1)\ne[n= 0,n= 3](both half-filled) , ρ(2)\ne[n=\n1,n= 2] (both half filled) and ρ(3)\ne[n= 2,n= 3](both half\nfilled) are the excited-state densities of the potential Vpro-\nduced as the ground state density of the potentials V1,V2\nandV3. (b) Shows all the four potentials V,V1,V2andV3of\n(a). (c)ρe[n= 0,n= 3] (both half filled) is the excited-state\ndensity of the potential Vproduced in an alternative excited\nstate configuration [ n= 0,n= 2] (V1). (d) Shows both the\npotentials of (c).\nActually, the generalized CS approach gives us a strong\nbasis in choosing a potential out of several multiple po-\ntentials for a fixed ground/excited state density. In our\npropositions, we have strictly defined the bi-density func-\ntionals for a fix pair of ground and excited-state densities\nin order to establish the density-to-potential mapping.Not only that, the theory also gives us a clear definition\nof excited-state KS systems through the comparison of\nkinetic and exchange-correlation energies w.r.t. the true\nsystem. It does takes care the stationarity and orthog-\nonality of the quantum states. So everything fits quite\nnaturally into the realm of modern DFT.\nTo conclude, we have demonstrated density-to-\npotential mapping for non-interacting fermions. For in-\nteracting case the GAC can be used to formulate all the\ntheoretical and numerical contents in a similar way. We\nare working along this direction for strictly correlated\nfermions and the results will be reported in future. 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We present an overview of di\u000berent tomo-\ngraphic methods for determining the quantum-mechanical density matrix of a single qubit: (scaled)\ndirect inversion, maximum likelihood estimation (MLE), minimum Fisher information distance, and\nBayesian mean estimation (BME). We discuss the di\u000berent prior densities in the space of density\nmatrices, on which both MLE and BME depend, as well as ways of including experimental errors\nand of estimating tomography errors. As a measure of the accuracy of these methods we average the\ntrace distance between a given density matrix and the tomographic density matrices it can give rise\nto through experimental measurements. We \fnd that the BME provides the most accurate estimate\nof the density matrix, and suggest using either the pure-state prior, if the system is known to be in\na rather pure state, or the Bures prior if any state is possible. The MLE is found to be slightly less\naccurate. We comment on the extrapolation of these results to larger systems.\nCONTENTS\nI. Introduction 1\nII. Tomographic methods 3\nA. Direct inversion tomography 3\nB. Distance minimization to rd 3\n1. Minimum p-distance of the Bloch vectors 3\n2. Minimum Schatten p-distance of the\ndensity matrices 3\n3. Maximum \fdelity 4\n4. Kullback{Leibler divergence and the\nmaximum likelihood estimate 4\n5. Fisher information distance 6\nC. Maximum-likelihood estimate with radial\nprior 6\n1. Radial prior densities of quantum states 7\n2. Maximum likelihood estimates with\ndi\u000berent priors 8\nD. Bayesian mean estimate 8\nIII. Error considerations 9\nA. Including experimental errors 9\nB. Estimating the tomographic uncertainty 10\nIV. Comparison of tomography methods 10\n1. Extrapolation to larger systems 13\nAcknowledgments 13\nA. Choice of measurement axes 14\n1. Direct inversion (\fltered backprojection) 14\n2. Maximum likelihood estimation 14\nReferences 14\n\u0003roman.schmied@unibas.chI. INTRODUCTION\nQuantum state tomography is the attempt to discover\nthe quantum-mechanical state of a physical system, or\nmore precisely, of a \fnite set of systems prepared by\nthe same process [1]. The experimenter acquires a set\nof measurements of di\u000berent non-commuting observables\nand tries to estimate what the density matrix of the\nsystems must have been before the measurements were\nmade, with the goal of being able to predict the statistics\nof future measurements generated by the same process.\nIn this sense, quantum state tomography characterizes\na state preparation process that is assumed to be stable\nover time [2].\nIn the context of the generation and characterization\nof non-classical states of Bose{Einstein condensates with\ninternal degrees of freedom [3], the system under study\nis known to be in a totally symmetric state because of\nits Bose symmetry. These states are usually described\nin terms of total-spin observables, with the e\u000bective spin\nlength equal to half the atom number. In this restricted\nframework, quantum-state reconstruction is much more\nfeasible than for general many-particle systems; for this\nreason, the reconstruction of spin (or pseudo-spin) den-\nsity matrices is an important real-world case for quantum\nstate tomography. In practice, there are many di\u000berent\nmathematical methods for determining a density matrix\nfrom a given experimental data set, yielding sometimes\nvery di\u000berent results, and it is not obvious which of these\nis objectively better, even when opinions and philosoph-\nical arguments are seemingly clear.\nIn order to see these methods more clearly and com-\npare them, we apply them to the simplest possible\nquantum-mechanical problem of determining the density\nmatrix of a two-level system (a qubit, or a spin of length\n1/2), and compare the obtained results. We \fnd that for\nqubits in general, Bayesian mean estimates (section II D)\nare most accurate at determining a density matrix, in\nagreement with general statements of Refs. [4, 5]. We\ngenerally consider mixed qubit states; for a review ofarXiv:1407.4759v3  [quant-ph]  8 Feb 20162\npure qubit state estimation, see Ref. [6].\nThe quantum-mechanical state of any two-level system\ncan be expressed as a 2 \u00022 density matrix\n^\u001a=1\n2(1+x^\u001bx+y^\u001by+z^\u001bz) =1\n2(1+r\u0001^\u001b) (1)\nin terms of the Pauli matrices\n^\u001bx=\u0012\n0 1\n1 0\u0013\n^\u001by=\u0012\n0\u0000i\ni 0\u0013\n^\u001bz=\u0012\n1 0\n0\u00001\u0013\n1=\u0012\n1 0\n0 1\u0013\n(2)\nand the vectors r= (x;y;z )2R3and^\u001b= (^\u001bx;^\u001by;^\u001bz).\nSince the eigenvalues of ^ \u001aare\u0015\u0006=1\n2(1\u0006p\nx2+y2+z2)\nand must both be nonnegative, a Bloch vectorronly rep-\nresents a physical (positive semi-de\fnite) state if krk2=\nx2+y2+z2\u00141. The three-dimensional unit sphere of\nBloch vectors, where every physically possible qubit den-\nsity matrix can be represented as a point in space, is an\nappealing and convenient representation and will be used\nthroughout this paper.\nAn alternative representation of a qubit density matrix\nis the spherical Wigner function [3, 7]\nW(#;') =1 +p\n3(sin#cos';sin#sin';cos#)\u0001rp\n8\u0019(3)\nde\fned as a pseudo-probability density on the surface of\nthe unit sphere. It encodes the direction of the vector\nrin the angular distribution and the length of rin the\namplitude of the pseudo-probability density. This repre-\nsentation is convenient for longer spins, where the Bloch\nvector representation is unavailable.\nMany characteristics of a qubit state ^ \u001acan be ex-\npressed in terms of the length r=krkof its Bloch vector\nalone, for example the quantum Fisher information [8]\nFQ(^\u001a) =r2, the purity Tr(^\u001a2) = (1 +r2)=2, or the von\nNeumann entropy\nS(^\u001a) =\u00001 +r\n2ln\u00121 +r\n2\u0013\n\u00001\u0000r\n2ln\u00121\u0000r\n2\u0013\n:(4)\nIn what follows we consider only Stern{Gerlach type\nmeasurements on a single qubit: a projective measure-\nment along an axis n(withknk= 1) is represented by\nthe observable ^ \u001bn=n\u0001^\u001b, and has an expectation value\nh^\u001bni= Tr(^\u001bn^\u001a) =n\u0001r: (5)\nThe probabilities for detecting the qubit in the \\up\" state\njn\"isatisfying ^\u001bnjn\"i= +jn\"i, or in the \\down\" state\njn#isatisfying ^\u001bnjn#i=\u0000jn#i, are\np\"(n) =1 +n\u0001r\n2; p#(n) =1\u0000n\u0001r\n2;(6)\nrespectively.\nIf we identically prepare Nnqubits and measure the\nobservable ^ \u001bnon each one, we will \fnd Nn\"qubits inthejn\"istate andNn#qubits in thejn#istate, giving\nan estimate of the expectation value (sample mean)\nhh^\u001bnii=Nn\"\u0000Nn#\nNn\"+Nn#=Nn\"\u0000Nn#\nNn: (7)\nA statistical estimate of the error of this expectation\nvalue is given by the width of a binomial distribution\nwith the same expectation value,\n\u0001hh^\u001bnii=2p\nNn\"Nn#\nN3=2\nn: (8)\nThis error measure will be justi\fed below through\nEq. (17).\nIn the absence of prior knowledge about the experi-\nmental system's state, the precision of the tomographic\nmethods of section II is highest if the measurement axes\nare arranged uniformly on the sphere. As detailed in\nappendix A, for a spin-1/2 system any angular distribu-\ntion is considered uniform if its quadrupolar component\nvanishes. Examples of uniform sampling strategies ac-\ncording to this criterion are equal sampling along the\nthree Cartesian axes, the four axes through the vertices\nof a tetrahedron [9], or a completely uniform distribution\nof measurement axes over the entire sphere. In what fol-\nlows, we assume that the experimenter performs the same\nnumber of single-qubit measurements along each of the\nthree Cartesian axes n=ex,ey,ez, which is the simplest\ncomplete and uniform measurement strategy [10]. Using\ndi\u000berent axes or more than three axes generally makes all\nof the following tomography schemes more complicated,\nand if the measurements are not uniformly distributed\n(for example by making more measurements along ez\nthan alongexorey) the tomographic result will gen-\nerally be less precise or even biased. However, if each\nmeasurement axis is adaptively chosen depending on the\nprevious measurement results, e\u000eciency can be improved\nover the Cartesian axes [11{13]. Also, multi-qubit joint\nmeasurements may yield information faster than sequen-\ntial single-qubit measurements [14, 15]. Such adaptations\nare not considered in the present work.\nIf our qubits are all in the state of Eq. (1) and we\nperformNxmeasurements along the x-axis,Nyalong\nthey-axis, andNzalong thez-axis, the probability of\ngetting a certain set of results is\nP(Nx\";Nx#;Ny\";Ny#;Nz\";Nz#j^\u001a) =\n\u0012Nx\nNx\"\u0013\u00121 +x\n2\u0013Nx\"\u00121\u0000x\n2\u0013Nx#\n\u0002\u0012Ny\nNy\"\u0013\u00121 +y\n2\u0013Ny\"\u00121\u0000y\n2\u0013Ny#\n\u0002\u0012Nz\nNz\"\u0013\u00121 +z\n2\u0013Nz\"\u00121\u0000z\n2\u0013Nz#\n;(9)\nwhereNx=Nx\"+Nx#etc., and we will assume Nx=\nNy=Nzbelow. In such a setup, the problem of quan-\ntum state tomography is to invert Eq. (9): given a set of3\nexperimental results, what can we say about the qubits'\ndensity matrix that has given rise to these results? In\nwhat follows, we \frst present several tomographic meth-\nods and apply them to a single qubit (section II), make\nsome comments about experimental and tomographic er-\nrors (section III), and then compare the accuracies of the\ndi\u000berent methods (section IV).\nII. TOMOGRAPHIC METHODS\nA. Direct inversion tomography\nThe simplest tomographic method, called a direct in-\nversion , assumes that the sample mean hh^\u001bniiis a good\nand unbiased estimate of the population mean h^\u001bni[16].\nCombining Eqs. (5) and (7) along the three Cartesian\naxes fully de\fnes an estimate of the qubits' Bloch vector,\nrd=\u0012Nx\"\u0000Nx#\nNx\"+Nx#;Ny\"\u0000Ny#\nNy\"+Ny#;Nz\"\u0000Nz#\nNz\"+Nz#\u0013\n:(10)\nWe note that rdis the global maximum of Eq. (9), which\nis a de\fnition of rdthat is readily extensible to di\u000berent\nmeasurement schemes.\nIn \fgures 1 and 2 this direct inversion Bloch vector is\nshown as a blue dot for two sets of experimental results,\nboth found by performing 30 Stern{Gerlach measure-\nments along each Cartesian axis. While in \fgure 2 the\nBloch vector is physically valid since krdk\u00141, the Bloch\nvector in \fgure 1 is invalid and points out a fundamen-\ntal problem with the direct inversion method. Eq. (10)\ncan be seen as three individual parameter estimations for\nthe three Cartesian components of the Bloch vector, and\neven though each parameter estimate is unconstrained on\nits own, the three estimates must satisfy the joint con-\nstraintx2\nd+y2\nd+z2\nd\u00141. For any given state rof the\nqubit and for any number of measurements ( Nx;Ny;Nz),\nthere is a \fnite probability that direct inversion tomog-\nraphy will \fnd a physically invalid Bloch vector that vi-\nolates this joint constraint. For example, for the com-\npletely mixed state ^ \u001a=1\n21withr= (0;0;0), measuring\nNx=Ny=Nz= 30 times along each Cartesian direc-\ntion, the probability of \fnding an unphysical rdis only\n3\u000210\u00007; but if we do the same measurements on the\npure state ^\u001a=jz\"ihz\"jwithr= (0;0;1), the chance of\n\fnding an unphysical rdis 98%. For higher-dimensional\nquantum systems, this problem becomes even more se-\nvere (see section IV 1). It has been argued recently [17]\nthat the direct inversion method provides more accurate\nresults because it is less biased than other methods (see\ntable I for an example of such biases); but we side with\nRef. [18] in preferring physically valid density matrices\ndespite their bias, and do not report direct inversion re-\nsults in our comparison of methods. Many interesting\nquantities derived from the density matrix, particularly\nones that go beyond linear operator expectation values\nand involve the entire density matrix, cannot be de\fnedproperly for density matrices that are not positive semi-\nde\fnite.\nNevertheless, rdis an important starting point for\nmany other tomographic techniques. In what follows, we\nbroadly distinguish between tomographic methods that\nminimize some distance between rdand the space of\nphysically valid tomographic Bloch vectors (sections II B\nand II C), and methods not based on rdat all (sec-\ntion II D).\nB. Distance minimization to rd\nIn order to \fnd a valid tomographic density matrix\neven ifkrdk>1, we search for a modi\fed Bloch vector\nrtomo that (i) is physically valid, krtomok\u00141, and that\n(ii) lies closest tordin terms of a distance to be de\fned.\nIn \fgure 1 the three dashed lines emanating from the\nblue dot indicate the locations of the points that mini-\nmize three types of distances to rdon concentric spher-\nical shells around the origin ( r= 0); their intersections\nwith the unit sphere surface (red circle), among others,\nprovide useable tomographic Bloch vectors, and are dis-\ncussed in detail below.\n1. Minimum p-distance of the Bloch vectors\nThe simplest family of distances between two Bloch\nvectors are thep-distanceskr\u0000r0kp= (jx\u0000x0jp+jy\u0000\ny0jp+jz\u0000z0jp)1=pforp\u00151. Even though the direct inver-\nsion Bloch vector rdcan be located anywhere in the unit\ncube, the space of physically valid Bloch vectors has an\nintrinsic spherical symmetry around the fully mixed state\nr= 0, which suggests that only the Euclidean distance\np= 2 is to be used. In this case, the scaled direct inver-\nsion Bloch vector minimizing the Euclidean distance to\nrdover the space of physically valid Bloch vectors is\nrsd=(\nrd ifkrdk\u00141,\nrd=krdkifkrdk>1.(11)\nRadial scaling is shown in \fgures 1 and 2 as a red line,\nwithrsdindicated as a red dot.\n2. Minimum Schatten p-distance of the density matrices\nThe simplest family of distances between two density\nmatrices are the Schatten p-distancesk^\u001a\u0000^\u001a0kp. They\ninclude the trace distance ( p= 1) and the Frobenius\nor Hilbert{Schmidt distance ( p= 2). The Schatten p-\ndistance between two qubit density matrices ^ \u001a=1\n2(1+\nr\u0001^\u001b) and ^\u001a0=1\n2(1+r0\u0001^\u001b) isk^\u001a\u0000^\u001a0kp= 21=p1\n2kr\u0000\nr0k, proportional to the Euclidean distance between their\nBloch vectors. The minimum of any Schatten p-distance\nbetween the direct inversion tomography and the space4\nFIG. 1. Example of a qubit tomography, assuming that 30 ideal measurements ( \u0011= 1, see section III A) along each Cartesian\nquantization axes have resulted in ( Nx\";Nx#;Ny\";Ny#;Nz\";Nz#) = (29;1;25;5;15;15). All quantities are restricted to the\nz= 0 plane. The blue dot shows the zero of the Kullback{Leibler divergence (log-likelihood) at rd= (14\n15;2\n3;0) from Eq. (10),\nwhich is outside of the physically allowed region krk\u00141 indicated by the red circle . The straight red line (sections II B 1\nand II B 2) connects rdwith the totally mixed state r= 0; the red dot shows the result of linear scaling rsd= (0:814;0:581;0)\n[Eq. (11)]. The gray contours of the Kullback{Leibler divergence are at DKL(rdjr) = 10n=4forn= 1:::10 (outward from\nthe blue dot). The gray line (section II B 4) traces the constrained likelihood maximum, Eq. (15), as a function of krk; the\nlikelihood maximum must be on this line if the prior depends only on krk(Haar measure), such as the gray dot showing the\nmaximum of the likelihood at r1<k\u00142\nMLE =rCh\nMLE = (0:848;0:530;0), or the black dots showing the maximum of the likelihood\nwith entropy weight (24) for the Hilbert{Schmidt prior ( k= 2) at (0:800;0:494;0), the Bures prior ( k=3\n2) at (0:827;0:513;0),\nand the Cherno\u000b-information prior at (0 :832;0:517;0). The green contours of the Fisher information distance (section II B 5)\nare atDF(r\u0000rd) = 10n=4forn= 1:::11; the green dot shows the point with minimum Fisher information distance\natrFi= (0:866;0:500;0), located on the green line of points tracing the minimum of the Fisher information distance as a\nfunction ofkrk. The orange ellipses show the Bayesian means and variances of rweighted by the likelihood (section II D): from\nleft to right, they use radial priors with k= 2 (Hilbert{Schmidt measure), k=3\n2(Bures measure), the Cherno\u000b-information\nmeasure (23), and k= 1 (pure states only); the cyan ellipses show the same with entropy weight (24): from left to right, they\nuse radial priors with k= 2 (Hilbert{Schmidt measure), k=3\n2(Bures measure), and the Cherno\u000b-information measure.\nof physically valid density matrices is therefore given by\nEq. (11).\n3. Maximum \fdelity\nThe \fdelity F(^\u001a;^\u001a0) = Tr(pp^\u001a\u0001^\u001a0\u0001p^\u001a) is a fre-\nquently used measure of the overlap between two qubit\ndensity matrices [17]. Since it does not break the spher-\nical symmetry of the space of Bloch vectors, maximizing\nthe \fdelity between two density matrices necessarily re-\nduces to the purely radial scaling of Eq. (11).4. Kullback{Leibler divergence and the maximum likelihood\nestimate\nBayes' theorem states that if we are\ngiven a set of experimental measurements\n(Nx\";Nx#;Ny\";Ny#;Nz\";Nz#), the likelihood that a\ncertain density matrix ^ \u001a=1\n2(1+r\u0001^\u001b) was at the source\nof these data is\nL(^\u001ajNx\";Nx#;Ny\";Ny#;Nz\";Nz#)\n/C(^\u001a)\u0002P(Nx\";Nx#;Ny\";Ny#;Nz\";Nz#j^\u001a);(12)\nwhereC(^\u001a) is a prior density in the space of density ma-\ntrices, vanishing whenever krk>1. Choosing a prior5\nFIG. 2. Same as \fgure 1 but for ( Nx\";Nx#;Ny\";Ny#;Nz\";Nz#) = (26;4;23;7;15;15). The global likelihood maximum rd=\n(0:733;0:533;0) (blue dot) now lies inside the unit sphere and coincides with the red, gray, and green dots of \fgure 1.\ndensity can be a matter of taste or actual prior knowl-\nedge; however, in almost all cases the prior density will\ndepend only onkrkbut not on the direction of r(i.e., it\nis a Haar measure with respect to the spherical symmetry\ngroup).\nIn this section we only use the Hilbert{Schmidt mea-\nsure\nCHS(r) =(\nconst. ifkrk\u00141,\n0 ifkrk>1(13)\nas a prior density, which is uniform when viewed as\nthe density of Bloch vectors within the unit sphere (but\nnon-uniform when viewed in any other parametrization).\nWhile this is a simple and very common (often tacit)\nchoice, it is not the most natural prior density; in sec-\ntion II C we discuss di\u000berent prior densities and their\napplication.\nA popular tomography method is to search for the\nmaximum of the likelihood (12) with CHS(r) [19]. Since\nthe global maximum of the probability P, Eq. (9), is\natrd, we see that whenever krdk\u0014 1 the maximum-\nlikelihood estimate (MLE) of the Bloch vector is simply\nrMLE =rd. Ifkrdk>1, on the other hand, we de\fne\nthe scaled log-likelihood, relative entropy, or Kullback{\nLeibler divergence [19, 20]\nDKL(rdjr) = ln\u0014P(rd)\nP(r)\u0015\n=Nx\"ln\u00121 +xd\n1 +x\u0013\n+Nx#ln\u00121\u0000xd\n1\u0000x\u0013\n+Ny\"ln\u00121 +yd\n1 +y\u0013\n+Ny#ln\u00121\u0000yd\n1\u0000y\u0013\n+Nz\"ln\u00121 +zd\n1 +z\u0013\n+Nz#ln\u00121\u0000zd\n1\u0000z\u0013\n(14)and minimize this distance over the space of physically\nvalid density matrices krk\u00141 [21]. Especially for large\nnumbers of experimental data, the log-likelihood is eas-\nier to calculate in practice than the likelihood, as its dy-\nnamic range is much smaller; since the logarithm is mono-\ntonic, maximizing Pis equivalent to minimizing DKL. In\n\fgures 1 and 2, the gray contours show the Kullback{\nLeibler divergence, and the gray dot in \fgure 1 gives the\nlikelihood maximum within the unit sphere. The gray\nline emanating from the blue dot is found by maximizing\nEq. (12), or minimizing Eq. (14), for constant krk, as-\nsuming that the prior depends only on krk: we \fnd that\nthese extrema are located at\nrMLE(\u000b) =\u0002\nJ\u000b=Nx(xd);J\u000b=Ny(yd);J\u000b=Nz(zd)\u0003\n(15)\nwith the analytic function Ju(t) de\fned piecewise,6\nJu(t) =8\n>>>>>>>>>>>><\n>>>>>>>>>>>>:sign(t) for u!\u00001\n2q\nu+1\n3usign(t) cosh\n1\n3cos\u00001\u0010\n3\n2jtjq\n3u\n(u+1)3\u0011i\nifu<\u00001 (branch cut at t= 0)\nsign(t)jtj1=3ifu=\u00001\n2q\nu+1\n\u00003usinhh\n1\n3sinh\u00001\u0010\n3\n2tq\n\u00003u\n(u+1)3\u0011i\nif\u00001<u< 0\nt ifu= 0\n2q\nu+1\n3usinh\n1\n3sin\u00001\u0010\n3\n2tq\n3u\n(u+1)3\u0011i\nifu>0\n0 for u!+1.(16)\nThe gray line given by Eq. (15) has the following prop-\nerties as a function of the Lagrange multiplier \u000b:\n\u000frMLE(\u000b) maximizes\nP(Nx\";Nx#;Ny\";Ny#;Nz\";Nz#j^\u001a) and mini-\nmizesDKL(rdjr) under the constraint that\nkrk=krMLE(\u000b)k,\n\u000fkrMLE(\u000b)kdecreases monotonically with \u000b2R,\n\u000flim\u000b!\u00001rMLE(\u000b) = (sign[xd];sign[yd];sign[zd]),\n\u000frMLE(0) =rd,\n\u000flim\u000b!1rMLE(\u000b) = 0.\nSinceJu(t) has a branch cut discontinuity at t= 0 for\nu<\u00001, we must be careful when evaluating Eq. (15) if\nany of the ( xd;yd;zd) are zero (see below).\nThus the maximum likelihood method for Cartesian-\naxes qubit tomography is simpler than the ^R\u0001^\u001a\u0001^Rit-\neration used for larger systems [19], and consists of the\nfollowing steps:\n1. Calculaterdfrom Eq. (10).\n2. Ifkrdk\u00141, setrMLE=rd.\n3. Ifkrdk>1, \fnd\u000b>0 such thatkrMLE(\u000b)k= 1.\n5. Fisher information distance\nWhen the direct inversion Bloch vector rdis only\nslightly outside the unit sphere of physically valid states,\nit may be su\u000eciently accurate to minimize the quadratic\napproximation of the Kullback{Leibler divergence (14),\nDKL(rdjr) =1\n2\"\u0012x\u0000xd\n\u0001hh^\u001bxii\u00132\n+\u0012y\u0000yd\n\u0001hh^\u001byii\u00132\n+\u0012z\u0000zd\n\u0001hh^\u001bzii\u00132#\n+O[(r\u0000rd)3];(17)\ngiven in terms of the error estimates of Eq. (8). This ap-\nproximation, called the Fisher information distance [22],\nis easier to use than the Kullback{Leibler divergence\nwhile mostly giving comparable results (see table II). In\fgures 1 and 2 the green lines show the minima of the\nFisher information distance on concentric shells around\nthe originr= 0, and the green dot in \fgure 1 minimizes\nthis distance between rdand the space of physically valid\nstates. In analogy to Eq. (15), the green line is given by\nrFi(\u000b) =\u0012xd\n1 +\u000b[\u0001hh^\u001bxii]2;yd\n1 +\u000b[\u0001hh^\u001byii]2;zd\n1 +\u000b[\u0001hh^\u001bzii]2\u0013\n(18)\nand has similar properties, so that the three-step recipe\nof section II B 4 can still be used. There are situations\nwhere no\u000bexists that satis\fes krFi(\u000b)k= 1, but we\nhave found that they are very unlikely to occur in an\nexperiment (see tables I and II).\nC. Maximum-likelihood estimate with radial prior\nThe Bayesian prior density C(^\u001a) used in Eq. (12) con-\ntains two components that are sometimes di\u000ecult to dis-\ntinguish. On the one hand, it contains a measure on the\nspace of density matrices, which is a way of saying how\n\\\fnely grained\" this space is in its di\u000berent regions, or\nfrom what distribution a purely random density matrix\nshould be drawn in the absence of concrete knowledge\nabout the system [23]. This \frst part is likely invariant\nunder unitary transformations ( i.e., a Haar measure). On\nthe other hand,C(^\u001a) can contain prior knowledge about\nthe particular situation in which we are determining den-\nsity matrices, gained for example from previous experi-\nments. This second part need not be invariant under uni-\ntary transformations. Expressed in a given parametriza-\ntion, which in our case is the Bloch vector rand Eq. (1),\nthe prior density C(r) is the product of the measure ex-\npressed in terms of rand the density gained from prior\nknowledge. It is important to note that concrete prior\nknowledge in the absence of a measure on the space of\ndensity matrices is useless.\nIn the previous section we have used the Hilbert{\nSchmidt measure on the space of Bloch vectors (13) be-\ncause of its simplicity, ubiquity, and geometric appeal.\nHowever, this prejudice is misleading, and the Hilbert{\nSchmidt measure is neither the only nor the most natural7\ndensity of quantum states of a qubit. In this section we\ndiscuss di\u000berent density-matrix measures, and then use\nthese to generalize the maximum-likelihood method to\nnon-trivial priors.\n1. Radial prior densities of quantum states\nThere is much freedom in de\fning a measure on the\nspace of Bloch vectors. In order to focus on more natural\nmeasures, we use a physical argument for de\fning such\na measure: a constructive procedure related to quantum\nstate puri\fcation [4, 23, 24].\nWe start from the observation that the density of pure\nstates of a d-dimensional quantum system is uniquely\nde\fned as a Haar measure over the unitary group U(d);\nthat is, since every pure state is related to every other\npure state by a unitary transformation, and since all uni-\ntary transformations can be parametrized as points on\nthe surface of a ( d2\u00001)-dimensional hypersphere, we can\nuse the geometric measure on this hypersphere's surface\nas the natural measure in the space of pure states.\nNext, we consider the joint tensor-product quantum\nstate of our two-dimensional qubit ( D= 2) and a k-\ndimensional ancillary system, for a total dimension d=\nD+k. For every pure state of this (2 + k)-dimensional\nsystem, we can trace out the ancillary dimensions to \fnd\na reduced qubit density matrix (1). The reverse is also\ntrue, called quantum state puri\fcation: for every qubit\ndensity matrix (1) we can \fnd a pure state of a system of\nd\u00152D= 4 dimensions, of which our state is the partial\ntrace. This partial trace operation therefore constructs\na unique measure of qubit density matrices, depending\nonly on the ancilla dimension k. Expressed as a density\nin the space of qubit Bloch vectors (the unit sphere), the\nresulting density (measure) is\nCk(r) =(\u0000(k+1\n2)\n\u00193=2\u0000(k\u00001)(1\u0000krk2)k\u00002ifkrk<1\n0 if krk>1(19)\nfork >1, where \u0000( z) is the Euler gamma function. As\nexpected, this measure only depends on the length of the\nBloch vector but not on its direction. The mean squared\nBloch vector of this measure is hkrk2i= 3=(2k+ 1): for\nlarger values of k, mixed states carry more weight than\npure states.\nHow can we choose a value for the ancillary dimension\nk? While the derivation of Eq. (19) assumes that kis\nan integer, we can use the resulting prior density for any\nvalue ofk. Not all values of kare equally natural; we\ndeem the following choices meaningful:\nk= 1pure states: In the limit k!1+the mea-\nsure (19) becomes fully concentrated on the surface\nof the unit sphere ( krk= 1), meaning that only\npure states have a nonzero likelihood in Eq. (12).\nWhile this is not a natural choice, as (strictly speak-\ning) pure states do not exist in nature, it can be ofinterest for theoretical considerations or in cases\nwhere the state purity is known to be very high.\nk=3\n2Bures measure: In general, the Bures mea-\nsure [23, 25, 26] is considered the most natural\ndensity of mixed states [23], as it is the Je\u000breys\nprior [27]. For qubits, its distribution is formally\nthat of tracing over k=3\n2ancillary dimensions,\nand has the radial density [28][29]\nCB(r) =C3\n2(r) =(1\n\u00192p\n1\u0000krk2ifkrk<1,\n0 ifkrk>1,(20)\nshown as a solid blue line in \fgure 3. If nothing at\nall is known about the expected tomographic den-\nsity matrix, then this Je\u000breys prior density should\nbe used.\nNote that for systems with Hilbert space dimension\nD > 2, the Bures measure cannot be constructed\nby choosing a particular value of k.\nIn the sphere of Bloch vectors r, the Bures measure\nassigns a higher density of states to purer states\n(larger=krk) than to more mixed states (small\nr). We can introduce a transformed radial coordi-\nnates=\u00022\n\u0019\u0000\nsin\u00001(r)\u0000rp\n1\u0000r2\u0001\u00031=3, in terms of\nwhich the Bures measure is homogeneous:\nCB(s) =(\n3\n4\u0019ifksk<1,\n0 ifksk>1.(21)\nThis shows that the \ratness of the measure depends\non the chosen parametrization, and cannot be used\nas a criterion to prefer one measure over another.\nk= 2Hilbert{Schmidt measure: The previously\nused Hilbert{Schmidt measure of Eq. (13) is found\nby setting the ancilla dimension equal to the\nsystem dimension, k=D= 2. It is equal to the\nEuclidean measure in the unit sphere of Bloch\nvectors, meaning that it gives every Bloch vector\nequal a priori weight in the simplest geometric\nsense (solid red line in \fgure 3). This prior is\nused very frequently in practice, mainly due to\nits mathematical simplicity; but it must be noted\nthat it does not represent the natural density of\nqubit states [23].\nk\u001d2highly mixed states: For large ancilla dimen-\nsions the density matrix measure becomes Gaussian\nand peaked around the fully mixed state,\nCk(r)\u0019\u0012k+1\n2\n\u0019\u00133\n2\ne\u0000(k+1\n2)krk2fork\u001d1 (22)\nThis measure can be used for tomographies where\nthe state is known to be highly mixed.8\n00.2 0.4 0.6 0.8 1 ||\nr||00.10.20.30.40.5C(r)\nHilbert-Schmidt x  entropyBures \nx  entropyChernoff \nx  entropyHilbert-Schmidt measure\nBures measure\nChernoff measure\nFIG. 3. A few spherically symmetric prior densities of\nBloch vectors. Solid red line: the Hilbert{Schmidt measure\nCHS(r) =C2(r), Eq. (13). Solid blue line: the Bures measure\nCB(r) =C3\n2(r), Eq. (20). Solid green line: the Cherno\u000b-\ninformation measure CCh(r), Eq. (23). The dashed lines are\nthe same measures weighted by the entropy as in Eq. (24),\nand normalized.\nThere are other ways of de\fning a measure on the space\nof qubit density matrices. As an example, the Cherno\u000b-\ninformation measure [28]\nCCh(r) =(\n(1\u0000krk2)\u00001\n2\u00001\n2\u0019(\u0019\u00002)krk2ifkrk<1\n0 if krk>1(23)\nfollows from the experimental distinguishability of den-\nsity matrices, and is shown as a green line in \fgure 3.\nOnce a measure has been chosen for the space of den-\nsity matrices, the prior density in Eq. (12) can be taken\ndirectly from Eq. (19), (23), or other, or it can be fur-\nther multiplied by a weight of our choice, for example\nrepresenting concrete prior knowledge. As an example,\nwe may use the entropy (4) as a radial weight, in combi-\nnation with an underlying state measure:\nC(r)/Ck(r)\u0002S(r); (24)\nbiasing the likelihood (12) towards less pure states. Fig-\nure 3 shows a few examples of prior densities, including\nentropy weights.\n2. Maximum likelihood estimates with di\u000berent priors\nThe maximum-likelihood estimate of section II B 4 is\neasily adapted to any spherically symmetric prior den-\nsityC(^\u001a) =C(krk). Since Eq. (15) maximizes the like-\nlihood on each concentric shell krk=krMLE(\u000b)k, max-\nimizing the likelihood globally thus means \fnding the\nvalue of\u000b2Rthat maximizes the likelihood L[rMLE(\u000b)],\nEq. (12). For this maximization we can distinguish dif-\nferent classes of priors:\npure or pure-peaked: If the prior density is singular\natkrk= 1, for example Eq. (19) with 1 \u0014k <2or Eq. (23), it is su\u000ecient to look for the value of\n\u000bfor whichkrMLE(\u000b)k= 1. Two special cases are\nimportant: if two or three components of rdare\nzero, then the likelihood maximum is not unique\nand the MLE should not return a value; the same\nis true if only one component of rdis zero and the\ndetermined value of ( \u0000\u000b) is larger than the number\nof measurements along this axis [30].\nmonotonic pure-biased: We distinguish three cases:\nkrdk>1:\fnd the value of \u000b > 0 for which\nkrMLE(\u000b)k= 1.\nkrdk= 1:the likelihood maximum is at rd.\nkrdk<1:\fnd the value of \u000b < 0, with\nkrMLE(\u000b)k\u00141, that maximizes the likelihood\nL[rMLE(\u000b)].\nuniform (Hilbert{Schmidt): see section II B 4 for an\nextended discussion.\nmonotonic mixed-biased: \fnd the value \u000b > 0 for\nwhichrMLE(\u000b) maximizes Eq. (12).\nnon-monotonic: \fnd the value \u000b, withkrMLE(\u000b)k\u00141,\nfor whichrMLE(\u000b) globally maximizes Eq. (12).\nIn \fgures 1 and 2 the likelihood maxima are shown\nfor several di\u000berent prior densities. We can see that for\nmany priors and experimental results, the MLE is rank-\nde\fcient (krMLEk= 1), which is a serious drawback of\nthis method [4].\nD. Bayesian mean estimate\nInstead of reporting only the maximum of the likeli-\nhood (12), we can interpret the likelihood as a density in\nthe state space and use it to calculate a weighted mean\nstate. This Bayesian mean estimate [4] is\n^\u001aBME =R\n^\u001aL(^\u001a)D^\u001aR\nL(^\u001a)D^\u001a; (25)\nwhereD^\u001arepresents the chosen measure on the space\nof density matrices, and L(^\u001a) contains the experimental\nknowledge including prior knowledge (see the discussion\nof section II C on the two components of the prior den-\nsity). In practice this integral is done by averaging the\ncomponents of the Bloch vector,\nrBME =R\nkrk\u00141rL(r)d3r\nR\nkrk\u00141L(r)d3r; (26)\nwhere the measure on the space of density matrices is\nnow included in the de\fnition of the likelihood (12) and\nexpressed in terms of the geometric Bloch vector measure\nd3r, as in Eq. (19). The Bayesian mean is generally more\nplausible than the likelihood maximum [4] because it is\nnever rank-de\fcient.9\nThis method can be naturally extended to higher mo-\nments of the density matrix, from which we can calculate\na covariance matrix: for example, with\nx2BME =R\nkrk\u00141x2L(r)d3r\nR\nkrk\u00141L(r)d3r(27)\nwe can de\fne the variance (\u0001 xBME)2=x2BME\u0000x2\nBME,\nand similarly the entire covariance matrix for the compo-\nnents ofrBME. In \fgures 1 and 2 we show these covari-\nances as orange and cyan ellipses around the Bayesian\nmean estimates for di\u000berent choices of the prior density\nC(r).\nAt this point we need to distinguish between two kinds\nof uncertainty: \frstly, there is the quantum-mechanical\nuncertainty within a single density matrix (Bloch vec-\ntor), which is typically a statistical mixture of pure\nstates, and leads to the well-known stochastic outcomes\nof observables through Born's rule; and secondly, there\nis the uncertainty in the density matrix (Bloch vector)\nparametrization coming from tomographic uncertainties,\ngiven above by the covariance matrix of the components\nof the Bloch vector.\nWhen we calculate the linear expectation value and\nvariance of an operator ^A, these two types of uncertainty\ncannot be distinguished, and the expectation value and\nvariance are estimated with\nh^Ai=R\nkrk\u00141Tr[^\u001a(r)^A]L(r)d3r\nR\nkrk\u00141L(r)d3r= Tr[ ^\u001aBME^A];\n(28a)\n(\u0001A)2=h^A2i\u0000h ^Ai2\n= Tr[ ^\u001aBME^A2]\u0000Tr[^\u001aBME^A]2; (28b)\nwhere the average density matrix is given in Eqs. (25)\nand (26). In this sense, the mean density matrix ^\u001aBME =\n^\u001a(rBME) represents the statistical mixture containing\nboth the quantum uncertainty in each Bloch vector and\nthe uncertainty of the parametrization of the Bloch vec-\ntor itself.\nThe situation is di\u000berent when we estimate the\nBayesian mean value of a non-linear quantity such as\nthe mean purityhTr(^\u001a2)ior the mean entropy hS(^\u001a)i: in\nthese cases, the covariance of the Bloch vector, Eq. (27),\nbecomes important. For example,\nhS(^\u001a)i=R\nkrk\u00141S[^\u001a(r)]L(r)d3r\nR\nkrk\u00141L(r)d3r6=S(^\u001aBME);(29a)\n[\u0001S(^\u001a)]2=R\nkrk\u00141S2[^\u001a(r)]L(r)d3r\nR\nkrk\u00141L(r)d3r\u0000hS(^\u001a)i2(29b)\nmust be calculated by taking the uncertainty in the Bloch\nvector parametrization, given by L(r), into account.III. ERROR CONSIDERATIONS\nA. Including experimental errors\nIn a real experiment, the outcomes of Stern{Gerlach\nmeasurements are never perfect. For simplicity, we as-\nsume that independently of the measurement direction,\nevery measurement has a probability \u0011of giving the cor-\nrect result and a probability 1 \u0000\u0011of giving a random\nresult ( i.e., passing through a depolarizing channel [31]),\nor equivalently, a probability of (1 + \u0011)=2 of giving the\ncorrect result and (1 \u0000\u0011)=2 of giving the wrong result.\nMany sources of experimental errors can be expressed\nin this form, apart from simple detection errors. For ex-\nample, if the experimental Stern{Gerlach axes \ructuate\naround their respective mean directions with a variance\n4hsin2(\u001f=2)i(with\u001fthe angle between the desired axis\nand the true experimental axis), the experimental error\ncan be described by \u0011=hcos(\u001f)i= 1\u00002hsin2(\u001f=2)i.\nIf several independent sources of errors \u00111;\u00112;::: are\npresent, the total error is described by their product\n\u0011=\u00111\u00112\u0001\u0001\u0001.\nIn the presence of such experimental errors, the prob-\nability of measuring a certain data set is modi\fed from\nEq. (9) to\nP\u0011(Nx\";Nx#;Ny\";Ny#;Nz\";Nz#j^\u001a) =\n\u0012Nx\nNx\"\u0013\u00121 +\u0011x\n2\u0013Nx\"\u00121\u0000\u0011x\n2\u0013Nx#\n\u0002\u0012Ny\nNy\"\u0013\u00121 +\u0011y\n2\u0013Ny\"\u00121\u0000\u0011y\n2\u0013Ny#\n\u0002\u0012Nz\nNz\"\u0013\u00121 +\u0011z\n2\u0013Nz\"\u00121\u0000\u0011z\n2\u0013Nz#\n(30)\nwith\u00112[0;1]. For\u0011= 1 the measurements are perfect\nand we recover Eq. (9); for \u0011= 0 the measurements\ncontain no information about ^ \u001a.\nThe form of Eq. (30) is strictly that of Eq. (9) where\nthe Bloch vector ris replaced by \u0011r. All the tomographic\nmethods of section II can therefore be used to determine\nthe vector\u0011r, with the caveat that the prior density de-\npends onrand not on \u0011r. The direct inversion Bloch\nvector (10), which does not depend on the prior density,\nis nowrd(\u0011) =rd=\u0011: it contains more structure than rd,\nsincekrd(\u0011)k\u0015krdk, in order to compensate for the loss\nof information during the measurement. This observa-\ntion remains true for the more complicated tomographic\nmethods discussed above, and invites the following dis-\ntinction:\n\u000fFor\u0011 <1 we can interpret any tomographic r(\u0011)\nas aplatonic state representing the ideal of the sys-\ntem, which is poorly measured in our experiment\nusing Eq. (30). If we could perform a more ac-\ncurate measurement, we would \fnd a state more\nclosely resembling this r(\u0011).10\n-1.0-0.50.00.51.0-1.0-0.50.00.51.0\nxy\nFIG. 4. Bootstrapping (section III B) the state r=\n(13=15;0;0) (yellow dot) via scaled direct inversion tomogra-\nphy (section II B 1). To generate this graphic, all 29 791 possi-\nble experimental outcomes ( Nx\";Nx#;Ny\";Ny#;Nz\";Nz#) of\nperforming Nx=Ny=Nz= 30 Stern{Gerlach measure-\nments on the state ^ \u001a(r) were subjected to a tomographic state\nreconstruction, and the resulting Bloch vectors (projected into\nthexyplane) were plotted in a 2D histogram using weights\nfrom Eq. (9), in 31 \u000231 bins. The mean reconstructed state\nvector is at (0 :862\u00060:086;0\u00060:180;0\u00060:180), as listed in\ntable I. The rms trace distance to r, Eq. (31), is 0.135.\n\u000fWe can de\fne a positivist state ~r(\u0011) =\u0011r(\u0011) that\nalready includes the e\u000bects of imprecise measure-\nments; experimental outcomes can be predicted in\nterms of perfect measurements of this positivist\nstate, using Eq. (9).\nThe author believes that platonic ideals such as r(\u0011)\nshould be discouraged in quantum mechanics, as they\ndo not represent what can currently be measured, but\ninstead hypothesize knowledge that may forever remain\nout of experimental reach. Instead, we suggest using the\nstate ~r(\u0011) as a fair representation of the experimenter's\ncurrent and actual knowledge about the system.\nB. Estimating the tomographic uncertainty\nFor every tomographic reconstruction, it is important\nto be able to give an estimate of the uncertainty of the\nresulting density matrix [32]. While the Bayesian mean\nof section II D gives such an estimate via Eq. (27), shown\nas orange and cyan ellipses in \fgures 1 and 2, the other\nmethods presented here do not give natural error esti-\nmates.\nA widely used method for nonetheless \fnding such an\nerror bar, called bootstrapping [33] or case resampling ,\ngoes as follows: once we have tomographically deter-\nmined a density matrix from experimental data, we canuse this density matrix to generate new \\fake\" data sets\nusing the same measurement operators and the proba-\nbilities of Eq. (9) or (30); the argumentation is that in\nprinciple, each one of these fake data sets could have been\nmeasured, instead of the set we have measured in reality.\nFor each such fake data set we can then do a tomography,\nand \fnally average any observables (or just the density\nmatrix) over these tomographies. As is shown in \fg-\nure 4 and table I, this procedure can be used to calculate\nthe covariance matrix of the components of the tomo-\ngraphic Bloch vector. While these covariances correctly\nestimate the uncertainty we are looking for [33], the boot-\nstrap method contains systematic biases for the di\u000berent\ntomographic methods, as shown in table I. Ideally, the\nweighted mean of all fake-data tomographies would be\nequal to the input state, such that we can use this tech-\nnique to extract a covariance matrix without introducing\na bias; however, this is not the case [17]. Nevertheless,\nthe covariance matrix found in this way can still be used\nin order to get an idea of the tomographic uncertainty.\nWe \fnd that the bootstrapped covariances of the Bloch\nvector components are slightly smaller than the covari-\nances estimated with the Bayesian mean (section II D) for\na single experimental data set. The similar magnitudes of\nthese two sets of error estimates lead us to the conclusion\nthat the bootstrap method can be a valid tool for estimat-\ning tomographic uncertainties. We believe that the cau-\ntionary footnote of Ref. [4] concerning the absurd results\nof bootstrapping for, e.g., ( Nx;Ny;Nz) = (0;0;1) do not\napply when several non-commuting observables are mea-\nsured, as we do in this text with Nx=Ny=Nz\u00151.\nConcerning the choice of input state rfor the gen-\neration of fake data sets, the non-parametric bootstrap\nmethod (direct re-sampling of measured data) requires\nus to use the direct-inversion Bloch vector rd, Eq. (10),\neven ifkrdk>1 is unphysical. While this rd-based non-\nparametric bootstrap is closest to the experimental data\nand may therefore be expected to be least biased, it is\nunrealistic in the case krdk>1 because it neglects the\nphysical condition that any density matrix used for pre-\ndicting experimental outcomes, including generating fake\ndata sets, must be positive semi-de\fnite. If we use a dif-\nferent Bloch vector, for example the maximum-likelihood\nestimaterMLE, the method is called a parametric boot-\nstrap and is physically better justi\fed, albeit biased.\nIV. COMPARISON OF TOMOGRAPHY\nMETHODS\nIn this section we quantify the performance of the\ndi\u000berent tomographic methods discussed above. We\nuse the following procedure, similar to suggestions in\nRefs. [4, 5, 19], to calculate an accuracy measure for each\nmethod:\n1. For a given input state rand a desired num-\nber of measurements along the Cartesian axes,11\nTABLE I. The bootstrapping covariance ellipses of \fgure 4 for the di\u000berent tomography methods, using the exemplary input\nstate r= (13=15;0;0) andNx=Ny=Nz= 30 measurements. For each method, the corresponding color in \fgure 1 is indicated.\nThe ellipses are centered at hrtomoi=fhxtomoi;0;0gand have the given radii f\u0001x;\u0001y;\u0001zgin the Cartesian directions. \u0001 tomo\ngives the mean accuracy in terms of the rms trace distance to r, Eq. (31). The most accurate methods (and up to 5% higher)\nare highlighted in green, and poorly performing methods are highlighted in red. The last column gives the failure rate for\nmethods that do not always give a well-de\fned result.\nmethod and prior density hxtomoi\u0001x\u0001y;z\u0001tomoPfail\nscaled direct inversion (Sec. II B 1) \u000f0.862 0.086 0.180 0.135\nFisher information distance (Sec. II B 5) \u000f0.866 0.091 0.168 0.127 5 \u000210\u000010\nMLE: 1\u0014k<2 and Cherno\u000b measure \u000f0.924 0.045 0.269 0.193 3%\n(Sec. II C) k= 2 (Hilbert{Schmidt) \u000f0.864 0.088 0.174 0.131\nCherno\u000b with entropy weight \u000f0.853 0.084 0.165 0.124\nk=3\n2with entropy weight \u000f0.844 0.085 0.160 0.122\nk= 2 with entropy weight \u000f0.816 0.083 0.149 0.116\nBME: k= 1 (pure states) \r0.907 0.044 0.224 0.161\n(Sec. II D) Cherno\u000b measure \r0.842 0.101 0.167 0.129\nk=3\n2(Bures measure) \r0.830 0.077 0.162 0.122\nk= 2 (Hilbert{Schmidt measure) \r0.797 0.077 0.148 0.117\nCherno\u000b with entropy weight \r0.790 0.084 0.146 0.118\nk=3\n2with entropy weight \r0.781 0.076 0.142 0.116\nk= 2 with entropy weight \r0.756 0.075 0.136 0.117\nTABLE II. Comparison of the accuracies \u0001 tomo(r) of various tomography methods, Eq. (31). For each method, the correspond-\ning color in \fgure 1 is indicated. For a given input state (Bloch vector r),Nx=Ny=Nz= 30 measurements are simulated\nalong each Cartesian quantization axis, and all 29 791 possible experimental outcomes N= (Nx\";Nx#;Ny\";Ny#;Nz\";Nz#) are\nfed into each tomography method (see \fgure 4 for an example), in the same way as bootstrapping (section III B and table I);\n\fnally, the root-mean-square (rms) of the trace distances1\n2kr\u0000rtomokof the results (see section II B 1) are computed with\nweights from Eq. (9). Smaller values indicate better accuracy of the tomography method; the most accurate methods for each\ninput state (and up to 5% higher) are highlighted in green, and poor methods are highlighed in red. We show exemplary results\nfor six input states r, as in table I, and the last column gives the rms of these accuracies averaged over all possible input states\nrwith the given method's prior density using Eq. (32). (\u0003) The probability for these methods to give an ill-de\fned result was at\nmost 0:2%, except where noted; the given mean values only include well-de\fned results. The strongly mixed states with large\nfailure rates only contribute minimally to \u0016\u0001tomo due to the given prior density weighting.x) For the six pure states along the\nCartesian axes, this method always gives exactly the correct result. (y) Averages were done with the Hilbert{Schmidt measure.\nmethod and prior density r= (0;0;0) (0;0;1\n2) (0;0;0:9) (0;0;1) (1;1;0)=p\n2 (1;1;1)=p\n3\u0016\u0001tomo\nscaled direct inversion (Sec. II B 1) \u000f 0.158 0.151 0.132 0.123 0.116 0.114 0.137y\nFisher information distance (Sec. II B 5)\u0003\u000f 0.158 0.151 0.119 0x0.126 0.123 0.139y\nMLE: k= 1 (pure states)\u0003\u000f37% failure 19% failure 3% failure 2% failure 0.113 0.107 0.111\n(Sec. II C) Cherno\u000b measure\u0003(23)\u000f37% failure 19% failure 3% failure 2% failure 0.113 0.107 0.167\nk=3\n2(Bures measure)\u0003(20)\u000f37% failure 19% failure 3% failure 2% failure 0.113 0.107 0.179\nk= 2 (Hilbert{Schmidt) (13) \u000f 0.158 0.151 0.125 0.087 0.117 0.118 0.137\nCherno\u000b with entropy weight \u000f 0.158 0.150 0.118 0.087 0.118 0.121 0.135\nk=3\n2with entropy weight \u000f 0.156 0.148 0.116 0.087 0.120 0.124 0.135\nk= 2 with entropy weight \u000f 0.150 0.142 0.112 0.090 0.127 0.133 0.135\nBME: k= 1 (pure states) \r 0.443 0.306 0.145 0.086 0.111 0.109 0.110\n(Sec. II D) Cherno\u000b measure \r 0.316 0.634 0.118 0.089 0.124 0.133 0.274\nk=3\n2(Bures measure) \r 0.154 0.149 0.116 0.090 0.121 0.125 0.126\nk= 2 (Hilbert{Schmidt) \r 0.148 0.141 0.112 0.095 0.131 0.136 0.131\nCherno\u000b with entropy weight \r 1.65 1.53 0.112 0.097 0.139 0.161 1.08\nk=3\n2with entropy weight \r 0.146 0.139 0.112 0.099 0.136 0.142 0.132\nk= 2 with entropy weight \r 0.141 0.134 0.115 0.106 0.144 0.151 0.133\nin our case Nx=Ny=Nz= 30, we enu-\nmerate all 313= 29 791 possible experimental\noutcomes N= (Nx\";Nx#;Ny\";Ny#;Nz\";Nz#) of\nStern{Gerlach measurements, together with their\nprobabilitiesP(Njr) from Eq. (9).2. For each possible experimental outcome Nwe re-\nconstruct the tomographic Bloch vector rtomo(N).\nThis step is done di\u000berently for the various meth-\nods discussed in section II. As an example, \fgure 4\nshows a 2D histogram of the Bloch vectors recon-\nstructed with scaled direct inversion, binned with12\ntheir weight given in Eq. (9). Table I compares the\nperformance of the di\u000berent tomographic methods\nwhen applied to the example of \fgure 4.\n3. For each possible experimental outcome Nwe\nquantify the tomographic error through the trace\ndistance1\n2kr\u0000rtomok(see section II B 2). We\nchoose the trace distance here because it quanti-\n\fes the experimental distinguishability of the two\ninvolved density matrices; but since all Schatten\np-distances are equivalent for qubits, this is the\nsame as the Hilbert{Schmidt distance quanti\fer of\nRef. [19].\n4. We calculate the weighted root-mean-square (rms)\ntrace distance of all possible experimental outcomes\nNwith\n\u0001tomo(r) =1\n2\"X\nNP(Njr)\u0002krtomo(N)\u0000rk2#1=2\n:\n(31)\nIn table II these accuracies are shown for several\ntomographic methods and for several input states\nr.