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Physical review. B, Condensed matter

Random one-body approximation to the Hubbard model. I. Formalism

Eleftherios N. Economou, Carter T White, R. R. Demarco

Published Oct 15, 1978• 43 citations
Publisher

Abstract

An approximation scheme is presented for treating the Hubbard Hamiltonian through an effective selfconsistent random potential. The random variables determining the effective potential are statistically correlated to simulate many-body correlations. The formalism is developed in detail for the half-filled Hubbard model; it is shown that the proposed scheme reproduces exactly the metallic and atomic limits and interpolates between the two.

DOI
10.1103/physrevb.18.3946

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An approximation scheme is presented for treating the Hubbard Hamiltonian through an effective selfconsistent random potential. The random variables determining the effective potential are statistically correlated to simulate many-body correlations. The formalism is developed in detail for the half-filled Hubbard model; it is shown that the proposed scheme reproduces exactly the metallic and atomic limits and interpolates between the two.
The effective independent-particle (mean-field) approximation of the Fermi–Hubbard Hamiltonian is described in a many-body basis to develop a formal comparison with the exact diagonalization of the full Fermi–Hubbard model using small atomic chain as test systems. This allows for the development of an intuitive understanding of the shortcomings of the mean-field approximation and how critical correlation effects are missed in this popular approach. The description in the many-body basis highlights a potential ambiguity related to the definition of the density of states. Specifically, satellite peaks are shown to emerge in the mean-field approximation, in departure from the common belief that they characterize correlation effects. The scheme emphasizes the importance of correlation and how different many-body corrections can improve the mean-field description. The pedagogical treatment is expected to make it possible for researchers to acquire an improved understanding of many-body effects as found in various areas related to the electronic properties of molecules and solids.
We develop a new mean field-theory for systems with on-site correlations, like the Hubbard and Anderson models. It consists in a generalization of the saddle-point approximation to the functional integral representation proposed by Kotliar and Ruckenstein to more than one saddle point. It contains also the alloy analog approximation proposed by Hubbard as a particular case. For a half-filled Hubbard model and a model density of states appropriate for three dimensions, we obtain a metal to insulator transition as U increases. The effective Hamiltonian for the insulating phase contains two split bands (rather than only one with an extremely heavy mass) which reproduce correctly the Green’s functions of the atomic limit.
A solution for the Hubbard Hamiltonian in the half-filled-band case is presented, which takes into account the degeneracy of the ground state in the atomic limit and also correctly interpolates to the limit Coulomb energy U to 0. It is shown explicitly which approximations lead to Hubbard's improved decoupling scheme and other existing approximate solutions. Taking into account the antiferromagnetic character of the ground state the splitting of the two sub-bands in the case of large U is enhanced. The band gap is shown to decrease with increasing temperatures.
The ground-state energy E and momentum distribution n k of the electrons are expanded from the atomic limit for the half-filled Hubbard model. The coefficients of expansion are represented by the spin correlation functions of the spin 1 / 2 Heisenberg model. Using the spin-wave theory, approximate values of coefficients are calculated for the square lattice and the simple cubic lattice. In one dimension, the theory shows a good agreement with the exact solution. An effective spin Hamiltonian for the half-filled Hubbard model with arbitrary hopping integrals is obtained up to the fifth order. It is shown that the fourth term contains four spin interactions.
The Hubbard model is investigated in the framework of lattice density functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable of the many-body problem. A new approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ is proposed which is based on its scaling properties and which recovers exactly the limit of strong electron correlations at half-band filling. In this way, a more accurate description of W is obtained throughout the domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}},$ including the crossover from weak to strong correlations. As examples of applications results are given for the ground-state energy, charge-excitation gap, and charge susceptibility of the Hubbard model in one-, two-, and three-dimensional lattices. The performance of the method is demonstrated by comparison with available exact solutions, with numerical calculations, and with LDFT using a simpler dimer ansatz for W. Goals and limitations of the different approximations are discussed.