Order parameter, correlation functions and fidelity susceptibility for the BCS model in the thermodynamic limit

TL;DR

This paper investigates the exact ground state of the reduced BCS Hamiltonian in the thermodynamic limit, comparing it with the BCS ansatz and revealing both agreements and persistent discrepancies in correlation functions and fidelity susceptibility.

Contribution

It provides exact numerical analysis of the BCS model's ground state for large systems, highlighting non-perturbative corrections to BCS predictions.

Findings

Order parameter equals the largest eigenvalue of Yang's reduced density matrix.

Ground state energy and level occupations converge to BCS values for small systems.

Discrepancies remain in pair-pair correlations and fidelity susceptibility even at large sizes.

Abstract

The exact ground state of the reduced BCS Hamiltonian is investigated numerically for large system sizes and compared with the BCS ansatz. A "canonical'' order parameter is found to be equal to the largest eigenvalue of Yang's reduced density matrix in the thermodynamic limit. Moreover, the limiting values of the exact analysis agree with those obtained for the BCS ground state. Exact results for the ground state energy, level occupations and a pseudospin-pseudospin correlation function are also found to converge to the BCS values already for relatively small system sizes. However, discrepancies persist for a pair-pair correlation function, for inter-level correlations of occupancies and for the fidelity susceptibility, even for large system sizes where these quantities have visibly converged to well-defined limits. Our results indicate that there exist non-perturbative corrections to the BCS predictions in the thermodynamic limit.