\n5. Theser-dependent accuracies are further aver-\naged using the prior density appropriate for each\nmethod,\n\u0016\u0001tomo =\"Z\nkrk\u00141C(r)\u00012\ntomo(r)d3r#1=2\n: (32)\nThis is a single number characterizing a given com-\nbination of a tomographic method and a prior den-\nsity, without any further parameters. In the last\ncolumn of table II we show these averaged accu-\nracies for the methods considered here, where for\nmethods with no inherent prior we use the Hilbert{\nSchmidt priorCHS(r), Eq. (13).\nThese tomographic accuracy quanti\fers contain vari-\nance contributions from both the tomographic method in\nquestion and the randomness of the measurement process\n(quantum projection noise). However, as the latter is in-\ndependent of the tomographic method, we can nonethe-\nless use Eqs. (31) and (32) to compare the accuracies\nof di\u000berent tomography methods with each other. Even\nthough the di\u000berences in the accuracies are sometimes\nsmall, we therefore compare them carefully and hope to\nextrapolate the \fndings to higher-dimensional quantum\nstate tomographies.\nWe make the following observations for the di\u000berent\ntomographic methods:\nScaled direct inversion: While this method never\ngives the most accurate results, it is very simple,\nnever fails, and provides a baseline against which\nwe can compare the more complex methods. The\noverall performance of this method is comparableto that of the maximum likelihood estimate with\nHilbert{Schmidt prior, since their results are very\noften the same.\nFisher information distance: This method gives\ngood results for mixed states, but for pure states\nits results are worse than those of the scaled direct\ninversion, except on the Cartesian axes where the\nresults are perfect. This inconsistent behavior\nleads us to discourage the use of this method.\nMaximum likelihood estimate: The maximum likeli-\nhood method fails for priors that are singular for\npure states ( k= 1, the Bures prior, and the Cher-\nno\u000b information prior) by not giving unique results.\nThe frequently used Hilbert{Schmidt prior, on the\nother hand, performs well, comparable to the scaled\ndirect inversion. When mixed states are known to\npredominate, adding an entropy weight gives even\nslightly more accurate results.\nIt is often argued that the maximum likelihood\nmethod with Hilbert{Schmidt prior (section II B 4)\nis the \\best\" method since we cannot gain by giv-\ning an answer that is less likely, such as we do when\ngiving a Bayesian mean estimate [19]. This argu-\nment is misleading, however. Firstly, the question\nof which prior density C(^\u001a) to use in the de\fni-\ntion of the likelihood (12) is not answered to our\nsatisfaction by tacitly using the Hilbert{Schmidt\nprior (13), especially in situations where the exper-\nimenter knows a priori that the generated states\nare nearly pure. The argument that the \ratness\nof the Hilbert{Schmidt prior makes it most natu-\nral is contingent on the chosen parametrization, see\nEq. (21) and Ref. [34]. While the more natural Bu-\nres prior (20) fails to give satisfactory results, see\ntable II, other priors are possible, and this degree\nof freedom casts at least some uncertainty on the\noptimality of the MLE method with HS prior. Sec-\nondly, as discussed below, other methods give on\naverage more accurate tomographic results accord-\ning to our quanti\fers (31) and (32).\nBayesian mean estimate: The BME is not only more\nplausible than the MLE because it is of full rank [4],\nbut according to our overall quanti\fer (32) the\nBME is the most accurate method studied here\n(except when using the Cherno\u000b information mea-\nsure, see below). As we can see in \fgures 1 and 2,\nthe BME is strongly in\ruenced by the choice of\nthe prior density C(^\u001a): in general, the MLE is not\neven contained within the corresponding BME un-\ncertainty ellipsoid. In table II we see that in ex-\nperiments where pure states are expected, using a\npure-state ( k= 1) prior gives the best results of\nour study; if the purity of the state is not known\na priori , using a Bures or Hilbert{Schmidt prior is\nthe optimal choice.13\nIt may be surprising that we \fnd the Bayesian mean\nestimate to be more accurate on average than the like-\nlihood maximum, even though the former su\u000bers from\nrather strong prior-dependent biases and the latter has\nbeen shown to be the most e\u000ecient estimation strat-\negy [19]. This discrepancy comes from the observation,\nseen in table II, that while the BME is generally more\naccurate for mixed states, the MLE method is more accu-\nrate for pure states; however, in our averaging procedure,\nEq. (32), mixed states carry much more weight than the\npure states near the surface of the sphere of Bloch vec-\ntors. In real experiments, the experimenter often tries to\ngenerate rather pure quantum states, and for these the\nMLE method indeed does give more accurate results if\nthe Hilbert{Schmidt prior is assumed. However, we ar-\ngue that in this case the Hilbert{Schmidt prior is not the\ncorrect one to use, but either the pure-state prior C1(r)\nor the Bures prior CB(r), which correctly prioritize pure\nstates; and in these cases, the BME does out-perform\nthe MLE method [4], quantitatively for C1(r) and qual-\nitatively forCB(r) (since in this case the MLE method\nalways returns a pure state, which is not justi\fed a pri-\nori).\nWe make the following observations for the di\u000berent\nprior densities:\nk= 1pure states: If we can be sure a priori that the\nexperimental data has been generated by measure-\nments on a pure state, then the BME method with\npriorC1(^\u001a) is slightly more accurate on average\nthan the MLE; also, the MLE method has a small\nprobability of not giving a unique result at all.\nTherefore, the BME is preferred in this case, keep-\ning in mind that the BME is never a pure state.\nk=3\n2Bures measure: On average, the BME method\ngives much more accurate results than the MLE\nmethod, and it never fails. For pure states, how-\never, the MLE is more accurate than the BME; but\nin this case the pure-state prior is more appropri-\nate.\nk= 2Hilbert{Schmidt measure: On average, the\nBME method gives slightly more accurate results\nthan the MLE method. Again, for pure states the\nMLE is more accurate than the BME; but in this\ncase the Hilbert{Schmidt measure is an inappropri-\nate choice.\nCherno\u000b information measure: While the Cherno\u000b\ninformation measure gives good results for the\nMLE, comparable to those of the Bures measure, it\ngives very poor results for the BME of mixed states.\nThe reason for this is that the Cherno\u000b information\nmeasure is strongly peaked at pure states; but even\nfor the BME of pure states its results are less accu-\nrate than those obtained with the Bures measure.\nFor this reason we discourage the use of the Cher-\nno\u000b information measure.Entropy weights: In general, both the MLE and the\nBME give very good results with entropy-weighted\nprior densities. An exception is the entropy-\nweighted Cherno\u000b information measure, which\ngives very poor results for mixed states.\nWe conclude that the Bayesian mean estimate is the\npreferred method for single-qubit quantum state tomog-\nraphy. The instances where the maximum-likelihood\nmethod performs better, namely when pure states are\nreconstructed with the Hilbert{Schmidt or Bures prior,\nare not well justi\fed since these priors are ill adapted to\nthe experimental situation concerning pure states.\n1. Extrapolation to larger systems\nFor systems with larger Hilbert spaces (dimension\nD\u001d2), the direct inversion result is very likely to be\nnon-physical. The reason for this is that experimental\nquantum states are mostly of very low rank, and the to-\nmographic estimates of the zero eigenvalues of the density\nmatrix are statistically scattered around zero [35], with\nthe probability of all of them being positive becoming\nexponentially small with the system dimension.\nFor this reason, the MLE becomes independent of the\nchoice of prior, since any measure Ckfork\u0014D, as well\nas the Bures measure, will yield the same rank-de\fcient\nresult (the equivalent of a Bloch vector on the surface of\nthe unit sphere for the single-qubit case, the gray dot in\n\fgure 1). In this sense, the usual choice of the Hilbert{\nSchmidt measure k=D[19] is valid.\nForD\u001d2, the BME becomes computationally dif-\n\fcult to evaluate and must be calculated with a Monte\nCarlo algorithm. While in principle the BME is still the\npreferred method [4], such practical di\u000eculties may dis-\ncourage its use in large systems.\nIn our experimental practice with two-component\nBose{Einstein condensates [3, 36, 37] we apply these\ninsights and use the MLE with Hilbert{Schmidt prior\nfor quantum state reconstruction, as is done in other\ngroups [38]. We have found that this is the only computa-\ntionally feasible and physically valid method for systems\ncomprising hundreds or even thousands of particles, and\nconsider it fortuitous that the present comparative study\ndeems it appropriate. The problem of its general rank-\nde\fciency is considered acceptable.\nACKNOWLEDGMENTS\nThe author would like to thank Philipp Treutlein, An-\ndreas Nunnenkamp, and Christoph Bruder for valuable\ndiscussions and criticism. The Centro de Ciencias de\nBenasque Pedro Pascual has provided a very stimulating\nenvironment for \fnishing this manuscript. This work was\nsupported by the Swiss National Science Foundation and\nby the EU project QIBEC.14\nAppendix A: Choice of measurement axes\nIn this section we motivate the set of Cartesian mea-\nsurement axes used for spin-1 =2 tomography in this work\n(see section I). We \fnd the conditions under which a set\nof measurement axes can be used for e\u000ecient and accu-\nrate quantum state tomography.\n1. Direct inversion (\fltered backprojection)\nThe direct inversion (\fltered backprojection) tech-\nnique of Ref. [3] is the simplest general tomographic\nmethod for quantum-mechanical spins of arbitrary length\nj. Given a true density matrix with spherical tensor co-\ne\u000ecients\u001akqand a set of Mmeasurement axes ( #n;'n)\nwith measurement weights cn, the average tomograph-\nically reconstructed spherical coe\u000ecients of the density\nmatrix (averaged over all possible measurement results)\nare found by inserting Eq. (6) into Eq. (4) of Ref. [3],\nh\u001a(fbp)\nkqi\n= (2k+ 1)MX\nn=1cnDk\nq0('n;#n;0)jX\nm=\u0000jpm(#n;'n)tjmm\nk0\n=kX\nq0=\u0000k\u001akq0\u00024\u0019MX\nn=1cn[Yq\nk(#n;'n)]\u0003Yq0\nk(#n;'n):\n(A1)\nThis is the correct result h\u001a(fbp)\nkqi=\u001akqif the measure-\nment axis orientations ( #n;'n) and their weights cnsat-isfy\n4\u0019MX\nn=1cn[Yq\nk(#n;'n)]\u0003Yq0\nk(#n;'n) =\u000eqq0 (A2)\nfor allk= 0;1;:::; 2jandq;q0=\u0000k;\u0000k+ 1;:::; +k. If\nwe decompose the angular density of measurement axes\ninto a sum of spherical harmonics with coe\u000ecients\nskq=MX\nn=1cn[Yq\nk(#n;'n)]\u0003; (A3)\nthen Eq. (A2) is satis\fed whenever skq= 0 for all k=\n2;4;6;:::; 4j.\nForj= 1=2, as used in this work, this implies that\ndirect inversion tomography [3] is correct on average if\nthe distribution of measurement axes satis\fes s2;\u00002=\ns2;\u00001=s2;0=s2;1=s2;2= 0, i.e., if there is no\nquadrupolar anisotropy in the distribution of measure-\nment axes. The smallest set of measurement axes that\nsatis\fes these conditions is the Cartesian-axes tomogra-\nphy set with equal numbers of measurements along each\naxis, as used in this text.\n2. Maximum likelihood estimation\nIn contrast to direct inversion (section A 1), maximum\nlikelihood estimation is intrinsically biased due to the\nphysicality constraint on the density matrix (see table I).\nWe can nonetheless ask: what conditions must a set of\nmeasurement axes ful\fl so that the mean tomographic\nestimate is closest to the true density matrix?\nFor the tomographic reconstruction of a completely\nmixed spin-1/2 state (Bloch vector r= 0), we can quan-\ntify this question by calculating the variance hkrMLEk2i\nof the resulting Bloch vector, averaged over all possible\nsets of experimental results. We \fnd numerically that\nhkrMLEk2iis smallest if the distribution of measurement\naxes hass2;\u00002=s2;\u00001=s2;0=s2;1=s2;2= 0, which\nis the same vanishing-quadrupole condition as found in\nsection A 1.\n[1] M. Paris and J. \u0014Reh\u0013 a\u0014 cek, eds., Quantum State Estima-\ntion, Lect. Notes Phys., Vol. 649 (Springer, Berlin Hei-\ndelberg, 2004).\n[2] S. J. van Enk and R. Blume-Kohout, New J. Phys. 15,\n025024 (2013).\n[3] R. Schmied and P. Treutlein, New J. Phys. 13, 065019\n(2011).\n[4] R. Blume-Kohout, New J. Phys. 12, 043034 (2010).\n[5] H. K. Ng and B.-G. Englert, Int. J. Quantum Inform. 10,\n1250038 (2012).\n[6] E. Bagan, A. Monras, and R. Mu~ noz-Tapia, Phys. Rev.\nA71, 062318 (2005).\n[7] J. P. Dowling, G. S. Agarwal, and W. P. Schleich, Phys.\nRev. A 49, 4101 (1994).[8] L. Pezz\u0013 e and A. Smerzi, Phys. Rev. Lett. 102, 100401\n(2009).\n[9] J. \u0014Reh\u0013 a\u0014 cek, B.-G. Englert, and D. Kaszlikowski, Phys.\nRev. A 70, 052321 (2004).\n[10] A. J. Scott, J. 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A 82, 042312 (2010).\n[29] The Bures measure for qubits (20) is the spherical equiv-\nalent of the Je\u000breys prior of the Bernoulli trial, C(p) =\n1=[\u0019p\np(1\u0000p)].\n[30]Ju(t) has a branch cut discontinuity at t= 0 foru<\u00001,\nand any results based on these values are ill-de\fned.\n[31] M. A. Nielsen and I. L. Chuang, Quantum Computation\nand Quantum Information (Cambridge University Press,\nCambridge, 2000).\n[32] M. Christandl and R. Renner, Phys. Rev. Lett. 109,\n120403 (2012).\n[33] B. Efron and R. J. Tibshirani, Stat. Sci. 1, 54 (1986).\n[34] J. Shang, H. K. Ng, A. Sehrawat, X. Li, and B.-G. En-\nglert, New J. Phys. 15, 123026 (2013).\n[35] E. P. Wigner, Ann. Math. 67, 325 (1958).\n[36] M. F. Riedel, P. B ohi, Y. Li, T. W. H ansch, A. Sinatra,\nand P. Treutlein, Nature 464, 1170 (2010).\n[37] C. F. Ockeloen, R. Schmied, M. F. Riedel, and P. Treut-\nlein, Phys. Rev. Lett. 111, 143001 (2013).\n[38] H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B.\nHume, L. Pezz\u0012 e, A. Smerzi, and M. K. Oberthaler, Sci-\nence345, 424 (2014)."    },    {        "title": "1407.4872v2.Roles_of_Hund_s_rule_coupling_in_excitonic_density_wave_states.pdf",        "content": "arXiv:1407.4872v2  [cond-mat.str-el]  5 Jan 2015Roles of Hund’s rule coupling in excitonic density-wave sta tes\nTatsuya Kaneko and Yukinori Ohta\nDepartment of Physics, Chiba University, Chiba 263-8522, J apan\n(Dated: Received 17 July 2014)\nExcitonic density-wave states realized by the quantum cond ensation of electron-hole pairs (or\nexcitons) are studied in the two-band Hubbard model with Hun d’s rule coupling and the pair\nhopping term. Using the variational cluster approximation , we calculate the grand potential of\nthe system and demonstrate that Hund’s rule coupling always stabilizes the excitonic spin-density-\nwave state and destabilizes the excitonic charge-density- wave state and that the pair hopping term\nenhances these effects. The characteristics of these excito nic density-wave states are discussed using\nthe calculated single-particle spectral function, densit y of states, condensation amplitude, and pair\ncoherence length. Implications of our results in the materi als’ aspects are also discussed.\nPACS numbers: 71.10.Fd, 71.35.-y, 75.30.Fv, 71.30.+h\nI. INTRODUCTION\nThe excitonic phases, often referred to as the excitonic\ninsulator states or excitonic density-wave states, are de-\nscribed by the quantum condensation of excitons, which\nwere predicted to occur in a small band-gap semiconduc-\ntor or a small band-overlap semimetal.1–3The exciton\ncondensation in semimetallic systems can be described\nin analogy with the BCS theory of superconductors, and\nthat in semiconductingsystemscan be discussedin terms\nof the Bose-Einstein condensation (BEC) of preformed\nexcitons.4The crossover phenomena between the BCS\nand the BEC states are then expected to produce rich\nphysics in the field of quantum many-body systems. The\nexcitonic phases are characterized by an order parameter\n/angb∇acketleftc†\nk+Qσfkσ′/angb∇acket∇ight, wherec†\nkσandf†\nkσare the creation opera-\ntors of an electron in the conduction and valence bands,\nrespectively. If the valence-band top and conduction-\nband bottom are separated by the wave vector Q, the\nsystem shows the density wave with modulation Q.2,3\nA number of candidate materials for the excitonic\nphases have been discovered. It was claimed that\nTm(Se,Te) shows a transition into the excitonic insu-\nlator state by applying pressure.5The weak ferromag-\nnetism of Ca 1−xLaxB6was interpreted in terms of a\ndoped spin-triplet excitonic insulator.6–8Recently, the\nphase transition of a layered chalcogenide Ta 2NiSe5has\nbeen attributed to a realization of the spin-singlet ex-\ncitonic insulator and has attracted much experimental\nand theoretical attention.9–12The charge-density wave\n(CDW) of 1 T-TiSe2has also been claimed to be of the\nexcitonic origin.13–16The spin-density wave (SDW) state\nof iron pnictide superconductors has sometimes been ar-\ngued to be of the excitonic origin as well.17–19It was\nproposed that the condensation of spin-triplet excitons\ncan occur in the proximity of a spin-state transition;20\nPr0.5Ca0.5CoO3is an example.21\nIn this paper, motivated by the above development\nin the field, we study the stability of the excitonic\ndensity-wave states in the two-band Hubbard model\nwhere Hund’s rule coupling J, the pair hopping term\nJ′, as well as the interorbital Coulomb repulsion U′are\ntaken into account in addition to the standard intraor-\nbital Hubbard repulsion U. It is known that the interor-\nbital repulsion U′induces the excitonic instability in the\nsystem,19,22but the condensationsofthe spin-singlet and\nspin-tripletexcitonsareexactlydegenerateunlessHund’s\nrule coupling, the pair hopping term, or electron-phononcoupling are taken into account. Thus, we here study the\nroles of Hund’s rule coupling and the pair hopping term\nplayed in the excitonic density wave of which not much\nis known so far.\nWe first rewrite the interorbital interaction terms of\nthe Hamiltonian in terms of the creation and annihila-\ntion operators of the spin-singlet and spin-triplet exci-\ntons. We then show that the interorbital repulsion U′\nactually leads to the exciton formation in both the spin-\nsingletand the spin-triplet channelsand that Hund’s rule\ncoupling always lowers the energy of the spin-triplet ex-\nciton and raises the energy of the spin-singlet exciton.\nThe variationalcluster approximation(VCA)23,24is then\nusedtostudy thetwo-bandHubbard modelin detail, and\nwe show that Hund’s rule coupling and the pair hopping\nterm always stabilize the excitonic SDW state and desta-\nbilize the excitonic CDW state. The characteristics of\nthese excitonic density-wave states will moreover be ex-\namined using a variety of calculated physical quantities,\nincluding the single-particle spectral function, density of\nstates (DOS), condensation amplitude, and pair coher-\nence length. Consequences of the present results on the\nexcitonicdensity-wavestatesofa varietyofmaterialswill\nalso be discussed.\nThispaperisorganizedasfollows: InSec.II,themodel\nand method of calculations will be given. In Sec. III, the\nresults of calculations for various physical quantities will\nbe presented. Summary and discussion will be given in\nSec. IV.\nII. MODEL AND METHOD\nA. The two-band Hubbard model\nWe consider the two-band Hubbard model defined by\nthe Hamiltonian,\nH=−t/summationdisplay\n/angbracketlefti,j/angbracketright/summationdisplay\nσ/summationdisplay\nαα†\niσαjσ−D/summationdisplay\ni(nif−nic)\n+U/summationdisplay\ni/summationdisplay\nαniα↑niα↓+U′/summationdisplay\ninifnic\n−2J/summationdisplay\ni/parenleftbigg\nSif·Sic+1\n4nifnic/parenrightbigg\n−J′/summationdisplay\ni/parenleftBig\nf†\ni↑f†\ni↓ci↑ci↓+c†\ni↑c†\ni↓fi↑fi↓/parenrightBig\n,(1)2\nwhereα†\niσ(=f†\niσ,c†\niσ) denotes the creation operator of\nan electron with spin σ(=↑,↓) on theα(=f,c) orbital\nat siteiandniα=niα↑+niα↓=α†\ni↑αi↑+α†\ni↓αi↓.tis\nthe hopping integral between the same orbitals on the\nneighboring sites, and Dis the level splitting between\nthe two orbitals. UandU′are the intra- and interorbital\nCoulomb repulsions, respectively, between electrons and\nJandJ′are the strengths of Hund’s rule coupling and\nthe pair hopping term, respectively. Note that J=J′\nin the standard two-orbital Hubbard model, but when\nnecessary we retain only Hund’s rule coupling by setting\nJ′= 0 to examine the role of the pair hopping term.\nThroughout the paper, we assume the filling of two elec-\ntrons per site (or half-filling), i.e., /angb∇acketleftnf/angb∇acket∇ight+/angb∇acketleftnc/angb∇acket∇ight= 2, where\n/angb∇acketleftnα/angb∇acket∇ight=/summationtext\ni,σ/angb∇acketleftniασ/angb∇acket∇ight/N.\nThe Hamiltonian Eq. (1) in the spinless case with\nU=J=J′= 0 is equivalent to the extended Falicov-\nKimball model with dispersive candfelectrons of which\nthe excitonic insulator state has been studied much in\ndetail.25–29The excitonic phases in the two-band Hub-\nbard model without Hund’s rule coupling and the pair\nhopping term ( J=J′= 0) have also been studied\nrecently19,22where it was shown that the model exhibits\nthree ground-state phases: (i) the band insulator (at\nU′,D≫U,J), where /angb∇acketleftnf/angb∇acket∇ight= 2 and /angb∇acketleftnc/angb∇acket∇ight= 0, (ii) the an-\ntiferromagnetic Mott insulator (at U,J≫U′D), where\n/angb∇acketleftnf/angb∇acket∇ight=/angb∇acketleftnc/angb∇acket∇ight= 1, and (iii) the excitonic density-wave\nstate between the above two phases, where 2 >/angb∇acketleftnf/angb∇acket∇ight>\n1>/angb∇acketleftnc/angb∇acket∇ight>0. However,althoughmanystudies havebeen\nperformedon the multiband Hubbard models, recentlyin\nrelation to iron pnictide superconductors,30–33the exci-\ntonicdensity-wavestatesinthetwo-bandHubbardmodel\nwith Hund’s rule coupling have not been studied in de-\ntail; only a recent dynamical mean-field theory (DMFT)\ncalculation20,21is noticed.\nTo see the stability of the spin-singlet and spin-triplet\nexcitons, let us introduce the creation operators of the\nspin-singlet and spin-triplet excitons, respectively, which\nare defined as\nA0\ni†=1√\n2/summationdisplay\nσc†\niσfiσ,Ai†=1√\n2/summationdisplay\nσσ′c†\niσσσσ′fiσ′,(2)\nwhereσis the vector of the Pauli matrices. Using the\nspin-singlet and spin-triplet exciton operators thus de-\nfined, the interorbital Coulomb repulsion term can be di-\nvided exactly into the spin-singlet and spin-triplet terms\nas\nU′nifnic=−U′A0\ni†A0\ni−U′A†\ni·Ai.(3)\nTherefore, the formation of excitons lowers the energy of\nthe system in both the spin-singlet and the spin-triplet\nchannels by the same amount. Hund’s rule coupling and\nthe pair hopping terms can also be rewritten exactly as\n−JSif·Sic=3J\n4A0\ni†A0\ni−J\n4A†\ni·Ai,(4)\n−J′c†\ni↑c†\ni↓fi↑fi↓=J′\n4A0\ni†A0\ni†−J′\n4A†\ni·A†\ni.(5)\nTherefore, due to the Hund’s rule coupling term, the for-\nmation of the spin-triplet (spin-singlet) excitons always\nlowers (raises) the energy of the system, thus lifting the\ndegeneracy that occurs at J=J′= 0. The pair hoppingterm can also be divided into the spin-singlet and spin-\ntriplet terms as in Eq. (5), which are of the off-diagonal\nform.\nB. Variational cluster approximation\nWe use the VCA,23,24which is a quantum cluster\nmethod based on the self-energy functional theory,34\nand solve the quantum many-body problem defined in\nEq. (1). Note that, unlike in DMFT, we can taken into\naccountthe effects ofshort-rangespatialelectron correla-\ntions preciselyin this approach. The VCAintroduces the\ndisconnected finite-size clustersthat aresolvedexactlyto\nobtain the exact self-energy of the clusters ˆΣ′with which\na superlattice is formed as a reference system. The ma-\ntrices are indicated by a ˆ hereafter. By restricting the\ntrial self-energy to ˆΣ′, we obtain an approximate grand\npotential of the original system,\nΩ = Ω′−1\nN/contintegraldisplay\nCdz\n2πi/summationdisplay\nK,σlndet/bracketleftbigˆI−ˆVσ(K)ˆG′\nσ(z)/bracketrightbig\n,(6)\nwhere Ω′is the grand potential of the reference system,\nˆIis the unit matrix, ˆVis the hopping matrix between\nadjacent clusters, and ˆG′is the exact Green’s function\nof the reference system. The Ksummation is performed\nin the reduced Brillouin zone of the superlattice, and the\ncontourCof the frequency integral encloses the negative\nreal axis. Details of the VCA can be found in Refs. [35,\n36].\nTo study the symmetry-breaking phases in the VCA,\nwe introduce the Weiss fields as variational parameters.\nThe variational Hamiltonian for the excitonic CDW and\nSDW states are then defined as\nH′\nCDW=H+∆′\n0/summationdisplay\ni,σeiQ·ri/parenleftbig\nc†\niσfiσ+H.c./parenrightbig\n,(7)\nH′\nSDW=H+∆′\nz/summationdisplay\ni,σσeiQ·ri/parenleftbig\nc†\niσfiσ+H.c./parenrightbig\n,(8)\nrespectively, where ∆′\n0is the Weiss field for condensa-\ntion of the spin-singlet excitons and ∆′\nzis thezcom-\nponent of the Weiss field for condensation of the spin-\ntriplet excitons. The variational parameters ∆′\n0and\n∆′\nzare optimized on the basis of the variational prin-\nciple, i.e.,∂Ω/∂∆′\n0= 0 for the excitonic CDW state and\n∂Ω/∂∆′\nz= 0 for the excitonic SDW state. The solutions\nwith ∆′\n0/negationslash= 0 and ∆′\nz/negationslash= 0 correspond to the excitonic\nCDW and SDW states, respectively.\nWe solve the eigenvalue problem H′|ψ0/angb∇acket∇ight=E0|ψ0/angb∇acket∇ight\nof a finite-size ( Lcsites) cluster to obtain the ground\nstate, and we calculate the trial Green’s function by the\nLanczos exact-diagonalization method. Using the ba-\nsisΨ†\ni= (f†\niσ,c†\niσ), the Green’s-function matrix ˆG′\nσin\nEq. (6) may be written as\nˆG′\nσ(ω) =/parenleftbiggˆG′ff\nσ(ω)ˆG′fc\nσ(ω)\nˆG′cf\nσ(ω)ˆG′cc\nσ(ω)/parenrightbigg\n, (9)\nwhereˆG′αβ\nσis anLc×Lcmatrix and each matrix element3\nis defined as\nG′αβ\nij,σ(ω) =/angb∇acketleftψ0|αiσ1\nω−H′+E0β†\njσ|ψ0/angb∇acket∇ight\n+/angb∇acketleftψ0|β†\njσ1\nω+H′−E0αiσ|ψ0/angb∇acket∇ight.(10)\nThe matrix ˆVin Eq. (6) is given as\nˆVσ(K) =/parenleftbiggˆT(K)−∆′\nσˆI\n−∆′\nσˆIˆT(K)/parenrightbigg\n, (11)\nwhere ˆT(K) is the intercluster hopping ma-\ntrix with the matrix elements Tij(K) =\n−t/summationtext\nX,xeiK·Xδi+x,jδR+X,R′. Here, xdenotes the\nneighboring site of the ith site, and Xdenotes the\nneighboring cluster of the Rth cluster. ∆′\nσ= ∆′\n0for the\nexcitonic CDW state, and ∆′\nσ=σ∆′\nzfor the excitonic\nSDW state.\nIn our VCA calculation, we assume the two-\ndimensional square lattice and use an Lc= 2×2 = 4 site\n(eight-orbital) cluster as the reference system. We set\nD/t= 3.2 so that the noninteracting tight-binding band\nstructure is a semimetal with a small band overlap. The\nband structure has an electron pocket at k= (0,0) and\na hole pocket at k= (π,π) of the Brillouin zone. Hence,\nthe modulation vector of the density waves is given by\nQ= (π,π). Throughout the paper, we assume the rela-\ntionU′= (U+J)/2 between the interaction parameters\nso that the Hartree shift can be suppressed. The stan-\ndard choice U′=U−2J(withJ′=J) valid in the\natomic limit37–39has also been used to check that the\nessential features obtained in the present paper do not\nchange (see the Appendix). We moreover assume the\nvalueU/t= 8 at which the excitonic density-wave state\nis stabilized between the band-insulator and the Mott-\ninsulator states.22\nIII. RESULTS OF CALCULATION\nA. Stability of the excitonic density waves\nFirst, letusexaminethestabilityoftheexcitonicCDW\nand SDW states using the grand potential. In Fig. 1(a),\nwe show the calculated grand potentials of the excitonic\nCDW and SDW states as a function of the variational\nparameter ∆′. We find that the grand potential has\nthe stationary points at ∆′= 0 and ∆′/negationslash= 0, and the\nlatter is lower in energy, indicating that the excitonic\ndensity-wave states are thermodynamically stable. At\nJ=J′= 0, the grand potentials of the excitonic CDW\nand SDW states are exactly degenerate [see Fig. 1(a)],\nbut Hund’s rule coupling Jand pair hopping term J′lift\nthis degeneracy. The optimized values of the grand po-\ntential as a function of J(=J′) are shown in Fig. 1(b)\nwhere we find that, with increasing J(andJ′), the en-\nergyoftheexcitonicSDWstatedecreases,but the energy\nof the excitonic CDW state increases and approaches the\nenergy of the normal semimetallic state. Therefore, the\nexcitonic SDW (CDW) state is stabilized (destabilized)\nbyJandJ′. In Fig. 1(c), we show the optimized values\nof the grand potentials in the presence ( J′=J) and ab-\nsence (J′= 0) of the pair hopping term where we find\nFIG. 1: (Color online) (a) Calculated grand potentials for t he\nexcitonic CDW and SDW states as a function of the vari-\national parameter ∆′(= ∆′\n0,∆′\nz) atJ/t=J′/t= 0, 0.25,\n0.5, 0.75, and 1. Ω 0is the grand potential in the normal\n(semimetallic) state. The crosses and dots indicate the sta -\ntionary points of the excitonic CDW and SDW states, respec-\ntively. (b) J(=J′) dependence of the grand potential at the\nstationary point for the normal (or semimetallic), exciton ic\nCDW, and SDW states. (c) Optimized values of the grand\npotentials in the presence ( J′=J) and absence ( J′= 0) of\nthe pair hopping term. (d) Jdependence of the order param-\neters of the excitonic CDW and SDW states in the presence\n(J′=J) and absence ( J′= 0) of the pair hopping term.\nthat the stability of the excitonic SDW (CDW) state is\nenhanced (suppressed) by pair hopping term J′.\nWe also calculate the order parametersof the excitonic\nCDW and SDW states. Here, we introduce the quanti-\nties Φ 0and Φ zfor the excitonic CDW and SDW order\nparameters, respectively, which are defined as\nΦ0=1\n2N/summationdisplay\nk/summationdisplay\nσ/angb∇acketleftc†\nk+Qσfkσ/angb∇acket∇ight, (12)\nΦz=1\n2N/summationdisplay\nk/summationdisplay\nσσ/angb∇acketleftc†\nk+Qσfkσ/angb∇acket∇ight. (13)\nThe calculated results for Φ 0and Φ zare shown in\nFig. 1(d) in the presence ( J′=J) and absence ( J′= 0)\nof the pair hopping term. We find that Φ zis enhanced\nwithJ(andJ′) and Φ 0is suppressed with J(andJ′),\nwhichareinaccordancewiththe stabilityofthe excitonic\nCDW and SDW states evaluated from the behaviors of\nthe calculated grand potentials. Thus, we may state that\nHund’s rule coupling stabilizes the excitonic SDW state\nand destabilizes the excitonic CDW state. As seen in\nFigs. 1(c) and 1(d), we may moreover state that the pair\nhoppingtermenhancesthestabilityoftheexcitonicSDW\nstate and suppresses the stability of the excitonic CDW\nstate.4\nFIG. 2: (Color online) Single-particle spectral function\nA(k,ω) and DOS Nα(ω) calculated by CPT at J/t=J′/t=\n0.5. We show the results for the excitonic CDW state\n(metastable) in (a) and (c) and for the excitonic SDW state\n(stable) in (b) and (d). In (c) and (d), the solid, dashed,\nand dotted lines indicate the forbital,corbital, and total\nDOSs, respectively. The artificial Lorentzian broadening o f\nη/t= 0.15 is used for A(k,ω) andη/t= 0.05 is used for\nNα(ω). The Fermi level is located at ω= 0.\nB. Single-particle spectral function\nNext, let us calculate the Green’s function at the op-\ntimized values of the variational parameters using the\ncluster perturbation theory (CPT).40The Green’s func-\ntion is defined as\nˆGσ(k,k′,ω) =1\nLcLc/summationdisplay\ni,j=1ˆGCPT\nij,σ(k,ω)e−ik·ri+ik′·rj,(14)\nwhereˆGCPT\nσ(k,ω) =/bracketleftbigˆG′−1\nσ(ω)−ˆVσ(k)/bracketrightbig−1. Using this\nGreen’s function, the single-particle spectral function is\ndefined as\nA(k,ω) =−1\nπ/summationdisplay\nα,σImGαα\nσ(k,k,ω+iη),(15)\nwhereηgives the artificial Lorentzian broadening to the\nspectrum. We also calculate the DOS for the α(=f,c)\norbital, which is defined as\nNα(ω) =−1\nπN/summationdisplay\nk/summationdisplay\nσImGαα\nσ(k,ω+iη).(16)\nIn Fig. 2, we show the calculated single-particle spec-\ntral function A(k,ω) and DOS Nα(ω); the results for\nthe metastable CDW state [see Figs. 2(a) and 2(c)] and\nstable SDW state [see Figs. 2(b) and 2(d)] obtained at\nJ/t=J′/t= 0.5 are shown. We find that, although a\nsemimetallic state with a small band overlap is assumed\nas the noninteracting band structure, the valence band\naroundk= (π,π)ishybridizedwiththeconductionband\naroundk= (0,0) due to the spontaneous c-fhybridiza-\ntion (or exciton condensation), leading to the opening\nFIG. 3: (Color online) Calculated DOSs for the (a) excitonic\nCDW state and (b) excitonic SDW state at J/t=J′/t= 0.5.\nSolid and dashed lines indicate the DOSs of the A and B\nsublattices, respectively. The Lorentzian broadening of η/t=\n0.05 is used. The vertical line indicates the Fermi level.\nof the band gap at the Fermi level. At J=J′= 0,\nthe single-particle excitation gap ∆ gis estimated to be\n∆g/t= 1.47. We find that, in agreementwith the change\nin the order parameters, the single-particle gap in the ex-\ncitonicCDWstate, e.g.,∆ g/t= 0.76atJ/t=J′/t= 0.5,\nis suppressedin comparisonwith the J=J′= 0case [see\nFigs. 2(a) and 2(c)]. We also find that the single-particle\ngap in the excitonic SDW state, e.g., ∆ g/t= 1.81 at\nJ/t=J′/t= 0.5, is enhanced in comparison with the\nJ=J′= 0 case [see Figs. 2(b) and 2(d)]. We moreover\nfind in Figs. 2 (c) and 2(d) that the sharp coherencepeak\nappears at the edges of the gap and that the coherence\npeak of the SDW state is sharper than that of the CDW\nstate, indicating that the spontaneous c-fhybridization\nin the excitonic SDW (CDW) state is enhanced (sup-\npressed) by Hund’s rule coupling and the pair hopping\nterm. We note that no significant differences are found\nin the behaviors of Nα(ω) discussed above, even if we\nswitch off the pair hopping term, retaining only Hund’s\nrule coupling.\nIn order to see the character of the excitonic density-\nwave states, we calculate the DOS of the A and B sub-\nlattices. The sublattice Green’s function is given by\nˆGXσ(k,ω) =2\nLc/summationdisplay\ni,j∈XˆGCPT\nij,σ(k,ω)e−ik·(ri−rj)(17)\nwithX= A or B. Using this sublattice Green’s function,\nthe DOS of the A or B sublattices is defined as\nNXσ(ω) =−1\nπN/summationdisplay\nk/summationdisplay\nα,βImGαβ\nXσ(k,ω+iη).(18)\nIn Fig. 3(a), we show the calculated DOS for the ex-\ncitonic CDW state at J/t=J′/t= 0.5. We note that,\nbelow the Fermi level ( ω <0), the up- and down-spin\nDOSs are the same and the DOS of the A sublattice is\nlargerthanthat oftheBsublattice: NA↑(ω) =NA↓(ω)>\nNB↑(ω) =NB↓(ω). We alsonote that NAσ(ω)≃NBσ(ω)\nfarawayfromtheFermilevelandthatthecoherencepeak\nappears in the DOS of the A sublattice just below the\nFermi level, where NAσ(ω)> NBσ(ω). Using the order\nparameter Φ 0, the local number of electrons is given by\nni= 1+2Φ 0cos(Q·ri). AtJ/t=J′/t= 0.5, we have5\nFIG. 4: (Color online) Condensation amplitude F(k) [=F0(k) orFz(k)] calculated by CPT. We show the results for (a) the\nCDW state at J/t=J′/t= 1.0 (metastable), (b) the CDW/SDW states at J=J′= 0 (degenerate), and (c) the SDW state\natJ/t=J′/t= 1.0 (stable).\nΦ0= 0.11, and thus the local numbers ofthe electrons on\neach sublattice are given by nA↑=nA↓= 1+2Φ 0= 1.22\nandnB↑=nB↓= 1−2Φ0= 0.78. We therefore find\nthat, due to the effect of Hund’s rule coupling and the\npair hopping term, Φ 0is suppressed and thus the ex-\ncitonic CDW modulation in real space becomes rather\nweak.\nIn Fig. 3(b), we show the calculated DOS for the ex-\ncitonic SDW state at J/t=J′/t= 0.5. We note that,\nbelow Fermi level, the up-spin DOS of the A (B) sublat-\ntice is equal to the down-spin DOS of the B (A) sublat-\ntice and that the up-spin DOS of the A (B) sublattice\nis larger (smaller) than the down-spin DOS of the A (B)\nsublattice: NA↑(ω) =NB↓(ω)>NA↓(ω) =NB↑(ω). We\nalso note that the DOS has a large gap and a sharp co-\nherence peak appears at the edge of the DOS. Using the\norder parameter Φ z, the local magnetization is given by\nmi= 2Φzcos(Q·ri). AtJ/t=J′/t= 0.5, we have\nΦz= 0.20, and thus the local numbers of electrons on\neach sublattice are given by nA↑=nB↓= 1+2Φ z= 1.40\nandnA↓=nB↑= 1−2Φz= 0.60. We therefore find\nthat, due to the effect of Hund’s rule coupling and the\npairhoppingterm, Φ zisenhanced, andthustheexcitonic\nSDW modulation in real space becomes rather strong.\nC. Condensation amplitude and coherence length\nIn order to see the character of the exciton conden-\nsation in momentum space, we calculate the condensa-\ntion amplitude (or the anomalous momentum distribu-\ntion function). Using the off-diagonal (or anomalous)\nGreen’s function given in Eq. (14), the condensation am-\nplitudes for the spin-singlet and spin-triplet excitons are\ndefined as\nF0(k) =1\n2/summationdisplay\nσ/contintegraldisplay\nCdz\n2πiGcf\nσ(k,k+Q,z),(19)\nFz(k) =1\n2/summationdisplay\nσσ/contintegraldisplay\nCdz\n2πiGcf\nσ(k,k+Q,z),(20)\nrespectively. Note that we here use the term “anoma-\nlous” to indicate that the number of electrons on each of\nthecandforbitals is not conserved due to the excitonic\ncondensation, although the total number of electrons is\nconserved.We show the calculated results in Fig. 4 for the exci-\ntonic CDW and SDW states. We find that, with increas-\ningJ(=J′), the peak of F0(k) at the Fermi momentum\nkFbecomes sharper in the CDW state [see Fig. 4(a)] and\nthat the peak of Fz(k) atkFbecomes broaderin momen-\ntum space in the SDW state [see Fig. 4(c)]. The sharp\n(broad) peak of F(k) [=F0(k) orFz(k)] in momentum\nspace indicates that the spatial extension of the electron-\nhole pair becomes large (small) in real space. We note\nthat no significant differences are found in the behavior\nofF(k), even if we set J′= 0 retaining only Hund’s rule\ncoupling.\nUsingF(k), we evaluate the pair coherence length ξ,\nwhich corresponds to the spatial size of the electron-hole\npair and may be defined by26,28,29\nξ2=/summationtext\nk|∇kF(k)|2\n/summationtext\nk|F(k)|2. (21)\nIn Fig. 5, we show the calculated results for the spin-\nsinglet excitons ( ξ0) and spin-triplet excitons ( ξz) as a\nfunction of J. We find that, with increasing J(=J′),ξ\nforthespin-singlet(triplet)excitonsincreases(decreases)\nmonotonically. Thus, the size of the spin-singlet exciton\nbecomes larger than the lattice constant ( ξ0>1) for\nlargerJvalues, indicating the crossover from the tightly\npaired BEC state to the weakly paired BCS state. The\nspin-triplet excitons, on the other hand, are paired more\ntightly, and the size is always smaller than the lattice\nconstant in the parameter space examined. We also find\nin the inset of Fig. 5 that the above tendencies induced\nby Hund’s rule coupling are again enhanced by the pair\nhopping term.\nIV. SUMMARY AND DISCUSSION\nTosummarize,wehavestudiedthestabilityoftheexci-\ntonicdensity-wavestatesinthetwo-bandHubbardmodel\nwith the interorbital Coulomb interaction U′, Hund’s\nrule coupling J, pair hopping term J′, as well as the\nintraorbital Hubbard interaction U. We have rewrit-\nten the interorbital interactions of the Hamiltonian in\nterms of the creation and annihilation operators of the\nspin-singlet and spin-triplet excitons and examined the\nroles of these interactions. We have thereby shown that\ntheU′term drives the formation of excitons in both\nthe spin-singlet and the spin-triplet channels, and the6\nFIG. 5: (Color online) Calculated pair coherence length ξ\nin units of the lattice constant. J(=J′) dependence of ξis\nshown for the spin-singlet (open circles) and spin-triplet (solid\ncircles) exciton condensations. The inset shows the result s in\nthe absence of the pair hopping term J′= 0 (open and solid\nsquares), which are compared with the results in the presenc e\nof the pair hopping term J′=J(open and solid circles).\nJterm stabilizes (destabilizes) the formation of the spin-\ntriplet (spin-singlet) excitons. Using the VCA to calcu-\nlate the grand potential of the system in the thermody-\nnamic limit, we have moreover shown that Hund’s rule\ncoupling always stabilizes the excitonic SDW state and\ndestabilizes the excitonic CDW state of which the ten-\ndencies are enhanced by the pair hopping term. A vari-\nety of physical quantities has also been calculated, which\ninclude the single-particle spectral function, density of\nstates, anomalous Green’s functions, condensation am-\nplitude, and pair coherence length. We have thus char-\nacterized the excitonic CDW and SDW states in detail.\nFinally, let us discuss the experimental implications of\nour results obtained in this paper. At first sight, the con-\ndensations of the spin-singlet excitons possibly observed\nin 1T-TiSe2(Refs. [13–16]) and Ta 2NiSe5(Refs. [9–12])\nseem to contradict the stability of the spin-triplet exci-\ntons in the presence of Hund’s rule coupling. However,\nin these materials, the valence and conduction bands are\nformed by the orbitals located on different atoms, i.e.,\nthe 4porbitals of Se ions for the valence bands and the\n3dorbitals of Ti ions for the conduction bands in 1 T-\nTiSe2,13and the 3dorbitals of Ni ions for the valence\nbands, and the 5 dorbitals of Ta ions for the conduction\nbands in Ta 2NiSe5,11and therefore Hund’s rule coupling\nacting between electrons on different orbitals in a single\nion does not work to stabilize the condensation of the\nspin-triplet excitons. We anticipate that in these materi-\nals the electron-phonon coupling should work to stabilize\nthe condensation of the spin-singlet excitons as was dis-\ncussed in Refs. [11,15,16]. In the excitonic SDW states\npossibly observed in, e.g., iron pnictide superconductors\nand Co oxide materials, on the other hand, Hund’s rule\ncouplingratherthantheelectron-phononcouplingshould\nwork to stabilize the condensation of the spin-triplet ex-\ncitons as we have shown in this paper. We may there-\nfore suggest that the competition between Hund’s rule\ncoupling and electron-phonon coupling in the stability of\nexcitonic condensations (or excitonic density-wave for-\nFIG.6: (Color online)(a)Thenumberoftheconduction-band\nelectrons /angbracketleftnc/angbracketright(or the valence-band holes) as a function of J/t\nin the normal state (or ∆′= 0), which is obtained using the\natomic-limit relation U′=U−2JwithU/t= 5,D/t= 2, and\nJ′= 0. (b) Calculated grand potentials of the excitonic CDW\nand SDW states as a function of the variational parameter ∆′\n(= ∆′\n0,∆′\nz), which are obtained using the atomic-limit rela-\ntionU′=U−2JwithU/t= 5,J/t=J′/t= 0.5 andD/t= 2.\nThe crosses and circles indicate the stationary points of th e\nexcitonic CDW and SDW states, respectively.\nmations) will be of great interest in future studies.\nAcknowledgments\nT. K. acknowledges support from the JSPS Research\nFellowship for Young Scientists. This work was sup-\nported, in part, by a Kakenhi Grant No. 26400349 from\nJSPS of Japan.\nAppendix A: Use of the atomic-limit relation\nIn the main text, we have assumed the relation U′=\n(U+J)/2 between the interaction parameters. In this\nAppendix, we presentsomeresults obtainedin adifferent\nchoice of the relation, i.e., U′=U−2J, which is valid\nin the atomic limit,37–39and show that the essential fea-\ntures of our results do not alter.\nIn the BCS-likemean-field theoryapplied to ourmodel\nEq. (1), the diagonal terms of the mean-field Hamilto-\nnian are given by εf(k) =−εc(k) =−2t/summationtextd\nicoski−D+\nn(U/2−U′+J/2) withn=/angb∇acketleftnifσ/angb∇acket∇ight − /angb∇acketleftnicσ/angb∇acket∇ight, and the\noff-diagonal term gives the spontaneous c-fhybridiza-\ntion (or excitonic condensation).11The Hartree shift\nn(U/2−U′+J/2) appears in this expression. Depending\non the values of U,U′andJ, we therefore find, e.g., the\nMott-insulator state at U′≪(U+J)/2 and the band-\ninsulator state at U′≫(U+J)/2,19,22which are due\nsimply to the effect of the Hartree shift.\nTheeffectsofthisHartreeshift canbe suppressedcom-\npletely if we assume the relation U′= (U+J)/2 as in\nthe main text. However, if we assume the atomic-limit\nrelationU′=U−2J, the changein the parametervalues,\ne.g.,J, leads to the change in the overlap of the valence\nand conduction bands and hence to the change in the\nnumber of conduction-band electrons (and valence-band\nholes) as shown in Fig. 6(a). 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Rev.\nLett84, 522 (2000)."    },    {        "title": "1408.7083v1.Truncated_Moment_Problem_for_Dirac_Mixture_Densities_with_Entropy_Regularization.pdf",        "content": "Truncated Moment Problem for Dirac Mixture Densities\nwith Entropy Regularization\nUwe D. Hanebecka\naIntelligent Sensor-Actuator-Systems Laboratory (ISAS)\nInstitute for Anthropomatics and Robotics\nKarlsruhe Institute of Technology (KIT), Germany\nAbstract\nWe assume that a finite set of moments of a random vector is given. Its underlying density is unknown.\nAn algorithm is proposed for efficiently calculating Dirac mixture densities maintaining these moments\nwhile providing a homogeneous coverage of the state space.\n1. Introduction\nx →f(x) →Densities: True and Dirac mixture approximation, L=20, M=6\n−4−3−2−1 0 1 2 3 400.10.20.30.40.50.6\nFigure 1: Maximum entropy Dirac mixture density (purple)\nwith 20 components and prescribed moments up to order 6.\nThe underlying continuous density (yellow) for generating\nthe moments is unknown.We consider a sequence of mappings\nek=/integraldisplay\nXmk(x)˜f(x)dx (1)\nfrom a probability density function ˜f(x) to the\nso-called moments ekfork= 0,1,..., whereXis\na Polish space. Examples for Xare the set of real\nnumbers R, theN-dimensional Euclidean space\nRN, the unit interval [0 ,1],C,CN, the unit circle\nS1={z∈C:|z|= 1}, and so forth. Here, we\nfocus on the N-dimensional Euclidean space RN.\nWe are interested in the inverse problem of\ndeducing the probability density function ˜f(x)\nfrom these mapping given the moment sequence.\nThis is called the moment problem . Here, we focus\non the case that a finite moment sequence ekfor\nk= 0,1,...,K is given. The problem is then\ncalled the truncated moment problem .\nVarious types of moments can be considered depending upon the functions mk(x),k= 0,1,...,K .\nThis includes the common power moments and trigonometric moments, which are useful for periodic\nstate spaces such as the unit circle. Here, we focus on power moments.\nWe can now ask several fundamental questions such as: Does a density ˜f(x) exist for the given\nmoment sequence ek,k= 0,1,...,K ? When a density ˜f(x) exists, is it uniquely defined by the\nmoment sequence? When it is not uniquely defined, how is the set of densities with the given moment\nsequence characterized?\nSo far, we did not pose restrictions on the probability density function ˜f(x) to be reconstructed from\nthe moment sequence. So, ˜f(x) can be selected from the space of density functions, which lead to an\ninfinite-dimensional problem. More practical questions on existence, uniqueness, and characterization\ncan be asked, however, when we restrict ourselves to finite-dimensional approximations of the underlying\nEmail address: uwe.hanebeck@ieee.org (Uwe D. Hanebeck)arXiv:1408.7083v1  [cs.SY]  29 Aug 2014true density ˜f(x). We consider specific densities f(x) with a finite-dimensional parametrization, where\nwe have to select a density type with a given structure. In some cases, it is also useful to consider\nrestrictions on their parameter sets. For example, we could ask whether a Gaussian mixture density\nwith three components exists for a given moment sequence and whether it is unique.\nThere are many types of parametric densities available such as Gaussian densities, Gaussian\nmixture densities, and exponential densities. In this paper, we consider Dirac mixture densities f(x)\nfor approximating the underlying true density ˜f(x).\nUp to now, the only information available about the underlying true density ˜f(x) was the moment\nsequence of length K+ 1. This restricts the number of (independent) parameters of the approximating\ndensityf(x) to be less than or equal to K+ 1. Restricting the number of parameters to K+ 1 is\nespecially problematic as typically the number of available moments is itself limited. For the case\nof power moments of up to a certain order M, the number of moments quickly increases with the\nnumber of dimensions Nand the order M. Even for a moderate number of dimensions, calculating\nhigher-order moments becomes intractable.\nA good coverage of the important regions of the state space with the approximating density f(x)\nis mandatory in many applications. For Dirac mixture densities, this means that we need a large\nnumber of components, equivalent to a large number of parameters to be determined. When more\ndensity parameters than given moments have to be determined, we face an underdetermined inverse\nproblem with an infinite solution set. In that case, additional information about the underlying true\ndensity ˜f(x) such as its support, shape, symmetries, or its smoothness is required. Alternatively, we\nhave to directly impose additional assumptions on the approximating density f(x). This information\ncan be used to define a regularizer for picking out a single solution with the desired properties.\nAn interesting border case is the availability of the full underlying true density ˜f(x) together with\na few of its moments. In that case, we want to find an approximating density f(x) that maintains the\ngiven moments and is in some way as close as possible to the true density.\nA detailed problem formulation is given in the following section including a compact representation\nof given moments up to a certain order and some words on regularization. Sec. 3 gives an overview of\nthe state of the art.\n2. Problem Formulation\nA random vector x=/bracketleftBig\nx1,x2,...,xN/bracketrightBigT∈RNis characterized by a finite set of moments only.\nThe underlying true probability density function ˜f(x) ofxis unknown. The true density ˜f(x) can be\na continuous density or a discrete density on the continuous domain RN.