We analyze the effects of the on-site Coulomb repulsion U on a band insulator
using dynamical mean field theory (DMFT). We find the surprising result that
the gap is suppressed to zero at a critical Uc1 and remains zero within a
metallic phase. At a larger Uc2 there is a second transition from the metal to
a Mott insulator, in which the gap increases with increasing U. These results
are qualitatively different from Hartree-Fock theory which gives a
monotonically decreasing but non-zero insulating gap for all finite U.Comment: 4 pages, 5 figure

Arti Garg

H. R. Krishnamurthy

Mohit Randeria

T. Egami

English

arXiv

We analyze the effects of the on-site Coulomb repulsion U on a band insulator using dynamical mean field theory (DMFT). We find the surprising result that the gap is suppressed to zero at a critical U_c1 and remains zero within a metallic phase. At a larger U_c2 there is a second transition from the metal to a Mott insulator, in which the gap increases with increasing U. These results are qualitatively different from Hartree-Fock theory which gives a monotonically decreasing but nonzero insulating gap for all finite U

Garg, Arti

Krishnamurthy, H. R.

Randeria, Mohit

KnowledgeBank at OSU

Can Correlations Drive a Band Insulator Metallic?Arti Garg,1 H. R. Krishnamurthy,2 and Mohit Randeria31Department of Theoretical Physics, Tata Institute of Fundamental Research, Mumbai 400005, India2Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560 012,and Condensed Matter Theory Unit, JNCASR, Jakkur, Bangalore 560 064, India3Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA(Received 14 November 2005; published 26 July 2006)We analyze the effects of the on-site Coulomb repulsion U on a band insulator using dynamical meanfield theory (DMFT). We find the surprising result that the gap is suppressed to zero at a critical Uc1 andremains zero within a metallic phase. At a larger Uc2 there is a second transition from the metal to a Mottinsulator, in which the gap increases with increasing U. These results are qualitatively different fromHartree-Fock theory which gives a monotonically decreasing but nonzero insulating gap for all finite U.DOI: 10.1103/PhysRevLett.97.046403 PACS numbers: 71.30.+h, 71.27.+aThere are many examples of interactions opening a gapin a metallic state: the Mott metal-insulator transition, aswell as spin or charge density wave gaps on nested Fermisurfaces. However, the possibility of interactions suppress-ing the gap in a band insulator all the way down to zeroseems, at first sight, to be counterintuitive. In this Letter,we show that this does occur in a simple model treatedwithin a well-tested approximation scheme, namely, dy-namical mean field theory (DMFT) [1,2].Specifically, we start with a simple tight-binding bandinsulator with two bands, one filled and the other unfilled,and turn on an on-site Coulomb repulsion, the Hubbard U.We treat the effect of U using the DMFT within theparamagnetic sector, and show that with increasing U thegap of the band insulator first collapses to zero at a Uc1 andremains zero, leading to a metallic phase, for a finite rangeof U. This may be intuitively understood in terms ofcorrelations screening out the one-body potential respon-sible for the gap in the band insulator. Eventually, with afurther increase in U, there is a second phase transition at aUc2 to a Mott insulator. Our calculated phase diagram isshown in Fig. 1.The model we consider has tight-binding electrons on abipartite lattice (sublattices A and B) described by H  t Xi2A;j2B;cyicj  H:c:  Xi2Ani Xi2BniUXini"ni# Xini; (1)where t is the nearest neighbor hopping, U the Hubbardrepulsion, and  a one-body potential which doubles theunit cell. The chemical potential is chosen [3] to be  U=2, so that the average occupancy is hnAi  hnBi=2  1(‘‘half filling’’). The Hamiltonian (1) is sometimes calledthe ‘‘ionic Hubbard model’’ with  the ‘‘ionic’’ potential,and has