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Journal of Physics Conference Series

Bethe Ansatz solutions for two qudits

M. Łabuz, Tadeusz Lulek

Published Mar 1, 2010• 0 citations
Publisher

Abstract

We use the famous Weyl recipe to introduce Bethe Ansatz (BA) solutions for the two-magnon sector in the case of the Heisenberg magnet consisting of a small number N of nodes. In accordance with the Weyl recipe, the obvious symmetry group of the model is the translation group C N , whereas the group of all automorphisms Aut C N constitutes the hidden symmetry, and, in fact, manifests important symmetry properties of the system. With the use of the action of the Aut C N on the set of eigenvalues of BA, we propose the way of drawing energy bands for the problem being discussed.

DOI
10.1088/1742-6596/213/1/012019

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We use the famous Weyl recipe to introduce Bethe Ansatz (BA) solutions for the two-magnon sector in the case of the Heisenberg magnet consisting of a small number N of nodes. In accordance with the Weyl recipe, the obvious symmetry group of the model is the translation group C N , whereas the group of all automorphisms Aut C N constitutes the hidden symmetry, and, in fact, manifests important symmetry properties of the system. With the use of the action of the Aut C N on the set of eigenvalues of BA, we propose the way of drawing energy bands for the problem being discussed.
Bethe Ansatz is the famous method of determination of eigenstates and eigenenergies for a wide range of quantum problems, e.g. for the Heisenberg XXX s = 1/2 model.The Bethe equations applied to solve the problem of N nodes and r overturned spins on a magnetic chain are labeled by sets of winding numbers {ni}, however the condition for admissible sets give an overcomplete number of results.On the other hand, combinatorial objects, so called paths, give the exact number of eigenvectors for the problem described by (N, r) values.The paper presents the method of determining the set of winding numbers from the appropriate path for the sector of r = 2 spin deviations.
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.
The kinematics of the one-dimensional finite Heisenberg magnetic ring with impurities is discussed in the light of the general recipe of Weyl (1952). The cyclic group generating the ring, and the group of all its automorphisms, play the roles of the obvious and hidden symmetry respectively. The hidden symmetry group imposes a property of the distribution of quantum states of a translationally invariant ensemble of magnets over the finite Brillouin zone. Inhomogeneities of the distribution arise from irregular orbits of the action of the cyclic group on the set of all magnetic configurations of the ensemble. Size-dependent effects, like rarefied bands, are classified using the arithmetic structure of integers, (prime numbers, socles and arithmetic exponents), which is more appropriate than the linear order in the ring of integers.
In the present paper we explore some useful devices for discussing the exact solutions of XXX isotropic Heisenberg Hamiltonian for the single node spin equal to 1/2 as the tensor product states. Our aim is to presents the monodromy matrix and the Lax operator in the contex of the Bethe Ansatz, signed from now on BA. We construct the former, for N equal to 4 nodes of the magnet, using the so called auxiliary space which is taken as a copy of 2 . The form of this matrix in the basis of orbits of the translation group C 4 reveals its block structure of all possible nonzero elements. Each block has its meaning in the language of creation and anihilation of the magnon. This fact implies, that one can think about appropriate operators and create the theory very similar to that of the quantum oscillator.
Abstract Bethe Ansatz provides an exact classification of eigenstates of the Heisenberg Hamiltonian for a finite magnetic ring, consisting of N nodes, each with the spin s = ½ (with some extensions to an arbitrary spin s) in terms of rigged string configurations. The latter are some combinatorial objects which serve as classification labels for solutions of Bethe equations. An astonishing feature is existence of Robinson–Schensted (RS) and Kerov–Kirillov–Reshetikhin (KKR) bijections between sets of (i) all magnetic configurations, (ii) all pairs of standard Young and Weyl tableaux of N boxes and n = 2s + 1 rows, (iii) all rigged string configurations, for given N and s. These bijections allow to point out an exact correspondence between physically admissible solutions of highly nonlinear Bethe Ansatz equations and the initial basis of quantum calculations – magnetic configurations which are just possible distributions of spins over the nodes. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The 1D Heisenberg spin model with an anisotropy of the XXZ type is analysed in terms of the symmetry given by the quantum Galilei group Gamma q (1). For a chain with an infinite number of sites the authors show that the magnon excitations and the s=1/2, n-magnon bound states are determined by the algebra. In this case the Gamma q (1) symmetry provides a description naturally compatible with the Bethe ansatz. The recurrence relations determined by Gamma q (1) permit one to express the energy of the n-magnon bound states in a closed form in terms of Tchebischeff polynomials.