Recent experiments have emphasized that our understanding of the interplay of
electron correlations and randomness in solids is still incomplete. We address
this important issue and demonstrate that particle-hole (ph) symmetry plays a
crucial role in determining the effects of disorder on the transport and
thermodynamic properties of the half-filled Hubbard Hamiltonian. We show that
the low-temperature conductivity decreases with increasing disorder when
ph-symmetry is preserved, and shows the opposite behavior, i.e. conductivity
increases with increasing disorder, when ph-symmetry is broken. The Mott
insulating gap is insensitive to weak disorder when there is ph-symmetry,
whereas in its absence the gap diminishes with increasing disorder.Comment: 4 pages, 4 figure

A. M. Finkel'stein

A. M. Goldman

D. L. Shepelyansky

E. Abrahams

F. J. Wegner

G. Blatter

M. A. Paalanen

M. P. A. Fisher

M. Ulmke

N. Trivedi

P. J. H. Denteneer

P. W. Brouwer

R. Gade

R. N. Bhatt

R. T. Scalettar

S. R. White

T. Senthil

T. Vojta

X. Waintal

Y. Otsuka

English

arXiv

Denteneer, P. J. H.

Scalettar, R. T.

Trivedi, N.

Eldorado - Ressourcen aus und für Lehre, Studium und Forschung

Particle-Hole Symmetry and the Effect of Disorder on the Mott-Hubbard Insulator

The understanding of the interplay of electron correlations and randomness in solids is enhanced by demonstrating that particle-hole ( p−h) symmetry plays a crucial role in determining the effects of disorder on the transport and thermodynamic properties of the half-filled Hubbard Hamiltonian. We show that the low-temperature conductivity decreases with increasing disorder when p−h symmetry is preserved, and shows the opposite behavior, i.e., conductivity increases with increasing disorder, when p−h symmetry is broken. The Mott insulating gap is insensitive to weak disorder when there is p−h symmetry, whereas in its absence the gap diminishes with increasing disorder.Theoretical Physic

Denteneer, P.J.H.

Scalettar, R.T.