been extensively studied especially in one dimen-sion [4–10], although the existence of an intermediatemetallic state between the band and Mott insulators isapparently not settled conclusively even in the 1D case.In any case, both our motivation and methodology are verydifferent from previous studies.We begin by considering various limiting cases at halffilling. In the noninteracting (U  0) limit the system is aband insulator with a band gap of . However, a verydifferent picture emerges in the atomic limit (t  0).Here we obtain a ‘‘band insulator’’ for U=2< with siteoccupancies nA  0 and nB  2 and a gap U=2 foreither injecting or extracting an electron. This gap de-creases with increasing U and collapses to zero at U=2  when the singly occupied states becomes degeneratewith the empty one at site A and with the doubly occupiedone at site B. Beyond this point, for U=2>  we get a‘‘Mott insulator’’ with nA  nB  1, and a gap that goeslike U when U  . Hence, in this trivially soluble t  0limit, we clearly see that the interaction U suppresses thegap of (the state adiabatically connected to) the bandinsulator all the way down to zero, but only at a single 0 1 2 3 4 5 6 7 0  0.5  1  1.5  2 0  2  4  6  0   0.02   0.04   U/t∆/tU/t∆/tBandInsulatorMottInsulatorBIMIMetalMETALFIG. 1. The T  0 phase diagram of model of Eq. (1) at halffilling obtained using DMFT for the Bethe lattice. The phaseboundary Uc1 is denoted by open circles while Uc2 by filledcircles. Inset: Detail of lower left-hand corner of the phasediagram.PRL 97, 046403 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending28 JULY 20060031-9007=06=97(4)=046403(4) 046403-1 © 2006 The American Physical Societypoint U  2. A question of considerable interest iswhether this metallic point obtained in the atomic limitcan broaden into a metallic phase when t is nonzero.Here we address this question using the DMFT ap-proach. The DMFT approximation is exact in the limit oflarge dimensionality [1,2] and has been demonstrated to bevery successful in understanding the metal-insulator tran-sition [1,2] in the usual Hubbard model, which is the  0 limit of Eq. (1). We focus in this Letter on the paramag-netic sector of Eq. (1), for which it is convenient to in-troduce the matrix Green’s function G^ k; i!n  Ak; i!n kk Bk; i!n 1; (2)where ,  are sublattice (A, B) indices, k belongs to thefirst Brillouin Zone (BZ) of one sublattice, i!n  2n1T, and T is the temperature. The kinetic energy isdescribed by the dispersion k and AB 	 i!n ABi!n. Within the DMFT approach the self-energy is purely local [1]. Thus the diagonal self-energiesi!n are k independent and the off-diagonal self-energies vanish (since the latter would couple the A andB sublattices).We note that within the Hartree-Fock (HF) approxima-tion the self-energy is HF Uhni=2 and the difference infilling factors of the two sublattices n  hnBi  hnAi=2is given by the T  0 self-consistent equation n  Un=2PkE1k , where Ek 2k  Un=22qis theHF excitation spectrum. Thus, within HF theory there is agap (Un=2) for all   0 and the system is a bandinsulator for all U. (For   0, one gets n  0 and thesystem is a metal for all U). However, a much richer,qualitatively different phase diagram emerges when weinclude fluctuations beyond HF, even at the level of theDMFT.The DMFT approach includes local quantum fluctua-tions by mapping [1,2] the lattice problem onto a single-site or ‘‘impurity’’ with local interaction U hybridizingwith a self-consistently determined bath as follows.(i) We start with a guess for ! and n and computethe local Gi!n PkGk; i!n rewritten as Gi!n    i!nZ 11d	0Ai!nBi!n  2(3)with   AB and   BA, where 	0 is the baredensity of states (DOS) for the lattice considered (seebelow). We actually need to solve the problem for onlyone sublattice and use the relations GAi!n GBi!n and Ai!n  U Bi!n valid at halffilling. (ii) We next determine the ‘‘host Green’s function’’[1,2] G0 from the Dyson equation G10 i!n G1 i!n  i!n. (iii) We solve the impurity problemto obtain i!n  G0i!n. (iv) We iterate steps(i), (ii), and (iii) until a self-consistent solution is obtained.We use as our ‘‘impurity solver’’ in step (iii) a general-ization of the iterated perturbation theory (IPT) [1,11]scheme which has the merit of giving semianalytical re-sults directly in the real frequency (!  ! i0) do-main. The IPT ansatz IPT !  HF  A2 ! isconstructed to be (a) exact for U=t  1, (b) exact fort=U  0, and (c) exact in the large ! limit for all U=t,which imposes various exact sum rules. Here HF is the HFself-energy with n  2R01 ImG!d!=, and 2 !  U2Z 11Y3i1di~	i N1; 2; 3!  1  2  3 :(4)This has the form of the second order self-energy with~	i  Im~G0i =, where ~G10 ! G10 !  HF is the Hartree corrected host Green’sfunction and N1;2;3f1f2f3f1f2f3, where f is the Fermi function. From con-dition (c) above we find that An1n=2=n01n0=2 with n0  2R01 Im~G0!d!=. Note thatA is same for both the sublattices. The results of thissimple approximate method are expected to be in semi-quantitative agreement with those of more exact but nu-merically intensive methods [1,2,12]. For simplicity, herewe present the results for the T  0 solution of DMFTequations on a Bethe lattice of connectivity z ! 1. Thehopping amplitude is rescaled as t ! t= zp to get a non-trivial limit and the bare DOS is then given by 	0 4t2  2p=2t2 which greatly simplifies the integral inEq. (3). We have also solved the DMFT equations on the2D square lattice and found that the results are qualitativelythe same [13]. 0  0.4  0.8 -2 -1.5 -1 -0.5  0  0.5  1  1.5  2ρ(ω)ωBand Insulator U=0.1t 0  0.4  0.8 -4 -2  0  2  4ρ(ω)ωMetal U=3.25t 0  0.4  0.8 -6 -4 -2  0  2  4  6ρ(ω)ωMott Insulator U=7.5tFIG. 2. The density of states 	! (in units of 1=t) plotted as afunction of ! (in units of t). The three panels are all at   0:1tfor various values of U one for each phase. The gaps are clearlyvisible in the top (band insulator at U  0:1t) and bottom (Mottinsulator at U  7:5t) panels, while the middle panel for metallicstate at U  3:25t shows gapless excitations.PRL 97, 046403 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending28 JULY 2006046403-2The phase diagram of Fig. 1 has been obtained from theGreen’s function, and the (renormalized) density of states(DOS), given by 	!  PkImTrG^k; !=, calcu-lated using the above procedure. Although the sublatticeDOS 	 are not individually symmetric, 	B! 	A! and thus 	!  	A!  	B! is symmetricfor half filling. Figure 2 shows how 	! evolves as afunction of U for a fixed . At small U <Uc1 there is a gapin the spectrum, with the DOS at higher energies similar tothe noninteracting (semicircular) result. We call this state aband insulator since it is adiabatically connected to theU  0 band insulator. At intermediate U the gap collapsesto zero [14], and we find a metallic state. The effects ofcorrelations also show up in the precursors to the upper andlower Hubbard bands at higher energies. At sufficientlylarge U >Uc2, the DOS again shows a gap and the systemis a Mott insulator (adiabatically connected to the well-studied Mott insulator at   0). The metallic phase,sandwiched between Uc1 and Uc2 in Fig. 1, shrinksas  increases, consistent with there being a single metallic‘‘point’’ U  2 in the atomic limit. We return below tothe question of the scale of the critical Uc’s as a function of.The difference in filling n  nB  nA=2 and theenergy gap are plotted in Figs. 3 and 4 as functions of Ufor fixed . For n the DMFT results are qualitativelysame as the HF results, though the n values within DMFTare much smaller than the HF results and become verysmall (of the order of error in evaluating n) for largevalues of U [15]. On the other hand, the DMFT result forthe energy gap is qualitatively very different from thesimple HF result (monotonically decreasing, nonzero forall U), and vanishes continuously [14] at both metal-insulator transitions, at Uc1 and Uc2.Further insight into the scale of the gap and its non-monotonic variation with U is obtained by examining theself-energy !  