Excitation Scattering in Integrable Models and Hall-Littlewood-Kerov Polynomials

TL;DR

This paper connects integrable (1+1)-dimensional models' excitation scattering to geometric problems using generalized Hall-Littlewood-Kerov polynomials, revealing new mathematical structures and physical insights.

Contribution

It introduces a novel link between scattering matrices in integrable models and generalized Hall-Littlewood-Kerov polynomials, extending Macdonald polynomials with unlimited parameters.

Findings

Scattering matrices derived from asymptotic behavior of Kerov's polynomials.

Scattering processes reduced to geometric s-wave scattering on quantum-symmetric spaces.

Integrable models characterized by geometric scattering of excitations.

Abstract

The S-matrices for the scattering of two excitations in the XYZ model and in all of its SU(n)-type generalizations are obtained from the asymptotic behavior of Kerov's generalized Hall-Littlewood polynomials. These physical scattering processes are all reduced to geometric s-wave scattering problems on certain quantum-symmetric spaces, whose zonal spherical functions these Hall-Littlewood-Kerov polynomials are. Mathematically, this involves a generalization with an unlimited number of parameters of the Macdonald polynomials. Physically, our results suggest that, of the (1+1)-dimensional models, the integrable ones are those, for which the scattering of excitations becomes geometric in the sense above.