Anti-Drude Metal of Bosons
AAnti-Drude Metal of Bosons
Guido Masella, Nikolay V. Prokof’ev,
2, 1 and Guido Pupillo ISIS (UMR 7006) and icFRC, University of Strasbourg and CNRS, 67000 Strasbourg, France Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA (Dated: February 17, 2021)In the absence of frustration, interacting bosons in the ground state exist either in the superfluidor insulating phases. Superfluidity corresponds to frictionless flow of the matter field, and in opticalconductivity is revealed through a distinct δ -functional peak at zero frequency with the amplitudeknown as the Drude weight. This characteristic low-frequency feature is instead absent in insulatingphases, defined by zero static optical conductivity. Here we demonstrate that bosonic particles indisordered one dimensional, d = 1, systems can also exist in a conducting, non-superfluid, phasewhen their hopping is of the dipolar type, often viewed as short-ranged in d = 1. This phase ischaracterized by finite static optical conductivity, followed by a broad anti-Drude peak at finitefrequencies. Off-diagonal correlations are also unconventional: they feature an integrable algebraicdecay for arbitrarily large values of disorder. These results do not fit the description of any knownquantum phase and strongly suggest the existence of a novel conducting state of bosonic matter inthe ground state. Quantum phases of matter are distinguished by theirstatic and dynamical properties, quantified by correla-tion functions. For interacting bosonic matter in onedimension, the superfluid phase is characterized by anon-integrable algebraic decay of static one-body (off-diagonal) correlations as a function of distance and bya δ -functional peak at zero frequency in the optical con-ductivity, respectively. The latter is reflecting a singu-lar response to a weak externally applied field. Strongenough disorder can induce a quantum phase transitionfrom the superfluid to an insulating phase, known asthe Bose glass [1]. In this phase, off-diagonal correla-tions decay exponentially with distance and the opticalconductivity starts from zero at zero-frequency, reflect-ing the absence of long-lived collective modes at low-energy. These two phases exhaust the known possibilitiesfor disordered bosons in one dimension in the absence offrustration, where by frustration we understand a situ-ation when the path-integral representation of quantumstatistics in imaginary time is not sign-positive. In thiswork, we provide numerical evidence for the existence ofa novel disorder-induced phase that is neither superfluidnor insulating. Despite featuring an algebraic decay ofoff-diagonal correlations, it has zero superfluid densityand its optical conductivity is finite at zero frequency.The latter is followed by a broad peak at a finite fre-quency of the order of the nearest-neighbor hopping en-ergy. Because of this characteristic ”anti-Drude” behav-ior of optical conductivity, with finite minimum insteadof maximum at zero frequency, we term this novel phasean anti-Drude metal of bosons (aDMB).The aDMB phase is a result of interplay between inter-actions, disorder, and particle hopping, which we chooseto be of the dipolar type. The latter is usually consideredas short-ranged in d = 1 [2]. For non-interacting modelswith short-range hopping, disorder is generally expectedto localize all wave-functions exponentially (Anderson lo- calization) [3]. However, recent theoretical works havedemonstrated that single particle states can localize al-gebraically in the presence of couplings that decay withdistance as a power-law [4–8]. What happens in stronglyinteracting systems remained an open question, and thiswork provides the first answers with the discovery of theaDMB ground state.Dipolar couplings have been already experimentally re-alized for internal excitations of cold magnetic atoms [9–12], Rydberg excited atoms [13–15], ions [16, 17], andmolecules [18]. The propagation of excitations with dipo-lar couplings in the presence of disorder is also highly rele-vant for a variety of solid-state systems, including nuclearspins [19], nitrogen-vacancy centers in diamonds [20], ortwo-level emitters placed near a photonic crystal waveg-uide [21].We note that the existence of a metallic bosonic phasehas