\nOur goal is to represent the unknown probability density function ˜f(x) of the random vector x\nby an approximate density f(x) that has the desired moments. For the approximation, we focus on\ndiscrete probability density functions f(x) on the continuous domain RN. Here, we use a so called\nDirac mixture density f(x) withLDirac components given by\nf(x) =L/summationdisplay\ni=1fi(x) =L/summationdisplay\ni=1wi·δ(x−ˆxi), (2)\nwith positive weights, i.e., wi>0 fori= 1,...,L , that sum up to one, i.e.,/summationtextL\ni=1wi= 1, and locations\nˆxiwith components ˆxkifor dimension kwithk= 1,...,N andˆxi/negationslash=ˆxjfori= 1,...,L ,j= 1,...,L ,\ni/negationslash=j. The locations are collected in a matrix ˆX=/bracketleftBig\nˆx1,ˆx2,..., ˆxL/bracketrightBig\n∈RN×L.\nOur goal is to systematically find a Dirac mixture density f(x) in (2) that maintains the moments\nof the true density ˜f(x) by adjusting its parameters, i.e., its weights wiand its locations ˆxifor\ni= 1,...,L . In this paper, we focus on adjusting the locations only. The component weights are all\nequal. In addition, we assume that a solution exists, i.e., the number of components Lis selected to\nbe large enough so that locations exist that fulfill the given moments defined in the next subsection.\n2We define a parameter vector η∈S=RL·Ncontaining the parameters as\nη=/bracketleftBig\nˆxT\n1,ˆxT\n2,..., ˆxT\nL/bracketrightBigT(3)\nand writef(x) =f(x,η).\n2.1. Given Moments\nWe consider power moments for characterizing the random vector x, so we will now specify\nconcrete functions mk(x) in (1). For denoting the moment order, we employ a multi-index notation\nwithκ=/parenleftBig\nκ1,κ2,...,κN/parenrightBig\ncontaining non-negative integer indices for every dimension. We define\n|κ|=κ1+κ2+...+κN,κ+i=/parenleftBig\nκ1+i,κ2+i,...,κN+i/parenrightBig\nwithi∈Zsuch thatκ+i≥0, and\nxκ=xκ1\n1·xκ2\n2···xκN\nN. For a scalar c, the expression κ≤cis equivalent to κk≤cfork= 1,2,...,N .\nThe power moments of a random vector xwith density f(x) are given by\neκ=/integraldisplay\nRNxκf(x)dx (4)\nforκ∈NN\n0. For zero-mean random vectors x, the moments coincide with the central moments.\nMoments of a certain order mare given by eκfor|κ|=m. We define a multi-dimensional matrix\nEMof moments of up to order Mas\nEM(κ+ 1) =/braceleftBigg\neκ|κ|≤M\nunspecified elsewhere, (5)\nwithκ≤Mandeκfrom (4). We increase the multi-index κby 1, so that the matrix EM∈\nR(M+1)×(M+1)×...×(M+1)is indexed from 1 to M+ 1 in every dimension.\nThe matrix EMcontains unspecified elements for |κ|≤Mthat can either be set to zero in a\nfull matrix or omitted in sparse matrices (when the chosen matrix implementation supports sparse\nmatrices). WithKNMthe set of all valid index sequences given by\nKNM={κ:|κ|≤M} (6)\nthe number of specified elements, i.e., the number of moments of up to order M, is defined as\nPNM=|KNM|and is given next.\nLemma 2.1. For anN-dimensional random vector, the number of moments up to order Mis\nPNM=(M+N)!\nM!N!\nProof. Elementary.\nForN= 10 dimensions, the number of moments up to order M= 3 isPNM= 286, forM= 5\nalreadyPNM= 3003.\nOf course, there is no need to specify all possible moments for |κ|≤M. In a practical application,\nthere will generally be a lot more unspecified elements.\n2.2. Regularization\nWhen the length PL=L·Nof the parameter vector ηis larger than the number of given moments,\nthe parameters of f(x,η) are redundant and a regularizer for f(x,η) is required. Here, regularization\nis performed by selecting the least informative Dirac mixture, e.g., the one having maximum entropy.\nAs the Shannon entropy for Dirac mixture densitys is not well defined, we use the entropy of a\ncorresponding piecewise constant density. This results in a constrained optimization problem, where\nthe most homogeneous Dirac mixture approximation f(x,η) is desired that fulfills the given moments\nandmaximizes the entropy.\n33. State of the Art\nFor determining PLparameters of a Dirac mixture density in (2) from a set of PNMmoments, we\nhave to distinguish three cases:\ni)PL<PNM, the overdetermined case, i.e., the number of parameters is smaller than the number\nof moments.\nii)PL=PNM, the fully determined case, i.e., the number of parameters is equal to the number of\nmoments.\niii)PL>PNM, the underdetermined case, i.e., the number of parameters is larger than the number\nof moments.\n3.1.PL<PNM, the Overdetermined Case\nFor the overdetermined case, no literature seems to be available. This case is interesting from a\ntheoretical point of view and it will be discussed in more detail later in this paper. From a practical\npoint of view, it makes sense when for some reason the redundancy in the moments can be used to\nbetter estimate the parameters of the desired Dirac mixture density. On the other hand, as discussed\nabove, many Dirac components are required to cover the interesting parts of the state space, which\nleads to a large amount of parameters. It might then be impractical to calculate more moments than\nparameters.\n3.2.PL=PNM, the Fully Determined Case\nThe fully determined case has been treated a lot in literature in the context of nonlinear Kalman\nfiltering. Moment-based approximations of Gaussian densities are the basis for Linear Regression\nKalman Filters (LRKFs) [ 1]. Examples are the Unscented Kalman Filter (UKF) in [ 2] and its scaled\nversion in [3].\nThe case of maintaining the first two moments received most attention. Moments of up to second\norder can be maintained by a Dirac mixture density with two Dirac components per dimension with\nan optional additional point placed at the mean. This has the huge advantage that the number of\ncomponents only grows linearly with the number of dimensions.\nMaintaining higher-order moments is important for two reasons: First, even for Gaussian densities,\nit makes sense to explicitly consider the higher-order moments as the simplest Dirac mixture (the one\nwith two points per dimension) does not possess the same higher-order moments as a Gaussian density.\nSecond, for non-Gaussian densities higher-order moments are essential for capturing asymmetry,\nmultimodality, and so forth.\nThird-order moments are considered in [ 4]. A Dirac mixture density with 2 N+ 2 weighted\ncomponents is designed that maintains moments of up to second order and, in addition, minimizes the\nthird-order moments. Minimizing the third-order moments is motivated by an underlying Gaussian\ndensity, but is not useful for general densities, where asymmetries can lead to nonzero third-order\nmoments.\nUnder several strong assumptions, Dirac mixture densities with prescribed higher-order moments\nhave been derived in [ 5]. Assumptions include a placement of components on the coordinate axes only\nand symmetric densities, so that all odd moments are set to zero. For the actual derivations, Gaussian\ndensitys were assumed and the point sets were limited to 4 N+ 1 and 6N+ 1 samples with Nthe\nnumber of dimensions.\n[6] proposes two methods for constructing scalar Dirac mixture densities with arbitrary first\nthree moments. The first method is a direct approach based on solving for the parameters of a Dirac\nmixture density with three weighted components given the first three moments under certain symmetry\nconditions. Existence of a solution is not guaranteed. The second method is an indirect approach,\nwhere a Gaussian mixture density with two components is matched to the given moments, where\ntwo degrees of freedom remain to be set by the user. In a second step, two Dirac mixture densities\nwith three components are calculated matching the first two moments of the individual Gaussian\n4components of the Gaussian mixture density. This results in a final Dirac mixture density with six\ncomponents matching the given three moments.\nDirac mixture approximation of circular probability density functions analogous to the UKF\nfor linear quantities is introduced in [ 7] for the von Mises distribution and the wrapped Normal\ndistribution. Based on a closed-form expression for matching the first circular moment, three Dirac\ncomponents are systematically placed by exploiting symmetry. In [ 8], a closed-form solution is derived\nfor a symmetric wrapped Dirac mixture density with five components based on matching the first\ntwo circular moments. This Dirac mixture approximation of continuous circular probability density\nfunctions has already been applied to sensor scheduling based on bearings-only measurements [ 9]. The\nresults have also been used to perform recursive circular filtering for tracking an object constrained to\nan arbitrary one-dimensional manifold in [10].\n3.3.PL>PNM, the Underdetermined Case\nWe will now consider the case of more parameters than given moments. As the solution of this\ninverse problem is not unique, it either requires more information about the underlying density to be\nreconstructed or assumptions on the desired Dirac mixture density. In either case, we will perform\nregularization to guarantee a unique solution with the desired properties. We will consider prior\ninformation in two different ways: Either the full density is given or we just know that the underlying\ntrue density is smooth.\n3.3.1. Full Density Available\nWhen the full underlying true density is given, the most basic problem is to not maintain any\nmoment. Regularization is performed by minimizing a distance between the underlying true density\nand its Dirac mixture approximation. As distances between continuous densities and discrete densities\non continuous domains are difficult to define, the densities are typically transformed to a different\nrepresentation before the distance computation is performed.\nMethods for Dirac mixture approximation of scalar continuous densities based on a distance\nbetween cumulative distributions with no moment constraint are introduced in [ 11,12] for a given but\narbitrary number of components. An algorithm for sequentially increasing the number of components\nis given in [13] and applied to recursive nonlinear prediction in [14].\nFor arbitrary multi-dimensional Gaussian densities, Dirac mixture approximations are systemati-\ncally calculated in [ 15]. The comparison of densities is performed by comparing probability masses\nunder kernels of arbitrary location and size. For this purpose, the so called Localized Cumulative\nDistribution (LCD) is introduced in [ 16]. A modified Cram´ er-von Mises distance is then defined based\non the LCDs. For the case of standard normal distributions with a subsequent transformation, a more\nefficient method is given in [17].\nFor multidimensional densities with given mean and given variances in every dimension, a method\nfor placing an arbitrary number of Dirac components along the coordinate axes is introduced in [ 18].\nThe multi-dimensional problem is broken down into one-dimensional problems that are solved by\nminimizing the distance between cumulative distributions given the two moment constraints.\nMultidimensional Dirac mixture approximations of arbitrary densities with an arbitrary component\nplacement and arbitrary higher-order moment constraints are calculated in [ 19]. Compared to [ 15], a\nfaster but suboptimal distance comparison is used. Instead of comparing the probability masses on all\nscales as in [ 15], repulsion kernels are introduced to assemble an induced kernel density and perform\nthe comparison of the true density with its Dirac mixture approximation. This method is adapted\nto Gaussian densities in [ 20], where a closed-form expression for the distance measure is derived. In\naddition, a randomized optimization method is employed instead of a quasi-Newton method. This has\nthe advantage of being easier to implement with only a slight decrease in performance. An approach\nsimilar to the one proposed in [ 19] has been derived for Dirac mixture approximation of circular\nprobability density functions with an arbitrary number of Dirac components in [21].\n53.3.2. Smoothness Constraint\nWhen it is only known that the underlying density is a smooth continuous density, the first idea\nthat might come to mind is to use an indirect approach. In a first step, a continuous density with the\ndesired moments is found. This can be any smooth parametric density from the exponential family\nor from a mixture family such as a Gaussian mixture density. In a second step, a Dirac mixture\napproximation of this continuous density is performed. This Dirac mixture approximation can be\nperformed with methods discussed before that calculate the Dirac mixture closest to the given density\nwhile simultaneously maintaining the given moments.\nThe indirect approach has several disadvantages. It is difficult to take care of the given smoothness\ncondition by finding a parametric continuous density first as this includes finding both an appropriate\ntype of density with an appropriate structure and appropriate parameters. This step will most likely\nintroduce unwanted artifacts. In addition, the approximation becomes unnecessarily complicated as\nwe now have to solve two moment problems, a moment problem for the continuous density in the first\nstep and a moment problem for its Dirac mixture approximation in the second step.\nWe prefer a direct approach, where the smoothness constraint is fulfilled by finding the most\nhomogeneously distibuted Dirac mixture approximation under the given moment constraints. To the\nauthor’s knowledge, no solution to this problem exists for the case of multi-dimensional densities with\nan arbitrary placement of Dirac components.\n3.4. Contribution of this Paper\nWe consider the finite moment problem of calculating parameters of a Dirac mixture density with\na given number of components and prescribed moments. The true underlying density is unknown. We\nfocus on redundant problems, where the number of parameters is (much) larger than the number of\ngiven moment constraints. This is an underdetermined problem with an infinite solution set, so that a\nregularizer is required to exploit redundancy. Here, we desire a Dirac mixture density with the most\nhomogeneous coverage under the given moment constraints.\nFor regularization, the entropy of the Dirac mixture density could be used. However, the Shannon\nentropy is not well defined for discrete densities on continuous domains. Here, we propose to use\nthe entropy of the corresponding maximum entropy piecewise constant density approximation. This\napproximation has a nice interpretation and, for given Dirac components, is given as the solution of a\nconvex optimization problem with linear inequality constraints. Regularization is now performed by\nselecting the components of the Dirac mixture density in such a way that the entropy of this piecewise\nconstant density approximation is maximized.\nThe remainder of this paper is structured as follows. The piecewise constant density used for\nguaranteeing a homogeneous coverage of the final Dirac mixture density is introduced in Sec. 4.\nCalculating the Dirac mixture density with given moments and homogeneous coverage is discussed in\nSec. 5. Implementation details are given in Sec. 6. An evaluation is conducted in Sec. 7. Conclusions\nare drawn in Sec. 8.\n4. Piecewise Constant Density Approximation\nIn this section, we derive a piecewise constant approximation of the given Dirac mixture. Its\nparameters are calculated in such a way that the Shannon entropy is maximized. We now define the\nspecific form of piecewise constant density used in this paper.\nDefinition 4.1 (Piecewise constant density) .We define a piecewise constant density as a mixture\nwithLcomponents\np(x) =L/summationdisplay\ni=1R(x,ˆxi,di),\nwhere each component R(x,ˆxi,di),i= 1,...,L is constant within a sphere of radius diand given by\nR(x,ˆxi,di) =/braceleftBigg\nhifor/bardblx−ˆxi/bardbl≤di\n0 elsewhere,\n6withdi>0 and the constant heights hifor each component to be determined. In addition, we desire\nthat the components are disjoint according to\ndi+dj</bardblˆxi−ˆxj/bardbl (7)\nholds for all i= 1,...,L ,j= 1,...,L withi/negationslash=j.\nRemark 4.1.The diameters di,i= 1,...,L will be collected in a vector d=/bracketleftBig\nd1,...,dL/bracketrightBigT.\nRemark 4.2.By exploiting symmetry, (7) needs to be checked only for i= 1,...,L−1,j=i+1,...,L .\nThis results in a total of ( L+ 1)L/2 inequality constraints.\nWhen the piecewise constant density is used as a representation of a given Dirac mixture with\nweightswi,i= 1,...,L , we can calculate appropriate values for the constant heights hi.\nLemma 4.1. The constant height for each component hi,i= 1,...,L is given by\nhi=wi\nVN(di).\nwithVN(.)the volume of an N-dimensional hyper-sphere given by\nVN(d) =πN\n2\nΓ/parenleftBig\nN\n2+ 1/parenrightBigdN.\nProof. As the components of the piecewise constant density p(x) according to Definition 4.1 are\ndisjoint, it is possible to treat the individual components separately. We require that the probability\nmass of the individual component iis equal to the mass of the corresponding Dirac component, which\nis written as /integraldisplay\nRNR(x,ˆxi,di)dx=wi.\nBy evaluating the left-hand-side, we obtain an integral over the N-dimensional hyper-sphere SN(di)\nwith radius dias\n/integraldisplay\nSN(di)R(x,ˆxi,di)dx=/integraldisplay\nSN(di)hidx\n=hi/integraldisplay\nSN(di)dx\n=hi·VN(di) =wi,\nwhich concludes the proof.\nWe are now interested in piecewise constant densities that are as homogeneously distributed as\npossible under certain constraints. For that purpose, we will maximize its Shannon entropy.\nLemma 4.2. The Shannon entropy h(p)of the piecewise constant density p(.)defined in Definition 4.1\nis given by\nh(p) =cN−L/summationdisplay\ni=1wilog/parenleftBigg\nwi\ndN\ni/parenrightBigg\n,\nwherecNis a constant depending on the number of dimensions N.\nProof. The Shannon entropy of the piecewise constant density p(x) according to Definition 4.1 is\ndefined as\nh(p) =E{−log(p)}=−/integraldisplay\nRNp(x) log (p(x))dx.\n7Again, as the components of the piecewise constant density p(x) are disjoint, we can exchange\nsummation and integration, which gives\nh(p) =−L/summationdisplay\ni=1/integraldisplay\nRNR(x,ˆxi,di) log (R(x,ˆxi,di))dx\nor\nh(p) =−L/summationdisplay\ni=1/integraldisplay\nSNR(x,ˆxi,di) log (R(x,ˆxi,di))dx\n=−L/summationdisplay\ni=1/integraldisplay\nSNhilog(hi)dx\n=−L/summationdisplay\ni=1hilog(hi)/integraldisplay\nSN(di)dx\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nVN(di).\nWithhifrom Lemma 4.1, we obtain\nh(p) =−L/summationdisplay\ni=1wilog/parenleftbiggwi\nVN(di)/parenrightbigg\n=L/summationdisplay\ni=1wi\nlog\nπN\n2\nΓ/parenleftBig\nN\n2+ 1/parenrightBig\n−log/parenleftBigg\nwi\ndN\ni/parenrightBigg\n\n= log\nπN\n2\nΓ/parenleftBig\nN\n2+ 1/parenrightBig\n−L/summationdisplay\ni=1wilog/parenleftBigg\nwi\ndN\ni/parenrightBigg\n,\nwhich gives the desired result.\nLemma 4.3. For a given Dirac mixture according to (2) with weights wiand components ˆxi,i=\n1,...,L , the corresponding maximum entropy piecewise constant density in Definition 4.1 is the\nsolution to the optimization problem\nd= arg max\nd/parenleftBig\nh/parenleftBig\np(η,d)/parenrightBig/parenrightBig\ns.t. di>0 fori= 1,...,L\ndi+dj</bardblˆxi−ˆxj/bardblfori= 1,...,Lwithi/negationslash=j.j= 1,...,L\nRemark 4.3.The optimization (maximization) problem in Lemma 4.3 is characterized by a concave\nobjective function as\n∂h/parenleftBig\np(η,d)/parenrightBig\n∂di=−Nwi\ndi\nfori= 1,...,L and subject to linear inequality constraints. It remains to be investigated whether the\ninequality constraints form a convex set, which would make the optimization problem convex.\nRemark 4.4.In general, the optimization problem in Lemma 4.3 requires a total of L(L+ 1)/2\nlinear inequality constraints: di>0 fori= 1,...,L , which gives Linequality constraints, and\ndi+dj</bardblˆxi−ˆxj/bardblfori= 1,...,L ,j= 1,...,L , andi/negationslash=j, which together with symmetry gives\nanotherL(L−1)/2 inequalities. The scalar case N= 1 is an exception as the positions ˆxi,i= 1,...,L\ncan be ordered1. In that case, we have a total of 2 L−1linear inequality constraints: di>0 for\ni= 1,...,L and ˆxi+di<ˆxi+1−di+1fori= 1,...,L−1.\n1W.l.o.g., we can assume distinct positions, i.e., ˆxi /negationslash=ˆxjfori= 1, . . . , L ,j= 1, . . . , L , and i /negationslash=j. Otherwise, the\nnumber of points would have been reduced accordingly.\n85. Homogeneous Dirac Mixture Approximation with Given Moments\nOur goal is an algorithm for the efficient calculation of a Dirac mixture density with given moments\nand a homogeneous coverage of the state space. We will start with taking a look at the given moments.\nThey will act as constraints for the desired Dirac mixture density.\nGiven moments up to an order Mare stored in the matrix EMfrom (5). These given moments\nwill be denoted as ˜EM. The actual moments of the Dirac mixture density in (2) are just denoted by\nEM, so that our moment constraints can be written as\nEM!=˜EM,\nwhere the left hand side depends on η, i.e., we have EM=EM(η).\nDepending on the number of given moments compared to the number of parameters of the desired\nDirac mixture density, we have to distinguish between three different cases:\nCase 1: In the first case, the number of given moments is larger than the number of parameters.\nNow, the system of nonlinear equations given by\nEM(η)−˜EM!= 0\nis overdetermined as it has more equations than unknowns, i.e., number of parameters or length\nof the parameter vector η. In that case, we can calculate a least-squares solution as\nη= arg max\nη∈S/vextenddouble/vextenddouble/vextenddouble˜EM−EM/vextenddouble/vextenddouble/vextenddouble2\nF,\nwhere/bardbl./bardblFis the Frobenius norm defined for a matrix H∈RM1×M2with elements hij,i=\n1,...,M 1,j= 1,...,M 2as\n/bardblH/bardblF=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtM1/summationdisplay\ni=1M2/summationdisplay\nj=1|hij|2.\nA generalization would be to introduce weighting factors for moments of different orders.\nCase 2: In the second case, the number of given moments is exactly equal to the number of parameters.\nIn this case, we have to solve the system of nonlinear equations given by\nEM(η)−˜EM!= 0.\nfor the parameter vector η. In the case of power moments, the left hand side leads to a system of\nmulti-dimensional higher-order polynomials. Analytic solutions are only available in rare special\ncases, so that numerical root-finding technqies have to be applied. In addition, the solution is\nnot necessarily unique.\nCase 3: The third and last case is the one that we will pursue further. Here, the number of moment\nconstraints is smaller than the number of parameters, so calculating the desired Dirac mixture\ndensity is underdetermined. Just considering the moment constraints would give an infinite\nsolution set for the desired parameter vector η.\nWe will now pursue the third case. In that case, the solution, the parameter vector of the Dirac\nmixture density with the desired moments, is underdetermined. Thus, we have redundancy available\nthat can be exploited, so that we can impose additional constraints on the desired Dirac mixture\ndensity to perform a regularization and to finally end up with a unique solution.\nHere, we want the final Dirac mixture density to be not more informative as already specified by\nthe given moments: It should be as uninformative as possible within the given moment constraints. The\n9density in a Dirac mixture is encoded by both the weights and the “density” of its components, that is\ntheir relative spacing. When we do not want to favor certain regions of the state space in terms of their\ndensity, the Dirac mixture should be as homogeneous as possible in terms of weight distribution and\nlocation distribution. Intuitively, for equally weighted components we somehow desire equal distances\nbetween neighboring components or equivalently equal free spaces around each component. As this is\nnot very precise and does not hold for unequally weighted components, we need a formal definition of\nhomogeneity.\nAs a convenient, effective, and intuitive regularizer we use the corresponding piecewise constant\ndensity for a Dirac mixture density as introduced in Sec. 4. The most homogenous Dirac mixture from\nthe infinite solution set then is defined as the one that maximizes the entropy of the corresponding\npiecewise constant density.\nFor performing the optimization, we again couple the piecewise constant density with the Dirac\nmixture in such a way that the locations ˆxiof the Dirac mixture are the midpoints of the spherical\nsupport for each component of the piecewise constant density. The probability mass wiof the Dirac\nmixture determines the height hiof each component of the piecewise constant density. In contrast to\nSec. 4, now both the diameters diand the locations ˆxiare variables that are simultaneously optimized.\nThe optimization result provides the locations ˆxiof the most homogenous Dirac mixture. The optimal\ndiametersdiof the maximum entropy piecewise constant density are a by-product and can be used\nfor visualization.\nTheorem 5.1. For given moments collected in the moment matrix ˜EM, a Dirac mixture density with\nthe same moments and a corresponding maximum entropy piecewise constant density is the solution\nto the optimization problem\n[η,d] = arg max\nη,d/parenleftBig\nh/parenleftBig\np(η,d)/parenrightBig/parenrightBig\ns.t. EM=˜EM\ndi>0 fori= 1,...,L\ndi+dj−/bardblˆxi−ˆxj/bardbl<0fori= 1,...,Lwithi/negationslash=j.j= 1,...,L\nOf course, compared to the sole optimization of the diameters of the piecewise constant density\nin Sec. 4, this optimization problem now has nonlinear constraints: The equality constraints for\nmaintaining the desired moments are typically nonlinear. Also the inequality constraints for avoiding\ncollisions of the spherical supports of the piecewise constant densities are now nonlinear as the locations\nˆxinow are variables.\nRemark 5.1.It is important to note that all the required statistics such as the moments are directly\ncalculated from the Dirac mixture. They are notcalculated from the corresponding piecewise constant\ndensity as this one solely serves regularization purposes.\n6. Implementation and Complexity\nGiven the algorithm derived in the previous section, our goal is the efficient calculation of a Dirac\nmixture density with given moments and a homogeneous coverage of the state space. Homogeneity is\nachieved by a regularizer that picks out the most homogeneous solution from the infinite solution set\nthat we have in case of more parameters than moment constraint.\n6.1. Symmetric Densities\nOften, more information besides the moments is available about the true density ˜f(x), such as\ninformation on its support or on given symmetries. Even when information of this type is unavailable,\nanalogous assumptions could be made about the approximating density f(x).\n10Here, we consider a special case of symmetric densities, Dirac mixture densities f(x) that are\nsymmetric with respect to their expected value Ef{x}. We only have to specify Lmaster components\nf1,...,fLand implicitly end up with 2 Lcomponents, i.e., f1,...,f 2L, where the master components\nf1,...,fLcontrol slave components fL+1,...,f 2L. The slave components are symmetric copies of the\nmaster components in the sense, that\nˆxi+L−Ef{x}=−(ˆxi−Ef{x})\nholds fori= 1,...,L . The weights wiand the diameters diare simply copied according to wi+L=wi\nanddi+L=difori= 1,...,L .\nExploiting symmetries in this form has two major advantages. First, complexity is reduced as only\nhalf as many variables have to be optimized (this does not depend upon the number of dimensions).\nSecond, the prescribed expected value is automatically maintained without an explicit constraint.\nRemark 6.1.The master components are not confined to specific parts of the state space. They can\nbe located anywhere, which simplifies the implementation.\n6.2. Complexity\nRegularization is performed by maximizing the Shannon entropy of the corresponding piecewise\nconstant density, which requires complying with a number of constraints quadratic in the number of\nDirac components. We will now take a look at the complexity, especially the number of constraints to\nmaintain. The number of equality constraints (the moment constraints) is prespecified and for a given\norderMbounded by (6). For the inequality constraints, we have Llinear positivity constraints for the\nradiidi,i= 1,...,L of the piecewise constant density and ( L−1)L/2 nonlinear collision constraints.\nThe latter ones are the most critical and will be investigated further, where we distinguish the scalar\nor 1-dimensional case and the N-dimensional case with N > 1.\nIn the scalar case, the number of collision constraints is reduced by maintaining an ordered list of\nDirac components. Then, only the distances between neighbors have to be considered, which reduces\nthe number of collision constraints from ( L−1)L/2 toL−1.\nFor the general multi-dimensional case with N > 1, the number of collision constraints does not\ndepend upon the dimension. However, it depends quadratically upon the number of components. For\na small number of components, say up to 20, this poses no problem. For more components, however,\nthe number of constraints needs to be reduced.\nReducing the number of collision constraints while still guaranteeing a correct solution will be\npursued in a follow-up paper. The key is to exploit the fact that simpler constraints can be devised\nto check for collisions of the support spheres onto the coordinate axes. This is much simpler and\nleads to fewer constraints. Non-colliding projections are a sufficient, but not a necessary, condition for\nnon-colliding supports. Based on this insight, expensive collision constraints have to be considered for\nfar fewer components. Details are given in the conclusions.\n7. Evaluation\nWe will now demonstrate the performance of the proposed Dirac mixture approximation method\nby some examples. One-dimensional and two-dimensional densities will be considered.\nRemark 7.1.It is important to note, that in all cases where we consider underlying continuous densities,\nthese are only used for generating the moments. They are not known to the approximation methods!\nFor comparison, we employ a solver that directly finds a root of the underdetermined system of\nequations given by the moment constraints. We use a Levenberg-Marquardt method for that purpose,\nthat we from now on call Levenberg-Marquardt Dirac mixture approximation method. Matlab provides\nan implementation by calling fsolve with the appropriate options. This solver does neither provide\nunique nor reproducible results.\nIn the simulations, both optimization methods, the Levenberg-Marquardt Dirac mixture approx-\nimation method and the proposed maximum entropy Dirac mixture approximation method, are\ninitialized with a random parameter vector ηdrawn from a standard normal distribution.\n11x →f(x) →Densities: True and Dirac mixture approximation, L=6, M=2\n−4−3−2−1 0 1 2 3 400.20.40.60.81\nx →F(x) →Distributions: True and Dirac mixture approximation, L=6, M=2\n−4−3−2−1 0 1 2 3 400.20.40.60.81\nx →f(x) →Densities: True and Dirac mixture approximation, L=6, M=2\n−4−3−2−1 0 1 2 3 400.20.40.60.81\nx →F(x) →Distributions: True and Dirac mixture approximation, L=6, M=2\n−4−3−2−1 0 1 2 3 400.20.40.60.81Figure 2: Comparison of two Dirac mixture approximation methods for an underlying Gaussian density. (Top row)\nResult of solving the underdetermined moment constraints with the Levenberg-Marquardt Dirac mixture approximation\nmethod. (Bottom row) Result of the proposed maximum entropy Dirac mixture approximation method. (Left column)\nComparison of the densities, where the (unknown) underlying Gaussian densities are shown in yellow and the Dirac\nmixture densities in purple. (Right column) Comparison of the cumulative distributions, where the (unknown) underlying\nGaussian distribution is shown in yellow and the distributions of the Dirac mixtures in purple.\n7.1. Examples for the One-dimensional Case\nWe begin with the simplest case of generating moments from a Gaussian density that are then used\nfor characterizing a Dirac mixture density. For a standard normal distribution, we calculate the first\ntwo moments e1,e2. Then, we employ the Levenberg-Marquardt Dirac mixture approximation method\nand the maximum entropy Dirac mixture approximation method to find Dirac mixture densities with\nL= 6 components having exactly these moments. A comparison of densities and distributions is\nshown in Fig. 2, where the top two figures show the result of the Levenberg-Marquardt Dirac mixture\napproximation. This is just one representative result, as the solution is not unique and changes for\nevery optimization performed. The bottom row in Fig. 2 shows the result of the maximum entropy\nDirac mixture approximation. Here, the coverage is much more homogeneous. In addition, it becomes\nclear that it comes close to the underlying Gaussian as a maximum entropy solution is considered\nand the Gaussian is the continuous density with the highest entropy given a certain variance. This\nconvergence becomes even clearer when taking a look at Fig. 3. When the number of Dirac components\nincreases, in that case to L= 10 andL= 15, the generated Dirac mixture converges to the underlying\nGaussian although this density is not known to the algorithm .\nA similar evaluation is now performed by using moments obtained from a Gaussian mixture\ndensity with two components, weights w1= 0.4,w2= 0.6, meansm1=−1.5,m1= 1.5, and standard\ndeviationsσ1=σ2= 0.7. Moments e0= 1,e1,...,e 4up to fourth order are calculated according to\nthe appendix and used for generating a Dirac mixture with these moments. Fig. 4 shows the results\n12x →F(x) →Distributions: True and Dirac mixture approximation, L=10, M=2\n−4−3−2−1 0 1 2 3 400.20.40.60.81\nx →F(x) →Distributions: True and Dirac mixture approximation, L=15, M=2\n−4−3−2−1 0 1 2 3 400.20.40.60.81Figure 3: Dirac mixture approximation for an underlying Gaussian density with an increasing number of components L.\nOnly moments up to order M= 2 are maintained. (left) L= 10. (Right) L= 15. The Dirac mixture density quickly\napproaches the underlying Gaussian, although this density is not known to the approximation method.\nforL= 10 components, where the top row shows a comparison of densities and distributions for\nDirac mixtures obtained with Levenberg-Marquardt Dirac mixture approximation. Again, this is\nonly one possible result as the solution is not unique. The bottom row shows the result obtained\nwith maximum entropy Dirac mixture approximation, which is very homogeneous and close to the\nunderlying Gaussian mixture density in terms of its distribution.\nFig. 5 then shows that, for M= 6 moments, the maximum entropy Dirac mixture approximation\nquickly converges to the underlying Gaussian mixture density as demonstrated for L= 15 andL= 25.\n7.2. Examples for the Two-dimensional Case\nFor the two-dimensional case N= 2, we consider Dirac mixture densities maintaining moments up\nto second order, i.e., e00= 1 (the normalization constant), e01,e10,e11,e02, ande20are prespecified.\nFor the specific choice of moments corresponding to the axis-aligned normal distribution e00= 1,\ne01= 0,e10= 0,e11= 0,e02= 3, ande20= 1, we obtain the results shown in Fig. 6. Here, symmetry\nis enforced as introduced in Subsec. 6.1. As a result, only 20 master components are optimized and 20\nslave components follow accordingly.\nIt is obvious from Fig. 6 that the resulting Dirac mixture densities converge to a Gaussian density\nwith the given moments when the number of components increases. Again, the underlying density\nshape, in this case the Gaussian, is not known to the Dirac mixture approximation method.\n8. Conclusion\nThis paper provides an efficient algorithm for calculating Dirac mixture densities with given\nmoments and a homogeneous coverage of the state space. We focus on the case of fewer moment\nconstraints than parameters. Ensuring a unique solution and exploiting the redundancy by optimizing\nthe component arrangement is performed by regularization with respect to a corresponding piecewise\nconstant density. The most homogeneous Dirac mixture density is obtained by maximizing the entropy\nof this piecewise constant density.\n8.1. Applications\nThe proposed maximum entropy Dirac mixture approximation method will be used for generalizing\nthe Progressive Gaussian Filter introduced for generative system models in [ 22] and for systems with\ngiven likelihoods in [ 23]. So far, the Progressive Gaussian Filter maintains moments up to second\norder when progressively performing a measurement update. In addition, a Gaussian assumption is\n13x →f(x) →Densities: True and Dirac mixture approximation, L=10, M=4\n−4−3−2−1 0 1 2 3 400.10.20.30.40.50.6\nx →F(x) →Distributions: True and Dirac mixture approximation, L=10, M=4\n−4−3−2−1 0 1 2 3 400.20.40.60.81\nx →f(x) →Densities: True and Dirac mixture approximation, L=10, M=4\n−4−3−2−1 0 1 2 3 400.10.20.30.40.50.6\nx →F(x) →Distributions: True and Dirac mixture approximation, L=10, M=4\n−4−3−2−1 0 1 2 3 400.20.40.60.81Figure 4: Comparison of two Dirac mixture approximation methods for an underlying Gaussian mixture density. (Top row)\nResult of solving the underdetermined moment constraints with the Levenberg-Marquardt Dirac mixture approximation\nmethod. (Bottom row) Result of the proposed maximum entropy Dirac mixture approximation method. (Left column)\nComparison of the densities, where the (unknown) underlying Gaussian mixture densities are shown in yellow and the\nDirac mixture densities in purple. (Right column) Comparison of the cumulative distributions, where the (unknown)\nunderlying Gaussian mixture distributions are shown in yellow and the distributions of the Dirac mixtures in purple. In\nboth cases, moments up to order M= 4 are maintained and the Dirac mixture comprises L= 10 components.\nmade. Using the proposed maximum entropy Dirac mixture approximation, higher-order moments\nwill be propagated without any density assumption.\nThe piecewise constant density for a given Dirac mixture density as derived in Sec. 4 might also\nbe useful by itself in other contexts than providing a convenient density for plug-in estimation of the\nentropy of a Dirac mixture density.\n8.2. Extensions\nNow, we will discuss several extensions to the basic algorithm described in this paper. First, we\ndiscuss optimizing the weights in addition to the locations of the components of the considered Dirac\nmixture density. Second, we focus on the complexity and how to decrease it. Third, we consider spaces\ndifferent from the Euclidean space RN.\nWeights. In this paper, we focused on optimizing the locations of a Dirac mixture density only. Also\noptimizing the weights gives another L−1 degrees of freedom for Lcomponents (as the weights have\nto sum up to one). This gives the advantage of potentially maintaining more moments for the same\nnumber of components.\nThe downside of optimizing the weights is obvious: Now there are components of different\nimportance. Components with a small weight are almost negligible and do not carry much information\n14x →F(x) →Distributions: True and Dirac mixture approximation, L=15, M=6\n−4−3−2−1 0 1 2 3 400.20.40.60.81\nx →F(x) →Distributions: True and Dirac mixture approximation, L=25, M=6\n−4−3−2−1 0 1 2 3 400.20.40.60.81Figure 5: Dirac mixture approximation for an underlying Gaussian mixture density with an increasing number of\ncomponents L. Moments up to order M= 6 are maintained. (left) L= 15. (Right) L= 25. The Dirac mixture density\nquickly approaches the underlying Gaussian mixture, although this density is not know to the approximation method.\nalthough optimizing them is as costly as for large components. This is aggravated when components\nare fed through a nonlinear system in order to calculate the output density. In that case, the weights\nremain unchanged and the output weights are identical to the input weights. As the computational\ncost of propagating components does not depend on the weight, this implies that lots of computational\npower is spent for small output components with questionable usefulness.\nThe disadvantage of unequally weighted Dirac mixture densities becomes even more obvious\nwhen considering multiplication with a likelihood function in a Bayesian filter step. Already small\ncomponents can potentially be weighted down even more making them useless, while for equally\nweighted components there is more leeway before components degenerate.\nComplexity. Let us briefly sum up the complexity of the algorithm described in this paper: For the\nscalar case, the complexity of the optimization problem is low. We have to maintain M+ 1 moment\nconstraints (when considering moments up to order M) and 2L−1 inequality constraints to optimize\nforLlocation parameter of the desired Dirac mixture density. For the multi-dimensional case, we face\ntwo problems. First, the number of equality constraints corresponding to the number of moments up\nto orderMquickly increases with the number of dimensions Nso that calculating these moments for\na Dirac mixture density soon becomes intractable. Second, the number of inequality constraints grows\nas (L+ 1)L/2 with the number of components.\nFor the multi-dimensional case, the first problem can be coped with by considering only the most\nrelevant moments, which depends upon the application. The second problem, that is the quadratic\ngrowth of the inequality constraints, will be attacked by a two-step constraint hierarchy. The first step\nuses more conservative dimension-wise collision constraints, which results in N(L−1) constraints. In\nthe second step, multi-dimensional constraints are only assembled for the remaining Dirac components\nthat need further attention. This is expected to result in an additional O(L), saykL, constraints.\nTogether with the additional positivity constraint for the radii di,i= 1,...,L , we obtain a number\nof aboutkL+N(L−1) +Lconstraints. Hence, we trade an algorithm with a total number of\n(L+ 1)L/2 constraints for an algorithm with a number of kL+N(L−1) +Lconstraints, which is\nmore efficient when N <L (L−1−2k)/(2(L−1)). This is already the case for k= 1,N= 2, and\nL= 7. Implementation of these hierarchical constraints, however, is a challenge as the number of\nconstraints varies during runtime of the optimization procedure. Available optimization routines do\nnot seem be able to cope with a varying number of constraints.\n15x1 →x2 →Dirac mixture approximation, L=16, M=2\n−6−4−2 0 2 4 6−4−3−2−101234\nx1 →x2 →Dirac mixture approximation, L=20, M=2\n−6−4−2 0 2 4 6−4−3−2−101234\nx1 →x2 →Dirac mixture approximation, L=30, M=2\n−6−4−2 0 2 4 6−4−3−2−101234\nx1 →x2 →Dirac mixture approximation, L=40, M=2\n−6−4−2 0 2 4 6−4−3−2−101234Figure 6: Plots of Dirac mixture densities maintaining moments up to second order M= 2 with different numbers of\ncomponents. (top left) L= 16. (top right) L= 20. (bottom left) L= 30. (bottom right) L= 40. Symmetry is enforced,\nso that only 20 master components are optimized.\nAlternative Spaces. The techniques presented in this paper for the case that no underlying density is\navailable, will be generalized to different Polish spaces X, especially to periodic spaces such as the\nunit circleS1as it has been done in [21] for known circular probability density functions.\nReferences\n[1]T. Lefebvre, H. Bruyninckx, and J. De Schutter, “The Linear Regression Kalman Filter,” in Nonlinear Kalman\nFiltering for Force-Controlled Robot Tasks , ser. Springer Tracts in Advanced Robotics, 2005, vol. 19.\n[2]S. Julier, J. Uhlmann, and H. F. Durrant-Whyte, “A New Method for the Nonlinear Transformation of Means and\nCovariances in Filters and Estimators,” IEEE Transactions on Automatic Control , vol. 45, no. 3, pp. 477–482, Mar.\n2000.\n[3]S. J. Julier, “The Scaled Unscented Transformation,” in Proceedings of the 2002 IEEE American Control Conference\n(ACC 2002) , vol. 6, Anchorage, Alaska, USA, May 2002, pp. 4555– 4559.\n[4]S. Julier and J. Uhlmann, “Reduced Sigma Point Filters for the Propagation of Means and Covariances Through\nNonlinear Transformations,” in Proceedings of the 2002 American Control Conference. , vol. 2, 2002, pp. 887–892\nvol.2.\n[5]D. Tenne and T. Singh, “The Higher Order Unscented Filter,” in Proceedings of the 2003 IEEE American Control\nConference (ACC 2003) , vol. 3, Denver, Colorado, USA, Jun. 2003, pp. 2441–2446.\n[6]O. Straka, J. Dunik, and M. Simandl, “Measures of Non-Gaussianity in Unscented Kalman Filter Framework,” in\nProceedings of the 17th International Conference on Information Fusion (Fusion 2014) , Salamanca, Spain, Jul.\n2014.\n[7]G. Kurz, I. Gilitschenski, and U. D. Hanebeck, “Recursive Nonlinear Filtering for Angular Data Based on Circular\nDistributions,” in Proceedings of the 2013 American Control Conference (ACC 2013) , Washington D. C., USA, Jun.\n2013.\n16[8]——, “Deterministic Approximation of Circular Densities with Symmetric Dirac Mixtures Based on Two Circular\nMoments,” in Proceedings of the 17th International Conference on Information Fusion (Fusion 2014) , Salamanca,\nSpain, Jul. 2014.\n[9]I. Gilitschenski, G. Kurz, and U. D. Hanebeck, “Circular Statistics in Bearings-only Sensor Scheduling,” in\nProceedings of the 16th International Conference on Information Fusion (Fusion 2013) , Istanbul, Turkey, Jul. 2013.\n[10] G. Kurz, F. Faion, and U. D. Hanebeck, “Constrained Object Tracking on Compact One-dimensional Manifolds\nBased on Directional Statistics,” in Proceedings of the Fourth IEEE GRSS International Conference on Indoor\nPositioning and Indoor Navigation (IPIN 2013) , Montbeliard, France, Oct. 2013.\n[11] O. C. Schrempf, D. Brunn, and U. D. Hanebeck, “Density Approximation Based on Dirac Mixtures with Regard to\nNonlinear Estimation and Filtering,” in Proceedings of the 2006 IEEE Conference on Decision and Control (CDC\n2006) , San Diego, California, USA, Dec. 2006.\n[12] ——, “Dirac Mixture Density Approximation Based on Minimization of the Weighted Cram´ er-von Mises Distance,”\ninProceedings of the 2006 IEEE International Conference on Multisensor Fusion and Integration for Intelligent\nSystems (MFI 2006) , Heidelberg, Germany, Sep. 2006, pp. 512–517.\n[13] U. D. Hanebeck and O. C. Schrempf, “Greedy Algorithms for Dirac Mixture Approximation of Arbitrary Probability\nDensity Functions,” in Proceedings of the 2007 IEEE Conference on Decision and Control (CDC 2007) , New\nOrleans, Louisiana, USA, Dec. 2007, pp. 3065–3071.\n[14] O. C. Schrempf and U. D. Hanebeck, “Recursive Prediction of Stochastic Nonlinear Systems Based on Optimal\nDirac Mixture Approximations,” in Proceedings of the 2007 American Control Conference (ACC 2007) , New York,\nNew York, USA, Jul. 2007, pp. 1768–1774.\n[15] U. D. Hanebeck, M. F. Huber, and V. Klumpp, “Dirac Mixture Approximation of Multivariate Gaussian Densities,”\ninProceedings of the 2009 IEEE Conference on Decision and Control (CDC 2009) , Shanghai, China, Dec. 2009.\n[16] U. D. Hanebeck and V. Klumpp, “Localized Cumulative Distributions and a Multivariate Generalization of the\nCram´ er-von Mises Distance,” in Proceedings of the 2008 IEEE International Conference on Multisensor Fusion and\nIntegration for Intelligent Systems (MFI 2008) , Seoul, Republic of Korea, Aug. 2008, pp. 33–39.\n[17] I. Gilitschenski and U. D. Hanebeck, “Efficient Deterministic Dirac Mixture Approximation,” in Proceedings of the\n2013 American Control Conference (ACC 2013) , Washington D. C., USA, Jun. 2013.\n[18] M. F. Huber and U. D. Hanebeck, “Gaussian Filter based on Deterministic Sampling for High Quality Nonlinear\nEstimation,” in Proceedings of the 17th IFAC World Congress (IFAC 2008) , vol. 17, no. 2, Seoul, Republic of Korea,\nJul. 2008.\n[19] U. D. Hanebeck, “Kernel-based Deterministic Blue-noise Sampling of Arbitrary Probability Density Functions,” in\nProceedings of the 48th Annual Conference on Information Sciences and Systems (CISS 2014) , Princeton, New\nJersey, USA, Mar. 2014.\n[20] ——, “Sample Set Design for Nonlinear Kalman Filters viewed as a Moment Problem,” in Proceedings of the 17th\nInternational Conference on Information Fusion (Fusion 2014) , Salamanca, Spain, Jul. 2014.\n[21] U. D. Hanebeck and A. Lindquist, “Moment-based Dirac Mixture Approximation of Circular Densities (to appear),”\ninProceedings of the 19th IFAC World Congress (IFAC 2014) , Cape Town, South Africa, Aug. 2014.\n[22] U. D. Hanebeck, “PGF 42: Progressive Gaussian Filtering with a Twist,” in Proceedings of the 16th International\nConference on Information Fusion (Fusion 2013) , Istanbul, Turkey, Jul. 2013.\n[23] J. Steinbring and U. D. Hanebeck, “Progressive Gaussian Filtering Using Explicit Likelihoods,” in Proceedings of\nthe 17th International Conference on Information Fusion (Fusion 2014) , Salamanca, Spain, Jul. 2014.\nAppendix A. Moments of Scalar Gaussian Density\nWhen mean and standard deviation of a Gaussian random variable are given, allhigher–order\nmoments and central moments can be deduced analytically. The central moments are given by\nCi=Ef/braceleftBig\n(x−m)i/bracerightBig\n=\n\ni−1/productdisplay\nj=1\njoddjσiieven,\n0iodd,\nthe moments by\nEi=Ef/braceleftBig\nxi/bracerightBig\n=i/summationdisplay\nk=0/parenleftBigg\ni\nk/parenrightBigg\nCi−kmk\nwithC0(the zeroth central moment, the area under the density) is defined as C0= 1.\nExample Appendix A.1 (First Moments and Central Moments of Gaussian Density) .The first\neight moments Ei,i= 1,..., 8, and central moments Ci,i= 1,..., 8, of a Gaussian density with\n17meanmand standard deviation σare given by\nC1= 0,\nC2=σ2,\nC3= 0,\nC4= 3σ4,\nC5= 0,\nC6= 15σ6,\nC7= 0,\nC8= 105σ8,E1=m ,\nE2=m2+σ2,\nE3=m3+ 3mσ2,\nE4=m4+ 6m2σ2+ 3σ4,\nE5=m5+ 10m3σ2+ 15mσ4,\nE6=m6+ 15m4σ2+ 45m2σ4+ 15σ6,\nE7=m7+ 21m5σ2+ 105m3σ4+ 105mσ6,\nE8=m8+ 28m6σ2+ 210m4σ4+ 420m2σ6+ 105σ8.\nAppendix B. Moments of Mixture\nWe consider scalar mixtures of the form\nf(x) =P/summationdisplay\nk=1wkfk(x)\nwith\nwk≥0\nfork= 1,...,P and\nP/summationdisplay\nk=1wk= 1.\nWhen the moments of the individual densitiesfk(.) in the mixture are given by E(k)\nifork= 1,...,P ,\nthe individual moments can now be added up to the moments of the mixture denoted by EM\ni\nEM\ni=P/summationdisplay\nk=1wkE(k)\ni.\nFinally, central moments of the mixture are obtained as\nCM\ni=i/summationdisplay\nj=0/parenleftBigg\ni\nj/parenrightBigg\nEM\ni−j/parenleftBig\n−EM\n1/parenrightBigj.