Particle-hole duality, integrability, and Russian doll BCS model

TL;DR

This paper explores the particle-hole symmetry in the integrable Russian doll BCS model, deriving exact relations between Bethe roots in different representations and revealing simplified forms at specific filling and energy level conditions.

Contribution

It introduces exact relations between Bethe roots in particle and hole representations of the Russian doll BCS model, enhancing understanding of its symmetry and integrability.

Findings

Particle-hole symmetry exists at the Hamiltonian level.

Derived exact relations between Bethe roots in different representations.

Simplified relations at half-filling and symmetric energy levels.

Abstract

We address a generalized Richardson model (Russian doll BCS model), which is characterized by the breaking of time-reversal symmetry. This model is known to be exactly solvable and integrable. We point out that the Russian doll BCS model, on the level of Hamiltonian, is also particle-hole symmetric. This implies that the same state can be expressed both in the particle and hole representations with two different sets of Bethe roots. We then derive exact relations between Bethe roots in the two representations, which can hardly be obtained staying on the level of Bethe equations. In a quasi-classical limit, similar identities for usual Richardson model, known from literature, are recovered from our results. We also show that these relations for Richardson roots take a remarkably simple form at half-filling and for a symmetric with respect to the middle of the interaction band distribution of one-body energy levels, since, in this special case, the rapidities in the particle and hole representations up to the translation satisfy the same system of equations.