Boundary anomaly detection in two-dimensional subsystem symmetry-protected topological phases

TL;DR

This paper develops a numerical method based on topological response theory to detect boundary anomalies in 2D subsystem symmetry-protected topological phases, distinguishing different phases and revealing new intrinsic phases.

Contribution

It generalizes the topological response approach to subsystem symmetries, enabling detection of boundary anomalies and identification of new SSPT phases, including in mixed states.

Findings

Identified strong and weak $Z_2^ au\times Z_2^\sigma$ SSPT phases using tensor networks.

Discovered an intrinsic $Z_2$ SSPT phase with degenerate entanglement spectrum.

Extended anomaly detection to mixed states, showing anomalies persist under disorder.

Abstract

We generalize the topological response theory to detect the boundary anomalies of linear subsystem symmetries. This approach allows us to distinguish different subsystem symmetry-protected topological (SSPT) phases and uncover new ones. We focus on the cases where the mixed anomaly exists within the adjacent subsystems. Using numerical simulations, we demonstrate the power of this method by identifying strong and weak $Z_2^\tau\times Z_2^\sigma$ SSPT phases in a tunable tensor network state. Our analysis reveals an intrinsic $Z_2$ SSPT phase characterized by its degenerate entanglement spectrum. Furthermore, we extend the anomaly indicator to mixed-state density matrices and show that quantum anomalies of subsystem symmetry can persist under both uniform and alternating disorders. This finding establishes a connection between boundary quantum anomalies in pure and mixed states. Our work provides a numerical method to detect quantum anomalies of subsystem symmetries, offering new insights into the study of topological quantum phases.