Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface

TL;DR

This paper links constraints from the Lieb-Schultz-Mattis theorem and surface states of 3D topological insulators to develop a framework for understanding symmetry-enriched topological phases with translational and on-site symmetries, revealing new constraints and phenomena.

Contribution

It introduces a unified framework for symmetry-enriched topological phases considering both translational and on-site symmetries, including the analysis of symmetry defects and fractionalization constraints.

Findings

Constraints on symmetry fractionalization in 2D systems with fractional spin or charge

Conditions for the existence of spinon excitations in topological phases

Introduction of the concept of anyonic spin-orbit coupling

Abstract

The Lieb-Schultz-Mattis theorem and its higher dimensional generalizations by Oshikawa and Hastings require that translationally invariant 2D spin systems with a half-integer spin per unit cell must either have a continuum of low energy excitations, spontaneously break some symmetries, or exhibit topological order with anyonic excitations. We establish a connection between these constraints and a remarkably similar set of constraints at the surface of a 3D interacting topological insulator. This, combined with recent work on symmetry-enriched topological phases (SETs) with on-site unitary symmetries, enables us to develop a framework for understanding the structure of SETs with both translational and on-site unitary symmetries, including the effective theory of symmetry defects. This framework places stringent constraints on the possible types of symmetry fractionalization that can occur in 2D systems whose unit cell contains fractional spin, fractional charge, or a projective representation of the symmetry group. As a concrete application, we determine when a topological phase must possess a "spinon" excitation, even in cases when spin rotational invariance is broken down to a discrete subgroup by the crystal structure. We also describe the phenomena of "anyonic spin-orbit coupling", which may arise from the interplay of translational and on-site symmetries. These include the possibility of on-site symmetry defect branch lines carrying topological charge per unit length and lattice dislocations inducing on-site symmetry protected degeneracies.