Ground-state energy of the Hubbard model in the BCS approximation

Jacek Wojtkiewicz, Piotr H. Chankowski

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.

W. D. Langer, Daniel Charles Mattis

F. Marsiglio

The ground-state energy and energy gap to the first excited state are calculated for the attractive Hubbard model in one dimension using both the Bethe ansatz equations and the variational BCS wave function. Comparisons are provided as a function of coupling strength and electron density. While the ground-state energies are always in very good agreement, the BCS energy gap is sometimes incorrect by an order of magnitude, particularly at half-filling. Finite-size effects are also briefly discussed for cases where an exact solution in the thermodynamic limit is not possible. In general, the BCS result for the energy gap is poor compared to the exact result.

Robert Seiringer, Jun O. S. Yin

K. Tanaka, F. Marsiglio

We test the canonical BCS wave functions for fixed number of electrons for the attractive Hubbard model. We present results in one dimension for various chain lengths, electron densities, and coupling strengths. The ground-state energy and energy gap to the first excited state are compared with the exact solutions obtained by the Bethe ansatz as well as with the results from the conventional grand-canonical BCS approximations. While the canonical and grand-canonical BCS results are both in very good agreement with the exact results for the ground state energies, improvements due to conserving the electron number in finite systems are manifest in the energy gap. As the system size is increased, the canonical results converge to the grand-canonical ones. The ``parity'' effect that arises from the number parity (even or odd) of electrons are studied with our canonical scheme.