Model Realization and Numerical Studies of a Three-Dimensional Bosonic Topological Insulator and Symmetry-Enriched Topological Phases

Abstract

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.

Composite Dirac Liquids: Parent States for Symmetric Surface Topological Order

“…In both cases we recovered topological orders in agreement with previous works that treated isotropic systems with local time-reversal symmetry [17,55,56]. Given these successes it is natural to ask when a symmetric surface can be reliably distilled to a quasi-1D setup with reduced symmetry while preserving the structure of possible surface phases.…”

“…A rich story is now emerging [58]. Previous work has demonstrated that interactions allow one to (i) smoothly connect certain phases that are topologically distinct in the free-particle limit [33,[59][60][61][62][63][64][65][66][67], (ii) generate new shortrange-entangled topological phases with no noninteracting counterpart, for both bosonic and fermionic systems [17,68,69], and (iii) form gapped topological orders at the boundary of a 3D topological phase that cannot exist in strictly 2D systems with the same symmetry [13][14][15][16][17]55,56,[65][66][67]70,71]. To this list we have added a new possibility: Strong interactions can nucleate exotic gapless composite Dirac liquids at the surface of a 3D electronic TI.…”

“…We will treat two distinct bosonic topological phases-one protected by timereversal and charge-conservation symmetries, the other just time reversal-and recover the surface topological orders obtained by different means in Refs. [17,55,56].…”

Half-filled Landau level, topological insulator surfaces, and three-dimensional quantum spin liquids

“…On the other hand, with interactions, several groups [23][24][25][26] described how a symmetry preserving gapped surface can emerge for the bulk topological insulator. Inspired by similar constructions [27][28][29][30][31] for bosonic analogs of the topological insulators these papers showed that such a symmetry preserving gapped surface requires the kind of topological order familiar from discussions of the fractional quantum Hall effect and some quantum spin liquids. However, the symmetry is implemented in this topologically ordered state in a manner that is forbidden ("anomalous") in a strictly two-dimensional system.…”

Fractional Topological Insulators: A Pedagogical Review

“…Two dimensional systems are a natural playground for fractional statistics, since they provide a classification of trajectories of point particles in terms of winding numbers. However, the expansion of the world of topological systems from two to three dimensions, which was initiated by the discovery of the 3D topological insulators, naturally suggests a search for 3D fractional topological insulators 32,[58][59][60][61][62][63][64] . Defining characteristics of such a phase may be expected to be a degeneracy of the ground state on the boundary-less three dimensional torus, a surface quantum Hall effect of ν = (pe 2 )/2qh (with p/q being a fraction), fractionally charged excitations in the bulk, mutual statistics between bulk point and loop particles and a coupling of fractional charges to monopoles in the bulk.…”