Spin chains with dynamical lattice supersymmetry

TL;DR

This paper introduces a framework for spin chains with dynamical lattice supersymmetry, establishing local criteria for nilpotency, and explores specific models including the Fateev-Zamolodchikov chain with detailed analysis of supersymmetry singlets.

Contribution

It formulates a local criterion for lattice supersymmetry and demonstrates its application to models like the Fateev-Zamolodchikov chain, connecting supersymmetric states to combinatorial enumeration.

Findings

Lattice supersymmetry exists for a class of spin chains at arbitrary spin.

The Fateev-Zamolodchikov chain admits supersymmetry across all coupling constants.

Supersymmetry singlets relate to weighted enumeration of alternating sign matrices.

Abstract

Spin chains with exact supersymmetry on finite one-dimensional lattices are considered. The supercharges are nilpotent operators on the lattice of dynamical nature: they change the number of sites. A local criterion for the nilpotency on periodic lattices is formulated. Any of its solutions leads to a supersymmetric spin chain. It is shown that a class of special solutions at arbitrary spin gives the lattice equivalents of the N=(2,2) superconformal minimal models. The case of spin one is investigated in detail: in particular, it is shown that the Fateev-Zamolodchikov chain and its off-critical extension admits a lattice supersymmetry for all its coupling constants. Its supersymmetry singlets are thoroughly analysed, and a relation between their components and the weighted enumeration of alternating sign matrices is conjectured.