# Detecting subsystem symmetry protected topological order via entanglement entropy

## TL;DR

This paper introduces a method to detect subsystem symmetry protected topological order through a constant correction to entanglement entropy, using tensor networks, and explores its robustness and extensions in various models.

## Contribution

It provides an analytic and numerical framework to identify SSPT order via entanglement entropy corrections, including a new understanding of its persistence beyond the cluster phase.

## Key findings

- The correction to the entanglement area law reliably detects SSPT order.
- The SSPT order persists under certain external fields and interactions.
- The correction behavior extends to three-dimensional cluster states.

## Abstract

Subsystem symmetry protected topological (SSPT) order is a type of quantum order that is protected by symmetries acting on lower-dimensional subsystems of the entire system. In this paper, we show how SSPT order can be characterized and detected by a constant correction to the entanglement area law, similar to the topological entanglement entropy. Focusing on the paradigmatic two-dimensional cluster phase as an example, we use tensor network methods to give an analytic argument that almost all states in the phase exhibit the same correction to the area law, such that this correction may be used to reliably detect the SSPT order of the cluster phase. Based on this idea, we formulate a numerical method that uses tensor networks to extract this correction from ground-state wave functions. We use this method to study the fate of the SSPT order of the cluster state under various external fields and interactions, and find that the correction persists unless a phase transition is crossed, or the subsystem symmetry is explicitly broken. Surprisingly, these results uncover that the SSPT order of the cluster state persists beyond the cluster phase, thanks to a new type of subsystem time-reversal symmetry. Finally, we discuss the correction to the area law found in three-dimensional cluster states on different lattices, indicating rich behavior for general subsystem symmetries