Table Of Content

Static Screening and Delocalization Effects in the Hubbard-Anderson Model Peter Henseler and Johann Kroha Physikalisches Institut, Universität Bonn, Nußallee 12, D-53115 Bonn, Germany Boris Shapiro Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel We study the suppression of electron localization dueto the screening of disorder in a Hubbard- Andersonmodel. WefocusonthechangeoftheelectronlocalizationlengthattheFermilevelwithin 8 astaticpicture,whereinteractionsareabsorbedintotheredefinitionoftherandomon-siteenergies. 0 Two differentapproximations are presented,eitheroneyielding anonmonotonic dependenceof the 0 localization length on the interaction strength, with a pronounced maximum at an intermediate 2 interaction strength. In spite of its simplicity, our approach is in good agreement with recent n numerical results. a J PACSnumbers: 71.10.Fd,73.20.Fz,72.15.Rn,71.30.+h 5 2 I. INTRODUCTION work). Furthermore, we will show that, for fixed disorder, the ] n localization length ξ is a nonmonotonic function of the n Understanding the interplay between disorder and HubbardinteractionenergyU,withamaximumforsome - s electron-electron interactions remains one of the major intermediate value of U. This is because for strong i challenges in modern condensed matter physics, experi- interactions, the Mott-Hubbard physics of interaction- d . mental as well as theoretical. The research in this field suppressed hopping dominates, leading to the forma- t a has been stimulated by the possible metallic behavior tion of two disorder broadened Hubbard bands with m in two-dimensional disordered interacting systems.1 A a reduced average density of states at the Fermi level - metallic phase at zero temperature in two or less dimen- and, as a consequence, increasing effectively the disor- d sionswouldbeincontrasttothepredictionofthescaling der strength. In spite of the simplicity of the approach, n theory of Anderson localization.2 The possible existence the results are in good agreement with recent numerical o of a metallic phase, induced by interactions, is a long studies,5,7,8,9,10,13,14 in the appropriaterange of parame- c [ standing3 and still controversial problem, discussed by ters. many authors during the last three decades.4 The evaluation of the localization length within our 2 One of the ideas proposed and discussed by several au- presentstudyislimitedtoonedimension. However,since v thorsisthatinteractionsleadtoapartialscreeningofthe the competition of screening on the one hand and Mott- 1 0 random potential and, thus, reduce the effect of local- Hubbard physics on the other hand operates in any di- 3 ization. In particular, the 2d disordered Hubbard model mension,thenonmonotonicdependenceofξ onU should 1 (theHubbard-Andersonmodel)hasbeenstudied,mostly also hold in two and three dimensions, as argued below. . numerically, and it was demonstrated that repulsive in- 7 0 teractions can have a delocalizing effect.5,6,7,8,9,10,11 7 In this paper, we present an analytical study of the II. ATOMIC-LIMIT APPROXIMATION 0 screening effect, focussing on the case of strong disor- : der at zero temperature, when the Hubbard-Anderson In this paper, we consider the Hubbard-Anderson v i model is in the regime of an Anderson insulator. Our model,withon-siterepulsionandon-sitedisorder,atzero X approach is based on an exact treatment of the atomic temperature. The corresponding Hamiltonian is r limit, followedby ”switchingon” the intersite hopping t, a under the assumption that the atomic-limit occupation H = H0 + Hkin + He-e numbers do not change. Let us emphasize that in our = ε −µ c† c − t c† c staticapproach,theinteractionsonlychangetheoriginal i iσ iσ i,σ jσ on-site energies. Therefore, we are left with a single- Xi,σ (cid:16) (cid:17) <iX,j>,σ particle Anderson Hamiltonian, and the question is how +U ni↑ni↓. (1) thelocalizedsingle-particlestatesmaychangeduetothe i X new, renormalized probability distribution of the on-site As usual, c† (c ) denote fermion creation (destruction) energies. In this sense, the approach is close, although iσ iσ not identical, to the Hartree-Focktreatment. A compar- operatorsofanelectronatsiteiwithspinσ,n =c† c , iσ iσ iσ ison of both methods will be presented. Although this t is the nearest-neighbor hopping amplitude, U is the approach is formulated for an arbitrary filling factor, it on-site repulsion, µ is the chemical potential, and {ε } i becomes inadequate close to half filling where magnetic are the on-site energies. The latter are assumed to be effects dominate12 (sucheffectsarenotconsideredinour independent and uniformly distributed over the interval 2 ∆ ∆ ∆ 2 2 2 pA(ε) 3/2∆ µ µ U 1/∆ U U µ−U ε −∆/2 −∆/2+U µ−U µ µ+U ∆/2 −∆2 −∆2 −∆2 (a) a) b) c) pH(ε) FIG.1: Siteoccupationintheatomicgroundstatewithdou- blyoccupied(black),singlyoccupied(gray)andemptystates 2/∆ (white). (a) weak interaction, (b) strong interaction for less than half filling (i.e., ρ < 1), and (c) strong interaction for more than half filling (ρ>1). 1/∆ ε [−∆,∆], with the disorder parameter ∆. To focus on −∆/2 −∆/2+U µ−U/2 µ µ+U/2 ∆/2 2 2 thescreeningeffectinthecaseofstronglocalization,i.e., (b) ∆≫t,theinteractiontermwillbeabsorbedintotheon- site energies, yielding a renormalized distribution of the FIG.2: Therenormalizedon-siteenergyprobabilityfunctions ε . This results in an effective single-particle problem i (solidline)forweakrepulsionU: (a)atomic-limitapproxima- with a probability function p (ε ) which is derived as A i tion, and (b) Hartree-Fock approximation. For comparison, follows. also theoriginal function is shown (dashed lines). Intheatomiclimit(t=0),thegroundstateofthesystem can be solved exactly for an arbitrary filling factor ρ = Ne, whereN ,Narethe numbersofelectronsandlattice sNites,respecteively.9Thechemicalpotentialµandthesite- by U. The next two terms correspond to singly occupied sites. In the absence of spin polarization, hn i = dependentoccupationnumbershn i canbeexpressedas i,σ 0 i 0 hn i ,halfoftheseon-siteenergiesareagainshiftedby functions of ρ, ∆, and U: All sites with on-site energies i,−σ 0 U whereasthe otherhalfremainunchanged. Finally,the belowµ−U aredoublyoccupied,allsiteswithin[µ−U,µ] lastterm correspondsto the unoccupied sites, whose en- aresinglyoccupied,andallothersitesareempty(seeFig. ergies also remain unchanged. Combining Eqs. (3) and 1). Thus, the total occupation number of site i and the (4), the rule for replacing the bare site energy ε by a chemical potential are, respectively, i renormalized one is 2 , ε ≤µ−U i ε +U if ε ≤µ−U i i hn i = 1 , µ−U <ε ≤µ (2) i 0  i ε +U each with 0 , ε >µ ε 7→ i if µ−U <ε ≤µ (5)  i i  εi i (cid:18)prob. of 21 (cid:19) and15  εi if εi >µ. 1(∆ρ−∆+U) , ρ<1, U <∆ρ For aweakto intermediaterepulsionU these shifts lead 2 ∆(ρ− 1) , ρ<1, U ≥∆ρ to a raise of the lowest lying on-site energies towards µ =  1(∆ρ−∆+U2) , ρ≥1, U <2∆−∆ρ (3) the Fermi level µ resulting in a renormalizedprobability  2 ∆(ρ− 3)+U , ρ≥1, U ≥2∆−∆ρ. function pA(ε)with a reducedwidth and modifiedshape 2 (see Fig. 2(a)). The renormalized site energies depend on the site occu- In the atomic-limit approximation the Hamiltonian pationnumbersandcanbereadofffromthepolesofthe Eq. (1) is replaced by the effective single-particle An- (time-ordered) single-particle propagator, derson Hamiltonian hni,σi0hni,−σi0 H = εi −µ c†iσciσ −t c†iσcjσ (6) G (ω) = iσ ω−(εi−µ+U)−i0+ Xiσ (cid:16) (cid:17) <Xi,j>σ hni,σi0(1−hni,−σi0) (1−hni,σi0)hni,−σi0 withon-siteenergyprobabilityfunctionpA(ε),wherethe + + ω−(ε −µ)−i0+ ω−(ε −µ+U)+i0+ two-particle interaction U enters only as a (screening) i i parameter in p (ε). Note that, due to the asymmetry (1−hn i )(1−hn i ) A + i,σ 0 i,−σ 0 . (4) of pA(ε), the average value of the renormalized on-site ω−(ε −µ)+i0+ i energies is ThefirstterminEq.