Effects of nanoscale spatial inhomogeneity in strongly correlated systems

M. F. Silva,1 N. A. Lima,2 A. L. Malvezzi,3 and K. Capelle4,*1Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05315-970 São Paulo, SP, Brazil

2Colegiado de Engenharia de Produção, Fundação Universidade Federal do Vale do São Francisco, Caixa Postal 252,56306-410 Petrolina, PE, Brazil

3Departamento de Física, Faculdade de Ciências, Universidade Estadual Paulista, Caixa Postal 473, 17015-970 Bauru, SP, Brazil4Departamento de Física e Informática, Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369,

13560-970 São Carlos, SP, BrazilsReceived 10 January 2005; published 29 March 2005d

We calculate ground-state energies and density distributions of Hubbard superlattices characterized by pe-riodic modulations of the on-site interaction and the on-site potential. Both density-matrix renormalizationgroup and density-functional methods are employed and compared. We find that small variations in the on-sitepotentialvi can simulate, cancel, or even overcompensate effects due to much larger variations in the on-siteinteractionUi. Our findings highlight the importance of nanoscale spatial inhomogeneity in strongly correlatedsystems, and call for a reexamination of model calculations assuming spatial homogeneity.

DOI: 10.1103/PhysRevB.71.125130 PACS numberssd: 71.10.Fd, 71.10.Pm, 71.15.Mb, 71.27.1a

A large part of the complexity of strongly correlated sys-tems arises from the multiple phases that coexist or competein their phase diagrams. Metallic and insulating phases areseparated by metal-insulator transitions, and subject to theformation of various types of long-range order, such as anti-ferromagnetism, superconductivity, and charge- or spin-density waves. The relative stability of such phases is deter-mined by differences in appropriate thermodynamicpotentials, or, at zero temperature, in their ground-state ener-gies. Identification of the appropriate order parameters andcalculation of the ground-state energies of the various phasesis a complicated problem, and the nature of the phase dia-gram of many strongly correlated systems is still subject toconsiderable controversy. It is widely believed, however, thata minimal model containing the essence of strong correla-tions, and displaying many of the above-mentioned phases,is the homogeneous Hubbard model, which in one dimensionand standard notation reads

Hhom= − toi,s

scis† ci+1,s + H.c.d + Uo

i

ci↑† ci↑ci↓

† ci↓. s1d

Much theoretical effort is thus going into the analysis of thehomogeneous Hubbard model and the clarification of the na-ture of its ground state.

In a parallel development, nanoscale spatial inhomogene-ity has been observed experimentally to be a ubiquitious fea-ture of strongly correlated systems,1–8 but although its impor-tance is widely recognized, the consequences of suchinhomogeneity are still insufficiently understood. Thepresent paper investigates the effects of, and the competitionbetween, two different manifestations of nanoscale inhomo-geneity in strongly correlated systems, namely local varia-tions in the on-site potential and in the on-site interaction.We base our analysis on theinhomogeneousHubbard model

Hinhom= − toi,s

scis† ci+1,s + H.c.d + o

i

Uici↑† ci↑ci↓

† ci↓

+ ois

vicis† cis, s2d

which differs from the homogeneous models1d by allowingfor spatial variations in the on-site interactionUi and thepresence of the on-site potentialvi. Variations inUi and vimay arise, e.g., due to inequivalent sites in the natural unitcell, modulation of system parameters in artificial hetero-structures, or self-consistent modulations in local systemproperties due to formation of charge-ordered states. In thispaper, we are specifically concerned with one-dimensionalsuperlattice structures in which bothUi andvi vary periodi-cally on a length scale comparable to, or somewhat largerthan, the lattice constant. Such superlattices have recentlyattracted much attention due to their complex ground-stateand transport properties.9–18 Our results, reported below,have a direct bearing on the investigation of such superlat-tices. However, for our present purposes the most importantaspect of superlattice structures is that they constitute a rep-resentative system in which the consequences of nanoscalespatial variations of system parameters in the presence ofstrong Coulomb correlations can be explored systematically.Accordingly, we expect our main conclusions to hold also inmany other spatially inhomogeneous correlated systems.

