Abstract
Bosonic topological insulators (BTIs) in three dimensions are symmetry-protected topological phases protected by time-reversal and boson number conservation symmetries. BTIs in three dimensions were first proposed and classified by the group cohomology theory, which suggests two distinct root states, each carrying a index. Soon after, surface anomalous topological orders were proposed to identify different root states of BTIs, which even leads to a new BTI root state beyond the group cohomology classification. In this paper, we propose a universal physical mechanism via vortex-line condensation from a 3D superfluid to achieve all three root states. It naturally produces a bulk topological quantum field theory description for each root state. Topologically ordered states on the surface are rigorously derived by placing topological quantum field theory on an open manifold, which allows us to explicitly demonstrate the bulk-boundary correspondence. Finally, we generalize the mechanism to symmetries and discuss potential symmetry-protected topological phases beyond the group cohomology classification.
- Received 3 December 2014
DOI:https://doi.org/10.1103/PhysRevX.5.021029
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Published by the American Physical Society
Popular Summary
Three-dimensional topological insulators represent a new type of insulator made of electrons whose Hamiltonian and ground state respect time-reversal symmetry. The two-dimensional surfaces of three-dimensional topological insulators support anomalous metallic physics compared with ordinary two-dimensional metals. Such surface novelty is protected by time-reversal symmetry and U(1) fermion number conservation symmetry. Here, we consider the interplay of symmetry and topology by extending the idea of topological insulators to consider bosonic topological insulators that are formed by bosons and respect time-reversal symmetry and U(1) boson number conservation symmetry. We construct a bulk field theory using a condensation of topological line defects, i.e., vortex lines in a three-dimensional superfluid. We also construct nontrivial bosonic topological phases protected by Ising symmetry in three dimensions; previous claims stated that these phases were always trivial in three dimensions.
In contrast to topological insulators where electrons are either free or weakly interacting, interactions in bosonic topological insulators are vital since bosons generically form a Bose-Einstein condensate or superfluid if interactions are not sufficiently strong. As such, bosonic topological insulator physics is unfortunately a strongly correlated problem, in contrast to topological insulators that have well-defined noninteracting limits. It has been shown that this complexity of bosonic topological insulators creates surface physics that is much more profound than anomalous metallic surface physics. More precisely, it has been found that the surfaces of bosonic topological insulators support topological order that cannot survive on a two-dimensional plane alone unless symmetry is explicitly broken.
Our study of topological quantum field theory in three-dimensional materials reveals new topological field theories that resemble the cosmological constant term of loop quantum gravity; these theories have not yet been widely explored in condensed matter physics. We expect that our results will motivate future studies with ensembles of ultracold atoms.
