Classification and Description of Bosonic Symmetry Protected Topological Phases with semiclassical Nonlinear Sigma models

TL;DR

This paper classifies bosonic symmetry protected topological phases across all dimensions using semiclassical nonlinear sigma models with topological terms, linking physical boundary phenomena to mathematical group cohomology classifications.

Contribution

It introduces a unified NLSM framework for classifying SPT phases in all dimensions, connecting physical models with mathematical group cohomology and boundary physics.

Findings

Unified classification of SPT phases using O(d+2) NLSMs with topological terms

Explicit boundary physics and defect constructions for SPT phases

Consistency with mathematical group cohomology classification

Abstract

In this paper we systematically classify and describe bosonic symmetry protected topological (SPT) phases in all physical spatial dimensions using semiclassical nonlinear Sigma model (NLSM) field theories. All the SPT phases on a $d-$dimensional lattice discussed in this paper can be described by the same NLSM, which is an O(d+2) NLSM in $(d+1)-$dimensional space-time, with a topological $\Theta-$term. The field in the NLSM is a semiclassical Landau order parameter with a unit length constraint. The classification of SPT phases discussed in this paper based on their NLSMs is consistent with the more mathematical classification based on group cohomology. Besides the classification, the formalism used in this paper also allows us to explicitly discuss the physics at the boundary of the SPT phases, and it reveals the relation between SPT phases with different symmetries. For example, it gives many of these SPT states a natural "decorated defect" construction.