Figure 1 shows the density profile of a typical superlatticestructure in which the on-site interactionUi is modulated ina repeated pattern of repulsivesUi =3d and noninteractingsUi =0d “layers” with LU and L0 sites, respectively, and theon-site potentialvi is taken to be constant at all sites. Thetwo curves shown were obtained with different many-bodytechniques. The full curve was obtained using the density-matrix renormalization groupsDMRGd,19,20 while the dottedcurve was obtained from density-functional theorysDFTdwithin the Bethe-Ansatz local-density approximationsBA-LDA d.21–23 In view of the complexity of the problem

PHYSICAL REVIEW B 71, 125130s2005d

1098-0121/2005/71s12d/125130s5d/$23.00 ©2005 The American Physical Society125130-1

and the surprising nature of some of our conclusions, wefound it advisable to bring two independently developed andimplemented many-body methods to bear on the problem.

DMRG is a well-established numerical technique, whoseprecision can be improved systematically, at the expense ofincreased computational effort.19,20 In our DMRG calcula-tions, truncation errors were kept of the order of 10−6 orsmaller, and increasing the precision beyond this did not af-fect any of our conclusions. BA-LDA is a more recentdevelopment21–23 salthough the original LDA concept is, ofcourse, widely used inab initio calculationsd. In LDA calcu-lations, the final precision is ultimately limited by the local-ity assumption inherent in the LDA, and improvements mustcome from the development of better functionals. This intrin-sic limitation of LDA is offset by its applicability to verylarge and inhomogeneous systems, at much reduced compu-tational effort: Calculations for the type of superlattice struc-tures investigated here typically take only seconds to minuteswith BA-LDA, regardless of the type of boundary conditionused.24 Final BA-LDA results for densities and energies typi-cally agree with DMRG ones to within&3%, the agreementbeing slightly better for energies than for densities.25 Herewe consider both methods as complementary. All essentialconclusions reported below were obtained on the basis ofindependently implemented and performed BA-LDA andDMRG calculations. As an illustration, Table I comparesground-state energies obtained with both methods for onemuch larger and one much more rapidly modulated superlat-tice than the one shown in Fig. 1.

Inspection of Fig. 2 shows that an attractive potential onthe repulsively interacting sites can completely reverse theeffect of the Coulomb repulsionUi and draw a substantialnumber of electrons to the interacting sitesscircles in Fig. 2d.While this might have been anticipated qualitatively as aresult of the competition between an attraction and a repul-sion, it comes as a surprise that the effect of thesoften ne-glectedd variations in the on-site potential is much strongerthan the one of variations in the on-site interaction: already a

very weak attractive potential suffices to smooth out the den-sity distribution, resulting in an essentially homogeneouscharge profilestriangles in Fig. 2d. Although we have takensuperlattices as our example, the effect is clearly not depen-dent on periodicity of the modulations inUi and vi, and isexpected to show up rather generally.

Figure 1 and Table I represent superlattices in which onlythe on-site interaction is spatially modulated, which is thecase most studied in the literature.9–17 Many important fea-tures of superlattice structures are already apparent in thistype of model. However, in a real system it is impossible tomodulate the on-site interaction without simultaneouslymodulating the on-site potential as well, i.e., without creatinginequivalent sites. Such a double modulation is found, e.g.,in artificially grown layered structures, in impurity systems,and in periodic arrays of Fermi-liquid leadsscorrespondingto approximately noninteracting sitesd and quantum wires/dots scorresponding to interacting sitesd. Figure 2 illustratesthe consequences a modulation of the on-site potential has onthe density profile of a system in which both interaction andpotential vary.

In Fig. 3, we compare the Friedel oscillations arising fromthe system boundaries in a homogeneous system with theones arising in a superlattice of the same size and with thesame number of fermions, but subject to periodic modula-tions of Ui and vi, chosen such that both density profilesbecome similar. We have deliberately not chosen modulationparameters that optimize the agreement between both curves,because had we done so they would be visually indistin-guishable on this scale.

Figure 2 shows that a small value ofvi can have strongereffects than a larger value ofUi, while Fig. 3 shows that

FIG. 1. Density profile of a one-dimensional superlattice withL=60 sites,N=30 fermions, open boundary conditions, and a su-perlattice structure consisting of a periodic sequence ofLU=3 in-teractingsUi =3d andL0=2 noninteractingsUi =0d sites. Full curve:DMRG calculation. Dotted curve: DFT/BA-LDA calculation.

