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.
Highlights
The celebrated Lieb-Schultz-Mattis (LSM) theorem [1,2], including the higher-dimensional generalizations by Oshikawa [3] and Hastings [4], shows that translationally invariant spin systems with an odd number of S 1⁄4 1=2 moments per unit cell cannot have a symmetric, gapped, and nondegenerate ground state
III, we review the classification of 3D symmetry-protected topological (SPT) phases and describe the Künneth formula decomposition of the SPT class in terms of invariants associated with stacking, packing, and filling of lower-dimensional SPT phases
Which can be shown to be a 4-cocycle. (In this paper, the braiding R symbols are written with two superscript topological charge labels, in contrast to the global symmetry operators Rg, which are written with a subscript group element.) We emphasize that Eq (44) depends only on the symmetry fractionalization class and the F and R
Summary
The celebrated Lieb-Schultz-Mattis (LSM) theorem [1,2], including the higher-dimensional generalizations by Oshikawa [3] and Hastings [4], shows that translationally invariant spin systems with an odd number of S 1⁄4 1=2 moments per unit cell cannot have a symmetric, gapped, and nondegenerate ground state. A consequence of this bulk-boundary correspondence is that the physical properties associated with the 3D SPT invariants, namely the nature of the emergent boundary modes, imply which SET obstruction classes can be physically realized in purely 2D systems in a local and symmetry preserving manner. VI, we discuss the bulk-boundary correspondence and anomaly matching in detail and formulate an understanding of the theory of defects and their obstruction classes for 2D SET phases with translational and on-site unitary symmetries We consider such 2D SETs in systems with projective representations per unit cell and fractional charge per unit cell in detail. VII, we conclude with a discussion of further directions and open questions
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