Intrinsically gapless symmetry-protected topological phases (igSPT) are gapless systems with SPT edge states with properties that could not arise in a gapped system with the same symmetry and dimensionality. igSPT states arise from gapless systems in which an anomaly in the low-energy (infrared) symmetry group emerges from an extended anomaly-free microscopic (ultraviolet) symmetry. We construct a general framework for constructing lattice models for igSPT phases with emergent anomalies classified by group cohomology, and establish a direct connection between the emergent anomaly, group-extension, and topological edge states by gauging the extending symmetry. In many examples, the edge-state protection has a physically transparent mechanism: the extending UV symmetry operations pump lower-dimensional SPTs onto the igSPT edge, tuning the edge to a (multi)critical point between different SPTs protected by the IR symmetry. In two- and three-dimensional systems, an additional possibility is that the emergent anomaly can be satisfied by an anomalous symmetry-enriched topological order, which we call a quotient-symmetry-enriched topological order (QSET) that is sharply distinguished from the nonanomalous UV SETs by an edge phase transition. We construct exactly solvable lattice models with QSET order.
We consider topological phases in periodically driven (Floquet) systems exhibiting many-body localization, protected by a symmetry
G. We argue for a general correspondence between such phases and topological phases of undriven systems protected by symmetry $\mathbb{Z}\ensuremath{\rtimes}G$ where the additional
Z accounts for the discrete time-translation symmetry. Thus, for example, the bosonic phases in
d spatial dimensions without intrinsic topological order [symmetry-protected topological (SPT) phases] are classified by the cohomology group ${H}^{d+1}[\mathbb{Z}\ensuremath{\rtimes}G,\mathrm{U}(1)]$. For unitary symmetries, we interpret the additional resulting Floquet phases in terms of the lower-dimensional SPT phases that are pumped to the boundary during one time step. These results also imply the existence of novel symmetry-enriched topological (SET) orders protected solely by the periodicity of the drive.
We study a topological phase of interacting bosons in (3 þ 1) dimensions that is protected by charge conservation and time-reversal symmetry.We present an explicit lattice model that realizes this phase and that can be studied in sign-free Monte Carlo simulations.The idea behind our model is to bind bosons to topological defects called hedgehogs.We determine the phase diagram of the model and identify a phase where such bound states are proliferated.In this phase, we observe a Witten effect in the bulk whereby an external monopole binds half of the elementary boson charge, which confirms that it is a bosonic topological insulator.We also study the boundary between the topological insulator and a trivial insulator.We find a surface phase diagram that includes exotic superfluids, a topologically ordered phase, and a phase with a Hall effect quantized to one-half of the value possible in a purely two-dimensional system.We also present models that realize symmetry-enriched topologically ordered phases by binding multiple hedgehogs to each boson; these phases show charge fractionalization and intrinsic topological order as well as a fractional Witten effect.