Exact solution of the two-axis countertwisting Hamiltonian

TL;DR

This paper presents an exact solution for the two-axis countertwisting Hamiltonian using algebraic Bethe ansatz, revealing complete solutions and degeneracy properties for integer angular momentum quantum numbers.

Contribution

It introduces an exact solution method for the Hamiltonian leveraging SU(1,1) algebra and Bethe ansatz, demonstrating completeness and degeneracy features.

Findings

Exact solutions derived for integer angular momentum J.

Solutions correspond to zeros of Heine-Stieltjes polynomials.

Degeneracy in energy levels occurs in the large J limit.

Abstract

It is shown that the two-axis countertwisting Hamiltonian is exactly solvable when the quantum number of the total angular momentum of the system is an integer after the Jordan-Schwinger (differential) boson realization of the SU(2) algebra. Algebraic Bethe ansatz is used to get the exact solution with the help of the SU(1,1) algebraic structure, from which a set of Bethe ansatz equations of the problem is derived. It is shown that solutions of the Bethe ansatz equations can be obtained as zeros of the Heine-Stieltjes polynomials. The total number of the four sets of the zeros equals exactly to $2J+1$ for a given integer angular momentum quantum number $J$, which proves the completeness of the solutions. It is also shown that double degeneracy in level energies may also occur in the $J\rightarrow\infty$ limit for integer $J$ case except a unique non-degenerate level with zero excitation energy.