Topological Phases Protected By Reflection Symmetry and Cross-cap States

TL;DR

This paper introduces a method using cross-cap states to analyze (2+1)D topological phases protected by reflection symmetry, revealing their classification through symmetry-induced anomalies on unoriented spacetime surfaces.

Contribution

It develops a novel approach employing cross-cap states to diagnose reflection-symmetry-protected topological phases, connecting geometric twisting to phase classification.

Findings

Reproduces known Z_8 classification of topological crystalline superconductors.

Identifies anomalous phases in cross-cap states for non-trivial SPTs.

Provides a geometric framework for understanding symmetry anomalies in topological phases.

Abstract

Twisting symmetries provides an efficient method to diagnose symmetry-protected topological (SPT) phases. In this paper, edge theories of (2+1)-dimensional topological phases protected by reflection as well as other symmetries are studied by twisting reflection symmetry, which effectively puts the edge theories on an unoriented spacetime, such as the Klein bottle. A key technical step taken in this paper is the use of the so-called cross-cap states, which encode entirely the unoriented nature of spacetime, and can be obtained by rearranging the spacetime geometry and exchanging the role of space and time coordinates. When the system is in a non-trivial SPT phase, we find that the corresponding cross-cap state is non-invariant under the action of the symmetries of the SPT phase, but acquires an anomalous phase. This anomalous phase, with a proper definition of a reference state, on which symmetry acts trivially, reproduces the known classification of (2+1)-dimensional bosonic and fermionic SPT phases protected by reflection symmetry, including in particular the Z_8 classification of topological crystalline superconductors protected by reflection and time-reversal symmetries.