Electron structure of the Falicov-Kimball model with a magnetic field
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Electron structure of the Falicov-Kimball model with a magnetic field
Minh-Tien Tran
Institute of Physics, Vietnamese Academy of Science and Technology, P.O. Box 429, 10000 Hanoi, Vietnam
The two-dimensional Falicov-Kimball model in the presence of a perpendicular magnetic fieldis investigated by the dynamical mean-field theory. Within the model the interplay between elec-tron correlations and the fine electron structure due to the magnetic field is essentially emerged.Without electron correlations the magnetic field induces the electron structure to the so-called Hof-stadter butterfly. It is found that when electron correlations drives the metal-insulator transition,they simultaneously smear out the fine structure of the Hofstadter butterfly. In a long-range or-dered phase, the electron correlation induced gap preserves the fine structure, but it separates theHofstadter butterfly into two wings.
PACS numbers: 71.27.+a, 71.10.Fd, 71.70.Di, 67.85.-d
I. INTRODUCTION
The problem of electrons moving under an externalmagnetic field has attracted a lot of attention since thebeginning of quantum mechanics. Two dimensional elec-tron gas under a perpendicular magnetic field creates thequantum Hall effect.
It comprehensively deals withthe interplay of electron correlations and magnetic field.The integer quantum Hall effect is due to the quantiza-tion of the energy levels of free electrons under a magneticfield, while the fractional quantum Hall effect is essen-tially due to the electron interaction under a magneticfield.
The picture totally becomes complexity whenelectrons move additionally on a lattice or under a pe-riodic potential. The simultaneous presence of magneticfield and lattice potential gives the spectra of two di-mensional noninteracting electrons a fine structure of thefamous Hofstadter butterfly. The Hofstadter butterflydisplays a recursive structure over rational gauge fieldand a Cantor set at any irrational gauge field. The Hallconductance of the noninteracting Bloch electrons is stillquantized with an integer number when the Fermi en-ergy lies within a gap of the Hofstadter butterfly. Whenelectron correlations are included, the effect of simultane-ous presence of magnetic field and electron interaction onthe lattice band energy remains interesting. Without theexternal field electron correlations can induce differentphenomena, for instance, the metal-insulator transition,long-range ordered phases. When electron correlationsare absent, the magnetic field creates the Hofstadter but-terfly of the electron structure. A simultaneous presenceof electron correlations and magnetic field would inducea complexity of the electron structure. The exact diago-nalization of finite lattices shows that the electron corre-lations smear out the Hofstadter butterfly. However, itis not easy to distinguish the fine electron structure dueto the presence of magnetic field from the discrete energylevels due to the finite-size effect of the exact diagonal-ization. The Hartree-Fock mean field calculations revealthe electron structure with the Hofstadter butterfly andan additional correlation-induced gap.
It gives rise toan interest in the study of the interplay between electron correlations and magnetic field in a two-dimensional lat-tice beyond the Hartree-Fock approximation and in thethermodynamic limit. In experimental aspect, with therapid development of ultracold technique some funda-mental models of many quantum particles can be real-ized by loading ultracold particles into optical lattices(see, for example, the review in Ref. 10). In particular,by using the technique of laser-assisted tunneling or oflattice rotating artificial gauge fields can be created inoptical lattices. As a results it is possible to realize theeffect of magnetic field on the Bloch electrons by load-ing ultracold particles into optical lattices with artificialgauge field. Indeed, recently several models of opticallattices were proposed to study the Hofstadter butterflyof ultracold particles. In the present paper we theoretically study the effectof electron correlations on the Hofstadter butterfly of theelectron structure under a magnetic field. The electroncorrelations are modelled by the Coulomb interaction ofthe Falicov-Kimball model (FKM). It is a local repul-sive interaction of mobile electrons and massive localizedparticles. The FKM was originally introduced to describea metal-insulator transition in transition-metal oxides. Itcan be viewed as a simplified Hubbard model where elec-trons with down spin are frozen and do not hop. TheFKM was also used as a starting point to investigate dif-ferent electron correlation phenomena, for instance themixed valence or the electronic ferroelectricity. TheFKM can also be incorporated into different models tostudy various aspects of electron correlations such as thecharge ordered phase in manganese compounds orelectron localization.
Much progress has been madeon solving the FKM in both exact and approximationways, where all properties of the conduction electronsare well known.
In the homogeneous phase theFKM displays a metal-insulator transition. When theCoulomb interaction is strong, it prevents the mobilityof itinerant electrons by forming the Mott-Hubbard gap.At low temperature a charge ordering occurs. For halffilling the charge ordering gap opens at the Fermi leveland drives the system into an insulating phase. Onemay expect that the electron-correlation induced gapsof the Mott-Hubbard type and of long-range charge or-dering may have different effects on the fine structure ofthe Hofstadter butterfly. A realization of the FKM wasalso proposed as an optical lattice of a mixture of lightfermionic atoms (e.g., Li) and heavy fermionic atoms(e.g., K).
