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arXiv:solv-int/9705005

Some eigenstates for a model associated with solutions of tetrahedron equation. V. Two cases of string superposition

I. G. Korepanov

Published May 6, 1997• 0 citations
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Abstract

In paper IV (solv-int/9704013) we have considered a string living in the infinite lattice that was, in a sense, generated by a "particle". Here we show how to construct multi-string eigenstates generated by several particles. It turns out that, at least in some cases, this allows us to bypass the difficulties of constructing multi-particle states. We also present and discuss the "dispersion relations" for our particles-strings.

DOI
10.48550/arxiv.solv-int/9705005

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In paper IV (solv-int/9704013) we have considered a string living in the infinite lattice that was, in a sense, generated by a "particle". Here we show how to construct multi-string eigenstates generated by several particles. It turns out that, at least in some cases, this allows us to bypass the difficulties of constructing multi-particle states. We also present and discuss the "dispersion relations" for our particles-strings.
We give a review of some recent work on generalization of the Bethe ansatz in the case of 2 + 1-dimensional models of quantum field theory.As such a model, we consider one associated with the tetrahedron equation, i.e. the 2+1-dimensional generalization of the famous Yang-Baxter equation.We construct some eigenstates of the transfer matrix of that model.There arise, together with states composed of point-like particles analogous to those in the usual 1 + 1-dimensional Bethe ansatz, new string-like states and string-particle hybrids.
The model which is the generalization of the one-dimensional XY-spin chain for the case of the two-dimensional square lattice is considered. The subspace of the "string" states is studied. The solution to the eigenvalue problem is obtained for the single "string" in cases of the "string" with fixed ends and "string" of types (1, 1) and (1, 2) living on the torus. The latter case has the features of a self-interacting system and does not seem to be integrable while the previous two cases are equivalent to the free-fermion model.
We investigate solutions to the Bethe equations for the isotropic S = 1/2 Heisenberg chain involving complex, string-like rapidity configurations of arbitrary length. Going beyond the traditional string hypothesis of undeformed strings, we describe a general procedure to construct eigenstates including strings with generic deformations, discuss general features of these solutions and provide a number of explicit examples including complete solutions for all wavefunctions of short chains. We finally investigate some singular cases and show from simple symmetry arguments that their contribution to zero-temperature correlation functions vanishes.