Compressibility enhancement in an almost staggered interacting Harper model
CCompressibility enhancement in an almost staggered interacting Harper model
Bat-el Friedman and Richard Berkovits
Department of Physics, Jack and Pearl Resnick Institute, Bar-Ilan University, Ramat-Gan 52900, Israel
We discuss the compressibility in the almost staggered fermionic Harper model with repulsiveinteractions in the vicinity of half-filling. It has been shown by Kraus et al. [33] that for spinlesselectrons and nearest neighbors electron-electron interactions the compressibility in the central bandis enhanced by repulsive interactions. Here we would like to investigate the sensitivity of thisconclusion to the spin degree of freedom and longer range interactions. We use the Hartree-Fock (HF)approximation, as well as density matrix renormalization group (DMRG) calculation to evaluate thecompressibility. In the almost staggered Harper model, the central energy band is essentially flatand separated from the other bands by a large gap and therefore, the HF approximation is ratheraccurate. In both cases the compressibility of the system is enhanced compare to the non-interactingcase, although the enhancement is weaker due to the inclusion of Hubbard and longer rangedinteractions. We also show that the entanglement entropy is suppressed when the compressibilityof the system is enhanced.
PACS numbers: 71.23.Ft, 73.21.Hb, 73.23.Hk, 37.10.Jk
INTRODUCTION
The interplay between electron-electron (e-e) inter-actions and quasi-disorder has drawn much excitementsince the discovery of quasi-crystals [1, 2]. Much of thework has focused on a specific model of a one-dimensional(1D) quasi-crystal, namely the Harper (or Aubry-Andr´e)model [3, 4]. One of the main attractions of this modelis that contrary to conventional 1D disordered systemswhich are localized for any amount of disorder [5], theHarper model exhibits a metal-insulator transition asfunction of the quasi-disordered potential strength, evenin the absence of interactions [4, 6–11]. The influence ofe-e interactions on the metal-insulator transition of theHarper model was studied in several publications [12–14].Interest in the Harper model has lately peaked after it hasbeen shown that for an irrational modulation, the Harpermodel may be a 1D topologically nontrivial system, andhave topological boundary states [15–23]. This property,coupled with the fact that the Harper model may be re-alized in the context of cold atoms and molecules [24, 25]added to the excitement surrounding the Harper model.Recently, an additional aspect of the model has beeninvestigated, namely the inverse compressibility, whichmeasures the change in the chemical potential when anelectron is added to the system. In the context of dis-ordered quantum dots this has become a very popularmeasurement to extract information on the role of e-e in-teractions in these systems [26–28]. For a finite system of N particles, ∆ ( N ), is defined as the change in the chem-ical potential due to the insertion of the N th particle i.e.,∆ ( N ) = µ ( N ) − µ ( N − µ ( N ) is the chemicalpotential for N particles. Since µ ( N ) = E ( N ) − E ( N − E ( N ) is the system’s many-body ground-state en-ergy with N particles), ∆ ( N ) is given by:∆ ( N ) = E ( N ) − E ( N −
1) + E ( N − , (1) For non-interacting systems at zero temperature,∆ ( N ) = E N − E N − = ∆( N ) , (2)where E N is the N th single-particle eigenenergy and∆( N ) is the single-particle level spacing.How do the e-e interactions affect the inverse com-pressibility? The conventional wisdom leads to the con-stant interaction (CI) model [28, 29], which essentiallyassumes that the interactions between the electrons arewell described by mean-field. This leads to the conclu-sion that the effect of interactions on the inverse par-ticipation given by ∆ ( N ) = ∆( N ) + e /C , where C is the total