When the optical lattice modelling theFKM is established, it is also possible to create an ar-tificial magnetic field. In the present paper the two-dimensional square lattice with a perpendicular mag-netic field is considered. The dynamical mean-field the-ory (DMFT) is employed to calculate the electron struc-ture of the considered model. The DMFT is widely andsuccessfully applied to study strongly correlated electronsystems.
It gives the exact solutions in infinite di-mensions. However, for two-dimensional systems theDMFT is just an approximation. It neglects nonlocalcorrelations. However, the applications of the DMFT toFKM show that the approximation is still accurate in twodimensions.
We find that the electron correlation ef-fect on the Hofstadter butterfly depends on the natureof the correlated phase when the magnetic field is ab-sent. When a long-range order is absent, electron correla-tions only induce the metal-insulator transition and theysmear out the fine structure of the Hofstadter butterfly.In a long-range ordered phase, the electron correlationinduced gap preserves the fine structure, but separatesthe Hofstadter butterfly into two wings.The plan of the present paper is as follows. In Sec. IIwe describe the FKM with a perpendicular magnetic fieldon a square lattice. We also present the DMFT for calcu-lating the Green function in this section. In Sec. III thenumerical results are presented. Finally, the conclusionand remarks are presented in Sec. IV.
II. THE FALICOV-KIMBALL MODEL WITH APERPENDICULAR MAGNETIC FIELD ANDTHE DYNAMICAL MEAN-FIELD THEORY
In this section we present the DMFT for the Falicov-Kimball model in the presence of a magnetic field. Themodel is described by the following Hamiltonian H = − X t ij c † i c j − µ X i c † i c i + E f X i f † i f i + U X i c † i c i f † i f i , (1)where c † i ( c i ), f † i ( f i ) are the creation (annihilation) op-erators for itinerant and localized electrons at site i , re-spectively. t ij is the hopping integral of itinerant elec-trons between site i and j . U is the local interaction ofitinerant and localized electrons. µ is the chemical po-tential for itinerant electrons and E f is the energy levelof localized electrons. We will consider only the half fill-ing case, where µ = − E f = U/
2. In the presence of amagnetic field the hopping integral acquires the Peierls phase factor t ij = t exp (cid:18) i πφ R j Z R i A · d l (cid:19) , (2)where φ = hc/e , and A is the vector potential. For aconstant magnetic field perpendicular to the square lat-tice, the Landau gauge can be chosen for the vector po-tential A = (0 , Bx, B is the magnetic fieldstrength. With this Landau gauge the hopping integralin the x direction is just t , while in the y direction itacquires additional phase factor t exp( ± i παx i ), where α = Ba /φ , a is the lattice constant, and x i is the lat-tice site position in the x -axis. In the following we will set a = 1. Parameter α is just the magnetic flux per unit cellin the units of the flux quantum φ . It is clear that theHamiltonian is invariant with the translation α → α + m ,where m is a integer. Therefore it is only necessary toconsider 0 ≤ α ≤ α = p/q , where p , q are two coprime integers. Thetranslation operator that moves q lattice spacing in the x direction leaves the Hamiltonian unchanged. We dividethe lattice into q penetrating sublattice in x direction, i.e,each lattice site can be indexed by a number n , and itscoordinates x , y , where n = mod( R x , q ), (1 ≤ n ≤ q ). Wetake the Fourier transformation from the direct lattice tothe reciprocal lattice c n k = 1 p N/q X xy c nxy e ik x x + ik y y , where N is the number of lattice sites. The wave vectors k x , k y are restricted to the reduced Brillouin zone − πq ≤ k x ≤ πq , π ≤ k y ≤ π. The hopping part of Hamiltonian in Eq. (1) can be rewrit-ten as H t = X k ˆ X † k ˆ E ( k ) ˆ X k , (3)where ˆ X † k = ( c † k , . . . , c † q k ), andˆ E ( k ) = − t ε k e − ik x . . . e ik x e ik x ε k e − ik x . . . e ik x ε k e − ik x . . . . . . ... ... e − ik x . . . e ik x ε q k , (4)with ε n k = cos( k y + ( n − πp/q ).We apply the DMFT to the calculation of the Greenfunction of itinerant electrons in the matrix formˆ G ( k , ω ) = hh ˆ X k | ˆ X † k ii ω = (cid:2) ω + µ − ˆ E ( k ) − ˆΣ( ω ) (cid:3) − , (5)where ˆΣ( ω ) is the self energy. Within the DMFT the selfenergy is independent on momentum. Moreover, it is alsodiagonal, i.e., Σ nm ( ω ) = δ nm Σ n ( ω ). This formulation issimilar to the DMFT applications for the antiferromag-netic or checkerboard charge ordered phases. Basically,the DMFT is exact in infinite dimensions. However, itsapplication for two-dimensional systems is just approx-imation. The approximation neglects nonlocal correla-tions which exist as the momentum dependence and off-diagonal elements of the self energy. The DMFT calcu-lations for two-dimensional FKM without the magneticfield show that the approximation still accurate for theelectron dynamics.