(4)correspondstothedoublyoccu- 1 hεi = p (ε)εdε = ρU. (7) pied sites showing that these on-site energies are shifted A A 2 Z 3 In deriving Eq. (6), it was assumed that in the case of strong disorder, the occupation numbers in the atomic ground state, Eq. (2), are close to the occupation numbers hn i in the true ground state, with finite hopping i amplitude t. This assumption is based on the results from the single-particle theory of localization,16 where it is known that for strong disorder, an electron, once located at any site i, will stay at that site with high probability. More precisely, the change of the occupation number of site i, δhn i ≡ hn i−hn i , is of order i i i 0 t2/∆. The same estimate holds if one considers hopping of a single electron on the background of the other elec- trons, which are assumed to be immobile. Furthermore, forstrongdisorder,aperturbativeexpansionint,around the atomic ground state of Eq. (1), is possible.17 Thus, (a) with the same reasoning as in the noninteracting case, the leading self-energy corrections are again only of sec- ond order. One measure of localization is the localization length ξ whichgovernstheexponentialdecayofthesingle-particle wave functions ψ(r) at distances far away from its localization center. It can be calculated from the probability for a transition from site i to site j as18 log GR(E) 2 1 ij − = lim , (8) ξ(E) |xi−xj|→∞ 2D|(cid:12)(cid:12)xi−xj|(cid:12)(cid:12) E where GR(E) is the retarded propagator for a particle ij with energy E from site j to site i. In general, it is not possible to deduce ξ from the probability distribu- (b) tionanalytically,butincaseofaone-dimensionallattice, there exists a relatively simple relation18 between ξ and FIG.3: (Coloronline)LocalizationlengthξattheFermilevel hD(ω)i, the disorder averageddensity of states, as a function of repulsion U and lattice filling ρ for ∆ = 15: (a)atomic-limitapproximationand(b)Hartree-Fockapprox- ∞ imation. ξ−1(E) = hD(ε)i log|E−ε| dε. (9) Z −∞ (Here and in the following, we choose units where t = 1 The reason for this nonmonotonic behavior is simple: and measure ξ in units of the lattice spacing.) A weak to intermediate repulsion U changes the on-site In the strong disorder limit, hD(ε)i can be replaced by energy distribution from a rectangular distribution to a p (ε+µ), so that the inverse localization length at the narrower one by shifting low on-site energies toward the A Fermi level (E =0) is given by Fermi level (screening), see Fig. 2(a). In contrast, a very strong on-site repulsion enhances the localization ∞ by a large broadening of the probability density. There- ξ−1 = pA(ε) log|ε−µ| dε . (10) fore, in between, there will be some value UξA for which the screening is optimal and the localization length ac- Z −∞ quires a maximum. Such behavior is expected, since a A plot of ξ as a function of U and ρ, for fixed disorder strongrepulsion effectively suppresses hopping processes strength ∆ = 15, is shown in Fig. 3(a). It can be seen and leads to an accumulation of spectral weight in the that for each given filling factor, the localization length upper Hubbard band. exhibits apronouncedmaximum. Thismaximumcanbe InFig. 4(a),the localizationlengthξ isshownasafunc- calculated to appear at tion of U for ρ = 21 (quarter filling). A similar, nonmonotonic behavior was also found in recent quantum ∆ Monte Carlo simulations5,9,13,14 and a most recent sta- UξA = 3 1+3ρ(2−ρ)−1 tistical dynamical mean field theory evaluation10 of the ∆(cid:16)p (cid:17) problem, where the conductivity and the inverse partic- ≈ ρ(2−ρ) + O(ρ2(2−ρ)2). (11) ipation ratio7 of a finite system were calculated, respec- 2 4 tively. Identifying a maximum of conductivity with a maximum of localization length, we find in all cases a good qualitative agreement with our results. Further- more, we even find a reasonable quantitative agreement withourresultsforthepointsofmaximaldelocalization, UA, Eq. (11), and UH, defined in Eq. (17) below. Thus, ξ ξ thereisastrongcorrelationbetweenthedegreeofscreen- ing and the conductivity: optimal screening corresponds to maximal conductivity. In addition, our analytical results indicate that much of this physics of the nonmono- (a) tonic behavior can already be understood on the levelof a static screening approximation. Our result appears to be at odds with the statement9 that screening alone cannot account for the nonmonotonic behavior of the conductivity. We will discuss this point in the