Quantum Invariants of the Pairing Hamiltonian

TL;DR

This paper identifies quantum invariants in the degenerate orbit pairing problem, diagonalizes them using Bethe ansatz, and explores their relation to Gaudin magnet Hamiltonians, revealing a shared algebraic structure.

Contribution

It introduces quantum invariants for the degenerate orbit pairing problem and demonstrates their diagonalization via Bethe ansatz, linking them to Gaudin algebra.

Findings

Quantum invariants are identified in the degenerate orbit limit.

These invariants are diagonalized using Bethe ansatz.

A symmetry relating eigenvalues for different pair numbers is discussed.

Abstract

Quantum invariants of the orbit dependent pairing problem are identified in the limit where the orbits become degenerate. These quantum invariants are simultaneously diagonalized with the help of the Bethe ansatz method and a symmetry in their spectra relating the eigenvalues corresponding to different number of pairs is discussed. These quantum invariants are analogous to the well known rational Gaudin magnet Hamiltonians which play the same role in the reduced pairing case (i.e., orbit independent pairing with non degenerate energy levels). It is pointed out that although the reduced pairing and the degenerate cases are opposite of each other, the Bethe ansatz diagonalization of the invariant operators in both cases are based on the same algebraic structure described by the rational Gaudin algebra.