Canonical BCS Approximation for the Attractive Hubbard Model

TL;DR

This paper evaluates the effectiveness of canonical BCS wave functions in modeling the attractive Hubbard model, comparing results with exact solutions and highlighting improvements in energy gap predictions due to fixed electron number.

Contribution

It introduces a canonical BCS approach for the attractive Hubbard model and demonstrates its accuracy and convergence properties compared to grand-canonical methods.

Findings

Canonical BCS results agree well with exact solutions for ground state energy.

Energy gap improvements are observed when conserving electron number.

Results converge to grand-canonical results as system size increases.

Abstract

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.