Particle-hole symmetry and the Pfaffian state

We consider the properties of the Moore-Read Pfaffian state under particle-hole conjugation. We show that the particle-hole conjugate of the Pfaffian state - or "anti-Pfaffian" state - is in a different universality class from the Pfaffian state, with different topological order. The two states can be distinguished by both their bulk and edge physics though the difference is most dramatic at the edge: the edge of the anti-Pfaffian state has a composite structure that leads to a different thermal Hall conductance and different tunneling exponents than the Pfaffian state. At the same time, the two states are exactly degenerate in energy for a $\nu = 5/2$ quantum Hall system in the idealized limit of zero Landau level mixing. Thus, both are good candidates for the observed $\nu = 5/2$ quantum Hall plateau.