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.
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.
It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear $\ensuremath{\sigma}$ models $(\mathrm{NL}\ensuremath{\sigma}\mathrm{Ms})$ of the symmetry group
G with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology
Hd(G,R/Z). But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by $G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}) \mathrm{NL}\ensuremath{\sigma}\mathrm{Ms}$ with various topological terms. Through the group cohomology ${\mathcal{H}}^{d}[G\ifmmode\times\else\texttimes\fi{}SO(\ensuremath{\infty}),\mathbb{R}/\mathbb{Z}]$, we find that the set of our constructed SPT phases in
d-dimensional space-time are described by ${E}^{d}(G)\ensuremath{\rtimes}{\ensuremath{\bigoplus}}_{k=1}^{d\ensuremath{-}1}{\mathcal{H}}^{k}(G,{\text{iTO}}_{L}^{d\ensuremath{-}k})\ensuremath{\bigoplus}{\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$ where
G may contain time reversal. Here
iTOLd is the set of the topologically ordered phases in
d-dimensional space-time that have no topological excitations, and one has
iTOL1=iTOL2=iTOL4=iTOL6=0,iTOL3=Z,iTOL5=Z2,iTOL7=2Z. For $G=U(1)\ensuremath{\rtimes}{Z}_{2}^{T}$ (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond ${\mathcal{H}}^{d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},\mathbb{R}/\mathbb{Z}]$ are described by
Z2 in 3 + 1D,
Z in 4 + 1D,
3Z2 in 5 + 1D, and
4Z2 in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.
Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry
G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by
SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by
U(1) and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct
d-dimensional bosonic SPT phases with on-site symmetry
G (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in
H1+d[G,UT(1)], the Borel
(1+d)-group-cohomology classes of
G over the
G module
UT(1). Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological $\ensuremath{\theta}$ term in continuous nonlinear $\ensuremath{\sigma}$ models to lattice nonlinear $\ensuremath{\sigma}$ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear $\ensuremath{\sigma}$ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear $\ensuremath{\sigma}$ models. As a result, the symmetry
G must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use ${\mathcal{H}}^{1+d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},{U}_{T}(1)]$ to obtain interacting bosonic topological insulators (protected by time reversal
Z2T and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of
H1+d[Z2T,UT(1)], which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use ${\mathcal{H}}^{1+d}[U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T},{U}_{T}(1)]$ to construct the SPT phases of integer spin systems with time-reversal and
U(1) spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: $({G}_{H},{G}_{\ensuremath{\Psi}},{\mathcal{H}}^{1+d}[{G}_{\ensuremath{\Psi}},{U}_{T}(1)])$, where
GH is the symmetry group of the Hamiltonian and ${G}_{\ensuremath{\Psi}}$ the symmetry group of the ground states.
It has been shown that the bosonic symmetry-protected-trivial (SPT) phases with pure gauge anomalous boundary can all be realized via nonlinear σ models (NLσMs) of the symmetry group G with various topological terms. Those SPT phases (called the pure SPT phases) can be classified by group cohomology H[superscript d](G,R/Z). But there are also SPT phases with mixed gauge-gravity anomalous boundary (which will be called the mixed SPT phases). Some of the mixed SPT states were also referred as the beyond-group-cohomology SPT states. In this paper, we show that those beyond-group-cohomology SPT states are actually within another type of group cohomology classification. More precisely, we show that both the pure and the mixed SPT phases can be realized by G × SO(∞) NLσMs with various topological terms. Through the group cohomology H[superscript d][G × SO(∞),R/Z], we find that the set of our constructed SPT phases in d-dimensional space-time are described by E[superscript d](G) ⋊ ⊕[d−1 over k=1]H[superscript k](G,iTO[d−k over L]) ⊕ H[superscript d](G,R/Z) where G may contain time reversal. Here iTO[d over L] is the set of the topologically ordered phases in d-dimensional space-time that have no topological excitations, and one has iTO[1 over L] = iTO[2 over L] = iTO[4 over L] = iTO[6 over L] = 0, iTO[3 over L] = Z, iTO[5 over L] = Z[subscript 2], iTO[7 over L] =2Z. For G=U(1) ⋊ Z[ T over 2] (charge conservation and time-reversal symmetry), we find that the mixed SPT phases beyond H[superscript d][U(1) ⋊ Z[T over 2],R/Z] are described by Z[subscript 2] in 3 + 1D, Z in 4 + 1D, 3Z[subscript 2] in 5 + 1D, and 4Z[subscript 2] in 6 + 1D. Our construction also gives us the topological invariants that fully characterize the corresponding SPT and iTO phases. Through several examples, we show how the universal physical properties of SPT phases can be obtained from those topological invariants.