Particle Hole Symmetry Breaking In the Fractional Quantum Hall Effect at $\nu = \frac{5}{2}$

The fractional quantum Hall effect (FQHE) is widely studied because of its exotic topological phases. The FQHE at $\nu = \frac{5}{2}$ is interesting because it supports excitations of non-abelian quasiparticles. These non-abelian particles are one possible candidate for use as qubits in topological quantum computations. The leading theoretical description of the FQHE at $\nu = \frac{5}{2}$ is the Moore-Reed Pfaffian and its particle hole conjugate the anti-Pfaffian. The Pfaffian and the anti-Pfaffian are the exact ground states of a three body Hamiltonian ($H_3$) and its particle hole conjugate ($H^\prime_3$), respectively. The Pfaffian breaks particle hole symmetry (PHS) explicitly while the physical interaction (Coulomb) is largely PHS. We define a PHS Hamiltonian (H$_2$) and ask is PHS breaking necessary in order to produce a Pfaffian ground state? To answer, we study $H(\alpha)= (1-\alpha)H_3 + \alpha H_2$ and tune alpha from 0 to 1. We show that the ground and low energy states for $H_2$ and $H_3$ remain adiabatically connected. This adiabatic connection shows the low energy states for $H_2$ and $H_3$ are in the same universality class.