Dynamical Mean Field Study of the Two-Dimensional Disordered Hubbard Model

0.48 MB·English

by  Yun Song

Dynamical Mean Field Study of the Two-Dimensional Disordered Hubbard Model
Yun Song1,2, R. Wortis1, W. A. Atkinson1
1Department of Physics and Astronomy, Trent University,
1600 West Bank Dr., Peterborough ON, K9J 7B8, Canada
2Department of Physics, Beijing Normal University, Beijing 100875, China
(Dated: February 1, 2008)
WestudytheparamagneticAnderson-Hubbardmodelusinganextensionofdynamicalmean-field
theory(DMFT), known as statistical DMFT, that allows usto treat disorder and strongelectronic
correlations on equal footing. An approximate nonlocal Green’s function is found for individual
disorder realizations and then configuration-averaged. We apply this method to two-dimensional
8
lattices with up to 1000 sites in the strong disorder limit, where an atomic-limit approximation is
0
0 made for the self-energy. We investigate the scaling of the inverse participation ratio at quarter-
2 and half-filling and find a nonmonotonic dependence of the localization length on the interaction
strength. For strong disorder, we do not find evidence for an insulator-metal transition, and the
n
disorder potential becomes unscreened near the Mott transition. Furthermore, strong correlations
a
suppressthe Altshuler-Aronovdensity of states anomaly near half-filling.
J

8
2 I. INTRODUCTION ForU =0,itiswellunderstoodthatthesingle-particle

] eigenstatesofEq.(1)areAndersonlocalizedforW >Wc,
l Thephysicalpropertiesofinteractingdisorderedmate- where Wc is the critical disorder and Wc = 0 in two
e
and fewer dimensions. For W = 0, and at half-filling
- rialsareoftenqualitativelydifferentfromtheir noninter-
r (ie. n = 1, where n is the charge density), there is a
t actingcounterparts. Forexample,ithaslongbeenknown
s that noninteracting quasiparticles in two-dimensional critical interaction strength Uc such that the model is
.
t (2D)materialsarelocalizedbyarbitrarilyweakdisorder; a gapped Mott insulator for U > Uc and (neglecting
a
possible broken-symmetry phases) a strongly-correlated
m however,there is evidence that interactions can drive an
insulator-metal transition in 2D.1 Similarly, there is a metal for U < Uc or for n 6= 1. There is evidence
- that the Mott transition is fundamentally different in
d growing awareness that disorder can fundamentally al-
n ter the physical properties of interacting systems. This the presence of disorder. For example, some work has
o arisesinanumberoftransition-metaloxides,2,3,4,5 where shownthatUc =0incleanlow-dimensionalsystemswith
c nested Fermi surfaces,6,7,8,9 while in the disordered case
the predominantly d-orbital character of the conduction
[
U is not only nonzero,but rapidly becomes of order the
electronsresultsinalargeintra-orbital(on-site)Coulomb c
2 interaction relative to the bandwidth. These materials bandwidth as a function of W.9,10,11 In fact, it has be-
v are of interest because they have an interaction-driven comeclearthroughnumerousstudiesthatthegeneralU-
1 W-n phase diagram is complicated and also potentially
insulating(Mott-insulating)phaseandbecauseoftheva-
9 contains superconducting, antiferromagnetic, and spin
riety of exotic phases, such as high temperature super-
7 glass phases.11,12,13,14 Much of the recent progress has
conductivity, which appear near half-filling. However,
0
been for infinite-dimensional systems14,15,16,17,18,19,20,21
. transition metal oxides are typically doped by chemical
7 and the applicability of this work to two and three di-
substitution and, with few exceptions, are intrinsically
0
mensions is not well established. A recent focus has
disordered. At present, there is little consensus on the
7
been the extent to which interactions screen the disor-
0 effects of this disorder, particularly near the transition
der potential19,21,22,23 and whether screening may lead
: to the Mott-insulating phase.