been suggested previously [22–24]; e.g., in the con-text of finite-temperature strange metal behavior of high-temperature superconductors [23, 25] and as a possibleground state in lattice models with multi-particle inter-actions [24, 26, 27]. However, up to date, the existenceof a metallic phase of bosons has not been confirmed byexact methods in any physical system. Since frustratedspin systems featuring a variety of spin-liquids phases canbe always re-formulated in terms of strongly interactingbosons, we exclude frustrated models from this discus-sion.We consider the following Hamiltonian for hard-corebosons confined to one dimension H = − t (cid:88) i 75 1 . 00 1 . 25 1 . W − . . . . D W E − D W E L m a x FIG. 1. Mean-squared winding numbers (cid:10) W (cid:11) as func-tions of the disorder strength W for lattice sizes L = 64(blue circles), 96 (orange squares), 128 (green diamonds), 192(red hexagons), 256 (purple stars). Inset highlights the areanear the phase transition, showing crossing points between thecurves within the interval W c = 1 . ± . 15; the curve corre-sponding to the largest size ( L = 256) is subtracted from alldata for clarity. the lattice spacing, a , are taken as units of energy andlength, respectively. Hopping amplitudes between sites i and j decay with the distance between them as r − ij , and (cid:15) i are random on-site energies uniformly distributed be-tween − W and W . In spin language, Eq. (1) is equivalentto an XY Hamiltonian with dipolar couplings, which, inthe absence of disorder, can be realized in experimentswith cold polar molecules [18], trapped ions [16, 17] andRydberg atoms [13, 14, 28], with the latter also in thepresence of disorder [15]. Recent theoretical works pro-vide strong evidence that Eq. (1) supports a many-bodylocalized (MBL) phase at finite energy [18, 29–32]. Ourresult then implies that the MBL transition out of aDMBtakes place as the temperature is increased. In a systemwith an upper bound on the maximal energy per particlethis result is not that surprising [33].In the following, we determine the ground-state quan-tum phases of Eq. (1) using large scale path-integralquantum Monte-Carlo simulations based on the Wormalgorithm [34]. Without loss of generality, we focus onparticle density ρ = 1 / ρ s ,that characterizes the response to twisted boundarycondition caused by an external vector potential field.It can be conveniently computed within quantumMonte-Carlo, see Methods, through the statistics of ‘ − − − G ( ‘ ) ≡ h a † i a i + ‘ i W = 0 . W = 4 . W = 8 . L = 256 L = 128 L = 64 f ( x ) ∝ /‘ α FIG. 2. One-body density matrix G ( (cid:96) ) as a function of thedistance (cid:96) for different system sizes L = 64 (blue solid lines),128 (yellow dashed lines), and 256 (green dotted lines), andvalues of the disorder strength W = 0 . 5, 4 . 0, and 8 . /(cid:96) α with α = 3 . L = 256 and W = 8 . winding numbers, W , using the Pollock-Ceperley re-lation ρ s ∝ (cid:104)W(cid:105) [35][36]. However, it is well knownthat the superfluid density of this system is immediatelysuppressed by any finite strength of disorder W , dueto Anderson localization [1]. Dipolar hopping changesthis picture entirely, by allowing for pair-wise bosonicexchanges, somewhat similar to soft-core particles.One then expects superfluidity to be robust againstweak disorder, and, possibly, undergo a quantum phasetransition to a non-superfluid phase when disorderexceeds some critical value W c .Figure 1 shows numerical results for the statistics ofmean-squared winding numbers (cid:10) W (cid:11) as a function ofthe disorder strength W for different lattice sizes L .Mean-squared winding numbers are expected to be scaleinvariant at a continuous phase transition, regardlessof the system dimension. This allows one to identifythe critical disorder strength W c where superfluidityis lost by the crossing point of the (cid:10) W (cid:11) -vs- W curvesfor different values of L . The figure shows that allsizes larger than L > 64 cross at W c = 1 . ± . W < W c to a quantum phase that is notsuperfluid for W > W c . In the following, we focus oncharacterising the properties of this non-superfluid phasewith W > W c by studying its correlation functions andoptical conductivity.The one-body density matrix G ( (cid:96) ) = (cid:68) b † i b i + (cid:96) (cid:69) isexpected to decay algebraically as a function of distance − − ω . . . . h σ i (a) W = 4 . L = β ω . . h σ i L = 64 L = 128 L = 256 − − ω . . . h σ i (b) W = 6 . L = β ω . . h σ i L = 64 L = 128 L = 256 FIG. 3. Disorder-averaged optical conductivity σ as a function of the frequency ω for W = 4 (Panel a, left) and W = 6(Panel b, right), at different system sizes L = 64 (blue circles), 128 (orange squares), and 256 (green diamonds). Data in themain plots is shown on the logarithmic scale for the frequency, highlighting the behaviour for small ω . Insets show data on thelinear scale. (cid:96) for a one-dimensional superfluid ground state, while inan insulating phase it is expected to decay exponentially,e.g. in a crystalline phase or Bose glass. Figure 2 shows G ( (cid:96) ) for the Hamiltonian Eq. (1), for chosen values of thedisorder strength W . The figure shows that in the su-perfluid phase with W = 0 . < W c , G ( (cid:96) ) displays a slowalgebraic decay, as expected. Surprisingly, we find thatan initial exponential decay of G ( (cid:96) ) is followed at largedistances (cid:96) by an algebraic decay in the non-superfluidphase for W > W c . The large-distance decay is welldescribed by the G ( (cid:96) ) ∼ /(cid:96) dependence. This behavioris at odds with known results for insulating many-bodyphases with short-range hopping [1], indicating thatother physical properties may also be unconventional.We thus proceed with analysing the optical conductivityof the non-superfluid phase at W > W c .The optical conductivity σ ( ω ) relates the current den-sity J to the strength of an externally applied elec-tric field E as J ( ω ) = σ ( ω ) E ( ω ), with ω the field fre-quency. We obtain the optical conductivity σ ( ω ) withinthe linear response theory by first computing the current-current correlation function χ ( ıω n ) = (cid:104) j ( τ ) j (0) (cid:105) ıω n atMatsubara frequencies ω n = 2 πnT using the Worm al-gorithm, followed by its numerical analytic continuation(see Methods). Here j is the lattice current operator de-fined as j = ıt (cid:80) i 0) is absent. However, the numerical results also showtwo striking features: (i) The zero-frequency responseis finite and system size independent within the (rela-tive large) error bars; (ii) Unlike in usual conductorsfeaturing a Drude peak (maximum at ω = 0), theoptical conductivity has a minimum at zero frequencyfollowed by a large peak at frequency ω (cid:39) t , whichprovides a large response at energies of the order ofthe nearest-neighbor hopping amplitude. This peakbroadens with increasing W , providing a large responseup to frequencies ω (cid:39) t . These results for the averagedconductivity demonstrate the existence of a conducting,non superfluid phase of bosons in the ground state.This conducting behaviour is not due to well defineddelocalized quasiparticle states as in typical Drude-typemetals; rather, it is an ”anti-Drude metal” , where thelargest response occurs at a small but finite frequency.Figure 4(a) shows selected results for σ ( ω ) in theaDMB phase for individual realizations of disorder, i.e.without averaging. We find that at frequencies ω > t the optical conductivity behavior is rather robust andsample-to-sample fluctuations are not substantial. Thesame cannot be said about the low-frequency part thatwildly fluctuates from sample to sample - whilst someof the samples are metallic, the majority display aninsulating behavior. This suggests that static σ is infact not a self-averaging quantity in our system. Thesefluctuations will be reflected in similar fluctuations inexperiments.The discovery of the aDMB phase is particularlysurprising as the dipolar hopping term in Eq. (1) isusually considered to be short ranged in one dimension. − − ω . . . . σ W = 6 . L = β = 64 (a) Sample 1Sample 2Sample 3Sample 4Averaged − L − L L L r . . . . | r | G ( β , r i , r i + r ) (b) Sample 4 FIG. 4. Panel a: Optical conductivity σ as a function of the frequency ω for different disorder realizations. The blackcontinuous line represents the average over all 384 disordered samples. Panel b: Correlation function | r | G τ = β ( r i , r i + r ) as afunction of r , sampled for imaginary time difference τ = β between the two end points on the