\n18"    },    {        "title": "1409.2110v1.Optoelectronically_probing_the_density_of_nanowire_surface_trap_states_to_the_single_state_limit.pdf",        "content": "1 \n Optoelectronically p robing  the density of nanowire surface trap state s to the \nsingle state  limit   \nYaping Dan  \nUniversity of Michigan – Shanghai Jiao Tong University Joint Institute, Shanghai, China 200240  \nCorrespondence should be address ed to: yaping.dan@sjtu.edu.cn  \nDue to the large surface -to-volume ratio , surface trap states play a  dominant role in the \noptoelectronic properties of nanoscale devices (1-6). Understanding the surface trap states \nallows us to properly engineer the device surfaces for better performance. But \ncharacterization of  surface trap states at nanoscale has been a formidable challenge  using the \ntraditional  capacitive  techniques based on  metal -insulator -semiconductor (MIS) structures (7) \nand deep level transi ent spectroscopy (DLTS) (8-11). Here, we demonstrate a simple but \npowerful optoelectronic method to probe the density of nanowire surface trap states  to the \nlimit of a single trap state . Unlike traditional capacitive t echniques  (Fig1a) , in this method w e \nchoose to tune t he quasi -Fermi level across the bandgap  of a silicon nanowire \nphotoconductor , allow ing for capture and emission of photogenerated charge carriers by \nsurface trap states  (Fig1b) . The experimental data sho w that the energy density of nanowire \nsurface trap states is in a range from 109cm-2/eV at deep levels to 1012cm-2/eV in the middle \nof the upper half bandgap.  This optoelectronic method allows us to conveniently probe trap \nstates of ultra -scaled nano/quantum devices at extremely high  precision.  \nTo understand this method, let us use p-type semiconductor s as an example. A highly doped p -\ntype semiconductor has an extremely high concentration of holes but a very low concentration \nof electrons. Under illumination, excess electron s and hole s are generated in the conduction  \nand valence band, respectively. At small injection condition,  these excess carriers are negligibly \nlow in concentration  compared to the majority holes, but often orders of magnitude larger than \nthe minority electrons. Consequently, the quasi -Fermi level of electrons  (   ) shifts away \nsignificantly from the Fermi level at equilibrium  (  ), while the quasi -Fermi level o f holes (   ) \nremains nearly unchanged as   , as shown in Fig.1b. In general, t he shift of quasi -Fermi levels \nleads to filling  the trap states  below     (but above   ) with  electrons , and those above     (but \nbelow   ) with holes. Clearly,  for a highly doped p -type semiconductor  at small injection \ncondi tion, only the minority electrons  get involved in the capture -emission process of trap \nstates . For every photogenerated electron captured by trap states,  there is one corresponding \nhole remai ning in the valence band to contribute to photoconduc tance . The shift of     will \nallow a large number of electrons to be trapped if the density of trap states , at deep levels in \nparticular,  is high, leading to the giant gain  in photoconductance that has been widely \nobserved (1-3).  2 \n  \nFigure 1 . Methods for probing the density of surface trap states. (a) Metal -Insulator -Semiconductor (MIS) \ncapacitor. The gate voltage drives the Fermi level to sweep across the bandgap on the Si/SiO 2 interface  to \nallow for capture and emission of charges carriers by surface trap states . (b) Quasi-Fermi level can be tuned \nacross the bandgap by changing the intensity of optical illumination. Δn is the excess electron concentration \nin the conduction band,  and Δnt the excess electron concentration captured by the surface trap states . The \nexcess carriers in the valence band  can be divided into two parts.  One part (Δp 1) corresponds to the trapped \nelectrons Δnt and the other ( Δp2) is equal to Δn. We have Δnt + Δn = Δp 1 +  Δp2. \n \nWhen  the illumination is turned off, the excess electrons in the conduction band  and those \ncaptured by the trap states will all recombine with the excess holes in the valence band \neventually, but the recombination processes are different. For the electrons in the conduction \nband, the recombination occurs almost immediately through surface recombination centers ( ~ \n100 picoseconds  for unpassivated nanow ires(12, 13 )). In contrast, the electrons in the trap \nstates  are first emit ted to the conduction  band at a much lower rate  (milliseconds to tens of \nseconds)  and then recombine with the corresponding  holes .  With this contrast in time  domain , \nwe can decouple these two processes and separately extract  photogenera ted electrons in the \nconduction  band  Δn and those in the trap states Δnt by tuning the modulation frequency of \nillumination . At high frequencies, for instance, Δ nt will not be able to keep up the pace of the \nmodulation, leaving  only Δn to contribute to the photoconductance. The location of     can be \nfound f rom Δn. Differentiating Δnt with respect to     will allow us to obtain  information about \nthe energy density of trap states.  \nFollowing the argument above, w e first cond uct four -probe measurements on a single \nnanowire device  (L=2.4 μm long and 80 nm in diameter)  that is Ohmically contacted  by four \nmicroelectrodes  (Fig.2a) . The nanowire is p-type and the doping concentration is ~1018 cm-3. \nThen w e place the nanowire device under illumination of a monochromatic green light beam (λ \n3 \n ~ 500nm) that is modulated on/off by a mechanical chopper. A lock-in amplifier  is employed to \npick up t he period ic ac photocurrent . The transi ent behavior of the nanowire conductance \nunder the periodic on/off illumination can be described by the red curve in Fig.2b. During the \nfirst half period which is under illumination, the conductance starts with an instantaneous jump \nby σ ph followed by a re latively slow electron capture  process. During the second half period, the \nlight is cut off. The conductance immediately dips down by σ ph due to the fast surface \nrecombination , and then slowly decays by emitting electrons from the surface trap states.  \nAnalyt ically, one period of the red curve   (  ) in Fig. 2b can be described by Eq. (1) below.  \n (  ) {   [     (        (   )     )]                    \n     (          )                                                                … Eq.(1) \nwhere     is the capture time constant ,    the emission time  constant , and the constant s   and   (see the \nsupplementary information  for their expression) are set to ensure that the above equation is continuous. σ T \nis contributed by Δp 1 which is the excess holes left in the valence band after  the corresponding electrons \ncaptured by trap states.  σph is contributed by  Δn and Δp 2 in equilibrium. σ d is the nanowire conductance in \ndark and  ω=2π/T  with T as the chopping period.  \nAccording to advanced mathematics , any periodic function can be decomposed into a n infinite  \nsummation of sine and cosine functions  in the following form , so can eq.(1) : \n (  )   \n  ∑[      (   )       (   )] \n     … eq.(2) \nWhen the signal of above equation is fed into the lock -in amplifier , the amplifier  multipl ies the \nequation  by      (  ) where ω=2πf  and f is the chopping frequency  (=1/T) . All the terms are \nstill in a wave form  after the multiplication, except       (   ) with n=1 which contains  a dc \ncomponent  b1. The internal filters inside the lock -in amplifier will filter out all the wave  term s, \nallow ing for only   , the amplitude of the fundamental term     (  ), to be picked up . But the \nlock-in amplifier is designed to display the root mean square  (RMS) instead of  the amplitude  of \nthis term , meaning that the reading on the amplifier is      √ . After transforming eq.(1) into \nthe form of eq.(2) , we find that    can be expressed  in the following form  (see Section 1  of the \nsupplementary materials ): \n    \n   [ ( \n     \n   )  ( \n     \n   )]  (      )\n … eq. (3)  \nwhere  (   ) [    ][    ]\n[      ][( \n ) \n  ] \nIt is widely known that, for trap states in silicon (8), the electron capture process is significantly \nfaster than the electron emission process . In this case, we assume     , leading to   4 \n      \n [     (  \n   )]\n( \n    ) \n    (      )\n  …  eq. (4)  \n \n \nFigure 2. (a) Diagram of experimental setup. ( b) Transit signal input to the lock -in amplifier. ( c) Nanowire ac \nphotocondance as a function of chopping  frequency  under illumination of 12.2 W/m2. (d) Frequency \ndependent photoconductance under illumination of differe nt light intensity.  \n \nBy fitting      √  to the experimental data in Fig. 2c, we find σph = 0.8 nS and σT = 3.3 nS. The \nphotoconductance σ ph comes from both the photoexcited electrons (Δn in Fig.1b) and their \ncounterpart holes (Δp 2). But the trap -induced photoconductance σ T is only contributed by part \nof the total excess holes ( Δp1) in the valence band , because their corresponding photoexcited  \nelectrons  are captured by the surface trap states.  This fact allows us to conclude that, under \nillumination of a light intensity 12.2 W/m2, the electron quasi -Fermi level     of the nanowire \nshifts from the original location 0.47eV below to 0.23eV above the mid -bandgap energy level   , \n5 \n and that  accordingly, a concentration 3.3 x1015 cm-3 of elec trons is captured by the trap states \nwith in the range of     shift. The conclusion is based on the fact that the hole mobility μ p is \nestimated to be ~30 cm2/Vs which is close to the hole mobilities  of silicon nanowires found in \nliterature (14, 15 ). The electron mobility μ n is assumed to be 3 times of this value . \nTo find the energy density of trap states, we sweep  the quasi -Fermi energy across the bandgap  \nby tuning the illumination intensity . The ac photoconductance as a function of the chopping \nfrequency at different light intensity is illustrated  in Fig.2d . All these curves are fit ted by \n     √   of eq.(4) without any visible deviation, indicating that other effects such as local  \nheating  by illumination  are negligibly small. From the fittings, we find a  set of σph and σT, and \nplot them in Fig.3 a and b, respe ctively . σph is linear with the light intensity, which shows  that \nsmall injection condition is always maintained  in the experiments . That is,  Δn and Δp are both \nnegligibly  small compared to the majority holes, and therefore     remains almost the same as \n  . Base d on this, we conclude that only electrons are involved in the capture and emission \nprocess es of the surface trap states . The linearity of σ ph results in a logarithm dep endence of     \non the illumination intensity , as shown in Fig.3a . It is evident that  a small injection of \nillumination  will shift     over a wide range in the bandgap. If the density of trap states in this \nrange is high, then a large number of carriers will potentially be trapped to induce a high \nphotocurrent gain. This could be achiev ed by properly engineering the device surfaces.  \nFig.3b shows  the trap -induced photoconductance σ T, which is a little nonlinear with the light \nintensity but always remains approximately 4 times of σ ph for the whole range. This \nphotoconductance gain is mainly due to the fact that the trap states on the nanowire surfaces \ncapt ure a large number of electrons.  The same number of holes is left in the valence band to \ncontribute to  this amplified photoconductance. The capture of electrons on the nanowire \nsurfac es will inevitably modulate the energy band of the silicon nanowire, introducing an \nadditional change in photoconductance. Clearly, w e cannot exclude the possibility that this \n“gating effect”  (16) also plays a role in our device.  The nonlinearity of σ T could be due to this \neffect. The “gating effect”  induced by surface trap states has two properties  that need to be \npointed out. First, in time domain its modulation of photoconductance  is similar to  that of  the \ncapture -emission process by the trap states  (Fig.2b ). Second, how strong  the “gating effect”  is \ndepends on the density of surface trap state s, the initial net charge on the device surface and \nthe size of the device  (surface -to-volume ratio) (16), all of which are part of device surface \nproperties . These facts  make  us believe that it is more appropriate to incorporate “the gating \neffect”  as part of the capture -emission mechanism  stated at the beginning of this Letter . By \ndoing so, we can establish a unified model and quantitatively identify the nominal density of \nsurface  trap states  that reflects all the properties of the device surfaces . The photoconductance \ngain induced  by the “gating effect” is equivalent to an amplification of the density of surface \ntrap states.  6 \n  \nFigure 3. (a) Photoconductance σph and quasi -Fermi energy of electrons vs illumination intensity. (b) Trap -\nstates -induced p hotoconductance σT as a functio n of illumination intensity.  (c) Surface concentration of \ntrapped electrons as quasi Fermi level of electrons moves up . Inset: a close -up of the first four data points.  (d) \nCalculated density  of surface trap states of our nanowire device.  \nFrom Fig.3b, we calculate the number of excess holes (Δp 1) that contribute  to the trap -induced \nphotoconductance  σT. The same number of  electron s is captured by the surface trap states of \nthe silicon nanowire. We plot the surface concentration of the trapped electrons  as a function \nof     shown in  Fig.3c. The concentration follow s an exponential  curve . Amazingly, t he first data \npoint (            ) has a concentration almost equal to the unit concentration (1.66 ×108 \ncm-2) when only one electron is trapped on our nanowire surface s (2.4μm long and 80nm in \ndiameter) , as shown in the inset . This indicates that there is one electron captured by the \nsurface trap states . Accordingly,  one counterpart  hole is left in the valence to contribute to the \nphotoconductance σ T in Fig.3b (see Section 2 of the supplementary materials) . The electron \nconcentration jumps up to only 3 times of this unit concentrat ion (3 electrons captured)  after  \n     increases  by more than 110 meV  (>4kT, k the Boltzman n constant and T=300K) to the \nsecond data point . This indicates that the captured electrons follow the Fermi -Dirac distribution \ninstead of Boltzmann distribution. So it is most likely that there is one trap state located below \n7 \n     of the first data point , and that two trap states are located within th e gap between the first \nand second data point. A gap of 110 meV  is wide enough to allow these two trap states to be  \nspaced  far apart so that they can be  filled with electrons one by one. This is the reason why we \nobserve d the quantization in the charge trapping.  At high energy levels, the trap states are \nlocated more closely next to each other . The quantization of charge trapping  becomes \nincreasingly less likely (see the inset).  \nWe calculate  the density  of surface trap st ates  by simply  differentiat ing the surface \nconcentration of trapped electrons over the quasi Fermi energy , as shown in Fig.4d . The trend \nis the same with what is obtained by the C -V characteristics of nanowire MIS capacitors (7), but \nour obtained density  is 2 orders of magnitude more precise. At deep levels, the density is \naccurate since the filling of trap states is quantized. At higher energy levels where the energy \ngaps between trap states are narrower, the density is somewhat overestimated since the \nelectrons have high probabilities to be captured by the states above    . An accurate  solution is \nalways difficult to find since the related equation is ill -defined  (see Section 3  of the \ncomplementary materials for details ). \nIn conclusion , we have demonstrated a simple but powerful method to accuratel y measure the \ndensity of surface trap states at single nanowire level. At room temperature, we observed that \nthe charge trapping by  the trap  states is quantized, indicating that the states are filled with  \nelectrons one by one.  This method allows us to conv eniently probe the density of trap states in \nultra -scaled nano/quantum devices at very high precision , which is not possible by traditional \ncapacitive techniques . It opens up opportunities to develop high -performance nanodevices  by \nengineering the device surfaces . \n \nMethods  \nThe silicon nanowires were synthesized by the typical gold -catalyzed vapor -liquid -solid (VLS) \nprocess in a mixture of H 2-balanced SiH 4 and BH 3. To make contacts to single nanowires, we \nfirst prepared a nanowire  suspension by sonicating the as -grown nanowire samples in \nisopropanol  (IPA) followed by a proper cleaning process (17). A drop of the nanowire suspension \nwas then applied onto a quartz substrate with prefabricated alignment  markers. The existence \nof these markers enables us to locate individual nanowires under optical microscope. Ohmic \ncontacts were made to the individual nanowires by multiple microelectrodes (200nm Pd with \n3nm Cr as the adhesion layer) formed in the process es of photolithography and metallization. \nAfter proper wire -bonding, we placed the dc biased nanowire device under illumination of a \nmonochromatic light beam ( λ ≈ 500 nm) that is modulated ON/OFF periodically by a mechanical \nchopper. The illumination inten sity on the nanowire device is difficult to accurately estimate , 8 \n and are subject to a variation of a few times  in the related figures . But this variation does not \naffect the conclusion we reached in  this Letter. A Labview  script was used to tune the \nmodulation frequency from 200 Hz to 2000 Hz, and the ac photocurrent was picked up by a \nlock-in amplifier. We performed four probe measurements on the nanowire device  to exclude \nthe contact resistance .  \nAcknowledgement s \nWe thank  Prof. Ali Javey for providing us silicon nanowires, and Prof. Abdelmadjid Mesli and Dr. \nSiew Li Tan  for useful discussions . The work reported here is supported by the national “1000 \nYoung Scholar s” program of the Chinese central government, the National Science Foundation \nof China ( 61376001 ) and the “Pujiang Talent Program” of the Shanghai municipal government . \nCompleting financial interests  \nThe author declares no completing financial interests.  \nReferences  \n1. H. Kind, H. Q. Yan,  B. Messer, M. Law, P. D. Yang, Nanowire ultraviolet photodetectors and \noptical switches. Advanced Materials  14, 158 -+ (2002).  \n2. C. Soci  et al. , ZnO nanowire UV photodetectors with high internal gain. Nano Letters  7, 1003 -\n1009 (2007).  \n3. A. Zhang, H. Kim, J. Cheng, Y. H. 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Henini, Deep level transient spectroscopy of gallium arsenide. Deep leve l transient \nspectroscopy of gallium arsenide ,  (1984).  \n10. S. Carapezzi, A. Castaldini, A. Irrera, A. Cavallini, in AIP Conference Proceedings . (2014), vol. 1583, \npp. 263 -267.  \n11. A. Mesli, J. C. Muller, P. Siffert, Deep levels subsisting in ion implanted silicon after various \ntransient thermal annealing procedures. Applied Physics A (Solids and Surfaces)  A31, 147 -152 \n(1983).  \n12. Y. Dan  et al. , Dramatic Reduction of Surface Recombination by in Situ Surface Passivation of \nSilicon Nanowires. Nano Letters  11, 2527 -2532 (2011).  \n13. J. E. Allen  et al. , High -resolution detection of Au catalyst atoms in Si nanowires. Nature \nNanotechnology  3, 168 -173 (2008).  9 \n 14. J. Goldberger, A. I. Hochbaum, R. Fan, P. D. Yang, Silicon vertically integrated nanowire field \neffect tr ansistors. Nano Letters  6, 973 -977 (2006).  \n15. M. T. Bjork, O. Hayden, H. Schmid, H. Riel, W. Riess, Vertical surround -gated silicon nanowire \nimpact ionization field -effect transistors. Applied Physics Letters  90, 3 (2007).  \n16. C.-J. Kim, H. -S. Lee, Y. -J. Cho, K. Kang, M. -H. Jo, Diameter -Dependent Internal Gain in Ohmic Ge \nNanowire Photodetectors. Nano Letters  10, 2043 -2048 (2010).  \n17. Y. Dan, Y. Cao, T. E. Mallouk, A. T. Johnson, S. Evoy, Dielectrophoretically assembled polymer \nnanowires for gas sensing. Sensors and Actuators B -Chemical  125, 55-59 (2007).  \n "    },    {        "title": "1409.6010v1.The_role_of_fluctuations_across_a_density_interface.pdf",        "content": "arXiv:1409.6010v1  [physics.flu-dyn]  21 Sep 2014Under consideration for publication in J. Fluid Mech. 1\nThe role of fluctuations across a stable density\ninterface\nA. VENAILLE1L. GOSTIAUX2ANDJ. SOMMERIA3\n1CNRS, Laboratoire de Physique, ENS de Lyon, France,2CNRS, LMFA, École Centrale de\nLyon, France,3CNRS, LEGI, Université de Grenoble, France\n(Received May 20, 2018)\nA statistical mechanics theory for a fluid stratified in densi ty is presented. The pre-\ndicted statistical equilibrium state is the most probable o utcome of turbulent stirring. It\nresults from a competition between turbulent transport and sedimentation of fluid par-\nticles by buoyancy effect. An approximate equipartition bet ween kinetic and available\npotential energy is obtained. The slow temporal evolution o f the vertical density profile\nis then related to the presence of irreversible mixing, whic h alters the global distribution\nof density levels. We propose a model in which the vertical de nsity profile evolves through\na sequence of statistical equilibrium states.\nThe theory is then tested with laboratory experiments in a tw o-layer stably stratified\nfluid forced from below by an oscillating grid. The turbulenc e produced by the grid\nspreads upwards in the lower region and is blocked at the inte rface between the dense\n(salty) and the light (fresh) water. The interface slowly mo ves upward by entraining\nfresh water in the turbulent region. Quantitative measurem ents of density fluctuations\nare made by planar laser induced fluorescence.\nThe density fluctuations across the interface are splitted i n a \"wave\" part and a \"tur-\nbulent\" part. Temporal and spatial spectra of the wave part o f the density field are well\ndescribed by a previous theory due to Phillips.We argue that statistical mechanics pre-\ndictions apply for the turbulent part of the density field. As suming a two level global\ndensity distribution, the theory predicts a hyperbolic tan gent shape for the mean ver-\ntical density profile, in agreement with experimental obser vations. The theory predicts\nthat the interface thickness is inversely proportional to t he Richardson number, which is\nalso consistent with experimental observations. The densi ty fluctuations obtained after\nremoval of the wave part fit well with the statistical equilib rium theory in the interface\nregion. However inside the mixed layer density fluctuations are instead controlled by a\nbalance between eddy flux downward and dissipation by cascad e to small scales. We re-\nport exponential tails for the density pdf in this region, si milar to previously observed\ntemperature pdf in high Rayleigh number convection experim ents.\n1. Introduction\nA remarkable property of strongly stratified turbulent flows is their propensity to form\ndensity staircases with relatively thin interfaces separa ting regions of homogenised density\n(Ruddick et al.1989; Park & Gnanadeskian 1994). Such staircases are routin ely observed\nin the thermocline structure of the ocean. I that case, they o ften can be attributed to\ndouble diffusive convective instabilities involving tempe rature and salt (Schmitt 1994).\nHowever, staircases also occur in situations where the dens ity variations are due solely to\nthe temperature, as for instance in lakes (Simpson & Woods 19 70), and we will restrict\nourself in this paper to cases without double diffusion. The e mergence of sharp density\ninterfaces involves small scale mixing process, and yet the ir presence strongly affects the2 A. Venaille, L. Gostiaux and J. Sommeria\nlarge scale transport of tracers such as pollutants or nutri ents. Understanding what sets\nthe interface shape and its evolution is therefore a challen ging problem.\nA satisfactory model for the interface shape remains elusiv e. Phillips (1972); Posmentier\n(1977) proposed a dynamical equation for the mean vertical d ensity profile by mod-\nelling small scale turbulence with an eddy diffusivity depen ding on the local density\ngradient. The key assumption was to assume that the eddy diffu sivity coefficient de-\ncreases sufficiently fast with the local density gradients ab ove a given threshold, so that\nthe eddy flux decreases with increasing density gradient. Th is approach explains the\nemergence of a sharp interface when stirring an initial stab ly (strongly) stratified linear\ndensity profile, but also predicts that the interface thickn ess becomes infinitely small\n(only limited by molecular diffusivity). By contrast, the de nsity interfaces observed by\n(Park & Gnanadeskian 1994) (with a permanent stirring) were found to be larger than\nthe diffusive length scale. In order to overcome this difficult y, Balmforth et al.(1998) pro-\nposed a model in which the eddy diffusion of density is coupled to a dynamical equation\nfor the turbulent kinetic energy – also based on eddy viscosi ty. Under some assumptions\non the forcing terms in the energy equation, Balmforth et al. (1998) obtained solutions\nfor the density profile showing the formation of homogeneous layers separated by sharp\nbut finite interfaces.\nHere we propose a radically different, but complementary app roach: the emergence of a\nsharp but finite interface is interpreted as the most probabl e outcome of turbulent stirring\nbetween two regions of homogenised density, just as for the v orticity field in freely evolving\ntwo-dimensional turbulence (Miller 1990; Robert & Sommeri a 1991). We show that the\nmost probable state results from a competition between turb ulent transport that tends to\nincrease the interface thickness, and buoyancy effects that tend to sharpen the interface\nby the vertical drift of fluid elements. This approach does no t require any eddy diffusivity\nhypothesis: it relies on the assumption that the system suffic iently explores the phase\nspace available under the energy conservation constraint. We will build upon the work\nof Tabak & Tal (2004); Venaille & Sommeria (2010), which were to our knowledge the\nonly attempts to generalise the equilibrium statistical me chanics approach to stratified\nturbulence.\nWe will use this statistical mechanics theory to interpret e xperimental observations of\ndensity fluctuations across a turbulent density interface. We will consider for that purpose\nan experimental idealisation of a mixed layer originally pr oposed by Rouse, H. & Dodu, J.\n(1955). An oscillating grid is localised at the bottom of a ta nk initially filled with a two\nlayer stratified fluid. The grid-generated turbulence sprea ds in the homogeneous bottom\nlayer and is blocked at the interface, where lighter fluid is s lowly entrained in the tur-\nbulent layer. Since stirring is present only on the lower par t, entrainment induces a slow\ndepletion of the layer at rest and a progression of the mixed l ayer thickness. Here we dis-\ntinguish two questions: i/ what sets the interface shape ? ii / what sets the entrainment\nacross the interface ?\nMost previous studies have been focused on the second questi on, see e.g. Linden (1979);\nFernando (1991) for a review. Turner (1968) addressed the ro le of molecular diffusivity\non the entrainment velocity. Here we will assume that both Re ynolds number and the\nPeclet numbers are very large. The key parameter of the probl em is then the Richard-\nson number Ri=g′Lt/ec, whereLtandecare the integral length scale and kinetic\nenergy of the turbulent velocity field, and g′the reduced gravity at the interface. De-\npending on this Richardson number, different flow regime have been identified close to\nthe interface, see e.g. McGrath et al.(1997) and references therein. Each of these regimes\nhas led to a different entrainment model: i/ At moderate Richa rdson numbers, entrain-\nment is mostly due to coherent vortices of the mixed layer imp inging on the densityFluctuations across a density interface 3\ninterface (Linden 1973; Sullivan 1972). ii/ At larger Richa rdson number, entrainment is\ncontrolled by the breaking of Kelvin-Helmholtz instabilit ies (Mory 1991). iii/ At even\nlarger Richardson number, entrainment is dominated by inte rmittent interfacial wave\nbreaking (Fernando & Hunt 1997).\nBy contrast with these studies, the equilibrium statistica l mechanics approach does\nnot rely on a particular physical mechanism, but rather on th e assumption that the flow\nsufficiently explores the phase space. We will show that this t heory provides predictions\nfor the interface shape, but that it must be supplemented by a model for irreversible\nmixing in order to describe entrainment across the interfac e. In real flows, there may\nexist dynamical regimes preventing the relaxation toward a n equilibrium state, but the\nstatistical mechanics approach provides at least an intere sting limit case that has yet not\nbeen explored.\nThere are only few experimental and theoretical studies dea ling with the vertical den-\nsity profile across the interface, which is most often taken a s given for theories on the\nentrainment velocity. In addition, most studies are limite d to the estimate of the interface\nthickness. Crapper & Linden (1974); Fernando & Long (1985) f ound that the interface\nthickness did not vary significantly with the Richardson num ber, but Hopfinger & Toly\n(1976); Hannoun & List (1988) observed that the interface th ickness decreases with in-\ncreasing Richardson number. E & Hopfinger (1986) found a simi lar result, but with a\nfinite asymptotic thickness depending on the Peclet number a nd the Reynolds number.\nHere we present novel measurements of density fluctuations a cross the interface in these\nexperiments, using planar laser induced fluorescence techn iques, with index matching.\nThis technique was already used by Hannoun & List (1988); McG rathet al. (1997) in\na similar experimental setting, or by Guyez et al. (2007) in the case of two layers sta-\nbly stratified Taylor-Couette experiments, but these studi es did not focus on the role of\ndensity fluctuations.\nAs pointed out by Hannoun & List (1988) , the density field of a s napshot and of a\ntime (or spatial) averaged of the density field may be very diff erent. We will address this\nimportant issue. In particular, we will show that the densit y interface can be decomposed\ninto a large scale wave motion whose amplitude is well descri bed by a theory due to\nPhillips (1977), and that removing this wave motion allows u s to recover the turbulent\ndensity field predicted by statistical mechanics. Hannoun & List (1988) already gave\nexperimental evidence for Phillips theory concerning the i nterfacial wave amplitude. Here\nwe confirm these observations and provide additional suppor t for the theory. We note\nhowever that the Phillips approach is limited because it ass umes that the velocity close\nto the interface is entirely due to the interface motion, whi le turbulent velocities in the\nmixed layer are in reality of the same order (or even larger) a nd may play an important\nrole (McGrath et al. 1997). We will also show that the mean density profile which is a\npriori prescribed in the Phillips theory can be accounted for by the statistical mechanics\ntheory.\nImportantly, the experiments presented in this paper are pe rformed with a permanent\nforcing mechanism, which allows us to sustain a quasi-stati onary turbulent field. We will\ninterpret this turbulent field as an effective heat bath that s ets the level of fluctuation\nfor the density field. We will neglect the direct effect of buoy ancy on the turbulent ed-\ndies. Although this may be true sufficiently far form the densi ty interface, the interplay\nbetween turbulence and stratification may be important in re al flows (Hopfinger & Toly\n1976; Mcdougall 1979). In addition, we consider an experime ntal setting with only one\nturbulent layer. However, previous work do not show qualita tive differences between the\nsingle stirred and the double stirred case (McGrath et al. 1997). Note that the evolution\nof the vertical density profiles in the presence of a permanen t forcing differs from the4 A. Venaille, L. Gostiaux and J. Sommeria\nstudies by Linden (1980); Whitehead & Stevenson (2007), who considered the evolution\nof the mean vertical density profile between isolated mixing events, obtained either by\ndropping an horizontal grid in the flow, or by moving horizont ally a vertical rod, and\nwaiting for the turbulence to decay between each mixing even t.\nThe paper is organised as follows. The equilibrium statisti cal mechanics theory is in-\ntroduced and discussed in the second section. It is shown in a third section that one must\nadd to this equilibrium theory a model for irreversible mixi ng in order to describe en-\ntrainment across the interface. The experimental setting i s presented in a fourth section.\nThe fifth section contains experimental observations on int erfacial waves, and statistical\nmechanics predictions for the mean vertical density profile . It is also shown in this section\nthat the density fluctuations far from the interface are intr insically out of equilibrium,\nand their pdf display exponential tails. We conclude and sum marise the main results in\nthe sixth section.\n2. Turbulent density interfaces as a statistical equilibri um state\n2.1.Boussinesq equations and their conservation laws\nConsider a flow in the Boussinesq approximation, taking plac e in a domain V. At each\ntimetand each point x= (x,y,z)∈ Vthe system is described by the reduced density\nb=g(ρ−ρ0)/ρ0and the velocity field u. Hereρis the fluid density, gthe gravity and\nρ0a reference density. Note that with our convention the densi tybis the opposite of the\nbuoyancy, and this field will be simply referred to as the dens ity field in the remaining\nof this paper. The velocity field is non-divergent:\n∇u= 0. (2.1)\nThe flow dynamics is expressed by the advection and molecular diffusion of density b\n∂tb+u∇b=κ∆b, (2.2)\nand by the momentum equation\n∂tu+u∇u=−∇P−bez+ν∆u+F, (2.3)\nwhereFis a mechanical forcing and k,lis the vertical unit vector.\nWe consider here a case without forcing and dissipation ( F= 0,κ=ν= 0). If the\nvelocity field remains differentiable within the domain Vwhere the flow takes place, the\ntotal energy of the flow\nE=/integraldisplay\nV/parenleftbigg1\n2u2+bz/parenrightbigg\ndx (2.4)\nis a dynamical invariant. In addition, as a consequence of th e density advection equation,\nthe global distribution (i.e. histogram) of density levels\ng(σ) =1\n|V|/integraldisplay\nVδ(b−σ)dx (2.5)\nis also conserved. Note that g(σ)can be related to the sorted density profile bs(z), ob-\ntained by allowing each fluid particle to settle down to its po sition at rest (lower po-\ntential energy), using −g(bs)dbs=Adz, whereAis the horizontal domain area. There-\nfore the function gis proportional to the inverse of the derivative of this sort ed profile,\ng(bs) =−A(dbs/dz)−1.\nBoussinesq equations admit other dynamical invariants cal led Casimir functionals,Fluctuations across a density interface 5\nwhich are related to the conservation of Ertel potential vor ticity, see e.g. Salmon (1998).\nHowever, in this paper, we will not take into account these in variants to compute the\nequilibrium state. The fact that in the absence of rotation, statistical equilibria are not\naffected by dynamical invariants related to potential vorti city conservation has been re-\ncently discussed in detailed in the framework of the equilib rium statistical mechanics of\nthe shallow water system (Renaud et al. 2014).\n2.2.Equilibrium states\nHere we proceed by analogy with statistical mechanics of two dimensional Euler or\nquasi-geostrophic flows originally proposed by Miller (199 0); Robert & Sommeria (1991),\nsee Sommeria (2001); Majda & Wang (2006); Bouchet & Venaille (2012); Lucarini et al.\n(2013) for recent reviews.\nWe define a microscopic state of the system as a given fine grained density field b(x,y,z)\nand a non-divergent velocity field u(x,y,z) = (u,v,w). These microscopic fields are\nrelevant phase space variables because they satisfy a Liouv ille theorem, stating that the\nflow in phase space is non-divergent. Then Liouville theorem can be written formally as\n/integraldisplay\nVdxdydz/parenleftbiggδ∂tb\nδb+δ∂tu\nδu+δ∂tv\nδv+δ∂tw\nδw/parenrightbigg\n= 0. (2.6)\nThe proof of such a Liouville theorem for the velocity field (u,v,w)is a classical result\nfor 3D Euler dynamics. It can for instance be obtained by deco mposing the velocity\nfield into Fourier modes, see e.g. Bouchet & Venaille (2012) a nd references therein. The\ngeneralisation to the Boussinesq system is straightforwar d, since the only difference with\n3d Euler is the presence of an additional term linear in bin the momentum equation,\nas well as an additional equation describing the pure advect ion of density bby the non-\ndivergent velocity field.\nAmacroscopic state of the system is defined by the probability density field ρ(x,σ,v)\nthat describes the probability to measure a given density le velb=σand a given velocity\nvalueu=vin the vicinity of the point x.\nThe conservation laws can be expressed as constraints on the se probability field. Indeed,\nthe energy defined in Eq. (2.4) and the global distribution of density levels defined in Eq.\n(2.5) can be expressed as a functionals of ρ,\nE[ρ] =/integraldisplay\nVdx/integraldisplay+∞\n−∞dv/integraldisplay+∞\n−∞dσ ρ/parenleftbiggv2\n2+σz/parenrightbigg\n, (2.7)\nHσ[ρ] =/integraldisplay\nVdx/integraldisplay+∞\n−∞dv/integraldisplay+∞\n−∞dσ ρ . (2.8)\nFinally, the probability density field ρis normalised at each point x:\n/integraldisplay+∞\n−∞dv/integraldisplay+∞\n−∞dσ ρ(x,σ,v) = 1. (2.9)\nJust as in the case of the vorticity field in 2D turbulence, we a nticipate that a typical\nmicroscopic state b,upicked at random for a given set of constraint is characteris ed by\nlarge scale variations superimposed with wild fluctuations at small scales. This is what\nmotivates the introduction of the probability field ρto describe the field at a macroscopic\nlevel: the large scale flow will be obtained by computing the m ean quantities\nb(x)≡/integraldisplay+∞\n−∞dv/integraldisplay+∞\n−∞dσσρ(x,σ,v),u(x)≡/integraldisplay+∞\n−∞dv/integraldisplay+∞\n−∞dσvρ(x,σ,v),(2.10)6 A. Venaille, L. Gostiaux and J. Sommeria\nwhile the local fluctuations around this mean flow will be stat istically described by the\ndistribution ρ(x,σ,v). Classical counting arguments allow us to estimate the numb er of\nmicroscopic states associated with a given macroscopic fiel dρ, and to show that the most\nprobable state is the one that maximises the mixing entropy\nS=−/integraldisplay\nVdx/integraldisplay+∞\n−∞dv/integraldisplay+∞\n−∞dσ ρlnρ , (2.11)\namong all the states that satisfy the constraints of the prob lemE[ρ] =EandHσ[ρ] =\ng(σ), see e.g. Miller (1990); Robert & Sommeria (1991). This vari ational problem can be\nwritten in the compact form:\nS(E,g(σ)) = max\nρ{S[ρ]| E[ρ] =E,Hσ[ρ] =g(σ)}, (2.12)\nwhereS(E,g(σ))is the equilibrium entropy, and where the maximum is searche d over\nall the density probability fields ρwhich are normalised at each point x.\n2.2.1. Determination of the equilibrium\nThe first step to find the equilibrium state is to compute criti cal points of the variational\nproblem, i.e. to find the field ρsuch that first variations of the mixing entropy around\nthis state do vanish, given the constraints of the problem. O ne needs for that purpose to\nintroduce the Lagrange multipliers βandγ(σ)associated respectively with the energy\nconstraint (2.7) and with the constraints of the global dens ity distribution (2.8), and\nthen compute first variations with respect to ρ:\nδS −βδE+/integraldisplay\nγ(σ)δHσdσ= 0. (2.13)\nEquation (2.13) with the normalisation constraint Eq. (2.9 ) yield the following necessary\nand sufficient condition for ρto be a critical point of the variational problem:\nρ(σ,x,v) =/parenleftbiggβ\n2π/parenrightbigg3/2\ne−βv2\n2ρb(z,σ), ρb(z,σ)≡e−βσz+γ(σ)\n/integraltext+∞\n−∞dσ e−βσz+γ(σ). (2.14)\nThe values of the Lagrange multipliers βandγ(σ)are (implicitly) determined by the\nexpression of the constraints E[ρ] =EandHσ[ρ] =g(σ), given by Eq. (2.7) and Eq.\n(2.8) , respectively. Relaxation equations towards these e quilibria (for a given global\ndistribution of density level, and for either a prescribed e nergy and a prescribed in-\nverse temperature) are proposed in Venaille & Sommeria (201 0), with an application to\nrestratification problems. Three remarkable properties ar e satisfied by the equilibrium\nstate.\nFirst, the probability density field (2.14) is expressed as a product of the probabilities\nfor density σand velocity v, which means that bandvare two independent quantities\nat equilibrium.\nSecond, the predicted velocity distribution is Gaussian, i sotropic and homogeneous in\nspace: The local kinetic energy ecof the equilibrium state is therefore homogeneous in\nspace, with\nec=1\n2/integraldisplay\ndσdv v2ρ=3\n2β−1. (2.15)\nThe inverse of βdefines an effective “temperature” of the turbulent field. Acc ording to\nEq. (2.15), this temperature is proportional to the varianc e of the velocity fluctuations.\nThird, the distribution of density levels ρb(z,σ)depends only on the vertical coordinateFluctuations across a density interface 7\nz. Using Eq. (2.14) and the definition of the averaged density b(z)given in Eq. (2.10) ,\nthe expression of the distribution of density levels can be w ritten as\nρb(σ,z) =ρb(σ,0)e−β/parenleftbig\nσz−/integraltextz\n0dz′b(z′)/parenrightbig\n, (2.16)\nwhereρb(σ,0)must be determined using the constraints Hσ[ρ] =g(σ)given Eq. (2.8).\nAny moments of the density distribution can be computed usin g\nbn=/integraldisplay\nσnρdσ. (2.17)\nApplying this expression to the second moment, we can easily show from Eq. (2.14) that\nthe density variance is proportional to the vertical gradie nt of the mean density, with a\ncoefficient of proportionality given by the inverse temperat ure:\ndb\ndz=β/parenleftBig\nb2−b2/parenrightBig\n. (2.18)\nTo conclude, the kinetic energy of the equilibrium state is h omogeneous in space, and the\nvariance of the density fluctuations is proportional to this kinetic energy, with a coefficient\nproportional to the vertical mean density gradient. It is sh own in Appendix A that Eq.\n(2.18) implies equipartition between kinetic energy and av ailable potential energy in a low\nenergy limit, which allows to interpret the widely reported mixing efficiency coefficient of\n0.25as a consequence of the rapid relaxation of the system toward s statistical equilibrium.\n2.3.Sharp interfaces as a Fermi-Dirac distribution\nLet us consider the particular case of an initial state compo sed of two density levels in\nequal proportion. This would for instance be the global dist ribution of a tank initially\nfilled with two uniform density layers of equal thickness. No te that one can always choose\nthe reference density such that the two levels are symmetric around 0, writing:\nσ∈ {−Σ,Σ}. (2.19)\nWe introduce the probability pto measure the level Σat height z:\nρb=p(z)δ(σ−Σ)+(1−p(z))δ(σ+Σ) (2.20)\nAccording to equation (2.14) and using the fact that the two d ensity level are in equal\nproportions to eliminate γ(σ), we obtain\np(z) =e−βΣz\ne��βΣz+eβΣz, (2.21)\n(to avoid a constant of integration, we have chosen the origi n of thez-axis at the interface\nbetween the two layers at rest). This is reminiscent of a Ferm i-Dirac distribution. Indeed,\nfor the two level system, the incompressibility constraint plays the same role as the\nexclusion principle for the statistics of Fermions. Follow ing this analogy, the density field\nis a collection of fluid particles with energy ep=σz, with a Fermi level εf= 0, in thermal\ncontact with a heat bath characterised by the inverse temper atureβ. Using Eq. (2.21)\nand Eq. (2.15), the mean density profile b= Σp−Σ(1−p)can be expressed in term of\nthe density jump ∆b= 2Σ and the local kinetic energy ec:\nb(z) =−∆b\n2tanh/parenleftBigz\n∆h/parenrightBig\n,∆h≡4ec\n3∆b, (2.22)\nwhere we have introduced the interface thickness ∆h. This tanh profile was previously\nobtained by Tabak & Tal (2004) using similar arguments. Howe ver, Tabak & Tal (2004)8 A. Venaille, L. Gostiaux and J. Sommeria\ndid not relate the inverse temperature βto an effective turbulent heat bath.\nAccording to Eq. (2.22), the Richardson number based on the i nterface thickness\nRi∆h≡3∆h∆b/(4ec)is always equal to one. This means that the interface thickne ss\nis the typical vertical length of an overturning of the stabl e interface provided that an\norder one fraction of the kinetic energy inside the overturn ing region is transferred into\npotential energy.\nThe interface thickness can also be expressed in term of a glo bal Richardson number\nbased on the layer thickness H.\n∆h=2H\nRiH, RiH≡H∆b\n2ec/3. (2.23)\nNote that the term 3ec/2is the square of a typical turbulent velocity. The interface is\nsharp when ∆h≪H, i.e. whenever RiH≫1. In that case the potential energy associated\nwith an overturning of the density field at the domain scale His much larger than the\ntotal kinetic energy. By contrast, the density field becomes homogeneous in the limit\nfor which the total kinetic energy is much larger than the pot ential energy associated\nwith an overturning of the density field at the domain scale H, in which case RiH≪1.\nMore generally, it is clear from Eq. (2.16) that the mean dens ity profile and the statistics\nof density levels result from a competition between turbule nt transport and buoyancy\nrepelling. Turbulent transport is related to the turbulent kinetic energy, which increases\nwhen the inverse temperature βdecreases. Buoyancy repelling is related to the difference\nbetween values of the density levels σinitially present in the flow. In the high energy limit\n(βσ→0), the density statistics are homogeneous in space, and ther e is no stratification.\n2.4.Turbulent velocity fluctuations as a \"heat bath\"\nThe previous results have been obtained in a microcanonical framework, assuming that\nthe total energy is conserved. However the total energy is no t actually conserved. Given a\ngiven cut-off length scale, kinetic energy is indeed transfe rred to scales smaller than this\ncut-off, no matter how small this cut-off length scale. This ha s two consequences: i/ in the\npresence of viscosity, the fluctuations will be dissipated, no matter how small is the viscos-\nity ii/ even in the absence of viscosity, the velocity field ma y become non-differentiable,\nhence breaking energy conservation. This contrasts sharpl y with equilibrium states of the\n2D Euler equations, in which case small scale vorticity fluct uations do not contribute to\nthe total energy, which belongs entirely to a large scale flow structure.\nTherefore a forcing term is needed in order to maintain a stat istically steady state. But\nin that case the system is out-of equilibrium. However, we pr opose here a phenomenolog-\nical interpretation of the results obtained in previous sub sections by assuming that the\nobserved density field remains close to an equilibrium state , even when the velocity field\nis strongly out of equilibrium. We assume that just as in the e quilibrium case above, the\nkinetic energy is homogeneous in space and is related to the ( dynamical) ’temperature’\nthrough Eq. (2.15). This temperature is set by a balance betw een forcing and dissipa-\ntion in the momentum equation. Since βis given, the relevant statistical ensemble is the\ncanonical one, and the equilibrium state is the minimiser of the free energy\nF[ρ]≡ −S[ρ]+βE[ρ]. (2.24)\nAccording to Eq. (2.14), the total mixing entropy can be expr essed as the sum of the\nmixing entropy associated with the density field and the mixi ng entropy associated withFluctuations across a density interface 9\nthe velocity field:\nS=Sb−3\n2lnβ,Sb≡ −/integraldisplay\nVdx/integraldisplay+∞\n−∞dσ ρblnρb, (2.25)\nand according to Eq. (2.15) the total energy is\nE[ρb] =Ep[ρb]+3\n2β,withEp[ρb] =/integraldisplay\nVdx/integraldisplay+∞\n−∞dσ σzρb. (2.26)\nSinceβis given, minimising the functional Fdefined Eq. (2.24) is equivalent to minimis-\ning\nFb[ρb]≡ −Sb[ρb]+βEp[ρb]. (2.27)\nFinally, we obtain a variational problem in the canonical en semble:\nF(β,g(σ)) = min\nρb{−Sb[ρb]+βEp[ρb]| Hσ[ρb] =g(σ)}, (2.28)\nwhich means that we look for the probability density field ρbthat minimises a free energy\nwhile conserving the global distribution of density levels .\nTo conclude, the density field characterised by its potentia l energy and its global dis-\ntribution of density levels can be considered as a subsystem in thermal contact with an\neffective heat bath provided by the turbulent velocity field. Note that forcing is required\nto maintain this turbulent velocity field, but we still assum e that no forcing and no dissi-\npation is present in the dynamics of the density field. The effe ct of including dissipation\nin the density dynamics is discussed in the next section.\n3. Entrainment and irreversible mixing\nIn the previous section, it was assumed that the global distr ibution of density levels\nis conserved. There is in that case no temporal evolution of t he vertical mean density\nprofile once the equilibrium state is reached. This is becaus e the statistical mechanics\napproach does not take into account irreversible mixing thr ough turbulent cascade. This\nirreversible mixing process changes the global distributi on of density over time. If this\nglobal distribution of density levels evolves on a sufficient ly slow time scale, one may\nassume that this evolution occurs through a sequence of equi librium states. We show in\nFig. 1 the sequence of vertical density profiles in the case of a two level configuration b∈]−\nΣ,Σ[with decreasing values of Σfor a fixed inverse temperature β= 3/(2ec)(prescribed\nby a turbulent heat bath). This shows a trend towards complet e homogenisation of the\ndensity field with a persistence of the density interface.