Supersymmetric quantum spin chains and classical integrable systems

TL;DR

This paper establishes a deep connection between supersymmetric quantum spin chains and classical integrable systems, introducing a master T-operator that links quantum spectra to classical tau-functions.

Contribution

It introduces the master T-operator for supersymmetric spin chains, revealing their eigenvalues as tau-functions of the classical mKP hierarchy, thus bridging quantum and classical integrable models.

Findings

Eigenvalues of the master T-operator are tau-functions of the mKP hierarchy.

The spectrum of spin chain Hamiltonians satisfies algebraic equations derived from classical integrable systems.

The approach provides a new method to analyze quantum spectra using classical integrable system techniques.

Abstract

For integrable inhomogeneous supersymmetric spin chains (generalized graded magnets) constructed employing Y(gl(N|M))-invariant R-matrices in finite-dimensional representations we introduce the master T-operator which is a sort of generating function for the family of commuting quantum transfer matrices. Any eigenvalue of the master T-operator is the tau-function of the classical mKP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0-th time of the hierarchy. This implies a remarkable relation between the quantum supersymmetric spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, we obtain a system of algebraic equations for the spectrum of the spin chain Hamiltonians.