An integrable deformed Landau-Lifshitz model with particle production?

TL;DR

This paper explores a non-Hermitian deformation of the Heisenberg XXX spin chain, leading to a non-unitary Landau-Lifshitz model with particle production, challenging traditional integrability assumptions.

Contribution

It introduces a non-Hermitian, non-diagonalisable spin chain deformation and derives its continuum limit as a non-unitary Landau-Lifshitz model with novel particle production features.

Findings

The deformed model is a Drinfeld twist of the XXX chain.

The continuum limit yields a non-unitary Landau-Lifshitz model.

The model exhibits a non-zero 1→2 S-matrix with particle production.

Abstract

We discuss the continuum limit of a non-Hermitian deformation of the Heisenberg XXX spin chain. This model appeared in the classification of $4\times4$ solutions of the Yang--Baxter equation and it has the particular feature that the transfer matrix is non-diagonalisable. We show that the model is given by a Drinfeld twist of the XXX spin chain and its continuum limit is a non-unitary deformation of the Landau-Lifshitz model. We compute the tower of conserved charges for this deformed Landau-Lifshitz model and show that they are generated by a boost operator. We furthermore show that it gives a non-vanishing $1\to 2$ S-matrix, where one of the outgoing particles has vanishing energy and momentum, and thus it does not fulfil the usual "no particle production" condition of integrability. We argue that this result is natural when looked from the point of view of the non-diagonalisability of the spin chain.