\nLet us assume that the density field is anti-symmetric (in a st atistical sense) with\nrespect to an interface located at z= 0, which is the case for instance if the initial\ncondition is made of two layers of homogeneous fluid with equa l depthH. We define the\nentrainment velocity as the relative temporal variation of the averaged density in the\nlower layer:\nUe=−H\n2dt/angbracketleftbig\nb/angbracketrightbig\n/angbracketleftbig\nb/angbracketrightbig,with/angbracketleftbig\nb/angbracketrightbig\n≡1\nH/integraldisplay0\n−Hdzb . (3.1)\nThis definition is consistent with Eq. (1) in Turner (1968).10 A. Venaille, L. Gostiaux and J. Sommeria\n3.1.A simplified model in the two-level case\nLet us now come back to the two level configuration b∈]−Σ,Σ[in order to devise a\nsimple model for irreversible mixing, assuming that the dyn amics goes through a sequence\nof equilibria. It amounts to find a dynamical equation for the levelΣ(t).\nWe propose to model the temporal evolution of the density lev elΣas a simple linear\nrelaxation process towards the averaged density in the lowe r layer introduced Eq. (3.1):\n∂tΣ =−s(Σ−∝an}b∇acketle{tb∝an}b∇acket∇i}ht), (3.2)\nwheresis a mixing rate, i.e. the inverse of a relaxation time, which can be interpreted as\na typical stretching time or a cascade rate. The physical mot ivation for this model is that\nthe interface acts as a barrier for irreversible mixing, in s uch a way that the turbulence\ntends to homogenise the fluid independently in each layer. Co nsidering that bis given\nby the equilibrium profile Eq. (2.22), using Eq. (2.23), and t aking the limit of a sharp\ninterface RiH≫1, Eq. (3.2) becomes\n∂tΣ\nΣ=slog2\nRiH+o/parenleftbig\nRi−1\nH/parenrightbig\n. (3.3)\nStill in the high Richardson limit ( RiH≫1), we get ∂t/angbracketleftbig\nb/angbracketrightbig\n//angbracketleftbig\nb/angbracketrightbig\n=∂tΣ/Σ +o/parenleftbig\nRi−1\nH/parenrightbig\n,\nwhich, using Eq. (3.3) and Eq. (3.1), yields\nUe=sH\nRiHlog2\n2+o/parenleftbig\nRi−1\nH/parenrightbig\n. (3.4)\nAssuming that the velocity field is not affected by stratificat ion, the straining rate s\ncan be obtained on dimensional ground as ∝e1/2\nc/LtwhereecandLtare the turbulent\nkinetic energy and length scale in the absence of stratificat ion. We recover in that case\nthe classical result Ue∼Ri−1\nHinitially proposed by Rouse, H. & Dodu, J. (1955) who\nassumed first the presence of a sharp interface between two ho mogeneous layers, and\nsecond that the increase of the potential energy is proporti onal to the energy production\nby mechanical stirring. However, many experimental observ ations suggest that for very\nlarge values of RiHthe power relation between entrainment velocity and Richar dson\nnumber is steeper that −1. Different arguments have been proposed to account for these\nobserved power law, see e.g. Fernando (1991) and references therein. The simplified model\npresented above translates the scaling for the entrainment velocity Ue∼Ri−n\nHinto a\nscaling for the cascade rate s∼Ri−n+1\nH.\nTo conclude, we have devised a toy model in which the mean vert ical profile evolves\nthrough a sequence of equilibrium states described by a tanhprofile until the flow is fully\nhomogenised. The main caveat of the model is that it assumes a two level distribution\nfor the global density distribution, while mixing through t urbulent cascade leads to the\ncreation of a continuum of density levels between its extrem al values. This aspect will be\ndiscussed in more details in the experimental part of this pa per.\n3.2.Comparison with a model based on turbulent diffusion\nIn order to appreciate the difference between the model propo sed in the previous subsec-\ntion and other approaches based on an effective turbulent diff usivity, it is instructive to\nconsider the simple case of the temporal evolution of an init ial step function through the\nheat equation ∂tb=K∂zzbwith an homogeneous diffusion coefficient Kand no density\nflux at the upper and lower boundary ∂zb|z=0,H= 0. The sequence of vertical density\nprofiles from the initial condition to the final homogeneous s tate is shown on the right\npanel of Fig. 1. We clearly see that the route toward complete homogenisation is differentFluctuations across a density interface 11\n−1 10H\nMean density profileHeight za) Sequence of equilibrium states\n0H\n−1 1Height z\nMean density profileb) Constant turbulent diffusion\nFigure 1. a) Sequence of statistical equilibrium states in the case of a two level global density\ndistribution ( b∈ {−Σ(t),Σ(t)}), at fixed temperature β(prescribed by a turbulent heat bath).\nThe value of the density level Σ(t)is decreasing from 1(plain green line) to 0(dashed red line).\nb) purely diffusive relaxation of an initial step function to wards an homogeneous density profile.\nThe diffusion coefficient is homogeneous in space.\nin the diffusive case and in the quasi-equilibrium case, for w hich the interface thickness\nremains quasi-constant through the homogenisation proces s.\nDensity interfaces are sometimes fitted with error function s, see e.g. Crapper & Linden\n(1974); Linden (1980); Whitehead & Stevenson (2007). The er ror function is the solution\nof the heat equation for a constant diffusion coefficient, in th e case of an initial step\nfunction in an unbounded domain. In the case a bounded domain , this error function\nis a good fit for the density profile as long as the interface thi ckness remains much\nsmaller than the domain size. With a proper rescaling of the zaxis and of the density\naxis, the error function and the hyperbolic tangent functio ns are hardly discernible. We\nnote however that the physical mechanisms underlying the ch oice of one function rather\nthan the other to fit experimental data are drastically differ ent. Indeed, the choice of an\nerror function result from a model based on a local turbulent diffusivity hypothesis. By\ncontrast, there is no such assumption required for the choic e of a tanh profile: the density\nprofile is interpreted in that case as the equilibrium state o f a two level system, which\nresults from the competition between turbulent transport a nd buoyancy.\n4. Two-layer stratified fluid forced by an oscillating grid\n4.1.Experimental setting\nA tank with horizontal cross section 40×40cm2is filled with a layer of dense fluid below\na layer of light fluid, see Fig. 2-a. The density is homogeneou s in each layer, and each layer\ndepth is initially around 40cm . This experimental setting is similar to the one describe d\nin Hopfinger & Toly (1976).The novelty comes from measuremen ts techniques. We use\nPlanar LASER Induced Fluorescence (PLIF) with index matchi ng between both layer in\norder to observe quantitatively density fluctuations. The l ower layer contains water, salt\nand rhodamine. The upper layer contains water and ethanol, s uch that the optical index\nis the same in each layer. The density difference is imposed by the concentration in salt\nand ethanol in both layers.\nTurbulence is generated by an horizontal grid oscillating v ertically at 5Hz, with a grid12 A. Venaille, L. Gostiaux and J. Sommeria\nFigure 2. a) Experimental setting. The tank is filled below with a layer of water with salt and\nrhodamine, and above with water, ethanol and rhodamine. The LASER sheet illuminates the\ncentre of the tank. The density field is observed in the 25×25cm2window represented with a\nblack line. b) Vertical variation of the turbulent kinetic e nergy in the 25×25cm2window when\nthere is no stratification (PIV measurements).\nmesh of10cm (including the 2cm thickness of the grid bars), and an amplitude of 8cm\n(crest to crest). The same forcing is used for all the experim ent.\nTurbulence properties have been characterised using PIV me asurements in a case with-\nout stratification. We observed in that case an exponential d ecay of the kinetic energy,\nwith an e-folding depth Lt= 10cm interpreted as the integral length scale of turbulence,\nwhich is of the order of the grid mesh, see Fig. 2-b and Appendi x B. We see that typ-\nical turbulent velocities close to the density interface ar e of the order of U∼1cm.s−1.\nThis corresponds to a Reynolds number Re=LtU/ν≈103associated with moderate\nturbulence close to the interface at the beginning of the exp eriment. This means that\nviscous effect may be important once the interface has moved u p by around 10 cm. Note\nhowever that deeper in the mixed layer, the Reynolds number i ncreases by two order of\nmagnitude. The Peclet number is Pe=LtU/κ≈106, withκthe salt diffusivity, also of\nthe same order for alcohol and Rhodamine.\nThe main control parameter is the density jump ∆ρ/ρvarying from 0.01%to0.8%,\nsee Tab. 1. The initial interface height denoted His slightly different from one exper-\niment to another. Therefore, each experiment is characteri sed by two non-dimensional\nparameters, namely, the bulk Richardson number Riand the Richardson number based\non the interface height RiH, respectively defined by\nRi=Lt∆b\n2et/3,andRiH=H∆b\n2ec/3, (4.1)\nwhereHis the initial interface height, ecthe turbulence kinetic energy measured at\nz=Hin the homogeneous case and Ltthe integral length scale of turbulence in the\nhomogeneous case.\nWe see Tab. 1 that the bulk Richardson number varies from 1to150. In practice, a\nwell defined, sharp interface was only observed for Ri≥10. For lower bulk RichardsonFluctuations across a density interface 13\nFigure 3. a) Snapshot of the density field at the centre of the tank, norm alised between 0(light\nfluid, blue color) and 1(dense fluid, red color) . b) Same density field, but for each va lue ofx\nthe vertical density profile is sorted with denser fluid below , see subsection 4.3. The black line\nis the interface defined as the height of the intermediate den sity level (the contour α= 1/2) in\nthe sorted field. The white line is a fit of this interface heigh t with an order 2 polynomial.\nExperiment EXP1 EXP2 EXP3 EXP4 EXP5 EXP6 EXP7\n∆ρ/ρ(%)0.33 0.3 0.8 1.3 0.1 0.4 0.11\nRi 28 28 84 144 0.9 3.5 12\nRiH 127 125 370 640 3.5 14 55\nTable 1. Parameters for the different experiments. The density jump ∆ρ/ρis estimated as\nthe beginning of each experiment. See Eq. (4.1) for the defini tion ofRiandRiH.\nnumber, the density field did not reach a quasi-stationary st ate presenting a turbulent\ndensity interface. For this reason, we will mostly focus on e xperiments characterised by\nRi >10in order to test statistical mechanics predictions.\nA snapshot of the density field is shown Fig. 3-a. The density fi eld is observed in a\n25×25cm2frame centred 10cm above the initial density interface at the beginning of\nthe experiment, in the central part of the tank, see Fig. 2. Fo r each density snapshot,\nlight adsorption is compensated, as well as the presence of p ossible imperfections in the\nLASER sheet, as dark bands due to bubbles or dust in the optica l path.\nEach experiment is performed during 800 second, during whic h a snapshot of the den-\nsity field is recorded every second. The grid oscillation sta rts after t= 5seconds, and\nstopped at t= 700 seconds. Then the relaxation to rest is observed. The tempor al evolu-\ntion of the x-averaged density is shown Fig 4 for three different experime nts associated\nwith decreasing Richardson numbers from panel a to c. It alwa ys takes around 20 seconds\nbefore the turbulence reaches the interface. Then the avera ged density of the lower layer\ndecreases while the interface height increases slowly. In t he three experiments presented14 A. Venaille, L. Gostiaux and J. Sommeria\nFigure 4. Temporal evolution of the density fields averaged in the hori zontal direction for 3\ndifferent experiments, see Tab. 1: a) ∆b= 0.13m.s−2(EXP4) b) ∆b= 0.033m.s−2(EXP1)\nc)∆b= 0.011m.s−2(EXP7). The black line represents the interface (see text). The density is\nnormalised such that it varies between 0and1at the initial time t= 0for each experiment.\nin Fig.4, we see qualitatively that the interface associate d with the x-averaged density\nremains sharp, and its thickness increases with decreasing Richardson numbers.\n4.2.Relation with the statistical mechanics model\nSeveral assumptions are required to interpret this experim ent in the framework of the\nequilibrium statistical mechanics theory introduced in se ction 2. The source of kinetic\nenergy is localised at the grid position in the experiment, i mplying a vertical decay of\nthe kinetic energy. This contrasts with the statistical equ ilibrium stating that the kinetic\nenergy (or the effective temperature) is homogeneous in spac e. Our working hypothesis\nis that prediction from equilibrium statistical mechanics for the density field may be\napplied near the density interface, by considering that the effective temperature of the\nequilibrium state is provided by velocity fluctuations that would be observed at the\ninterface in the absence of stratification.\nA second difficulty is that there must be sufficient mixing in pha se space for the\nsystem to reach the equilibrium state. It is clear that the in terface motion at large scale is\ndominated by waves, for which nonlinear effects driving this mixing are inhibited. We shall\ntherefore assume that the turbulent fluctuations involved i n the statistical equilibrium\nare limited to small scales, after exclusion of a \"wave motio n \" of the density interface.\nA third limitation for the applicability of the theory is the irreversible dissipation\nof density fluctuations through turbulent cascade, changin g the global distribution of\ndensity levels with time. As discussed in section 3, this effe ct is actually related to en-\ntrainment across the interface. We will assume that the glob al distribution of density\nlevels evolves on a time scale longer than the one required fo r the system to reach the\nequilibrium state.Fluctuations across a density interface 15\nA fourth limitation of the present experiment comes from the asymmetric forcing:\nonly the lower layer is turbulent. A direct consequence of th is asymmetric forcing is\nthat the turbulent layer is actually penetrating into the qu iescent layer: contrary to the\nhomogeneous case discussed in previous section, the interf ace is shifting vertically in the\nexperiment. Testing equilibrium statistical mechanics pr edictions requires in that case\nto be in the reference frame moving with the interface, assum ing that the fluid evolves\nthrough a sequence of stationary states. It is thus necessar y to define precisely what is\nthe interface height in the experiment.\n4.3.Definition of the interface height\nOne major difficulty associated with the definition of the inte rface height and thickness\nstems from the fact that instantaneous density profiles may b e very different from tem-\nporal or spatial averages, see e.g. Hannoun & List (1988). In order to define the interface\nheight at each time tand location x, Hannoun & List (1988); McGrath et al. (1997)\nconsidered iso-density contours parametrised by\nα≡ρ−ρmin\nρmax−ρmin, (4.2)\nand defined the interface height h(x,t)as the height of the contour α= 1/2As noticed\nby Hannoun & List (1988); McGrath et al. (1997), this method cannot be applied when\ninterface overturns. This occurs for instance with wave bre aking. As an example, one\nclearly sees on the snapshot of Fig. 3-a that iso-density con tours do not define singled-\nvalued functions for the interface height h(x,t).\nWe propose and discuss in this paper a method to define the loca tion of a corrugated\ninterface, which is well defined even in the presence of overt urning events in he density\nfield. This method allows us to distinguish interfacial wave s form turbulent fluctuations\naround the interface. In order to find the interface height h(x,t)on the density field\nsnapshot Fig. 3-a, a \"sorted density field\" is computed Fig. 3 -b: at each horizontal point\nx, the vertical sequence of nzpixels is sorted so that the density is decreasing with\nincreasing height z. As a consequence of this sorting procedure, iso-density co ntours are\nalways single-valued functions of the horizontal xcoordinate. The interface h(x,t)is then\ndefined as the height of the intermediate density contour α= 1/2of the sorted density\nfield, which is represented as a thick black line in Fig. 3-a,b .\n4.4.Estimate of the entrainment velocity and test of PLIF calibra tion\nThe spatial and temporal variability of the interface heigh th(x,t)will be explored in\nnext section. Here we consider the slow temporal evolution o f the x-averaged interface\nheighth(t)shown Fig. 5-a. We assume that there is a time scale separatio n between a\nfast temporal variability of the interface height and a slow penetration of the turbulent\nlayer into the layer at rest. The corresponding entrainment velocityUeis estimated by\nconsidering a linear fit of the interface height h(t)between t= 100 andt= 300 seconds\nfor each experiment.\nThe PLIF calibration is checked on Fig. 5-a by comparing the t emporal evolution of\nthe x-averaged interface h(t)with the estimate h0< b0> / < b > , where< b >(t)is\nthe averaged density in the turbulent layer, below the inter face, and h0,< b0>are the\ninitial interface height and density in the lowest layer. De spite the temporal variability of\nh(t)associated with interfacial waves, there is a good agreemen ts between the estimate of\nthe interface elevation h(t)using the sorting algorithm, and the estimate of the interfa ce\nelevation h0< b0> / < b > using mass conservation. A systematic drift is observed onl y16 A. Venaille, L. Gostiaux and J. Sommeria\nFigure 5. a) mass conservation and validation of PLIF measurements. T he thin plain line\nrepresents the temporal evolution of the x-averaged interf ace height h(t)normalised by the\ninitial height h0, and obtained with the sorting algorithm. The \" +\" symbols represent the\ntemporal evolution of the averaged density < b > in the mixed layer, normalised by the initial\ndensity< b0>in this layer. b) Entrainment coefficient as a function of the a verage Richardson\nnumber for each experiment. The entrainment coefficient is de fined as the ratio between the\ninterface velocity and the turbulent velocity. The interfa ce velocity is obtained by a linear fit\nof the curve of the left panel over the interval t= 100−300s. The turbulent velocity around\nthe interface height is estimated by using the PIV measureme nts performed in the case without\nstratification.\nin the strongly stratified case, probably due to light absorp tion that was not completely\ncancelled with our data analysis procedure for this particu lar experiment.\nThe entrainment coefficient E=Ue/(2ec/3)1/2is defined as the ratio between the\nentrainment velocity and the rms turbulent velocity. This r ms turbulent velocity were\nestimated at the interface height z=h, using PIV measurements in a case without\nstratification. The variations of the entrainment coefficien tEwith the bulk Richardson\nnumberRidefined Eq. (4.1) is plotted Fig. 5-b. We find a power law E∼Ri−nwith\nthe exponent nbetween 1and3/2, consistently with previous observations in the same\nrange of Richardson numbers, either in similar experimenta l setting (Fernando 1991), or\nin Taylor-Couette experiments (Guyez et al. 2007). In the remaining of this paper, we\ndo not further investigate entrainment mechanism, which wo uld require more detailed\nmeasurements of the velocity field close to the interface in o rder to better characterise\nvertical density fluxes. We rather assume that the system evo lves slowly through a se-\nquence of statistically steady states, and we focus on the pr operties of the density fields\nassociated with these states.\n5. Characterisation of interfacial waves, interface shape a nd density\nfluctuations across the interface\nOn the one hand, it is clear from Fig. 3 that the interface is co rrugated and present\nsmall scale structures, consistent with the statistical me chanics approach. On the other\nhand, low frequency oscillations of the interface height ar e clearly visible in Fig. 4. These\nlow frequency oscillations cannot be explained with the sta tistical mechanics approach.\nWe show in the first subsection that some properties of the obs erved interfacial waves\nare well described by a heuristic theory due to Phillips (197 7). The characterisation\nof interfacial waves will allow us to propose a criterion to d istinguish a \"wave part\"\nand a \"turbulent part\" for the fluctuations of the density fiel d close to the interface.\nThe \"turbulent part\" of the density field will be considered i n the second subsection inFluctuations across a density interface 17\nFigure 6. a) Interface height spectrum for EXP1 ( ∆b= 0.03m.s−2). For each frequency f,\nthe spectrum is normalised by its maximal value. The black li ne is the dispersion relation for\ninterfacial waves given Eq. (5.2). b) Same plot for EXP4 ( ∆b= 0.13m.s−2).\norder to test statistical mechanics predictions for the int erface shape. Finally, we show in\nthe last subsection that density fluctuations within the mix ed layer and sufficiently far\nfrom the interface are much larger than expected from equili brium statistical mechanics\narguments, and present exponential tails.\n5.1.Interfacial waves\nThe spectrum of the interface elevation in a two layer fluid su bject to an external forcing\nwas predicted by Phillips (1977) with heuristic arguments. In this framework, the pre-\ndicted spectrum does not depend on the forcing mechanism. Th e only input of the theory\nis the interface thickness ∆h. We give in the following some experimental evidence for\nPhillips’ theory, assuming that the interface thickness is the one predicted by statistical\nmechanics, and also discuss limitation of this approach.\n5.1.1. Dispersion relation for interfacial waves\nPhillips (1977) considered a sharp density interface separ ating two homogeneous layers\nof height H/2≫∆h, with a density jump ∆b=g∆ρ/ρ. The buoyancy frequency inside\nthe interface can be estimated as\nN∼/radicalbigg\n∆b\n∆h. (5.1)\nLet us first assume that the interface elevation is a monochro matic wave characterised by\nthe frequency ωand the wavenumber modulus K. Let us consider in addition that the\nwavenumber modulus is such that K≪2π/(∆h), so that the interface can be considered\ninfinitely thin at lowest order, and K≥2π/H, so that the limit of deep water can be\nconsidered. Under these assumptions, the dispersion relat ion is\nω=/radicalbigg\n∆bK\n2forN≪ω≤/radicalbigg\n∆bπ\nH. (5.2)\nThe presence of interfacial waves is revealed Fig. 6 by the sp atial-temporal spectrum of\nthe interface elevation, denotedˆˆh(f,k), whereh(x,t)is the observed interface elevation\ndefined with the sorting algorithm introduced in subsection 4.3.\nFor each frequency f=ω/2π, the spectra shown in Fig. 6 have been normalised by their\nmaximum value over the horizontal wavenumbers k. We see that significant contributions\nto the spectrum are always located inside the region delimit ed by the dispersion relation18 A. Venaille, L. Gostiaux and J. Sommeria\nEq. (5.2). Note that the interface variations are measured o n a line in the xdirection and\nnot in the horizontal (x,y)plane, so any wave numbers |k|<2ω2/∆bmay correspond\nto an interfacial wave, according to Eq. (5.2). This is why th e spatio-temporal spectra of\nFig. 6 are not merely peaked around the dispersion relation.\n5.1.2. Prediction of the interface elevation amplitude for a monoch romatic wave\nFor a given interfacial wave with wavenumber modulus K, the interface elevation\namplitude is denoted aK. We assume that the flow around the interface is entirely due t o\nthe potential flow associated with the interface deformatio n. This allows us to estimate\nthe velocity field close to the interface as\nUK∼aKω . (5.3)\nGiven that the flow in the mixed layer is strongly turbulent, t his hypothesis may be ques-\ntioned, and we will provide further discussion on this point at the end of this subsection.\nWhen the interface is infinitely sharp ( ∆h= 0), the horizontal velocity field due to the\nvariation of the interface elevation is discontinuous acro ss the interface, with a velocity\njump given by ∆UK∼aKω. Let us now consider that the interface is characterised by a\nsmall but non-zero thickness ( ∆h∝ne}ationslash= 0) Using the estimate of the velocity jump obtained\nin the limit of an infinitely sharp interface, the vertical gr adient of the horizontal velocity\nfield is estimated as\n∂zU∼∆UK\n∆h∼aKω\n∆h. (5.4)\nDefining the local (or gradient) Richardson number inside th e interface as\nRiloc=N2\n(∂zU)2∼∆b∆h\na2\nKω2∼∆h\na2\nKK, (5.5)\na sufficient condition for stability of the flow inside the thin interface is Riloc>1/4\n(Miles 1961). The key idea of Phillips (1977) is then to assum e i/ that the flow is ac-\ntually unstable whenever Riloc<1/4; ii/ that this instability eventually leads to wave\nbreaking, which limits the growth of the wave amplitude aK; iii/ that this is the dominant\nmechanism to extract energy from the interfacial wave; iv/ t hat an external mechanism\nconstantly supplies energy to the wave. Then a steady state c an be reached, and the sat-\nurated wave amplitude aKis such that the condition of criticality Riloc= 1/4is satisfied.\nInjecting Eq. (5.2-5.4) in Eq. (5.5), this condition for cri ticality yields\na2\nK∼∆h\nK. (5.6)\n5.1.3. Spatial power spectrum of the interface elevation\nLet us now assume that the interface is an (isotropic) collec tion of waves with wavenum-\nber(k,l)(and wavenumber modulus K=√\nk2+l2), and that these waves do not interact.\nLet us write Ψ(k,l)the spatial power spectrum of the interface elevation. The v ariance\nof interface elevation at wave number Kis related to the power spectrum through\na2\nK=/integraldisplay\n√\nk′2+l′2>Kdk′dl′Ψ(k′,l′)∼K2Ψ(k,l). (5.7)\nInjecting then Eq. (5.6) in Eq. (5.7) yields\nΨ(k,l)∼∆h\nK3. (5.8)Fluctuations across a density interface 19\nThe experimental spatial power spectra are obtained by meas uring the interface elevation\nalong a line in the xdirection. Let us call Ψx(k)the power spectrum of the interface\nelevation along this direction. It is given by\nΨx(k)≡/integraldisplay+∞\n−∞dlΨ(k,l)∼∆h\nk2. (5.9)\nThe experimental observation of the interface elevation po wer spectrum Ψx(k)/∆his\nshown Fig. 7-a. The spectral amplitudes have been normalise d by the interface thickness\npredicted Eq. (2.22) with statistical mechanics arguments , i.e. by ∆h∼ec/∆b. We\nsee that the Phillips prediction of a −2slope for this spectrum is consistent with the\nbehaviour of the experimental spectrum at low wave numbers i n Fig. 7-a. However, some\ncare must be taken to interpret these spatial spectra for lar ge wavenumbers. Indeed, the\nmeaning of the small scale spatial fluctuations of the interf aceh(x,t)defined with the\nsorting algorithm is not clear. These small scales may be dom inated by the presence\nof turbulent fluctuations in the density field rather than by i nterfacial waves. For these\nreasons, we will consider in the remaining the interface hinterp(x,t), obtained for each\ntimetby fitting the interface elevation h(x,t)with a third order polynomial. Finally,\nrescaling the spatial power spectrum shown Fig. 7-a by the he ight∆hallows us to obtain\na reasonable collapse of the three different experiments, co nsistently with Eq. (5.9).\n5.1.4. Temporal power spectrum of the interface elevation\nLet us now introduce Φ(ω)the temporal power spectrum of the interface elevation.\nThe variance of interface elevation at frequency ωis\na2\nK=/integraldisplay+∞\nωdω′Φ(ω′)∼ωΦ(ω). (5.10)\nInjecting Eq. (5.10) in Eq. (5.6), and using the dispersion r elation Eq. (5.2) yields\nΦ(ω)∼∆b∆h\nω3. (5.11)\nConsidering the statistical mechanics prediction in Eq. (2 .22) for the interface thickness\n∆h, one gets\nΦ(ω)∼ec\nω3. (5.12)\nThis means that the amplitude of the interface displacement frequency spectrum is in-\ndependent from the density jump ∆b. It only depends on the local kinetic energy. To our\nknowledge, this simple but important consequence of an inte rface thickness scaling as\nthe inverse of the Richardson number ( Ri∼Lt∆b/ecorRiH∼H∆b/ec) has not been\ndiscussed previously.\nExperimental observations of the interface elevation temp oral power spectrum are\nshown Fig. 7-b. The agreement between the theoretical predi ctions and experimental\nresults is good. For frequency higher than the gravest linea r mode of interfacial waves\n(represented as vertical dashed lines), the spectrum slope is consistent with the −3slope\npredicted by Phillips theory. Note that the maximum observe d frequencies are always\nlarger than the buoyancy frequency N=/radicalbig\n∆b/∆h. Perhaps more strikingly, no rescaling\nhave been used to plot the spectra, and yet they all collapse o n the same curve in the\nregime where the −3slope is observed. Since the interface elevation is roughly similar\nfor all the experiment , the kinetic energy ecaround the interface within the mixed layer\nis not expected to vary significantly from one experiment to a nother. The collapse of all\nthe experiments on the same curve confirms therefore the pred iction of Eq. (5.12). This20 A. Venaille, L. Gostiaux and J. Sommeria\nFigure 7. a) Spatial power spectrum. Blue diamonds: ∆b= 0.011m.s−2(EXP7); Green circles:\n∆b= 0.03m.s−2(EXP1); Red squares: ∆b= 0.13m.s−2(EXP4); b) Temporal power spectra\nof the x-averaged interface height time series. The vertica l dashed lines represent the location\nof the gravest linear interfacial mode for each experiment, given by ω0=/radicalbig\nπ∆b/L, whereLis\nthe lateral extension of the tank.)\nequation was obtained under the assumption that the interfa ce thickness varies as the in-\nverse of the Richardson number, consistently with statisti cal mechanics predictions. The\ncollapse of the spectra on a single curve at high frequency is therefore an indirect test\nof these statistical mechanics prediction for the interfac e thickness of the mean vertical\ndensity profile.\nPrevious experimental observations of a −3slope of the temporal power spectrum of the\ninterface elevation were provided by Hannoun & List (1988). This slope is also consistent\nwith the experiments by McGrath et al. (1997). However, Hannoun & List (1988) found\nthat the amplitude of the frequency power spectra of the inte rface elevation scaled as\nRi−1. Their scaling amounted to an interface thickness decreasi ng asRi−2. By contrast,\nMcGrath et al. (1997) found an interface thickness that was not varying sig nificantly\nwith the Richardson number. Our result is intermediate betw een both cases.\nFinally, we observed in the experiments at high Richardson n umbers the presence of a\nwell identified peak in the frequency spectrum. The frequenc y of this peak were slightly\nsmaller than the gravest linear interfacial mode represent ed as vertical dashed lines on\nFig. 7-b. These sloshing frequencies may probably be attrib uted to nonlinear interactions\nbetween interfacial waves, and may also be associated with t he presence of solitons. To\nour knowledge, there were no previous observation of such sl oshing dynamics in similar\nexperiments. Understanding this phenomenon will require f urther work.\n5.1.5. Consistency of the approach\nA key assumption of Phillips theory is that close to the inter face, the velocity field UK\nat scaleKis entirely due to the potential flow created by the variation s of the interface\nelevation. In the experiment, the energy source for the wave s is the turbulent velocity\nfield. Let us call Uturb,K the rms turbulent velocity at scale K, close to the interface,\nwithin the turbulent mixed layer. This velocity contains th e contribution of all wave\nnumbers lager than K. Phillips approach is consistent at scales such that UK≫Uturb,K .\nIn order to obtain a simple estimate for Uturb,K , we assume that the turbulent ve-\nlocity field is unaffected by stratification, and well describ ed by the phenomenology of\nthree dimensional homogeneous isotropic turbulence. Of co urse, this hypothesis is tooFluctuations across a density interface 21\nsimplistic, but it can be used as a lowest order estimate. A mu ch more detailed anal-\nysis and discussion on the coupling between turbulent and st ratification is provided in\nFernando & Hunt (1997). For the sake of simplicity, we assume in the following that\nthere is no turbulent motion at scales larger the integral le ngth scale Lt†. According\nto previous notations, the velocity field at the integral len gth scale is Uturb,K∼e1/2\nc.\nAt scales K≫2π/LT, assuming an inertial range, the turbulent velocity is obta ined\nby dimensional analysis: Uturb,K∼ǫ1/3K−1/3, withǫ∼e3/2\nc/Ltthe energy dissipation\nrate. Following Carruthers & Hunt (1986), we assume that tem poral fluctuations of the\nvelocity field at a given point are given by the random advecti on of turbulent eddies by\nthe integral scale eddies. This yields ω∼e1/2\ncK, andUturb,K∼ǫ1/3e1/6\ncω−1/3.\nInjecting Eq. (5.12-5.10-5.2) in Eq. (5.3), we obtain the es timateUK∼e1/2\ncfor the\nvelocity close to interface associated with the variation o f the interface elevation. In other\nwords, whatever the scale Ksuch that 2π/∆h≪K≤2π/H, the potential flow created\nby the interface elevation variations is of the order of the r ms turbulent velocity e1/2\nc.\nWe stress that this result relies on the assumption h∼ec/∆b, which was done based\non the statistical mechanics result Eq. (2.22). We see that t he condition for consistency\nUturb,K≪UKis valid for sufficiently small scales ( K≫2π/Lt), or for sufficiently high\nfrequency ( ω≫2πe1/2\nc/Lt). The cut-off frequency fc=e1/2\nc/Ltis of the order of 0.1s−1\nin the experiment. The maximum observed frequency is fmax= 1s−1on Fig. 7-b. The\ncriterion f≫fcis therefore only marginally satisfied in the range of the obs erved−3\nslope.\n5.2.The distribution of density levels\nNow that we have characterised the properties of the interfa ce elevation, we focus in this\nsubsections on the statistical properties of density fluctu ations. The temporal evolution\nof the distribution of density levels ρb(z,σ,t)is shown Fig. 8. Each plotted distribution\nis obtained by building a normalised histogram of density le vels for each depth z, using a\nsequence of 100 images separated by one second, and each succ essive plots of Fig. 8 are\nseparated by 100 seconds. The maximal density levels decrea se slowly with time in the\nlower layer, with a concomitant increase of the interface he ight. An experiment with weak\nstratification is shown on Fig. 8-d. In that case the flow is rap idly fully homogenised, the\ninterface is not well defined, and the density field does not ev olves through a sequence of\nstationary states.\nAround the density interface, the density distributions of Fig. 8 is closed to a double\npeaked function. However, the fluctuations of the interface elevation have not be removed\nto obtain these statistics. The doubled peaked function for the density distribution around\nthe interface is therefore mostly due the rapid motion of int erfacial waves around a slowly\nevolving mean interface height. The sloshing dynamics of th e interface only affects the\nthe density field close to the interface. Far from the interfa ce, the motion of the density\nfield is not affected by the variations of the interface elevat ion and the method used to\nbuild the histograms of Fig. 8 is relevant to describe densit y fluctuations.\nWe see that separating the part of the density statistics due to the wave motion of\nthe interface from the actual turbulent fluctuations is diffic ult in practice, and we will\npropose in the following a rudimentary decomposition of the density fields into \"waves\"\nand \"turbulence\" close to the interface.\n†This assumption can not be fully valid in our experiment, sin ce the presence of large scale\nflow structure filling the whole domain are often reported in c onfined turbulent flows).22 A. Venaille, L. Gostiaux and J. Sommeria\nFigure 8. Temporal evolution of the distribution of density levels: a ) EXP4; b) EXP1; c) EXP7;\nd) EXP6. For each experiment the density levels are normalis ed from 0 (light fluid, blue color)\nto 1 (dense fluid, red color), and a logarithmic scale has been chosen to visualise fluctuations\nfar from the interface.Fluctuations across a density interface 23\n5.2.1. The mean vertical profile after removing the effect of large sca le interfacial waves\nWe present here an experimental test of the statistical mech anics predictions for the\nmean density profile around the interface. As a starting poin t, we note that the global\ndistribution in the experimental of density levels is close to a double delta function, for\nRichardson numbers from 10to150, since the ratio of the interface width with the layer\ndepth is much smaller than one. Equilibrium statistical mec hanics theory predicts in that\ncase atanhshape for the x-averaged density profile, see Eq. (2.22) in subsection 2.3.\nIn order to test the statistical mechanics prediction for th e density interface, it is\nnecessary to cancel the spurious effect of sloshing dynamics on density statistics close\nto the interface. We rebuild for that purpose the histograms of density levels obtained\ninitially Fig. 8 by considering a frame of reference followi ng the interface height for each\nimage. The underlying assumption is that the wave motion of t he interface is dominated\nat lowest order by spatial mode with typical length scale lar ger that the horizontal size\nof the image.\nAssuming that the system is in a quasi-stationary state on ti me intervals of 200seconds\naftert= 100 s for each experiment, the x-averaged vertical profile of density levels bexp(z)\nis fitted with the function\nbfit(z) =∆bfit\n2/parenleftbigg\n1−tanh/parenleftbiggz−h\n∆hfit/parenrightbigg/parenrightbigg\n. (5.13)\nAccording to the statistical mechanics prediction in Eq. (2 .22), all experimental vertical\nmean density profile should collapse on the same curve if the v ertical axis is redefined\nbyz∗= (z−h)/∆hfitand if the density is rescaled as b∗=b/∆bfitRemarkably, all\nthe vertical profile collapse on a curve that is very close to t he predicted tanhrelation\n(black curve) on Fig. 9-a. We note that the fit is better above t he interface than below\nthe interface †. A possible reason is that the tanh-profile corresponds to an equilibrium\nstate for a two level system, while irreversible mixing thro ugh turbulent cascade leads\nto the creation of new intermediate density levels close to t he interface. The effect of\nirreversible mixing through turbulent cascade is discusse d in more detailed in the next\nsubsection.\n5.2.2. Variation of the interface thickness with the Richardson numb er\nAccording to the statistical mechanics prediction Eq. (2.2 2), the interface thickness and\nthe Richardson number RiHintroduced Eq. (4.1) are related through ∆h= 2H/RiH.\nWe check Fig. 9-b that the observed interface width ∆hfitis inversely proportional to\nthe Richardson number RiH. However, the coefficient of proportionality is larger than\nthe one predicted by the theory: we observe ∆hfit≈3H/RiH. A possible reason for this\ndiscrepancy may be attributed to our estimate of the rms kine tic energy in the Richard-\nson number RiH. Indeed, ecwere estimated by considering the rms turbulent kinetic\nenergy measured in a case without stratification at the heigh t of the interface. Assum-\ning that the factor ∆hfit/∆h≈3/2may be attributed to the modification of turbulent\nproperties due to the stratification, and that the statistic al mechanics predictions for the\ninterface height are correct, we define an effective Richards on number Rifit\nHsuch that\n∆hfit= 2H/Rifit\nH, and we use this Richardson number to estimate the effective e nergy\nefit\nc= (3/2)H∆b/Rifit\nH. This effective energy can then be used to define the effective\n†As explained in subsection 3.2, a fit with the error function w ould be as good as the fit with\nthetanhfunction: the two function would be indiscernible on Fig. 2. 22-b.24 A. Venaille, L. Gostiaux and J. Sommeria\ntemperature close to the interface\nβfit=3\n2efit\nc=1\n∆hfit∆bfit. (5.14)\nSince the pioneering work of Crapper & Linden (1974); Hopfing er & Toly (1976), the\nvariations of the interface thickness with the Richardson n umber has remained highly de-\nbated, partly because different experimental settings and m easurement techniques have\nbeen used, partly because different definitions for the inter face thickness have been con-\nsidered. Crapper & Linden (1974); Wolanski (1975); Fernand o & Long (1985) found that\nthe interface thickness was independent from the Richardso n number based not he tur-\nbulence length scale ( Ri= ∆bLt/(2ec/3)). By contrast, Hopfinger & Toly (1976) distin-\nguished a static thickness hsmeasured after stopping mechanical stirring from a dynam-\nical thickness hdmeasured in the presence of turbulence. They observed that t he static\nthickness hswas independent from the Richardson number, while (hd−hs)/hs∼Ri−1.\nUsing PLIF measurements, Hannoun & List (1988) observed tha t the mean interface\nthickness was decreasing with the Richardson number, but wi th a different scaling ( hs∼\nLtRi−2) and that the interfacial wave amplitude was decreasing as hd∼LtRi−1for suf-\nficiently large Ri. However, McGrath et al. (1997) found hd∼Ltusing a similar method\n(but better spatial resolution) to determine the interface thickness. Hannoun & List\n(1988); McGrath et al. (1997) defined the interface thickness ∆h(x,t)as the height dif-\nference between two prescribed iso-density contours (the h eight difference between the\ncontours α= 0.2andα= 0.8, whereαis defined Eq. (4.2)). Just as in the case of the\ninterface height h(x,t), this method can not be applied to a density field with a strong ly\ncorrugated interface, presenting overturning events ever ywhere.\nHere we have presented two different experimental results su pporting a scaling of the\ninterface thickness with the inverse of the Richardson numb er. First, the amplitude of\nthe frequency spectra do collapse on the same curve at high fr equency, which is predicted\nwith Phillips theory and the additional assumption that the interface thickness scales as\nRi−1. Second, the fit of the interface shape obtained in a frame of r eference following the\nx-averaged interface elevation also yields a similar scali ng. This second test is not fully\nsatisfactory: indeed, in the presence of a perfectly thin in terface∆h≈0, with interfacial\nwaves of wavelength smaller than the windows of observation and wave amplitude scaling\nasRi−1, our procedure to obtain the interface thickness would lead to a scaling ∆h∼\nRi−1. We expect that the scaling ∆h∼Ri−1predicted by statistical mechanics is valid\nfor large but moderate Richardson numbers ( Ri∼10), when the interface is permanently\nbreaking, while it is not valid for very high Richardson numb ers, when the interface is\nonly breaking intermittently. In this case, the interface a cts as a mixing barrier that\nprevent mixing in physical space, and in phase space.\n5.2.3. Vertical profile of the variance of density levels\nStatistical mechanics predicts not only the mean vertical d ensity profile but also the\npresence of density fluctuations across the interface. Thes e fluctuations can be related to\nthe mean density profile and to the effective temperature of th e turbulent flow through\nEq. (2.18). We assume that the inverse temperature is given b yβfit=/parenleftbig\n∆hfit∆bfit/parenrightbig−1\ndefined Eq. (5.14). We also assume that the mean density profil eb(z)is well described\nby the tanh profile defined Eq. (5.13). Then Eq. (2.18) yields\nb2−b2\n(∆bfit)2=1\n2/parenleftbigg\ncosh/parenleftbiggz−h\n∆hfit/parenrightbigg/parenrightbigg−2\n. (5.15)Fluctuations across a density interface 25\nThe observed vertical variation of the variance of density fl uctuations is plotted in Fig. 9-c\nfor different experiments (corresponding to different Richa rdson numbers). The vertical\ncoordinate z∗= (z−h)/∆hfithas been rescaled by the interface thickness for each\nexperiment, and we consider the frame of reference followin g the interface elevation, just\nas in Fig.9-a. The variance of density fluctuations is rescal ed by/parenleftbig\n∆bfit/parenrightbig2.\nThe thin black line is the statistical mechanics prediction given by Eq. (5.15), This\ntheoretical prediction is qualitatively correct sufficient ly close to the interface. Far above\nthe interface, the statistical mechanics theory overestim ates the fluctuations : these fluc-\ntuations are absent in the upper layer since the turbulence i s located in the lower layer\nand around the interface. Far below the interface, within th e mixed layer, equilibrium\ntheory underestimate the density fluctuations: filaments of light fluids entrained in the\nmixed layer are not stirred as much as would be predicted by th e equilibrium theory: the\ndensity field is strongly out of equilibrium in this region.\nAlthough the observed vertical profiles of density fluctuati ons close to the interface are\nclose to the predicted vertical profile, their amplitude rem ains smaller than the amplitude\npredicted by the equilibrium theory. The main reason for thi s discrepancy is that the the-\nory does not take into account irreversible mixing of densit y, which tends to decrease the\ndensity fluctuations. In addition, we clearly see on Fig. (8) that the density distributions\nclose to the interface contains a continuum of density level s between the extremal values.\nThe presence of these intermediate density levels which wer e not initially present in the\ntwo layer density field are also evidence for irreversible mi xing by turbulent cascade. This\neffect can not be captured in the framework of the simple two le vel system. Modelling the\ncombined effect of irreversible mixing (through turbulence cascade) and of the relaxation\ntoward equilibrium has partly been addressed by Venaille & S ommeria (2010) and will\nbe the object of future work\nTo conclude, one can distinguish two regions for the density field: i/ The region close\nto the density interface, which may be interpreted as an equi librium state once the effect\nof interfacial waves and interface increase due to entrainm ent are removed. The theory\npredict correctly the mean vertical density profile, but ove restimate the fluctuations,\nand the observed density distribution is different from the i nitially postulated two level\ndistribution. ii/ The region far from the interface ( z≫∆h), which is strongly out of\nequilibrium since the observed variance of density fluctuat ions is much larger than the\none predicted by the equilibrium theory. The aim of the next s ection is to describe in\nmore details the properties of density fluctuations in this o ut-of equilibrium region.\n5.3.Exponential shape of the tracer distribution far from the int erface\nSufficiently far from the interface, the fluid motion is not infl uenced by the motion induced\nby the interfacial waves, and the distribution of density le vels observed in Fig. 8 (obtained\nwithout change of reference frame) can directly be interpre ted as turbulent fluctuations.\nWe see on Fig. 10-a the distribution of density levels plotte d for each experiment at five\ndifferent depth corresponding to five prescribed values of th e density variance (relative\nto the density variance at the interface for each experiment ). The density levels on the\nhorizontal axis are normalised by ∆bfit. According to Eq. (5.13), ∆bfitis the maximal\nvalue of the x-averaged vertical profile of density for each experiment. S ince the initial\ndensity jump is ∆b0>∆bfit, density levels larger than ∆bfitcan be observed.\nThe depth and normalised density variance corresponding to each plotted density dis-\ntribution are shown Fig. 10-b. We see on Fig. (8) that a given v alue of the density variance\nis associated with a well defined depth when sufficiently close to the interface, but that26 A. Venaille, L. Gostiaux and J. Sommeria\nFigure 9. a) Vertical profile of the x-averaged density. The thin black line is the tanh profile\npredicted by equilibrium statistical mechanics in the two l evel case. Red circle: EXP7; black\ntriangle: EXP1; magenta triangle: EXP2; green star: EXP3; b lue diamond: EXP4. b) Variation of\nthe interface thickness ∆hfitwith the Richardson number. c) Vertical variations of the va riance\nof the density distribution.The thin black line in the stati stical mechanics prediction Eq. (5.15).\nthe depth associated with a given value of the variance are mo re scattered far from the\ninterface.\nStrikingly, the density distributions of Fig. 10-a do colla pse qualitatively well, and are\ncharacterised by exponential tails, with an e-folding dept h that decreases at increasing\ndistances from the interface.\nExponential tails in the distribution of a tracer in turbule nt flow have been previously\nreported either in the case of an isolated source dischargin g the tracer into an infinite\n(unconfined) medium (Duplat et al. 2010), or in the case a confined medium with large\nscale inhomogeneities due the injection of the tracer at the boundaries. This is for in-\nstance the case in convection experiments, where exponenti al tails in the temperature\ndistribution have been reported for very high Reynolds numb ers (Castaing et al. 1989).\nIn these convection experiments, one can consider that the t emperature in the bulk is\nstatistically homogeneous. The statistically steady dist ributions result from a competi-\ntion between turbulent cascade that tends to dissipate temp erature fluctuations and the\ntracer fluxes at the boundaries that inject fluctuations in th e bulk (Pumir et al. 1991).\nIn this approach, the density is considered as a passive trac er\nAs far as the distribution of density levels is concerned, th e mixed layer in the present\nstably stratified experiment is analogous to the mixed layer in the convection experiment\nat very high Reynolds number. The only difference is that the s ource of density fluctua-\ntions in the stably stratified experiment comes only from the interface at the top of the\nmixed layer, while the injection of density fluctuations com es from both the upper and\nthe lower layer in convection experiments. This asymmetry i n the injection of density\nfluctuations explains why only one tail of the density distri bution present an exponential\nshape in the case of the stably stratified experiment.