next section. For that discussion, we need thevarianceoftherenormalizedon-siteenergieswhichis given by σ2 ≡ hε2i −hεi2 (12) A A A 1−ρÛˆ +ρÛˆ2+3Uˆ3 , ρ<1, Uˆ <ρ ∆2 1−2ρ′Uˆ +ρÛˆ2 , ρ<1, Uˆ ≥ρ (b) =  12  1−ρÛˆ1−+2ρˆρU˜Ûˆ2++ρ3ÛUˆˆ23 ,, ρρ≥≥11,, UUˆˆ ≥<22−−ρρ, FIG. 4: (a) Localization length ξ at the Fermi level, normalized by its noninteracting value, as a function of the on-site with  repulsion U for ∆ = 15 and ρ = 21. (b) Variance of the renormalized on-site energy distribution, also normalized by Uˆ = U/∆, ρˆ = 3ρ(2−ρ), ρ′ = 3ρ(1−ρ), its noninteracting value. The solid curves show the results fortheatomic-limitapproximationandthedashedcurvesare ρ˜ = 3 1 −(ρ− 3)2 . (13) 4 2 theHartree-Fock results. Here, σ2 has a minimum(cid:0) at (cid:1) A ∆ ρˆ2+9ρˆ−ρˆ , 1 .ρ. 5 UA = 9 3 3 (14) which in Eq. (1) corresponds to the replacement, σ ( (cid:16)p∆ |1−ρ| (cid:17) , otherwise, 1+|1−ρ| Un n → U hn in +hn in which can be seen in Fig. 4(b) for ρ= 1 and ∆=15. i↑ i↓ i↑ i↓ i↓ i↑ 2 U (cid:16) U (cid:17) → hn i n +n ≈ hn i n +n . (16) So far, our calculations were restricted to one di- 2 i i↓ i↑ 2 i 0 i↓ i↑ mension because a generalization of relation (9) to two (cid:16) (cid:17) (cid:16) (cid:17) orthreedimensionsis,ingeneral,notpossible. However, Here, the absence of any kind of magnetization was as- in the limit of strong disorder, Gij(E) can be calculated sumed. Note that for a local, energy independent inter- in good approximation by taking into account only the action U, the Hartree and the Hartree-Fock approxima- direct path19 from i to j. Therefore, Eq. (8) can be tions are identical. The average on-site occupation hn i i considered as a one-dimensional problem, leading again wastakentobetheoneoftheatomicgroundstate,hn i , to Eqs. (9) and (10), respectively. Hence, we conjecture i 0 Eq. (2), according to the assumption of a stable atomic that the general result, i.e., the nonmonotonic behavior configuration. of localization, holds also in two and three dimensions. As in the atomic-limit approximation, the shift of the occupied on-site energies leads to a renormalized distribution, with probability function p (ε) (Fig. 2(b)). The III. SITE-DEPENDENT HARTREE-FOCK H screening effect now is even more pronounced because APPROXIMATION all singly occupied states are shifted by U yielding a 2 smaller width and a stronger increase of the probability Thesecondmodelwhichwillbediscussedinthispaper tofindastatearoundtheFermilevel. Theresultingeffec- is the site-dependent Hartree-Fock approximation.6,9 In tivesingle-particleHamiltonianisagaingivenbyEq.(6), this single-particle approximation, each site has a single however,with the probability function p (ε). H renormalized energy level, given by The corresponding plots for the localization length in U Hartree-FockapproximationareshowninFigs. 3(b)and εi 7→ εi + 2hnii0, (15) 4(a), respectively. Again, a pronounced maximum of ξ 5 arises. In this case, it is found at the value ∆ UH = ρ(2−ρ), (17) ξ 2 which up to order O ρ2·(2−ρ)2 coincides with the resultfromtheatomic-limitapproximationEq.(11). The (cid:0) (cid:1) increase of the localization length is considerably more pronounced in the Hartree-Fock approximation due to its narrower probability distribution and especially its largerprobabilitydensityaroundtheFermilevel. Inboth approaches,theeffectofscreeningbecomesstrongerwith (a) decreasing disorder. However, for small values of ∆, the stability of the atomic ground state becomes doubtful. Theaverageandthe varianceofthe renormalizedon-site energy distribution are, respectively, 1 hεi = hεi = ρU, (18) H A 2 1−ρÛˆ +ρÛˆ2 , ρ<1, Uˆ <ρ ∆2 1−2ρ′Uˆ +ρ′Uˆ2 , ρ<1, Uˆ ≥ρ σ2 =  (19) H 12  11−−2ρρˆ˜UUˆˆ ++ρρ˜ÛUˆˆ22 ,, ρρ≥≥11,, UUˆˆ ≥<22−−ρρ, withthesameabbreviationsasinEq.