The self energy Σ n ( ω ) is self consistently determinedfrom the dynamics of a single interaction site embeddedin an effective mean-field medium. For the FKM theeffective single-site problem can be solved exactly. Weobtain the Green function of the single site problem G n ( ω ) = W n G − n ( ω ) + W n G − n ( ω ) − U , (6)where G n ( ω ) is the Weiss field for sublattice n . Theweight factors W n and W n can be calculated from theWeiss field W n = f ( e E n ) , (7) W n = 1 − W n , (8)where f ( ω ) = 1 / (exp( ω/T ) + 1) is the Fermi-Dirac dis-tribution function, and e E n = E f − Z dωπ f ( ω )Im log (cid:18) − U G n ( ω ) (cid:19) . (9)The Weiss field Green function G n ( ω ) also satisfies theDyson equation of the effective single site problem, i.e., G − n ( ω ) = G − n ( ω ) − Σ n ( ω ) . (10)The self-consistent condition requires that the Greenfunction obtained from the effective single-site problemmust coincide with the local Green function, i.e., G n ( ω ) = 1 N/q X k G nn ( k , ω ) . (11)With this self-consistent condition the system of equa-tions for the self energy is closed. We can solve the sys-tem of equations by iterations as usual. III. NUMERICAL RESULTS
In this section we present the numerical results ob-tained by solving the DMFT equations by iterations. Wetake t = 1 as the energy unit. The magnetic parameter α = p/q varies with p = 0 , , . . . , q . We take q = 40.When p and q are not coprime, we can reduce them to FIG. 1: (Color online) The density plot of the DOS of itin-erant electrons of the high temperature phase ( T = 1) forvarious values of U . The magnetic field parameter α = p/q varies with p = 0 , , . . . , q , and q = 40. The color schemecorresponds to the rainbow color scheme, i.e. the red (violet)color corresponds to the largest (smallest) value of the DOS. coprime integers. This also reduces the matrix dimensionof the Green function, and saves the computation time.First we consider the high temperature phase, where themetal-insulator transition of the Mott-Hubbard type oc-curs. We emphasize that in this phase temperature doestnot affect on the electron structure. The phase is justthe homogeneous solution of the DMFT equations andit is stable at high temperature. The electron structurecan be imaged by using the density plotting of the den-sity of states (DOS) of the itinerant electrons. In Fig. 1we plot the image of the DOS for various values of U .It shows that the electron structure mimics the Hofs-tadter butterfly when the electron correlations are in-cluded. For weak interactions the fine structure of theHofstadter butterfly still survives. However, the elec-tron correlations already smear out it. Some fine gaps inthe structure of the Hofstadter butterfly are closed. Asthe value of U increases, the smearing becomes stronger.For strong interactions all fine gaps are closed. How-ever, a middle rough gap opens for U > U c ≈ t . Thisgap is essentially the Mott-Hubbard gap, which opensin the insulating phase. Without the magnetic field themetal-insulator transition occurs at U c . In the presenceof the magnetic field the metal-insulator transition stilloccurs, but the lower and upper bands mimic the Hof-stadter butterfly. The electron structure is symmetry U=0 U=2 =2/5=1/2=3/8 =0 U=5 -6 -4 -2 0 2 4 60.000.250.50 -6 -4 -2 0 2 4 60.00.10.2 -6 -4 -2 0 2 4 60.00.10.2 FIG. 2: The DOS of itinerant electrons of the high tempera-ture phase ( T = 1) for various values of α and U . in respect to lines ω = 0 and α = 1 / ρ ( ω, α ) = ρ ( − ω, α ) = ρ ( ω, − α ),where ρ ( ω, α ) = − P n Im G n ( ω ) /qπ is the DOS of itiner-ant electrons.In Fig. 2 we plot the DOS of itinerant electrons forvarious values of α and U . When the electron correla-tions are absent ( U = 0) the number of bands is just q . One can observe that the q bands can be groupedinto subgroups of bands which are separated by moder-ate gaps. Within a band subgroup the bands are alsoseparated by fine gaps. For example, when α = 3 / ω = 0, and twoothers around ω = ±
2. When the interaction is included,the band number is reduced by closing the gaps. Asthe interaction increases, first the fine gaps within theband subgroups are closed, and then the gaps betweenthe band subgroups are closed. In the insulating phaseall the gaps in the Hofstadter butterfly are closed, but
FIG. 3: (Color online) The density plot of the DOS of itiner-ant electrons of the charge ordered phase at low temperature( T = 0 .
