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Physical review. B, Solid state

Atomic Limit and Projected Hubbard Models for a Linear Chain

Douglas J. Klein

Published Oct 1, 1973• 55 citations
Publisher

Abstract

The Hubbard model in the limit of infinite intrasite interaction and a related projected Hubbard model, which includes this extreme limit, are considered. This projected model is that portion of the full Hubbard model which preserves the number of sites of each possible occupancy, 0, 1, and 2. For the linear case, this projected model is solved exactly, both for the eigenvalue spectrum and the static thermodynamic properties. A second-order phase transition is found.

DOI
10.1103/physrevb.8.3452

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The Hubbard model in the limit of infinite intrasite interaction and a related projected Hubbard model, which includes this extreme limit, are considered. This projected model is that portion of the full Hubbard model which preserves the number of sites of each possible occupancy, 0, 1, and 2. For the linear case, this projected model is solved exactly, both for the eigenvalue spectrum and the static thermodynamic properties. A second-order phase transition is found.
The linear Hubbard model is considered when the intrasite Coulomb repulsion is very large compared to the intersite charge-transfer interaction. The solution for a general density of electrons per site is carried out by a perturbative treatment which relates this Hubbard model to a generalized Heisenberg spin Hamiltonian describing effective exchange interactions between (partially) delocalized electrons. Accurate numerical results for the linear Heisenberg model are employed to obtain good estimates of the static thermodynamic properties for Hubbard models near the atomic limit. Some modifications of the simple Hubbard model are also briefly discussed.
The extended Hubbard model in the zero-bandwidth limit is studied.The eective Hamiltonian consists of (i) on-site U interaction, (ii) intersite densitydensity interaction W , and (iii) Ising-like magnetic exchange interaction J between the nearest-neighbors.We present rigorous (and analytical) results obtained within the transfer-matrix method for 1D chain in two particular cases: (a) W = 0 and n = 1; (b) U → +∞ and n = 1/2 (W = 0, J = 0).We obtain the exact formulae for the partition functions which enables to calculate thermodynamic properties such as entropy, specic heat (c), and double occupancy per site.In both cases the system exhibits an interesting temperature dependence of c involving a characteristic two-peak structure.There are no phase transitions at nite temperatures and the only transitions occur in the ground state.
We present the exact solution of the one-dimensional extended Hubbard model in the atomic limit within the Green's function and equations of motion formalism. We provide a comprehensive and systematic analysis of the model by considering all the relevant response and correlation functions as well as thermodynamic quantities in the whole parameters space. At zero temperature we identify four phases in the plane (U,n) ( U is the on-site potential and n is the filling) and relative phase transitions as well as different types of charge ordering. These features are endorsed by investigating at T=0 the chemical potential and pertinent local correlators, the particle and double occupancy correlation functions, the entropy, and by studying the behavior in the limit T-->0 of the charge and spin susceptibilities. A detailed study of the thermodynamic quantities is also presented at finite temperature. This study evidences that a finite-range order persists for a wide range of the temperature, as shown by the behavior of the correlation functions and by the two-peak structure exhibited by the charge susceptibility and by the entropy. Moreover, the equations of motion formalism, together with the use of composite operators, allows us to exactly determine the set of elementary excitations. As a result, the density of states can be determined and a detailed analysis of the specific heat allows for identifying the excitations and for ascribing its two-peak structure to a redistribution of the charge density.
Within the one-dimensional Hubbard model linear closed chains with various numbers of sites are considered in the self-consistent random phase approximation (SCRPA). Excellent results with a minimal numerical effort are obtained for (2+4n)(2+4n)(2+4n)-site cases, confirming earlier results with this theory for other models. However, the 4n4n4n-site cases need further consideration. The SCRPA solves the two-site problem exactly. It therefore contains the two-electron and high-density Fermi gas limits correctly.
The atomic limit of the Hubbard model is a simple single-site problem which can be solved exactly, and all one- and two-particle Green's functions can be obtained analytically. These solutions can thus serve as a means of critiquing the success of various approximate theories which might be applied to the full Hubbard model. In particular, we have examined the TTT-matrix approximation for the attractive Hubbard model in the atomic limit, which should give reasonable results at low electronic densities, if one can avoid the spurious phase transition that results when a fully non-self-consistent TTT-matrix approximation is employed---previously we have shown that any level of self-consistency guarantees that this phase transition is correctly suppressed to zero temperature in two dimensions or less. Here, a minimally self-consistent TTT-matrix approximation is shown to be successful in reproducing the exact results for the atomic limit, while fully self-consistent TTT-matrix results do not agree with the known solutions. Of particular note is that the minimally self-consistent TTT-matrix approximation reproduces not only one- and two-particle (static) thermodynamic quantities, but it also exactly reproduces the one-particle spectral function at low but nonzero temperatures. We also make a comparison to the two-particle self-consistent approach of Vilk and Tremblay, and find that the minimally self-consistent TTT-matrix theory can give better results over a broader temperature range.