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Open Systems & Information Dynamics

Arithmetic of Partial Fibres in Relative Position Space of Bethe Ansatz

Jan Milewski, Grzegorz Banaszak, Tadeusz Lulek

Published Mar 1, 2010• 3 citations
Publisher

Abstract

We consider fibration of the relative position space for the system of r Bethe pseudoparticles on a ring of N nodes. We define primary and secondary configurations in the relative position space. We study cardinalities of the set of primary configurations and of the partial fibres of the fibration.
DOI
10.1142/s1230161210000072

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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)
It is well known that the space of all quantum states of the XXX model for a magnetic ring of N nodes, each with the spin 1/2, decomposes into sectors with r spin deviations, r = 0,1, 2,..., N [1, 2, 3, 4]. The sectors r and N – r are related mutually by the particle-hole transformation which exchanges the signs + and – on each node. We discuss here effects of this transformation on the formalism of algebraic Bethe Ansatz, in particular on the form of the monodromy matrix, the main tool of this formalism. We derive explicitly appropriate transformation rules for C N - orbits of magnetic configurations and the corresponding Fourier transformations within the bases of wavelets. In particular, we point out some important phase relations between orbits on both sides of the equator.