We know that conformal field theories are closely related to two-dimensional topological orders via edge-boundary correspondence. An Ising topological order can be obtained by gauging the fermion parity from a $p+ip$ superconductor. The anyon fusion rule $\sigma\times \sigma = 1+ \psi$ (where $\sigma$ is the vortex excitation that binds a Majorana zero mode, from which the above fusion rule is easily identified) indicates its relation with the free (Majorana) fermion CFT with $c=1/2$.

Indeed, on its edge, there is a chiral mode with $c=1/2$. Besides the fermion mode, such a CFT has a twist operator $\sigma$ with $h=1/16$. My question is what is this operator in the context of $p+ip$ superconductor? What is its relation with the Majorana-binding vortex in the bulk? How do I understand its fusion rule $\sigma\times \sigma = 1+ \psi$ in the edge CFT sense?


p+ip superconductor is an invertible topological order whose intrinsic bulk excitations are fermions. There is no non-abelian anyons. The vortex with Majorana zero mode is not an intrinsic bulk excitation.

The $SU(2)_2$ QH state $\chi_1(z_i)\chi_2^2(z_i)$ and the Paffian QH state have Ising topological order. They have $\sigma$ non-abelian particle as intrinsic bulk excitations. (Here $\chi_n$ is the many-fermion wave function with $n$ filled Landau levels.)

  • $\begingroup$ Thanks! Can't I gauge some symmetries in the p+ip SC to make the Majorana binding vortex an intrinsic excitation? $\endgroup$ – pathintegral Aug 2 '17 at 19:48
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    $\begingroup$ Yes. Just gauge $Z_2^f$ fermion-parity symmetry. $\endgroup$ – Xiao-Gang Wen Aug 3 '17 at 0:35
  • $\begingroup$ So this was exactly what I said in the original question :) I was just wondering what the twist operator in the edge CFT mean in the context of the p+ip SC. $\endgroup$ – pathintegral Aug 3 '17 at 1:04

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