classical capacitance. Thus, the e-e inter-actions increase the inverse compressibility compared toits non-interacting value. This description fits well theexperimental measurements in quantum dots [28].However, the CI mean-field description doesn’t hold atcertain conditions. It has been shown [30–32] that closeto the Mott metal-insulator transition occurring at half-filling of a clean the Hubbard model, the inverse com-pressibility may decreases with the Hubbard interactionsstrength. Recently, it has been shown [33] that for thealmost staggered Harper model of spinless electrons withnearest-neighbors e-e interactions, close to half-filling,the system becomes more compressible as the interac-tions are increased, although no metal-insulator transi-tion occurs there. This counter intuitive behavior stemsfrom the properties of the electronic bands and densityfor the almost staggered Harper model. Under these con-ditions the non-interacting Harper model has an almostflat narrow band around zero energy, separated from theother bands by large gaps. The density of the narrowband around half-filling is anti-correlated with the on-site potential, whereas the density of the lower occupiedbands follows the potential. Therefore, once e-e inter-action is introduced, the electrons in the lower occupiedbands squeeze out the states in the narrow central band, a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r resulting in a narrower central band. This flattening ofthe central band due to the interaction with the lowerband electronic density results in an increase of the com-pressibility.In this paper we address the question whether this in-crease of the compressibility is the result of the partic-ular model studied in Ref. [33]. Specifically, we shallsee what happens to the compressibility when the spindegree of freedom is taken into account, or equivalentlywhen considering a spinless two legged ladder. Anothercase which we explore is when next nearest neighbors in-teractions are included. To study the compressibility wemainly rely on the HF approximation, which has beenshown to be extremely accurate for this model [33] dueto the large gap between the flat central band and thelower band and to the localized nature of the states inthe narrow band. We will also compare some of these re-sults to density matrix renormalization group (DMRG)numerical calculations, which for these 1D systems areessentially exact [34, 35], and describe very well the de-pendence of the ground state energy on the number ofparticles [36]. Using DMRG we also show that the en-hancement of the compressiblity is accompanied by thesuppression of the entanglement entropy. HUBBARD INTERACTION
In this section we discuss the influence of the spin de-gree of freedom on the compressibility in the staggeredHarper model close to half-filling. The clearest differ-ence between spin-polarized (spinless) and non-polarizedelectron is the fact that for non-polarized (spinfull) elec-trons there are Hubbard interactions. The on-site po-tential is spatially modulated with a frequency of almosttwo lattice-sites period (i.e., staggered), correspondingto fast modulation with a slow envelope. The interac-tion terms are repulsive and short ranged (on-site andnearest-neighbors (n.n.)-interactions). We assume thatin the limit of weak Hubbard interactions no spin polar-ization occurs, i.e., the total S z = 0 for even filling and S z = ± / H = (cid:88) s (cid:54) = s (cid:48) = ↑ , ↓ L (cid:88) j =1 (cid:2) t ( c † j,s c j +1 ,s + h.c. ) + t (cid:48) c † j,s c j,s (cid:48) + λ cos(2 πbj + φ ) n j,s + U n j,s n j +1 ,s + U (cid:48) n j,s n j,s (cid:48) ) (cid:3) . (3)where c j,s is the single particle annihilation operator atsite j with spin s and n j,s = c † j,s c j,s is the number op-erator. t, t (cid:48) ∈ R are the site hopping and spin flippingamplitudes, respectively. λ > b and a