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Journal of Physics A Mathematical and Theoretical

A complete set of commuting operators for the Bethe ansatz

Tadeusz Lulek, S. Topolewicz

Published Nov 11, 2010• 0 citations
Publisher

Abstract

We have derived an explicit formula for a complete set of commuting operators for the XXX model of the Heisenberg magnetic ring, using an algebraic Bethe ansatz approach of Faddeev and Takhtajan. Each operator turns out to be the sum of increasing l -cycles in the symmetric group acting on the set of nodes of the ring. It is demonstrated that the resulting algebra of operators encloses the total spin S , the Hamiltonian and the quasimomentum. We point out that it is a maximal Abelian subalgebra in the algebra of the symmetric group, associated with the basis of exact Bethe ansatz eigenstates, the latter classified by rigged string configurations of Kerrov, Kirillov and Reshetikhin. This algebra is also conjugated to the Jucys–Murphy algebra, responsible for the Young orthogonal basis of standard tableaux along the Schur–Weyl duality.

DOI
10.1088/1751-8113/43/49/495201

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We have derived an explicit formula for a complete set of commuting operators for the XXX model of the Heisenberg magnetic ring, using an algebraic Bethe ansatz approach of Faddeev and Takhtajan. Each operator turns out to be the sum of increasing l -cycles in the symmetric group acting on the set of nodes of the ring. It is demonstrated that the resulting algebra of operators encloses the total spin S , the Hamiltonian and the quasimomentum. We point out that it is a maximal Abelian subalgebra in the algebra of the symmetric group, associated with the basis of exact Bethe ansatz eigenstates, the latter classified by rigged string configurations of Kerrov, Kirillov and Reshetikhin. This algebra is also conjugated to the Jucys–Murphy algebra, responsible for the Young orthogonal basis of standard tableaux along the Schur–Weyl duality.
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)
Quantum kinematics of the linear Heisenberg ring, consisting of N crystal nodes, each with spin s, is presented in terms of the Weyl duality between actions of the symmetric and unitary groups in the space of quantum states of the magnet. This space is spanned on magnetic configurations which gives rise to an application of the combinatorial Robinson–Schensted–Knuth algorithm for a unique classification of irreducible basis of the duality of Weyl in terms of a pair of tableaux: a standard Young tableau in the alphabet of nodes, accompanied by a semistandard Weyl tableau in the alphabet of spins. Similarities and distinctions between various group-theoretic and combinatorial objects are discussed within the context of the magnetic interpretation. In particular, the role of the spectrum of Jucys–Murphy operators in the clasification and construction of magnetic eigenstates corresponding to Young tableaux is illustrated.
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 Robinson-Schensted and Kerov-Kirillov-Reshetikhin (RSKKR) bijections allow us to confirm the completeness of solutions of the eigenproblem of the one-dimensional Heisenberg Hamiltonian in a purely combinatorial manner, by studying the structure of the classical configuration space of the system. The combined bijection relates two sets, namely the basis of magnetic configurations and the set of combinatorial objects called rigged string configurations. The former serve as the initial basis for quantum computations, whereas the latter classify the exact Bethe Ansatz (BA) eigenstates. We discuss in this report the application of this bijection in the procedure of construction of two-particle states within Clebsch-Gordan scheme. This bijection provides the irreducible bases for the decomposition of the tensor product of transitive representations R ( Σ N :(Σ N-1 × Σ 1 ) within the scheme of a Hopf algebra of symmetric groups. We point out the role of true physical BA eigenstates, as well as fictituous configurations corresponding to doubly occupied sites (the diagonal of the cartesian square of the one-magnon classical configuration space) and to antisymmetric states.
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.