Lattice Translation Modulated Symmetries and TFTs

TL;DR

This work develops a categorical, tensor-network framework for lattice translation modulated symmetries in $1+1$D, extending to a modulated SymTFT bulk in $2+1$D. Modulations are encoded by a monoidal autoequivalence $F_T:\mathcal{C}\to\mathcal{C}$ acting on the internal symmetry category, with boundary data carried by $F_T$-twisted $\mathcal{C}$-module autoequivalences. The authors classify modulated SPT and weak SPT phases in $1+1$D, including mixed anomalies described by cohomology groups $H^k$, and provide explicit lattice and bulk constructions, culminating in a foliated BF theory in the continuum limit for the abelian dipole case. This framework unifies invertible and non-invertible modulated symmetries, connects tensor-network realizations to domain-wall (bimodule) constructions, and offers a pathway to higher-dimensional and higher-form generalizations with potential applications to fracton-like systems.

Abstract

Modulated symmetries are internal symmetries that are not invariant under spacetime symmetry actions. We propose a general way to describe the lattice translation modulated symmetries in 1+1D, including the non-invertible ones, via the tensor network language. We demonstrate that the modulations can be described by some autoequivalences of the categories. Although the topological behaviors are broken because of the presence of modulations, we can still construct the modulated version of the symmetry TFT bulks by inserting a series of domain walls described by invertible bimodule categories. This structure not only recovers some known results on invertible modulated symmetries but also provides a general framework to tackle modulated symmetries in a more general setting.

Figures (6)

• Figure 1: We construct the "modulated" MPO with a site-dependent form.
• Figure 2: We construct the "modulated" MPS with a site-dependent form.
• Figure 3: Modulated TFT is constructed by inserting a series of domain walls described by an invertible bimodule category $\mathcal{T}$.
• Figure 4: An anyonic excitation $Z\in\mathcal{Z}(\mathcal{D})$ passes through the domain wall $\mathcal{T}$ shown in domain wall.
• Figure 5: Stabilizers of the modulated bulk model.