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

TL;DR

This work presents a comprehensive, physics-driven classification of bosonic SPT phases across dimensions using semiclassical $O(d+2)$ nonlinear sigma models with topological $\Theta$-terms. By analyzing edge theories, RG flows, and decorated-defect constructions, it yields explicit root phases and boundary properties that reproduce the group cohomology classifications for a wide set of symmetries, including $Z_m$, $U(1)$, and $SO(3)$, among others. The approach clarifies when $\Theta=2\pi k$ and $\Theta=0$ are connected, explains decorated-domain-wall and decorated-vortex pictures, and connects 1d, 2d, and 3d SPTs through a unified field-theoretic framework. It also highlights beyond-cohomology states and potential generalizations to larger symmetry groups and lattice symmetries, underscoring the practical impact for constructing and understanding robust bosonic topological phases.

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 $Θ-$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.