Matrix-product operator dualities in integrable lattice models

TL;DR

This work develops a unified framework for matrix-product operator (MPO) dualities in integrable lattice models, clarifying how dualities modify the local Yang-Baxter data while preserving bulk integrability. By classifying dualities into invertible on-site, invertible with MPO inverse, and non-invertible (discrete gauging), it shows that an extended or modified RLL/Yang-Baxter structure under dualities can still yield commuting transfer matrices. Two XXZ-based case studies illustrate the principles: (i) a cluster entangler realized as an MPO that maps the XXZ chain to a cluster-type Hamiltonian and (ii) a Kramers-Wannier-type non-invertible duality that converts a vertex model into a face model with a KW MPO intertwinement. Overall, the results provide a practical route to engineering integrable transitions between phases (including SPT and gauged phases) via MPO transformations and offer a local-structure perspective via extended R-matrices and modified YB algebras.

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.

Figures (3)

• Figure 1: A reference matrix-product operator (MPO). Each four-index tensor $M_j$, that may also depend on some parameters, has two physical indices (vertical lines) and two virtual indices (horizontal lines). For fixed virtual indices, $M_i$ gives a linear transformation between physical Hilbert spaces $V_i^{}\rightarrow V_i'$. These are referred to as 'quantum space'(s) in some integrability literature. Fixing the physical indices $M_i$ gives a matrix mapping $W_{i-1}\rightarrow W_i$. The $V_i$ are referred to in the literature as bond, virtual or auxilliary spaces. We will often take translation invariant MPOs acting on a fixed Hilbert space. In this case all $M_i=M$, all virtual spaces are isomorphic, and, taking the trace over the virtual space $V_0\equiv W_j$, we have an operator from $V^{\otimes L}$ to itself.
• Figure 2: 'Scattering' YBE for $R$-matrix. The labels $1$, $2$ and $3$ stands for the three vector spaces in $V \otimes V \otimes V$. Arrows indicate the operators acting from right to left.
• Figure 3: 'Circuit' YBE for $\check{{R}}$-matrix. This is \ref{['eq:checkYBE']} where we specialize to the case where $\check{{R}}$ acts on two neighbouring sites. This moreover represents \ref{['eq:check_R_YBE']} when we take the representation over a tensorized Hilbert space where $\check{{\mathcal{R}}}_j \mapsto \check{R}_{j,j+1}$.