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Journal of Physics Condensed Matter

Exact solution of the extended Falicov-Kimball model in the limit of zero conduction band width

Canio Noce, Alfonso Romano

Published Nov 6, 1989• 8 citations
Publisher

Abstract

A method for solving exactly the Falicov-Kimball model in the limit of vanishing conduction band is proposed. By means of the path integral formalism, a general procedure for the evaluation of the exact n-point Green functions at a finite temperature is developed. The results obtained here represent a starting point for a perturbation expansion in the conduction electron band width.

DOI
10.1088/0953-8984/1/44/007

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A method for solving exactly the Falicov-Kimball model in the limit of vanishing conduction band is proposed. By means of the path integral formalism, a general procedure for the evaluation of the exact n-point Green functions at a finite temperature is developed. The results obtained here represent a starting point for a perturbation expansion in the conduction electron band width.
We perform a perturbative analysis for the nonequilibrium Green's functions of the spinless Falicov-Kimball model in the presence of an arbitrary external time-dependent but spatially uniform electric field. The conduction-electron self-energy is found from a strictly truncated second-order perturbative expansion in the local electron-electron repulsion UUU. We examine the current at half-filling and compare to both the semiclassical Boltzmann equation and exact numerical solutions for the contour-ordered Green's functions from a transient-response formalism (in infinite dimensions) on the Kadanoff-Baym-Keldysh contour. We find that a strictly truncated perturbation theory in the two-time formalism cannot reach the long-time limit of the steady state; instead, it illustrates pathological behavior for times larger than approximately 2∕U2∕U2∕U.
In order to contribute to understand well the finite-temperature properties of the Falicov-Kimball model, we study the temperature dependence of the specific heat, internal energy, entropy, and mean square fluctuation of the difference between the ion and electron numbers, and of the correlation functions of the one-dimensional half-filled-band Falicov-Kimball model. For this purpose we used the method of small-cluster exact-diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain. Our results show that the examined Falicov-Kimball model exhibits behavior of alternate electron-ion short-range ordering at low temperature, and gradual metal-insulator transition at high temperature, but, does not exhibit the crossing point of the specific heat curves, characteristic of the Hubbard model.
We discuss the infinite dimension limit of the Hubbard model by means of a perturbative expansion of the one-particle Green's function around the atomic limit. The diagrammatic structure is simplified in this limit, allowing a formal resummation that reproduces a previously proposed mapping to a single-site mean-field problem. The method provides a natural way of addressing this effective problem by means of a perturbative expansion in the local mean field. This gives the correct exact result for the Falicov-Kimball model, which is used as a starting point to study the Hubbard case. \textcopyright{} 1996 The American Physical Society.
We derive an alternative representation for the f -electron spectral function of the Falicov-Kimball model from the original one proposed by Brandt and Urbanek.In the new representation, all calculations are restricted to the real time axis, allowing us to go to arbitrarily low temperatures.The general formula for the retarded Green's function involves two determinants of continuous matrix operators that have the Toeplitz form.By employing the Wiener-Hopf sum equation approach and Szeg ö's theorem, we can derive exact analytic formulas for the large-time limits of the Green's function; we illustrate this for cases when the logarithm of characteristic function (which defines the continuous Toeplitz matrix) does and does not wind around the origin.We show how accurate these asymptotic formulas are to the exact solutions found from extrapolating matrix calculations to the zero discretization size limit.
The extended Falicov-Kimball model is analyzed exactly in the ground state at half filling in the limit of large dimensions. In the model the on-site and the intersite density-density interactions between all particles are included. We determined the model's phase diagram and found a discontinuous transition between two different charge-ordered phases. Our analytical calculations show that the ground state of the system is insulating for any nonzero values of the interaction couplings. We also show that the dynamical mean-field theory and the static broken-symmetry Hartree-Fock mean-field approximation give the same results for the model at zero temperature. In addition, we prove using analytical expressions that at infinitesimally small, but finite, temperatures the system can be metallic.