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arXiv:1705.04117 · Journal of Statistical Mechanics Theory and Experiment

Bethe ansatz for two-magnon scattering states in 2D and 3D Heisenberg–Ising ferromagnets †

P. N. Bibikov

Published Apr 1, 2018• 4 citations
arXivHTMLPDF

Abstract

Abstract Two different versions of Bethe ansatz are suggested for evaluation of scattering two-magnon states in 2D and 3D Heisenberg–Ising ferromagnets on square and simple cubic lattices. It is shown that the two-magnon sector is subdivided on two subsectors related to non-interacting and scattering magnons. The former subsector possess an integrable regular dynamics and may be described by a natural modification of the usual Bethe Ansatz. The latter one is characterized by a non-integrable chaotic dynamics and may be treated only within discrete degenerative version of Bethe Ansatz previously suggested by the author. Some of these results are generalized for multi-magnon states of the Heisenberg–Ising ferromagnet on a D dimensional hyper cubic lattice.

DOI
10.1088/1742-5468/aab84e

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Abstract Two different versions of Bethe ansatz are suggested for evaluation of scattering two-magnon states in 2D and 3D Heisenberg–Ising ferromagnets on square and simple cubic lattices. It is shown that the two-magnon sector is subdivided on two subsectors related to non-interacting and scattering magnons. The former subsector possess an integrable regular dynamics and may be described by a natural modification of the usual Bethe Ansatz. The latter one is characterized by a non-integrable chaotic dynamics and may be treated only within discrete degenerative version of Bethe Ansatz previously suggested by the author. Some of these results are generalized for multi-magnon states of the Heisenberg–Ising ferromagnet on a D dimensional hyper cubic lattice.
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Bethe Ansatz approach to the three-magnon problem in an infinite XXXXXX spin chain with Izing next neighboring ferromagnetic interaction
Using the traditional and degenerative discrete-diffractive versions of Bethe Ansatz we construct a rude set of three-magnon states in the infinite XXXXXX spin chain with Izing next neighboring ferromagnetic term.
A three-magnon problem for an exactly rung-dimerized spin ladder is brought up separately in all total spin sectors. At first, a special duality transformation of the Schrödinger equation is found within the general outlook. Then the problem is treated within the coordinate Bethe ansatz. A straightforward approach is developed to obtain pure scattering states. At values S = 0 and S = 3 of total spin, the Schrödinger equation has a form inherent in the XXZ chain. At S = 1, 2, solvability holds only in five previously found completely integrable cases. Nevertheless, even in a general non-integrable case, there are some special Bethe solutions in both S = 1 and S = 2 sectors. Pure scattering states in all total spin sectors are presented explicitly.
XXX Heisenberg s1{2 model has been examined in detail during last decades, however, recently one may find some new insights into that issue.Among several approaches describing the eigenproblem for the finite case, a close look into the structure of Bethe equations (BE) for the two-magnon sector case seems to be particularly interesting.BE enable us to evaluate parameters labeling eigenstates of a magnet, however to find appropriate sets of winding numbers, which parametrize BE, one has to apply the Inverse Bethe Ansatz method.On the other hand, one may choose a different -combinatoric approach -which also parametrizes Bethe eigenstates, with the use of rigging numbers describing string configurations.We present an idea of comparison of the concepts mentioned above for the particular case of two-spin deviations sector.
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