The Phase Diagram of the $\nu=5/2$ Fractional Quantum Hall Effect: Effects of Landau Level Mixing and Non-Zero Width*

We study the phase diagram of the $\nu=5/2$ state by exactly diagonalizing an effective Hamiltonian describing the fractional quantum Hall effect of electrons under realistic conditions in GaAs semiconductors. The effective Hamiltonian takes Landau level mixing into account to lowest-order perturbatively in $\kappa$, the ratio of the Coulomb energy scale to the cyclotron gap and we incorporate non-zero width $w$ of the quantum well and sub-band mixing. Using the torus and sphere, we analyze the non-trivial competition between candidate ground states with 4 criteria: overlaps with trial wave functions; the size of energy gaps; the sign of an order parameter for particle-hole symmetry breaking; and entanglement spectrum. We find the ground state is in the universality class of the Moore-Read Pfaffian state, rather than anti-Pfaffian, for $\kappa<\kappa_c(w)$, where $0.6< \kappa_c (w) <1$. Landau level mixing and non-zero width suppress the excitation gap with Landau level mixing having a larger effect. Our findings have important implications for the identification of non-Abelian fractional quantum Hall states.