Leiden University Scholary Publications

VOLUME 87, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 1 OCTOBER 200114Particle-Hole Symmetry and the Effect of Disorder on the Mott-Hubbard InsulatorP. J. H. DenteneerLorentz Institute, Leiden University, 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 400005, India(Received 5 June 2001; published 18 September 2001)The understanding of the interplay of electron correlations and randomness in solids is enhancedby demonstrating that particle-hole (p-h) symmetry plays a crucial role in determining the effects ofdisorder on the transport and thermodynamic properties of the half-filled Hubbard Hamiltonian. Weshow that the low-temperature conductivity decreases with increasing disorder when p-h symmetry ispreserved, and shows the opposite behavior, i.e., conductivity increases with increasing disorder, whenp-h symmetry is broken. The Mott insulating gap is insensitive to weak disorder when there is p-hsymmetry, whereas in its absence the gap diminishes with increasing disorder.DOI: 10.1103/PhysRevLett.87.146401 PACS numbers: 71.10.Fd, 71.30.+h, 72.20.– iIntroduction.—The interplay of disorder and interac-tions is at the heart of many interesting and unexplainedphenomena in condensed matter physics. For example,the effects of disorder and interactions in two-dimensional(2D) electronic systems acting separately lead to insulat-ing behavior of the Anderson and Mott kinds, respectively.However, experimental findings on silicon metal-oxide-semiconductor field-effect transistors (MOSFETs) showthe occurrence of a conducting phase [1], which is the re-sult of the combined importance of randomness and inter-actions [2,3]. Other examples of situations in which bothdisorder and interactions are crucial, yet incompletely un-derstood, include the formation of local moments and thebehavior of the susceptibility in doped semiconductors [4],the superconductor-insulator transition and universal con-ductivity in thin metallic films [5,6], and the pinning offlux lines in type-II superconductors [7].In recent years, it has become increasingly clear that fornoninteracting electrons the presence or absence of certainsymmetries is crucial in determining the effect of disorderon both transport and thermodynamic properties, as well ascritical properties of the localization transition [8]. Recentexamples where symmetry considerations are importantare given in the context of quantum wires [9] and disor-dered superconductors [10,11], where chiral, time-reversal,and spin-rotation symmetries play an important role.In this paper, we examine the effect of different typesof disorder on both the dynamic and equilibrium thermo-dynamics of the 2D Hubbard model in the vicinity of halffilling, electron density n  1. Our results suggest thatthe presence or absence of particle-hole symmetry deter-mines the effect of randomness on the conductivity and theMott gap.The model and computational approach.—We considerthe following 2D Hubbard Hamiltonian,6401-1 0031-90070187(14)146401(4)$15.00H  2Xij,stijcyiscjs 2Xik,st0ikcyiscks1 UXjµnj" 212∂ µnj# 212∂2Xj,smjnjs . (1)Here tij is a bond-dependent hopping matrix element onnearest-neighbor sites ij, t0ik is a bond-dependent hop-ping matrix element on next-nearest-neighbor sites ik,U is an on-site repulsion, and mj is a site-dependentchemical potential. We choose Ptij  1Dt for tij [t 2 Dt2, t 1 Dt2, and zero otherwise, with t  1 toset our scale of energy. Similarly, Pt0ik  1D0t for t0ik [t0 2 D0t2, t0 1 D0t2, and Pmj  1Dm for mj [2Dm2, 1Dm2, so that the various D measure thedisorder strength. We will focus on half filling where theeffects of interactions are most prominent, as evidencedby the formation of antiferromagnetic correlations and aMott-Hubbard charge gap at low temperatures.Our computational technique is determinant quantumMonte Carlo (QMC) [12], an approach which allows us tostudy much larger numbers of particles than those exploredwith exact diagonalization (and the two-electron problem)[13,14]. In this method the electron-electron interactionsare replaced by a space- and imaginary-time-dependentHubbard-Stratonovich field. The integral over possiblefield configurations, which exactly retains the interactionsin the problem, is done stochastically and allows us tocalculate static and dynamic (Matsubara time) correlationfunctions at a fixed temperature T . All our results are forlattices of 8 3 8 spatial sites and coupling U  4t. Weaveraged over up to 16 disorder realizations; the error barsindicate the statistical fluctuations from disorder sampling.The disordered Hubbard model in Eq. (1) is particle-hole (p-h) symmetric when t0ik  mj  0. That is, under© 2001 The American Physical Society 146401-1VOLUME 87, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 1 OCTOBER 2001the transformation cyis! 21icis the Hamiltonian is un-changed, and the system is precisely half filled for all val-ues of the parameters in H and also for all T . Therefore,while nearest-neighbor bond and local site disorder