# Symmetry-Protected Topological Interfaces and Entanglement Sequences

## TL;DR

This paper explores how gapped interfaces in 2D Abelian topological phases host sequences of 1D symmetry-protected topological states, linking entanglement entropy corrections to these states and their symmetry properties.

## Contribution

It introduces the Boundary Topological Entanglement Sequence, connecting entanglement entropy corrections to classes of 1D SPT states at topological interfaces.

## Key findings

- Interfaces support a sequence of 1D SPT states.
- Entanglement entropy correction signals SPT emergence.
- Domain walls host parafermion-like excitations.

## Abstract

Gapped interfaces (and boundaries) of two-dimensional (2D) Abelian topological phases are shown to support a remarkably rich sequence of 1D symmetry-protected topological (SPT) states. We show that such interfaces can provide a physical interpretation for the corrections to the topological entanglement entropy of a 2D state with Abelian topological order found in Ref.~\onlinecite{cano-2015}. The topological entanglement entropy decomposes as $\gamma = \gamma_a + \gamma_s$, where $\gamma_a > 0$ only depends on universal topological properties of the 2D state, while a correction $\gamma_s > 0$ signals the emergence of the 1D SPT state that is produced by interactions along the entanglement cut and provides a direct measure of the stabilizing symmetry of the resulting SPT state. A correspondence is established between the possible values of $\gamma_s$ associated with a given interface - which is named Boundary Topological Entanglement Sequence - and classes of 1D SPT states. We show that symmetry-preserving domain walls along such 1D interfaces (or boundaries) generally host localized parafermion-like excitations that are stable to local symmetry-preserving perturbations.