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Physical Review B

Ground states of the Falicov-Kimball model with hybridization

L. G. Sarasúa, M. A. Contínentino

Published Feb 20, 2004• 15 citations
Publisher

Abstract

We obtain exact ground states for the spinless Falicov-Kimball model with hybridization in one dimension $(d=1).$ Ground-state energies and their corresponding wave functions are given for a restricted region of the parameter space in the limit of strong interactions for the half-filling condition. We find that the ground states possess a diagonal long range order and constitute a condensation of excitons in a zero momentum state. On the other hand, the correlation function $A=〈{d}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{f}_{i}〉$ is purely imaginary, indicating that the ground state does not exhibit spontaneous ferroelectricity.

DOI
10.1103/physrevb.69.073103

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We obtain exact ground states for the spinless Falicov-Kimball model with hybridization in one dimension (d=1).(d=1).(d=1). Ground-state energies and their corresponding wave functions are given for a restricted region of the parameter space in the limit of strong interactions for the half-filling condition. We find that the ground states possess a diagonal long range order and constitute a condensation of excitons in a zero momentum state. On the other hand, the correlation function A=〈di\ifmmode†\else\textdagger\fifi〉A=〈{d}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{f}_{i}〉A=〈di\ifmmode†\else\textdagger\fi​fi​〉 is purely imaginary, indicating that the ground state does not exhibit spontaneous ferroelectricity.
We present exact results for the extended periodic Anderson model with Falicov-Kimball interactions in one dimension (d=1).(d=1).(d=1). Ground-state energies and their corresponding wave functions are obtained for a restricted region of the parameter space of the model Hamiltonian. In particular, two ground states are obtained, one containing one electron per site and the other three electrons per site. The expectation values of the occupation numbers and of the excitonic correlation functions 〈cf+〉〈{\mathrm{cf}}^{+}〉〈cf+〉 are calculated using the exact ground state wave functions. Finally we discuss the possibility of a ferroelectric ground state in the Falicov-Kimball model.
The extrapolation of small-cluster exact-diagonalization calculations is used to examine the possibility of electronic ferroelectricity in the one-dimensional spinless and spin-one-half Falicov-Kimball model (FKM). It is found that neither the spinless nor the spin-one-half version of the FKM allows for a ferroelectric ground state with a spontaneous polarization, i.e., there is no nonvanishing 〈d+f〉〈{d}^{+}f〉〈d+f〉 expectation value for vanishing hybridization V.
The extrapolation of small-cluster exact-diagonalization calculations is used to examine the possibility of electronic ferroelectricity in the one dimensional spinless and spin-one-half Falicov-Kimball model (FKM). It is found that neither spinless nor spin-one-half version of the FKM does not allow for a ferroelectric ground state with a spontaneous polarization, i.e. there is no nonvanishing $$-expectation value for vanishing hybridization VVV.
The density matrix renormalization group method is used to examine possibilities of electronic ferroelectricity in the spinless Falicov-Kimball model. The model is studied for a wide range of parameters including weak and strong interactions as well as the symmetric and unsymmetric case. In all examined cases the 〈d+f〉〈{d}^{+}f〉〈d+f〉-expectation value vanishes for vanishing hybridization V, indicating that the spinless Falicov-Kimball model does not allow for a ferroelectric ground state with a spontaneous polarization.
We consider the excitonic insulator state (often associated with electronic ferroelectricity), which arises on the phase diagram of an extended spinless Falicov-Kimball model (FKM) at half-filling. Within the Hartree--Fock approach, we calculate the spectrum of low-energy collective excitations in this state up to second order in the narrow-band hopping and/or hybridization. This allows to probe the mean-field stability of the excitonic insulator. The latter is found to be unstable when the case of the pure FKM (no hybridization with a fully localized band) is approached. The instability is due to the presence of another, lower-lying ground state and not to the degeneracy of the excitonic phase in the pure FKM. The excitonic phase, however, may be stabilized further away from the pure FKM limit. In this case, the low-energy excitation spectrum contains important information about the properties of the excitonic condensate (including the strongly suppressed critical temperature).
Ground-state properties of the asymmetric Hubbard model are studied on one-dimensional clusters (rings) with N=6, 8, and 10 sites. The ground-state energy, correlation functions, and phase diagrams are determined for various sets of model parameters. Computations are performed both by exact diagonalization (for N=6) and by an approximate method by which correlation effects can be examined on larger clusters than exact diagonalization allows. In the limiting cases (one limit corresponds to the spinless Falicov-Kimball model and the other to the Hubbard model) our results agree quite well with those obtained analytically for infinite systems.