The Tamm-Dancoff Approximation as the boson limit of the Richardson-Gaudin equations for pairing

TL;DR

This paper establishes a theoretical link between the Tamm-Dancoff Approximation and the Richardson-Gaudin equations for pairing, using a parametrized algebra that interpolates between quasi-spin and bosonic limits.

Contribution

It introduces a parametrized algebra that connects the exact eigenstates of the BCS Hamiltonian with the Tamm-Dancoff Approximation, revealing their relationship in a unified framework.

Findings

Derived Richardson-Gaudin equations from the algebraic framework.

Showed the Tamm-Dancoff Approximation as a bosonic limit of the Richardson-Gaudin equations.

Provided an in-depth example illustrating the theoretical connection.

Abstract

A connection is made between the exact eigen states of the BCS Hamiltonian and the predictions made by the Tamm-Dancoff Approximation. This connection is made by means of a parametrised algebra, which gives the exact quasi-spin algebra in one limit of the parameter and the Heisenberg-Weyl algebra in the other. Using this algebra to construct the Bethe Ansatz solution of the BCS Hamiltonian, we obtain parametrised Richardson-Gaudin equations, leading to the secular equation of the Tamm-Dancoff Approximation in the bosonic limit. An example is discussed in depth.