v to an insulator-metal transition.12,22,23,24 In particular,
Here, we discuss the effects of strong electronic corre-
i
X lationsondisorderedtwo-dimensional(2D)materialsvia some calculations show perfect screening near the Mott
transition.19,21
r a numerical study of the Anderson-Hubbard model,
a
Part of the confusion surrounding the Anderson-
Hˆ =−tXXc†iσcjσ +X(Unˆi↑nˆi↓+ǫinˆi), (1) Hubbard phase diagram stems from the variety of
hi,ji σ i theoretical approaches that have been applied. Self-
consistent Hartree-Fock (HF) calculations11,12,13 treat
where hi,ji refers to nearest neighbor lattice sites i and the disorder potential exactly but do not capture the
j, σ =↑,↓ is the spin index, and nˆi = nˆi↑ +nˆi↓ where strong-correlation physics of the Mott transition. Dy-
nˆ = c† c is the local charge density operator. The namical mean-field theory (DMFT) approaches con-
iσ iσ iσ
model has four parameters: the kinetic energy t, the tain the necessary strong-correlation physics but are
intra-orbitalCoulombinteractionU,the width W ofthe based on a local approximation that generally pre-
disorder-potential distribution, and the chemical poten- cludes exact treatment of disorder. A variety of
tial µ. Disorder is introduced through randomly chosen coherent-potential-approximation (CPA) and CPA-like
site energies ǫ , which in this work are box-distributed approximationshave been employedin conjunction with
i
according to P(ǫ )=W−1Θ(W/2−|ǫ |). DMFT.14,15,16,17,18,19,20 WhiletheCPAreproducessome
i i
2
disorder-averaged quantities accurately, eg. the density G (ω)= G0(ω)−1−ΣHI(ω) −1 where
i (cid:2) i i (cid:3)
ofstates (DOS) in the noninteracting limit, it fails to re-
produce quantities that depend on explicit knowledge of n U2ni(1− ni)
ΣHI(ω)=U i + 2 2 , (3)
spatial correlations between lattice sites. As a notable i 2 ω−ǫ −U(1− ni)
example, the CPA fails to predict the Altshuler-Aronov i 2
DOS anomaly which appears at the Fermi energy in dis-
and n ≡ hnˆ i is self-consistently determined for each
i i
ordered metals.25 Quantum Monte Carlo22,23 and exact site. The HI approximationis the simplest improvement
diagonalization24 (ED) methods treat both disorder and
over Hartree-Fock (HF) that generates both upper and
interactionsexactly,butsufferfromseverefinite-sizelim- lower Hubbard bands. When either the HF or HI ap-
itations, typically generate only equal-time correlations,
proximationsare usedasa solver,the DMFT resultis in
and (in the case of quantum Monte Carlo) suffer from fact equivalent to the result obtained directly from the
the fermion sign problem.
corresponding approximation. However, the statistical-
In this work,we use an extensionof DMFT, knownas DMFT procedure outlined above, applied to a physical
statistical DMFT,26 that incorporates both strong cor- lattice, and hence implemented through a matrix inver-
relations and an exact treatment of the disorder poten- sion [Eq. (2)], is very computationally intensive. In par-
tial. By varying U for a fixed disorder strength, we are ticular, it can be difficult to achieve a converged self-
abletomovesmoothlyfromthewell-understoodweakly- consistentsolutiontotheN coupledequationsforΣ (ω).
i
correlatedregimeintotheunknownterritoryofstrongly- This work, aside from being a simple improvement be-
correlateddisorderedsystems. Because our approachre- yond HF, represents an important proof of principle re-
tains spatial correlations between the local self-energy gardingthefeasibilityofthisstatisticalDMFTapproach.
