Translational Symmetry and Microscopic Constraints on Symmetry-Enriched Topological Phases: A View from the Surface

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 with on-site unitary symmetries, enables us to develop a framework for understanding the structure of symmetry-enriched topological phases 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 degeneracies protected by onsite symmetry.

Intrinsic and emergent anomalies at deconfined critical points

“…It is also consistent with the results of Ref. [10] obtained in the context of topologically ordered 2+1D phases with crystalline symmetries.…”

“…To see this, we note that the microscopic symmetry group generated by T x , T y , and This leaves the question: If we allow for weak Lorentz breaking perturbations to the CP 1 model consistent with S = 1 square lattice symmetry, can a trivial gap be opened? 10 For instance, we can envision a perturbation…”

“…Furthermore, it was noted that the impossibility of a trivial gap is very reminiscent of the situation occurring on the boundary of a topological insulator, or a more general symmetry-protected topological (SPT) phase. In fact, one may view a system with S = 1/2 per unit cell as a boundary of a crystalline SPT phase protected by a combination of translational symmetry and SO(3) s [10]. Such a crystalline SPT can be constructed as an array of 1+1D Haldane chains; then the boundary is an array of "dangling" spin-1/2's.…”

Anyon condensation and a generic tensor-network construction for symmetry-protected topological phases

“…In Ref. 65, the authors find that weak indices can be elegantly incorporated into the cohomology formulation by treating translation in the same way as the on-site symmetry. Weak indices can be explicitly calculated using Künneth formula.…”

Anyonic entanglement and topological entanglement entropy

“…This includes on-site symmetry defects [65] and translational symmetry defects [66]. In such cases, the topological defects in the system have fusion and associativity properties that are precisely the same as that of quasiparticles, and they have a generalization of braiding that incorporates the symmetry action.…”