Evaluation of the BCS Approximation for the Attractive Hubbard Model in One Dimension

TL;DR

This paper evaluates the accuracy of the BCS approximation for the attractive Hubbard model in one dimension, comparing it to exact solutions across various parameters, and finds significant discrepancies in the energy gap predictions.

Contribution

It provides a detailed comparison between BCS and exact solutions for the 1D attractive Hubbard model, highlighting limitations of BCS in predicting the energy gap.

Findings

Ground state energies agree well between methods

BCS energy gap often significantly inaccurate

Finite size effects are briefly discussed

Abstract

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 wavefunction. 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.