0!  i00!. From Fig. 5 wesee that 00A! vanishes for j!j  3Egap in both the insu-lating phases. This can be understood from the imaginarypart of 2 of Eq. (4), which comes from a three fermionfinal state. In the metallic phase 00A! !2 near !  0,characteristic of a Landau Fermi liquid. In all the threephases, 0A! can be written at low ! as a Taylor expan-sion 0!  00  1 Z1! . . . , where Z canbe shown to be independent of . The spectral functionis defined by A;!  1= ImG;!. In theinsulating phases, where 00  0 for j!j  3Egap, we findfrom Eq. (2) that A;!  r!  2 with r! !0A!!  0B!. As  isreal, !’s which satisfy r!< 0 lie within the gap. Theenergy gap is then given by rEgap  0 which, using thelow-energy form of 0 given above and the Kramers-Kronig relation, leads to the result Egap  ZjUn=2 Sj; (5)where S  PR11 d!00A!=!. We note that the energygap obtained from the DOS (filled circles in Fig. 4) is inexcellent agreement with that given by Eq. (5) (opensquares in Fig. 4), where the error bars on the latter areobtained from the estimated numerical error in 00 at small! [14]. In particular we have checked very carefully thatthe gap indeed vanishes in the entire metallic phase withinthe limits of our numerical accuracy.We note that Eq. (5) helps greatly in clarifying severalfeatures of our results [13]. First, it shows explicitly howcorrelations ‘‘screen’’ the one-body potential  respon-sible for the gap in the band insulator and lead to a gapwhich is always less than the HF result in the band insulat-ing phase, The dominant role in this being played by S<0and by Z<1. Second, we estimate the location Uc1 of theband insulator to metal transition using Egap  0, whichimplies U  2j SUj=nU. We find from our nu-merics that although jSUj increases with U, it is alwaysmuch smaller than . Thus Uc1 ’ 2=nUc1 2=n0, since nU decreases with increasing U (see 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 1  1.5  2  2.5  3  3.5  4  4.5δnU/t∆/t=0.025DMFTHFFIG. 3. Difference in filling factor n  nB  nA=2 for  0:025t plotted as a function of U=t: The DMFT result (filledcircles) is much smaller than the HF result (triangles). 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 1  1.5  2  2.5  3  3.5  4  4.5  5GapU/t∆/t=0.025EgapDMFTHFFIG. 4. The energy gap for   0:025t plotted as a function ofU=t. The DMFT gap (dots) from the DOS and Egap (opensquares) from Eq. (5) are in excellent agreement. Note that thegap vanishes in the metallic phase at intermediate U’s. (See textfor discussion of error bars). In contrast, the HF results (tri-angles) do not show a metallic phase.PRL 97, 046403 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending28 JULY 2006046403-3Fig. 3). Thus for small , we find that the very small nleads to Uc1  ; e.g., for   0:01t we get Uc1 100 ’ t, while for   0:025t, Uc1  20 ’ 0:5t.Finally, the eventual increase in the gap with U in theMott phase, can be also traced to the U dependences ofZ and S in Eq. (5).In the metallic phase [13] the low-energy spectral func-tion A;!  ImTrG^;! ’ Z! Z. Thusthe Fermi surface is given by   0, i.e., the same as forthe metal with U    0, as a consequence of the kindependence of DMFT self-energy. In this phase Z hasthe meaning of quasiparticle residue, which is also theinverse mass renormalization. Within the metallic phase, Zdecreases with increasing U for a given , and withincreasing  for fixed U.In conclusion, we have analyzed a simple model of aband insulator with on-site Coulomb repulsion U. At theHartree-Fock level the gap is reduced but the insulatingbehavior persists. However, when we treat correlationeffects using DMFT we find the surprising result that thegap is suppressed to zero and there is a band insulator tometal transition at a critical Uc1. At a larger Uc2 there is asecond transition from the