We study the temperature-dependent conductivity $\sigma(T)$ and spin
susceptibility $\chi(T)$ of the two-dimensional disordered Hubbard model.
Calculations of the current-current correlation function using the Determinant
Quantum Monte Carlo method show that repulsion between electrons can
significantly enhance the conductivity, and at low temperatures change the sign
of $d\sigma/dT$ from positive (insulating behavior) to negative (conducting
behavior). This result suggests the possibility of a metallic phase, and
consequently a metal-insulator transition,in a two-dimensional microscopic
model containing both interactions and disorder. The metallic phase is a
non-Fermi liquid with local moments as deduced from a Curie-like temperature
dependence of $\chi(T)$.Comment: 4 pages; 4 postscript figures; added (1) a new figure showing
  temperature dependence of spin susceptibility; (2) more references. accepted
  for publication in Phys. Rev. Let

A. M. Finkel'stein

B. L. Altshuler

C. Castellani

D. Belitz

D. J. Scalapino

D. L. Shepelyansky

D. Popović

D. Simonian

E. Abrahams

F. Wegner

K. B. Efetov

M. A. Paalanen

M. Ulmke

M. Y. Simmons

N. Trivedi

P. A. Lee

P. J. H. Denteneer

P. Phillips

R. T. Scalettar

S. Chakravarty

S. R. White

S. V. Kravchenko

T. Vojta

V. Dobrosavljević

W. Krauth

Y. Hanien

English

arXiv

Crossref

Conducting Phase in the Two-Dimensional Disordered Hubbard Model

Denteneer, P.J.H.

Scalettar, R.T.

Trivedi, N.

