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arXiv:hep-th/9403107 · Journal of Physics A Mathematical and General

Completeness of Bethe's states for the generalizedXXZmodel

Anatol N. Kirillov, Nadejda A Liskova

Published Feb 21, 1997• 35 citations
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Abstract

We study the Bethe ansatz equations for a generalized XXZ model on a one-dimensional lattice. Assuming the string conjecture we propose an integer version for vacancy numbers and prove a combinatorial completeness of Bethe's states for a generalized XXZ model. We find an exact form for the inverse matrix related with vacancy numbers and compute its determinant. This inverse matrix has a tridiagonal form, generalizing the Cartan matrix of type A.

DOI
10.1088/0305-4470/30/4/022

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We propose a Bethe-Ansatz-type solution of the open spin-1/2 integrable XXZ quantum spin chain with general integrable boundary terms and bulk anisotropy values i \\pi/(p+1), where p is a positive integer. All six boundary parameters are arbitrary, and need not satisfy any constraint. The solution is in terms of generalized T - Q equations, having more than one Q function. We find numerical evidence that this solution gives the complete set of 2^N transfer matrix eigenvalues, where N is the number of spins.
In this paper we count the numbers of real and complex solutions to Bethe constraints in the two-particle sector of the XXZ\mathrm{XXZ}XXZ model. We find the exact number of exceptions to the string conjecture and total number of solutions that is required for completeness.
The main purpose of this report is a thorough analysis of completeness of solutions of the one-dimensional Heisenberg Hamiltonian through the hypothesis of strings. A somehow astonishing conclusion emerges from studying of the structure of the classical configuration space of this system. Namely, all allowed information concerning quantum states, which are exact solutions of the Bethe equations, encoded in quantum numbers, are predictable via a bijection between the set of the magnetic configurations and the string configurations. This startling and beautiful observation constitutes the proof of the completeness of the eigenstates of the Heisenberg Hamiltonian, deduced in a purely combinatorial way. We interpret the set of all magnetic configurations with a fixed number r of spin deviations as the classical configuration space of a hypothetic system of r Bethe pseudoparticles, which move, in a stroboscopic manner, on the magnetic ring. The geometry of this configuration space, induced by the action of Heisenberg Hamiltonian and the translation symmetry group of the ring, implies the structure of a locally r -dimensional hypercubic lattice with well defined F -dimensional boundaries, 1 ⩽ F ⩽ r . We demonstrate that rigged string configurations originate from these boundaries, depending upon the island structure of spin deviations. We show that a relatively simple combinatoric definition of rigged strings reproduces completely exact results of Bethe Ansatz. It is expressed in terms of a combined bijection: Robinson-Schensted with Kerov- Kirillov-Reshetikhin (RSKKR) which produces a geography of exact Bethe Ansatz solutions on the classical configuration space.
The authors analyse the Bethe ansatz equations for the two-particle sector of the spin 1/2 Heisenberg XXX model on a one-dimensional lattice of length N. They show that, beginning at a critical lattice length of N=21.86, new pairs of real solutions develop, whereas complex solutions start to disappear. The integers (that appear in the logarithmic form of the Bethe equations) of the new solutions do not fit into the conventional classification scheme. The total number of solutions in the two-particle sector remains unchanged and is in agreement with the claim that the SU(2) extended Bethe ansatz gives a complete set of 2 N eigenstates.