TABLE I. Ground-state energy of two open superlattices withmodulated on-site interactionUi and spatially constant on-site po-tential vi, obtained with DMRG and with DFT/BA-LDA. Upperpart: large lattice withL=300 sites.LU=10 interacting sitessUi

=3d alternate withL0=10 noninteracting sitessUi =0d. Lower part:strongly modulated lattice withL=100 sites.LU=1 interacting sitesUi =6d alternates withL0=1 noninteracting site.N is the number offermions, and the column labeledD% contains the absolute percentdeviation of the DMRG from the DFT/BA-LDA values. The agree-ment between DMRG and BA-LDA is slightly better for the moreslowly modulated lattice.25

N E0DMRG/ t E0

BA-LDA / t D%

50 −97.994 −98.342 0.35

100 −185.66 −187.32 0.89

150 −255.62 −258.65 1.17

200 −302.10 −305.68 1.17

250 −321.09 −324.02 0.90

300 −310.35 −311.87 0.49

40 −69.822 −71.110 1.81

50 −82.078 −83.730 1.97

75 −100.14 −103.11 2.88

80 −101.81 −104.94 2.98

120 −79.947 −79.323 0.79

SILVA et al. PHYSICAL REVIEW B 71, 125130s2005d

125130-2

essentially the same modulation pattern in the density profilecan be obtained from eitherv or U. These are unexpectedfindings. Normally it is assumed that in systems modeled bythe Hubbard model, the particle-particle interactionU ismuch more important than the on-site potential, which ismostly taken to be spatially constant, or, if it varies, to pro-duce only minor modifications in systems whose physics isgoverned byU.

To investigate in more detail this competition betweenon-site interaction and on-site potential, we need to establishcriteria for comparing the consequences ofvi and ofUi. Mo-tivated by the experimental observation of nanoscale densityvariations and by the theoretical importance of ground-stateenergies for analyses of phase diagrams, we adopt two dis-tinct criteria. Criterion sid consists in searching for thatmodulation of the on-site potentialvi in a doubly modulatedlattice that cancels the effect of the modulation ofUi on thedensity distribution, i.e., smoothes out the oscillations, mak-ing the net density homogeneous. A particular example ofthis cancellation is given by the triangles in Fig. 2. Criterionsii d consists in searching for that modulation of the on-sitepotential vi in the doubly modulated lattice that yields thesame ground-state energyE0 as in a homogeneous latticewith vi =0 andUi =U at all sites.27 Our results, displayed inFig. 4,24 show that, regardless of whether one adopts thedensity or the energy criterion,the modulation of the on-sitepotential required to cancel the effect of the modulation ofthe on-site interaction is up to an order of magnitude smallerthan U. For open boundary conditions, we have obtained thesame conclusion also from DMRG calculations. Changes inthe modulation pattern do not change the order of magnitudeof the ratio ofuvu to U appreciably.

A semiquantitative explanation for this relation ofuvu to Ucan be given within DFT by considering the effective poten-tial entering the Kohn-Sham equations for the Hubbardmodel, vef f,i =vext,i +vH,i +vc,i. For unpolarized systemssn↑,i

=n↓,i =ni /2d, the Hartree potentialvH,i can be writtenvH,i

=Uini /2. Within BA-LDA DFT, the density and total energyare thus calculated from an effective Hamiltonian containingthe modulated interaction and external potential only via thecombinationvext,i +Uini /2+vc,isni ,Uid. Since the correlationpotentialvc,i is typically about an order of magnitude smaller

FIG. 2. Density profiles, obtained with BA-LDA, of anL=100site system ofN=50 fermions with periodically modulated on-siteinteraction of amplitudeU=3 sLU=L0=10d and periodic boundaryconditions.26 Squares: no modulation in on-site potentialsvi =0d.Circles: on-site potential modulated such thatvi =−2 on the inter-acting sites andvi =0 on the noninteracting sites. The density profileis inverted, indicating overcompensation ofUi by vi. Triangles: on-site potential modulated such thatvi =−0.555 on the interacting sitesand vi =0 on the noninteracting sites. The superlattice structure iserased from the density profile. The lines are guides for the eye.

