Bosonic Anomalies, Induced Fractional Quantum Numbers and Degenerate Zero Modes: the anomalous edge physics of Symmetry-Protected Topological States

Abstract

The boundary of symmetry-protected topological states (SPTs) can harbor new quantum anomaly phenomena. In this work, we characterize the bosonic anomalies introduced by the 1+1D non-onsite-symmetric gapless edge modes of 2+1D bulk bosonic SPTs with a generic finite Abelian group symmetry (isomorphic to $G={\prod }_i{Z}_{N_i}=Z_{N_1}\times Z_{N_2}\times Z_{N_3}\times ...$). We demonstrate that some classes of SPTs (termed "Type II") trap fractional quantum numbers (such as fractional $Z_N$ charges) at the 0D kink of the symmetry-breaking domain walls; while some classes of SPTs (termed "Type III") have degenerate zero energy modes (carrying the projective representation protected by the unbroken part of the symmetry), either near the 0D kink of a symmetry-breaking domain wall, or on a symmetry-preserving 1D system dimensionally reduced from a thin 2D tube with a monodromy defect 1D line embedded. More generally, the energy spectrum and conformal dimensions of gapless edge modes under an external gauge flux insertion (or twisted by a branch cut, i.e., a monodromy defect line) through the 1D ring can distinguish many SPT classes. We provide a manifest correspondence from the physical phenomena, the induced fractional quantum number and the zero energy mode degeneracy, to the mathematical concept of cocycles that appears in the group cohomology classification of SPTs, thus achieving a concrete physical materialization of the cocycles. The aforementioned edge properties are formulated in terms of a long wavelength continuum field theory involving scalar chiral bosons, as well as in terms of Matrix Product Operators and discrete quantum lattice models. Our lattice approach yields a regularization with anomalous non-onsite symmetry for the field theory description. We also formulate some bosonic anomalies in terms of the Goldstone-Wilczek formula.

29 pages, 12 Figures. v3 clarification to be accessible for both HEP and CMT. Thanks to Roman Jackiw for introducing new Refs

References in corpus (14)

• Many-Body Physics with Ultracold Gases
• Topological Field Theory of Time-Reversal Invariant Insulators
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• A classification of symmetry enriched topological phases with exactly solvable models
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond
• Fractional topological superconductors with fractionalized Majorana fermions
• Anomalies of discrete symmetries in three dimensions and group cohomology
• Quantum Phase Transition between Integer Quantum Hall States of Bosons
• Symmetry protected Spin Quantum Hall phases in 2-Dimensions
• Conflicting Symmetries in Topologically Ordered Surface States of Three-dimensional Bosonic Symmetry Protected Topological Phases
• Microscopic Realization of 2-Dimensional Bosonic Topological Insulators
• Interacting and fractional topological insulators via the ${\mathbb{Z}}_2$ chiral anomaly

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