phase factor φ . U > U (cid:48) > λ < t , which is the metallic regime [4]. We furtherassume that b mod 1 = 1 / (cid:15) , (cid:15) (cid:28) / (cid:15) ∈ R is non-rational so thatthe system is disordered.Let us first discuss the non-interacting Hamiltonian,i.e., set U, U (cid:48) = 0 in Eq. (3). A numerical solution in thiscase reveals the existence of an almost flat central energyband (see Fig. 1), splitted due to the spin flip matrix el-ement to a lower and higher central band. We are mostlyinterested in the central band energy spectrum, and sincethese energy states which are close to zero minimize bothkinetic and potential energy, we conclude that the mostimportant contribution comes from states localized in thepotential valleys, i.e. states localized around the position l z corresponding to 2 π(cid:15)l z + φ = ( Z + ) π [33]. In the val-ley, we can approximate cos(2 π(cid:15)j + φ ) ≈ π | (cid:15) | ( j − l z ) s z ,and s z = − sign (sin(2 π(cid:15)l z + φ )) = ±
1. The effectiveHamiltonian describing the central band is H val = (cid:88) s (cid:54) = s (cid:48) = ↑ , ↓ L (cid:88) j =1 (cid:2) t ( c † j,s c j +1 ,s + h.c. )+ t (cid:48) c † j,s c j,s (cid:48) + 2 π(cid:15)λs z ( − j ( j − l z ) c † j,s c j,s (cid:3) = (cid:88) s,s (cid:48) = ↑ , ↓ L π π (cid:90) Ψ † k,s (cid:2) (2 t cos( k ) σ x +2 π | (cid:15) | s z λ (ˆ p k − l z ) σ z ) δ ss (cid:48) + t (cid:48) (1 − δ ss (cid:48) )]Ψ k,s (cid:48) (4)where ψ k,s = (cid:18) c ek,s c ok,s (cid:19) , is the sub-lattice pseudo-spinor that splits the lattice intoeven and odd sites, according to c ek,s = L (cid:80) L/ j =1 e ik j c j,s and c ok,s = L (cid:80) L/ j =1 e ik (2 j − c j − ,s . ˆ p k ≡ i∂ k and σ x , σ z are the 2 × FIG. 1: Energy bands of the free Hamiltonian (U=U’=0).The central band is splitted due to the gap created by thespin flipping amplitude t (cid:48) . The parameters used through thefigures are t = 1; t (cid:48) = 0 . λ = 0 . φ = 0 . π ; b = √ (cid:15) = − . L = 200 . The isolated points correspond to pro-tected edge (topological) states in the Harper model and arenot discussed in this paper.
Diagonalizing the spin degrees of freedom (which areindependent of k-space), we get ψ k, = 1 √ (cid:18) c ek, ↑ + c ek, ↓ c ok, ↑ + c ok, ↓ (cid:19) ψ k, = 1 √ (cid:18) c ek, ↑ − c ek, ↓ c ok, ↑ − c ok, ↓ (cid:19) . This representation allows us to write the Hamiltonianas a sum of two distinct subspaces, each relates to a dif-ferent spin eigenstate. The subspaces depend only onthe momentum k , and therefore can be solved using thesame methods used for spinless fermions [33]. Thus, theeigenenergies for the Hamiltonian of the potential val-leys are E val = ±√ n tξ + t (cid:48) , and E val = ±√ m tξ − t (cid:48) ,where m, n ∈ { , , ... } , and ξ = tπλ | (cid:15) | . E val , E val cor-respond to the spin states 1 , t (cid:48) .The eigenfunctions for the states belonging to the split-ted central band are: | l z,i > ≈ ( πξ ) − L (cid:88) j =1 ( s z ) j S j e − ( j − lz )22 ξ | j, i >, (5)where | j, i > = √ ( c † j, ↑ ± c † j, ↓ ) |∅ > , where |∅ > is the vac-uum state. These wavefunctions are Gaussians of width ξ around l z . In the limit of small t (cid:48) our assumptions holdand this result is a good approximation of the real groundstate.These states form a basis for the central band, definedby m, n = 0, since < l z,i | l z ± ,i > = 0 , < l z, | l z, > = 0,and | < l z,i | l z (cid:48) ,i > | ≤ e − ( lz − lz (cid:48) )22 ξ (cid:28) < l z,i | H | l z (cid:48) ,j > , are not negligi-ble only between nearest neighbors states | z − z (cid:48) | = 1.Thus, the central band states follow an effective Hamil-tonian: H central = − ¯ t L z (cid:88) z =1 (cid:88) i =1 , ( − z c † l z ,i c l z +1 ,i + h.c. + t (cid:48) c † l z ,i c l z ,i . (6)Diagonalizing this Hamiltonian yields the eigenstates | k, i > = L − / z L z (cid:88) z =1 S z e ikz | l z,i > (7)with eigenvalues E central ( k ) = − t cos( k ) ± t (cid:48) .Now, let us focus on the case where the Hubbard in-teractions in the Hamiltonian Eq. (3) are turned on( U (cid:48) (cid:54) = 0), but no longer range interactions are yet con-sidered ( U = 0). For U (cid:48) → ∞ the model can be solvedanalytically. In that limit only the interaction term isimportant. The eigenenergies are therefore E = 0 and E = U (cid:48) . The latter case occurs when two particles withopposite spins occupy the same site. This will cost infi-nite energy and therefore such states are decoupled fromthe theory. The remaining states contain a single particleper site.Next, we consider the case where U (cid:48) is much biggerthan the other energy scales in the theory, i.e. U (cid:48) (cid:29) t, t (cid:48) , λ . Using perturbation theory with t as the pertur-bation parameter on the Hubbard model reveals that fer-romagnetism is the lowest energy state. Adding t (cid:48) to thetheory will not change the ground state, since the correc-tion in t (cid:48) will be of at least third order in perturbationtheory.As is discussed in Ref. 33, because the central band isessentially protected by the large gaps to the other bands,the HF approximation results are very accurate. There-fore, we approximate the Hubbard interaction using theHF method for interaction strength values smaller thanthese gaps U (cid:48) (cid:28) √ tξ . (cid:88) j n j, ↑ n j, ↓ ≈ (cid:88) j [ < n j, ↑ > n j, ↓ + n j, ↑ < n j, ↓ > − < n j, ↑ >< n j, ↓ > ] . (8)Rewriting the Hamiltonian in Eq. (3) with U = 0, andignoring the constant term which is simply a shift in theenergy, results in H = (cid:88) s (cid:54) = s (cid:48) = ↑ , ↓ L (cid:88) j =1 (cid:2) t ( c † j,s c j +1 ,s + h.c. ) + t (cid:48) c † j,s c j,s (cid:48) + (cid:0) λ cos(2 πbj + φ ) + U (cid:48) < n j,s (cid:48) > (cid:1) n j,s (cid:3) . (9)We find that the averaged electronic density be-tween the valleys of potential is < n j,s > ≈ − ( − j ¯ n ( λ t ) cos(2 π(cid:15)j + φ )) , with ¯ n ( x ) = xπ √ x K ( x ), and K is the complete elliptical in-tegral of the first kind. Hence H HF = (cid:88) s (cid:54) = s (cid:48) = ↑ , ↓ L (cid:88) j =1 (cid:2) t ( c † j,s c j +1 ,s + h.c. )+ t (cid:48) c † j,s c j,s (cid:48) + (cid:0) λ eff cos(2 πbj + φ ) + 14 U (cid:48) (cid:1) n j,s (cid:3) , (10)where λ eff = λ − U (cid:48) ¯ n ( λ t ).The solutions of H HF are closely related to the solu-tions of H in the non-interacting case. Yet, the width ofthe valley states, ξ , has changed due to the change in λ .Moreover, for n.n.-interactions ( U (cid:54) = 0) it is possi-ble to use the HF approximation, and obtain the HFeigenstates and eigenvalues, which are identical to thenon-interacting solutions, up to the modified parame-ters ˜ t , and ˜ λ [33]. The many-body density and the ex-change terms are proportional to those obtained alreadyfor the spinless case [33] up to a proportionality constantof 1 /
2, due to the spin degrees of freedom. Therefore, < p j,s > ≈ ¯ p ( λ t cos(2 π(cid:15)j + φ )). Between the potentialvalleys this can be approximated by < p j,s > ≈ ¯ p ( λ t ).We can now write the HF. Hamiltonian with both Hub-bard and n.n. interactions: H HF = (cid:88) s (cid:54) = s (cid:48) = ↑ , ↓ L (cid:88) j =1 (cid:2) t eff ( c † j,s c j +1 ,s + h.c. )+ t (cid:48) c † j,s c j,s (cid:48) + (cid:0) λ eff cos(2 πbj + φ )+12 U + 14 U (cid:48) (cid:1) n j,s (cid:3) , (11)with t eff = t + U ¯ p ( λ t ) and λ eff = λ + (2 U − U (cid:48) )¯ n ( λ t ).We again can solve the system with the modified pa-rameters, and obtain the HF eigenvalues and eigenstates, E HFval = ±√ n t eff ξ ± s t (cid:48) + 12 ( U + 12 U (cid:48) ) | l z,i > ≈ ( πξ ) − L (cid:88) j =1 ( s z ) j S j e − ( j − lz )22 ξ | j, i >, (12)where the Gaussian decay parameter ξ = ξ ( t eff λ eff ) is mod-ified due to the effective values taken by λ and t . ξ is multiplied by a numerical constant equal to 1 .