FIG. 3. Full curve: density profile, obtained with BA-LDA, ofan L=160 site homogeneous system with open boundary condi-tions,Ui =2, N=80, vi =0. Friedel oscillations arising from the sys-tem boundaries are clearly visible.26 Dotted curve: density profile ofthe same system but subject to modulations periodically alternatingLU=6 sites withUi =2 andvi =−0.2, with L0=10 sites withUi =vi

=0. The superlattice structure due to the presence of both modula-tions is completely erased from the density profile, while the Friedeloscillations arising from the boundary remain prominent.

FIG. 4. Open squares: Amplitude of the modulation of the at-tractive on-site potential that cancels as much as possible the effectof the modulation of the repulsive on-site interaction, leading to ahomogeneous density profilefcriterion sidg. Full circles: Amplitudeof the modulation of the attractive on-site potential that reproducesin the doubly modulated system the energy of the homogeneoussystemfcriterion sii dg. System parameters:L=160 sites,N=80 fer-mions,LU=6 interacting sites, alternating withL0=10 noninteract-ing sites.27

EFFECTS OF NANOSCALE SPATIAL INHOMOGENEITY… PHYSICAL REVIEW B 71, 125130s2005d

125130-3

than vH,i, the modulated interactionUi enters the effectiveHamiltonian approximately on the same footing as themodulated potentialvext,i, but renormalized by the factorni /2. In the above calculations, the average densityN/L=0.5, and the local densityni is not very different. The up-shot is that self-consistent screening of the particle-particleinteraction effectively reduces the modulation in the interac-tion by a factor,4, compared to modulations in the poten-tial, in good agreement with the numerical results in Fig. 4.28

Of course, in a real system one cannot adjustvi at will,and the precise fine-tuning required to obtain smooth densityprofiles, or energies identical to the ones found in homoge-neous systems, is not expected to occur frequently in nature.The main implication of these criteria is rather that they es-tablish a scale for comparison ofv and U, indicating thateven weak spatial variations ofv can be more important thanmuch stronger ones inU. This observation flags a warningsignal to the use of homogeneous Hubbard modelssor onesin which only U is modulatedd in the analysis of situationscharacterized by nanoscale spatial inhomogeneity, such asthe pseudogap phase of cuprates1–8 or superlattices and simi-lar heterostructures.9–18

We conclude that even in the presence of strong correla-

tions, spatial variations of the on-site potentialvi are not aminor complication in a system dominated by the on-siteinteractionUi, but a major effect, which crucially contributesto observables, and can mask or overcompensate the effect ofthe interaction on the density profile, ground-state energy,and other quantities. For thedensity profile, this means thatattempts to model the microscopically inhomogeneouscharge distribution, seen experimentally,1–8 by Hubbardmodels that are homogeneous or that modulate only the in-teractionUi, cannot lead to conclusive results. The influenceof modulations invi on theground-state energy, on the otherhand, implies that an analysis of the relative energetic stabil-ity of the various phases appearing in strongly correlatedsystems is incomplete, and potentially misleading, if the ef-fects of spatial inhomogeneity in these phases are not takeninto account. All this calls into question the common practiceto employ the homogeneous Hubbard model to model spa-tially inhomogeneous many-body systems, and demands areconsideration of the role of nanoscale spatial inhomogene-ity in strongly correlated systems.1–18

This work was supported by FAPESP and CNPq.

*Electronic address: capelle@if.sc.usp.br1Intrinsic Multiscale Structure and Dynamics in Complex Elec-

tronic Oxides, edited by A. R. Bishop, S. R. Shenoy, and S.SridharsWord Scientific, New Jersey, 2003d.

2K. McElroy, D.-H. Lee, J. E. Hoffman, K. M. Lang, E. W. Hud-son, H. Eisaki, S. Uchida, J. Lee, and J. C. Davis, e-print cond-mat/0404005.

3D. J. Derro, E. W. Hudson, K. M. Lang, S. H. Pan, J. C. Davis, J.T. Markert, and A. L. de Lozanne, Phys. Rev. Lett.88, 097002s2002d.

4A. Yazdani, C. M. Howald, C. P. Lutz, A. Kapitulnik, and D. M.Eigler, Phys. Rev. Lett.83, 176 s1999d.

