A Framework for Predicting Entanglement Spectra of Gapless Symmetry-Protected Topological States in One Dimension

TL;DR

This paper develops a systematic framework to predict the entanglement spectra of gapless symmetry-protected topological states in one dimension using boundary conformal field theories and quantum channels.

Contribution

It introduces a method to relate the entanglement spectra of non-trivial gSPT states to trivial states via boundary-modifying quantum channels.

Findings

The reduced density matrix of gSPT states can be obtained from trivial states using a quantum channel.

The framework predicts boundary conformal field theories describing entanglement spectra.

Analysis of various symmetry-protected gSPT states demonstrates the method's versatility.

Abstract

The concept of gapped symmetry-protected topological (SPT) states has been generalized to gapless SPT (gSPT) states. Similar to gapped SPT states, gSPT states in one dimension exhibit universal degeneracies in their entanglement spectra. The entanglement spectra of gSPT states are further described by boundary conformal field theories, whose systematic prediction is a key open question. To address this problem, we focus on the class of gSPT states that are obtained by applying unitary SPT entanglers to trivial, critical states in one dimension. We find that the reduced density matrix of a non-trivial gSPT state can be obtained, either exactly or approximately, by applying a quantum channel to the reduced density matrix of the trivial gSPT state. This quantum channel acts only near the entanglement cut and modifies its corresponding conformal boundary condition, allowing us in turn to predict the boundary conformal field theory describing the entanglement spectra. We apply this framework to gSPT states protected by various symmetries and having different central charges, and further analyze the stability of boundary conditions of the entanglement cut. Our work thereby provides a framework for systematically analyzing and understanding the entanglement spectra of gSPT states.