at different lattice sites, but can also be applied to rea- Thefactthatwearestudyingdisorderedsystemsbears
sonably large lattices in finite dimensions, it provides a on two important and related issues: (i) the accuracy
bridgebetweenthevariousmethodsdescribedabove. Up of the HI approximation and (ii) the validity of single-
to now, statistical DMFT has been applied only on a site (as opposed to cluster or cellular) DMFT in a finite
Bethe lattice. Here, we work with a two-dimensional dimensional system. The strengths and weaknesses of
squarelatticeandpresentresultsforthedensityofstates, the HI approximation are well documented in the clean
scalingoftheinverseparticipationratio,andthescreened limit: it is exact in the atomic limit (U/t ≫ 1) but is
potential. nonconservingandfails to satisfy Luttinger’s theorem.27
However, it is uniquely effective in low-dimensional dis-
ordered systems because W/t ≫ 1 corresponds to the
II. METHOD atomic limit for arbitrary U in systems with finite coor-
dinationnumber. Thisisinapparentcontradictiontothe
We focus on paramagnetic solutions on the 2D square generalresultthatDMFTisonlyexactininfinitedimen-
lattice, for which the noninteracting bandwidth is D = sions, while nonlocal terms in the self-energy, neglected
8t. OnanN-sitelattice,thesingle-particleGreen’sfunc- inDMFT,canhaveadramaticeffectontheMotttransi-
tion can be expressed as an N ×N matrix in the site- tion in 2D.28,29 In the disorderedcase, however,we have
index: comparedourresultswithEDstudiesonsmallclusters30
and have found qualitative agreement for W = 12t (ie.
G(ω)=[ωI−t−ǫ−Σ(ω)]−1 (2) W = 1.5D), which is the focus of the current work. By
pushing the system towardsthe atomic limit, strong dis-
withItheidentitymatrix,tthematrixofhoppingampli- order both enhances the accuracy of the HI approxima-
tudes, ǫ the diagonalmatrix of site energies ǫ and Σ(ω) tion and also reduces the importance of nonlocal terms
i
the matrix of self-energies. The matrix t has nonzero in the self-energy.
matrix elements t = −t for i and j corresponding to
ij
nearest-neighbor sites. We assume that Σ(ω) is local,
having only diagonal matrix elements Σ (ω). III. RESULTS
i
TheiterationcyclebeginswiththecalculationofG(ω)
from Eq. (2). For each site i, one defines a Weiss mean The local density of states (LDOS) is extracted from
field G0(ω)=[G (ω)−1+Σ (ω)]−1 where G (ω) arethe the Green’s function as ρ(r ,ω) = −π−1ImG (ω) and
i ii i ij i ii
matrixelementsofG(ω). AsintheconventionalDMFT, the density of states (DOS) is ρ(ω) = N−1 ρ(r ,ω).
Pi i
one then solves for the full Green’s function G (ω) of an Figures1(a)-(c)showtheevolutionoftheDOSasafunc-
i
Anderson impurity whose noninteracting Green’s func- tionofU forfixedW athalf-filling. Thereisatransition
tion is G0(ω). The self-energy is then updated according from a single band to a gapped Mott-insulating state at
i
to Σnew(ω) = G0(ω)−1 −G (ω)−1 and the iteration cy- a critical interaction U ≈ 12.5t. In contrast, the LDOS
i i i c
cle is restarted. In the disorder-free case, this algorithm [Figs.1(e)-(g)]developsalocalMottgapatmuchsmaller
reduces to the conventional DMFT. values of U. We note that, at most sites, the noninter-
We have used the Hubbard-I (HI) approximation as actingLDOShasasinglestrongresonancenearthe bare
a solver for G (ω), for which, in the paramagnetic case, site energy [Fig.1(e)] indicating proximityto the atomic
i
3
0.12
0.1 (a) U=0.0 (b) U=8.0 (c) U=16.0 (d)
0.08
S U=8.0
O0.06
D
0.04
0.02
0
3
(e) (f) (g) (h)
2.5
ε=5.40 5.59 5.65 5.61
2
DOS1.5 0.14 -0.08 -0.39 0.06 FIG. 2: (color online) Local density of states for a single
L
1 disorder realisation at ω = µ, W = 12t, for (a) U = 0, (b)
0.5 -5.02 -5.58 -5.48 -5.58 U =8t,(c)U =12tforanN =32×32sitelatticeandn=1.