(13). Itsminimum (b) is found at the value FIG.5: Inverselocalization lengthξ−1 asafunction of∆for ∆/2 , 2 <ρ< 4 ρ= 12 and (a) U =7 in the atomic-limit approximation and UH = 3 3 (20) (b) U =14 in the Hartree-Fock approximation, respectively. σ ∆ , else. (cid:26) Asmentionedabove,itwasarguedinRef.9thatthepic- ture of screening would be too primitive to explain the picture would necessarily predict a monotonic increase nonmonotonic behavior and the evidence for a metallic of the inverse participation ratio with increasing ∆, ex- state found in the conductivity simulations by varying cluding screening as a possible explanation. Our model, the repulsion strength U. The argumentation was based contrary to that statement, exhibits such nonmonotonic onthe observation,which the variancewas a featureless, behavioraswell,asshowninFig. 5,whereξ−1 isplotted monotonicallydecreasingfunction of U aroundthe tran- as a function of the disorder strength ∆, for some fixed sitionpoint. Our resultsshowthat this reasoningis gen- values of ρ and U. This nonmonotonicity is caused by erallynotconclusive. Althoughthestatic,single-particle the crossover from the regime of disorder screening by treatmentdoesnotallowfor the occurrenceofametallic interaction,∆≫U,totheregimeofinteraction-reduced state, we find a strong enhancement and nonmonotonic hopping, ∆ ≪ U, which is controlled by the parameter behavior of the localization length ξ as function of U, U/∆. However, the exact position of the nonmonotonic- whereasthevarianceisalsoonlyamonotonicallydecreas- ity depends alsoont/∆, ρ,and the probability function. ing function aroundthe pointof maximaldelocalization. For strong disorder anda distribution which is not char- acterized by a single parameter (like in the present case, IV. CONCLUSION cf. Fig. 2), there is no simple relation between ξ and the variance σ2. Especially for lower fillings, our results, Eqs. (11), (14) and (17), (20), respectively, show that We examined the effect of static disorder screening the values of U for which the maximum of ξ and the by on-site repulsion in the one-dimensional Hubbard- minimum of σ2 occur, can be separated systematically. Anderson model for strong disorder. We presented two Moreover, we find that the atomic-limit approximation different approximation schemes by absorbing the inter- and the Hartree-Fock approximation do yield close val- actions into a redefinition of the single-particle on-site ues of ξ, although the variances can differ strongly (see energies. In both approaches a renormalized probability Fig. 4). distribution with an enhanced probability of finding site In Refs. 8 and 13, the inverse participation ratio was energies close to the Fermi level was obtained. We cal- calculated as a function of the disorder strength ∆, and culated the localization length at the Fermi energy for a nonmonotonic behavior with evidence for a metallic these single-particle problems and found a pronounced state was found. It was argued there that the screening maximum of the localization length for some interme- 6 diate value of the repulsion strength. This can be un- localizedsystem, whose Fermi energy is sufficiently close derstood as a consequence of the fact that the increase tothemobilityedge,willbecomemetallicuponswitching of the localization length ξ for small U (U < U ) and on interactions by shifting the mobility edge across the ξ the decrease of ξ for large U (U > U ) have different Fermilevel. Furthermore,whentheinteractionsexceeda ξ physical origins, namely disorder screening and reduced certainstrength,thesystemwouldreenterthe insulating hopping, respectively. Similarly, a change of the ”bare” phaseinanalogytothenonmonotonicbehaviorofthelo- disorder ∆, for fixed repulsion, resulted in a nontrivial, calization length in one dimension.21 (Let us emphasize nonmonotic dependence ofthe localizationlengthonthe that throughout this paper we do not discuss the case disorder strength. In contrast to the case of weak dis- ofhalffilling,withits characteristicMott’sphysics.12,22) order, we found no significant correlation between the Asimilar,andexperimentallymorerelevant,effectcould variance of the effective on-site energy distribution and also happen under a change of the bare disorder ∆, as thelocalizationlength. Ourresults,especiallyinthecase suggestedby Fig. 5, since the relevant dimensionless pa- oftheHartree-Fockapproximation,areinqualitativeand rameter is U/∆. to some