Richardson-Gaudin integrability in the contraction limit of the quasispin

TL;DR

This paper explores the critical behavior of Richardson-Gaudin equations in the BCS Hamiltonian using a pseudo-deformation approach, revealing how the Pauli principle influences integrability and collective states.

Contribution

It introduces a pseudo-deformation of the quasispin algebra to connect Bethe Ansatz states with bosonic states, clarifying the physical interpretation of Richardson-Gaudin variables.

Findings

Critical behavior observed in Richardson-Gaudin variables.

Bosonic multiphonon states change nature at critical points.

Pauli exclusion principle drives the critical behavior.

Abstract

Background: The reduced, level-independent, Bardeen-Cooper-Schrieffer Hamiltonian is exactly diagonalizable by means of a Bethe Ansatz wavefunction, provided the free variables in the Ansatz are the solutions of the set of Richardson-Gaudin equations. On the one side, the Bethe Ansatz is a simple product state of generalised pair operators. On the other hand, the Richardson-Gaudin equations are strongly coupled in a non-linear way, making them prone to singularities. Unfortunately, it is non-trivial to give a clear physical interpretation to the Richardson-Gaudin variables because no physical operator is directly related to the individual variables. Purpose: The purpose of this paper is to shed more light on the critical behavior of the Richardson-Gaudin equations, and how this is related to the product wave structure of the Bethe Ansatz. Method: A pseudo-deformation of the quasi-spin algebra is introduced, leading towards a Heisenberg-Weyl algebra in the contraction limit of the deformation parameter. This enables an adiabatic connection of the exact Bethe Ansatz eigenstates with pure bosonic multiphonon states. The physical interpretation of this approach is an adiabatic suppression of the Pauli exclusion principle. Results: The method is applied to a so-called "picket-fence" model for the BCS Hamiltonian, displaying a typical critical behavior in the Richardson-Gaudin variables. It was observed that the associated bosonic multiphonon states change collective nature at the critical interaction strengths of the Richardson-Gaudin equations. Conclusions: The Pauli exclusion principle is the main responsible for the critical behavior of the Richardson-Gaudin equations, which can be suppressed by means of a pseudo deformation of the quasispin algebra.