Different Approximations for Hubbard Model

Nathan Salwen, Steven A. Sheets, Stephen R. Cotanch

We extend previous studies of the (BCS) canonical approach for the attractive Hubbard model. A derivation of the BCS formulation is presented for both the Hubbard model and a simpler reduced Hamiltonian. Using direct diagonalization, exact one- and two-dimensional solutions for both Hamiltonians are compared to BCS variational calculations. Approximate and exact ground state energies and energy gaps are computed for different electron number systems as well as correlation observables not previously predicted. Reproducing published one-dimensional findings, the BCS approach is an excellent approximation for the Hubbard ground state energy but not the energy gap, a finding that remains true in two dimensions. Propagators and correlators are found more sensitive to wave functions and appreciable differences are computed with the Hubbard model exhibiting a weaker degree of superconductivity than the BCS. Even for weak coupling pronounced differences are found between the BCS and Hubbard on-site correlators. Significantly, in the extreme strong coupling limit the Hubbard model exhibits no dimensional crossover, or density induced transitions, to long-range order in two dimensions. However, for the reduced Hamiltonian the BCS approach is an excellent approximation for all observables in both one and two dimensions.

Cary Mcilwaine Maguire

K. Elk

Abstract Considering a very general derivation of Dyson's equation an approximation procedure is discussed, which exhibits a summary of different known methods. Concerning the Hubbard model a generalization of Roth's procedure follows. In particular it is possible to obtain a self‐consistent method.

A. Sherman

J. E. Hirsch

A. Ziegler

The deficiencies of the fermion cluster approximation by Caron and Pratt out-lined in a previous paper are corrected and the approximation is then applied to the one-site Hubbard model. Because the approximation in its standard form is unable to describe energy bands the boundary condition integration technique of C. Gros for free clusters is generalized to embedded clusters by averaging over the solutions to different self-consistency equations.