Particle-Hole Symmetry and the Pfaffian State

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 easily by their edge physics: their edges differ in both their thermal Hall conductance and their tunneling exponents. 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 ${\sigma }_{xy}=\frac{5}{2}(e^2/h)$ quantum Hall plateau.