5E. W. Hudson, S. H. Pan, A. K. Gupta, K.-W. Ng, and J. C. Davis,Science285, 88 s1999d.

6K. M. Lang, V. Madhavan, J. E. Hoffman, E. W. Hudson, H.Eisaki, S. Uchida, and J. C. Davis, NaturesLondond 415, 412s2002d.

7S. H. Pan, J. P. O’Neal, R. L. Badzey, C. Chamon, H. Ding, J. R.Engelbrecht, Z. Wang, H. E. Eisaki, S. Uchida, A. K. Gupta,K.-W. Ng, E. W. Hudson, K. M. Lang, and J. C. Davies, NaturesLondond 413, 282 s2001d.

8T. Cren, D. Roditchev, W. Sacks, and J. Klein, Europhys. Lett.54, 84 s2001d.

9A. L. Malvezzi, T. Paiva, and R. R. dos Santos, Phys. Rev. B66,064430s2002d; fVirtual Journal of Nanoscale Science & Tech-nology Vol. 6 s2002d swww.vinano.orgdg.

10J. Silva-Valencia, E. Miranda, and R. R. dos Santos, Phys. Rev. B65, 115115s2002d.

11J. Silva-Valencia, E. Miranda, and R. R. dos Santos, J. Phys.:Condens. Matter13, L619 s2001d.

12T. Paiva and R. R. dos Santos, Phys. Rev. B65, 153101s2002d;62, 7007s2000d; 58, 9607s1998d.

13T. Paiva and R. R. dos Santos Phys. Rev. Lett.76, 1126s1996d.14T. Paiva, M. El. Massalami, and R. R. dos Santos, J. Phys.: Con-

dens. Matter15, 7917s2003d.15D. Góra, K. J. Rościszewski, and A. M. Oleś, J. Phys.: Condens.

Matter 10, 4755s1998d.16M. Noh, D. C. Johnson, and G. S. Elliott, Chem. Mater.12, 2894

s2000d.17L. Bauernfeind, W. Widder, and H. F. Braun, Physica C254, 151

s1995d.18S. G. Ovchinnikov, Phys. Usp.46, 21 s2003d.19S. R. White, Phys. Rev. Lett.69, 2863s1992d; 77, 3633s1993d.20A. L. Malvezzi, Braz. J. Phys.33, 55 s2003d.21N. A. Lima, M. F. Silva, L. N. Oliveira, and K. Capelle, Phys.

Rev. Lett. 90, 146402s2003d.22N. A. Lima, L. N. Oliveira, and K. Capelle, Europhys. Lett.60,

601 s2002d.23K. Capelle, N. A. Lima, M. F. Silva, and L. N. Oliveira, inThe

Fundamentals of Density Matrix and Density Functional Theoryin Atoms, Molecules, and Solids, edited by N. Gidopoulos and S.Wilson sKluwer, Dordrecht, 2003d.

24The computational efficiency of BA-LDA is crucial for the pro-duction of Fig. 4, which requires for each value ofUi manycalculations with different values ofvi, until thevi that smoothesthe density or equalizes the energy is found.

25Improved LDA functionals, currently under construction, have inpreliminary calculations shown even smaller deviations fromDMRG data.

26For visual clarity, Figs. 2–4 contain only curves obtained withBA-LDA, but as Fig. 1 and Table I show, the difference betweenthe DMRG and DFT/BA-LDA results is much smaller than thedifference between the effect ofUi and that ofvi, which isrobust and not dependent on computational methodology.

SILVA et al. PHYSICAL REVIEW B 71, 125130s2005d

125130-4

27Note that one cannot use the Hohenberg-Kohn theorem to arguethat the potential that produces the same density distribution inboth systems must be the same one that yields the same ground-state energy. The ground-state energy is a unique functional ofthe density only for fixed interaction, whereas in the present case

the two superlattices that are being compared have different in-teractions. Hence the two criteria are different, and the two setsof potentials in Fig. 4 are not identical.

28K.C. thanks Ferdinand Evers and Erik Koch for useful discus-sions of this issue.

EFFECTS OF NANOSCALE SPATIAL INHOMOGENEITY… PHYSICAL REVIEW B 71, 125130s2005d

125130-5