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Journal of Physics Conference Series

A fibration of the hypercubic lattice in Bethe ansatz

Jan Milewski, E Ambrożko

Published Mar 1, 2008• 6 citations
Publisher

Abstract

We show that the configuration space in the Bethe Ansatz is a locally hypercubic lattice with some boundary. The lattice admits a fibre structure. The base of the fibration has a meaning of the mass centre space (with a suitable scale; indeed, we use the sum of position of the deviations) of configurations and the fibre is the relative space of configurations. The structure of the fibre depends on the point in the base (it means that on the position of the mass centre).

DOI
10.1088/1742-6596/104/1/012040

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We show that the configuration space in the Bethe Ansatz is a locally hypercubic lattice with some boundary. The lattice admits a fibre structure. The base of the fibration has a meaning of the mass centre space (with a suitable scale; indeed, we use the sum of position of the deviations) of configurations and the fibre is the relative space of configurations. The structure of the fibre depends on the point in the base (it means that on the position of the mass centre).
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.
We demonstrate that the seminal one-dimensional model of the Heisenberg magnet, consisting of N spins 1/2 with the nearest-neighbour isotropic interaction, solved exactly by Bethe ansatz, admits an interpretation of a system of r = N/2 -M pseudoparticles (spin deviations) which are indistinguishable, have hard cores and move on the chain by local hoppings.Such an approach allows us to construct a manifold with some boundaries, which is genericly r-dimensional, and whose F -dimensional regions, 0 < F < r, point out all l-strings.The latter classify, in terms of rigged string configurations of Kerov, Kirillov and Reshetikhin, all exact Bethe eigenfunctions.In this way, we interpret these eigenfunctions in terms of the classical configuration space, in particular on the structure of islands of adjacent spin deviations, in a way independent of the size N .
The method of q -oscillator lattices, proposed recently in Bazhanov and Sergeev 2005 ( Preprint hep-th/0509181), provides the tool for a construction of various integrable models of quantum mechanics in (2 + 1)-dimensional spacetime. In contrast to any one-dimensional quantum chain, its two-dimensional generalizations—quantum lattices—admit different geometrical structures. In this paper, we consider the q -oscillator model on a special lattice. The model may be interpreted as a two-dimensional lattice Bose gas. The most remarkable feature of the model is that it allows the coordinate Bethe ansatz: the p -particles' wavefunction is the sum of plane waves. Consistency conditions is the set of 2 p equations for p one-particle wave vectors. These 'Bethe ansatz' equations are the main result of this paper.