Exactly soluble models for fractional topological insulators in two and three dimensions

Abstract

We construct exactly soluble lattice models for fractionalized, time reversal invariant electronic insulators in 2 and 3 dimensions. The low energy physics of these models is exactly equivalent to a non-interacting topological insulator built out of fractionally charged fermionic quasiparticles. We show that some of our models have protected edge modes (in 2D) and surface modes (in 3D), and are thus fractionalized analogues of topological insulators. We also find that some of the 2D models do not have protected edge modes -that is, the edge modes can be gapped out by appropriate time reversal invariant, charge conserving perturbations. (A similar state of affairs may also exist in 3D). We show that all of our models are topologically ordered, exhibiting fractional statistics as well as ground state degeneracy on a torus. In the 3D case, we find that the models exhibit a fractional magnetoelectric effect.

Topological insulators avoid the parity anomaly

“…There are four flux choices for which the system is time-reversal invariant (Φ 1 , Φ 2 ) = (0, 0), (π, 0), (0, π), and (π, π). If the many-body ground state is Kramers degenerate (i.e., orthogonal to its timereversed conjugate, which is necessarily a state of the same energy) in an odd number of the 4 flux sectors, the surface spectrum has 2+1 dimensional gapless surface states (in the thermodynamic limit) [22]. These gapless states cannot be eliminated without breaking the Kramers degeneracy, and therefore breaking time-reversal symmetry; hence the system is a STI.…”

“…While these surface properties are well established (or at least well believed) theoretically for the case of topological band insulators which need not be coupled to a fluctuating gauge field in order to be defined, our analysis is equally applicable to the case of more exotic fractional topological insulators, such as those described in [20][21][22][23], in which the presence of a fluctuating "internal" gauge field is inevitable. Thus, our result is important in clarifying why P and T invariant gapless boundary modes exist in these systems -as well as in understanding their topological order.…”

“…The eigenvalues for the scattering states may be obtained by considering (22) in the limit x 3 → ±∞. In this limit, translation invariance along the x 3 -direction is effectively restored and so we introduce an asymptotic momentum k 3 parametrizing the eigenvalues of (22). There are two pairs of eigenspinors ψ λ(k3)±,(i)…”

“…[22,44]. They consider the fate of a STI on the thickened spatial torus S 1 × S 1 × I, which has two disconnected toroidal surface boundaries, and two non-contractible curves through which we may insert magnetic flux.…”

Three-dimensional topological insulators and bosonization

“…• Such an approach actually corresponds to the BF bulk theory [28], that can be naturally extended to (3 + 1) dimensions [26,27], where it accounts for particlevortex braiding relations and other topological effects [56,57].…”