Dynamical susceptibilities of the Falicov-Kimball model with correlated hopping : general approach

The Falicov-Kimball model with correlated hopping is studied in the limit of infinite spatial dimensions. Dynamical susceptibilities are calculated using the generalization of the Dynamical Mean-Field Theory (DMFT), which is based on expansion over electron hopping around the single-site limit. A special case of semi-elliptic density of states and diagonal correlated hopping is considered numerically and the absence of phase transition for all temperatures except T = 0 is demonstrated, which corresponds to the known results.


Introduction
The Falicov-Kimball model was first introduced in [1] as a model of metal-insulator transition.It appears that Falicov-Kimball model is the simplest model of electron correlations which displays long-range ordered phases [2].It was applied to the investigations of such phenomena as valence transitions [3,4], phase separation and crystallization [5].Dynamical properties of the model, including electron susceptibilities [6,7] and Raman response [8,9] were also investigated.The model contains two types of Fermi-particles: itinerant (conductive) particles, referred to as d-electrons, and localized particles, f-electrons or "ions".On the other hand, the model can be interpreted as an asymmetric Hubbard model with infinite mass of electrons for one of the spin orientations.
Falicov-Kimball model is attractive not only due to its simplicity but also because many exact results were obtained for this model [2].Among them there are proofs of phase separation for all spatial dimensions of the lattice from one to infinity, theorems of ground-state properties in one and two dimensions.In addition to the results for low spatial dimensions, there exists an exact solution of the model in the infinite dimensions case [10] (see for review [2]).This solution is important in applying Dynamical Mean-Field Theory (DMFT).
DMFT is based on the self-consistent mapping of a lattice problem onto an auxiliary singlesite problem [11].This transformation is advantageous due to the local nature of self-energy in the limit of infinite spatial dimensions or large coordination.The development of the DMFT approach brought about a noticeable progress in studying strongly correlated systems, but in its traditional form (see, e.g.[11]) it cannot be applied to an important form of hopping, the so-called correlated hopping.Correlated hopping describes the case when the hopping integral depends on the occupation of neighboring sites.Though Falicov-Kimball model gives a reasonable physical description of many processes, it neglects all nonlocal dynamical interactions.Correlated hopping is a notable example of such an interaction.For the first time it was introduced in the well-known Schubin and Wonsowsky work [12], the value of that kind of hopping was estimated for the first time by Hubbard [13], then Hirsch pointed out that correlated hopping can play an important role in the appearance of superconductivity [14].The importance of the correlated hopping in ferromagnetism was shown in [15][16][17][18].Other reasons for considering the correlated hopping are based on a substantial Coulombian interaction between electrons, reported for some materials [19, c A.M.Shvaika, O.Ya.Farenyuk 20], dynamical part of which can be approximated by correlated hopping.When correlated hopping is present, the self-energy becomes non-local, and, as it was shown in earlier investigations [21], this non-locality can be presented as the next-nearest-neighbors hopping contribution to self-energy.But using expansions over electron hopping (Larkin approach, opposite to Dyson approach which is based on expansions over many-electron interactions and leads to the self-energy concept), it can be shown that such non-locality is just an artifact of the Dyson approach.Even for correlated hopping, the irreducible part of Green's function, which cannot be divided by cutting a hopping line, remains local which permits to construct the DMFT for such systems [22].
In this article we develop a general scheme of calculating dynamical susceptibilities for systems with correlated hopping, which is based on generalization of DMFT.It is used to investigate electron susceptibilities of the Falicov-Kimball model with correlated hopping.This method permits to obtain explicit expressions for dynamical susceptibilities.The limit of diagonal correlated hopping on Bethe lattice is studied using the obtained general expressions.

The model
Hamiltonian of the (spinless) Falicov-Kimball model can be written as: where H t is a hopping term for the d-particles, µ f and µ d are the chemical potentials for the fand d-particles, respectively, and U is the Coulomb repulsion of the particles at the same site.Correlated hopping term for the Falicov-Kimball model has the following form: Here t 1 describes the part of hopping between sites i and j, which does not depend on the occupation of the sites, t 2 takes into account the hopping between sites, one of which is occupied by the f-particle, and t 3 describes the hopping between the sites occupied by the f-particles.Such a Hamiltonian can be presented in terms of hopping between sites with fixed occupations: where are the projection operators and we introduce matrix annihilation operator and hopping matrix In fact, correlated hopping naturally leads to matrix generalization for all quantities [22].Both representations for the Hamiltonian are related by the following equations: (2.6) The components of the matrix Green's function are defined by the following formula: where Ω is a grand canonical potential functional.The total Green's function is a sum of all components of the matrix Green's function: (2.9) As it was shown in the reference [22], for the systems with correlated hopping, the strongcoupling approach, based on the expansions over electron hopping around the atomic limit, is more convenient.In this approach, the single-electron Green's function is presented as a solution of the matrix generalization of the Larkin equation: where Ξ ij (ω) is an irreducible part of the Green's function which cannot be divided into parts by cutting one hopping line.This irreducible part is local in the limit of infinite spatial dimensions even for the systems with correlated hopping [22,23].After the Fourier transformation to the momentum representation, this equation can be solved for the Green's function: with t k = a k , where k is an unperturbed band energy.Now we can apply the modified Dynamical Mean Field Theory (DMFT), which leads to exact results in the limit of infinite spatial dimensions D → ∞, when hopping integrals are scaled according to: t αβ → t αβ / √ D. In this limit, an irreducible part is local and can be obtained from the condition that single-site Green's function for the lattice and for the auxiliary single-site problem is the same [22].On the other hand, matrix Green's function of the single-site problem can be written in Larkin representation as: where J is the coherent potential matrix (λ-field) [22,10].For the Falicov-Kimball model, G imp can be calculated exactly: (2.14) Furthermore, itinerant electrons concentration can be calculated as: (2.15)

Susceptibilities from the DMFT point of view
In general, susceptibilities are defined by: For instance, itinerant electron charge susceptibilities for the Falicov-Kimball model with correlated hopping can be obtained from the following matrix of projected susceptibilities: Analyzing the diagrammatical series, obtained by using Wick theorem in the same way as in [7], one can see that many-particle Green functions can be written in the following form: where is a sum of the chains of hopping and arrow denotes lattice Green's function (2.11).Here , and L are irreducible parts that cannot be divided by cutting two hopping lines and It should be noted, that here correlated hopping leads to a Cartesian product over α = + and − in place of ordinary multiplication in the similar equations for systems without correlated hopping [7].
The full four-vertex in equation (3.3) is connected with the irreducible one L by the Bethe-Salpeter-like equation: All the introduced irreducible parts are local in the limit of infinite spatial dimensions [7] and can be calculated from the corresponding single-site many-particle Green functions: ) where Υ is 4 × 4 matrix, whose indices take the values "++", "−−", "+−", "−+"; Quantity (3.7) correspond to the following diagrammatical sequence: Here the thin wavy lines correspond to the coherent potential J, the thick one is the sum of chains of the wavy lines, and a line above another one should be treated as [see also (3.5)]: = − J ⊗ J. (3.13) From equations (3.10) and (3.6) the four-vertex can be obtained as (3.14) In the same way, equations (3.8) and (3.9) for other irreducible many-particle Green functions can be written down: .
After performing all the necessary substitutions into equation (3.3) one can obtain the following formula for the lattice susceptibility χ AB q (ω n ): correspond to the irreducible parts Ξ and + .(3.10)Now, using definition (3.4), one can state that: