2012

Classification and analysis of two dimensional abelian fractional topological insulators

Levin, Michael, Stern, Ady

Understand

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.

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It is not hard to show that e i Θ ( Λ ) e^{i\Theta(\Lambda)} and e i Θ ( Λ ′ ) e^{i\Theta(\Lambda^{\prime})} commute if Λ T 𝒦 Λ ′ \Lambda^{T}\mathcal{K}\Lambda^{\prime} is even and anti-commute if Λ T 𝒦 Λ ′ \Lambda^{T}\mathcal{K}\Lambda^{\prime} is odd, using [ ϕ I ( x ) , ϕ J ( y ) ] = π i 𝒦 I J − 1 sgn ( x − y ) [\phi_{I}(x),\phi_{J}(y)]=\pi i\mathcal{K}^{-1}_{IJ}\text{sgn}(x-y)
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