0
-20 -10 0 10 20 -10 0 10 -20 -10 0 10 20 -10 0 10 20
ω − µ ω − µ ω − µ ω − µ
ing since there is also no resonance in the clean limit in
FIG.1: (coloronline)DensityofStates. (a)-(c)TotalDOSfor 2D.29 However,thereasonforthis absenceappearstobe
W =12t, n=1 and different U. Results havebeen averaged different in the clean and disordered systems. The lack
over 1000 samples and are for an N = 20×20 site square ofaquasiparticleresonanceinclean2Dsystemshasbeen
lattice. ED results, averaged over 10000 samples, for a 4-site attributed to nonlocal terms in the self-energy,29 terms
lattice(red)arealsoshownin(c). (d)DOSforn=0.5. Also which are not included in HI. These nonlocal terms de-
shown (e)-(g) is the LDOS for three arbitrarily-chosen sites scribeshort-rangedantiferromagneticcorrelationswhose
(site energies indicated in the figure). Panel (e) is for the effect is to suppress U to zero in the clean limit.7,8,28 In
c
same parameters as (a), (f) as (b), etc. Energies are in units
ourwork,the absenceofaquasiparticlepeakstems from
of t.
thecalculationsbeingintheatomiclimit. Thatthesame
physics controls the ED results is supported by the fact
that U is nearly identical in HI and ED calculations. It
limitwhereHIbecomesexact. Thecoordinationnumber c
thereforeappearsasifthe limitations ofthe HI solverdo
of the lattice determines the disorder strength required
not hide key physics in the large disorder limit.
to approach the atomic limit. In lattices with infinite
The DOS at quarter-filling, Fig. 1(d), does have a
coordination number, each site couples to a continuum
peak at the Fermi energy, but the origin of this peak
of states and the atomic limit is never reached for finite
is unrelated to strong correlations. Paramagnetic HF
W. Thus, somewhat paradoxically, DMFT with the HI
calculations25 have shown that nonlocal charge-density
solverworksbest,indisorderedsystems,forlatticeswith
correlations lead to a positive DOS anomaly at the
low coordination number.
Fermi level in disordered metals when the interaction is
It is useful to compare the DOS evolution in Fig. 1
zero-range and repulsive. This Altshuler-Aronov DOS
with existing published work. First, we note that the
anomaly is absent at half-filling in our calculations, in
Mott transition occurs at U ≈ W in both our DMFT
c contradiction with the HF calculations, indicating that
and ED calculations. This is consistent with QMC re-
strong correlations suppress the peak. We will discuss
sultsforthreedimensions,9aswellasinfinite-dimensional
this point below.
DMFTresults,15butUcissomewhatlargerthanfoundin InFig.2,ρ(r,µ) is plotted for differentvalues ofU for
one dimension.10 Our U is also significantly larger than
c aparticulardisorderconfiguration. AtU =0,siteswhich
in unrestricted HF calculations for three dimensions;11
havesignificantspectralweightatω =µaretypicallyiso-
however, in HF calculations, the metal-insulator transi-
lated from one another, consistent with electrons being
tionisoftheSlater-typeandcouldthereforebeexpected
Anderson-localized. TheLDOSismorehomogeneousfor
to respond differently to disorder.
U = 8t than for U = 0, consistent with an interaction-
PreviouslypublishedresultsfortheDOSneartheMott
driven delocalizing effect. However, at U = 12t the
transition in the large-disorder limit, to our knowledge,
LDOS is again highly inhomogeneous.
