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 3D bosonic SPT phases with space group symmetry G is , where the H 1 term classifies phases beyond the Borel group cohomology proposal.
It is demonstrated by explicit construction that three-dimensional bosonic crystalline symmetry protected topological (cSPT) phases are classified by ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})\ensuremath{\bigoplus}{H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ for all 230 space groups
G, where ${H}_{\ensuremath{\phi}}^{n}(G;\mathbb{Z})$ denotes the
nth twisted group cohomology of
G with
Z coefficients, and $\ensuremath{\phi}$ indicates that $g\ensuremath{\in}G$ acts nontrivially on coefficients by sending them to their inverses if
g reverses space-time orientation and acts trivially otherwise. The previously known summand ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})$ corresponds only to crystalline phases built without the
E8 state or its multiples on 2-cells of space. It is the crystalline analog of the ``group cohomology proposal'' for classifying bosonic symmetry protected topological (SPT) phases, which takes the form ${H}_{\ensuremath{\phi}}^{d+2}(G;\mathbb{Z})\ensuremath{\cong}{H}_{\ensuremath{\phi}}^{d+1}(G;\text{U}(1))$ for finite internal symmetry groups in
d spatial dimensions. The new summand ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ classifies possible configurations of
E8 states on 2-cells that can be used to build crystalline phases beyond the group cohomology proposal. The completeness of our classification and the physical meaning of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ are established through a combination of dimensional reduction, surface topological order, and explicit cellular construction. The value of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ can be easily read off from the international symbol for
G. Our classification agrees with the prediction of the ``generalized cohomology hypothesis,'' which concerns the general structure of the classification of SPT phases, and therefore provides strong evidence for the validity of the said hypothesis in the realm of crystalline symmetries.
In this thesis, we study gapped topological phases of matter in systems with strong inter-particle interaction. They are challenging to analyze theoretically, because interaction not only gives rise to a plethora of phases that are otherwise absent, but also renders methods used to analyze non-interacting systems inadequate. By now, people have had a relatively systematic understanding of topological orders in two spatial dimensions. However, less is known about the higher dimensional cases. In Chapter 2, we will explore three dimensional long-range entangled topological orders in the framework of Walker-Wang models, which are a class of exactly solvable models for three-dimensional topological phases that are not known previously to be able to capture these phases. We find that they can represent a class of twisted discrete gauge theories, which were discovered using a different formalism. Meanwhile, a systematic theory of bosonic symmetry protected topological (SPT) phases in all spatial dimensions have been developed based on group cohomology. A generalization of the theory to group supercohomology has been proposed to classify and characterize fermionic SPT phases in all dimensions. However, it can only handle cases where the symmetry group of the system is a product of discrete unitary symmetries. Furthermore, the classification is known to be incomplete for certain symmetries. In Chapter 3, we will construct an exactly solvable model for the two-dimensional time-reversal-invariant topological superconductors, which could be valuable as a first attempt to a systematic understanding of strongly interacting fermionic SPT phases with anti-unitary symmetries in terms of exactly solvable models. In Chapter 4, we will propose an alternative classification of fermionic SPT phases using the spin cobordism theory, which hopefully can capture all the phases missing in the supercohomology classification. We test this proposal in the case of fermionic SPT phases with Z2 symmetry, where Z2 is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known fermionic SPT phases in space dimensions less than or equal to 3.
This works uses the two dimensional rotational invariant plane in bosonic symmetry protected topological phases with
C2 rotational symmetry in four dimensions to classify the phases and obtain a boundary field theory as well as an anomalous topological order for the phase that is not captured by the group cohomology classification.
Motivated by the recent discovery of higher-order topological insulators, we study their counterparts in strongly interacting bosons: ``higher-order symmetry-protected topological (HOSPT) phases.'' While the usual (first-order) SPT phases in
d spatial dimensions support anomalous $(d\ensuremath{-}1)$-dimensional surface states, HOSPT phases in
d dimensions are characterized by topological boundary states of dimension $(d\ensuremath{-}2)$ or smaller, protected by certain global symmetries and robust against disorders. Based on a dimensional reduction analysis, we show that HOSPT phases can be built from lower-dimensional SPT phases in a way that preserves the associated crystalline symmetries. When the total symmetry is a direct product of global and crystalline symmetry groups, we are able to classify the HOSPT phases using the K\"unneth formula of group cohomology. Based on a decorated domain-wall picture of the K\"unneth formula, we show how to systematically construct the HOSPT phases, and demonstrate our construction with many examples in two and three dimensions.