BCS and Attractive Hubbard Model Comparative Study

TL;DR

This study compares the BCS approximation to exact solutions of the attractive Hubbard model in one and two dimensions, revealing its strengths and limitations in predicting energies, gaps, and correlations.

Contribution

It provides a detailed comparison between BCS variational calculations and exact diagonalization for both the Hubbard and reduced Hamiltonian models in 1D and 2D.

Findings

BCS accurately predicts ground state energy in Hubbard model

BCS does not accurately predict energy gaps in Hubbard model

BCS is highly accurate for all observables in the reduced Hamiltonian model

Abstract

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 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 is an excellent approximation for the Hubbard ground state energy but not energy gap, a finding that remains true in two dimensions. Propagators and correlators are found more sensitive to wavefunctions and appreciable differences are computed with the Hubbard model exhibiting a weaker degree of superconductivity than the BCS. However for the reduced Hamiltonian model the BCS is an excellent approximation for all observables in both one and two dimensions.