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Physical Review B

Correlation effect of local electrons in a one-dimensional Falicov-Kimball model

Qing-Wei Wang, Yu-Liang Liu

Published Sep 28, 2006• 5 citations
Publisher

Abstract

With the eigenfunctional theory, we give a general exact expression of the local electron Green function in the Falicov-Kimball model, and apply it to study the correlation effects of the local electrons in one dimension. For the two local electrons case, the correlation exponent of the local electron Green function has weak even-odd oscillation with the distance between these two local electrons, and it approaches zero in the strong coupling limit. While, at half filling of the local electrons, the ground-state phase is complicated. When the conduction electron is near $0.5$, the ground state configuration is the chessboard phase for a considerable range of $U∕{t}^{*}$, and the correlation exponent increases from zero to a finite value as $U∕{t}^{*}$ increases. When the conduction electron is far away from $0.5$, the ground state configuration is the segregated phase, and the correlation exponent first increases and then jumps to zero when $U∕{t}^{*}$ is larger than a finite value which is a function of ${n}_{c}$. Our results are compared with previous numerical simulations.

DOI
10.1103/physrevb.74.125119

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With the eigenfunctional theory, we give a general exact expression of the local electron Green function in the Falicov-Kimball model, and apply it to study the correlation effects of the local electrons in one dimension. For the two local electrons case, the correlation exponent of the local electron Green function has weak even-odd oscillation with the distance between these two local electrons, and it approaches zero in the strong coupling limit. While, at half filling of the local electrons, the ground-state phase is complicated. When the conduction electron is near 0.50.50.5, the ground state configuration is the chessboard phase for a considerable range of U∕t∗U∕{t}^{*}U∕t∗, and the correlation exponent increases from zero to a finite value as U∕t∗U∕{t}^{*}U∕t∗ increases. When the conduction electron is far away from 0.50.50.5, the ground state configuration is the segregated phase, and the correlation exponent first increases and then jumps to zero when U∕t∗U∕{t}^{*}U∕t∗ is larger than a finite value which is a function of nc{n}_{c}nc​. Our results are compared with previous numerical simulations.
The extrapolation of small-cluster exact-diagonalization calculations is used to examine the influence of correlated hopping on valence and metal-insulator transitions in the one-dimensional Falicov-Kimball model. It is shown that in the half-filled band case the ground-state phase diagram as well as the picture of valence and metal-insulator transitions found for the conventional Falicov-Kimball model (without correlated hopping) are strongly changed when the correlated hopping term is added. The effect of correlated hopping is so strong that it can induce the insulator-metal transition. Outside half-filling correlated hopping stabilizes the segregated phase in the ground-state, however, the nature of the ground-state remains qualitatively unchanged.
By studying the Falicov--Kimball lattice model, we demonstrate that the correlation functions of local electrons can be written directly in the form of the partition function of conduction electrons with a time dependent external potential. These general expressions of the correlation functions are equivalent to the exact ones obtained by the dynamical mean field theory in infinite dimension limit. Moreover, we also prove that the asymptotic power-law behavior of the local electron correlation functions is universal in any dimensions, as long as the conduction electron excitation spectrum is gapless, regardless whether they are Landau Fermi liquid or not. This conclusion is also valid for other quantum many-particle systems, if the coupling between the conduction and local electrons is on-site or short-range Coulomb interaction.
The authors have studied energetic and structural ground-state properties of the one-dimensional Falicov-Kimball model in the strong-coupling limit. Using standard perturbation theory they recover some results already obtained (the phase segregation at large interactions) and present some new results concerning the ground state, e.,g., the analytic expression for the boundary, below which the segregated configuration cannot be the ground state, and the phase diagram of the model calculated for special periodic (aperiodic) configurations of ions and for the case when the ion concentration goes to zero.
Two-dimensional spinless Falicov-Kimball model (FKM) with correlated hopping is studied perturbatively in the limit of large on-site Coulomb interaction U. In the half filled case (i.e., ${\ensuremath{\rho}}_{i}+{\ensuremath{\rho}}_{e}=1,$ where ${\ensuremath{\rho}}_{i},{\ensuremath{\rho}}_{e}$ are densities of ions and electrons, respectively) the effective Hamiltonian in spin variables is derived up to terms proportional to ${U}^{\ensuremath{-}3}.$ Unlike the simplest FKM case, it contains odd parity terms (resulting from the correlated hopping) in addition to even parity ones. The ground-state phase diagram of the effective Hamiltonian is examined in the (a/t,h)(a/t,h)(a/t,h) plane, where a/ta/ta/t is a parameter characterizing strength of the correlated hopping and h is a difference of chemical potentials of two sorts of particles present in the system. It appears to be asymmetric with respect to the change $\stackrel{\ensuremath{\rightarrow}}{h}\ensuremath{-}h$ and an additional ordered phase is found for a certain interval of a/t.a/t.a/t.
A new numerical method, recently developed to study ground states of the Falicov-Kimball model (FKM), is used to examine the effects of correlated hopping on the ground-state properties of this model in two dimensions. It is shown that the ground-state phase diagram as well as the picture of metal-insulator transitions found for the conventional FKM (without correlated hopping) are substantially changed when the correlated hopping term is added. The effect of correlated hopping is so strong that it can induce the insulator-metal transition, even in the strong-coupling limit, where the ground states of the conventional FKM are insulating for all f-electron densities.
In this work, we study two models of strongly correlated fermions: Hubbard and FalicovKimball. We present a proof of the existence of real and ordered solutions to the nested Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any U > 0 to U = ∞ is also shown, where U is the strength of the interaction between fermions at the same site. We use this continuity property, combined with the proof that the norm of the wavefunction obtained with the generalized Bethe Ansatz is not zero, to prove that the solution gives us the ground state of the finite system, as assumed by Lieb and Wu. For the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the Lieb-Wu solution of the 1D Hubbard model. We also obtain a lower bound for the ground state energy of the Falicov-Kimball model. This bound is given by a bulk term, plus a term proportional to the boundary. A numerical value for the coefficient of the boundary term is determined. The explicit derivation is important in the proof of the conjecture of segregation of the two kinds of fermions in the Falicov-Kimball model, for sufficiently large interactions.