trajectory. Here, r i is chosen sothat G β ( r i , r i ) is maximum. This quantity allows one to visualize the main contributions to the current for a single disorderrealization when the particle starts from point r i (see main text and Methods). In both panels data is shown for L = β = 64and W = 6. Nevertheless, it leads to large de-localized contributionsto the current that can be visualized as follows. Thesingle particle propagator G τ ( r, r (cid:48) ) = (cid:68) b † r (cid:48) ( τ ) b r (0) (cid:69) encodes information for where a particle/hole injectedinto the system at site r can go in time τ (for hard corebosons points r and r (cid:48) are connected by a trajectory).By setting τ = β/ β → ∞ we gaininsight into properties of the ground state wave function.Since current operator between distant sites involves anadditional power of distance we multiply G β ( r, r (cid:48) ) by | r − r (cid:48) | to establish a quantitative measure for currentcontributions. Figure 4(b) visualizes the correlationfunction | r |G β ( r i , r i + r ) for a single conducting realiza-tion as a function of the distance r for a fixed value of r i that was chosen from the condition of maximum for G β ( r i , r i ). The figure makes it clear that large currentcontributions are present over a wide range of distancesof the order of ∼ L/ Acknowledgements – The authors acknowledge sup- port from the University of Strasbourg Institute of Ad-vanced Studies (USIAS). G. P. acknowledges additionalsupport from the Institut Universitaire de France (IUF)and LABEX CSC. N. P. acknowledges support from theMURI Program ”New Quantum Phases of Matter” fromAFOSR. Computing time was provided by the High Per-formance Computing Center of the University of Stras-bourg. Part of the computing resources were funded bythe Equipex EquipMeso project (Programme Investisse-ments d’Avenir) and the CPER Alsacalcul/Big Data. [1] T. Giamarchi, Quantum Physics in One Dimension (Ox-ford University Press, 2003).[2] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, andT. Pfau, The physics of dipolar bosonic quantum gases,Reports on Progress in Physics , 126401 (2009).[3] P. W. Anderson, Absence of Diffusion in Certain RandomLattices, Physical Review , 1492 (1958).[4] T. Botzung, D. Vodola, P. Naldesi, M. M¨uller, E. Erco-lessi, and G. Pupillo, Algebraic localization from power-law couplings in disordered quantum wires, Physical Re-view B , 155136 (2019).[5] X. Deng, V. E. Kravtsov, G. V. Shlyapnikov, and L. San-tos, Duality in Power-Law Localization in DisorderedOne-Dimensional Systems, Physical Review Letters ,110602 (2018).[6] P. A. Nosov, I. M. Khaymovich, and V. E. Kravtsov,Correlation-induced localization, Physical Review B ,104203 (2019).[7] F. A. B. F. de Moura, A. V. Malyshev, M. L. Lyra,V. A. Malyshev, and F. Dom´ınguez-Adame, Localiza-tion properties of a one-dimensional tight-binding model with nonrandom long-range intersite interactions, Phys-ical Review B , 174203 (2005).[8] G. L. Celardo, R. Kaiser, and F. Borgonovi, Shieldingand localization in the presence of long-range hopping,Physical Review B , 144206 (2016).[9] A. de Paz, A. Sharma, A. Chotia, E. Mar´echal, J. H.Huckans, P. Pedri, L. Santos, O. Gorceix, L. Vernac, andB. Laburthe-Tolra, Nonequilibrium Quantum Magnetismin a Dipolar Lattice Gas, Physical Review Letters ,185305 (2013).[10] S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz,Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, ExtendedBose-Hubbard models with ultracold magnetic atoms,Science , 201 (2016).[11] S. Lepoutre, J. Schachenmayer, L. Gabardos, B. Zhu,B. Naylor, E. Mar´echal, O. Gorceix, A. M. Rey,L. Vernac, and B. Laburthe-Tolra, Out-of-equilibriumquantum magnetism and thermalization in a spin-3many-body dipolar lattice system, Nature Communica-tions , 1714 (2019).[12] A. Patscheider, B. Zhu, L. Chomaz, D. Petter, S. Baier,A.-M. Rey, F. Ferlaino, and M. J. Mark, Controlling dipo-lar exchange interactions in a dense three-dimensional ar-ray of large-spin fermions, Physical Review Research ,023050 (2020).[13] D. Barredo, H. Labuhn, S. Ravets, T. Lahaye,A. Browaeys, and C. S. Adams, Coherent ExcitationTransfer in a Spin Chain of Three Rydberg Atoms, Phys-ical Review Letters , 113002 (2015).