# Entanglement Spectra of Stabilizer Codes: A Window into Gapped Quantum Phases of Matter

## TL;DR

This paper investigates the entanglement spectra of stabilizer codes to understand gapped quantum phases, revealing universal non-local features and the influence of subsystem symmetries, especially in fracton and SSPT phases.

## Contribution

It develops a framework for computing entanglement features of gapped topological phases in stabilizer models, including fracton and SSPT phases, highlighting the role of non-local surface stabilizers.

## Key findings

- Non-local surface stabilizers encode universal entanglement features.
- Subsystem symmetries induce geometric dependence in entanglement spectra.
- Edge-entanglement correspondence extends to 3D fracton models.

## Abstract

The entanglement spectrum (ES) provides a barometer of quantum entanglement and encodes physical information beyond that contained in the entanglement entropy. In this paper, we explore the ES of stabilizer codes, which furnish exactly solvable models for a plethora of gapped quantum phases of matter. Studying the ES for stabilizer Hamiltonians in the presence of arbitrary weak local perturbations thus allows us to develop a general framework within which the entanglement features of gapped topological phases can be computed and contrasted. In particular, we study models harboring fracton order, both type-I and type-II, and compare the resulting ES with that of both conventional topological order and of (strong) subsystem symmetry protected topological (SSPT) states. We find that non-local surface stabilizers (NLSS), a set of symmetries of the Hamiltonian which form on the boundary of the entanglement cut, act as purveyors of universal non-local features appearing in the entanglement spectrum. While in conventional topological orders and fracton orders, the NLSS retain a form of topological invariance with respect to the entanglement cut, subsystem symmetric systems---fracton and SSPT phases---additionally show a non-trivial geometric dependence on the entanglement cut, corresponding to the subsystem symmetry. This sheds further light on the interplay between geometric and topological effects in fracton phases of matter and demonstrates that strong SSPT phases harbour a measure of quasi-local entanglement beyond that encountered in conventional SPT phases. We further show that a version of the edge-entanglement correspondence, established earlier for gapped two-dimensional topological phases, also holds for gapped three-dimensional fracton models.