bothintroduce randomness into the system, they differ funda-mentally in that site disorder breaks p-h symmetry.In order to reveal the effects of disorder quantita-tively, we will examine the transport by evaluating thetemperature-dependent dc conductivity,sdc b2pLxxq  0, t  b2 (2)(with b 	 1kBT) as determined [15] from the current-current correlation function, Lxxq, t  jx q, tjx2q, 0. Here jxq, t, the q, t-dependentcurrent in the x direction, is the Fourier transform ofjxl  iPs tl1x̂,lcyl1x̂,scls 2 cylscl1x̂ ,s. From theone-electron Green function as a function of imaginarytime we extract the temperature-dependent density ofstates at the chemical potential Ne  0 [16]:N0  2bGr  0, t  b2p . (3)These two quantities allow a clear characterization of thetransport and thermodynamic properties of the system. Forsdc and N0, “Trotter” errors associated with the dis-cretization of imaginary time b are considerably less thanthe fluctuations associated with Monte Carlo and disorderaveraging.Results.— First we discuss the transport properties: inFig. 1, we exhibit the effect of nearest-neighbor hopping(bond) disorder on the conductivity. For all disorderstrengths Dt , at temperatures greater than a characteristictemperature T related to the Mott gap, the system showsmetallic behavior with sdc increasing upon lowering T .The conductivity turns down sharply as the temperatureFIG. 1. The effect of particle-hole-symmetry preserving(nearest-neighbor) bond disorder in the half-filled HubbardHamiltonian is to decrease the conductivity sdc. Data are forU  4t on a 8 3 8 square lattice; Dt measures the strength ofthe bond disorder.146401-2drops below T and the system shows insulating behaviorwith sdc decreasing upon lowering T . In the case of zerorandomness, the perfect nesting of the Fermi surface in 2Dleads to antiferromagnetic long range order (AFLRO) inthe ground state with an associated spin density wave gapfor arbitrarily small U evolving to a Mott gap at larger U.Hopping disorder reduces AFLRO via the formation ofsinglets on bonds with large hopping tij and hence largecoupling J  t2ijU and ultimately destroys it beyondDt  1.6t [17]. The fascinating result we have found isthat insulating behavior in the conductivity neverthelesspersists to much larger Dt. Moreover, from the shift ofthe maximum in Fig. 1 we deduce that the mobility gapin fact increases with increasing Dt.The situation is quite different in the case of site dis-order, as shown in Fig. 2: at fixed temperature T , as sitedisorder Dm is turned on, the conductivity increases; i.e.,the Mott insulating state is weakened [18]. At weak dis-order, the conductivity drops with decreasing T , reflect-ing again the presence of the Mott insulating phase. Asthe disorder strength becomes large enough to neglect U,one would expect a similar temperature dependence aris-ing from Anderson insulating behavior. We believe that inall cases the conductivity will ultimately turn over and goto zero at low T , but we are limited in these simulationsto temperatures T . W48 because of the fermion signproblem. Nevertheless, the data for site disorder offer adramatic contrast to that of bond disorder (Fig. 1) whererandomness decreases the conductivity.What is the underlying reason for the different effects ofbond and site disorder on conductivity? There are severalobvious differences in the effect of bond and site disorderon local and even longer range spin and charge correla-tions. Site disorder enhances the amount of double occu-pancy on the lattice, since the energy cost U of doubleFIG. 2. Canonical site disorder (with strength Dm) enhancesthe conductivity. Particle-hole–symmetric site disorder (withstrength D0m), as with bond disorder (Fig. 1), suppresses theconductivity. Other parameters are as in Fig. 1.146401-2VOLUME 87, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 1 OCTOBER 2001occupancy is compensated by differences in site energies.One explanation of why site disorder increases sdc is thatthe concomitant increase in empty sites leads to more mo-bility. This destruction of local moments ultimately alsoleads to the end of antiferromagnetic order. Surprisingly,we find in our simulations that bond disorder has a similardiminishing effect on local moments, suggesting that thedifference in the behavior of the conductivity arises froma different origin.We argue here that particle-hole symmetry is the uni-fying criterion which underlies and determines the effectof disorder. As emphasized above, site and bond disor-der have rather similar effects on the double occupancy.Moreover, the consequences of this effect for sdc are ex-pected to become visible only above a threshold value ofdisorder strength, whereas we observe effects on sdc al-ready for weak disorder. Instead, the key distinction is inthe presence or absence of p-h symmetry. In order to ex-plore this conjecture more fully, we have studied two othertypes of disorder: site disorder that preserves p-h symme-try and bond disorder that breaks p-h symmetry (by in-cluding