\n6. Conclusion\nWe have proposed a statistical mechanics interpretation of the formation of sharp but\nhighly corrugated density interface between region of homo geneous density in the pres-\nence of turbulence, building upon previous work by Tabak & Ta l (2004), which generalisesFluctuations across a density interface 27\nFigure 10. a) Probability distribution function of the density at incr easing distances from the\ninterface. Each symbol corresponds to a different experimen t (o: EXP7, ⋄: EXP2, ∗: EXP4). For\neach experiment, 5 different depth (associated with 5 differe nt colors on the plot) are considered.\nThose 5 depth are chosen such that the density variances of th e pdf normalised by the density\nvariance at the interface are the same for a given color (the r elation between depth and the\nprescribed values of density variance is shown on panel b). F or each experiment, the density\nlevels on the xaxis are normalised by the maximum value of the x-averaged de nsity, denoted\n∆bfit. b) Variation of the height (vertical axis) associated with prescribed values of relative\ndensity variance (/parenleftBig\nb2−b2/parenrightBig\n//parenleftBig\nb2−b2/parenrightBig\n0on the horizontal axis).\nthe Miller-Robert-Sommeria approach for the vorticity in t wo-dimensional turbulence to\nthe density in three dimensional stratified turbulence. The statistics of the density field\nis predicted as the most probable outcome of turbulent stirr ing. An effective \"heat\"\nbath is provided by the turbulent velocity field. The tempera ture of this \"heat bath\" is\nproportional to the turbulent eddy kinetic energy.\nIn the case of a system initially composed of two homogeneous layers with a stable\ndensity interface, the theory predicts a tanh-shape for the mean vertical density profile.\nThis equilibrium density profile is interpreted as the resul t of a competition between\nturbulent transport that tends to smooth out the interface a nd buoyancy forces that\ntend to sharpen the interface by the sorting of the fluid eleme nts by density. For large\nRichardson numbers, buoyancy takes over turbulent transpo rt and the interface is thin.\nMore precisely, the interface thickness is inversely propo rtional to the Richardson number.\nThe equilibrium theory alone cannot describe entrainment a cross the interface, which\nwould eventually lead to complete homogenisation of the den sity field. In the case of\nthe experiments presented in this paper, turbulence genera tion is limited to the lower\nlayer. The interface is progressively drifting upward by th e entrainment of fresh fluid, and\nthe lower layer density decreases due to this mixing process . Entrainment is related to\nirreversible mixing of density levels through turbulent ca scade, a process which changes\nthe global distribution of density levels. We propose a simp lified model which keeps a\ntwo level system at statistical equilibrium, accounting fo r the effect of entrainment by a\nprogressive decrease of the lower layer density.\nThe advantage of the statistical mechanics approach over pr evious models is that it\nprovides a prediction for finite interface thickness in the p resence of a source of turbu-28 A. Venaille, L. Gostiaux and J. Sommeria\nlence without assumption about turbulent diffusivity. In ad dition, it does not rely on a\nparticular mechanism at stake close to the interface (wave b reaking, Kelvin-Helmholtz\ninstability,...) that may depend on the Richardson number ( McGrath et al. 1997). The\nonly assumption is that the system sufficiently explores the p hase space.\nIn order to test the equilibrium theory, we assumed the exist ence of a statistically\nstationary state, and obtained statistics of the density fie ld close to the interface using a\nreference frame following the horizontally averaged inter face height. This method allowed\nus to get rid of the out-of equilibrium effects of the entrainm ent and of the gravest spatial\nmodes of the interfacial gravity waves (which were found to d ominate the temporal\nfluctuations of the interface elevation). We found that the s hape of the observed mean\nvertical density profile is well fitted by a hyperbolic tangen t function, as predicted by the\nequilibrium theory in the case of a two level system. Further more the interface thickness\nwas found to be inversely proportional to the Richardson num ber between Ri ∼10\nand Ri∼100, as expected form the theory. We do not know whether these pre dictions\nwould remain valid for higher Richardson numbers (still for a turbulent flow). Indeed,\nthe interface then becomes locally sharper than the resolut ion of our measurements,\nwith spatially and temporally intermittent mixing events a cross this interface, see also\nMcGrath et al. (1997). The statistical theory also provides a good predict ion for the\nvertical profile of the rms density fluctuation variance at th e interface. The value of this\nvariance is however smaller than predicted by theory. The di screpancy is probably due\nto the dissipation of density fluctuations by the turbulent c ascade.\nBelow the interface, the measured density fluctuations are b y contrast much stronger\nthan the statistical equilibrium prediction, which is very low in relation with the quasi-\nuniform mean density, according to Eq. (2.17). These densit y fluctuations are strongly\nout-of equilibrium: they are transported from the interfac e by turbulent transport pro-\ncesses. In this region, we observed exponential tails in the probability distribution func-\ntion of density levels. Such exponential tails can be attrib uted to isolated filaments that\nare entrained from the interface and then stirred through tu rbulent cascade in the lower\nlayer. The density then behaves as a passive scalar, and the s hape of its pdf results from\na competition between turbulent cascade and the injection o f light density filaments en-\ntrained from the interface. This situation is analogous to c onvection experiment at high\nReynolds numbers, where exponential tails in the density pd f have been also previously\nreported.\nDensity fluctuations discussed above correspond to rather s mall scales, for which tur-\nbulent motion prevails. At larger scale the interface fluctu ates as internal waves, for which\nwe were able to check the dispersion relation. We observe tha t the spatial and tempo-\nral spectra of these waves can be interpreted by a theory due t o Phillips (1977) which\nstates that the wave amplitude at each frequency is such that the induced flow is close\nto criticality for shear instability at the interface. In th e range of frequency much larger\nthan the gravest mode, and much smaller than the buoyancy fre quency characterising\nthe density gradient inside the interface, this theory pred icts a−3slope for the inter-\nface elevation frequency power spectrum. We confirm this res ult, which also agrees with\nprevious observations by Hannoun & List (1988). The value of the predicted spectral en-\nergy depends on the interface thickness as an input. We obser ved that the amplitude of\nthe waves in this range of frequency is independent from the R ichardson number, which\ncorresponds to the statistical equilibrium prediction of a n interface thickness varying as\nthe inverse of the Richardson number. This scaling is differe nt than the one obtained by\nHannoun & List (1988), and we do not know whether it would stil l be valid for higher\nRichardson number (beyond 100), as explained above. Note fin ally that the Phillipp’sFluctuations across a density interface 29\napproach is limited to frequencies for which the turbulent v elocities are negligible with\nrespect to the potential flow associated with the variations of the interface elevation. A\nmore detailed study of the coupling between interfacial wav es and turbulence has been\naddressed by Fernando & Hunt (1997); McGrath et al. (1997).\nWe focused in this paper to the visualisation of the density fi eld using PLIF technique\nwith index matching. The only PIV measurement presented in t his paper were performed\nin a case without stratification. The underlying assumption was that turbulence prop-\nerties were not much affected by the presence of stratificatio n (below the interface). We\nchose to use PLIF only in order to obtain quantitative measur ements of turbulent fluc-\ntuations in the density field (the presence of particle used t he PIV measurements alters\nthe quality of the visualisation). However, we hope to addre ss in future for a more de-\ntailed study of the buoyancy fluxes close to the interface usi ng Simultaneous PIV and\nPLIF measurement, which has recently been performed in the c ontext of entrainment in\ngravity currents (Odier et al. 2014).\nTo conclude, equilibrium statistical mechanics allows us t o interpret qualitatively the\nformation of thin but corrugated and turbulent density inte rface between regions of homo-\ngeneous density. The approach is limited by out-of equilibr ium effect such as irreversible\nmixing and interfacial wave excitation. We believe that com bining these different phe-\nnomenological approaches will lead to fruitful models in th e context of turbulent mixing\nacross a density interface, or more generally in any stratifi ed turbulence problem where\ndensity fluctuations play an important role, as for intense i n gravity currents (Odier et al.\n2009, 2012). It would be also interesting to check the releva nce of equilibrium statistical\nmechanics at the interface of two immiscible fluids. On the on e hand, this would avoid\nthe issue with irreversible mixing through turbulence casc ade, but on the other hand,\nsurface tension effects may be influential.\n7. Appendix A: Equipartition and mixing efficiency of 0.25in the low\nenergy limit\nEq. (2.18) allows to derive an interesting side result conce rning energy equipartition\nin the low energy limit. This result may in turn be used to pred ict a mixing efficiency\ncoefficient, which is a measure of the fraction of the energy in jected in a stratified fluid\nthat is actually used to irreversibly increase the potentia l energy of the flow.\nLet us consider a stratified fluid initially at rest, characte rised by its density profile\nbbg(z)(the index “bg” stands for “background”). The potential ene rgy of this state if\ndenoted Ebg,p≡/integraltextH\n0dz bz. Let us then assume that the fluid is isolated, that a given\namount of energy ∆Eis injected in the system, and that a statistical equilibriu m is\nreached on a time scale much shorter than the time scale for vi scous dissipation and for\nirreversible mixing of density. Let us call EcandEpthe kinetic and the potential energy\nof the equilibrium state. The available potential energy of the equilibrium state is\n∆Ep≡/integraldisplay /integraldisplay /integraldisplay\ndxdydz(b−bbg)z=/integraldisplay\ndz/parenleftbig\nb−bbg/parenrightbig\n. (7.1)\nThe second equality is obtained by noting that the equilibri um state is statistically invari-\nant on the horizontal, and that the horizontal integral amou nts to an ensemble average.\nFor the equilibrium state, the potential energy ∆Epmay be qualified as available since\none recover b=bbgjust by setting the effective temperature to 0 ( β= +∞). Since\neach fluid particle conserves its density in the absence of di ssipation, a fluid particle of\ndensitybat height zand time tcan be seen as a fluid particle initially at height zbg(b),30 A. Venaille, L. Gostiaux and J. Sommeria\ndisplaced by ξ(x,y,z) =z−zbg(b(x,y,z))in the vertical direction. Let us assume that\nthis displacement ξis sufficiently small, which is ensured by considering a low en ergy\nlimit. At lowest order, the density fluctuation defined by b′(x,y,z) =b(x,y,z)−bbg(z)\nis proportional to the vertical fluid particle displacement :b′=ξ∂zbbg, and the available\npotential energy can be expressed (still at lowest order) as\n∆Ep=1\n2/integraldisplay /integraldisplay /integraldisplay\ndxdydzb′2\n∂zbbg=1\n2/integraldisplay\ndzb2−b2\n∂zbbg. (7.2)\nAgain, the second equality is obtained by noting the horizon tal integration amounts to\nan ensemble average. Injecting then Eq. (2.18) in Eq. (7.2) g ives\n∆Ep=1\n2β/integraldisplay\ndz∂zb\n∂zbbg. (7.3)\nIn the low energy limit, we get at lowest order ∂zb≈∂zbbg, which yields ∆Ep=V/(2β),\nwhereVis the volume where the flow takes place. The inverse temperat ureβis related\nto the total kinetic energy ∆Ec=Vecthrough Eq. (2.15), which yields ∆Ep= ∆Ec/3.\nThis expresses equipartition of the energy between the avai lable potential energy and the\nthree degrees of freedom of the kinetic energy.\nFinally, the ratio of the available potential energy with th e total energy injected in the\nsystem is\nη≡∆Ep\n∆Ep+∆Ec=1\n4. (7.4)\nLet us now assume that once the equilibrium state is reached, the density fluctuations\nare smoothed out on each horizontal plane due to the combined effect of direct turbulent\ncascade and molecular diffusivity. Let us also assume that th e rate of kinetic energy dis-\nsipation is equal to the rate of dissipation for the density v ariance at each height. These\nhypothesis ensure that the rhs of Eq. (2.18) remains constan t through the flow evolution,\nand that the profile bremains the equilibrium state throughout the flow evolution . At\nsufficiently large time, once the fluctuations around bare irreversibly mixed, the flow is\nat rest and this bbecomes the new background density profile. The increase of p oten-\ntial energy ∆Epdefined by Eq. (7.1) accounts therefore for the irreversible increase of\npotential energy †. The mixing efficiency defined as the irreversible increase of potential\nenergy normalised by the total energy injected in the system is then simply given by Eq.\n(7.4).\nTo conclude, a mixing efficiency coefficient of 1/4can be interpreted as a result of\nenergy equipartition at equilibrium under the following as sumption: i/ a low energy limit\nii/ a time scale for the dissipation of horizontal density flu ctuations much larger than\nthe time scale to reach the equilibrium state iii/ a rate of di ssipation of the variance of\ndensity fluctuation at each height equal to the rate of dissip ation of the kinetic energy. If\nthese conditions are not fulfilled, then the mixing efficiency should be smaller than 0.25.\nMixing efficiency coefficients between 0.2 and 0.3 are widely us ed in modelling con-\ntext. However, experiments and simulations and observatio ns seem to show that there is\nno universal mixing efficiency in stably stratified turbulenc e Peltier & Caulfield (2003);\nIveyet al. (2008). Our result suggest that the value 0.25may be interpreted as a limit\ncase in the framework of equilibrium statistical mechanics .\n†Note that Eq. (7.2) is derived from Eq. (7.1) by assuming that each fluid particle conserves\nits density; these equations are no more equivalent once the fluctuations of density have been\nsmoothed out at each height z(in which case the available potential energy vanish). Howe ver,\nthe result ∆Ep= ∆Ec/3remains valid since it is obtained at equilibrium.Fluctuations across a density interface 31\n8. Appendix B : Turbulence properties in the homogeneous cas e\nWe performed one experiment without stratification in order to have a reference flow\nthat we analysed using PIV measurements. This allowed us to o btain some characteristics\nof the flow in the turbulent layer only, assuming that these flo w properties would not be\nmuch different in the presence of stratification provided tha t one consider the dynamics\nsufficiently below the interface. In particular, this method s allows to estimate the energy\nflux due to the grid forcing, and the decay of the turbulence st rength with altitude.\nLet us call Lt(z)the integral length scale of turbulence. Sufficiently close t o the source,\nthe integral length scale of turbulence if given by the grid m esh of the oscillating grid\nLt=Lf. Sufficiently far from the source, the only length scale of the problem is the\ndistance from the source and Lt∼z. The temporal evolution of the kinetic energy may\nbe modelled as\n∂tec=∂z(νt∂zec)−cd\nLte3/2\nc, (8.1)\nwhere the effect of turbulence is modelled as effective viscos ity, with a turbulent energy\nflux\nFec=−νt∂zec, νt=ae1/2\ncLt. (8.2)\nThe sink of energy is given by a Kolmogorov dissipation term. Sufficiently close to the\ngrid,Lt=Lfis a constant and stationary kinetic energy profile is expone ntial:\nec=e0\ncexp/parenleftbigg\n−z\nLt/parenrightbigg\n, Lt=/parenleftbigga\ncd/parenrightbigg1/2\nL (8.3)\nwithe0\ncthe kinetic energy at the source located in z= 0. Sufficiently far from the source\nLt∼zand stationary kinetic energy profile is given by a power law ec∼z−2, see e.g.\nHopfinger & Toly (1976).\nREFERENCES\nBalmforth, N.J., Llewellyn Smith, S.G. & Young, W.R. 1998 Dynamics of interfaces\nand layers in a stratified turbulent fluid. Journal of Fluid Mechanics 355, 329–358.\nBouchet, F. & Venaille, A. 2012 Statistical mechanics of two-dimensional and geophys ical\nflows. Physics reports 515(5), 227–295.\nCarruthers, D.J. & Hunt, J.C.R. 1986 Velocity fluctuations near an interface between a\nturbulent region and a stably stratified layer. Journal of Fluid Mechanics 165, 475–501.\nCastaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libc haber, A., Thomae,\nS., Wu, X.-Z., Zaleski, S. & Zanetti, G. 1989 Scaling of hard thermal turbulence in\nrayleigh-bénard convection. Journal of Fluid Mechanics 204, 1–30.\nCrapper, P.F. & Linden, P.F. 1974 The structure of turbulent density interfaces. Journal\nof Fluid Mechanics 65(01), 45–63.\nDuplat, J., Innocenti, C. & Villermaux, E. 2010 A nonsequential turbulent mixing pro-\ncess.Physics of Fluids (1994-present) 22(3), 035104.\nE, X. & Hopfinger, E. J. 1986 On mixing across an interface in stably stratified fluid. Journal\nof Fluid Mechanics 166, 227–244.\nFernando, H.J.S. & Long, R.R. 1985 On the nature of the entrainment interface of a two-\nlayer fluid subjected to zero-mean-shear turbulence. Journal of Fluid Mechanics 151, 21–53.\nFernando, H. J. S. 1991 Turbulent mixing in stratified fluids. Annual Review of Fluid Me-\nchanics 23, 455–493.\nFernando, H. J. S. & Hunt, J. C. 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Journal of Fluid Mechanics\n10(04), 496–508.\nMiller, J. 1990 Statistical mechanics of euler equations in two dimens ions.Physical review\nletters 65(17), 2137.\nMory, M. 1991 A model of turbulent mixing across a density interface i ncluding the effect of\nrotation. Journal of Fluid Mechanics 223, 193–207.\nOdier, P., Chen, J. & Ecke, R.E. 2012 Understanding and modeling turbulent fluxes and\nentrainment in a gravity current. Physica D: Nonlinear Phenomena 241(3), 260–268.\nOdier, P., Chen, J. & Ecke, R. E. 2014 Entrainment and mixing in a laboratory model of\noceanic overflow. Journal of Fluid Mechanics 746, 498–535.\nOdier, P., Chen, J., Rivera, M. K. & Ecke, R. E. 2009 Fluid mixing in stratified gravity\ncurrents: The prandtl mixing length. Physical review letters 102(13), 134504.\nPark, Y.-G.and Whitehead, J.A. & Gnanadeskian, A. 1994 Turbulent mixing in stratified\nfluids: layer formation and energetics. Journal of Fluid Mechanics 279, 279–311.\nPeltier, WR & Caulfield, CP 2003 Mixing efficiency in stratified shear flows. Annual review\nof fluid mechanics 35(1), 135–167.\nPhillips, O.M. 1972 Turbulence in a strongly stratified fluid: is it unstable ? InDeep Sea\nResearch and Oceanographic Abstracts , , vol. 19, pp. 79–81. Elsevier.\nPhillips, O.M. 1977 The dynamic of upper ocean .\nPosmentier, E.S. 1977 The generation of salinity finestructure by vertical di ffusion. Journal\nof Physical Oceanography 7(2), 298–300.\nPumir, A., Shraiman, B. I. & Siggia, E. D. 1991 Exponential tails and random advection.\nPhysical review letters 66, 2984–2987.\nRenaud, A., Venaille, A. & Bouchet, F. 2014 Equilibrium states and energy partition of\nthe shallow water model. preprint 00(00), 000.\nRobert, R. & Sommeria, J. 1991 Statistical equilibrium states for two-dimensional fl ows.\nJournal of Fluid Mechanics 229, 291–310.\nRouse, H. & Dodu, J. 1955 Turbulent diffusion across a density discontinuity. La Houille\nBlanche 4, 522–532.\nRuddick, B.R., McDougall, T.J. & Turner, J.S. 1989 The formation of layers in a uni-\nformly stirred density gradient. Deep Sea Research Part A. Oceanographic Research Papers\n36(4), 597–609.\nSalmon, R. 1998Lectures on geophysical fluid dynamics , , vol. 378. Oxford University Press\nOxford.\nSchmitt, R.W. 1994 Double diffusion in oceanography. Annual Review of Fluid Mechanics\n26(1), 255–285.Fluctuations across a density interface 33\nSimpson, J.H. & Woods, J.D. 1970 Temperature microstructure in a fresh water thermocli ne.\nNature .\nSommeria, J. 2001 Two-dimensional turbulence. In New trends in turbulence Turbulence: nou-\nveaux aspects , pp. 385–447. Springer.\nSullivan, P.J. 1972 The penetration of a density interface by heavy vortex r ings.Water, Air,\nand Soil Pollution 1(3), 322–336.\nTabak, E.G. & Tal, F.A. 2004 Mixing in simple models for turbulent diffusion. Communica-\ntions on pure and applied mathematics 57(5), 563–589.\nTurner, J. S. 1968 The influence of molecular diffusivity on turbulent entr ainment across a\ndensity interface. Journal of Fluid Mechanics 33, 639–656.\nVenaille, A. & Sommeria, J. 2010 Modeling mixing in two-dimensional turbulence and str at-\nified fluids. In IUTAM Symposium on Turbulence in the Atmosphere and Oceans: Proceed-\nings of the IUTAM Symposium on Turbulence in the Atmosphere a nd Oceans, Cambridge,\nUK, December 8-12, 2008 , , vol. 28, p. 155. Springer.\nWhitehead, J.A. & Stevenson, I. 2007 Turbulent mixing of two-layer stratified fluid. Physics\nof Fluids (1994-present) 19(12), 125104.\nWolanski, E.J .and Brush, L.M. 1975 Turbulent entrainment across stable density step\nstructures. Tellus 27(3), 259–268."    },    {        "title": "1412.1108v2.Phenomenological_QCD_equation_of_state_for_massive_neutron_stars.pdf",        "content": "Phenomenological QCD equation of state for massive neutron stars\nToru Kojo,1Philip D. Powell,1, 2Yifan Song,1and Gordon Baym1, 3\n1Department of Physics, University of Illinois at Urbana-Champaign,\n1110 W. Green Street, Urbana, Illinois 61801, USA\n2Lawrence Livermore National Laboratory, 7000 East Ave., Livermore, CA 94550\n3Theoretical Research Division, Nishina Center, RIKEN, Wako 351-0198, Japan\n(Dated: June 28, 2021)\nWe construct an equation of state for massive neutron stars based on quantum chromodynam-\nics phenomenology. Our primary purpose is to delineate the relevant ingredients of equations of\nstate that simultaneously have the required sti\u000bness and satisfy constraints from thermodynamics\nand causality. These ingredients are: (i) a repulsive density-density interaction, universal for all\n\ravors; (ii) the color-magnetic interaction active from low to high densities; (iii) con\fning e\u000bects,\nwhich become increasingly important as the baryon density decreases; (iv) nonperturbative gluons,\nwhich are not very sensitive to changes of the quark density. We use the following \"3-window\"\ndescription: At baryon densities below about twice normal nuclear density, 2 n0, we use the Akmal-\nPandharipande-Ravenhall (APR) equation of state, and at high densities, \u0015(4\u00007)n0, we use the\nthree-\ravor Nambu-Jona-Lasinio (NJL) model supplemented by vector and diquark interactions. In\nthe transition density region, we smoothly interpolate the hadronic and quark equations of state in\nthe chemical potential-pressure plane. Requiring that the equation of state approach APR at low\ndensities, we \fnd that the quark pressure in noncon\fning models can be larger than the hadronic\npressure, unlike in conventional equations of state. We show that consistent equations of state of\nsti\u000bness su\u000ecient to allow massive neutron stars are reasonably tightly constrained, suggesting that\ngluon dynamics remains nonperturbative even at baryon densities \u001810n0.\nI. INTRODUCTION\nAs the Relativistic Heavy Ion Collider and the Large\nHadron Collider continue to push the limits of our ex-\nperimental knowledge of hot dense quantum chromody-\nnamics (QCD), neutron stars are the only cosmic lab-\noratories in which we can study the structure of cold\ndense QCD [1{5]. The recent discoveries of neutron stars\nwith masses M'2M\f(M\fis the solar mass), includ-\ning the binary millisecond pulsar J1614-2230 with mass\n(1:97\u00060:04)M\f[6] and the pulsar J0348+0432 with mass\n(2:01\u00060:04)M\f[7] (also PSR J1311-3430 [8]), together\nwith recent simultaneous determinations of neutron star\nmasses and radii [9, 10] pose particular challenges to the\ntheoretical construction of the neutron star equation of\nstate.\nOn the one hand, the existence of these massive stars\nsuggests that the equation of state must be sti\u000ber than\nconventional hadronic descriptions of matter including\nhyperons. Furthermore, the central baryon density in\nneutron stars with masses \u00182M\fwell exceeds twice nu-\nclear matter density n0, and may reach as high as \u001810n0.\nTo understand why such high mass stars are stable re-\nquires a knowledge of the equation of state at baryon\ndensitiesnBover a range\u00181\u000010n0. However we can-\nnot at present reliably calculate the equation of state over\nsuch a range; the densities are too high to apply reliably\nconventional hadronic equations of state, and too low to\napply perturbative QCD.\nThis situation motivates us to investigate the prop-\nerties of strongly correlated quark matter, intermediate\nbetween the hadronic and perturbative QCD phases, and\nask how the properties of such matter is constrainedby neutron star observations. Using a schematic quark\nmodel, we manifestly take into account quark degrees of\nfreedom, while including interaction e\u000bects such as vec-\ntor repulsion between quarks, known from hadron spec-\ntroscopy, color-magnetic diquark interactions, and six-\nquark interactions arising from the axial anomaly. We\nexamine the roles of these interactions and \fnd that it\nis possible, within a reasonable parameter range, to con-\nstruct an equation of state that (i) is su\u000eciently sti\u000b\nto include stable stars with M\u00182M\f; (ii) satis\fes\nthe thermodynamic constraint that the baryon number\ndensity be an increasing function of the baryon chemi-\ncal potential, @nB=@\u0016B>0; and (iii) is consistent with\nthe (suggestive) causality constraint that the speed of\nsound (at zero frequency) not exceed the speed of light\n[11, 12]. While these conditions provide relatively tight\nconstraints on the quark matter equation of state, it is\nnonetheless possible to construct the desired equation of\nstate using quark model parameters compatible with the\nhadron spectroscopy.\nTo further motivate the picture of strongly correlated\nquark matter, we brie\ry review the domain of applicabil-\nity of hadronic and perturbative QCD equations of state.\nConventional hadronic equations of state are constrained\nby experimental data at low energy and density, e.g.,\ntwo-body hadronic scattering below the pion production\nthreshold, the masses and level structure of light nuclei,\nand nuclear matter around nuclear saturation density n0.\nWhile hadronic equations of state include the relevant\nphysics in the low density regime in which their param-\neters are \ft, with increasing density multiple meson ex-\nchanges, many-baryon interactions, and virtual baryonic\nexcitations become increasingly important. (The system-arXiv:1412.1108v2  [hep-ph]  19 Jan 20152\nFIG. 1: Pressure vs. quark chemical potential for several equations of state. The black line is the APR result [16] (A18+ \u000ev+UIX*\nwithout pion condensation)): the bold line for nB<2n0and thin dotted line for nB>2n0. Various e\u000bects are successively added to\nthe standard NJL model; (a) a repulsive density-density interaction, which sti\u000bens the NJL equation of state ; (b) the color-magnetic\ninteraction (diquark correlation), which reduces the average quark energy at all densities; and (c) con\fning e\u000bects, which suppress the\narti\fcially large pressure in NJL models at low density down to the APR pressure, discussed in the text.\natics can be most clearly seen in the chiral e\u000bective theory\napproach [13{15].) In nucleonic potential models [16], the\nthree-body nucleon interaction is crucial to reproducing\nnuclear matter properties at nB'n0, and its contribu-\ntion to the energy density can be even comparable (and of\nopposite sign) to that of the two-body force at nB\u00182n0.\nBeyond baryon densities nB\u001dn0a well de\fned expan-\nsion in terms of static two-, three-, or more, body forces\nno longer exists.\nThe equation of state of perturbative QCD [17{19] re-\nlies on a picture of weakly coupled quarks and gluons.\nA current state-of-the-art calculation in this regime, to\nsecond order in the strong interaction \fne structure con-\nstant\u000bs, with strange quark mass corrections [18], \fnds\na relatively strong dependence of the QCD equation of\nstate on the renormalization scale below the quark chem-\nical potential \u0016\u00181 GeV, which corresponds to a baryon\ndensity\u0018102n0. Such dependence indicates that non-\nperturbative e\u000bects remain quite important in the lower\ndensity range relevant to neutron stars.\nIn constructing a phenomenological QCD equation of\nstate here, we follow the spirit of the \\3-window\" ap-\nproach of Masuda, Hatsuda, and Takatsuka [20] that in-\nterpolates between a nuclear equation of state at low\ndensity and a quark equation of state at high density.\nAt densities below 2 n0we adopt the hadronic Akmal-\nPandaripande-Ravenhall (APR) equation of state [16]\n(denoted in their paper as A18+ \u000ev+UIX*). At densities\nabove 4-7n0, where a gas of baryons of radius 0.4-0.5\nfm would begin to percolate [21], we employ a three-\n\ravor Nambu{Jona-Lasinio (NJL) quark model includ-\ning vector and diquark interactions. While we use the\nNJL model to be speci\fc, our discussions are more gen-\neral. In the intermediate region, where purely hadronic\nor purely quark descriptions are not appropriate, we con-\nstruct an equation of state using a smooth polynomial\ninterpolation in the baryon chemical potential{pressure\n(\u0016,P) plane. In this plane the pressure must be a con-\ntinuous and monotonically increasing function of \u0016. One\ncannot rule out the possibility of a \frst order transitionas the baryon density increases; such a transition would\nappear as a discontinuity in the \frst derivative, @P=@\u0016 .\nThe 3-window approach is quite di\u000berent from the con-\nventional hybrid one in which one regards the quark and\nhadronic phases as distinct. In the latter, the quark pres-\nsure at given \u0016must, with increasing \u0016, intersect the\nhadronic pressure from below, and moreover must re-\nmain larger than the hadronic pressure at larger \u0016. By\nregarding the hadronic equation of state at density larger\nthan\u00182n0as a valid description of matter, such a hybrid\nconstruction implicitly selects out possible forms of quark\nequations of state; in order to have an intersection, the\nquark pressure must be larger than that of the hadronic\nphase as large \u0016. Such quark equations of state are typi-\ncally soft, and as a consequence hybrid stars with larger\nquark cores tend to have smaller masses. In contrast, in\nthe 3-window approach, we construct high density quark\nequations of state independently of assuming a trustwor-\nthy high density hadronic pressure; at high density, the\nresulting quark pressure at given \u0016does not have to grow\nfast and may remain smaller than the pressure extrapo-\nlated from the hadronic phase. Such quark equations of\nstate tend to be sti\u000b, and a star with large quark core\ncan have a large mass.\nWithin this schematic 3-window description, we aim\nto incorporate the following e\u000bects known from ob-\nserved hadronic spectroscopy: (i) A repulsive \ravor-\nindependent density-density interaction [22], which sti\u000b-\nens the equation of state (Fig.1a). (ii) The attrac-\ntive color-magnetic interaction, relevant at all densi-\nties, which reduces the average single quark energy\n(Fig.1.(b)). This e\u000bect is similar to that observed in the\nconstituent quark model [23], in which the average quark\nenergy in a nucleon is reduced from the constituent quark\nmass,\u0018340 MeV, to one-third of the nucleon mass,\n\u0018313 MeV. As we show, this e\u000bect plays an important\nrole in ensuring that the interpolated equation of state\nsatis\fes thermodynamic constraints. (iii) Con\fnement,\nwhich, at low densities, traps quarks into baryons and\nforbids quarks to contribute signi\fcantly to the pressure.3\nAs we describe, the NJL model, which does not include\ncon\fnement, has a higher pressure at low density than\nnuclear models. The requirement that the interpolated\npressure merges smoothly into APR at low densities (Fig.\n1c) e\u000bectively suppresses such excess pressure.\nThe present approach of interpolating between a\nhadronic and an NJL based quark picture makes the tacit\nassumption that the behavior of the gluon sector does not\nchange appreciably over the range 0 .nB.10n0; and in\nparticular, that the gluons do not add a bag constant, Bg,\nto the energy (and subtracted from the pressure) when\nthe nonperturbative gluons become perturbative. The\nbag constant measures the energy di\u000berence between the\ntrivial (perturbative vacuum) and the nonperturbative\nvacuum. With the zero-point of the energy set to make\nthe QCD (nonperturbative) vacuum energy zero, the per-\nturbative vacuum has positive energy. Thus, whenever\nwe consider the extreme conditions under which nonper-\nturbative e\u000bects disappear, we must include the bag con-\nstant in addition to the contributions of the perturba-\ntive e\u000bects. However, the stability of massive neutron\nstars does not permit gluon condensation at the QCD\nscale \u0003 QCD\u00180:2 GeV to produce a gluonic bag con-\nstant,Bg\u0018\u00034\nQCD, since such a term would, as we argue,\ntoo greatly soften the equation of state; we thus exclude\nthe possibility of such a term. On the other hand, a\nquark bag constant, Bqof order \u00034\nQCD associated with\nrestoration of chiral symmetry, is unavoidable in the NJL\nmodel [24].\nWe extrapolate NJL parameters obtained via hadron\nphenomenology at nB\u0018n0to high density quark mat-\nter (nB\u001810n0) [25, 26], an approach that is consistent\nwith the observation from analyses for a large number of\ncolors,Nc, that gluon dynamics is insensitive to quark\nloop e\u000bects [27].\nThis paper is organized as follows. In Sec. II, we brie\ry\ndescribe the hadronic and quark models adopted in this\nstudy. In Sec. III A, we examine interaction e\u000bects on the\nquark equation of state. In Sec. IV, we construct the in-\nterpolated equation of state. In particular, we explain the\ndi\u000berence between the present 3-window description and\nconventional equations of state which introduce a \frst or-\nder phase transition between hadronic and quark matter\nin [1, 28{31]. As we see the constraints from thermody-\nnamics and causality are quite important. In Sec. V, we\nsolve the Tolman-Oppenheimer-Volko\u000b (TOV) equation\nand examine the resulting mass-radius ( M-R) relation of\nneutron stars. Section VI is devoted to a summary and\noutlook.\nWe use the following conventions: g\u0016\u0017 =\ndiag(1;\u00001;\u00001;\u00001),\r5=\ry\n5, and the charge conju-\ngation operator Cisi\r0\r2. The \ravor and color U(3)\ngenerator matrices \u001ci(i= 0;\u0001\u0001\u0001;8) and\u0015a(a= 0;\u0001\u0001\u0001;8)\nare composed, respectively, of the identity and Gell-\nMann matrices, and are normalized as tr[ \u001ci\u001cj] = 2\u000eij\nand tr[\u0015a\u0015b] = 2\u000eab. We work in units in which cand\n~= 1.II. MODELS\nIn this section we brie\ry summarize the features of\nthe hadronic (APR) and quark (NJL) matter equations\nof state employed in this paper. APR will be used to\ndescribe low density matter, nB<2n0, while the NJL\nmodel will be used at high densities, nB>(4\u00007)n0.\nThe precise density beyond which we adopt a fully quark\ndescription of matter will depend upon details of the in-\nterpolation, as discussed in section IV.\nA. The APR equation of state\nIn this work we adopt the A18+ \u000ev+UIX\u0003version of the\nAPR equation of state to describe low density hadronic\nmatter [16]. This equation of state, based on the Argonne\nv18two-body potential, which \fts hadronic scattering\ndata very well, and the Urbana IX three-body interac-\ntion, which is important to explain nuclear saturation\nproperties, includes charge neutrality and \f-equilibrium.\nThe\u000evindicates the inclusion of relativistic corrections.\nFor simplicity, we adopt the APR version excluding neu-\ntral pion condensate, which emerges at nB\u00181:4n0. The\nAPR model includes only nucleonic degrees of freedom,\nand does not take into account hyperons, whose interac-\ntions with nucleons and among themselves are not well\ndetermined. Typical models of nucleon-hyperon interac-\ntions predict hyperon onset at a density nB\u00182\u00003n0.\nWe restrict our application of APR to nB<2n0.\nB. The NJL equation of state\n1. The Lagrangian\nIn descriptions of quark matter, we adopt a three \ravor\nNambu{Jona-Lasinio model with Lagrangian density\nL=q(i=@\u0000^m)q+L(4)+L(6); (1)\nwhereqis a quark \feld with color, \ravor, and Dirac in-\ndices, ^mis the quark current mass matrix, and L(4)=\nL(4)\n\u001b+L(4)\nV+L(4)\ndandL(6)=L(6)\n\u001b+L(6)\n\u001bdare four- and\nsix-quark interaction terms, respectively, chosen to re-\n\rect the symmetries of QCD. The four-quark interac-\ntions possess UL(3)\u0002UR(3) symmetry for \ravors, while\nthe six-quark interactions re\rect the UA(1) anomaly.\nThe \frst of the four-quark interactions describes spon-\ntaneous chiral symmetry breaking:\nL(4)\n\u001b=G8X\ni=0\u0002\n(q\u001ciq)2+ (qi\r5\u001ciq)2\u0003\n= 8Gtr(\u001ey\u001e);(2)\nwhereG > 0 (attractive), and \u001eij= (qR)j\na(qL)i\nais the\nchiral operator with \ravor indices i;j(summed over the\ncolor index a).4\nThe second of the four-quark terms [32],\nL(4)\nV=\u0000gV(q\r\u0016q)2; (gV>0) (3)\ndescribes the repulsive density-density interaction, anal-\nogous to!-meson exchange in nuclear matter.\nThe third of the four-quark terms,\nL(4)\nd=HX\nA;A0=2;5;7\u0002\u0000\nqi\r5\u001cA\u0015A0CqT\u0001\u0000\nqTCi\r5\u001cA\u0015A0q\u0001\n+\u0000\nq\u001cA\u0015A0CqT\u0001\u0000\nqTC\u001cA\u0015A0q\u0001\u0003\n;\n= 2Htr(dy\nLdL+dy\nRdR); (H > 0); (4)\ndescribes attractive diquark pairing, where \u001cAand\u0015A0\n(A;A0= 2;5;7) are the antisymmetric generators of U(3)\n\ravor and SU(3) color, respectively. The structure of\nthe interaction can be understood as the color-magnetic\ninteraction in the 2 !2 scatterings of quarks in s-wave,\nspin-singlet, \ravor- and color- anti-triplet channel. The\noperators (dL;R)ai=\u000fabc\u000fijk(qL;R)j\nbC(qL;R)k\ncare diquark\noperators of left- and right-handed chirality.\nNext we discuss the six-quark interactions responsible\nfor theUA(1) anomaly [33]. The \frst term involves the\nproduct of the chiral condensates of di\u000berent \ravors:\nL(6)\n\u001b=\u00008K(det f\u001e+ h.c.);(K > 0) (5)\nwhere det fdenotes the determinant with respect to \ravor\nindices. The second term couples the chiral and diquark\ncondensates [34],\nL(6)\n\u001bd=K0(tr[(dy\nRdL)\u001e] + h.c.);(K0>0):(6)\nAt tree level, these two interactions may be related via\na Fierz transformation, which leads to the conclusion\nK0=K. However, renormalization e\u000bects will, in gen-\neral, destroy this equality so that at the mean \feld level\nwe may treat KandK0as independent parameters, but\nwithK0\u0018K.\n2. Electric and color charge neutrality constraints\nIn order to avoid energetically expensive static long-\nrange electric Coulomb interactions and color \rux tube\ncon\fgurations in stable homogeneous quark matter, we\nimpose the local electric and color charge neutrality con-\nstraints\nnQ(x) =na(x) = 0:(a= 1;\u0001\u0001\u0001;8) (7)\nwherenQis the local electric charge density, and the\nnaare the local color densities. These conditions are\nenforced via standard Lagrange multipliers { with the\nappropriate chemical potentials coupled to the electric\nand color charge densities, respectively [35].\nIntroducing the charge chemical potential \u0016Q, we add\nto the Lagrangian the terms\nLQ=\u0016Q(qyQq\u0000ly\nili);(i: summed) (8)whereQ= diag(2=3;\u00001=3;\u00001=3) is the quark charge\noperator for ( u;d;s ) quarks, in units of the proton charge\ne. Theli= (e;\u0016;\u001c ) are lepton \felds and mithe lepton\nmasses. We may safely omit contributions from \u0016- and\n\u001c- leptons since their populations are vanishingly small\nin the density range of interest [20].\nColors and \ravors in dense matter are coupled through\nthe diquark interactions of L(4)\nd. Thus, an asymmetry in\nquark \ravor densities (e.g., a 2SC phase) leads to a cor-\nresponding net quark color density. Most generally case,\nwe should introduce eight independent color chemical\npotentials [36]. However, for the diquark pairing struc-\ntures considered in this paper, all color densities except\nn3=hqy\u00153qiandn8=hqy\u00158qiautomatically vanish [37].\nThus, we only need to add the terms\nL3;8=\u00163qy\u00153q+\u00168qy\u00158q; (9)\nto constrain the system. The values of ( \u0016Q;\u00163;\u00168) will\nbe tuned to satisfy the neutrality conditions.\n3. Mean \feld equation of state\nThe mean \felds for the chiral condensate and quark\ndensities are\n\u001bi=hqiqii; n =3X\ni=1hqy\niqii: (10)\nBelow we write ( \u001b1;\u001b2;\u001b3) = (\u001bu;\u001bd;\u001bs) for later conve-\nnience. For the diquark mean \felds, we write\ndi=hqTCi\r5Riqi; (11)\nwhere\n(R1;R2;R3)\u0011(\u001c7\u00157;\u001c5\u00155;\u001c2\u00152): (12)\nWith these de\fnitions, the diquark condensates\n(d1;d2;d3) correspond to ( ds;su;ud ) quark pairings, re-\nspectively.\nThe thermodynamic potential may be computed from\nthe mean \feld particle propagators in terms of these\nmean \felds; the inverse of the propagator S(k), can be\nread o\u000b from the mean \feld Lagrangian [34],\nS\u00001(k) =\u0012=k\u0000^M+ ^\u0016\r0i\r5\u0001iRi\n\u0000i\r5\u0001\u0003\niRi=k\u0000^M\u0000^\u0016\r0\u0013\n;(13)\nwhere the e\u000bective mass matrix has diagonal elements\nMi=mi\u00004G\u001bi+Kj\u000fijkj\u001bj\u001bk+K0\n4jdij2;(14)\nwhile the three diquark pairing amplitudes,\n\u0001i=\u00002di\u0012\nH\u0000K0\n4\u001bi\u0013\n; (15)\nand the e\u000bective chemical potential matrix,\n^\u0016=\u0016\u00002gVn+\u00168\u00158+\u0016QQ; (16)5\nare color- and \ravor-dependent.\nFor each momentum, the inverse propagator is a 72 \u000272\nmatrix. There is spin degeneracy, so maximally there are\n36-independent eigenstates in the presence of the \ravor\nand color asymmetry. In the Nambu-Gor'kov formalism,\nthe eigenenergies appear as pairs, ( \u000f;\u0000\u000f). The single par-\nticle contribution to the thermodynamic potential is\n\nsingle =\u0000218X\nj=1Z\u0003d3k\n(2\u0019)3\u0014\nTln\u0010\n1 +e\u0000j\u000fjj=T\u0011\n+\u0001\u000fj\n2\u0015\n;\n(17)\nwhere \u0001\u000fj=\u000fj\u0000\u000ffree\nj; here \u0003 is an ultraviolet cuto\u000b. The\n\u0016-dependence is hidden in the eigenvalue \u000fj. Because Eq.\n(13) cannot be inverted analytically, the eigenvalues must\nbe computed numerically for each momentum [38].\nIn order to remove the double-counting of interactions\ntypical of mean-\feld treatments, we must also include in\nthe thermodynamic potential the terms\n\ncond=3X\ni=1\u0014\n2G\u001b2\ni+\u0012\nH\u0000K0\n2\u001bi\u0013\njdij2\u0015\n\u00004K\u001b1\u001b2\u001b3\u0000gVn2: (18)\nThese terms are positive ( \u001bi<0), except for the \fnal\nterm.\nThe quark matter thermodynamic potential is \nbare\nq=\n\nsingle +\ncond:. However, there still remains the nontriv-\nial choice of the \\zero\" of the thermodyanmic potential.\nFor discussions of neutron star masses, this procedure is\nextremely important because in general relativitity, the\nabsolute energy density, as in the TOV equation, and\nnot simply its deviation from the QCD vacuum, is phys-\nically relevant. Given that the cosmological constant is\nextremely small compared to the QCD scale, we set the\norigin of the thermodynamic potential to zero at zero\nquark density and temperature. Thus in constructing the\nquark matter equation of state, we will use the renormal-\nized thermodynamic potential\n\nq(^\u0016;T)\u0011\nbare\nq(^\u0016;T)\u0000\nbare\nq(^\u0016=T= 0);(19)\nwhich vanishes at T= ^\u0016= 0.\nFinally, the electron contribution to the thermody-\nnamic potential is the standard\n\ne=\u00002TX\n\u0015=\u0006Zd3k\n(2\u0019)3ln\u0010\n1 +e\u0000(Ee+\u0015\u0016Q)=T\u0011\n;(20)\nwithEe=p\nk2+m2e, and where we recall that the elec-\ntron chemical potential is \u0016e\u0000=\u0000\u0016Q.\nWriting the total thermodynamic potential as \n =\n\nq+ \ne, the thermodynamic state of the system is de-\ntermined minimizing the free energy with respect to the\nseven condensates f\u001bi,di,ngunder the neutrality condi-\ntions\nnQ;3;8=\u0000@\n@\u0016Q;3;8= 0; (21)TABLE I: Three common parameter sets for the three-\ravor\nNJL model: the average up and down bare quark mass mu;d,\nstrange bare quark mass ms, coupling constants GandK0, and\n3-momentum cuto\u000b \u0003 [25, 39, 40], with \u0003, mu;d, andmsin MeV.\n\u0003mu;dmsG\u00032K\u00035\nHK [25] 631.4 5.5 135.7 1.835 9.29\nRHK [39] 602.3 5.5 140.7 1.835 12.36\nLKW [40] 750.0 3.6 87.0 1.820 8.90\nwhich yields the \\gap equations.\"\n0 =\u0000@\n@\u001bi=\u0000@\n@di; n =\u0000@\n@\u0016: (22)\nBelow we solve these self-consistent equations using the\nmethod outlined in [37]. Whenever we encounter regions\nin the solution suggestive of \frst order phase transitions,\nwe explicitly compare \n in the relevant phases to deter-\nmine which local minima is gives the lower free energy.\nFor the ground state we calculate at a nonzero but very\nsmall temperature \u00180:1me, which makes the numerical\ncalculations faster and more stable.\n4. The NJL parameters\nFor the model outlined above, we identify two distinct\nsets of parameters: (\u0003 ;mu;d;ms;G;K ) and (gV;H;K0).\nThe \frst set is \fxed by matching to QCD vacuum phe-\nnomenology. In this work we will use the set by Hatsuda\nand Kunihiro (HK)[25] (Table I), which gives the vac-\nuum e\u000bective masses for light \ravors, Mu;d'336 MeV,\nand strange quark, Ms'528 MeV. The second set of\nparameters does not manifestly a\u000bect the quantities in\nQCD vacuum at the mean \feld level; we therefore treat\nthem as free parameters, but of the same order of mag-\nnitude as the \frst set, based upon Fierz transformations\nconnecting the associated interaction vertices in the ab-\nsence of any known anomalous suppression. Brie\ry, we\nwill investigate gV= 1\u00002G,H= 1\u00001:5G, andK0= 0\nandK0=K. The choice of these values will be explained\nin Sec.III.\nIII. QUALITATIVE EFFECTS ON THE QUARK\nEQUATION OF STATE\nIn this section we examine a number of qualitative ef-\nfects related to the quark matter equation of state.\nA. When do the quark equations of state become\nsti\u000b?\nWe begin our discussion of sti\u000bening of the equation of\nstate by considering a schematic expression for the quark6\nsector equation of state (see also [29]). For simplicity, we\npresently consider only matter in a single phase, ignor-\ning any complications arising from phase transitions. In\nthis context, the energy density may be parameterized in\nterms of the quark density as\n\"(n) =c1n4=3+c2n2=3+c\u00002n2+B; (23)\nwheren\u0018p3\nFwithpFthe quark Fermi momentum. The\n\frst term is the kinetic energy contribution. The second\nterm contains contributions from both diquark pairing on\nthe Fermi surface ( \u0018\u0000p2\nF\u0001(pF)2) and mass corrections\n(\u0018+p2\nFM(pF)2), where we assume that \u0001 ;M\u001cpF.\n(The limit M\u001cpFis applicable for all quarks, even\nstrange, at high density, where chiral restoration occurs,\nand is applicable for u and d quarks at intermediate den-\nsity.) The third term represents the density-density in-\nteraction. The last term is the bag constant B(>0).\nWe neglect, for large pF, the density dependence of the\npairing gaps, as well as that of the bag constant.\nDi\u000berentiating (23) yields the chemical potential \u0016=\n@\"=@n , from which the pressure P=\u0016n\u0000\"is:\nP=\"\n3\u00002\n3c2n2=3+2\n3c\u00002n2\u00004\n3B: (24)\nThe \frst term is the kinetic pressure, while the remaining\nterms correspond to corrections arising from the mecha-\nnisms discussed above. For given \", the pressure becomes\nlarge when c2<0 andc\u00002>0. The former condition is\nmet when the quarks interact attractively near the Fermi\nsurface. The latter condition simply expresses the re-\nquirement that the density-density interaction should be\nrepulsive in order to sti\u000ben the equation of state. More\ngenerally, for sti\u000b equations of state, the coe\u000ecients cm\u00152\nshould be negative, while cm<0should be positive. Fi-\nnally, a smaller value of the bag constant also tends to\nsti\u000ben the equation of state.\nB. Quark and gluon bag constants: BqandBg\nTo consider the impact of the quark and gluon bag con-\nstants, we begin by supposing that both the quark and\ngluon sectors are weakly interacting and that all quark\nand gluon condensates are vanishingly small. In the ab-\nsence of perturbative corrections, the equation of state is\nthen\nP(\u0016) =c0\u00164\u0000B; \" (\u0016) = 3c0\u00164+B; (25)\nwherec0=NcNf=12\u00192is a function of the number of\nquark colors Ncand \ravors Nf, and the net bag constant\nis sum of quark and gluon contributions, B=Bq+Bg.\nThe existence of the bag constant changes the energy-\npressure relation from \"= 3Pto\"\u00004B= 3P. Therefore,\na smallerBenhancesPat given\", sti\u000bening the equation\nof state. In fact, for a three \ravor ideal quark gas with\na bag constant, the maximum neutron star mass scalesas [41, 42]\nMmax'1:78M\f\u0012155 MeV\nB1=4\u00132\n; (26)\nwhile the corresponding radius scales as\nR'9:5\u0012155 MeV\nB1=4\u00132\nkm: (27)\nThus, smaller values of Bgive rise to more massive, larger\nneutron stars.