Table Of Content

Nonlinear screening theory of the Coulomb glass Sergey Pankov Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 Rue Lhomond, 75231 Paris Cedex 05, France Vladimir Dobrosavljević 5 Department of Physics and National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306 0 (Dated: February 2, 2008) 0 2 Anonlinearscreeningtheoryisformulatedtostudytheproblemofgapformationanditsrelation to glassy freezing in classical Coulomb glasses. We find that a pseudo-gap (”plasma dip”) in a n single-particledensityofstatesbeginstoopenalreadyattemperaturescomparabletotheCoulomb a energy. This phenomenon is shown to reflect the emergence of short range correlations in a liquid J (plasma) phase, a process which occurs even in the absence of disorder. Glassy ordering emerges 3 whendisorderispresent,butthisoccursonlyattemperaturesroughlyanorderofmagnitudelower. Our result demonstrate that the formation of the ”plasma dip” at high temperatures is a process ] n distinct from the formation of the Efros-Shklovskii (ES) pseudo-gap, which in our model emerges n only within theglassy phase. - s i d The interplayofinteractionsanddisorderremainsone To address these important issues in a controlled and . of the mostimportant open problems in condensed mat- precisefashion,wemakethe followingobservationwhich t a ter physics. These effects are most dramatic in disor- is the main physical point of this letter. We stress that m deredinsulators,where the pioneeringworkofEfrosand the principal ES result - the emergence of a power-law - Shklovskii (ES) [1] emphasized the fundamental signifi- spectrum - is not specific to low dimensions! Its phys- d n canceofthelong-rangednatureofCoulombinteractions. ical origin and its relation to high temperature anoma- o ThisworkpresentedconvincingevidencethatatT =0a lies can, therefore, be investigated by the theoretical ap- c soft”Coulombgap”emergesinthesingle-particledensity proaches controlled in the limit of high spatial dimen- [ of states (DOS) which, in arbitrary dimension d, reads sions. Such a theory for the Coulomb glass (CG) model 4 is presented in this letter. Our main conclusions are as v follows: (1) A non-universalpseudo-gap in the DOS (we 6 g(ε) εd−1. (1) call it the ”plasma dip”) begins to emerge at temper- 0 ∼ atures of the order of the Coulomb energy. It reflects 4 ¿From a general point of view this result is quite sur- 6 strong short range correlations in the Coulomb plasma, prising. It indicates a power-law distribution of exci- 0 a feature that is most pronounced in absence of disorder 4 tation energies, i.e. the absence of a characteristic en- [8], but is unrelated to the Coulomb gap of ES. We ob- 0 ergy scale for excitations above the ground state. Such tain simple analytical results that in quantitative detail / behavior is common in models with broken continuous t describe the temperature evolution of the plasma dip, a symmetry, where it reflects the correspondingGoldstone m in excellent agreement with all existing simulations. (2) modes, but is generally not expected in discrete symme- The high temperature plasma (fluid) phase becomes un- - try models, such as the one used by ES. Here, it may d stable to ergodicity breaking at temperatures typically reflect unusually strong frustration behavior inherent to n ten times lower, as the system enters a glassy state. We o Coulomb interactions in presence of disorder. arguethatatrueESpseudo-gapemergesonlywithinthe c Indeed, the ES model seems to display several glassy glassy (nonergodic) phase, and that its scale-invariant : v features characterized by a large number of meta-stable formreflectsthemarginalstabilityofsuchaglassystate. i states andslow relaxation,as clearly seenin many simu- X lations [2, 3, 4, 5], and even in some experiments [7]. In- Nonlinear screening theory. The simplest many-body r a terestingly,a precursorofthe gapbegins to appear[4, 6] approach to Coulomb systems is the well-known Debye- already at relatively high temperatures, while glassy or- Huckeltheory,whichprovidesalinearscreeningdescrip- dering emerges only at much lower T. Similar behavior tionequivalenttoaGaussianapproximationfortheplas- has been identified evenin absence of randomness [8]. Is monmode. Thisformulation,however,failsbadlyatlow thephysicsoftheCoulombgapthusrelatedorisitunre- temperatures,where nonlinear effects leadto strong cor- latedtotheglassyfeaturesofthesystem? Theclosecon- relations in the plasma phase. Evenworse,such a Gaus- nectionbetweenthetwophenomenawasrecentlydemon- sian theory is unable to describe glassy freezing even in strated [9] for a mean-field model of interacting disor- thewell-understoodmean-fieldlimitcorrespondingtoin- deredelectronsinthelimitoflargecoordination,butthe finite range interactions. To overcome these difficulties, issue remains unresolved for Coulomb systems in finite we use the simplest theory of nonlinear screening given dimensions where the ES theory applies. bytheclassicallimitoftheso-calledextendeddynamical 2 mean-field theory [10] , which also describes the leading where D[x] (2π)−1/2exp x2/2 dx. The self- ≡ {− } order nontrivial correlations in the limit of large coor- consistency condition becomes dination. In this approach, the environment of a given 1 site is approximated by free collective modes (plasmons χ+q = , in our case), the dispersion of which is self-consistently 4 determined. Inrecentwork,aversionofthismethodhas χ= χ−1+∆ βV −1, c k − beensuccessfullyappliedtotheproblemofself-generated Xk (cid:0) (cid:1) glassiness [11] in systems with frustrated phase separa- q = qχ−2 ∆ χ−1+∆ βV −2, (5) c k tion [12] without disorder. Here, we apply it to the clas- − − (cid:0) (cid:1)Xk (cid:0) (cid:1) sical Coulomb glass model [1] given by the Hamiltonian The Eqs. (4,5) can easily be solved numerically for the parameters χ,q,∆ ,∆,W , to calculate the den- c eff 1 H = φ n + V (n K)(n K), (2) sity of states (DOS) function g(ǫ), to examine the sta- i i ij i j 2 − − Xi Xij bility of the RS solution, and to compute the entropy S. The above equations are written for the half filled where n = 0,1 is the electron occupation number, and caseinabsenceofchargeordering. Generalizationtouni- i φ isaGaussiandistributedrandompotentialofvariance formlyorderedphasesisastandardprocedure[13],where i W2. We expressthe CoulombinteractionVij =ε0/rij in the ordering transition is signaled by a divergence in units of the nearest-neighborrepulsionε0, and the inter- χk = χ−1+∆c βVk −1 at the orderingvector k =Q. − sitedistancerij inunitsofthelatticespacing. We adopt For si(cid:0)mplicity, most of(cid:1)our results are written in the ho- thefollowingnotationthroughoutthepaper. Forvectors mogeneousphaseathalf filling andthat willbe assumed and matrices in the replica space we use bold font and unless stated otherwise. thehatsymbolcorrespondingly. Inthereplicasymmetric Densityofstates. FortheCGmodel,thesingleparticle ansatz(RS)amatrixOˆ Oc,O isparameterizedbyits DOS (tunneling DOS) function g(ǫ) is simply given by ≡{ } connected part Oc and its off-diagonal part O. Thus for thedistributionofthelocalfields(energies)ǫi =∂H/∂ni: thedensitydensitycorrelatorweuseqˆ χ,q . Thermal ≡{ } and disorder averages are denoted as O and [O] T dis h i respectively, and O [ O ] . g(ǫ)= δ(ǫ ǫ ) . (6) h i≡ h iT dis h − i i Toderivethe desiredself-consistencyequationswe av- Xi erageoverdisorderusingthestandardreplicamethod[9], Integrating out all sites except one, we derive an expres- and use a cavity construction [9, 10], integrating out the degreesoffreedomonallsitesexcepttheconsideredone. sion