next-nearest-neighbor hopping).Particle-hole symmetric site disorder is introduced byadding random chemical potentials to the Hubbard modelwhich couple with opposite sign to the density of up anddown electrons, i.e., choose mj 	 mjs  smj in (1).This type of disorder represents a random (Zeeman) mag-netic field. For U  0 p-h symmetric site disorder hasprecisely the same effect as conventional site disorder,since moving in a given random chemical potential land-scape or one obtained by reversing all the site energies isentirely equivalent. However, the behavior of the conduc-tivity at finite U is dramatically different. Figure 2 showsthat p-h symmetric site disorder (with strength D0m) hasthe same effect on sdc as bond disorder, i.e., conductivitydecreases with increasing D0m.To seek final confirmation of our conjecture, we havealso explored the effect of next-nearest-neighbor (nnn)hopping and randomness therein. Such longer ranging hy-bridization breaks p-h symmetry on a square lattice, sinceit connects sites on the same sublattice. We find that suchdisorder has the same effect as conventional site random-ness, i.e., increases the conductivity at finite T . Thus inall four types of disorder, the behavior of the conductivityfalls into the appropriate class based on the preservation ordestruction of p-h symmetry, strengthening the evidencethat it is this symmetry which is playing the crucial rolein determining the effect of randomness on the transportproperties.We now turn to thermodynamic properties. The mostdirect measure of the Mott gap is from the compressibil-ity, or from the behavior of density n as a function ofchemical potential m, as shown in Fig. 3. The range of mwhere n is constant (and the system is incompressible) isa direct measure of the gap in the spectrum. Hopping andp-h–symmetric site disorder clearly stabilize the plateauof the density at half filling. On the other hand, conven-146401-3FIG. 3. The Mott gap is made more robust by the additionof bond disorder or particle-hole–symmetric site disorder (openand filled squares) of strength D  2t  U2, as indicated bythe response of the density to changes in the chemical poten-tial. For canonical site disorder (filled circle) the Mott gap ispractically unaffected by this strength of randomness [19]. Cal-culations are for T  t8  W64 on a 8 3 8 lattice.tional site disorder (with Dm  U2) has a compressibilitywhich is indistinguishable (within the computational pos-sibilities) from the clean system.The density of states (DOS) at the Fermi level N0gives valuable information on the effect of disorder on theMott gap. In the pure system, QMC studies have shownthat the DOS exhibits a clear Mott gap with N0 ! 0as T is lowered to zero. The nonzero values of N0 weobtain at nonzero T reflect the small residual slopes in theplateaus in the n vs m plot (cf. Fig. 3); at lower T , N0approaches zero just as the plateaus become perfectly flat.The behavior of N0 at a fixed low T as a function of thestrength of the various types of disorder is given in Fig. 4.N0 is rather insensitive to p-h symmetric disorder (Dtand D0m) and is even reduced by it: the Mott gap persists.On the other hand, p-h symmetry breaking disorder (Dmand D0t) clearly enhances N 0, i.e., fills up the Mott gap.Our results provide a clear numerical demonstration ofthe key role of particle-hole symmetry. The effects canalso be understood qualitatively as follows: In the cleancase, at n  1 and strong coupling, the DOS consists ofan occupied lower Hubbard band (LHB) and an unoccu-pied upper Hubbard band (UHB), separated by a chargegap of the order of U. In the case of p-h–symmetric dis-order, the effect of disorder on LHB and UHB is identical.Therefore the Fermi energy remains in the middle of thegap: this enables the insulating behavior and Mott gap tostay intact. A stabilized charge gap for p-h–symmetricsite disorder is evident since double occupation is stronglysuppressed. For nn-hopping disorder a simple argument isless obvious, but the data in Fig. 3 clearly show that thesetwo cases fall into the same class. When p-h symmetry isbroken, the LHB and the UHB will be affected differently;146401-3VOLUME 87, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 1 OCTOBER 2001FIG. 4. Behavior of the density of states pN0 at the Fermilevel and at fixed low temperature as a function of disorderstrength Dt for various types of disorder. All data are forT  t6, except data for randomness in next-nearest-neighborhopping (disorder strength D0t) which are at temperature T t5 (the value t0  0 is used) [19]. Other parameters are as inFigs. 1 and 2.different numbers of states will appear at either side of thegap. As a consequence, the Fermi energy ends up in oneof the tails of the DOS, resulting in an enhanced N0(cf. Fig. 4) and increased conductivity (Fig. 2). The factthat the states introduced by disorder are localized [20] willkeep the system in an insulating state (cf. Fig. 2).Conclusions.