Reflections on time-reversal in the Symmetry Topological Field Theory

TL;DR

This paper extends the Symmetry Topological Field Theory framework to include anti-unitary time-reversal symmetry, analyzing its implications for classifying (1+1)d gapped phases and string order parameters.

Contribution

It introduces a symmetry-enriched SymTFT approach incorporating time-reversal symmetry, providing a detailed analysis of boundary conditions and non-local order parameters in this context.

Findings

Characterization of (1+1)d gapped phases with time-reversal symmetry.

Analysis of non-local string-order parameters and their relation to SPT invariants.

Identification of conditions for end-point operators to correctly detect time-reversal charge.

Abstract

Symmetry under time-reversal appears in the microscopic description of many physical systems. In a quantum mechanical setting it acts as an anti-unitary operator, so does not fall under general analyses based on unitary symmetries. In classifying zero temperature phases of matter in (1+1)d lattice models, the role of anti-unitary symmetries is, however, well-understood. In recent years, the Symmetry Topological Field Theory (SymTFT) approach to this classification has given a general framework to understand symmetries as topological defects, but does not naturally include anti-unitary symmetries. Following recent proposals in the literature, we adopt a symmetry-enriched SymTFT for a theory with both internal and time-reversal symmetry. In particular, we take a standard SymTFT associated with an internal unitary symmetry that is then enriched by a background time-reversal symmetry. A detailed analysis of the topological boundary conditions of this enriched SymTFT allows us to characterize the corresponding (1+1)d gapped phases that preserve the enriching symmetry (i.e. those that do not spontaneously break this symmetry in the ground state). Line operators in the SymTFT approach are related to non-local string-order parameters (with charged end-point operators) for SPT phases. These are subtle in the anti-unitary case and we explore them both on the lattice and in the continuum. We include an analysis of unitary string order parameters that reveal the Klein bottle SPT invariant. On the lattice, we show that the correct end-point charge coincides with the time-reversal-charge only when the end-point operator is hermitian.