Minimalist approach to the classification of symmetry protected topological phases

TL;DR

<3-5 sentences>We address the classification of symmetry protected topological (SPT) phases by proposing a Generalized Cohomology Hypothesis: the full array of d-dimensional G-protected SPT phases across all d and all symmetry groups G is captured by an unreduced generalized cohomology theory h, with h^d(BG) encoding the classifications and the abelian group structure arising from stacking. This meta-framework unifies diverse proposals (Borel group cohomology, oriented cobordism, spin cobordism, group supercohomology, etc.) as special cases, and yields dimension- and symmetry-related relations that hold beyond concrete constructions. By exploiting the omega-spectrum interpretation, the authors derive exact decompositions for systems with translation and Floquet symmetries, including a hierarchical strong/weak topological index structure and pumping interpretations. As a concrete application, they predict the complete classification of 3D bosonic crystalline SPTs with space group G to be $H^4_{ m Borel}(G;U(1)) \oplus H^1_{ m group}(G;\mathbb{Z})$, revealing beyond-group-cohomology phases; they also discuss realizability and obstruction-free symmetry enlargement within this universal framework. This approach offers a universal, construction-agnostic lens for understanding SPT phases and guides future exploration of spatiotemporal and crystalline symmetries in interacting systems.

Abstract

A number of proposals with differing predictions (e.g. Borel group cohomology, oriented cobordism, group supercohomology, spin cobordism, etc.) have been made for the classification of symmetry protected topological (SPT) phases. Here we treat various proposals on an equal footing and present rigorous, general results that are independent of which proposal is correct. We do so by formulating a minimalist Generalized Cohomology Hypothesis, which is satisfied by existing proposals and captures essential aspects of SPT classification. From this Hypothesis alone, formulas relating classifications in different dimensions and/or protected by different symmetry groups are derived. Our formalism is expected to work for fermionic as well as bosonic phases, Floquet as well as stationary phases, and spatial as well as on-site symmetries. As an application, we predict that the complete classification of 3-dimensional bosonic SPT phases with space group symmetry $G$ is $H^4_{\rm Borel}\left(G;U(1)\right) \oplus H^1_{\rm group}\left(G;\mathbb Z\right)$, where the $H^1$ term classifies phases beyond the Borel group cohomology proposal.

Figures (16)

• Figure 1: (color online). Schematic illustration of the structure of the space of $d$-dimensional, $G$-symmetric, non-symmetry-breaking, local, gapped systems. Each deformation class, shown as a patch here, is called a $G$-protected topological phase. Each invertible (respectively non-invertible) class, shown as a gray or black (respectively pink) patch, is called an SPT (respectively SET) phase. The identity class, shown as a black patch, is called the trivial SPT phase. Dashed circles are meant to indicate, by forgetting the symmetry, that more systems will be allowed and that distinct phases can become one.
• Figure 2: (color online). Schematic illustration of the structure of the space of $d$-dimensional local, gapped systems. Each deformation class, shown as a patch here, is called a topological order. Each invertible (respectively non-invertible) class, shown as a gray or black (respectively pink) patch, is called an invertible (respectively intrinsic) topological order. The identity class, shown as a black patch, is called the trivial topological order, which is in particular invertible. A system is called SRE (respectively LRE) if it belongs to an invertible (respectively intrinsic) topological order.
• Figure 3: (color online). Stacking is associative. Given three systems, $a$ (green), $b$ (blue), and $c$ (orange), combining $a$ and $b$ first and then $c$ (upper panel) produces the same system as combining $b$ and $c$ first and then $a$ (lower panel) does.
• Figure 4: (color online). Given $G'\subset G$, a representative of a $d$-dimensional $G$-protected SPT phase can also be viewed as a representative of a $d$-dimensional $G'$-protected SPT phase, which in turn can be viewed as a representative of a $d$-dimensional invertible topological order, by forgetting first the symmetry operations outside $G'$ and then $G'$ itself. This defines a map from the set of $d$-dimensional $G$-protected SPT phases to the discrete abelian group of $d$-dimensional $G'$-protected SPT phases, and then to the set of $d$-dimensional invertible topological orders.
• Figure 5: (color online). The pumping interpretation of $\Omega$-spectrum. (a) A $(d+1)$-dimensional SRE state $f(a)_t$ constructed from a $d$-dimensional SRE state $a$. (b) The evolution of $f(a)_t$ as $t$ varies from 0 to 1. (c) An arbitrary one-parameter family of $(d+1)$-dimensional SRE states, $\mu(t)$ for $0 \leq t \leq 1$. (d) The pumping of a $d$-dimensional SRE state to the boundary of a $(d+1)$-dimensional system that is cut open, in the adiabatic evolution defined by $\mu$.

Theorems & Definitions (97)

• Definition 4.1
• Definition 4.2
• Proof 1
• Proof 2
• Proof 3
• Proof 4
• Proof 5
• Proof 6
• Proof 7
• Proof 8