Topological Phases Protected By Reflection Symmetry and Cross-cap States

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.

17 pages, 1 figure, published version to PRB, updated acknowledgement

References in corpus (4)

• Classification of topological insulators and superconductors in three spatial dimensions
• Topology of crystalline insulators and superconductors
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Cited by in corpus (10)

• Symmetric-Gapped Surface States of Fractional Topological Insulators
• Anyon condensation and a generic tensor-network construction for symmetry protected topological phases
• Microscopic Theory of Surface Topological Order for Topological Crystalline Superconductors
• Crosscap States in Integrable Field Theories and Spin Chains
• Extracting the Luttinger parameter from a single wave function
• Lieb-Schultz-Mattis Theorem for 1D Quantum Magnets with Antiunitary Translation and Inversion Symmetries
• Numerical extraction of crosscap coefficients in microscopic models for (2+1)D conformal field theory
• Many-Body Quadrupolar Sum Rule for Higher-Order Topological Insulator
• New Crosscap States
• The non-orientable spacetime of the eternal black hole