for g(ǫ) in terms of the local effective field ∆ˆ˜. The The resulting contribution to the local effective action is final result for the RS solution reads: computed in the Gaussian approximation, giving a term tohfatthethfeorlmoca−l21dδenns∆îtδyn-d,ewnhseitryecδonrr=elant−orhqnαiβ. =Byδrneqαuδinrβingc g(ǫ)=Z D[x]√2βπ∆c coschos12hxβ21Wβǫeff h i is correctly reproduced by the effective action, one ob- 1 1 tains a self-consistency condition. We get ×exp(cid:26)−2∆ (cid:20)4∆2c +(βǫ+xβWeff)2(cid:21)(cid:27). (7) c qαβ =hδnαδnβicSeff, In the cavity method langauge[14] this result can be in- S = 1δn∆ˆ˜δn, terpretedastheGaussiandistributionofthecavityfields eff −2 (different from the local fields) of variance Weff, modi- −1 fied by the the (Onsager)self reactionterm representing qˆ= qˆ−1+∆ˆ βV , (3) Xk (cid:16) − k(cid:17) theWpelagsemtaincsoigrrhetlaitnitoonst.he behavior of the DOS by con- sideringsome analyticallysolvablelimits. AtT 0, the where ∆˜αβ =∆αβ +β2W2, and Vk is the Fourier trans- DOS remains finite at the Fermi level, though i→t can be form of the interactionpotential. In case of the RS solu- exponentially small for weak disorder. This shows that tion∆ˆ˜ ∆ ,β2W2 ,where W = W2+β−2∆ is ≡{ c eff} eff the fluid (RS) solution does not capture the physics of the renormalized disorder. p the true ES gap, which emerges only within the glassy Fluidsolution. Toexaminetheevolutionofthesystem phase. On the other hand, as long as the RS solution is in the high temperature phase we first examine the RS stable,alarge”plasmadip”maydevelop,butitwillhave solution. In the n 0 replica limit we get no direct relationto the glassy physics or the ES gap. It → reflects strong short-range correlations in the Coulomb 1 1 q = D[x]tanh2 xβW , (4) plasma (fluid) phase, which are suppressed at large dis- eff 4Z (cid:18)2 (cid:19) order. Here, the RS DOS reduces to the bare disorder 3 distribution. In the opposite limit of vanishing disorder W 0 the DOS expression simplifies and for an ar- eff → bitrary filling reads: dd==33 β 1∆2+β2ǫ2 g(ǫ)= exp 4 c √2π∆ (cid:18)− 2∆ (cid:19) c c 1 1 cosh βǫ (2K 1)sinh βǫ . (8) ×(cid:20) 2 − − 2 (cid:21) To support the validity of our formulation, we com- pare our analytical results with available numerical simulations. InFig. 1(toppanel), weexamine the situation studied in Ref.[6], where calculations were done for a 3D CG on a cubic lattice, for a set of temperatures in the fluidphase. Thelowesttemperatureisveryneartheglass transition temperature (see our phase diagram, Fig. 2). As we can see, our theory captures in surprisingly quantitative detail the formation of the ”plasma dip” in the fluidphase. Inthepast,thisphenomenonhasoftenbeen confused with the formation of the true ES gap which, aswearguebelow,onlyemergeswithinthe glassyphase. Finally, we test limits of our theory by computing the DOS for the 2D CG in absence of disorder. Even in this extremecase,wereproducesemi-quantitativelyexactnu- merical results of Ref. [8]. FIG.1: Ouranalyticalpredictionsforthesingle-particledensity of states (full lines) are found to be in excellent quanti- Glassy ordering. To examine the stability of the fluid tativeagreement with simulation results (dashed lines), with phasetoglassyordering,weexaminetheBaymKadanoff no adjustable parameters. Shown are results for the three (BK) functional ΓBK[qˆ]. This is a functional of the cor- dimensional case studied in from Ref. [6], corresponding to relator qˆ, which yields the exact equations of motion at W = 1/(2√3), and temperatures T = 0.4,0.2,0.1,0.05 (top the saddle point, where it coincides with the exact free panel), and the two dimensional model of Ref. [8], corre- energy. Toobtainourself-consistencyconditions,alocal sponding to W = 0, T = 0.1, K = 0.2. All