The open supersymmetric Haldane-Shastry spin chain and its associated motifs

TL;DR

This paper analyzes the open supersymmetric Haldane-Shastry spin chain, deriving its spectrum using combinatorial motifs and providing explicit results for the su(1|1) case, advancing understanding of supersymmetric integrable models.

Contribution

It introduces a new combinatorial framework for the spectrum of the open supersymmetric Haldane-Shastry chain using extended motifs and skew super Schur polynomials.

Findings

Derived the partition function via the supersymmetric spin Sutherland model

Expressed the spectrum in terms of extended motifs and skew super Schur polynomials

Obtained an explicit Helmholtz free energy for the su(1|1) model in the thermodynamic limit

Abstract

We study the open version of the su$(m|n)$ supersymmetric Haldane-Shastry spin chain associated to the $BC_N$ extended root system. We first evaluate the model's partition function by modding out the dynamical degrees of freedom of the su$(m|n)$ supersymmetric spin Sutherland model of $BC_N$ type, whose spectrum we fully determine. We then construct a generalized partition function depending polynomially on two sets of variables, which yields the standard one when evaluated at a suitable point. We show that this generalized partition function can be written in terms of two variants of the classical skew super Schur polynomials, which admit a combinatorial definition in terms of a new type of skew Young tableaux and border strips (or, equivalently, extended motifs). In this way we derive a remarkable description of the spectrum in terms of this new class of extended motifs, reminiscent of the analogous one for the closed Haldane-Shastry chain. We provide several concretes examples of this description, and in particular study in detail the su$(1|1)$ model finding an analytic expression for its Helmholtz free energy in the thermodynamic limit.