are for infinite dimensions where CPA-like approxima-
The inverse participation ratio (IPR)
tionscanbemade. Themaindistinctionbetweenourre-
sultsandthoseforinfinite-dimensionsisthequasiparticle N ρ(r ,ω)2
resonance due to Kondo screening of the local moments I2(ω,N)= Pi=1 i 2. (4)
thatappearsattheFermienergyforU ∼U inthelatter N ρ(r ,ω)
case.15,18,19,20 (Thepeakatω =µforU =8ctcomesfrom hPi=1 i i
theoverlapofthelowerandupperHubbardbands.) The provides a quantitative measure of the inhomogeneity of
HI solver cannot give such a peak; however, ED calcu- ρ(r,ω). TheIPRcanalsobeusedtodistinguishextended
lations for small clusters30 also find no resonance at any and localized states since lim I (ω,N) = 0 for the
N→∞ 2
U. In particular, Fig. 1(c) shows ED results for parame- former and is nonzero for the latter. It is important to
terscorrespondingtoamaximumquasiparticleresonance notethatthefrequencyωusedinthecalculationofG(ω)
heightinRefs.[15,20]. Thedifferencebetweenourresults in Eq.(2) contains,by necessity, a smallimaginarycom-
and those for infinite-dimensions might seem unsurpris- ponent iγ. Scaling quantities such as the IPR that are
4
0.02 0.1 2
(a) (b) U=8 U=1.0 U=8.0
0.08 U=6 1.5
N) U=4
µI(,20.01 γ = 90/N 00..0046 UUU===210.5 ni 1
γ = 180/N 0.5
(a) (b)
0 0.02
0 0.005 0.01 0 0.01 0.02 0.03 0.04
0
1 / N 1 / N 1 U=1.0 U=8.0
15 10 4 2V
DMFT ∆
10 (c) Hartree-Fock (d) Vi 0 00 0.5 1 1.5
µ) 5 U/W
ξ( AI MI
5 -4 (c) (d)
0 0 -6 -4 -2 0 2 4 -4 -2 0 2 4 6
0 2 4 6 8 10121416 0 2 4 6 8 10 12 ε ε
U U i i
FIG. 3: (color online) IPR scaling with system size for W = FIG.4: (coloronline)(a),(b)Localchargedensityand(c),(d)
12t. Shownare(a)thedependenceoftheIPRonγforU =0, corresponding screened potentialas afunction of siteenergy.
(b) the scaling of the IPR with γ = 4t/N and n = 1, and The screened potential in thet=0 limit (red dashed) is also
the dependence of the localization length (defined in text) shown in (d). The relative variance of Vi (circles) is shown
on U at (c) n = 1 and (d) n = 0.5. All curves are for ω = (inset) along with thet=0 result (dashed curve).
µ. Anderson-insulating(AI)andMott-insulating(MI)phases
are indicated in (c).
|ǫ | < U/2, n ≈ 1 and χ = 0, such that these sites are
i i ii
derived from G(ω) generally depend on γ.31 In our scal- unscreened (ie. V =ǫ ). There are, therefore, two limits
i i
ing calculations, we have taken γ ∝ 1/N such that the in which screening is small at half-filling: U ≪W where
ratioofγtothelevelspacingremainsconstant. Asshown the interactionis too weakto screenthe impurity poten-
in Fig. 3(a) for the noninteracting case, the effect of γ is tial,andU ∼W wherestrongcorrelationsenforcesingle-
toreduceI (ω,N),suchthatourresultsrepresentalower occupancy of mostsites. We note that the absence ofan
2
bound onthe true (γ =0)IPR.The IPRscalingat half- Altshuler-Aronov DOS anomaly near half-filling can be
filling is shown in Fig. 3(b) for ω = µ. For each value understood in this context: strong correlations suppress
of U, we extrapolate a limiting value I (µ,∞). We then the local response of the charge density to the impurity
2
define a localization length ξ =I (µ,∞)−1/d, where d is potential.
2
the dimensionofthe system.32 Forfiniteγ,I2(µ,∞)−1/d A measure of the screening is given by the relative
gives an upper bound for ξ. variance ∆V2, defined as the variance of V divided by
i
Figures3(c)and(d)illustratetheeffectofstrongcorre- the variance of ǫ . Our numerical results, Fig. 4, show
i
lationsonlocalization. WhiletheHIresultsagreeclosely that ∆V2 is a nonmonotonic function of U obtaining a
with self-consistent HF calculations for small U, the dis- minimum at U ≈ W/2 and approaching ∆V2 = 1 for
crepancy between the methods grows as U is increased. U →0 and U &W. At a qualitative level, this is consis-
In the weakly-correlated small-U regime, ξ grows with tent with the nonmonotonic dependence ofξ onU, since
U, consistent with increased screening of the impurity one expects ξ to be large when ∆V2 is small. There
potential.19,23 For large U, however, ξ is a decreasing are,however,quantitativediscrepancieswhichshowthat
function of U and is smaller near the Mott transition ∆V2 does not tell the whole story. First, ξ does not ob-
than in the U = 0 case. As we discuss below, this can tain its maximum at U = W/2, where ∆V2 obtains its
be partially attributed to a decrease in screening due minimum. Second,ξ issmalleratlargeU thanatU =0,
to strong correlations,although screening no longer pro- indicating that states near the Mott transition are more
vides a complete framework for understanding the evo- strongly localized than at U =0.