[14] A. P. Orioli, A. Signoles, H. Wildhagen, G. G¨unter,J. Berges, S. Whitlock, and M. Weidem¨uller, Relax-ation of an Isolated Dipolar-Interacting Rydberg Quan-tum Spin System, Physical Review Letters , 063601(2018).[15] S. de L´es´eleuc, V. Lienhard, P. Scholl, D. Barredo, S. We-ber, N. Lang, H. P. B¨uchler, T. Lahaye, and A. Browaeys,Observation of a symmetry-protected topological phaseof interacting bosons with Rydberg atoms, Science ,775 (2019).[16] P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith,M. Foss-Feig, S. Michalakis, A. V. Gorshkov, andC. Monroe, Non-local propagation of correlations inquantum systems with long-range interactions, Nature , 198 (2014).[17] P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller,R. Blatt, and C. F. Roos, Quasiparticle engineering andentanglement propagation in a quantum many-body sys-tem, Nature , 202 (2014).[18] B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A.Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Observation ofdipolar spin-exchange interactions with lattice-confinedpolar molecules, Nature , 521 (2013).[19] G. A. ´Alvarez, D. Suter, and R. Kaiser, Localization-delocalization transition in the dynamics of dipolar-coupled nuclear spins, Science , 846 (2015).[20] G. Waldherr, Y. Wang, S. Zaiser, M. Jamali, T. Schulte-Herbr¨uggen, H. Abe, T. Ohshima, J. Isoya, J. F. Du,P. Neumann, and J. Wrachtrup, Quantum error correc-tion in a solid-state hybrid spin register, Nature , 204(2014).[21] C.-L. Hung, A. Gonz´alez-Tudela, J. I. Cirac, and H. J.Kimble, Quantum spin dynamics with pairwise-tunable,long-range interactions, Proceedings of the National Academy of Sciences , E4946 (2016).[22] M. V. Feigelman, V. B. Geshkenbein, L. B. Ioffe, and A. I.Larkin, Two-dimensional Bose liquid with strong gauge-field interaction, Physical Review B , 16641 (1993).[23] P. Phillips and D. Dalidovich, The Elusive Bose Metal,Science , 243 (2003).[24] O. I. Motrunich and M. P. A. Fisher, $ d $ -wave correlatedcritical Bose liquids in two dimensions, Physical ReviewB , 235116 (2007).[25] C. Yang, Y. Liu, Y. Wang, L. Feng, Q. He, J. Sun,Y. Tang, C. Wu, J. Xiong, W. Zhang, X. Lin, H. Yao,H. Liu, G. Fernandes, J. Xu, J. M. Valles, J. Wang,and Y. Li, Intermediate bosonic metallic state in thesuperconductor-insulator transition, Science , 1505(2019).[26] H.-C. Jiang, M. S. Block, R. V. Mishmash, J. R. Garri-son, D. N. Sheng, O. I. Motrunich, and M. P. A. Fisher,Non-Fermi-liquid d -wave metal phase of strongly inter-acting electrons, Nature , 39 (2013).[27] M. S. Block, D. N. Sheng, O. I. Motrunich, and M. P. A.Fisher, Spin Bose-Metal and Valence Bond Solid Phasesin a Spin- $ $ Model with Ring Exchanges on a Four-Leg Triangular Ladder, Physical Review Letters ,157202 (2011).[28] J. Zeiher, J.-y. Choi, A. Rubio-Abadal, T. Pohl, R. vanBijnen, I. Bloch, and C. Gross, Coherent Many-BodySpin Dynamics in a Long-Range Interacting Ising Chain,Physical Review X , 041063 (2017).[29] A. L. Burin, Localization in a random XY modelwith long-range interactions: Intermediate case betweensingle-particle and many-body problems, Physical Re-view B , 104428 (2015).[30] A. L. Burin, Many-body delocalization in a strongly dis-ordered system with long-range interactions: Finite-sizescaling, Physical Review B , 094202 (2015).[31] A. Safavi-Naini, M. L. Wall, O. L. Acevedo, A. M. Rey,and R. M. Nandkishore, Quantum dynamics of disor-dered spin chains with power-law interactions, PhysicalReview A , 033610 (2019).[32] X. Deng, G. Masella, G. Pupillo, and L. Santos, Univer-sal Algebraic Growth of Entanglement Entropy in Many-Body Localized Systems with Power-Law Interactions,Physical Review Letters , 010401 (2020).[33] Y. Kagan and L. A. Maksimov, Quantum diffusion ofatoms in a crystal localization and phonon-stimulateddelocalization, Physics Letters A , 242 (1983).[34] N. V. Prokof’ev, B. V. Svistunov, and I. S. Tupitsyn, Ex-act, complete, and universal continuous-time worldlineMonte Carlo approach to the statistics of discrete quan-tum systems, Journal of Experimental and TheoreticalPhysics , 310 (1998).[35] E. L. Pollock and D. M. Ceperley, Path-integral compu-tation of superfluid densities, Physical Review B , 8343(1987).[36] See Supplementary material.[37] B. V. Svistunov, E. Babaev, and N. V. Prokof’ev, Super-fluid States of Matter , 1st ed. (CRC Press, 2015).