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

TL;DR

This paper characterizes bosonic anomalies at the edges of 2+1D bosonic SPTs with finite Abelian symmetry, revealing fractional quantum numbers and zero modes linked to group cohomology, and connects these phenomena to physical models and continuum theories.

Contribution

It provides a detailed physical and mathematical characterization of bosonic anomalies, fractional quantum numbers, and zero modes at SPT edges, linking them to cocycles in group cohomology and offering lattice and continuum descriptions.

Findings

Certain SPTs trap fractional quantum numbers at domain wall kinks.

Some SPTs have degenerate zero energy modes protected by symmetry.

Edge mode spectra under flux insertion distinguish SPT classes.

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.