Derivation of Green's Function of Spin Calogero-Sutherland Model by Uglov's Method

TL;DR

This paper derives the Green's function for the spin Calogero-Sutherland model using Uglov's method, connecting eigenfunctions to Macdonald polynomials, and confirms the finite-size and thermodynamic limit expressions, including spectral singularities.

Contribution

It applies Uglov's method to derive the Green's function of the spin Calogero-Sutherland model, linking eigenfunctions to Macdonald polynomials and analyzing spectral properties.

Findings

Derived finite-size hole propagator expression in terms of renormalized momenta and spins.

Confirmed the thermodynamic limit matches previous results.

Discussed spectral singularities related to SU(3) Haldane-Shastry model.

Abstract

Hole propagator of spin 1/2 Calogero-Sutherland model is derived using Uglov's method, which maps the exact eigenfunctions of the model, called Yangian Gelfand-Zetlin basis, to a limit of Macdonald polynomials (gl_2-Jack polynomials). To apply this mapping method to the calculation of 1-particle Green's function, we confirm that the sum of the field annihilation operator on Yangian Gelfand-Zetlin basis is transformed to the field annihilation operator on gl_2-Jack polynomials by the mapping. The resultant expression for hole propagator for finite-size system is written in terms of renormalized momenta and spin of quasi-holes and the expression in the thermodynamic limit coincides with the earlier result derived by another method. We also discuss the singularity of the spectral function for a specific coupling parameter where the hole propagator of spin Calogero-Sutherland model becomes equivalent to dynamical colour correlation function of SU(3) Haldane-Shastry model.