Diagnosing 2D symmetry protected topological states via mixed state anomaly

TL;DR

This paper introduces a new diagnostic method for 2D symmetry-protected topological states by identifying a mixed state anomaly in the reduced density matrix, extending entanglement-based tools beyond the spectrum.

Contribution

It demonstrates that the edge anomaly in 2D SPT phases can be detected through symmetry-twisted mixed states, revealing new topological and symmetry-breaking features.

Findings

Mixed state anomaly encodes edge anomaly in reduced density matrix

Topological contribution to disorder parameter beyond area law

Spontaneous symmetry breaking in time-reversal symmetric case

Abstract

Symmetry-protected topological (SPT) phases are short-range entangled quantum states characterized by anomalous edge behavior, a manifestation of the bulk-boundary correspondence for topological phases. Moreover, the Li-Haldane conjecture posits that the entanglement spectrum exhibits the same anomaly as the physical edge spectrum, thereby serving as an entanglement-based fingerprint for identifying topological phases. In this work, we extend the entanglement-based diagnostic tools by demonstrating that the edge anomaly is manifested not only in the entanglement spectrum but also in the reduced density matrix itself, a phenomenon we refer to as the mixed state anomaly. Focusing on the two-dimensional $\mathbb{Z}_2$ SPT phase, we show that this anomaly is subtly encoded in symmetry-twisted mixed states, leading to a topological contribution to the disorder parameter beyond the area law, as well as a spontaneous-symmetry-breaking type long-range order when time reversal symmetry is present.