\nIn the NJL model, the quark bag constant at large\ndensity appears automatically when the gap in the Dirac\nsea is closed through chiral restoration. As a result, its\nvalue be computed explicitly as\nBNJL\nq\u0011[ \n(Me\u000b=m)\u0000\n(Me\u000b=M) ]T=\u0016=0;(28)\nwhereMe\u000bis the e\u000bective mass in the quark energy; Me\u000b\nbecomes the current quark mass ( m) in the perturbative\nvacuum and the dynamically generated mass ( M) in the\nchiral symmetry-broken vacuum. In the HK parameter\nset with vacuum e\u000bective masses Mu;d= 336 MeV and\nMs= 528 MeV, the bag constant is\nBNJL\nq'284 MeV=fm3= (219 MeV)4: (29)\nNaively substituting this value into Eq. (26), we obtain a\nmaximum neutron star mass \u00180:9M\f, which less than\n1/2 the mass of observed massive stars. This low maxi-\nmum mass indicates the importance of interaction e\u000bects\nin order to sustain massive neutron stars.\nTypical values of the bag constant used previously\n[16, 42] are in the range B1=4\u0018150\u0000200 MeV. The bag\nconstant used in [42] to construct strange quark stars,\nBsq'(155MeV)4, is one-fourth of the NJL bag con-\nstant,Bsq'BNJL\nq=4. For a three-\ravor free quark gas\nwithBsq, the star mass is relatively large, '1:78M\f,\nbut still does not reach \u00182M\f; to do so requires includ-\ning interactions. Since the value of the bag constant is\nnot precisely known, we employ the NJL value (29) for\nconsistency. As noted, we should also consider a gluon\nbag constant, Bg\u0018\u00034\nQCD when gluons become pertur-\nbative at some large quark density. However, because\nthe quark bag constant in the NJL model alone already\nprovides considerable softening of the equation of state,\na signi\fcant contribution from the gluonic bag constant\nis unlikely in our equation of state, as we show later in\nSec. IV C.\nOne might argue that considering a gluonic bag con-\nstant is unnecessary because the gluons are integrated\nout in determining the interactions in the NJL model,\nand thus the quark bag constant already contains the\ngluonic contributions. This is not quite correct. In the\nNJL model, the long-range components of the gluons\nsuch as those producing con\fnement are certainly not\ntaken into account; integrating out the long-range com-\nponents generally produces nonlocal interactions among\nquarks, which are not present in the NJL model. There-\nfore, we must consider the contributions from long-range\ngluons separately, and not simply ignore the gluonic bag\nconstant.7\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0  0.2  0.4  0.6  0.8  1  1.2  1.4P (GeV/fm3)\ne (GeV/fm3)H=K'=0,  gV/G = 0.0\ngV/G = 1.0\ngV/G = 2.0\nAPR (nB < 2n0)\nFIG. 2: Pressure vs. energy density for several NJL parameter\nsets and APR. The bold lines for the NJL equations of state indi-\ncate thatnB>4n0. AsgVincreases, the NJL equation of state\nbecomes noticeably sti\u000ber. APR is also plotted for comparison, in\nbold fornB<2n0, and as double dots in the region above, where\nwe do not use APR.\nC. Repulsive density-density interaction: gV\nThe repulsive quark vector interaction is inspired by\nthe repulsive density-density interaction in nuclear mat-\nter, described, e.g., by omega meson exchange [43]. Ex-\ntrapolating the picture of nuclear matter to the strongly\ncorrelated quark matter domain, we anticipate that the\nquark vector coupling is of similar magnitude to the\nhadronic coupling scale gV\u0018G.\nReference [44] demonstrated that the vector coupling\nshould begV\u00182Gin order to explain the lattice re-\nsults on the curvature of the chiral restoration line near\nzero density [45]. (Note that the coupling constant G\nhere corresponds to half that used in Ref. [44].) In the\nfollowing, we focus mainly on the value gV= 2G.\nThe inclusion of a vector coupling smooths out chiral\nsymmetry restoration [44] because the density-density re-\npulsion forbids a rapid increase of the baryon density,\nand as a result the chiral transition also does not occur\nrapidly. Indeed, beyond a particular critical coupling,\na \frst order chiral transition is turned into a smooth\ncrossover.\nIntuitively, an increasing repulsive vector force sti\u000b-\nens the equation of state, Pvs.\", as shown in Fig. 2.\nWhile the NJL equation of state is considerably softer\nthan APR for small vector couplings, when gVis suf-\n\fciently large the NJL equation of state can achieve a\nsti\u000bness on par with APR across a wide range of densi-\nties. By increasing gVsu\u000eciently, we can obtain an equa-\ntion of state within the present framework sti\u000b enough\nto support neutron star masses \u00182M\f.\nOn the other hand, increasing gVmakes it more di\u000e-\ncult to interpolate between the APR and NJL regimes.\nThis challenge is seen in the plots of nB(\u0016) andP(\u0016) in\nFig. 3, where for both PandnB, the APR and NJL\ncurves become more widely separated in \u0016asgVin-\ncreases. One might imagine that the matching could be\n 0 1 2 3 4 5 6 7 8\n 0.3  0.35  0.4  0.45  0.5  0.55  0.6  0.65nB / n0\nm (GeV)H=K'=0,  gV/G = 0.0\ngV/G = 1.0\ngV/G = 2.0\nAPR (nB < 2n0)\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0.3  0.35  0.4  0.45  0.5  0.55  0.6  0.65P (GeV/fm3)\nm (GeV)H=K'=0,  gV/G = 0.0\ngV/G = 1.0\ngV/G = 2.0\nAPR (nB < 2n0)FIG. 3: Normalized baryon density nB=n0(top panel) and pres-\nsureP(bottom panel) as a function of quark chemical potential\n\u0016for several NJL parameter sets and APR. Typically, the baryon\ndensity and pressure in the APR rise faster at lower chemical\npotential than in NJL which has a larger e\u000bective quark mass,\nM'336 MeV, than 1/3 of the nucleon mass.\nperformed rather simply by allowing a \frst order phase\ntransition in the interpolated region; however a \frst or-\nder transition, a sudden increase in nBis simply a kink\ninPvs.\u0016, which does not help the interpolation.\nA part of the di\u000eculty of interpolating between APR\nand NJL is that the constituent quark mass for light \ra-\nvors isMu;d'336 MeV, larger than the one-third of\nthe nucleon mass. Accordingly, the P(\u0016) curve in the\nNJL model tends be below that of APR. We next discuss\ntwo-body correlations mediated by the color magnetic in-\nteraction, which tend to shift the P(\u0016) curves to the left,\nrendering the interpolation procedure more feasible.\nD. Two-body correlations: the color magnetic\ninteraction H\nAt high density, quarks undergo BCS pairing (diquark\ncondensation) as a consequence of the color magnetic\ninteraction. Pairing reduces the energy density by an\namount\u000e\"\u0018\u0000p2\nF\u00012, or equivalently, enhances the pres-\nsure by\u000eP\u0018+p2\nF\u00012.\nWe expect correlation e\u000bects among the quarks to in-\ncrease with decreasing density. Eventually three-quark\ncorrelations must be dominant in the hadronic phase.\nOne path to three-quark correlations is increasing di-8\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8\n 0.3  0.35  0.4  0.45  0.5  0.55  0.6  0.65Gaps (GeV)\nm (GeV)Mu\nMd\nMs\nDds\nDsu\nDud\nFIG. 4: Chiral and diquark gaps as a function of quark chemical\npotential for K0= 0,gV= 2:0G,H= 1:5G. With increasing\ndensity, the system undergoes a \frst order phase transition from\nthe 2SC to the CFL phase. The gaps are shown as bold lines for\nnB>4n0.\n 0 1 2 3 4 5 6 7 8\n 0.3  0.35  0.4  0.45  0.5  0.55  0.6  0.65nB / n0\nm (GeV)K'=0, gV= 2.0G,  H=0.0\nH=1.0 G\nH=1.25 G\nH=1.5 G\nAPR (nB < 2n0)\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0.3  0.35  0.4  0.45  0.5  0.55  0.6  0.65P (GeV/fm3)\nm (GeV)K'=0, gV= 2.0G,  H=0.0\nH=1.0 G\nH=1.25 G\nH=1.5 G\nAPR (nB < 2n0)\nFIG. 5: Normalized baryon density nB=n0(top panel) and pres-\nsureP(bottom panel) as functions of quark chemical potential\n\u0016. We take the vector and axial anomaly couplings in NJL to be\ngV=G= 2:0 andK0= 0, and allow the diquark coupling to take\non the values H=G = 0, 1.0, 1.25, and 1.5. As Hincreases the\ncurves are shifted toward lower chemical potential.quark correlations plus diquark-single quark correlations\nbeyond that described by the standard choice of the di-\nquark coupling H= 0.75-1.0 G, based on Fierz trans-\nformation of the one-gluon exchange type vertex. In this\nrange, we do not \fnd signi\fcant e\u000bects of Hon the equa-\ntion of state in the density range of interest. A diquark\nmean \feld appears only when the Fermi surface becomes\nsu\u000eciently large to overcome chiral symmetry breaking\ne\u000bects. However, diquark correlations, which can exist\neven without a large Fermi sea, reduce the energy of a\npair to less than twice the e\u000bective quark mass. To simu-\nlate such e\u000bects within the present mean \feld approach,\nwe allow the diquark mean \feld to reduce the single quark\nenergy at all densities by exploring somewhat larger val-\nuesH= 1.0-1.5Gthan the standard.\nAs shown in Sec. III A, pairing tends to sti\u000ben the\nequation of state in the high density regime. Figure 4\nshows the development of constituent quark masses and\nmean \feld pairing gaps; as \u0016increases, quark matter\n\frst appears in a 2SC phase in which only up and down\nquarks are paired (\u0001 ud6= 0, \u0001us= \u0001ds= 0), and later\nevolves into a CFL phase in which all three quark \ravors\npair (\u0001ud;\u0001ds;\u0001su6= 0). AtT= 0 the 2SC-CFL transi-\ntion appears to be \frst order for all NJL parameter sets.\nHowever, given that this transition occurs at relatively\nlow density ( nB<4n0), the quark model results must be\ntreated with caution.\nTwo-body correlations are also important at low densi-\nties. For example, in the constituent quark model, color\nmagnetic interactions between quarks, in the presence of\ncon\fnement, reduces the nucleon mass from three times\nthe constituent quark mass, '3\u0002340 = 1020 MeV by\nsome 80 MeV to its physical value. Since con\fnement,\nby localizing the quarks into a spatial region \u0018\u0003\u00003\nQCD,\nincreases the quark kinetic energies, as well as adding\nthe energy of color \rux tubes { of typical length \u0018\u0003\u00001\nQCD\nand energy\u0018\u001b\u0003\u00001\nQCD, where\u001bis the string tension { the\nenergy gain from the color magnetic interaction must ex-\nceed\u001830 MeV.\nThe diquark interaction, H, treated in mean \feld,\nqualitatively simulates the reduction in the average quark\nenergy at low density that results from pairing e\u000bects.\nAs the magnitude of the diquark interaction increases,\nthe curves of the thermodynamic variables as functions\nof\u0016are shifted leftwards to lower chemical potential, as\nshown in Fig. 5. Thus, by including e\u000bects of pairing,\none is able to maintain the sti\u000b equation of state pro-\nduced by a relatively large vector coupling, while at the\nsame time enabling a smooth interpolation between the\nNJL and APR equations of state for all thermodynamic\nvariables.\nFigure 6 demonstrates the impact of pairing on the\nsti\u000bness of the NJL equation of state. The discontinuous\nchange of\"at \fxedPre\rects the \frst order 2SC-CFL\nphase transition. While for 0 <H < 1:5G, the equations\nof state exhibit softening immediately following the 2SC-\nCFL transition, as the quark density increases further,\npairing e\u000bects sti\u000ben the equation of state for all NJL9\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0  0.2  0.4  0.6  0.8  1  1.2  1.4P (GeV/fm3)\ne (GeV/fm3)K'=0, gV= 2.0G,  H=0.0\nH=1.0 G\nH=1.25 G\nH=1.5 G\nAPR (nB < 2n0)\nFIG. 6: Pressure as a function of energy density for the same\nparameter sets as in Fig. 5. The discontinuous change of \"at\n\fxedPre\rects the \frst order 2SC-CFL phase transition. For\n0< H < 1:5G, the equation of state is softened immediately\nfollowing the 2SC-CFL phase transition, but as density increases,\nthe equation of state eventually sti\u000bens relative to the H= 0\ncase. ForH= 1:5G, the equation of state is sti\u000ber than the\nunpaired case for all densities.\nparameter sets, relative to the no-pairing case. Moreover,\nwhen the pairing is su\u000eciently strong ( H > 1:5G), the\nequation of state is sti\u000ber than without pairing ( H= 0)\nacross the entire density range.\nWe note that for large H, the quark pressure at given\n\u0016exceeds the APR pressure even at very low densities.\nTaken at face value, this would suggest that even at very\nlow densities the ground state of QCD matter is quark\nrather than hadronic matter. However, as we discuss in\nSec. IV, this high pressure is an unphysical consequence\nof the NJL model not being con\fning at low densities.\nE. Chiral-diquark coupling: K0\nWe now turn to the axial anomaly-induced coupling\nbetween the chiral and diquark condensates. The im-\nportance of the anomalous coupling K0depends on the\nsize of the chiral and diquark condensates. For K0>0,\nthis coupling favors the coexistence of chiral and diquark\ncondensates [34]. Thus, as K0increases from zero the\ndiquark condensate emerges at lower chemical potential\nand the chiral condensate persists to higher chemical po-\ntential.\nFigure 7 shows the impact of K0on the NJL equation\nof state. We note that while increasing K0from 0 to\nKslightly sti\u000bens the equation of state, its impact is\nmuch smaller than that of gVorH. Since the K0term\nin the Lagrangian can be read as a diquark interaction\nwith an interaction strength proportional to the chiral\ncondensate, the e\u000bect of K0can be largely absorbed by\nthe variation of gVandH; thus in the following we do\nnot study the variation of K0in detail but take K0=K\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0  0.2  0.4  0.6  0.8  1  1.2  1.4P (GeV/fm3)\ne (GeV/fm3)gV= 2.0G, H=1.5G, K'=0\nK'=K\nAPR (nB < 2n0)FIG. 7: Pressure as a function of energy density in the presence\nof the chiral-diquark coupling, K0=K. LargerK0sti\u000bens the\nequation of state, but its impact is signi\fcantly smaller than that\nof variations of gVorH.\nas a canonical value.\nIV. INTERPOLATED EOS\nWe now discuss constructing an interpolated equation\nof state,P(\u0016) that smoothly joins the equations of state\nof the low density APR model to the high density NJL\nmodel. The \frst step in de\fning an interpolation method\nfor joining the hadronic and quark sectors is to determine\nthe \\overlap region\" in which the two equations of state\nwill be merged. Beginning in the low density hadronic\nregime described by APR, we expect that as density in-\ncreases, corrections from many-body forces, hyperon de-\ngrees of freedom, multiple meson exchanges and the like,\nwill become important above nB\u00182n0. Thus, we \fx the\nlower boundary of the interpolation to n<\u00112n0.\nAs the density decreases in the quark regime , con\fning\ne\u000bects, which trap quarks into baryons, become increas-\ningly important. Assuming that the radius of a typical\nbaryon isrB\u00180:4\u00000:5 fm, baryons begin to perco-\nlate at around nB\u00184\u00007n0. Thus, we set the upper\nboundary of the interpolation to n>\u00114\u00007n0, with the\nprecise location determined by the details of the given\nNJL parameter set being used.\nIn the intermediate density regime at a given chemi-\ncal potential, the pressure of the interpolated equation of\nstate should be lower than the NJL pressure extrapolated\nto lower densities (Fig.8). This follows from the fact that\nnoncon\fning models yield excess quark populations at\ngiven chemical potential, due to the unphysically small\nenergy cost of having quarks present. In other words,\nwere the con\fning e\u000bects of QCD incorporated in the\ndescription of the quark phase, the pressure, especially\nin dilute region, would be signi\fcantly suppressed. This\nsituation is quite analogous to the \\semi\"-QGP picture\nfor \fnite temperature QCD [46] in which an \\overpop-10\nFIG. 8: Schematic illustration of con\fning e\u000bects on the hy-\nbrid quark-hadron equation of state, P(\u0016), here normalized by\nthe Stefan-Boltzman gas for Nf= 3. E\u000bects of con\fnement\nare strong at lower density, suppressing the excess NJL pres-\nsure. The boundaries of the interpolated equation of state are\n(n<;n>) = (2:0;5:0)n0.\nulated\" quark pressure is suppressed by Polyakov loop\ne\u000bects, until thermal quarks and gluons exhibit quasipar-\nticle behavior at temperatures beyond \u00182\u00003Tc, where\nTcis the pseudocritical temperature for decon\fnement\n[47].\nThe present 3-window description is quite di\u000berent\nfrom the conventional one involving a \frst order hadron-\nquark phase transition [1]. In the latter case, the quark\npressure at low density of noncon\fning models must be\nsmaller than the hadronic one, in order to ensure the\nintersection of the quark and hadronic pressure at rea-\nsonable density. This is achieved either by restricting the\nquark model parameters or by introducing a bag constant\nto lower the quark pressure. Such choices, however, gen-\nerally a\u000bect the quark matter equation of state not only\nin the (presumably unreliable) low density limit, but also\nin the high density regime in which it should be reliable.\nA. Thermodynamic constraints\nHaving discussed the qualitative aspects of a hadron-\nquark interpolation, we now brie\ry review the thermo-\ndynamic constraints imposed on this interpolation which\nare necessary to ensure that the interpolated equation\nof state is physical. These constraints are as follows:\n(i) the pressure P(\u0016) must be continuous everywhere;\n(ii)nB(\u0016) = (@P=@\u0016 )=Ncmust be a monotonically\nincreasing function in order to ensure stability of the\nsystem with respect to phase separation: @2P=@\u00162=\nNc@nB=@\u0016 > 0. (iii) In addition one physically expects\nthat the speed of sound must be less than the speed of\nlight:c2\ns=@P=@\"< 1 [11, 12]. These conditions tightly\nconstrain possible interpolations of the equation of state.B. Interpolation method\nWe now describe a particular method for construct-\ning a phenomenological quark-hadron equation of state.\nTo interpolate in the variables \u0016-P, we employ a simple\npolynomial interpolation function, de\fning an Nth order\npolynomial interpolant for the pressure:\nP(\u0016) =NX\nm=0bm\u0016m;for\u0016<<\u0016<\u0016>; (30)\nwhere\u0016<and\u0016>are de\fned as the points where\nnB(\u0016<)\u0011n<= 2n0andnB(\u0016>)\u0011n>= 4\u00007n0. The\ncoe\u000ecients bmare chosen to satisfy the matching condi-\ntions at the boundaries of the interpolating interval. At\n\u0016=\u0016<:\nPAPR(\u0016<) =P(\u0016<);@PAPR\n@\u0016\f\f\f\f\n\u0016<=@P\n@\u0016\f\f\f\f\n\u0016<;\u0001\u0001\u0001(31)\nand at\u0016=\u0016>:\nPNJL(\u0016>) =P(\u0016>);@PNJL\n@\u0016\f\f\f\f\n\u0016>=@P\n@\u0016\f\f\f\f\n\u0016>;\u0001\u0001\u0001(32)\nThe number of derivatives that one matches at each\nboundary is a matter of choice. In general, matching\nmore derivatives results in a smoother interpolation, but\nat the same time increases the probability of producing\nunphysical artifacts in the interpolating region (e.g., in-\n\rection points in P(\u0016) which violate thermodynamic con-\nstraints). Here we match up to second order derivatives\nat each boundary, which ensures that the pressure, num-\nber density, and number susceptibility are continuous.\nCorrespondingly, these six boundary conditions require\nthat Eq. (30) has six terms ( N= 5).\nC. Interpolated EOS\nFigure 9 shows the interpolated equation of state\nP(\u0016), with the interpolation boundaries ( n<;n>) =\n(2:0;5:0)n0. For illustration, we consider two NJL pa-\nrameter sets:\n(gV;H) =\u001a\n(2:0;1:5)G (Set I)\n(2:0;0:0)G (Set II)(33)\nwithK0=Kin both cases. For n>'4nB, Set I satis-\n\fes all the conditions demanded by the thermodynamic\nconstraints, as we verify shortly. One cannot, however,\nwithin the present polynomial interpolation, construct a\nsensible interpolation for Set II, because at the interpo-\nlation boundaries the APR and NJL pressures are rather\nwidely separated in \u0016, a possibility noted in Sec. III D.\nThis wide separation in \u0016requires a small slope of the\ninterpolated pressure, but at the same time the slope\nmust be larger than the slope of the APR pressure at the\nlower boundary, because of the compressibility condition\n@2P=@\u00162>0. The interpolated equation of state has a11\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0.3  0.35  0.4  0.45  0.5  0.55  0.6  0.65  0.7P (GeV/fm3)\nm (GeV)APR\n(gV, H)=(2.0, 1.5)G\ninterpolated\n(gV, H)=(2.0, 0.0)G\ninterpolated\nfixed slope\nFIG. 9: Interpolated equations of state, P(\u0016), for two param-\neter sets, with up to second order derivatives matched at the\ninterpolation boundaries. As a guide to the eye, we also plot the\npressure with constant slope \fxed at n<. Thermodynamic stabil-\nity requires that the slopes in the interpolated equation of state\nincrease monotonically: @2P=@\u00162=Nc@nB=@\u0016 > 0, which is\nviolated for the equation of state constructed in the simple inter-\npolation with parameter Set II.\n 0 1 2 3 4 5 6 7 8\n 0.3  0.35  0.4  0.45  0.5  0.55  0.6  0.65  0.7nB / n0\nm (GeV)APR\n(gV, H)=(2.0, 1.5)G\ninterpolated\n(gV, H)=(2.0, 0.0)G\ninterpolated\nFIG. 10: Baryon density as a function of chemical potential for\nthe interpolated equations of state. The parameter set ( gV;H) =\n(2:0;0:0)Ghas an unstable region @nB=@\u0016 < 0 with the simple\nsix-term interpolating function.\nregion of@nB=@\u0016 < 0, as is clearly seen in the plot of\nnB(\u0016) in Fig. 10.\nThe result presented in Fig. 10 does not preclude con-\nstructing a sensible interpolated pressure for Set II. As\none sees in Fig. 10, it is possible to join the low and\nhigh baryon density curves with a nondecreasing func-\ntion of\u0016. The present exercise shows that the class of\ninterpolating functions for Set II is much more restric-\ntive than for Set I. For example, if there is a \frst order\nphase transition between the hadronic and quark regions,\nthe possible density discontinuities are smaller for Set II\nthan Set I. More detailed treatments of the interpolation\nregion are beyond the scope of the present work and we\nhenceforth restrict our consideration to the simple poly-\nnomial interpolation, rejecting NJL parameter sets, such\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6P (GeV/fm3)\ne (GeV/fm3)APR\n(gV, H)=(2.0, 1.5)G\ninterpolatedFIG. 11: Pressure vs. energy density for the interpolated\nequation of state for ( gV;H) = (2:0;0:0)G,K0=K, and\n(n<;n>) = (2:0;5:0)n0.\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6\n 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6cs2\ne (GeV/fm3)n> / n0 = 4.0\n5.0\n6.0\n7.0\nFIG. 12: The speed of sound c2\ns=@P=@\" as a function of \".\nThe NJL parameter set is the same as Fig. 11, for di\u000berent values\nof the high density boundary: n>=n0= 4.0, 5.0, 6.0, and 7.0. For\neach of these choices, the causality condition is satis\fed.\nas II, incapable of being joined with APR in a thermo-\ndynamically consistent manner.\nFigure 11 shows the pressure vs. energy density for\nparameter set I. In this case, the high density equation\nof state is as sti\u000b as APR extrapolated into the region\nbeyondnB= 2n0. From Fig. 12, we observe that the\ncausality constraint is satis\fed when the high density\nboundary, n>, is varied from 4 n0to 7n0. If we take\nn>.4n0, however, we \fnd that c2\ns>1, a putative vio-\nlation of causality.\nFinally, we consider the possible impact of a gluonic\nbag constant, Bg\u0018(200MeV)4'0:2 GeVfm\u00003, which\nshould be included when the gluon sector becomes per-\nturbative. This contribution reduces the pressure in the\nquark matter region by 30 \u000040%, which makes it ex-\ntremely di\u000ecult to interpolate between the hadronic and\nquark regimes without violating the condition @nB=@\u0016>\n0 (cf. Fig. 9). At the same time, the bag constant in-\ncreases the energy density by Bg, so the resulting equa-\ntion of state becomes signi\fcantly softer. Strictly speak-\ning, even in this situation it would be possible to con-12\nstruct equations of state by increasing gVandHsig-\nni\fcantly from our current choices; however the current\nchoices for these couplings are already relatively large\nand it is di\u000ecult to identify a mechanism that would sig-\nni\fcantly increase either coupling in the dense regime.\nThus, we conclude that Bgshould be very small in the\nquark matter equation of state, even at nB\u001810n0; the\ngluons remain condensed and nonperturbative.\nV. MASS-RADIUS RELATION\nBy solving the TOV equation for a given value of the\nbaryon density at the center, we construct a family of\nstars whose masses and radii are functions of nc\nB. The\nM-nc\nBrelation for the equation of state with NJL param-\neters (gV;H;K0) = (2G;1:5G;K ) is shown in Fig. 13,\nwhere we have taken n>= 5n0as in the previous sec-\ntion. To examine e\u000bects of the quark bag constant as-\nsociated with chiral restoration, we also show results\nfor the three-\ravor free quark gas with bag constants\nBq=BNJL= (219MeV)4andBq= (155MeV)4. In the\nNJL model by itself, the quark bag constant is so large\nthat neutron star masses are restricted to M < M\f.\nHowever, we note that as the bag constant decreases,\nneutron star masses rise, so that models yielding smaller\nbag constants allow more massive stars.\nWe see in Fig. 13 that the M(nc\nB) curves for our in-\nterpolated equation of state and APR are quite similar,\na not too surprising result given that our chosen NJL\nparameter set yields an equation of state quite similar\nto APR. However, while the thermodynamic properties\nof the two systems are similar, the underlying e\u000bective\ndegrees of freedom are quite di\u000berent. Indeed, APR is\nwell known for its extreme sti\u000bness at high densities, but\nthe e\u000bect of hyperonic degrees of freedom in the hadronic\nsector are expected to reduce this sti\u000bness. On the other\nhand, our NJL treatment of the quark sector includes\nstrange quarks from the beginning, and is capable of\nproducing a su\u000eciently sti\u000b equation of state to support\nstars whose masses exceed 2 M\f[20].\nFigure 14 shows the M-Rrelation for the interpolated\nequation of state with parameter set I. For most central\ndensities the stellar radius is \u001811\u000012 km, which is\ncompatible with observational data [9, 10]. However,\nthe radius at which the mass starts to rise in the M-R\nplot is sensitive to the properties of the hadronic equation\nof state for n0\u00181:0\u00002:0n0, with di\u000berent hadronic\nequations of state yielding radii di\u000bering by up to \u00182\nkm [20].\nVI. SUMMARY\nIn this paper we have constructed a phenomenologi-\ncal equation of state over the range of baryon densities\nnB\u00181\u000010n0by interpolating between the low density\nhadronic APR equation of state and the high density NJL\nFIG. 13: Neutron star mass as a function of central baryon\ndensity,nc\nB. We display curves corresponding to our interpo-\nlated equation of state for ( gV;H;K0) = (2G;1:5G;K ) and\nn>= 5n0, as well as APR with and without the three-nucleon\ninteraction, and the three-\ravor free quark gas with bag con-\nstantsBq=BNJL = (219MeV)4andBq= (155MeV)4. The\nvertical arrows indicate the boundaries of the interpolation region\nfor the interpolated equation of state.\nFIG. 14: Mass-radius relation for neutron stars with several\nequations of state (same as Fig. 13).\nquark model. In so doing, we explored a number of rele-\nvant ingredients of the equation of state needed to realize\nneutron stars of mass \u00182M\f, while satisfying necessary\nthermodynamic and causality constraints. These require-\nments constrain the form of the interpolated equation of\nstate and allow one to infer qualitative e\u000bects regarding\nthe intermediate density region between the hadronic and\nquark regimes. The repulsive density-density interaction,\ncolor-magnetic interaction, and con\fning e\u000bects play a\nvital role in determining the structure of the equation of\nstate. A crucial result is that the gluonic bag constant\nmust be small, i.e., the gluons must remain strongly cou-\npled throughout the density region of interest in massive\nneutron stars; the gluon sector remains nonperturbative\neven at\u001810n0. One reaches a similar conclusion from13\nstudies of quarkyonic matter [27, 48, 49], in which non-\nperturbative gluons in quark matter play a crucial role.\nWe have also emphasized why the three window model\nof interpolation [20] is capable of producing sti\u000ber equa-\ntions of state than conventional hybrid equations of state\ninvolving \frst order phase transitions. As opposed to\nconventional models, which require the intersection of\nquark and hadronic P(\u0016) curves, we propose that the\nquark pressure P(\u0016) based on noncon\fning models need\nnot (and even should not) intersect the hadronic equation\nof state before the inclusion of con\fning e\u000bects. While\nthis observation is not directly applicable at low density\n(where we did not use the NJL model), it results in a\nmuch wider range of possible high density quark equa-\ntions of state. As a result, we are able to explore a re-\ngion of parameter space that has been omitted from prior\nstudies, while still producing a sti\u000b equation of state re-\nquired to support massive neutron stars.\nIn this work we have not taken into account possible\nmeson condensed phases, by which we mean condensates\nin which the order parameter has the quantum number\nof a mesonic \feld. Such condensates have been studied\nin a nuclear context [50{52] as well in quark matter, e.g.,\ninhomogeneous diquark [4] and chiral [5] condensates. If\nextant, such exotic phases would likely occur in the the\nneither purely hadronic nor purely quark density region\nin which we have interpolated. Thus one cannot directly\ntake over previous results for meson condensates, includ-\ning the strength, or density discontinuity, of the \frst or-\nder phase transition to the condensed phase. For a given\nhadronic equation of state at low density and quark equa-\ntion of state at high density, the strength of such a phase\ntransition is bounded. Although we have, for simplic-\nity, considered only a smooth interpolation scheme, one\nshould more generally allow for such exotic phases; then\nthe smooth P(\u0016) curve used in this work would be re-\nplaced by one with a small kink, keeping the positive\ncurvature of P(\u0016) in the interpolation. We anticipate\nthat even with condensates at intermediate density, it\nwill still be possible to \fnd a reasonable parameter set for\nthe color-magnetic and vector interactions that is com-\npatible with the existence of massive stars. The issue of\nexotic phases remains open until we can reliably estimate\nthe high density quark equations of state. Further stud-ies are needed to understand better the impact of exotic\nphases on the in-medium NJL parameters and in turn,\ntheir implications for the description of massive neutron\nstars.\nIn order to improve the description of the intermediate\ndensity matter in neutron stars it is important to fur-\nther re\fne our understanding of the hadronic equation\nof state near nB\u0018(1\u00003)n0. In particular, a more care-\nful assessment of the importance of many-body interac-\ntions and the emergence of hyperons is required. Further\nconstraints may be obtained from heavy-ion collisions,\nincluding strangeness production [53], and lattice QCD\ncalculations of hyperon-nucleon interactions [54]. Stud-\nies of the density dependence of nuclear forces in terms\nof quarks and gluons play a crucial role in determining\nwhen (and why) quarks emerge as the proper degrees of\nfreedom at high density. It would be desirable in the fu-\nture to extend to \fnite temperature the present approach\nto the equation of state to enable us to address dynamical\nquestions such as applications to heavy ion collisions, and\nneutron star cooling. It is also important to obtain an\nimproved estimate of the quark bag constant, for were it\nmuch smaller than the NJL estimate used here, the soft-\nening associated with chiral restoration would be signif-\nicantly reduced and large vector and diquark couplings\nwould not be necessary to obtain a sti\u000b quark matter\nequation of state within the present context.\nVII. ACKNOWLEDGEMENTS\nAuthor T.K. thanks Y. Hidaka, A. Ohnishi, and R.\nPisarski for enlightening discussions during the Fourth\nAPS-JPS joint meeting HAWAII 2014, and M. 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Phys. 127(2012) 723\n[arXiv:1106.2276 [hep-lat]]."    },    {        "title": "1501.04135v2.Topology_of_density_matrices.pdf",        "content": "Topology of density matrices\nJan Carl Budich1, 2and Sebastian Diehl3\n1Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria\n2Institute for Quantum Optics and Quantum Information,\nAustrian Academy of Sciences, 6020 Innsbruck, Austria\n3Institute of Theoretical Physics, TU Dresden, D-01062 Dresden, Germany\n(Dated: May 1, 2015)\nWe investigate topological properties of density matrices motivated by the question to what extent phenomena\nsuch as topological insulators and superconductors can be generalized to mixed states in the framework of open\nquantum systems. The notion of geometric phases has been extended from pure to mixed states by Uhlmann in\n[Rep. Math. Phys. 24, 229 (1986)], where an emergent gauge theory over the density matrices based on their\npure-state representation in a larger Hilbert space has been reported. However, since the uniquely defined square\nrootp\u001aof a density matrix \u001aprovides a global gauge, this construction is always topologically trivial. Here,\nwe study a more restrictive gauge structure which can be topologically non-trivial and is capable of resolving\nhomotopically distinct mappings of density matrices subject to various spectral constraints. Remarkably, in\nthis framework, topological invariants can be directly defined and calculated for mixed states. In the limit of\npure states, the well known system of topological invariants for gapped band structures at zero temperature is\nreproduced. We compare our construction with recent approaches to Chern insulators at finite temperature.\nI. INTRODUCTION\nWith the identification of geometric phases in quantum\nphysics [1–3], an emergent classical gauge structure in\nelementary quantum mechanics has been revealed. Several\nfundamental discoveries of contemporary quantum physics\nsuch as the integer quantum Hall effect [4–7] and, more\nrecently, the periodic table of topological insulators and\nsuperconductors [8–12] can be theoretically described as\ntopological properties of such emergent gauge theories in the\nframework of Bloch bands: The local U(n)gauge degree\nof freedom there consists in the choice of an arbitrary or-\nthonormal basis of Bloch states ju\u000b\nkithat span the projection\nP(k) =Pn\n\u000b=1ju\u000b\nkihu\u000b\nkjonto thenoccupied bands of an in-\nsulating band structure at lattice momentum k. The covariant\nderivative for such gauge theories has been constructed by\nKato as early as 1950 in his proof of the adiabatic theorem\nof quantum mechanics [13]. Topological invariants of this\ngauge structure distinguish homotopically distinct mappings\nk7!P(k), i.e., topologically inequivalent band structures.\nThe consideration of generic symmetries [14] refines this\nhomotopy classification and leads to the periodic table of\ntopological invariants [8–12] (see Ref. [15] for a recent\nrigorous discussion).\nReal physical systems, however, are not perfectly isolated\nfrom their environment and are to be described by a mixed\nstate density matrix \u001arather than a pure state wave function.\nIt is hence natural to ask whether the concept of geometric\nphases in quantum physics can be generalized to the realm\nof density matrices and their quantum mechanical evolution.\nThis question has been answered in the affirmative in a series\nof papers by Uhlmann [16–18], who viewed the redundancy\nin the purification of density matrices, i.e., in the representa-\ntion of\u001aas a pure state in a larger Hilbert space, as a gauge\ndegree of freedom: \u001a=wwy= (wU)(wU)y, where the\nHilbert Schmidt operator wdenotes a purification of \u001aandU\nis a unitary operator. In this context, the Uhlmann-connectiondefining parallel transport of Hilbert Schmidt operators along\na patht7!\u001a(t)of density matrices is given by the constraint\n[16]\n_wyw\u0000wy_w= 0; (1)\nwhere the dot denotes the derivative with respect to t.\nThe purpose of our present work is to unravel the topolog-\nical properties of various gauge structures over the space of\nquantum mechanical density matrices. Since the purification\nw=p\u001a, uniquely defined for every density matrix \u001a, pro-\nvides a global gauge, the general purification scheme in terms\nof Hilbert Schmidt operators as considered by Uhlmann is\nalways topologically trivial. However, we would like to point\nout that this scheme can also be applied to density matrices of\npure states which then would also yield a topologically trivial\ngauge theory – despite the fact that pure states give rise to a\nrich spectrum of topological phenomena as mentioned above.\nThus, the fact that some topologically trivial gauge structure\ncan be constructed over the space of density matrices by no\nmeans implies that there are no topological features to be\ndiscovered. In fact, more generally speaking, every topo-\nlogical gauge theory can be embedded into a topologically\ntrivial theory with a larger gauge degree of freedom over\nthe same base space [19]. This raises the natural question\nwhether a more restricted notion than general purification\nof density matrices in terms of Hilbert Schmidt operators\ncould be employed to reveal topological aspects of families of\ndensity matrices \u001a(k)parameterized by a lattice momentum k.\nKey results – We investigate an ensemble of pure states\n(EOPS) scheme in terms of non-orthonormalized pure states\nj~ \u000bisatisfying\n\u001a=X\n\u000bj~ \u000bih~ \u000bj (2)\nas a gauge structure over the space of density matrices \u001a. This\nconstruction draws intuition from the entanglement theory ofarXiv:1501.04135v2  [quant-ph]  30 Apr 20152\nmixed states where, e.g., the entanglement of formation [20]\nis defined in terms of similar ensembles of pure states. Start-\ning from the spectral representation \u001a=P\n\u000bp\u000bj \u000bih \u000bj,\na natural EOPS is j~ \u000bi=pp\u000bj \u000bi. Under arbitrary uni-\ntary rotations of this frame, the pure states are no longer mu-\ntually orthogonal but still project down to the same density\nmatrix\u001avia Eq. (2). The resulting gauge theory of non-\northogonal frames over the density matrices is a generaliza-\ntion of the pure state case. For invertible density matrices\nwithout any further spectral assumptions, it is equivalent to\nUhlmann’s construction and thus topologically trivial. Inter-\nestingly, if assumptions on the spectral degeneracy of the den-\nsity matrices are made, the present scheme can accommodate\ntopologically non-trivial mixed states. Topologically inequiv-\nalent mappings k7!\u001a(k)in this framework cannot be contin-\nuously deformed into each other without either violating the\nspectral assumptions or breaking possible protecting physical\nsymmetries. Our construction formalizes the notion of a pu-\nrity gap proposed in Refs. [21, 22] – a purity gap closing\nis a level-crossing in \u001aand thus a violation of the underlying\nspectral assumptions that makes a change in the topology pos-\nsible. Under non-equilibrium conditions, a physical system\nmay be non-ergodic and may thus be described by a singular\ndensity matrix of rank n < N , whereNis the dimension of\nthe Hilbert space. Also in this case, the EOPS gauge struc-\nture can become topologically non-trivial. The topological\ninvariants associated with the proposed gauge structure are\nconstructed in terms of Uhlmann’s connection (see Eq. (1))\nwithout reference to the pure state case which is, however,\ncorrectly reproduced as a limiting case here.\nFinally, we discuss notable differences to recent publica-\ntions [23–25] where topological invariants for two-banded\none-dimensional systems and Chern insulators [26] at finite\ntemperature, respectively, have been proposed that are not\ncharacteristic classes of a gauge theory. We illustrate with\nan explicit example that a two-dimensional system at finite\ntemperature is in general not uniquely characterized by a\nsingle invariant of that kind.\nOutline – This article is structured as follows. In Section\nII, we review how a gauge structure emerges for both pure\nand mixed states in elementary quantum mechanics. In Sec-\ntion III, topological invariants for density matrices with vari-\nous spectral assumptions are defined. We compare the present\napproach to the construction of Uhlmann phase winding num-\nbers that have recently been proposed [24, 25] to classify ther-\nmal Chern insulators in Section IV. Concluding remarks are\npresented in Section V.\nII. EMERGENT GAUGE STRUCTURES FROM PURE TO\nMIXED STATES\nA. Adiabatic time evolution of pure states\nA natural and conceptually simple scenario for the occur-\nrence of geometric phases is the time evolution of a quan-\ntum mechanical system with a Hamiltonian H(R(t))that de-pends adiabatically on time via some control parameters R(t)\n[27]. As a side remark, we note that the concept of geometric\nphases has been generalized to non-adiabatic [28] and non-\ncyclic [29, 30] evolution. The non-degenerate ground state of\nH(R)is denoted byjRi. Since the system cannot leave its\ninstantaneous ground state when initially prepared in jR(t0)i,\nthe geometric constraint\nP(t)j\t(t)i=j\t(t)i (3)\nwithP(t) =jR(t)ihR(t)jthe projection onto the ground\nstate, is imposed on the solution j\t(t)iof the time dependent\nSchr ¨odinger equation in the adiabatic limit. This constraint,\nreflecting the absence of any dynamical level transitions, im-\nplies the formj\t(t)i=ei(\u001e(t)\u0000\u001eD(t))jR(t)i, wherejR(t)iis\na family of ground states with a smooth relative phase, i.e., a\ngauge of ground states. \u001eD(t) =Rt\nt0d\u001cE0(\u001c)with the ground\nstate energy E0is called the dynamical phase. The additional\nphase factor ei\u001e(t)of the state vector j (t)i=ei\u001eD(t)j\t(t)i\nrelative to the gauge jR(t)ireveals a deep geometric “prin-\nciple of least effort” that nature employs in the adiabatic\nlimit of quantum mechanics: j (t)iis the shortest path in\nHilbert space that satisfies Eq. (3), shortest as measured\nbyL=Rt\nt0d\u001cq\nh_ (\u001c)j_ (\u001c)iwith “velocity”j_ (\u001c)i, i.e.,\nby the metric induced by the inner product in Hilbert space.\nTo systematically construct this path, it is helpful to decom-\npose its tangent vector into two orthogonal components as\nj_ (t)i=P(t)j_ (t)i+ (1\u0000P(t))j_ (t)i. The first compo-\nnent is called the vertical part j_ iVand generates an evolu-\ntion that stays within the projection P(t), i.e., a mere change\nof phase ofj (t)i. The second part in contrast generates an\nevolution perpendicular to P(t)in the sense of the inner prod-\nuct and is consequently called the horizontal part j_ iH. From\nk_ k2=k_ Vk2+k_ Hk2, it is easy to see that the tangent\nvector to the shortest path must be purely horizontal, i.e.,\nP(t)j_ (t)i= 0; (4)\nor, equivalently for non-degenerate ground states,\nh (t)j_ (t)i= 0 . This parallel-transport prescription\nimmediately determines\n\u001e(t) =iZt\nt0d\u001chR(\u001c)jd\nd\u001cjR(\u001c)i: (5)\nThe quantityAd\ndt(R(t)) =hR(t)jd\ndtjR(t)itransforms un-\nder a gauge transformation jRi ! ei\u001f(R)jRiasAd\ndt!\nAd\ndt+id\ndt\u001f(R(t)), i.e., as aU(1)gauge field. The geomet-\nric phase\u001e\r=iR\n\rA(mod2\u0019)associated with a loop \rin\nparameter space is gauge invariant. This construction can be\nimmediately generalized to n-fold degenerate ground states\nwhere the projection P(t)in Eq. (4) becomes\nP(t) =f(t)fy(t) =nX\nj=1jRj(t)ihRj(t)j (6)\nwith the ground state manifold basis or frame f=\n(jR1i;:::;jRni). We note the independence of the projection3\nunderU(n)basis transformations f!fU. Thus, instead of\na mere phase factor, the local gauge degree of freedom is then\naU(n)basis transformation on the ground state manifold. In\nthis case, the geometric phase or, in mathematical terms, the\nU(1)holonomyei\u001e\ris replaced by the U(n)holonomy\nU\r=Te\u0000R\n\rA; (7)\nwhereAjl=hRjjdjRliis the non-Abelian gauge field or con-\nnection andTis the time ordering operator along the cyclic\npath\r.\nWe finally note that the connection can be expressed in\na manifestly gauge invariant or basis independent form as\nAK=\u0000[(dP);P][13]. In this case, the geometric phase\nin the ground state manifold for a loop \rstarting at time t0\nis described by the propagator of the Schr ¨odinger equation in\nthe adiabatic limit, or the holonomy operator\nU\r\nK=P(t0)Te\u0000R\n\rAKP(t0): (8)\nU\ras occurring in Eq. (7) is then just the gauge (basis)\ndependent representation matrix of U\r\nK.\nB. From geometric phases to topological band structures\nLet us consider an insulating band structure in dspatial di-\nmensions with noccupied Bloch states ju\u000b\nkibelow the Fermi\nenergy that span the projection P(k) =Pn\n\u000b=1ju\u000b\nkihu\u000b\nkjat lat-\ntice momentum k. If we identify the energy gap relevant for\nthe adiabatic approximation with the band gap of the insulator\nand the parameter manifold with the Brillouin zone (BZ) in\nwhich the lattice momentum is defined, a gauge structure can\nbe defined in complete analogy to Section II A. The projection\nP(k)defines ann-plane in the N-dimensional Hilbert space\ndefined at every momentum kin the BZ, where Nis the total\nnumber of bands (occupied and empty) of the model system.\nP(k)is thus by definition a point on the Grassmann mani-\nfoldGn(CN) =U(N)=(U(n)\u0002U(N\u0000n)). Topological\ninvariants of this gauge structure then distinguish topologi-\ncally inequivalent mappings k7!P(k), that is homotopically\ninequivalent mappings from the BZ torus TdtoGn(CN). For\nthe case of particle number conserving insulators without ad-\nditional symmetries outlined here, an integer invariant named\nthem-th Chern number [31] is defined for even spatial dimen-\nsionsd= 2mwhile all insulators are equivalent in odd spa-\ntial dimensions. Physical symmetries imply constraints on the\nform of the mapping k7!P(k). For example the anti-unitary\ntime reversal symmetry TyieldsP(k) =TP(\u0000k)T\u00001. Map-\npings (band structures) that would be topologically equiva-\nlent when breaking the relevant symmetries may be topolog-\nically inequivalent when maintaining the symmetries. This\nphenomenon is commonly referred to as symmetry protection\nof a topological state. Taking also into account superconduct-\ning band structures as well as all symmetries considered by\nAltland and Zirnbauer [14] significantly refines the system of\ntopological invariants and leads to a pattern which has beencoined the periodic table of topological insulators and super-\nconductors [8–12]. In this work, we would like to address the\nquestion to what extent such invariants can be generalized to\nlattice translation-invariant quantum many body systems in a\nmixed state. While the relevant physical symmetries can read-\nily be generalized to the realm of density matrices, the defini-\ntion of topological invariants for mixed states from a gauge\nstructure over the space of density matrices has not been dis-\ncussed so far. However, a topologically trivial gauge structure\nfor density matrices has been discovered by Uhlmann [16–18]\nas we will review now. Later on (see Section III C), a more re-\nstrictive gauge structure that can be topologically non-trivial\nwill be derived from Uhlmann’s general construction.\nC. Parallel transport and geometric phases for density\nmatrices\nA density matrix \u001ais a positive semi-definite operator with\nunit trace on a Hilbert space H. Here, we consider dim H=\nN <1. For the application to gapped band structures that\nwe have in mind here, Nplays the role of the total number of\nbands of a model system. \u001acan be represented as a pure state\nj ion an extended Hilbert space HA\nHBwithHA'HB'\nHsuch that Tr [O\n1j ih j] =TrA[O\u001a]for any operator O\nacting inH, a prescription referred to as state purification.\nEquivalently, the purification can be represented in terms of a\nHilbert Schmidt operator, i.e., a N\u0002Nmatrixwthat satisfies\n\u001a=TrB[j ih j] =wwy; (9)\nwhere the trace over the auxiliary Hilbert space HBbecomes a\nmatrix multiplication. This description contains a redundancy\nsince under w!wUwithU2U(N),\u001a!wUUywy=\u001a\nis unchanged. We note a formal analogy to the basis indepen-\ndence of the projection (6). Furthermore, the inner product\n(w;v) =Tr[wyv]for matrices again (cf. Section II A) defines\na natural means to measure the length L=Rt\nt0d\u001cp\n( _w;_w)\nof a patht7!w(t)of Hilbert Schmidt operators that purify a\npatht7!\u001a(t)in the sense of Eq. (9). The shortest possible\npath can again be obtained by decomposing the tangent vec-\ntor_winto a vertical part _wVand a horizontal part _wHwith\nthe help of the inner product. _wis purely horizontal for the\nshortest path. Vertical vectors can be defined in terms of Eq.\n(9) as tangent vectors to curves s7!w(s)that project to the\nsame\u001a=wwyfor alls. Vertical vectors are hence of the\nform _wV=d\ndswU(s)\f\f\ns=0=wuwithU(s) =eus2U(N).\nHorizontal vectors _wHare defined as orthogonal to all ver-\ntical vectors, i.e., Tr [ _wy\nHwu] = 0 for allU(N)-generators\nu=\u0000uy. Adding the hermitian conjugate to this condition\nyields Tr [u( _wy\nHw\u0000wy_wH)] = 0 . This can be true for every\nantihermitian uonly if the second antihermitian factor is zero\nby itself, i.e., the tangent vector _wis horizontal if [16–18],\n_wyw\u0000wy_w= 0:\nEq. (1), repeated here for convenience, is the density matrix\nanalog of Eq. (4).4\nLet us now construct the U(N)-gauge field associated with\nthis parallel transport prescription, in a form that is amenable\nto practical calculations. For a given \u001a=P\njpjjjihjj, the\nuniquely defined square rootp\u001a=P\njppjjjihjjdefines a\ngeneric purification w=p\u001a. In fact every purification can by\nmeans of a polar decomposition be written as\nw=p\u001aUwithU2U(N) (10)\nin some analogy to the polar decomposition of a complex\nnumber. Let us consider a loop \r:t7!