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Physical review. B, Condensed matter

Exact solution and thermodynamics of the Hubbard model with infinite-range hopping

P. G. J. van Dongen, D. Vollhardt

Published Oct 1, 1989• 34 citations
Publisher

Abstract

The Hubbard model with unconstrained hopping of the particles on a lattice is solved exactly. It is shown that in this case the kinetic energy commutes with the interaction part, i.e., the model is essentially trivial. The thermodynamics is worked out explicitly. One finds that the results of the quasichemical approximation for the occupation probability of lattice sites are exact for this model. The ground state is insulating at half-filling and U>0 and is conducting otherwise.

DOI
10.1103/physrevb.40.7252

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The Hubbard model with unconstrained hopping of the particles on a lattice is solved exactly. It is shown that in this case the kinetic energy commutes with the interaction part, i.e., the model is essentially trivial. The thermodynamics is worked out explicitly. One finds that the results of the quasichemical approximation for the occupation probability of lattice sites are exact for this model. The ground state is insulating at half-filling and U>0 and is conducting otherwise.
We present the exact solution of the simplified Hubbard model in which only one kind of electrons can hop and this quantum mechanical hopping of electrons is assumed to be unconstrained. It is shown that the model still behaves non-trivially, although it no longer depends on the lattice structure and the dimensionality of the system. For this case we find: (i) a gap in the ground state energy always exists at the half-filled band point (n = 1), (ii) a preferred magnetic state at n = 1 and large U is a total spin singlet, (iii) U-dependence of the ground state energy has qualitatively the same form as one of the conventional Hubbard model with the (t2/U)-behavior at large U. A phase diagram of the model is discussed.
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.