Matrix-product operator dualities in integrable lattice models

TL;DR

This paper investigates how matrix-product operator dualities affect the integrable structures of lattice models, revealing simple transformations of the R-matrix and establishing a modified algebra for dual models, with applications to the XXZ spin chain.

Contribution

It introduces a framework for understanding MPO dualities in integrable models, detailing the transformation of the Yang--Baxter structures and analyzing specific case studies.

Findings

The $reve{R}$-matrix transforms simply under MPO dualities.

A modified algebra for the Yang--Baxter R-matrix is identified.

Case studies include the cluster entangler and Kramers--Wannier duality.

Abstract

Matrix-product operators (MPOs) appear throughout the study of integrable lattice models, notably as the transfer matrices. They can also be used as transformations to construct dualities between such models, both invertible (including unitary) and non-invertible (including discrete gauging). We analyse how the local Yang--Baxter integrable structures are modified under such dualities. We see that the $\check{R}$-matrix, that appears in the baxterization approach to integrability, transforms in a simple manner. We further show for a broad class of MPOs that the usual Yang--Baxter $R$-matrix satisfies a modified algebra, previously identified in the unitary case, that gives a local integrable structure underlying the commuting transfer matrices of the dual model. We illustrate these results with two case studies, analysing an invertible unitary MPO and a non-invertible MPO both applied to the canonical XXZ spin chain. The former is the cluster entangler, arising in the study of symmetry-protected topological phases, while the latter is the Kramers--Wannier duality. We show several results for MPOs with exact MPO inverses that are of independent interest.