\u001a(t); t2[0;T]\nin the space of density matrices. We denote by t7!w(t) =p\n\u001a(t)U(t)the parallel-transport of an arbitrary initial purifi-\ncationw(0) =p\n\u001a(0)U(0)along this loop \rand call the geo-\nmetric phase H\r\nU=U(T)U(0)yits Uhlmann holonomy. With\nti=iT=M; M2N, we can express H\r\nUas\nH\r\nU= lim\nM!1MY\ni=1U(ti)Uy(ti\u00001); (11)\nwhere the product is ordered from right to left with increasing\ni. We now wish to obtain the analog of Eq. (7) in an explicit\nform. To this end, we first obtain the parallel-transported U(t)\nin Eq. (10) at infinitesimally neighbouring points in time. Eq.\n(1) can be shown to be equivalent to\nwy(t+\u000f)w(t) =wy(t)w(t+\u000f)\u00150 (12)\nto leading order in \u000f; in particular, the hermitian matrix\nwy(t+\u000f)w(t)is positive semi-definite. Defining V=wy(t+\n\u000f)w(t) =Uy(t+\u000f)p\n\u001a(t+\u000f)p\n\u001a(t)U(t)\u00150we immedi-\nately verify\np\n\u001a(t+\u000f)p\n\u001a(t) =HU(t+\u000f)Uy(t)\nwith the semi-positive hermitian matrix H=U(t+\n\u000f)V(t)Uy(t+\u000f). Hence, the singular value decomposition\np\n\u001a(t+\u000f)p\n\u001a(t) =LDRy\nwith unitary L;R yieldsH=LDLyand, more importantly\n[24],\nU(t+\u000f)Uy(t) =LRy: (13)\nWith this recipe the connection AU=\u0000_U(t)Uy(t) =\nlim\u000f!01\n\u000f(1\u0000U(t+\u000f)Uy(t))can be computed explicitly. The\ndesired holonomy then reads as\nH\r\nU=Te\u0000R\n\rAU: (14)\nWe note thatAUis invariant under a rescaling \u001a(t)!\n\u0015(t)\u001a(t)with a strictly positive real function \u0015(t)>0, which\naffects the singular values Dalone. Therefore, the normaliza-\ntion of the density matrix is not relevant for geometrical and\ntopological considerations.III. TOPOLOGICAL ASPECTS OF MIXED STATES\nIn order to discuss topological features of emergent gauge\ntheories for mixed states, it is natural to view the above de-\ncomposition of tangent vectors into vertical and horizontal\ncomponents as a connection on a principle fiber bundle (PFB)\n(see e.g., Ref. [32]).\nIt is well known that the PFB of general Hilbert-Schmidt\noperators is topologically trivial since \u001a7!p\u001ais a global\nsection (gauge). At first sight, this may seem discouraging.\nOn the other hand, even the special case of pure states can be\nviewed as a topologically trivial gauge structure in terms of\nHilbert Schmidt operators. In contrast, it is well known that\nthe gauge structure for pure states discussed in Section II A\ncan very well be topologically non-trivial, phenomena such\nas topological insulators being striking ramifications of this\npossibility. More concretely, the frame bundle of orthonormal\nframesfRspanning the projection P(R)can be topologically\nnon-trivial. However, it can always be viewed as a sub-bundle\nof the topologically trivial Hilbert-Schmidt bundle of arbitrary\nquadratic matrices w(R)that satisfyP(R) =w(R)wy(R). In\nother words, the level of description determines whether topo-\nlogical aspects of pure states can be resolved. More generally\nspeaking, every gauge theory (PFB) can be viewed as a sub-\nbundle of a topologically trivial gauge theory (PFB) with a\nbigger gauge degree of freedom but over the same base man-\nifold. As for mixed state density matrices key questions are\nthus whether topologically non-trivial features exist and, if so,\nto identify a gauge structure which is able to reveal their topo-\nlogical content. Here we consider several physically moti-\nvated constraints under which a topologically non-trivial PFB\nof EOPS will be defined.\nA. Triviality and structure of the Hilbert-Schmidt bundle\nLet us first review the case of arbitrary invertible N\u0002N\ndensity matrices \u001aas considered by Uhlmann [16], i.e., strictly\npositive matrices without any further constraints such as nor-\nmalization of the trace. The space MN(N)of all Hilbert\nSchmidt operators wpurifying such density matrices via Eq.\n(9) is then simply given by the group of all invertible matri-\ncesGL(C;N). This is because \u001a=wwyis strictly posi-\ntive and has the non-vanishing determinant jdet(w)j2if and\nonly ifw2GL(C;N).MN(N) =GL(C;N)is the to-\ntal space of the Hilbert-Schmidt bundle which projects via\n\u0005 :w7!wwyonto the base manifold of strictly positive\nmatricesDN(N) =GL(C;N)=U(N). The quotient form of\nDN(N)is rooted in the fact that any regular matrix Mcan\nbe uniquely written as M=p\nMMyU; U2U(N). For the\nsame reason, wwithwwy=\u001a, can be uniquely represented\nasw=p\u001aU; U2U(N). Hence the fiber over \u001ais given by\nG\u001a=\bp\u001aU:U2U(N)\t\n, which is manifestly isomorphic\ntoU(N).GL(C;N)\u0005!DN(N)thus defines a PFB. Due to\nthe existence of the global gauge or section \u001a!p\u001a, the PFB5\nis topologically trivial, i.e.,\nGL(C;N) =GL(C;N)=U(N)\u0002U(N): (15)\nNow we consider singular density matrices, i.e., semi-positive\nN\u0002Nmatrices with rank n < N . A naive analog of the\nregular case with fibers G\u001a=\bp\u001aU:U2U(N)\t\nwhere\u001a\nnow has rank ndoes not give a PFB. The reason is that the\nright-action w=p\u001aUofU(N)is no longer free, i.e., many\nU2U(N)give the same w=p\u001aUdue to theN\u0000n-\ndimensional null-space ofp\u001a. Simply restricting the Uto\nU(n)acting on the support ofp\u001adoes not resolve this issue\nsince this action would not be transitive, i.e., unable to reach\nallwwithwwy=\u001a. In Ref. [33], it has been shown that\nMN(n)\u0005!DN(n); n < N with the same projection \u0005 :\nw7!wwystill defines a PFB. However, by the same argument\nas before, namely by the existence of the global section \u001a7!p\u001a, also these bundles are topologically trivial.\nB. Ensemble of pure states gauge structure\nInstead of considering general Hilbert Schmidt operators,\nwe define a EOPS bundle in terms of nnon-orthonormalized\npure statesj~ \u000bithat project onto a given N\u0002Ndensity ma-\ntrix\u001aof ranknas (Eq. (2) is repeated for convenience)\n\u001a=nX\n\u000b=1j~ \u000bih~ \u000bj\nas a gauge structure over the space of density matrices \u001a. In-\nstead of viewing \u001aas a single pure state in a larger Hilbert\nspace, EOPS in this scheme are plausible ensembles of pure\nstates in the system Hilbert space. A generic choice j~ \u000bi=pp\u000bj \u000bialong these lines is obtained from the spectral rep-\nresentation\u001a=P\n\u000bp\u000bj \u000bih \u000bj. Under arbitrary unitary ro-\ntations\nj~'\u000bi=j~ \fiU\f\u000b; (16)\nthe density matrix \u001aobtained from the projection (2) is un-\nchanged. However, the pure states j~'\u000biare no longer mutu-\nally orthogonal. The matrix representation\nw=\u0010\nj~ 1i;:::;j~ ni;0;:::; 0\u0011\n(17)\nwithN\u0000nzero columns defines a natural embedding of\nthe spacePN(n)of non-orthonormal n-frames into the space\nMN(n)of Hilbert Schmidt operators of rank n[33].Uas\noccurring in Eq. (16) defines the local U(n)gauge degree\nof freedom of the PFB PN(n)\u0005!DN(n). For regular den-\nsity matrices ( n=N), it is easy to see that the space of all\nEOPS is identical to the space of all Hilbert Schmidt oper-\nators satisfying \u001a=wwy, i.e.,PN(N) =MN(N). Thus,\nfor invertible density matrices with no further spectral con-\nstraints, the EOPS scheme is equivalent to the purification by\npure state representation in a larger Hilbert space. From thisobservation, we immediately conclude that the EOPS bundle\nis topologically trivial for unconstrained regular density ma-\ntrices. In the following, we will consider several constraints\nunder which the EOPS bundle can become topologically non-\ntrivial and explicitly construct the corresponding topological\ninvariants.\nC. Gauge structure of spectrally constrained density matrices\nNon-degenerate density matrices – The situation becomes\nmore interesting if we impose certain conditions on the spec-\ntrum of the density matrix. Let us first consider a regular non-\ndegenerate density matrix \u001a=P\n\u000bp\u000bj \u000bih \u000bj, where we\ncan now without loss of generality assume p1> p2> ::: >\npN. For the hermitian operator \u001awith non-denerate eigenval-\nues, the projections P\u000b=j \u000bih \u000bjare mutually orthogonal\nand it is natural to order the columnspp\u000bj \u000biof a Hilbert\nSchmidt representation (17) with descending size of the spec-\ntral weights p\u000b. This ordering will only be maintained under\na subgroup of gauge transformations (16) that are a direct sum\nofN U(1)-transformations ei\u001e\u000bacting on the rays of eigen-\nstates associated with the eigenvalues p\u000b. Eq. (16) then sim-\nplifies to\nj~'\u000bi=ei\u001e\u000bj~ \u000bi; \u000b = 1;:::;N: (18)\nThe EOPS bundle thus constrained consists of N U (1)-\nbundlesPN(1)\u0005!DN(1)which are subject to the constraint\nNX\n\u000b=1j \u000bih \u000bj=1: (19)\nWhile the individual U(1)-bundles can be topologically non-\ntrivial, Eq. (19) enforces a zero sum rule for their topolog-\nical invariants. The reason for this is analogous to the pure\nstate case and can be intuitively understood as follows. The\nindividual subspaces may exhibit a topologically non-trivial\nwinding as a function of the control parameters (e.g. lattice\nmomentum) in the total Hilbert space. This total space serves\nas fixed reference and thus by definition does not exhibit any\n”net-winding”. Hence, the parameter dependences of the in-\ndividual subspaces have to compensate each other in order to\nspan the resolution of the identity (19) of the total embedding\nspace at every point in parameter space.\nThe spectral constraint forbidding any level degeneracy can\nbe somewhat relaxed by assuming that msubsets of cardinal-\nityn1;:::;nmwithN=Pm\nj=1njof levels are allowed to\nbe degenerate, but still have distinct eigenvalues from levels\noutside of their subset. Such constraints simply result in a\nrestricted gauge group of the form U(n1)\u0002U(n2)\u0002:::\u0002\nU(nm). Again, the individual U(nj)-structures may be topo-\nlogically non-trivial but obey a zero sum rule due to Eq. (19).\nPurity gaps – The notion of a purity gap [21, 22] can be\nrationalized in this framework. To this end, we consider a\nspectral constraint along the lines of the above discussion\nwhere thenlargest eigenvalues of \u001abelong to one subset\nbut are by assumption not degenerate with the (n+ 1) -th6\nlargest eigenvalue of \u001a. Under these circumstances we can\nuniquely define a pure state projection P=Pn\n\u000b=1j \u000bih \u000bj\nwhich is closest to the mixed state defined by \u001a. In Refs.\n[21, 22], the topological invariant of \u001ahas been defined as\nthe invariant of the corresponding pure state projection P.\nBelow, we directly define topological invariants for mixed\nstates in terms of the Uhlmann connection (12) restricted to\nthe subspace associated with the nlargest eigenvalues of \u001a.\nTheir value is equal to the one obtained by performing an\nadiabatic deformation into a pure state along the lines of Refs.\n[21, 22] but, quite remarkably, no reference to pure states is\nrequired here. A purity gap closing means that the n-th largest\neigenvalue and the (n+ 1) -th largest eigenvalue of \u001abecome\ndegenerate, a singular-point at which the spectral constraint\nis violated and the topological invariant associated with the\nsubspace of the nlargest eigenvalues is not well defined.\nAs a consequence, this subspace can exchange topological\ncharge with the eigenspace associated to the (n+1)-th largest\neigenvalue and the topological invariant for the subspace of\nthenlargest eigenvalues may have changed when the purity\ngap reopens – a topological phase transition by purity gap\nclosing.\nSingular density matrices – A physical system under non-\nequilibrium conditions may not be ergodic and can thus be\ndescribed by a singular density matrix, i.e., a density matrix\nof rankn<N . This constraint can be viewed as special case\nof the above discussion of spectral constraints: A subset of\nneigenvalues p\u000bis non-zero and hence not degenerate with\nthe complement of N\u0000nnon-zero eigenvalues. The U(n)\nEOPS bundle associated with the non-zero subspace can be\ntopologically non-trivial.\nD. Topological invariants\nThe gauge structure of spectrally constrained density matri-\nces defined in Section III C allows for the definition of topo-\nlogical invariants directly in terms of the Uhlmann connection\n(see Section II C), i.e., without reference to pure states. To\nthis end, we consider a subset of n < N eigenvalues (eigen-\nvectors) labeled by j= 1;:::;n that are by assumption non-\ndegenerate with (orthogonal to) the N\u0000nremaining eigenval-\nues (eigenvectors) of \u001a. We denote byP=Pn\nj=1j jih jjthe\nprojection onto the rank ndensity matrix ^\u001a=P\njj~ jih~ jj,\ni.e.,^\u001a=P\u001aP. Eq. (17) shows how the corresponding\nU(n)EOPS bundle can be embedded into the general PFB\nof Hilbert Schmidt operators considered by Uhlmann. This\nallows us to construct a curvature on the constrained U(n)\nEOPS bundle which is directly inherited from Uhlmann’s gen-\neral construction. By general definition, a curvature F\u0016\u0017is the\ngeometric phase per area associated with the parallel trans-\nport around and infinitesimal parallelogram \r\u0016\u0017spanned by\nthe vectors\u000e\u0016^e\u0016;\u000e\u0017^e\u0017in momentum space (or more gener-\nally in parameter space). To obtain the curvature of the con-\nstrained EOPS bundle, we have to project Uhlmann’s geo-\nmetric phase H\r\u0016\u0017\nU(see Eq. (14)) associated with the in-\nfinitesimal loop \r\u0016\u0017for^\u001aonto then-dimensional subspaceof interest by virtue of P. We denote by ^H\r\u0016\u0017\nUthe repre-\nsentation matrix of the geometric phase in this subspace, i.e.,\u0010\n^H\r\u0016\u0017\nU\u0011\nij=h ijH\r\u0016\u0017\nUj ji. With these definitions, the cur-\nvature of the U(n)EOPS bundle is given by\n^F\u0016\u0017= lim\n\u000e!01\u0000^H\r\u0016\u0017\nU\n\u000e\u0016\u000e\u0017; (20)\nwhere\u000e!0is shorthand for (\u000e\u0016;\u000e\u0017)!(0;0). In particular,\nthe gauge invariant Abelian curvature reads as\n^FA\n\u0016\u0017=Tr^F\u0016\u0017=\u0000lim\n\u000e!0log det( ^H\r\u0016\u0017\nU)\n\u000e\u0016\u000e\u0017; (21)\nwhere log det( ^H\r\u0016\u0017\nU)isitimes the argument of the Abelian\ngeometric phase factor. In terms of this curvature, stan-\ndard topological invariants for mixed states with spectral con-\nstraints can be defined. For example, if the constrained rank n\ndensity matrix ^\u001a(k)is parameterized by a lattice momentum\nkin a two-dimensional Brillouin zone (BZ), the first Chern\nnumber is given by\nC=Z\nBZi^FA\n2\u0019=Z\nBZd2ki^FA\nxy\n2\u0019: (22)\nThe generalization to other invariants and the implementation\nof physical symmetries is straight forward and analogous to\nthe pure state case. As a prominent example, we would like\nto mention the case of time reversal symmetry as a protect-\ning symmetry (see Section II B for a discussion of the pure\nstate analog). Instead of the projection onto the occupied\nbands, the density matrix \u001a(k)itself then obeys the symme-\ntryT\u001a(k)T\u00001=\u001a(\u0000k)with the anti-unitary time reversal\noperatorT. As in the pure state case, this symmetry carries\nover to arbitrary Uhlmann-holonomies and allows for the def-\ninition of topological invariants in analogy to Ref. [34].\nIV . UHLMANN PHASE WINDING NUMBERS\nIn two recent back-to-back publications [24, 25], a com-\nplementary approach towards the definition of topological in-\nvariants for mixed states has been reported, in particular for\nChern insulators in thermal states. Their approach is based on\nthe so called Uhlmann phase \u001e\r\nUassociated with a loop \rin\nmomentum space, defined as\nei\u001e\r\nU=Tr\u0002\nw(0)yw(T)\u0003\n=Tr\u0002\n\u001a(0)U(T)U(0)y\u0003\n=Tr[\u001a(0)H\r\nU]; (23)\nwhere\ris traversed between t= 0 andt=T,w(t) =p\n\u001a(t)U(t)is a parallel transport in the sense of Uhlmann’s\nconnection (1) and the Uhlmann holonomy H\r\nUhas been de-\nfined in Eq. (14). By construction, \u001e\r\nUbears some anal-\nogy with the Abelian Berry phase of a pure state defined\nas ei\u001e\r=h (0)j (T)i(cf. Section II A). However, there\nis one crucial difference in the mathematical structure of \u001e\r7\nand\u001e\r\nU: While ei\u001e\ris aU(1)-holonomy as is clear from the\nexplicit representation in terms of an integral over a U(1)-\ngauge field (see Eq. (5)), ei\u001e\r\nUdoes in general nothave\nthis property – in contrast to the full non-Abelian Uhlmann\nholonomyH\r\nU. This observation is not just a minor techni-\ncal point but can have drastic consequences. The key fea-\nture of aU(1)-holonomy is its additive group structure, i.e.,\n\u001e\r12=\u001e\r1+\u001e\r2(mod2\u0019), where\r12denotes the concatena-\ntion of the loops \r1and\r2. It is this very property that allows\nus to go from an infinitesimal geometric phase as measured\nlocally by a curvature (see Eq. (20)) to a quantized global\ntopological invariant represented as an integral over the entire\nparameter space (see, e.g., the definition of the Chern num-\nber in Eq. (22)). Again making use of this additive group\nstructure of the ordinary Berry phase, the Chern number C\ncan be represented as the change of the Berry phase \u001ekx=R\nS1dkyAy(kx;ky)associated with a ky-circle in the BZ at\nfixedkx. This reformulation is achieved by dividing the BZ\ntorus intoky-ringsRkxof infinitesimal width [kx;kx+dkx]\naroundkxand then using the Stokes theorem to transform\nthe area integral of the curvature over these rings into a\nline integral of the connection Aalong their boundaries, i.e.,R\nRkxd2kFxy=R\nS1dky[Ay(kx+dkx;ky)\u0000Ay(kx;ky)] =\u0010\n@\u001ekx\n@kx\u0011\ndkx. Explicitly, this procedure leads to the following\nwell known representation of the Chern number\nC=1\n2\u0019Z\nS1\u0012@\u001ekx\n@kx\u0013\ndkx; (24)\nwhere the circle S1is around the kxloop of the BZ. Equiv-\nalently, the role of kxandkycould be exchanged here, of\ncourse. This is again because of the holonomy group struc-\nture of the Berry phase which allows us to divide the BZ into\nstripes in an arbitrary direction.\nThe basic idea of Refs. [24, 25] is to take the right hand\nside of Eq. (24) but to replace the ordinary Berry phase by the\nUhlmann phase as defined in Eq. (23), i.e.,\nCU=1\n2\u0019Z\nS1 \n@\u001ekx\nU\n@kx!\ndkx: (25)\nWe note that Ref. [24] actually discusses several constructions\nin terms of the spectrum of the “holonomy matrix” \u001a(0)H\r\nU.\nSince the absence of a holonomy group structure which is at\nthe heart of our present discussion also pertains to \u001a(0)H\r\nU,\nthis distinction is not of central importance here as will be\naddressed more explicitly in our example below. As both kxin\nthe BZ and the phase \u001ekx\nUare defined on a circle, the mapping\nkx7!\u001ekx\nUis characterized by an integer quantized winding\nnumber which is exactly measured by Eq. (25).\nHowever, since the Uhlmann phase does not have an addi-\ntive group structure, CUcannot be represented as the integral\nof a curvature over the 2D BZ. It is hence not immediately\nclear to what extent CU, technically being a 1D winding num-\nber, can be seen as a unique property of the 2D system under\ninvestigation. In particular, it is not clear that CUas defined in\nEq. (25) is equal to ~CUobtained fromCUby exchanging kxandkyin all calculations, i.e.\n~CU=1\n2\u0019Z\nS1 \n@\u001eky\nU\n@ky!\ndky: (26)\n-3-2-1123kx\n-3-2-1123fUkx\n-3-2-1123ky\n-3-2-1123fUky\nFIG. 1. (color online) Momentum dependence of the Uhlmann phase\nfor a thermal state at \f= 1:3for the model defined in Eq. (27),\nnumerically obtained by discretizing the paths in momentum space\ninto 500 equidistant points. Left panel: \u001ekx\nUas a function of kx.\nRight panel: \u001eky\nUas a function of ky.\nHere, we demonstrate that indeed CU6=~CUcan occur for\na Chern insulator the Hamiltonian of which is not invariant\nunder the exchange of kxandky. To this end, we consider a\ntwo-banded fermionic Chern insulator on a 2D square lattice\nwith unit lattice constant defined by the Bloch Hamiltonian\nH(k) =~d(k)\u0001~ \u001b=3X\nj=1dj(k)\u001bj;\nd1(k) = sin(kx); d2(k) = 3 sin(ky); (27)\nd3(k) = 1\u0000cos(kx)\u0000cos(ky);\nwhere\u001bjare Pauli matrices. Note the anisotropy factor of\n3ind2(k)which is crucial here. For the two-banded model\n(27), the Uhlmann connection and the Uhlmann phase can be\nreadily calculated along the lines of Ref. [35]. Explicitly, the\nUhlmann connection AUin this special case reads as\nAU;\u0016=\u0000h\n(@k\u0016p\n\u001a(k));p\n\u001a(k)i\n: (28)\nAtT= 0, Eq. (28) concurs with the Kato connection [13]\n(see Eq. (8)) for the pure state case and the ordinary Chern\nnumber (24) reads as\nC=1\n4\u0019Z\nBZd2k\u0010\n^d(k)\u0001[(@kx^d(k))\u0002(@ky^d(k))]\u0011\nwith ^d=~d\nj~dj. Explicit calculation yields C=\u00001. Also, both\nCUand~CUtrivially concur with Csince the Uhlmann phase\nreduces to the Berry phase for pure states such that Eq. (24),\nEq. (25), and Eq. (26) become equivalent. At finite T, the\nsystem is in a thermal state defined by \u001a(k) =1\nZe\u0000\fH(k)\nwithZ=Tr[e\u0000\fH(k)]. Using the simplified form (28) of\nthe Uhlmann connection, the direct calculation of Uhlmann\nholonomies (14) for arbitrary loops in momentum space is\nstraightforward which in turn allows the direct calculation\nofCUand~CUfrom Eq. (25) and Eq. (26), respectively.8\nIn the infinite temperature limit, this calculation becomes\ntrivial as\u001a(k) =1independent of kand henceAU= 0 in\nEq. (28), resulting in CU=~CU= 0. Hence, the Uhlmann\nphase winding numbers have to jump from \u00001to0at some\ntemperature. Numerically performing this calculation for\nfinite temperature, we find that both CUand~CUjump to zero\nat finite critical temperatures \fc=1\nTcand~\fc=1\n~Tc, respec-\ntively. Remarkably, for the present model, \fc6=~\fc, i.e.,\nCUand~CUdo not concur at all temperatures. Numerically,\nwe find\fc= 0:874 and~\fc= 1:32. This discrepancy is\nillustrated In Fig. 1, where we visualize the winding numbers\nCU(left panel) and ~CU(right panel) for \f=1\nT= 1:3, i.e.,\n\fc< \f < ~\fc. Clearly,CU= 1 since\u001ekx\nUmonotonously\nincreases, interrupted by a jump from \u0019to\u0000\u0019atk= 0,\nby2\u0019askxcompletes the loop from kx=\u0000\u0019tokx=\u0019.\nIn contrast, \u001eky\nUdoes not reach all values in [\u0000\u0019;\u0019]which\nimplies ~CU= 0. On a more detailed note, we would like\nto point out that, for the parameters chosen here, CUis in\nthe well defined regime in the sense of Ref. [24]. This\nis because the “holonomy matrix” \u001a(kx;ky0)H\rkx;ky0\nU of\naky-loop with footpoint ky0at fixedkxis gapped for all\nkx;ky0, i.e., the difference of the absolute values of its\neigenvalues is finite and bounded from below by \u0001 = 0:268.\nComparing the construction in Refs. [24, 25] to our present\nanalysis in Section III, we would like to emphasize that the\nChern number defined through Eq. (20) and Eq. (22) does\nnot exhibit similar finite temperature transitions as CUand\n~CUbut can, for the model (28), be uniquely defined as \u00001\nfor the larger eigenvalue of the density matrix for all finite\ntemperatures. At infinite temperature, in contrast, the density\nmatrix does not obey any spectral constraints in the sense\ndiscussed in Section III C, rendering all topological invariants\ntrivial.V . CONCLUDING REMARKS\nWe have discussed how several assumptions regarding\nthe spectrum of a family of density matrices can lead to a\ntopologically non-trivial gauge structure. In this framework\ntopological invariants that are protected by these spectral\nassumptions have been defined for mixed states. Protected\nhere means that topologically inequivalent mappings from a\nparameter space into the density matrices can be continuously\ndeformed into each other only if the underlying spectral\nassumptions are violated. Identification of the parameter\nspace with the Brillouin zone of a lattice translation invariant\nsystem provides one way of generalizing topological band\nstructure invariants to the realm of mixed states. A non-trivial\nexample where going beyond pure states is crucial to obtain\na topologically non-trivial Chern insulator as a steady state\nin the framework of a non-equilibrium open quantum system\ndynamics has been reported in Ref. [36]. Additional physical\nsymmetries refining this system of topological invariants\ncan be considered in analogy to the pure state case. The\ntopological invariants defined here, being gauge invariant\nproperties of the density matrix, are in principle experimen-\ntally accessible via state tomography. However, their relation\nto natural observables such as response functions is not yet\nconclusively understood. First progress along these lines\nhas been reported in Ref. [37]. As for two-banded Chern\ninsulators such as the toy model in Eq. (27), any statistical\nmixture of bands with opposite Chern number will certainly\ncause deviations from the quantized Hall conductance.\nIn other situations where the physical ramification of the\ntopological invariant is related to a half-integer quantized po-\nlarization, a statistical mixture may still exhibit a quantization.\nVI. ACKNOWLEDGEMENTS\nSupport from the ERC Synergy Grant UQUAM, and the\nSTART Grant No. Y 581-N16 is gratefully acknowledged.\n[1] M. V . Berry, Proc. R. Soc. London A 392, 45 (1984).\n[2] B. Simon, Phys. Rev. Lett. 51, 2167 (1983).\n[3] F. Wilczek, and A. Zee, Phys. Rev. Lett. 52, 2111 (1984).\n[4] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.\n45, 494 (1980).\n[5] R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).\n[6] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den\nNijs, Phys. Rev. Lett. 49, 405 (1982).\n[7] M. Kohmoto, Annals of Physics 160, 343 (1985).\n[8] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig,\nPhys. Rev. B 78, 195125 (2008).\n[9] A. Kitaev, AIP Conference Proceedings 1134 , 22 (2009).\n[10] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig,\nNew J. Phys. 12, 065010 (2010).\n[11] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).\n[12] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).[13] T. Kato, J. Phys. Soc. Japan 5, 435 (1950).\n[14] A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 (1997).\n[15] R. Kennedy, M. R. Zirnbauer, arXiv:1409.2537 (2014).\n[16] A. Uhlmann, Rep. Math. Phys. 24, 229 (1986).\n[17] A. Uhlmann, Lett. Math. Phys. 21, 229 (1991).\n[18] A. Uhlmann, Rep. Math. Phys. 33, 253 (1993).\n[19] The mathematical background of this statement is that for every\nsmooth vector bundle, there is a “complement” such that the\ndirect sum of the two gives a trivial bundle over the same base\nmanifold.\n[20] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Woot-\nters, Phys. Rev. A 54, 3824 (1996).\n[21] S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Nature Physics\n7, 971 (2011).\n[22] C.-E. Bardyn, M. A. Baranov, C. V . Kraus, E. Rico, A.\nImamoglu, P. Zoller, S. Diehl, New J. Phys. 15, 085001 (2013).9\n[23] O. Viyuela, A. Rivas, M. A. Martin-Delgado, Phys. Rev. Lett.\n112, 130401 (2014).\n[24] Z. Huang, D. P. Arovas, Phys. Rev. Lett. 113, 076407 (2014).\n[25] O. Viyuela, A. Rivas, M. A. Martin-Delgado, Phys. Rev. Lett.\n113, 076408 (2014).\n[26] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).\n[27]R(t)may be thought of as external electric or magnetic fields\nfor example. A finite spacing between the energy levels at all\ntimes is assumed to define the adiabatic time scale.\n[28] Y . Aharonov, J. Anandan, Phys. Rev. Lett. 58, 1593 (1987).\n[29] S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).\n[30] J. Samuel and R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988).[31] S. S. Chern, The Annals of Mathematics, 47(1), 85 (1946).\n[32] M. Nakahara, Geometry, Topology and Physics , CRC Press\n(2003).\n[33] J. Dittmann, G. Rudolph, Journal of Geometry and Physics 10,\n93 (1992).\n[34] E. Prodan, Phys. Rev. B 83, 235115 (2011).\n[35] M. H ¨ubner, Phys. Lett. A 179, 226 (1993).\n[36] J. C. Budich, P. Zoller, S. Diehl, Phys. Rev. A 91, 042117\n(2015).\n[37] J. E. Avron, M. Fraas, G. M. Graf, Journal of Statistical\nPhysics, 148(5) , 800 (2012)."    },    {        "title": "1501.07049v1.Odd_Frequency_Density_Waves.pdf",        "content": "Odd Frequency Density Waves\nYaron Kedem\nCenter for Quantum Materials, Nordic Institute for Theoretical Physics (NORDITA),\nRoslagstullsbacken 23, S-106 91 Stockholm, Sweden\nAlexander V. Balatsky\nInstitute for Materials Science, Los Alamos National Laboratory, Los Alamos, NM 87545, USA and\nCenter for Quantum Materials, Nordic Institute for Theoretical Physics (NORDITA),\nRoslagstullsbacken 23, S-106 91 Stockholm, Sweden\nA new type of hidden order in many body systems is explored. This order appears in states which\nare analogues to charge density waves, or spin density waves, but involve anomalous particle-hole\ncorrelations that are odd in relative time and frequency. These states are shown to be inherently\ndi\u000berent from the usual states of density waves. We discuss two methods to experimentally observe\nthe new type of pairing where a clear distinction between odd and even correlations can be detected:\n(i) by measuring the density-density correlation, both in time and space and (ii) via the conduc-\ntivity which, according to the Kubo formula, is given by the current-current correlation. An order\nparameter for these states is de\fned and calculated for a simple model, illuminating the physical\nnature of this order.\nI. INTRODUCTION\nThe conventional way to consider ordered many body\nsystems is the Landau-Ginzburg formalism of phase tran-\nsition that is centered on order parameter as the attribute\nof order. That appearing order is classi\fed as the anoma-\nlous correlations that describe the emergent state. For\nexample, in charge density wave the order parameter is\n\u0001q(k) =hay\nk+q;\u000bak;\u000bi: (1)\nwherea(y)\nk;\u000bis creating (annihilating) an electron with mo-\nmentum kand spin\u000b. Similarly, the order parameter for\nspin density waves is given by\n~\u0001q(k) =hay\nk+q;\u000bak;\fi~ \u001b\u000b\f: (2)\nwhere~ \u001bis a vector of Pauli matrices. Summation over\nspin indices is implied, both in (1) and (1) and these\nanomalous correlations are taken at equal time for the\ninvolved operators. Static values as written are often\ntaken as the order parameters of the charge (spin) density\nwaves.1,2\nAs our view of correlations evolves, it becomes ap-\nparent that materials can exhibit composite meta or-\nder that signi\fcantly expands the old Ginzburg-Landau\nparadigm. Examples of new orders include topological\norder with no local order parameter3{6. Another ex-\ntension of the concept of order is the notion of odd\nfrequency or odd-time correlations, \frst proposed by\nBerezinskii.7,8Odd-frequency order in superconductors\nand super\ruids has been proposed for numerous real-\nizations and most likely to occur in superconducting\nheterostructures, in nanoscale devices9,10and in mulit-\nband superconductors11. In addition to superconductiv-\nity odd time orders were expanded to spin nematics12and\nBEC13. Here we wish to further advance the notion of\nodd time orders by considering the odd in time densitywave correlations. This odd time density waves would\ndescribe dynamic order that does not manifest itself in\nany static density wave. The situation can be viewed\nas a dynamic order that is hidden from the conventional\nspectroscopies of charge and spin density waves.\nII. INTRODUCING ODD TIME CDW\nThe odd time, or odd frequency, orders are character-\nized by correlations that vanish when hayaiexpectaion is\ntaken at equal time. We now focus on charge and spin\ndensity waves (CDW,SDW). Since the discussion, in our\ncontext, for these two types of states, is highly similar,\nfor the sake of simplicity, we will \frst treat only CDW,\ndescribed by eq. (1). Since spin indices are summed over,\nthey are suppressed in all expressions. In order to treat\nSDW, described by eq. (2) one simply has to reintro-\nduce the spin indices and multiply by ~ \u001bthroughout the\nderivation.\nWe can generalize the correlation (1) to include oper-\natorsak(\u001c) which act in Matsubara time \u001c. The result\nis the anomalous green function\n\u0001q(k;\u001c) =D\nT\u001ch\nay\nk+q(\u001c)ak(0)iE\n=\u0012(\u001c)D\nay\nk+q(\u001c)ak(0)E\n\u0000\u0012(\u0000\u001c)D\nak(0)ay\nk+q(\u001c)E\n(3)\nwhereT\u001cis the time ordering operator and \u0012(x) is the\nHeaviside step function. Since the system is assumed to\nbe invariant to time translation, \u0001 q(k;\u001c) is a function of\nthe relative time only.\nThere can be a few possibilities for the time depen-\ndency of \u0001 q(k;\u001c) at short times, \u001c!0. If there is a\nzero time order, i.e. a time independent part, then we\nrecover the usual form of CDW with non vanishing orderarXiv:1501.07049v1  [cond-mat.str-el]  28 Jan 20152\nparameter given by (1). Then the time dependence of\nthe correlator at nonzero times would re\rect the retar-\ndation e\u000bects that occur at a later time. If, on the other\nhand, the equal time part vanishes, but the \frst order\n\u0001q(k;\u001c)\u0018\u001cremains \fnite, the situation is di\u000berent. In\nsuch a case, all the well-known results regarding CDW\nmight not be valid, but nonetheless there will be non-\ntrivial density-density correlations. We are thus facing\nthe possibility to have a nontrivial correlations that have\nno equal time signature, i.e. a hidden order.\nTo capture the dynamic nature of the correlation we\nconsider a scenario is when \u0001 q(k;\u001c) is odd in time:\n\u0001q(k;\u001c) =\u0000\u0001q(k;\u0000\u001c): (4)\nIn this case it is clear that \u0001 q(k) = \u0001 q(k;0) = 0. This\nimplies that if the system is described by quantity obey-\ning a condition of the form (4), it is in a state that pos-\nsess a dynamic kind of order. This state might share\nmany of the characteristics, of the usual CDW, by it is\nfundamentally di\u000berent in that there is no static density\ncorrelations.\nIII. DETECTION OF ODD TIME CDW\nHere we outline experimental observation that would\ntest the odd frequency density order. The usual type\nof CDW, which is described by (1) is manifested with a\nmodulation of the density\nn(r) =h\ty(r)\t(r)i=n0+nqcos(qr+\u001e);(5)\nwhere \t( r) =R\ndkeikrak,n0is the average density, nq\nis the amplitude of the modulation with wave number q\nand\u001eis an arbitrary phase. When writing Eq. (5) we\nignore any density modulations which are not directly\nrelated to CDW, so we explicitly assume a uniform den-\nsity fornq= 0. Indeed, the uniform density n(r) =n0\nwould be the case when the equal time order parameter\nvanish, \u0001 q(k) = 0, as would occur, for example, in the\nodd frequency case. So one has to study higher order cor-\nrelations to reveal this kind of order. This can be done\nby looking at the density correlation, both in space and\ntime\nhT\u001c^n(r;\u001c)^n(r0;0)i=n2(6)\n+hT\u001c\ty(r;\u001c)\t(r0;0)ihT\u001c\ty(r0;0)\t(r;\u001c)i;\nwhere we used Wick's theorem and assumed that n=\nh^n(r;\u001c)iis spatially and temporally independent. This\nassumption involves two important aspects. First the\nspatial homogeneity means we do not have the usual\nCDW state, which is manifested by the modulations\n(5). Second the time \u001cinh^n(r;\u001c)iis the absolute\ntime, or \\center of mass\" time, as opposed to \u001cin\nhT\u001c^n(r;\u001c)^n(r0;0)iwhich represent a relative time, or\nsome correlation time. Dependency on the relative time\ncan comes from microscopical dynamics. Here we note\nEnergy\nk kF -kF2∆𝑞=2𝑎𝑘+𝑞+𝑎𝑘∆−\n𝑘𝐹→−𝑘𝐹\nq=-2𝑘𝐹 ∆+\n−𝑘𝐹→𝑘𝐹\nq=2𝑘𝐹Figure 1. (Color online) An illustration of the typical sit-\nuation of CDW. Correlations between electrons on opposite\nsides of the Fermi surface appear and a gap is opened, around\nthe Fermi energy, proportional to the correlation magnitude.\nSince the involved electrons should be in the vicinity of the\nFermi wave vector \u0006kF, the momentum di\u000berence, which also\ndetermine the wave vector of the density modulations, have\nto bejqj= 2kF. The correlation operator can be written\neither as annihilation of electron at kFand creation at\u0000kF\nso \u0001 q(k) =hay\n\u0000kFakFi, which make q=\u00002kF, or vice versa\n\u0001q(k) =hay\nkFa\u0000kFi, which makes q= 2kF. Note that for\nodd time CDW \u0001 q(k) = \u0001 q(k;\u001c= 0) = 0 and a gap might\nnot open at kF, but it is still reasonable to assume that cor-\nrelations with the same momentum properties appear. This\nassumption is represented in eq (8).\nthat there are discussions on states that depend on the\noverall time. Such a scenario can imply the system is\nnot in a steady state, or even the formation of a time\ncrystal14,15. We do not address those issues in this work.\nThe quantity of interest for us is the second term on the\nright hand side of (6), \u001f(r;r0;\u001c) =hT\u001c^n(r;\u001c)^n(r0;0)i\u0000\nn2. By going over to momentum space we can write it\nusing (3):\n\u001f(r;r0;\u001c) =Z\ndKe\u0000i(~ r~k+rq+r0q0)\u0001q(k;\u001c)\u0001q0(k0;\u0000\u001c);\n(7)\nwhere ~ r=r\u0000r0,~k=k\u0000k0andR\ndK=dkdk0dqdq0\nmeans integration on all momentum variables. Let us\nnow consider a case similar to the typical scenario for\nCDW, shown in Fig 1, where the abnormal correlations\noccur on the two opposite sides of a Fermi surface. This\nimplies q=\u00002k=\u00062kf, where kfis the Fermi wave\nvector. So the order parameter is given by\n\u0001q(k;\u001c) = \u0001 +(\u001c)\u000e(k\u0000kf)\u000e(q+ 2kf)\n+ \u0001\u0000(\u001c)\u000e(k+kf)\u000e(q\u00002kf): (8)\nwhere \u0001 +(\u001c) and \u0001\u0000(\u001c) are in principle independent\nquantities. Here, for simplicity we choose \u000e-function as\nthe form of \u0001 q(k;\u001c). A more realistic assumption would\nbe that it is sharply peaked function at those values, so\nthat \u0001 +(\u0000)(\u001c) =R\ndkdq\u0001q(k;\u001c) with integration inter-\nvals are around the points k= (\u0000)kf;\u0000q= (\u0000)2kf.3\nInserting (8) in (7) we have,\n\u001f(r;r0;\u001c) = \u0001 +(\u001c)\u0002\n\u0001+(\u0000\u001c)ei4Rkf+ \u0001\u0000(\u0000\u001c)\u0003\n+ \u0001\u0000(\u001c)\u0002\n\u0001\u0000(\u0000\u001c)e\u0000i4Rkf+ \u0001 +(\u0000\u001c)\u0003\n:(9)\nThe fact that \u001f(r;r0;\u001c) depends on the \\center of mass\"\ncoordinate R= (r+r0)=2 is a clear manifestation of\nthe broken translation symmetry associated with CDW.\nSince we are considering a situation where the average\ndensity is uniform, the broken symmetry that can be ob-\nserved in the density-density correlation is a strong indi-\ncation to a hidden order residing in the system.\nThe \frst two terms in (7) are oscillating with e\u000bec-\ntive wave vector of 4 kf. Such terms would appear for\nthe usual case of CDW, if one looks at density-density\ncorrelations, in addition to terms with a wave vector of\n2kf, which is also how the density is modulated. The\nmore rapid oscillations of might be di\u000ecult to observe in\ncertain systems. Neglecting the oscillating term we are\nleft with a constant, i.e. rindependent term. In the sim-\nple case where \u0001 +(\u001c) = \u0001\u0000(\u001c) = \u0001(\u001c), the sign of this\nterm is given by the time parity of \u0001( \u001c), regardless of the\nsign of \u0001(\u001c) itself. Thus, if \u0001( \u001c) =\u0000\u0001(\u0000\u001c), there will\nbe a negative contribution from \u001f(r;r0;\u001c) to the density-\ndensity correlation. We can consider a scenario, where\nthis fact can be the basis for experimental evidence for\nodd time correlation, when the density-density correla-\ntion is a measured function of some experimental param-\neter, like temperature. If the correlation is constant in\nsome region and then starts changing, the direction of\nthe change can tell us whether \u0001 q(k;\u001c) is odd or even in\ntime, i.e. a decrease in correlation would suggest an odd\ntime CDW.\nAnother quantity that can be of interest is the current-\ncurrent correlations, which according to the Kubo\nformula16can yield the conductivity. Assuming the av-\nerage current vanishes hji= 0, the current-current cor-\nrelations are given by (see appendix for details):\nhT\u001cji(r;\u001c)jj(r0;0)i=Z\ndKe\u0000i(~ r~k+rq+r0q0)\u0002\n(k+k0+q)i(k+k0+q0)j\n4m2\u0001q(k;\u001c)\u0001q0(k0;\u0000\u001c):\n(10)\nFollowing the same procedure we used above for\n\u001f(r;r0;\u001c), we consider the typical case, shown in Fig 1,\nwhere \u0001 q(k;\u001c) is given by (8). Inserting (8) into (10),\nwe get\nhji(r;\u001c)jj(r0;0)i=\n\u0000(kf)i(kf)j\nm2[\u0001+(\u001c)\u0001\u0000(\u0000\u001c) + \u0001\u0000(\u001c)\u0001+(\u0000\u001c)]:\n(11)\nIt is interesting that the rapidly oscillating terms, which\nappear in eq (7), vanish here because they correspond to\nzero momentum. The disappearance of a conductivityP T \u0001+\n+\u0001\u0000\n+\u0001+\n\u0000\u0001\u0000\n\u0000\u001f(R;\u001c)hjjim2\nk2\nf\n+ + 1 1 1 1 2 + 2 cos(4 Rkf) -2\n+ - 1 1 -1 -1\u00002 + 2 cos(4 Rkf) 2\n- + 1 -1 1 -1\u00002\u00002 cos(4 Rkf) 2\n- - 1 -1 -1 1 2\u00002 cos(4 Rkf) -2\nEntang 1 1 1 -1 \u00002isin(2Rkf) 0\nTable I. The values of the density-density and the current-\ncurrent correlation for di\u000berent parity behaviors of \u0001 q(k;\u001c)\naccording te eq. (9) and (11). \u0001\f\n\u000bdenotes \u0001 \u000b(\f\u001c),PandT\nare the reversal operator for time and momentum respectively.\nThe last line refer to a situation where PandTdo not have\na de\fned value each, but are interdependent, which is noted\nas Entang(led). From the observation of these values can one\ncan learn the type, and especially the parity, of electron-hole\ncorrelations present in a system.\nterm, accompanied by the appearance of oscillating den-\nsity terms seems similar, and might be closely related, to\nthe usual scenario of CDW where the conductivity vanish\ntogether with the appearance of density modulation.\nIn the same way we consider \u0001 q(k;\u001c) to be an odd\nor even function of time, we can also consider whether\nit is odd or even in momentum. For the case de-\nscribed by (8), even (odd) parity in momentum implies\n\u0001+(\u001c) = (\u0000)\u0001\u0000(\u001c). These properties of \u0001 q(k;\u001c) can\nbe formulated by considering \u0001 q(k;\u001c) to be a eigenfunc-\ntion of two operators: the time reversal T, de\fned as\nT\u0001q(k;\u001c) = \u0001 q(k;\u0000\u001c) and the momentum reversal, de-\n\fned as as P\u0001q(k;\u001c) = \u0001\u0000q(\u0000k;\u001c). For each oper-\nator, an eigenvalues of -1 or 1 implies the function is\nodd or even respectively. The values of \u001f(r;r0;\u001c) and\nhj(r;\u001c)j(r0;0)i, for the di\u000berent parity options are shown\nin Table I.\nApart from the four parity options for the operators P\nandT, there can be another kind of behavior. For ex-\nample, if \u0001 +(\u001c) is even in time and \u0001 \u0000(\u001c) is odd. Such\na situation, where the parity in one variable depends on\nthe sign of the other variable suggest that the time and\nmomentum parities are entangled (not necessarily in the\nquantum mechanics sense). Thus the parity of each indi-\nvidual operator is not de\fned and we refer to this state\nas entangled. There could be several possibilities for such\na state and the values of \u001f(r;r0;\u001c) andhj(r;\u001c)j(r0;0)i,\nfor one of those possibilities are shown in the last line of\nTable I.\nThe typical case of CDW require that \u0001 +(\u001c) = \u0001\u0000(\u001c),\ni.e.Peven, and independent of the sign of the mo-\nmentum. One can allow to have explicit non-trivial de-\npendence on the sign of the momentum i.e. \u0001 +(\u001c) =\n\u0000\u0001\u0000(\u001c), which means we are considering p-wave CDW.\nHence, we have both possibilities and Table I represent\na general symmetry classi\fcation.4\nIV. DEFINITION OF AN ORDER PARAMETER\nFOR ODD TIME CDW\nIf indeed this state represent a new phase, which pos-\nsess some order, it is described by an order parameter.\nSince the order parameter, as it is most commonly de-\n\fned, vanish in the case of odd time correlations, we\nde\fne a new order parameter. This can be done, for\nexample, by considering the time derivative of the corre-\nlation at equal time, i.e. \u001c= 0. Let us de\fne our order\nparameter as17,18\nDq(k) =d\nd\u001c\u0001q(k;\u001c)j\u001c=0: (12)\nWhileDq(k) does not depend on time it capture some of\nthe dynamics of the state. For any model Hamiltonian\nH, we can use the Heisenberg equationd\nd\u001cO(t) = [H;O ]\nto write this order parameter as\nDq(k) =D\n[H;ay\nk+q]akE\n: (13)\nIn case the part that is most signi\fcant is the correlations\nat close to equal time \u001c\u001c1, it is plausible to approxi-\nmate this correlations to \frst order as\n\u0001q(k;\u001c)'Dq(k)\u001c (14)\nand then we can learn from the order parameter (13)\nrelevant correlations that are responsible for odd time\norder (3).\nIt can be very instructive to calculate (13) for a spe-\nci\fc model and a good candidate would be the textbook\nmodel for CDW. However, our discussion of CDW, which\ninvolved a phenomenological approach and was focused\non electron correlation, is somewhat di\u000berent from the\ntraditional treatment of CDW, which typically includes\na microscopic model and is focused on average of phonon\noperators. Thus we brie\ry review the traditional formal-\nism and show how it connects to the one we used.\nThe traditional derivation of CDW1,2involves the\nelectron-phonon interaction\nHI=X\nk;qgq;k\u0010\nby\n\u0000q+bq\u0011\nay\nk+qak (15)\nwhereb(y)\nqis creating (annihilating) a boson with momen-\ntumqandgq;kis a coupling function. Using mean \feld\nmethods, a self consistent equation can be obtained for\nhbqiwith a solution having hbqi6= 0, which represent a\nperiodic distortion in the underlying lattice. The average\nhbqiis related to the gap that is opened in the electron\nspectrum and it is interpreted as an order parameter.\nFrom a phenomenological point of view, if one is inter-\nested in the electronic structure, it is not necessary to\nconsider the interaction (15), or even to involve phonons,\nin order to derive an order parameter. One can simply\nconsider periodic spatial modulation nqin the density ofelectrons, as given by (5), and write an expression for nq\nin term of averages on creation operators\nnq=ei\u001eZ\ndkhay\nk+qaki=Z\ndk\u0001q(k): (16)\nThe presence of modulations nq6= 0, implies a non\nvanishing order parameter regardless of the microscopic\nmodel but from (15) it is clear that an hbqi\u0018\u0001q(k).\nSo the traditional formalism can be regarded as a pos-\nsible recipe to construct a microscopic model which can\nexplain our phenomenological quantities.\nWe can use (15) for getting an explicit expression for\nDq(k). We have\nDq(k) =*X\nq0gq0;k+q\u0010\nby\n\u0000q0+bq0\u0011\nay\nk+q+q0ak+\n:(17)\nIn the typical case, which we discussed above, q=q0=\n2kf. Thus the order parameter is given by electron-hole\ncorrelation with a momentum di\u000berence of 4 kf. This ob-\nservation is consistent with the behavior of \u001f(r;r0;\u001c) and\nhj(r;\u001c)j(r0;0)icalculated above. Indeed this suggests\nthat the hidden order we are interested in is manifested\nin higher order correlation between electron and lattice\ndegrees of freedom.\nV. CONCLUSION\nIn conclusion, we present the odd time (frequency)\nCDW and SDW as an extension to the notion of den-\nsity waves considered to date. Main feature of these odd\ntime density waves is that there are no equal time modu-\nlation in the expectation values of spin and charge densi-\nties. Yet as is clear from the overall structure, there are\nnontrivial dynamic correlations. These nontrivial corre-\nlations, Eq. (9) and (11) could serve as a test for the\nodd time density wave. As such this work represents the\nextension of odd-frequency state classi\fcation to density\nwaves, beyond superconductors and BEC.\nIt remains to be seen what are the realistic fermion\ninteractions that would enable the odd time density cor-\nrelations. We also point out that these dynamic correla-\ntions with no equal time expectation value do correspond\nin a way to a \\hidden order\" that cannot be observed as\nthe traditional charge/spin modulation. As such it can be\na candidate for materials with hidden orders where one\nobserves thermodynamic features of a transition while\nno order parameter consistent with the transition can be\ninferred from the current measurements.\nAcknowledgements We acknowledge useful discussions\nwith Sergey Pershoguba and Jonathan Edge. This work\nwas supported by ERC DM-321031 and US DOE BES\nE304.5\nAppendix: Current-current correlation\nThe current operator is given by\nji(r) =1\n2mi\u0000\n\ty(r)ri\t(r)\u0000\t(r)ri\ty(r)\u0001\n=Z\ndkdk0eir(k0\u0000k)ay\nkak0k0+k\n2m: (A.1)\nThe current-current correlation is given by\nhT\u001cji(r;\u001c)jj(r0;0)i=\nZ\nKeir(k1\u0000k2)eir0(k3\u0000k4)(k1+k2)i\n2m(k3+k4)j\n2m\u0002\nhT\u001cay\nk1(\u001c)ak2(\u001c)ay\nk3(0)ak4(0)i: (A.2)\nwhereR\ndK=dk1dk2dk3dk4means integration on all\nmomentum variables.Using Wicks theorem we can write the average on four\noperators as the product of two averages on two opera-\ntors. Since we are not considering the situation of Cooper\npairs, averages containing two annihilation operators or\ntwo creation operators will vanish.\nThus we are left with two ways of pairing\nthe operators: hT\u001cay\nk1(\u001c)ak2(\u001c)ihT\u001cay\nk3(0)ak4(0)iand\nhT\u001cay\nk1(\u001c)ak4(0)ihT\u001cay\nk3(0)ak2(\u001c)i.\nThe \frst way of pairing will simply give us\nhj(r;\u001c)ihj(r0;0)i, and we can assume for simplicity that it\nvanish. The second way of pairing is related to \u0001 q(k;\u001c).\nWe can make this relation concrete with the mapping\nk=k4;\nq=k1\u0000k4;\nk0=k2;\nq0=k3\u0000k2:\nRewriting the current-current correlation above using\nthese variables yields eq (10) in the main text.\n1G. Gr uner, Density Waves in Solids (Addison-Wesley,\n1994).\n2G. Gr uner, Rev. Mod. Phys.66, 1 (1994).\n3M. K onig et al., Science 318, 766 (2007)\n4N. Nagaosa, Science 318, 758 (2007),\n5B. A. Bernevig, T. L. Hughes and S.-C. Zhang, Science\n314, 17571761 (2006)\n6J. E. 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