lution of ξ. Our results for the IPR are consistent with re-
Tounderstandbetterthelocalizingeffectofstrongcor- cent quantum Monte Carlo calculations of the dc
relations, we define a screened potential19,20,23 V based conductivity23,24 which find a similar nonmonotonic de-
i
on the HF site-energy: V =ǫ +Uni−n.23 Plots of both pendence on U. The results for the screened poten-
i i 2
n and V as a function of ǫ are shown in Fig. 4. In the tial, however, are inconsistent with infinite-dimensional
i i i
weakly-correlatedlimit (U =t), n is approximately lin- DMFT results.19,20 In Ref. [19], impurities are found to
i
ear in ǫ over a wide range and V ≈ ǫ (1−Uχ ) with be perfectly screened (ie. ∆V2 → 0) at the Mott tran-
i i i ii
χ =−dn /dǫ . Since χ depends only weakly on U, V sition, which would correspond to a divergent ξ in our
ii i i ii i
is a decreasing function of U. In the strongly-correlated calculations. Near the Mott transition, Ref. [21] found
limit (U = 8t), n is a nonlinear function of ǫ . For thatsiteswith|ǫ |<U/2areperfectly screened. Bothof
i i i
5
these results are opposite to what we have found here. In conclusion, we have studied the 2D Anderson-
There are two important distinctions between our cal- Hubbard model at half- and quarter-filling, in the limit
culation and those of Refs. [19,21], both of which con- of large disorder using statistical DMFT. We have cal-
tribute to these opposing results on screening. First, culated the localization length ξ from the inverse par-
Refs. [19,21] use an effective medium approach for the ticipation ratio, and find that it varies nonmonotoni-
disorder potential that results in metallic behavior for cally with the strength of the interaction: at small U,
smallU.ThisisreasonableinhighdimensionswhereAn- the interaction screens the impurity potential, but at
derson localization only occurs for strong disorder. Our large U strong correlations reduce the screening. As a
exact treatment of the disorder potential, on the other consequence,the Altshuler-AronovDOSanomalyissup-
hand,allowsusto describethe Andersonlocalizedphase pressedathalf-filling. Forstrongdisorder,wefindnoev-
thatoccursin2DforsmallU.WhereastheLDOSiscon- idenceforaninsulator-metaltransitionnorforenhanced
tinuous and relatively uniform in the metallic case, it is screening near the Mott transition.
inhomogeneous and dominated by small numbers of res-
onances in the Anderson insulating phase [cf. Fig. 1(f)].
The local charge susceptibility χ is suppressed for sites
ii
Acknowledgments
with small LDOS at the Fermi level, and screening in
the Anderson-insulating phase is consequently expected
to be less than in the metallic phase. The second dis- We thank R. J. Gooding and E. Miranda for help-
tinctionisthequasiparticleresonancewhicharisesinthe ful conversations. We acknowledge Trent University,
infinite-dimensional case, but does not occur in our cal- NSERC of Canada, CFI, and OIT for financial support.
culations. This resonance is a key factor in the perfect SomecalculationswereperformedusingtheHighPerfor-
screening found in Refs. [19,21]. mance Computing Virtual Laboratory (HPCVL).
1 S. V. Kravchenko and M. P. Sarachik, Rep. Prog. Phys. 17 M.S.Laad,L.Craco,andE.Mu¨ller-Hartmann,Phys.Rev.
67, 1 (2004). B 64, 195114 (2001).
2 D. D. Sarma, A. Chainani, S. R. Krishnakumar, 18 P.Lombardo, R.Hayn,and G. I.Japaridze, Phys.Rev.B
E.Vescovo,C.Carbone,W.Eberhardt,O.Rader,C.Jung, 74, 085116 (2006).