lines correspond totheplasmaphase,while T =0.05isveryclose totheglass approximation[13]ismadeonthetwoparticleirreducible transition temperature. partofΓ . Inthisformulation,thestabilityofourfluid BK RS solution can be obtained by a standard replica symmetry breaking(RSB) analysis [15]of the BK functional exact value[17] T = .129) the system enters the charge at the saddle point. The corresponding RSB instability c ordered phase. phase. Stronger disorder suppresses the criterion takes the form chargeordering,and the system canexist either in a liq- 1 1 1 uid phase (at higher temperature) or in the glass phase + =0. (9) [χ2] − [χ ]2 [χ ]2 (at lower T). The liquid is separated from the glass by ii dis ii dis j ij dis P the RSB line, also known as the Almeida-Thouless [15] Here,χij isthedensity-densitycorrelationfunctioncom- line. We emphasize that the ordering temperature we puted for a fixed realization of disorder, i.e. χij = predict is roughly an order of magnitude smaller then δniδnj T δni T δnj T. The left hand side of Eq. (9) the Coulomb energy, in remarkable quantitative agree- his nothinig−buht 1/i hj[χ2iji]dis, the inverse of the glass sus- ment with all available simulation results [3, 4, 5, 8]. ceptibility, a quaPntity which diverges at the transition. Thisinterestingfactcanbetraceddowntothescreen- In terms of the RS solution the RSB condition reads ing of the Coulomb interaction. Indeed, the overall en- q 1 −4 1 ergyscalecharacterizingthescreenedCoulombpotential = D[x]cosh xβW . (10) ∆ 16Z (cid:18)2 eff(cid:19) Vscr(r)=ε0exp{−r/ℓscr}/r is roughly an order of magnitude smaller then the bare Coulomb energy. The cor- As anillustration,wepresentresults forthe CGona3D respondingscreeninglengthℓscr =[(χ−1+∆c)/(βε0)]1/2 cubic lattice, and in Fig. 2 we plot the corresponding (shown for W = (2√3)−1 in the inset of Fig. 2) de- phasediagramsobtainedbynumericallysolvingourself- creases (albeit weakly) with temperature, and remains consistency conditions. At small disorder and tempera- short throughout the fluid (RS) phase. This observation tureT .95(whichisinsatisfactoryagreementwiththe also makes it clear why the true ES gap cannot emerge ≈ 4 form of the T = 0 Coulomb gap in the glassy phase. At W ε0, we find ≫ 1+1 −1 TG ε0 γW γ, (12) ∼ where the exponent γ is identical to that predicted by RR EE ES for the T =0 DOS in the CG DD RR OO g(ǫ) ε−1−γǫγ. (13) E E ∼ 0 GG RR For an interaction of a general power law form V(r) AA ∼ CHCH ε0/ra, the ES argument predicts γ = (d−a)/a in arbi- trarydimensiond. Theasymptoticregime,however,sets in at larger values of disorder, not shown in the Fig.(2). Let us explain the importance of Eq. (12). We have seen that for W = 0, the DOS remains finite in the RS 6 fluid phase, so a true ES pseudo-gap can emerge only FIG.2: 3DCoulombglassphasediagram. Thefullhorizontal duetoglassyordering. Foranotherelectronglassmodel, line indicates the RSB instability and the dotted line shows work of Ref. [9] has established that the emergence of where the RS entropy turns negative. The screening length a true pseudo-gap at T = 0 directly follows from the in the inset is plotted for the same disorder and the range of temperatures (fluid phase), as in Fig.1. marginal stability of the glassy state, and that its form alsodeterminesthelargedisorderasymptoticsofT (W). G Giventhe closesimilarityofourmean-fieldequationsfor in absence of glassy ordering. The screening mechanism the CG model to those examined in Ref. [9], we expect remains operative throughout the (ergodic) fluid phase, the same mechanism to apply here as well. Using the and thus the long-range character of the Coulomb in- expected form of the ES gap, we can estimate the glass teraction, which is crucial for the ES argument remains transition temperature