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arXiv:0807.4878 · Iranian Journal of Physics Research

Introduction to Hubbard Model and Exact Diagonalization

S. A. Jafari

Published Jul 30, 2008• 16 citations
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Abstract

Hubbard model is an important model in theory of strongly correlated electron systems. In this contribution we introduce this model along with numerically exact method of diagonalization of the model.
DOI
10.48550/arxiv.0807.4878

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Cluster perturbation theory is a technique for calculating the spectral weight of Hubbard models of strongly correlated electrons, which combines exact diagonalizations on small clusters with strong-coupling perturbation theory at leading order. It is exact in both the strong- and weak-coupling limits and provides a good approximation to the spectral function at any wave vector. Following the paper by S\'en\'echal et al. [Phys. Rev. Lett. 84,84,84, 522 (2000)], we provide a more complete description and derivation of the method. We illustrate some of its capabilities, in particular regarding the effect of doping, the calculation of ground state energy and double occupancy, the disappearance of the Fermi surface in the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model, and so on. The method is applicable to any model with on-site repulsion only.
This chapter provides an introduction to the theory of strongly correlated electrons and discusses some of the ideas that presently are being pursued to try to resolve this dilemma. The Hubbard Model is by now surely the most infamous or notorious model problem in solid state physics because the ratio of number of papers to understanding of this model must be an all time record. Yet it is a model which describes the competition between the kinetic energy and the Coulomb interaction in the simplest way. Its origin lies in work of Van Vleck. One tried and true method when have to deal with strong interactions is to look for the right mean-field or Hartree-Fock method. This method has of course been widely applied to the Hubbard Model. The one-dimesional Hubbard model with t not equal to zero for nearest neighbours was solved by Lieb and Wu using Bethe-Ansatz methods.
We present a method to compute eigenstates and eigenvalues of strongly correlated models on clusters. The calculation is performed applying the SU(2) symmetry, which allows a considerable reduction of the dimension of Hilbert space of the system. We illustrate the method classifying the states of the four-sites Hubbard model on a chain. This classification has been made using also the translational symmetry and the S\ifmmode \tilde{}\else \~{}\fi{}\ifmmode \tilde{U}\else \~{U}\fi{}(2) pseudospin symmetry. \textcopyright{} 1996 The American Physical Society.