Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology

Symmetry-protected topological (SPT) states are short-range entangled states with a symmetry G. They belong to a new class of quantum states of matter which are classified by the group cohomology $H^{d+1}(G,\mathbb{R}/\mathbb{Z})$ in d-dimensional space. In this paper, we propose a class of symmetry- protected topological invariants that may allow us to fully characterize SPT states with a symmetry group G (ie allow us to measure the cocycles in $H^{d+1}(G,\mathbb{R}/\mathbb{...

Quantum phase transitions out of a symmetry-protected topological (SPT) phase in (1+1) dimensions into an adjacent, topologically distinct SPT phase protected by the same symmetry or a trivial gapped phase, are typically described by a conformal field theory (CFT). At the same time, the low-lying entanglement spectrum of a gapped phase close to such a quantum critical point is known(Cho et al., arXiv:1603.04016), very generally, to be universal and described by (gapless) boun...

Symmetry protected topological (SPT) states are bulk gapped states with gapless edge excitations protected by certain symmetries. The SPT phases in free fermion systems, like topological insulators, can be classified by the K-theory. However, it is not known what SPT phases exist in general interacting systems. In this paper, we present a systematic way to construct SPT phases in interacting bosonic systems, which allows us to identify many new SPT phases, including three bos...

We explore QCD$_4$ quark matter, the $\mu$-T (chemical potential-temperature) phase diagram, possible 't Hooft anomalies, and topological terms, via non-perturbative tools of cobordism theory and higher anomaly matching. We focus on quarks in 3-color and 3-flavor on bi-fundamentals of SU(3), then analyze the continuous and discrete global symmetries and pay careful attention to finite group sectors. We input constraints from $T=CP$ or $CT$ time-reversal symmetries, implementi...

The classification and lattice model construction of symmetry protected topological (SPT) phases in interacting fermion systems are very interesting but challenging. In this paper, we give a systematic fixed point wave function construction of fermionic SPT (FSPT) states for generic fermionic symmetry group $G_f=\mathbb{Z}_2^f \times_{\omega_2} G_b$ which is a central extension of bosonic symmetry group $G_b$ (may contain time reversal symmetry) by the fermion parity symmetry...

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 (1st-order) SPT phases in d spatial dimensions support anomalous (d-1)-dimensional surface states, HOSPT phases in d dimensions are characterized by topological boundary states of dimension (d-2) or smaller, protected by certain global symmetries and robust against di...

A large class of symmetry-protected topological phases (SPT) in boson / spin systems have been recently predicted by the group cohomology theory. In this work, we consider SPT states at least with charge symmetry (U(1) or Z$_N$) or spin $S^z$ rotation symmetry (U(1) or Z$_N$) in 2D, 3D, and the surface of 3D. If both are U(1), we apply external electromagnetic field / `spin gauge field' to study the charge / spin response. For the SPT examples we consider (i.e. U$_c$(1)$\rtim...

We consider the classification problem of symmetry protected topological (SPT) phases on quantum spin systems. SPT phases are gapped short-range-entangled quantum phases with a symmetry $G$. We explain that in one and two-dimensional quantum spin systems, there are $H^{2}(G, U(1))$/ $H^{3}(G, U(1))$-valued invariant, confirming a physicists conjecture.

We describe recent progress in our understanding of the interplay between interactions, symmetry, and topology in states of quantum matter. We focus on a minimal generalization of the celebrated topological band insulators to interacting many particle systems, known as Symmetry Protected Topological (SPT) phases. In common with the topological band insulators these states have a bulk gap and no exotic excitations but have non-trivial surface states that are protected by symme...

The boundary of symmetry-protected topological states (SPTs) can harbor new quantum anomaly phenomena. In this work, we characterize the bosonic anomalies introduced by the 1+1D non-onsite-symmetric gapless edge modes of 2+1D bulk bosonic SPTs with a generic finite Abelian group symmetry (isomorphic to $G=\prod_i Z_{N_i}=Z_{N_1} \times Z_{N_2} \times Z_{N_3} \times ...$). We demonstrate that some classes of SPTs (termed "Type II") trap fractional quantum numbers (such as frac...