metal to a Mott insulator.Our work raises several important questions. First, ourresults are obtained in the paramagnetic sector. We expectthat the metallic phase will survive if antiferromagnetism[13] is suppressed due to frustration, in much the same wayas in the DMFT treatment in the   0 limit [1]. It wouldbe interesting to treat a model in which such frustration isexplicitly included. Second, it is important to ask if in finitedimensions, i.e., with k-dependent self-energies, otherbroken symmetry states might also appear, for example,the bond ordered state proposed as an intermediate phase inlow dimensional studies [8–10]. Finally, it would be mostinteresting to look for experimental systems, either tran-sition metal oxides [16] or fermions in optical lattices [17],where increasing correlations could drive a band-insulatormetallic. All these are questions for future work.M. R. would like to thank N. Trivedi for very usefulconversations.[1] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,Rev. Mod. Phys. 68, 13 (1996).[2] T. Pruschke, M. Jarrell, and J. K. Freericks, Adv. Phys. 44,187 (1995).[3] This can be shown by using particle-hole symmetry at halffilling.[4] J. Hubbard and J. B. Torrance, Phys. Rev. Lett. 47, 1750(1981).[5] T. Egami, S. Ishihara, and M. Tachiki, Science 261, 1307(1993).[6] G. Ortiz and R. Martin, Phys. Rev. B 49, 14 202 (1994).[7] R. Resta and S. Sorella, Phys. Rev. Lett. 74, 4738 (1995);82, 370 (1999).[8] M. Fabrizio, A. O. Gogolin, and A. A. Nersesyan, Phys.Rev. Lett. 83, 2014 (1999).[9] T. Wilkens and R. M. Martin, Phys. Rev. B 63, 235108(2001).[10] C. D. Batista and A. A. Aligia, Phys. Rev. Lett. 92, 246405(2004).[11] H. Kajueter and G. Kotliar, Phys. Rev. Lett. 77, 131(1996).[12] R. Bulla, T. A. Costi, and D. Vollhardt, Phys. Rev. B64,045103 (2001).[13] Further details will be published elsewhere; A. Garg, H. R.Krishnamurthy, and M. Randeria (unpublished).[14] The very low-energy results for the DOS and 00 arehighly susceptible to numerical errors, particularly in themetallic phase. We have checked that our results arerobust under increased numerical accuracy. In many caseswe have also compared results using two different tech-niques (fast Fourier transform and direct numerical inte-gration). This forms the basis for error bars on Egap inFig. 4. Within our numerical accuracy, we have found noevidence so far for coexisting phases or hysteretic behav-ior for nonzero .[15] For U  t, , the leading contribution to the energy ofmodel (1), comes from a Heisenberg term with super-exchange J  4t2U=U2  42 and we can show thatn t2=U3. Although the n calculated using DMFTIPT need not agree with this exact asymptotic form indetail, we nevertheless expect that it is nonzero for largeU=t values, where we cannot reliably estimate n fromour numerics.[16] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys.70, 1039 (1998).[17] M. Kohl, H. Moritz, T. Stoferle, K. Gunter, andT. Esslinger, Phys. Rev. Lett. 94, 080403 (2005).-0.0001-6x10-05-2x10-05 -0.4 -0.2  0  0.2  0.4Σ'' A(ω)ωBand InsulatorU=0.1t-0.006-0.004-0.002 0-0.1 -0.05  0  0.05  0.1Σ'' A(ω)ωU=3.25tMetal-3x10-06-2x10-06-1x10-06 0-0.4 -0.2  0  0.2  0.4Σ'' A(ω)ωU=7.5tMott InsulatorFIG. 5. Imaginary part of self-energy 00A! plotted as afunction of !, where both are measured in units of t. All threepanels are for   0:1t for various values of U. In the top (bandinsulator at U  0:1t) and bottom (Mott insulator at U  7:5t)panels 00A! vanishes identically for low energies j!j< 3Egap.The metallic phase in the middle panel at U  3:25t has aFermi-liquid-like 00A! !2.PRL 97, 046403 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending28 JULY 2006046403-4

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Can Correlations Drive a Band Insulator Metallic?

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Can correlations drive a band insulator metallic?

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