Leiden University Scholary Publications

VOLUME 83, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 29 NOVEMBER 19994Conducting Phase in the Two-Dimensional Disordered Hubbard ModelP. J. H. DenteneerLorentz Institute, University of Leiden, P.O. Box 9506, 2300 RA Leiden, The NetherlandsR. T. ScalettarPhysics Department, University of California, 1 Shields Avenue, Davis, California 95616N. TrivediDepartment of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400-005, India(Received 1 April 1999)We study the temperature-dependent conductivity sT and spin susceptibility xT of the two-dimensional disordered Hubbard model. Calculations of the current-current correlation function using aquantum Monte Carlo method show that repulsion between electrons can significantly enhance the con-ductivity, and at low temperatures change the sign of dsdT from positive (insulating behavior) tonegative (conducting behavior). This result suggests the possibility of a metallic phase, and conse-quently a metal-insulator transition, in a two-dimensional microscopic model containing both interac-tions and disorder. The metallic phase is a non-Fermi liquid with local moments as deduced from xT.PACS numbers: 71.10.Fd, 71.30.+h, 72.15.RnWhen electrons are confined to two spatial dimensionsin a disordered environment, common understanding un-til recently was that the electronic states would always belocalized and the system would therefore be an insulator.This idea is based on the scaling theory of localization[1] for noninteracting electrons and corroborated by subse-quent studies using renormalization group (RG) methods[2]. The scaling theory highlights the importance of thenumber of spatial dimensions and demonstrates that whilein three dimensions for noninteracting electrons there ex-ists a transition from a metal to an Anderson insulator uponincreasing the amount of disorder; a similar metal-insulatortransition (MIT) is not possible in two dimensions.The inclusion of interactions into the theory has beenproblematic, certainly when both disorder and interactionsare strong and perturbative approaches break down. Fol-lowing the scaling theory the effect of weak interactions inthe presence of weak disorder was studied by diagrammatictechniques and found to increase the tendency to localize[3]. Subsequent perturbative RG calculations, includingboth electron-electron interactions and disorder, found in-dications of metallic behavior, but also, for the case with-out a magnetic field or magnetic impurities, found runawayflows to strong coupling outside the controlled perturba-tive regime and therefore were not conclusive [4,5]. Theresults of such approaches therefore have not changed thewidely held opinion that in two dimensions (2D) the MITdoes not occur.The situation changed dramatically with the recent trans-port experiments on effectively 2D electron systems insilicon metal-oxide-semiconductor field-effect transistors(MOSFETs) which have provided surprising evidence thata MIT can indeed occur in 2D [6]. In these experiments thetemperature dependence of the conductivity sdc changesfrom that typical of an insulator (decrease of sdc upon610 0031-90079983(22)4610(4)$15.00lowering T ) at lower density to that typical of a conductor(increase of sdc upon lowering T ) as the density is in-creased above a critical density. The fact that the data canbe scaled onto two curves (one for the metal and one for theinsulator) is seen as evidence for the occurrence of a quan-tum phase transition with carrier density n as the tuningparameter. The possibility of such a transition has stimu-lated a large number of further experimental [7,8] and alsotheoretical investigations [9,10], including proposals thata superconducting state is involved [11]. Explanations interms of trapping of electrons at impurities, i.e., not requir-ing a quantum phase transition, have also been put forward[12]. While there is no definitive explanation of the phe-nomena yet, it is likely that electron-electron interactionsplay an important role.The central question motivated by the experiments is:Can electron-electron interactions enhance the conductiv-ity of a 2D disordered electron system, and possibly leadto a conducting phase and a metal-insulator transition? Itis this question that we address by studying the disorderedHubbard model which contains both relevant ingredients:interactions and disorder. While the Hubbard model doesnot include the long range nature of the Coulomb repul-sion, studying the simpler model of screened interactionsis an important first step in answering the central qualita-tive question posed above. We use quantum Monte Carlosimulation techniques which enable us to avoid the limi-tations of perturbative approaches (while of course beingconfronted with others). We mention that recent studiesusing very different techniques from ours have indicatedthat interactions may enhance conductivity: two interact-ing particles instead of one in a random potential have adelocalizing effect [13], and weak Coulomb interactionswere found to increase the conductance of spinless elec-trons in (small) strongly disordered systems [14].© 1999 The American Physical SocietyVOLUME 83, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 29 NOVEMBER 1999The disordered Hubbard model that we study is de-fined byĤ  2Xi,j,stijcyiscjs 1 UXjnj"nj# 2 mXj,snjs , (1)where cjs is the annihilation operator for an electron atsite j with spin s. tij is the nearest neighbor hoppingintegral, U is the on-site repulsion between electrons ofopposite spin, m is the chemical potential, and njs cyjscjs is the occupation number operator. Disorder isintroduced by taking the hopping parameters tij froma probability distribution Ptij  1D for tij [ 1 2D2, 1 1 D2, and zero otherwise. D is a measure forthe strength of the disorder [15].We use the determinant quantum Monte Carlo (QMC)method, which has been applied extensively to the Hub-bard model without disorder [16]. While disorder andinteraction can be varied in a controlled way and stronginteraction is treatable, QMC is limited in the size of thelattice, and the sign problem restricts the temperatureswhich can be studied. The sign problem is minimized bychoosing off-diagonal rather than diagonal disorder, as atleast at half filling n  1 there is no sign problem in theformer case, and consequently simulations can be pushedto significantly lower temperatures. For results away fromhalf filling we choose n  0.5 where the sign problem isless severe compared to other densities [16]. Also, inter-estingly, the sign problem is reduced in the presence ofdisorder [15].The quantity of immediate interest when studying apossible metal-insulator transition is the conductivity andespecially its T dependence. By the fluctuation-dissipationtheorem sdc is related to the zero-frequency limit of thecurrent-current correlation function. A complication ofthe QMC simulations is that the correlation functionsare obtained as a function of imaginary time. To avoida numerical analytic continuation procedure to obtainfrequency-dependent quantities, which would requireMonte Carlo data of higher accuracy than produced inthe present study, we employ an approximation that wasused and tested before in studies of the superconductor-insulator transition in the attractive Hubbard model [17].This approximation is valid when the temperature issmaller than an appropriate energy scale in the problem.Additional checks and applicability to the present problemare discussed below. The approximation allows sdc to becomputed directly from the wave vector q- and imaginarytime t-dependent current-current correlation functionLxxq, t:sdc b2pLxxq  0, t  b2 . (2)Here b  1T , Lxxq, t   jxq, tjx2q, 0, andjxq, t, the q, t-dependent current in the x direc-tion, is the Fourier transform of jx  iPs t1x̂, 3cy1x̂,scs 2 cysc1x̂,s (see also Ref. [18]).As a test for our conductivity formula (2) we first presentresults in Fig. 1(a) for sdcT at half filling for U  4 andvarious disorder strengths D. The behavior of the con-ductivity shows that as the temperature is lowered belowa characteristic gap energy the high T “metallic” behaviorcrosses over to the expected low T Mott insulating behav-ior for all D, thereby providing a reassuring check of for-mula (2) and our numerics.In Fig. 1(b), we show sdcT for a range of disorderstrengths at density n  0.5 and U  4. The figuredisplays a striking indication of a change from metallicbehavior at low disorder to insulating behavior above acritical disorder strength, Dc  2.7. If this persists toFIG. 