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Acta Physica Polonica A

Paths vs. Winding Numbers for the Two-Magnon Sector of the XXX Heisenberg Magnet

M. Łabuz

Published Aug 1, 2015• 2 citations
Publisher

Abstract

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.

DOI
10.12693/aphyspola.128.190

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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.
Abstract Every solution of the Bethe ansatz equations (BAEs) is characterized by a set of quantum numbers called the Bethe quantum numbers, which are fundamental for evaluating each Bethe root numerically. We rigorously derive the Bethe quantum numbers for the real solutions of the spin-1/2 massive XXZ spin chain in the two down-spin sector, assuming the existence of solutions to some form of BAE. In the sector the quantum numbers J 1 and J 2 were derived for complex solutions, but not for real solutions. We show the exact results in the sector as follows. (i) When two Bethe quantum numbers are different, i.e. for J 1 ≠ J 2 , we introduce a graphical method, which we call a contour method, for deriving the solution of BAE to a given set of Bethe quantum numbers. By the method, we can readily show the existence and the uniqueness of the solution. (ii) When two Bethe quantum numbers are equal, i.e. for J 1 = J 2 , we derive the criteria for the collapse of two-strings and the emergence of an extra two-string by an analytic method. (iii) We obtain the number of real solutions, which depends on the site number N and the XXZ anisotropy parameter ζ. (iv) We derive all infinite-valued solutions of BAE for the XXX spin chain in the two down-spin sector through the XXX limit. (v) We explicitly show the completeness of the Bethe ansatz in terms of the Bethe quantum numbers.
Abstract Every solution of the Bethe-ansatz equations (BAEs) is characterized by a set of quantum numbers, by which we can evaluate it numerically. However, no general rule is known how to give quantum numbers for the physical solutions of BAE. For the spin-1/2 XXX chain we rigorously derive all the quantum numbers for the complete set of the Bethe-ansatz eigenvectors in the two down-spin sector with any chain length N . Here we obtain them both for real and complex solutions. We also show that all the solutions associated with them are distinct. Consequently, we prove the completeness of the Bethe ansatz and give an exact expression for the number of real solutions which correspond to collapsed bound-state solutions (i.e., two-string solutions) in the sector: in terms of Gauss’ symbol. Moreover, we prove in the sector the scheme conjectured by Takahashi for solving BAE systematically. We also suggest that by applying the present method we can derive the quantum numbers for the spin-1/2 XXZ chain.
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)
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.
We investigate Bethe Ansatz equations for the one-dimensional spin-12\frac{1}{2}21​ Heisenberg XXX chain with a special interest in a finite system. Solutions for the two-particle sector are obtained. The ground state in antiferromagnetic case has been analytically studied through the logarithmic form of Bethe Ansatz equations.