Matrix Product States for Modulated Topological Phases: Crystalline Equivalence Principle and Lieb-Schultz-Mattis Constraints

TL;DR

This paper classifies one-dimensional topological phases protected by modulated and spatial symmetries using matrix product states, confirming the crystalline equivalence principle and deriving constraints like Lieb-Schultz-Mattis for such phases.

Contribution

It introduces a classification scheme for modulated symmetry-protected topological phases, linking them to internal SPT phases via the crystalline equivalence principle and spectral sequences.

Findings

Classified modulated SPT phases by cohomology group H^{2}(G,U(1)_s)

Derived a matrix product state proof of the Lyndon-Hochschild-Serre spectral sequence

Proved Lieb-Schultz-Mattis constraints for modulated symmetries and non-invertible reflection symmetries.

Abstract

Modulated symmetries are internal symmetries that act in a spatially non-uniform manner. Consequently, when a modulated symmetry $G_{\text{int}}$ is combined with a spatial symmetry $G_{\text{sp}}$, the total symmetry group takes the form of a semidirect product $G=G_{\text{int}}\rtimes G_{\text{sp}}$. Using matrix product states, we classify topological phases protected by modulated symmetries together with spatial symmetries in one spatial dimension. We show that these modulated symmetry-protected topological (SPT) phases are classified by $H^{2}(G,U(1)_s)$, in agreement with the crystalline equivalence principle, which states that SPT phases protected by symmetries involving spatial elements are in one-to-one correspondence with internal SPT phases protected by the same symmetries, viewed as acting internally. Furthermore, we provide a matrix product state derivation of the Lyndon-Hochschild-Serre spectral sequence for the corresponding internal SPT phases, which enables us to construct an explicit correspondence between modulated SPT phases and internal SPT phases. As applications of this classification, we prove a Lieb-Schultz-Mattis (LSM) theorem for modulated symmetries that forbids the existence of symmetric short-ranged entangled ground state, as well as an SPT-LSM constraint that enforces nontrivial entanglement in the SPT ground state. Finally, we use the classification to establish a similar LSM-type constraints for non-invertible Kramers-Wannier reflection symmetries.