C. Hellwig, W. Gudat, et al., Phys. Rev. Lett. 80, 4004 19 D. Tanaskovi´c, V. Dobrosavljevi´c, E. Abrahams, and
(1998). G. Kotliar, Phys. Rev.Lett. 91, 066603 (2003).
3 S. Nakatsuji, V. Dobrosavljevic, D. Tanaskovic, M. Mi- 20 M. C. O. Aguiar, V. Dobrosavljevic, E. Abrahams, and
nakata,H.Fukazawa,andY.Maeno, Phys.Rev.Lett. 93, G. Kotliar, Phys. Rev.B 73, 115117 (2006);
146401 (2004). 21 M.C.O.Aguiar,V.Dobrosavljevi´c,E.Abrahams,andG.
4 K.W.Kim,J.S.Lee,T.W.Noh,S.R.Lee,andK.Char, Kotliar, arXiv:0704.0450v1.
Phys. Rev.B 71, 125104 (2005). 22 B. Srinivasan, G. Benenti, and D. L. Shepelyansky, Phys.
5 J. Kim, J. Kim, B. G. Park, and S. J. Oh, Phys. Rev. B Rev.B 67, 205112 (2003).
73, 235109 (2006). 23 P.B.Chakraborty,P.J.H.Denteneer,andR.T.Scalettar,
6 Elliott H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 Phys. Rev.B 75, 125117 (2007).
(1968). 24 R. Kotlyar and S. Das Sarma, Phys. Rev. Lett. 86, 2388
7 J. E. Hirsch, Phys.Rev.B 31, 4403 (1985). (2001).
8 S. R. White, D. J. Scalapino, R. L. Sugar, E. Y. Loh, J. 25 B.L.AltshulerandA.G.Aronov,Electron-Electron Inter-
E. Gubernatis, and R. T. Scalettar, Phys. Rev.B 40, 506 actionsinDisordered Systems,vol.10ofModernProblems
(1989). in Condensed Matter Sciences (North-Holland, 1985).
9 Y.OtsukaandY.Hatsugai,J.Phys.Cond.Mat.12,9317 26 V. Dobrosavljevi´c and G. Kotliar, Phys. Rev. Lett. 78,
(2000) 3943(1997);V.Dobrosavljevi´candG.Kotliar,Phil.Trans.
10 Y. Otsuka,Y.Morita, and Y.Hatsugai, Phys. Rev.B 58, R. Soc. London 356, 57 (1998); E. Miranda and V. Do-
15314 (1998). brosavljevi´c, Rep.Prog. Phys. 68, 2337 (2005).
11 M. A. Tusch and D. E. Logan, Phys. Rev. B 48, 14843 27 F. Gebhard, The Mott Metal-Insulator Transition: Mod-
(1993). els and Methods, vol. 137 of Springer Tracts in Modern
12 D. Heidarian and N. Trivedi, Phys. Rev. Lett. 93, 126401 Physics (Springer,1997).
(2004). 28 S. Moukouri and M. Jarrell, Phys. Rev. Lett. 87, 167010
13 F. Fazileh, R. J. Gooding, W. A. Atkinson, and D. C. (2001)
Johnston, Phys.Rev.Lett. 96, 046410 (2006). 29 Y. Z. Zhang and Masatoshi Imada, Phys. Rev. B 76,
14 M. Ulmke, V. Jani˘s, and D. Vollhardt, Phys. Rev. B 51, 045108 (2007).
10411 (1995). 30 S. Bulut and W. A. Atkinson (unpublished).
15 K. Byczuk, W. Hofstetter, and D. Vollhardt, Phys. Rev. 31 Y.Song,W.A.Atkinson,andR.Wortis,Phys.Rev.B76,
Lett. 94, 056404 (2005). 045105 (2007); R.Wortis,YunSong,andW.A.Atkinson,
16 M. Balzer and M. Potthoff, Physica B 359-361, 768 arXiv:0709.0713.
(2005). 32 Y. V. Fyodorov and A. D. Mirlin, Phys. Rev. Lett. 69,
6
1093 (1992).