Exact solution of the two-axis countertwisting Hamiltonian for the half-integer $J$ case

TL;DR

This paper derives exact Bethe ansatz solutions for the two-axis countertwisting Hamiltonian for both integer and half-integer total angular momentum J, revealing symmetric energy levels and asymptotic quadratic energy relations.

Contribution

It provides the first exact solutions for half-integer J cases using Bethe ansatz and extended Heine-Stieltjes polynomials, expanding understanding of the Hamiltonian's spectrum.

Findings

Solutions for both integer and half-integer J obtained

Energy levels exhibit symmetry about E=0 for half-integer J

Excitation energies asymptotically follow quadratic functions of J

Abstract

Bethe ansatz solutions of the two-axis countertwisting Hamiltonian for any (integer and half-integer) $J$ are derived based on the Jordan-Schwinger (differential) boson realization of the $SU(2)$ algebra after desired Euler rotations, where $J$ is the total angular momentum quantum number of the system. It is shown that solutions to the Bethe ansatz equations can be obtained as zeros of the extended Heine-Stieltjes polynomials. Two sets of solutions, with solution number being $J+1$ and $J$ respectively when $J$ is an integer and $J+1/2$ each when $J$ is a half-integer, are obtained. Properties of the zeros of the related extended Heine-Stieltjes polynomials for half-integer $J$ cases are discussed. It is clearly shown that double degenerate level energies for half-integer $J$ are symmetric with respect to the $E=0$ axis. It is also shown that the excitation energies of the `yrast' and other `yrare' bands can all be asymptotically given by quadratic functions of $J$, especially when $J$ is large.