— In this Letter, we have shown that p-hsymmetry plays a decisive role in determining the effect ofrandomness on transport and thermodynamic properties ofthe half-filled Hubbard model. By exploring four differenttypes of disorder, of which two preserve p-h symmetry andtwo break it, we demonstrate that a classification by thissymmetry allows us to understand the effect of disorderon sdcT , the charge gap, and the compressibility. Thepresence of p-h symmetry is found to have a protectiveinfluence on the charge gap.A related example where symmetry plays a crucial rolein the effects of disorder is the case of localization in thesuperconducting phase, where the quasiparticles are de-scribed by a Bogoliubov–de Gennes Hamiltonian [11]. Inthis case, one can classify the system according to the pres-ence or absence of time reversal and spin rotation symme-tries, and it is found in one dimension that in the absenceof spin rotation symmetry, the conductance decays alge-braically with system size, while in the symmetric case itdecays exponentially. Therefore, in this situation as well,the extra spin rotation symmetry leads to a strengtheningof insulating behavior.The question of the behavior of the half-filled fermionHubbard model as disorder is added is furthermore remi-niscent of similar issues in the p-h–symmetric boson Hub-bard model [6]. At generic densities, it is believed that a146401-4new “Bose glass” phase arises to intervene in the originalground state phase diagram between superfluid and Mottinsulating phases, but the situation at the p-h–symmetrictip of the Mott lobe is uniquely different. Our work is afirst step in the analysis of the nature of the behavior of thefermionic model.The authors thank Piet Brouwer, Andreas Ludwig, andGeorge Sawatzky for helpful discussions. The research ofR. T. S. is supported by Grant No. NSF-DMR-9985978.[1] E. Abrahams, S. V. Kravchenko, and M. P. Sarachik, Rev.Mod. Phys. 73, 251 (2001), and references therein.[2] A. M. Finkel’stein, Zh. Eksp. Teor. Fiz. 84, 168 (1983)[Sov. Phys. JETP 57, 97 (1983)].[3] P. J. H. Denteneer, R. T. Scalettar, and N. Trivedi, Phys.Rev. Lett. 83, 4610 (1999).[4] R. N. Bhatt and P. A. Lee, Phys. Rev. Lett. 48, 344 (1982);M. A. Paalanen, S. Sachdev, R. N. Bhatt, and A. E. Ruck-enstein, Phys. Rev. Lett. 57, 2061 (1986); M. A. Paalanen,J. E. Graebner, R. N. Bhatt, and S. Sachdev, Phys. Rev. Lett.61, 597 (1988).[5] A. M. Goldman and N. Markovic, Phys. Today 51, No. 11,39 (1998).[6] M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S.Fisher, Phys. Rev. B 40, 546 (1989).[7] G. Blatter et al., Rev. Mod. Phys. 66, 1125 (1994).[8] F. J. Wegner, Z. Phys. B 35, 207 (1979); R. Gade and F. J.Wegner, Nucl. Phys. B360, 213 (1991).[9] P. W. Brouwer, C. Mudry, and A. Furusaki,cond-mat/0009198.[10] T. Senthil and M. P. A. Fisher, Phys. Rev. B 61, 9690(2000).[11] P. W. Brouwer, A. Furusaki, I. A. Gruzberg, and C. Mudry,Phys. Rev. Lett. 85, 1064 (2000), and references therein.[12] S. R. White, D. J. Scalapino, R. L. Sugar, E. Y. Loh, J. E.Gubernatis, and R. T. Scalettar, Phys. Rev. B 40, 506(1989).[13] T. Vojta, F. Epperlein, and M. Schreiber, Phys. Rev. Lett.81, 4212 (1998); X. Waintal, G. Benenti, and J.-L. Pichard,Europhys. Lett. 49, 466 (2000).[14] D. L. Shepelyansky, Phys. Rev. Lett. 73, 2607 (1994).[15] N. Trivedi, R. T. Scalettar, and M. Randeria, Phys. Rev. B54, R3756 (1996). The formula for sdc is expected to holdwhen T is lower than other energy scales in the problem(e.g., disorder).[16] N. Trivedi and M. Randeria, Phys. Rev. Lett. 75, 312(1995). The expression for N0 is valid at temperaturesbelow characteristic frequency scales of structures in Nv.[17] M. Ulmke and R. T. Scalettar, Phys. Rev. B 55, 4149(1997).[18] We have verified that beyond a certain large site disorder(Dm $ 4t) sdc decreases again.[19] Particle-hole symmetric disorder has no sign problems athalf-filling; data are better then, and simulations can go tolower T .[20] Y. Otsuka, Y. Morita, and Y. Hatsugai, Phys. Rev. B 58,15 314 (1998).146401-4

Particle-hole symmetry and the effect of disorder on the Mott-Hubbard insulator

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The understanding of the interplay of electron correlations and randomness in solids is enhanced by demonstrating that particle-hole ( p−h) symmetry plays a crucial role in determining the effects of disorder on the transport and thermodynamic properties of the half-filled Hubbard Hamiltonian. We show that the low-temperature conductivity decreases with increasing disorder when p−h symmetry is preserved, and shows the opposite behavior, i.e., conductivity increases with increasing disorder, when p−h symmetry is broken. The Mott insulating gap is insensitive to weak disorder when there is p−h symmetry, whereas in its absence the gap diminishes with increasing disorder.</p

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Particle-Hole Symmetry and the Effect of Disorder on the Mott-Hubbard
  Insulator

Particle-Hole Symmetry and the Effect of Disorder on the Mott-Hubbard Insulator

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