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

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.

LaTex 12 pages, 3 figures. Major cosmetic type revision. arXiv admin note: text overlap with arXiv:1609.05581

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• Squeezing and entanglement in a Bose-Einstein condensate
• Fisher Information and entanglement of non-Gaussian spin states
• Exact spectrum of the Lipkin-Meshkov-Glick model in the thermodynamic limit and finite-size corrections
• Gaudin models solver based on the Bethe ansatz/ordinary differential equations correspondence
• Exact solution of the two-axis countertwisting Hamiltonian

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