A Note on Dressed S-Matrices in Models with Long-Range Interactions

TL;DR

This paper evaluates dressed S-matrices for models with long-range interactions, revealing some as momentum-independent phases indicating ideal gas behavior, and others as nontrivial, with a crossover to trivial matrices in certain limits.

Contribution

It applies Korepin's method to compute dressed S-matrices in long-range interaction models, highlighting the transition from interacting to noninteracting quasiparticles.

Findings

S-matrix for 1/sin^2(r) interactions is a momentum-independent phase.

S-matrices for 1/sinh^2(r) interactions are generally nontrivial.

In the 1/r^2 limit, S-matrices become trivial, indicating a crossover.

Abstract

The {\sl dressed} Scattering matrix describing scattering of quasiparticles in various models with long-range interactions is evaluated by means of Korepin's method\upref vek1/. For models with ${1\over\sin^2(r)}$-interactions the S-matrix is found to be a momentum-independent phase, which clearly demonstrates the ideal gas character of the quasiparticles in such models. We then determine S-matrices for some models with ${1\over\sinh^2(r)}$-interaction and find them to be in general nontrivial. For the ${1\over r^2}$-limit of the ${1\over\sinh^2(r)}$-interaction we recover trivial S-matrices, thus exhibiting a crossover from interacting to noninteracting quasiparticles. The relation of the S-matrix to fractional statistics is discussed.