16 as in[33].Projecting the HF Hamiltonian on the central bandyields H HFcentral = − ¯ t HF L z (cid:88) z =1 (cid:88) s (cid:54) = s (cid:48) = ↑ , ↓ ( − z c † l z ,s c l z +1 ,s + h.c. + t (cid:48) c † l z ,s c l z ,s (cid:48) + 14 (2 U + U (cid:48) ) c † l z ,s c l z ,s . (13)The eigenvalues and the eigenstates of the central bandare then given by: E central ( k ) = ( − n +1 t HF cos( k ) + 14 ( ± t (cid:48) + 2 U + U (cid:48) ) , | k, i > = L − / z L z (cid:88) z =1 S z e ikz | l z,i >, k = 2 πnL z , n = 1 , .., L z , (14)with L z = (cid:98) | (cid:15) | L (cid:99) the number of valley states. The hop-ping amplitude ¯ t HF is given by¯ t HF ≈ e − ξ (cid:15) (cid:0) t eff e − ξ sinh (cid:0) ξ | (cid:15) | (cid:1) − λ eff e − ( π(cid:15)ξ ) (cid:1) . (15)Thus the inverse compressibility ∆ ( N ) can be calculatedusing (2) and the eigenvalues are presented in Eq. (14).As shown in Fig. 2, ∆ ( N ) decreases with the n.n.-interaction U , in agreement with the case of spinlessfermions [33]. However, the Hubbard interaction U (cid:48) en-hances ∆ ( N ). As was shown in Eq. (10), the Hubbardinteraction reduces the value of the effective Harper po-tential amplitude, λ eff . The decrease in λ eff increasethe width of the Gaussian wavefunctions. Thus, there ismore overlap between different states and therefore anychange of configuration in the system, such as addinganother particle, requires more energy. For U = 2 U (cid:48) thesystem returns to the non-interacting Hamiltonian valueof ∆ ( N ). The interplay between U and U (cid:48) determinesweather ∆ ( N ) will be larger ( U < U (cid:48) ) than its non-interacting value or smaller ( U > U (cid:48) ) than it.For an intuitive understanding let us revisit Fig. 1.The states which occupy the lowest energy band residein the valleys of potential. When the Hubbard inter-action is turned on, occupying these states become toocostly in energy for some of the spins. In order to avoidthe Hubbard interaction they tend to occupy the sur-roundings of potential peaks, where there are less spinsto interact with. This tendency delocalizes the Gaussianwavefunctions. However, since only half of the particlesparticipate in the interaction between the opposite spinsit is less significant (by a factor of ) than U .An exception to this behavior is found for the stateat edge of the lower splitted band. As detailed ear-lier, due to the spin flipping amplitude t (cid:48) , a gap of size2 t (cid:48) opens between the lower central band occupied by √ ( ↑ + ↓ ) states and the higher central band with states FIG. 2: The variation of the inverse compressibility ∆ ( N )of the lower central band states with the n.n. interaction ( U )and the Hubbard interaction ( U (cid:48) ). ∆ ( N ) decreases with U ,which is in line with the results of Ref. 33 and increases with U (cid:48) . Thus, the Hubbard interaction delocalizes the particles,smearing their wave functions and increasing the amount ofenergy needed for adding another particle to the system. corresponding to √ ( ↑ − ↓ ) . ∆ ( N ) decreases with U (cid:48) and increases with U at the edge, similar to the behaviorobserved close to the half-filling point of the 1D Hubbardmodel [30]. NEXT-NEAREST NEIGHBORS INTERACTIONS
In order to understand the behavior of the compress-ibility for a system with long range interactions, we con-sider here the influence of next nearest neighbors inter-action. For simplicity, we discuss spinless fermions. Theresults of this section can be easily extended for fermionswith spin using the methods described in the previoussection.The Hamiltonian is given by: H = L (cid:88) j =1 (cid:2) t ( c † j c j +1 + h.c. ) + λ cos(2 πbj + φ ) n j + U n j n j +1 + U n j n j +2 ) (cid:3) , (16)and the mean-field approximation yields L (cid:88) j =1 n j +2 n j ≈ L (cid:88) j =1 ( < n j +2 > + < n j − > ) n j − < n j >< n j +2 > − < ˜ p j > c † j +2 c j + h.c + | < ˜ p j > | , (17)where < n j > is the (already known) background density.The background exchange energy is < ˜ p j > ≡ < c † j c j +2 > . Here we ignore constant terms, since they do not con-tribute to ∆ . Using the known value of < n j > | (cid:15) =0 , L (cid:88) j =1 ( < n j +2 > + < n j − > ) = (cid:88) j (1 − n ( λ t ) cos(2 πbj + φ )) . (18)Interestingly, the exchange term disappears (the calcula-tion appears in the appendix) resulting in < ˜ p j > = 0 . (19)This structural robustness can be attributed to thesymmetry of the non-interacting Hamiltonian’s wave-functions used in the calculation. Thus, the additionalinteraction only changes the value of λ eff without chang-ing the structure of the HF Hamiltonian. The effectiveHamiltonian becomes H HFcentral = L z (cid:88) z =1 − ¯ t HF ( − z c † l z c l z +1 + h.c., (20)where ¯ t HF given by Eq. (15) with t eff = t + U ¯ p ( λ t )and λ eff = λ +(2 U − U )¯ n ( λ t ). Here we ignored on-siteterms, which just lead to an over all energy shift.We also calculate ∆ ( N ) using DMRG [34, 35], forthe following parameters: b = √
30 (corresponding to (cid:15) ≈ − . φ = 0 . π . The length of the system is L = 200, and we calculated the ground state energy E ( N )for each number of electrons N = 91 , , . . . , t =1, the potential amplitude was chosen as λ = 0 .
7, whichresults in a flat central band, with the typical ∆ greaterthan the numerical accuracy. Interaction strengths of U = 0, U = 0 and U = 0 . U = 0 , . , . , . is about ± · − t andthe discarded weight is ∼ − .The resulting change in the compressibility can beviewed in Fig. 3. Comparing the analytic values to theresults obtained using the numerical DMRG results, wefind good agreement between the two methods. Here theGaussian decay parameter ξ is modified according to ξ → . ξ (1 − . U ). The 1 .
16 factor arise from usingthe linear approximation of the potential also betweenthe valleys, leading to a too-fast decay of the wave func-tion as was discussed for the n.n interactions [33]. For then.n. interaction an additional linear dependence of ξ on U is needed. It seems that the longer-range interactionresults in an additional correction of the wave functionbehavior in the valleys.With the additional interactions the compressibility(1 / ∆ ) decreases. Intuitively, the increase in the value of FIG. 3: The variation of the inverse compressibility ∆ ( N ) inthe central band of the spinless Harper model with the n.n.interaction ( U ) and the next n.n. interaction ( U ). ∆ ( N )increases with the next n.n. interaction, since the interactionbroadens the Gaussian wave functions. Thus, adding a parti-cle to the system has a non-local effect, and therefore it costsmore energy. HF analytic results denoted by symbols, DMRGresults denoted by straight lines. The DMRG numerical re-sults are in agreement to the analytic results we get using theHF method. λ due to the interaction results a decrease in the Gaus-sian decay parameter ξ , which results in a greater over-lap between the wavefunctions. This can be interpretedas a change in the local nature of the system due tothe next n.n.-interactions which delocalizes the wavefunc-tions. Thus, adding another particle costs more energy.This additional energy cost is reflected in the growth of∆ ( N ).The opposite behavior between the n.n and next n.n.interactions can also be observed in the behavior of thebipartite entanglement entropy. The entanglement en-tropy of a system in a pure state | Ψ (cid:105) is defined as thevon Neumann entropy of the reduced density matrix ofregion A, ˆ ρ A = Tr B | Ψ (cid:105)(cid:104) Ψ | , where the degrees of free-dom of the rest of the system (region B) are traced out,resulting in S A = − Tr (ˆ ρ A ln ˆ ρ A ) (21)For the 1D Harper model the system is divided betweenregions A and B, where region A is of length L A whileregion B is the remaining L − L A sites.The entanglement entropy for a typical state in thecentral band is depicted in Fig. 4. The behavior of S A isnon-monotonous, quite different than the entanglemententropy of a clean wire, and has several intriguing fea-tures. Here we will