Classification and description of bosonic symmetry protected topological phases with semiclassical nonlinear sigma models

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 Completely Identical to the more mathematical classification based on group cohomology given in Ref. 1,2. 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.

Construction and field theory of bosonic-symmetry-protected topological states beyond group cohomology

“…There are roughly two types of BSPT states, their mathematical difference is whether they can be classified and described by group cohomology [1,2] and semiclassical nonlinear sigma model field theory [3]. For example, the well-known E 8 bosonic short range entangled (BSRE) state [18] [4,5] in 2d space [19], and its higher dimensional generalizations [6] cannot be classified by group cohomology.…”

Bilayer Graphene as a Platform for Bosonic Symmetry-Protected Topological States

“…Based on the formalism developed in Ref. 14,15, this state has a Z classification, i.e. with these symmetries there is an infinite set of non-trivial 2d BSPT classes, which are indexed by an integer k. Effective field theory descriptions of these BSPT states have been given in terms of Chern-Simon field theory [14] and a non-linear sigma model (NLSM) with a Θ-term [15,17].…”

“…1, 2 in dimensions higher than one all involve high order multiple spin interactions, and are thus unlikely to exist in realistic materials. Up to now, all approaches to classifying and characterizing BSPT states [1,2,[13][14][15][16] rely on mathematical or effective field theory descriptions, which shed little light on how to identify a realistic candidate BSPT state.…”

Symmetry-protected topologically ordered states for universal quantum computation

“…We follow the seminal paper by Chen, Gu, Liu and Wen [29]. We remark that the classification can actually be understood via other approaches: (1) anomalous symmetry action at the boundary [35], (2) the nonlinear sigma model with a topological theta term [36,37], and (3) gauge fields for symmetry twists [38], and (4) cobordism [39], all leading to SPT phases beyond cohomology [40]. Readers who are familiar with the canonical form of short-ranged entangled states in 2D and group cohomology can skip this section and go directly to Sec.…”