Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface (1511.02263v3)

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.

Insights on Symmetry-Enriched Topological Phases with Translational Symmetry

The paper entitled "Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface" explores the constraints imposed by microscopic translational symmetry on the manifestation of symmetry-enriched topological phases (SETs). Meng Cheng and colleagues employ insights garnered from considering surface states of three-dimensional topological insulators to derive a framework for understanding SETs that involve both translational and on-site symmetries. The research addresses a gap in the understanding of constraints affecting systems with fractional quantum numbers per unit cell, such as spin-liquid systems and fractional quantum Hall states.

Key Findings and Their Implications

1. Projective Representation Per Unit Cell: The authors generalize the Lieb-Schultz-Mattis (LSM) theorem for two-dimensional systems with $S=1/2$ spins per unit cell, establishing that those cannot have a symmetric, gapped, and non-degenerate ground state unless symmetry is broken or the system possesses non-trivial topological order. This is extended to any arbitrary on-site symmetry group by identifying projective representations per unit cell.
2. Anomalous Symmetry Fractionalization: By leveraging recent work, the paper shows that constraints similar to the surface states of three-dimensional symmetry-protected topological phases (SPTs) arise from weak SPT phases. These constraints impose nontrivial structure on 2D SETs and gauge anomalies that must be matched by corresponding bulk SPT phases classified in $H^{4},{G}$.
3. Theory of Symmetry Defects: The authors specify that the theory of extrinsic defects displayed in these topological phases must meet stringent conditions. Specifically, the obstruction class $[\mathscr{O}]$, derived from the SET phase's symmetry fractionalization pattern, must match the corresponding weak SPT phase's indices. This matches anomalies at the surface reflecting the underlying non-trivial bulk topology.
4. Constraints on Spinons: Additional results obtained elucidate when a topological phase must possess a "spinon" excitation, revealing that the symmetry fractionalization class manifested in anomaly constraints tightly bounds the permissible states. In scenarios with continuous symmetries or lattice rotational symmetries, constraints are notable.

Broader Theoretical and Practical Implications

The researchers elucidate the interplay between translational symmetries and on-site symmetries, proposing rigorous constraints on symmetry fractionalization in condensed matter systems. This work broadens the theoretical understanding required to pursue solutions to phenomena like high-temperature superconductivity and quantum magnetism. Practically, it suggests pathways to engineer materials pre-disposed to manifest exotic topological phases by manipulating symmetry via lattice structure.

Future Research Directions

The implications of this work suggest numerous avenues for future development. Expanding similar analyses to systems with non-symmorphic symmetries, such as those involving glide symmetries, is one pathway for further research. Another promising direction includes harnessing this rigorous structure for itinerant systems — where residual charge fluctuations further enrich symmetry properties. Additionally, examining gapless phases with fractional symmetry charge per unit cell may unveil new physics, perhaps generalizing the FL$^\ast$ framework.

In summary, this paper offers a rigorous conceptual framework for realizing SET phases in systems constrained by translational symmetry, expanding upon traditional paradigms in topological phases and setting a foundation for further exploitation in theoretical and applied physics.