Classifying phases protected by matrix product operator symmetries using matrix product states

TL;DR

<3-5 sentence high-level summary> This work advances the classification of one-dimensional quantum phases protected by matrix product operator (MPO) symmetries by introducing a local-to-global framework based on L-symbols that satisfy coupled pentagon equations. It shows that equivalence classes of L-symbols provide complete invariants for MPO-protected phases and connects these to the module-category structure of fusion categories, including a precise link to gapped boundaries in (2+1)d topological phases. The analysis recovers the standard symmetry-protected topological (SPT) classification for on-site group symmetries and extends it to general MPOs, including time-reversal interplay, with concrete group-based examples and explicit MPS/MPO constructions. The results offer both conceptual insight and practical tools for numerically studying MPO-symmetric systems and for understanding boundary phenomena in higher-dimensional topological orders.

Abstract

We classify the different ways in which matrix product states (MPSs) can stay invariant under the action of matrix product operator (MPO) symmetries. This is achieved through a local characterization of how the MPSs, that generate a ground space, remain invariant under a global MPO symmetry. This characterization yields a set of quantities satisfying the coupled pentagon equations, associated with a module category over the fusion category that describes the MPO symmetry. Equivalence classes of these quantities provide complete invariants for an MPO symmetry protected phase: they are robust under continuous deformations of the MPS tensor, and two phases with the same equivalence class can be connected by a symmetric gapped path. Our techniques match and extend the known renormalization fixed point classifications and facilitate the numerical study of these systems. For MPO symmetries described by a group, we recover the symmetry protected topological order classification for unique and degenerate ground states. Moreover, we study the interplay between time reversal symmetry and an MPO symmetry and we also provide examples of our classification, together with explicit constructions based on groups. Finally, we elaborate on the connection between our setup and gapped boundaries of two-dimensional topological systems, where MPO symmetries also play a key role.

Figures (1)

• Figure 1: (a) Virtual symmetry of the tensors of the PEPS, virtual level in black, in a small region under the MPO, virtual level in red, labeled by $a\in \mathcal{A}$. (b) The boundary state $|\psi\rangle$ is a MPS, virtual level in blue. (c) In the SPT case, the on-site symmetry of the PEPS is translated into an MPO acting at the virtual level. The boundary MPS $|\psi\rangle$ is invariant under the action of that MPO.

Theorems & Definitions (2)

• Definition 1
• Definition 2