TY - JOUR

T1 - Translational symmetry and microscopic constraints on symmetry-enriched topological phases

T2 - A view from the surface

AU - Cheng, Meng

AU - Zaletel, Michael

AU - Barkeshli, Maissam

AU - Vishwanath, Ashvin

AU - Bonderson, Parsa

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

KW - Condensed matter physics

UR - http://www.scopus.com/inward/record.url?scp=85008237802&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85008237802&partnerID=8YFLogxK

U2 - 10.1103/PhysRevX.6.041068

DO - 10.1103/PhysRevX.6.041068

M3 - Article

AN - SCOPUS:85008237802

VL - 6

JO - Physical Review X

JF - Physical Review X

SN - 2160-3308

IS - 4

M1 - 041068

ER -