We construct an exactly soluble Hamiltonian on the
D=3 cubic lattice, whose ground state is a topological phase of bosons protected by time-reversal symmetry, i.e., a symmetry-protected topological (SPT) phase. In this model, excitations with anyonic statistics are shown to exist at the surface but not in the bulk. The statistics of these surface anyons is explicitly computed and shown to be identical to the three-fermion
Z2 model, a variant of
Z2 topological order which cannot be realized in a purely
D=2 system with time-reversal symmetry. Thus the model realizes a novel surface termination for three-dimensional (3D) SPT phases, that of a fully symmetric gapped surface with topological order. The 3D phase found here was previously proposed from a field theoretic analysis but is outside the group cohomology classification that appears to capture all SPT phases in lower dimensions. Such phases may potentially be realized in spin-orbit-coupled magnetic insulators, which evade magnetic ordering. Our construction utilizes the Walker-Wang prescription to create a 3D confined phase with surface anyons, which can be extended to other topological phases.
We analyze
2+1d and
3+1d bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group
G by using dual bulk and boundary approaches. In the bulk, we study an effective field theory, which upon coupling to a background flat
G gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat
G gauge fields to obtain Dijkgraaf-Witten topological
G gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in
2+1d and condensing bosonic gauge charges in
3+1d. In the dual boundary approach, we study
1+1d and
2+1d quantum field theories that have
G 't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct
G characters that generate topological data for the bulk topological gauge theory. Finally, we study a
2+1d quantum field theory with a mixed ${\mathbb{Z}}_{2}^{T/R}\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ anomaly where
Z2T/R is time-reversal/reflection symmetry, and the
U(1) could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in
3+1d that cancels this anomaly. This signals the existence of SPTs in
3+1d protected by 0,1-form $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.
In two-dimensional topological phases, quasiparticle excitations can carry fractional symmetry quantum numbers. We generalize this notion of symmetry fractionalization to three-dimensional topological phases, in particular to loop excitations, and propose a partial classification for symmetry-enriched
Z2 toric code phase. We apply the results to the classification of fermionic symmetry-protected topological phases in three dimensions.
It is well known that symmetry-protected topological (SPT) phases host nontrivial boundaries that cannot be mimicked in a lower-dimensional system with a conventional realization of symmetry. However, for SPT phases of bosons (fermions) within the cohomology (supercohomology) classification the boundary can be recreated without the bulk at the cost of a non-onsite-symmetry action. This raises the following question: Can one also mimic the boundaries of SPT phases which lie outside the (super)cohomology classification? In this paper, we study this question in the context of
(2+1)-dimensional
[(2+1)D] fermion SPTs. We focus on the root SPT phase for the symmetry group $G={Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Starting with an exactly solvable model for the bulk of this phase constructed by Tarantino and Fidkowski, we derive an effective one-dimensional (1D) lattice model for the boundary. Crucially, the Hilbert space of this 1D model does not have a local tensor product structure, but rather is obtained by placing a local constraint on a local tensor product Hilbert space. We derive the action of the
Z2 symmetry on this Hilbert space and find a simple three-site Hamiltonian that respects this symmetry. We study this Hamiltonian numerically using exact diagonalization and DMRG and find strong evidence that it realizes an Ising conformal field theory where the
Z2 symmetry acts as the Kramers-Wannier duality; this is the expected stable gapless boundary state of the present SPT. A simple modification of our construction realizes the boundary of the
(2+1)D topological superconductor protected by time-reversal symmetry
T with ${\mathcal{T}}^{2}={(\ensuremath{-}1)}^{\mathcal{F}}$.
We discuss several bosonic topological phases in (3+1) dimensions enriched by a global
Z2 symmetry, and gauging the
Z2 symmetry. More specifically, following the spirit of the bulk-boundary correspondence, expected to hold in topological phases of matter in general, we consider boundary (surface) field theories and their orbifold. From the surface partition functions, we extract the modular
S and
T matrices and compare them with
(2+1)-dimensional toplogical phase after dimensional reduction. As a specific example, we discuss topologically ordered phases in
(3+1) dimensions described by the BF topological quantum field theories, with abelian exchange statistics between pointlike and looplike quasiparticles. Once the
Z2 charge conjugation symmetry is gauged, the
Z2 flux becomes non-Abelian excitation. The gauged topological phases we are considering here belong to the quantum double model with non-Abelian group in
(3+1) dimensions.
Topological phases of matter are usually realized in deconfined phases of gauge theories. In this context, confined phases with strongly fluctuating gauge fields seem to be irrelevant to the physics of topological phases. For example, the low-energy theory of the two-dimensional (2D) toric code model (i.e., the deconfined phase of
Z2 gauge theory) is a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ Chern-Simons theory in which gauge charges (i.e.,
e and
m particles) are deconfined and the gauge fields are gapped, while the confined phase is topologically trivial. In this paper, we point out a route to constructing exotic three-dimensional (3D) gapped fermionic phases in a confining phase of a gauge theory. Starting from a parton construction with strongly fluctuating compact $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ gauge fields, we construct gapped phases of interacting fermions by condensing two linearly independent bosonic composite particles consisting of partons and $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ magnetic monopoles. This can be regarded as a 3D generalization of the 2D Bais-Slingerland condensation mechanism. Charge fractionalization results from a Debye-H\"uckel--type screening cloud formed by the condensed composite particles. Within our general framework, we explore two aspects of symmetry-enriched 3D Abelian topological phases. First, we construct a new fermionic state of matter with time-reversal symmetry and $\mathrm{\ensuremath{\Theta}}\ensuremath{\ne}\ensuremath{\pi}$, the fractional topological insulator. Second, we generalize the notion of anyonic symmetry of 2D Abelian topological phases to the charge-loop excitation symmetry (
Charles) of 3D Abelian topological phases. We show that line twist defects, which realize
Charles transformations, exhibit non-Abelian fusion properties.