degree even quantitative agreement with recent numericalstudies. By ouranalytic approach,it waspos- sible to investigate the static screening effects separately from dynamical (inelastic) processes. Acknowledgments We gave an argument that the same behavior should also be found for strong disorder in two and three dimensions.20 In three dimensions there might be an This work was supported in part by the Deutsche interesting possibility of an interaction induced metal- ForschungsgemeinschaftthroughSFB608. P.H.acknowl- insulator transition. Such a possibility is based on the edges additional support by Deutscher Akademischer assumption that our results, obtained in the strongly lo- Austausch Dienst (DAAD). B.S. acknowledges the hos- calized regime, can be extrapolated up to the mobility pitality of Bonn University where the present work was edge. Underthisassumption,anoninteractingAnderson- initiated. 1 Forareview, seeE.Abrahams,S.V.Kravchenko,and M. lished); arXiv:0704.0450, for a recent reference on the P. Sarachik, Rev.Mod. Phys. 73, 251 (2001). Hubbard-Andersonmodel at half filling in 3d. 2 E. Abrahams, P. W. Anderson,D. C. Licciardello, and T. 13 N. Trivedi, P. J. H. Denteneer, D. Heidarian, and R. T. V. Ramakrishnan, Phys.Rev.Lett. 42, 673 (1979). Scalettar, Pramana-J. of Physics 64, 1051 (2005). 3 L.Fleishman andP.W.Anderson,Phys.Rev.B21, 2366 14 P. J. H. Denteneer and R. T. Scalettar, Phys. Rev. Lett. (1980). 90, 246401 (2003). 4 Recently, it has been demonstrated that interaction- 15 Athalffillingandstrongrepulsion,thechemicalpotential induced decay of single-particle excitations (Anderson lo- lies in the gap of the two Hubbard bands. To avoid an calized in the absence of interactions) can lead to the in- unnecessary case differentiation, in that case, we defined teresting phenomenon of a metal-insulator transition at µ = U −∆/2. µ defined in this way is at the top of the finite temperatures, see, I. V. Gornyi, A. D. Mirlin, and gap, in contrast to themore common definition,µ=U/2, D.G.Polyakov,Phys.Rev.Lett.95,206603 (2005); D.M. at the center of the gap. Physical quantities are of course Basko, I.L.Aleiner,and B.L. Altshuler,Annalsof Physics not affected by this definition. 321, 1126 (2006). 16 P. W. Anderson,Phys. Rev. 109, 1492 (1958). 5 P. J. H.Denteneer, R.T. Scalettar, and N.Trivedi, Phys. 17 M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D.S. Rev.Lett. 83, 4610 (1999). Fisher, Phys.Rev B 40, 546 (1989). 6 I. F. Herbut,Phys.Rev. B 63, 113102 (2001). 18 I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Intro- 7 B. Srinivasan, G. Benenti, and D. L. Shepelyansky, Phys. duction to the Theory of Disordered Systems (Wiley, New Rev.B 67, 205112 (2003). York,1988). 8 D. Heidarian and N. Trivedi, Phys. Rev. Lett. 93, 126401 19 E. Medina and M. Kardar, Phys.Rev.B 46, 9984 (1992). (2004). 20 We emphasize that throughout this paper, we only con- 9 P.B.Chakraborty,P.J.H.Denteneer,andR.T.Scalettar, siderthecaseofstrongdisorderawayfromhalffilling.For Phys. Rev.B 75, 125117 (2007). the question of disorder screening in the opposite limit of 10 Y. Song, R. Wortis, and W. A. Atkinson, arXiv:cond- disordered Mott insulators, see, e.g., D. Tanaskovic, V. mat/0707.0791. Dobrosavljevic, E. Abrahams, and G. Kotliar, Phys. Rev. 11 Such effect has also been observed in models with long- Lett. 91, 066603 (2003), as well as Ref. 12. range Coulomb interactions; see Z. A. Nemeth and J.-L. 21 This reentry resembles the double insulator-metal- Pichard, Eur. Phys. J. B45, 111 (2005), and references insulator transition, taking place in a disordered system therein. underachangeofthemagneticfield.SeeB.Shapiro,Phil. 12 See, e.g., M. C. O. Aguiar, V. Dobrosavljevic, E. Abra- Mag. B 50, 241 (1984), and the discussion in N. Mott, hams, and G. Kotliar,Proceedings of the Conference on Metal-Insulator Transitions (Taylorand Francis, 1990), p. Strongly Correlated Electron Systems SCES’07 (unpub- 160. 7 22 See K. Byczuk, W. Hofstetter, and D. Vollhardt, Phys. those discussed above, havebeen found. Rev. Lett. 94, 056404 (2005), where reentries, similar to