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Canadian Journal of Physics

Ground-state analysis of the Falicov-Kimball model on complete graphs

Oscar Bolina, Domingos H. U. Marchetti

Published Jan 1, 2000• 0 citations
Publisher

Abstract

The ground-state nature of the Falicov–Kimball model of mobile electrons and fixed nuclei on complete graphs is investigated. We give a pedagogic derivation of the eigenvalue problem and present a complete account of the ground-state energy both as a function of the number of electrons and nuclei and as a function of the total number of particles for any value of interaction U [Formula: see text] [Formula: see text]. We also study the energy gap and show the existence of a phase transition characterized by the absence of gap at the half-filled band for U < 0. The model in consideration was proposed and partially solved by Farkasovsky for finite graphs and repulsive on-site interaction U > 0. Contrary to his proposal, we conveniently scale the hopping matrix to guarantee the existence of the thermodynamic limit. We also solve this model on bipartite complete graphs and examine how sharp the Kennedy–Lieb variational estimate is as compared with the exact ground state. PACS Nos.: 71.2-b, 02.10Gd

DOI
10.1139/cjp-78-7-679

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The ground-state nature of the Falicov–Kimball model of mobile electrons and fixed nuclei on complete graphs is investigated. We give a pedagogic derivation of the eigenvalue problem and present a complete account of the ground-state energy both as a function of the number of electrons and nuclei and as a function of the total number of particles for any value of interaction U [Formula: see text] [Formula: see text]. We also study the energy gap and show the existence of a phase transition characterized by the absence of gap at the half-filled band for U < 0. The model in consideration was proposed and partially solved by Farkasovsky for finite graphs and repulsive on-site interaction U > 0. Contrary to his proposal, we conveniently scale the hopping matrix to guarantee the existence of the thermodynamic limit. We also solve this model on bipartite complete graphs and examine how sharp the Kennedy–Lieb variational estimate is as compared with the exact ground state. PACS Nos.: 71.2-b, 02.10Gd
The ground state nature of the Falicov-Kimball model with unconstrained hopping of electrons is investigated. We solve the eigenvalue problem in a pedagogical manner and give a complete account of the ground state energy both as a function of the number of electrons and nuclei and as a function of the total number of particles for any value of interaction U. We also study the energy gap and show the existence of a phase transition characterized by the absence of gap at the half--filled band for U<0. The model in consideration was proposed and solved by Farkasovsky for finite lattices and repulsive on-site interaction U>0. Contrary to his proposal we conveniently scale the hopping matrix to guarantee the existence of the thermodynamic limit. We also solve this model with bipartite unconstrained hopping matrices in order to compare with the Kennedy--Lieb variational analysis.
We consider the one-dimensional Falicov-Kimball lattice model as well as a continuum model, where the attractive potential between the electrons and the nuclei is a \ensuremath{\delta} function, with an equal number N of classical ``nuclei'' and fermion ``electrons.'' We find the exact leading behavior of the ground-state energy as a function of U for U>Uc{\mathit{U}}_{\mathit{c}}Uc​. The error term is rigorously estimated. For fixed N the system forms ``atoms'' which have an effective repulsion between them. From this we argue that when N becomes infinite the ground-state configurations of the nuclei should be periodic. The low-temperature structure of the model is also discussed.
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.
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.
The Falicov-Kimball model is a lattice model of itinerant spinless fermions ("electrons") interacting by an on-site potential with classical particles ("ions"). We continue the investigations of the crystalline ground states that appear for various filling of electrons and ions, for large coupling. We investigate the model for square as well as triangular lattices. New ground states are found and the effects of a magnetic flux on the structure of the phase diagram is studied. The flux phase problem where one has to find the optimal flux configurations and the nuclei configurations is also solved in some cases. Finaly we consider a model where the fermions are replaced by hard-core bosons. This model also has crystalline ground states. Therefore their existence does not require the Pauli principle, but only the on-site hard-core constraint for the itinerant particles.
We consider the Falicov-Kimball model in the case of an equal number of classical "nuclei" and quantum "electrons" with Fermi statistics. In one dimension we find an exact formula for the leading behavior of the ground-state energy as a function of the attractive potential U between electrons and nuclei. For U > U c the system forms "atoms" which have an effective repulsion between them. Similar results hold for a continuum model provided there is a sufficiently large hard core between the nuclei. Some low-temperature properties are also discussed.