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arXiv:1610.05905 · Journal of Statistical Mechanics Theory and Experiment

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

Feng Pan, Yao-Zhong Zhang, Jerry P Draayer

Published Feb 1, 2017• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-5468/aa5a28

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Abstract Bethe ansatz solution of the two-axis two-spin Hamiltonian is derived based on the Jordan–Schwinger boson realization of the SU(2) algebra. It is shown that the solution of the Bethe ansatz equations can be obtained as zeros of the related extended Heine–Stieltjes polynomials. Symmetry properties of excited levels of the system and zeros of the related extended Heine–Stieltjes polynomials are discussed. As an example of an application of the theory, the two equal spin case is studied in detail, which demonstrates that the levels in each band are symmetric with respect to the zero energy plane perpendicular to the level diagram and that excited states are always well entangled.
Abstract We present an exact solution to the asymptotic Bethe equation of weakly anisotropic Heisenberg spin chain, which is a set of non-linear algebraic equations. The solution describes the low-energy excitations above ferromagnetic ground state with fixed magnetisation, and it has a close relation to generalised Jacobi polynomial. It is equivalent to a generalised Stieltjes problem and in the continuous limit, it becomes a Riemann–Hilbert problem closely related to the finite-gap solutions of classical Landau–Lifshitz field theory.
Evidence is presented that the solutions of the Bethe ansatz equations for spin-½ isotropic Heisenberg chains in fixed total spin and momentum sectors are the roots of single variable polynomials with integer (or integer based) coefficients. Such solutions are used as a starting point for investigation of long chain (critical region) properties. In the total spin S = 0 sector I conjecture explicit formulae for the Bethe string configuration labelling of all left and right tower excitations in the k = 1, SU(2) Wess-Zumino-Witten model.
A new approach for solving the Bethe ansatz (Gaudin-Richardson) equations of the standard pairing problem is established based on the Heine-Stieltjes correspondence. For kkk pairs of valence nucleons on nnn different single-particle levels, it is found that solutions of the Bethe ansatz equations can be obtained from one $(k+1)\ifmmode\times\else\texttimes\fi{}(k+1)$ and one $(n\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(k+1)$ matrices, which are associated with the extended Heine-Stieltjes and Van Vleck polynomials, respectively. Since the coefficients in these polynomials are free from divergence with variations in contrast to the original Bethe ansatz equations, the approach provides an efficient and systematic way to solve the problem, which by extension, can also be used to solve a large class of Gaudin-type quantum many-body problems, including an efficient angular momentum projection method for multiparticle systems.
Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.