Matrix Product States for Modulated Symmetries: SPT, LSM, and Beyond

TL;DR

This paper extends the matrix product states framework to systems with modulated symmetries, enabling classification of SPT phases and formulation of LSM constraints in these more complex settings.

Contribution

It introduces a generalized MPS formalism for modulated symmetries, revises the symmetry push-through condition, and applies this to classify SPT phases and LSM constraints.

Findings

Generalized the MPS formalism for modulated symmetries.

Derived a revised symmetry push-through condition.

Classified 1D SPT phases with modulated symmetries.

Abstract

Matrix product states (MPS) provide a powerful framework for characterizing one-dimensional symmetry-protected topological (SPT) phases of matter and for formulating Lieb-Schultz-Mattis (LSM)-type constraints. Here we generalize the MPS formalism to translationally invariant systems with general modulated symmetries. We show that the standard symmetry "push-through" condition for conventional global symmetry must be revised to account for symmetry modulation, and we derive the appropriate generalized condition. Using this generalized push-through structure, we classify one-dimensional SPT phases with modulated symmetries and formulate LSM-type constraints within the same MPS-based framework.