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

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. (paper: quantum statistical physics, condensed matter, integrable systems)