[38] R. Levy, J. P. F. LeBlanc, and E. Gull, Implementation ofthe maximum entropy method for analytic continuation,Computer Physics Communications , 149 (2017).[39] N. V. Prokof’ev and B. V. Svistunov, Spectral analysisby the method of consistent constraints, JETP Letters , 649 (2013).[40] O. Goulko, A. S. Mishchenko, L. Pollet, N. Prokof’ev, and B. Svistunov, Numerical analytic continuation: An-swers to well-posed questions, Physical Review B ,014102 (2017).[41] M. Jarrell and J. E. Gubernatis, Bayesian inference andthe analytic continuation of imaginary-time quantumMonte Carlo data, Physics Reports , 133 (1996). METHODS We perform quantum Monte Carlo simulations ofHamiltonian Eq. (1) in the path-integral representationin the grand-canonical ensemble using the worm algo-rithm [34] for system sizes as large as L = 256 and tem-peratures as low as T /t = 1 / (cid:104) W i (cid:105) = µ = 0 for eachrealization, with µ the chemical potential. The resultingdensity is then (cid:104) ρ (cid:105) = when averaged over the disorderrealizations with tiny, i.e. 2 . 8% for L = 256 and W = 6 . t ij → e ıφr ij . An expansion of the phase factorup to the second order in φ leads to the current operatorfor the studied Hamiltonian j = ıt (cid:88) i In the regimeof weak field φ (linear response) it is sufficient to look atthe current-current correlation function χ ( ıω n ) = (cid:104) j ( τ ) j (0) (cid:105) ıω n (6)at Matsubara frequencies ω n = 2 πT n ( n > σ ( ω ). Here, − − ω . . . . . σ ( ω ) L = 64 β = 64 W = 6CCHistoric MEClassic MEBryan’s ME FIG. 5. Averaged optical conductivity σ as a function of thefrequency ω for different analytic continuation algorithms in-cluding, consistent constraints (solid blue), and three differentvariants of the maximum entropy method: historic (dashedyellow), classic (dash dotted green), and Bryan’s method (dot-ted red) [see [38]]. Data is shown for L = β = 64 and W = 6.The average is taken over all 384 disordered samples. the subscript ıω n denotes that the Fourier transform istaken of the corresponding correlation function (cid:104) j ( τ ) j (0) (cid:105) in imaginary time. Path integral representation of quan-tum statistics for the Hamiltonian Eq. (1) allows one tosample Fourier components of this correlation functiondirectly, and collect statistics for different Matsubara fre-quencies by using the estimator | (cid:80) k ır k e ıω n τ k | , whereagain the sum goes over all hopping transitions on thesystems worldlines. For zero-frequency ω n = 0, this es-timator is equivalent to measuring the winding numbersquared W while for large Matsubara frequencies it ap-proaches the constant value corresponding to the esti-mator for T . After computing statistical averages, wesubtract (cid:104)T (cid:105) from the data to obtain the current-currentcorrelation function. To suppress finite size effects asso-ciated with rare configurations with finite winding num-bers, we restrict the sampling of the correlation function χ ( ıω n ) to configurations W = 0. Analytic continuation Here, we are interested in com-puting the optical conductivity σ ( ω ), an observable thatcan be measured experimentally but not readily ac-cessible by numerical techniques. By the dissipation-fluctuation theorem χ ( ıω n ) = − π (cid:90) ∞ ω ω n + ω σ ( ω ) d ω . (7)Finding σ ( ω ) is thus a standard ill conditioned inverseproblem when small fluctuations of the input due to sta-tistical noise in the Monte Carlo sampling lead to largefluctuations in the output results. To solve this problemwe use a method of consistent constraints [39, 40]. Itallows us to restore the spectral density σ ( ω ) from the − − ω . . . . . σ ( ω ) (a) Sample 4 CCHistoric MEClassic MEBryan’s ME − − ω . . . σ ( ω ) (b) Sample 2 CCHistoric MEClassic MEBryan’s ME FIG. 6. Comparison of different analytic continuation algorithms for single disorder realizations for the optical conductivity σ ( ω ) in a system with L = 64 and β = 64. Each panel corresponds to different disorder realizations and different lines todifferent algorithms including, consistent constraints (solid blue), and three different maxent variants: historic (dashed yellow),classic (dash dotted green), and Bryan’s method (dotted red) [see [38]]. corresponding correlation function χ ( ıω nn