as the energy scale E charac- gap inoperative. In contrast, within the glass phase, follow- terizing the ”width” of the gap that opens in the low ing the arguments from Ref. [9], we expect the relevant 1+1 −1 temperature phase. We find Egap ε0 γW γ, coincid- zero-field cooled compressibility to decrease and vanish ∼ ing with Eq. (12). This result presents strong evidence at T =0. This mechanism opens a route for the screen- infavorofthe closerelationbetweenglassyorderingand ing to be suppressed at low temperatures, and the true the emergence of the ES gap. ES gap to emerge. Inconclusion,wehaveformulatedasimplemany-body To provide further evidence of the instability of the theory that is able to clarify the relation between the fi- fluid phase to glassy ordering, we also calculate the en- nite temperature formation of the Coulomb gap and the tropy in the fluid (RS) solution, which takes the form emergenceof glassyorderingin disorderedCoulombsys- tems in finite dimensions. This nonlinear screening ap- 1 proach is flexible enough to allow for future extensions S = D[x]ln 2cosh xβW eff Z (cid:20) (cid:18)2 (cid:19)(cid:21) to quantum models [9, 16], and to study the role of An- 1 1 derson and Mott localization [18] in Coulomb systems. + lnχ lnχ 2χβW . (11) 2 k− 2 − eff We thank G. Biroli, A. Georges, D. Grempel, M. Xk Muller, D. Popović, B. Shklovskii, T. Vojta, and G. Zi- It is well known that for standard mean-field glass mod- manyifor usefuldiscussions. This workatFSU wassup- els, the RS replica theory predicts negative entropy at ported by the NSF grant DMR-0234215. SP acknowl- T = 0, while the lower bound for the glass transition edges support from the CNRS, France. After this work temperature can be set where the RS entropy changes has been completed, the authors became aware of the sign. It is easy to show from Eq. (11) that our RS en- closely related calculation of M. Muller and L. B. Ioffe tropy proves strictly negative at T = 0 as well. In Fig. (cond-mat/0406324), where complementary results con- 2 we also plot (dashed line) a lower bound for the RSB sistent with our findings were presented. instabilitylinewheretheRSentropyturnsnegative,pro- viding further evidence that the fluid phase cannot sur- vive down to T =0. The glass phase and the Efros-Shklovskii gap. In this [1] A.L.EfrosandB.I.Shklovskii,J.Phys.C8,L49(1975). letter we do not explicitly examine the RSB solution of [2] J. H.Davies, et al.,Phys. Rev.Lett.49, 758 (1982). our model. Nevertheless, we follow arguments similar to [3] A. Möbius, et al.,Phys. Rev.B 45, 11568 (1992). those of Ref. [9], and use the large disorder asymptotics [4] E. R. Grannan and C. C. Yu, Phys. Rev. Lett. 71, 3335 of the glass transition line to determine the powerlaw (1993). 5 [5] D.Grempel, Europhys.Lett. 66, 854 (2004). [12] H. Westfahl, et al., Phys. Rev. B 68, 134203 (2003); S. [6] M. Sarvestani, et al., Phys.Rev.B 52, R3820 (1995). Wu, et al.,cond-mat/0305404. [7] A. Vaknin et al., Phys. Rev. Lett. 81, 669 (1998); ibid., [13] R. Chitra, and G. Kotliar, Phys. Rev. B 63, 115110 84, 3402 (2000); S. Bogdanovich and D. Popović, Phys. (2001);S.Pankov,etal.,Phys.Rev.B66,045117(2002). Rev.Lett. 88, 236401 (2002). [14] M. Mezard, et al.,Europhys.Lett. 1, 77 (1986). [8] A.L. Efros, Phys. Rev.Lett. 68, 2208-2211 (1992). [15] J. R. L. de Almeida, D.J. Thouless, J. Phys. A 11, 983 [9] A.A.Pastor,andV.Dobrosavljević,Phys.Rev.Lett.83, (1978). 4642 (1999). [16] F. Epperlein et al., Phys. Rev. B 56, 5890 (1997). [10] J.L.Smith,andQ.Si,Phys.Rev.B61,5184(2000);R. [17] A. Möbius and U.K. Rößler, cond-mat/0309001. Chitra,andG.Kotliar,Phys.Rev.Lett.843678(2000). [18] V. Dobrosavljević, et al., Phys. Rev. Lett. 90, 016402 [11] A. V. Lopatin, and L. B. Ioffe, Phys. Rev. B 66, 174202 (2003). (2002).