1. Conductivity sdc as a function of temperature T forvarious values of disorder strength D at U  4 for (a) halffilling n  1 and (b) n  0.5. Calculations are performedon an 8 3 8 square lattice; data points are averages over fourrealizations for a given disorder strength.4611VOLUME 83, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 29 NOVEMBER 1999T  0 and in the thermodynamic limit, it would describe aground state metal-insulator transition driven by disorder.In order to obtain a more precise understanding of therole of interactions on the conductivity, we compare inFig. 2 the results of Fig. 1(b) with the disordered non-interacting s0 [19]. The comparison is made at strongenough disorder D  2.0 such that the localization lengthis less than the lattice size and the noninteracting sys-tem is therefore insulating with ds0dT . 0 at low T .Interactions are found to have a profound effect on theconductivity: in the high-temperature metallic region, in-teractions slightly reduce s compared to the noninteractings0 behavior. On the other hand in the low-temperature“insulating” region of s0 the data show that upon turn-ing on the Hubbard interaction the behavior is completelychanged with dsdT , 0, characteristic of metallic be-havior. This is the regime of interest for the MIT.In order to ascertain that the phase produced by repulsiveinteractions at low T is not an insulating phase with alocalization length larger than the system size but a truemetallic phase we have studied the conductivity responsefor varying lattice sizes. We find a markedly different sizedependence for the U  0 insulator and the U  4 metal,resulting in a confirmation of the picture given above. ForU  0, the conductivity on a larger 12 3 12 system islower than that on a smaller 8 3 8 system (see Fig. 2),consistent with insulating behavior in the thermodynamiclimit, whereas for U  4 the conductivity on the larger8 3 8 system is higher than that on the smaller 4 3 4system (data not shown), indicative of metallic behavior.Thus the enhancement of the conductivity by repulsiveFIG. 2. Conductivity sdc as a function of temperature T com-paring U  4 and U  0 at n  0.5 and disorder strengthD  2.0. Data points are averages over many realizations forthis disorder strength (see text). Error bars are determined bythe disorder averaging and not the Monte Carlo simulation.4612interactions becomes more pronounced with increasedlattice size [20].Concerning finite-size effects for the noninteracting sys-tem we note that at lower values of D, where the local-ization length exceeds the lattice size, s0 shows metallicbehavior which is diminished upon turning on the interac-tions [21]. Based on our analysis above, we would predictthat at low enough T and large enough lattice size the con-ductivity curves for the noninteracting s0 and interactings cross with s . s0 at sufficiently low T .To obtain information on the spin dynamics of the elec-trons and because it is a quantity often discussed in connec-tion with the localization transition, we also compute thespin susceptibility x as a function of T [through xT  bS0T  where S0 is the magnetic structure factor at wavevector q  0]. Figure 3 shows two things: (i) xT  is en-hanced by interactions with respect to the noninteractingcase (at fixed disorder strength), and (ii) starts to divergewhen T is lowered, both on the metallic D  2 and in-sulating D  4 sides of the alleged transition. This isin agreement with experimental and theoretical work onphosphorus-doped silicon, where a (3D) MIT is known tooccur and the behavior is explained by the existence of lo-cal moments [22], and also with diagrammatic work on 2Ddisordered, interacting systems [23].In order to definitively establish the existence of a pos-sible quantum phase transition in the disordered Hubbardmodel requires the following: (i) extending the presentdata at T  0.1  W80, where W is the noninteractingbandwidth, to lower T , which is however difficult becauseof the sign problem, (ii) a more detailed analysis of theFIG. 3. Spin susceptibility x as a function of temperature Tat n  0.5 comparing interaction strengths U  0, 2, 4 anddisorder strengths D  2, 4. Calculations are performed on8 3 8 square lattices; error bars are from disorder averagesover up to eight realizations.VOLUME 83, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 29 NOVEMBER 1999scaling behavior in both linear dimension and some scaledtemperature, (iii) a more accurate analytic continuationprocedure to extract the conductivity. The condition forthe validity of the approximate formula (2) for sdcT ,requires that T be less than an appropriate energy scalewhich is fulfilled within the two phases, but breaks downclose to a quantum phase transition where the energy scalevanishes.In summary, we have studied the temperature-dependentconductivity sT  and spin susceptibility xT  of a modelfor two-dimensional electrons containing both disorder andinteractions. We find that the Hubbard repulsion can en-hance the conductivity and lead to a clear change in signof dsdT . More significantly, from a finite-size scal-ing analysis we demonstrate that repulsive interactions candrive the system from one phase to a different phase. Wefind that sT  has the opposite behavior as a function ofsystem size in the two phases indicating that the transi-tion is from a localized insulating to an extended metal-lic phase. The xT  data further suggest the formationof an unusual metal, a non-Fermi liquid with local mo-ments. While the simplicity of the model we study pre-vents any quantitative connection to recent experiments onSi-MOSFETs, there is nevertheless an interesting qualita-tive similarity between Fig. 1(b) and the experiments.Varying the disorder strength D at fixed carrier density n,as in our calculations, can be thought of as equivalent tovarying carrier density at fixed disorder strength, as in ex-periments, since in a metal-insulator transition one expectsno qualitative difference between tuning the mobility edgethrough the Fermi energy (by varying D) and vice versa(by varying n). Our work then suggests that electron-electron interaction induced conductivity plays a key rolein the 2D metal-insulator transition.We thank C. Huscroft for useful comments on themanuscript, H. V. Kruis for help with the calculations, andD. Belitz, R. N. Bhatt, C. Di Castro, T. R. Kirkpatrick,T. M. Klapwijk, S. V. Kravchenko, M. P. Sarachik, andG. T. Zimanyi for stimulating discussions. 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Conducting phase in the two-dimensional disordered Hubbard model

Conducting phase in the two-dimensional disordered Hubbard model

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