Matrix-product operator dualities in integrable lattice models

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 argue for a broad class of MPO dualities that the usual Yang--Baxter $R$-matrix should be extended. This extended $R$-matrix satisfies modified algebraic relations, including a modified Yang--Baxter algebra previously identified in the unitary case, that altogether give 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.