01) for various values of U . The magnetic field param-eter α = p/q varies with p = 0 , , . . . , q , and q = 40. The colorscheme corresponds to the rainbow color scheme, i.e. the red(violet) color corresponds to the largest (smallest) value ofthe DOS. U=0 U=2 =2/5=1/2=3/8 =0 U=4 -4 -2 0 2 40.000.250.50 -4 -2 0 2 40.00.51.0 -4 -2 0 2 401
FIG. 4: (Color online) The DOS of itinerant electrons of thecheckerboard charge ordered phase at low temperature ( T =0 .
01) for various values of α and U . The red and green colorlines correspond to the DOS of two penetrating sublattices ofthe checkerboard charge ordered phase. the Mott-Hubbard gap opens. The metal-insulator tran-sition occurs at the same value of U for all values of themagnetic field. Despite of closing of the fine gaps in theinsulating phase, the intensity of the DOS of itinerantelectrons still shows a mimic Hofstadter butterfly withsmearing out fine features, as shown in Fig. 1.Without the magnetic field the FKM also displays thecheckerboard charge ordering at low temperature for anyinteraction U = 0. We study the possibility of the chargeordering when the magnetic field is present. In this casewe additionally divide the lattice into two penetratingsublattices in the y direction. If q is odd integer, wedouble the value of p and q that α = 2 p/ q and thecheckerboard charge ordering is commensurate with themagnetic structure. In Fig. 3 we plot the image of the to-tal DOS of itinerant electrons in the checkerboard chargeordered phase. It shows that the fine structure of the Hof-stadter butterfly still remains, however, it is separatedby a middle gap. The middle gap is the charge orderinggap, which locks itinerant electrons into the checkerboardcharge pattern. As the interaction increases the chargeordering gap also increases, and the band width of thelower and upper bands are slightly reduced. The preser-vation of the fine structure of the Hofstadter butterflywas also observed within the Hartree-Fock mean fieldcalculations. However, the Hartree-Fock mean-field ap-proximation cannot find the smearing of the Hofstadterbutterfly due to electron correlations when a long-rangeorder is absent.In Fig. 4 we also plot the DOS of itinerant electronsof the two penetrating sublattices of the checkerboardcharge ordered phase for various values of U and α . Itshows that the magnetic field does not affect on thecharge ordering gap. The gap sorely depends on the in-teraction as in the case of absence of the magnetic field.When q is even, the number of bands is also q , like thenoninteraction case. However, unlike the noninteractioncase, the subgroup of bands at the Fermi level is sepa-rated by the charge ordering gap. When q is odd, theband at the Fermi level is split into two bands which arealso separated by the charge ordering gap, as shown inFig. 4 (the case α = 2 / IV. CONCLUSION
In the present paper we study the effect of electron cor-relations on the Hofstadter butterfly which is the electronstructure of the two-dimensional Bloch electrons under aperpendicular magnetic field. By employing the DMFTwe calculate the Green function of itinerant electronsin the case of rational magnetic field. Electron correla-tions exhibit different effects on the Hofstadter butterflydepending on the nature of the correlated phase whenthe magnetic field is absent. In the absence of a long-range order, electron correlations also induce the metal-insulator transition when the magnetic field is present.However, the electron correlations smear out the finestructure of the Hofstadter butterfly. The number ofbands is reduced as the interaction increases. In the in-sulating phase, all fine gaps of the Hofstadter butterfly are closed, but the Mott-Hubbard gap opens at the Fermilevel. In a long-range ordered phase, the electron correla-tion induced gap, such as the checkerboard charge order-ing gap in the FKM, preserves the fine structure of theHofstadter butterfly. However, the Hofstadter butterflyis separated into two wings by the long-range orderinggap.In the present paper we have only considered the ra-tional magnetic field. In the noninteraction case an ir-rational magnetic field induces the Hofstadter butterflyin the form of a Cantor set. The magnetic structure isincommensurate with the lattice structure as well as withthe checkerboard charge ordering pattern. The effect ofelectron correlations on the such Hofstadter butterfly re-mains open, and we leave it for further study.
Acknowledgments
This work was supported by the Vietnamese NAFOS-TED.
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