Classification and analysis of two-dimensional Abelian fractional topological insulators

We present a general framework for analyzing fractionalized, time-reversal invariant electronic insulators in two dimensions. The framework applies to all insulators whose quasiparticles have Abelian braiding statistics. First, we construct the most general Chern-Simons theories that can describe these states. We then derive a criterion for when these systems have protected gapless edge modes, that is, edge modes that cannot be gapped out without breaking time-reversal or charge-conservation symmetry. The systems with protected edge modes can be regarded as fractionalized analogues of topological insulators. We show that previous examples of 2D fractional topological insulators are special cases of this general construction. As part of our derivation, we define the concept of “local Kramers degeneracy” and prove a local version of Kramers theorem.

Figure 1

The flux insertion argument for the $\nu =k$ toy model: we start with the ground state ${\Psi }_0$ and adiabatically inset $\pm {\Phi }_0/2$ flux through the cylinder, obtaining two states ${\Psi }_1\text{and}{\Psi }_2$. If $k$ is odd, then ${\Psi }_1$ has a Kramers degeneracy at the two ends of the cylinder, and is therefore degenerate in energy with three other states, one of which is ${\Psi }_2$. If we start in one of these three degenerate states and then adiabatically reduce the flux to 0, we obtain an excited state ${\Psi }_{ex}$ whose energy gap $\Delta$ vanishes in the thermodynamic limit.

Figure 2

(a) A schematic portrait of the ground state ${\Psi }_0$ of the $\nu =1$ toy model. Working in the Landau gauge, the single-particle states consist of spin-up and spin-down Landau orbitals. In the time-reversal invariant ground state, the spin-up and spin-down orbitals are equally occupied. After inserting ${\Phi }_0/2$ flux, the spin-up and spin-down orbitals shift in opposite directions, resulting in the state ${\Psi }_1$. (b) The state ${\Psi }_1$ has a Kramers degeneracy on both ends of the cylinder and is therefore degenerate with its time-reversed partner (c) ${\Psi }_2$ as well as two other states, (d) ${\Psi }_3$, and (e) ${\Psi }_4$.