Matrix-product operator dualities in integrable lattice models

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.

💡 Research Summary

The paper investigates how matrix‑product operators (MPOs) can be used as duality transformations for integrable lattice models, focusing on the spin‑½ XXZ chain as a canonical example. After reviewing the standard integrable framework—R‑matrix, (\check R)‑matrix, Yang‑Baxter equation (YBE), and the construction of commuting transfer matrices—the authors introduce three classes of MPO‑induced dualities: (i) invertible unitary MPOs (including matrix‑product unitaries, MPUs), (ii) non‑invertible MPOs that implement discrete gauging (e.g., Kramers‑Wannier), and (iii) MPOs possessing an exact finite‑dimensional MPO inverse.

A central technical result is that under any such MPO transformation the original (\check R)‑matrix is conjugated by the MPO, (\check R’ = M,\check R,M^{-1}), and satisfies a modified Yang‑Baxter algebra. This algebra, previously identified for unitary MPUs, contains additional projector‑type terms that encode the MPO’s virtual bond structure. Consequently, the transformed transfer matrix (\tilde T(u)=\operatorname{Tr}_0