concentrate on the feature directlypertaining to the compressibility. The most obvious fea-ture are the peaks appearing in S A ( L A ). It is apparent FIG. 4: The entanglement entropy, S A , as function of thebisection point L A , for the ground-state with 102 particles(i.e., the N = 7 state in the central band of the spinlessHarper model with 200 sites) with different n.n. interactions( U ) and the next n.n. interactions ( U ). It is apparent thatmaximums in S A correspond to the Gaussian localized states,or the edge states, and that S A around the maximums isnot influenced by the interactions. The entanglement mini-mum between the Gaussian states though are influenced bythe interactions. When U increases (resulting in a decreasein ∆ ( N )) the entanglement decreases. On the other hand,when U increases (resulting in an increase in ∆ ( N )) the en-tanglement is enhanced. A zoom into the central minimum ispresented in the inset. that the positions of the peaks not immediately adjacentto the edges correspond to the positions of the centralband states | l z,i > . These peaks are very robust anddo not change when the interaction strength is changed.On the other hand, the entanglement of the minimum be-tween the peaks are influenced by the interactions. Whenn.n. interactions ( U ) are introduced the entanglement inthe minimum regions are suppressed (this is clearly seenin the enlarge segment in Fig. 4). When next n.n. in-teractions ( U ) are added, the entanglement minimumremains closer to its non-interacting value. This followsexactly the pattern exhibited by the inverse compress-ibility (∆ ( N )) is reduced. One can speculate that theentanglement is related to the extension of the band state | l z,i > into its nearest neighbor, and thus the suppressionof ∆ ( N ) is related to the suppression of the entangle-ment. It is interesting whether it might be possible todirectly relate the compressibility to the entanglement ina similar manner to the relation between fluctuations inthe number of particles and entanglement [37]. This isleft for further study. DISCUSSION
In this paper we considered the variation of the inversecompressibility ∆ ( N ) with respect to repulsive Hubbardinteraction and next n.n.-interaction in the central bandof the almost staggered fermionic Harper model in thevicinity of half-filling. The behavior of the central bandstates is studied using the HF approximation, justified bythe flatness of this band and its isolation from the otherbands. For the next n.n.-interaction we also calculated∆ ( N ) using DMRG. The comparison between the twomethods promise reliable results. We found both for theHubbard interaction and for the next n.n. interactionsan increase in ∆ ( N ), which corresponds to a decrease inthe compressibility of the system. Thus, the increase inthe compressibility due to the n.n. interactions is some-what suppressed once Hubbard or next n.n. interactionsare considered. It is interesting to note the different roleplayed by the Hubbard interactions for the clean 1D Hub-bard model and the Harper model. For the clean Hub-bard model close to to the metal-insulator phase transi-tion at half-filling of 1D systems, the Hubbard interac-tion enhance compressibility [30]. This behavior is alsomanifested for the Harper model close to the edge of thelower central band. On the other hand, for the rest of thecentral band, the Hubbard term effectively reduces thestrength of the on-site potential in the system ( λ eff < λ )and thus the energy gaps become smaller, weakening theenhancement of compressibility. Open questions, such asthe classification of interaction terms (which terms leadto delocalization and decrease in ∆ ( N ), and which lo-calize the wavefunctions and increase ∆ ( N )) and thefull understanding of the non monotonous entanglemententropy, remain for further study.Financial support from the Israel Science Foundation(Grant 686/10) is gratefully acknowledged. APPENDIX
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