  Interesting non-Abelian states, e.g., the Moore-Read Pfaffian and the
anti-Pfaffian, offer candidate descriptions of the $\nu = 5/2$ fractional
quantum Hall state. But the significant controversy surrounding the nature of
the $\nu = 5/2$ state has been hampered by the fact that the competition
between these and other states is affected by small parameter changes. To study
the phase diagram of the $\nu = 5/2$ state we numerically diagonalize a
comprehensive 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. We also incorporate non-zero width $w$ of the quantum well and
sub-band mixing. We find the ground state in both the torus and spherical
geometries as a function of $\kappa$ and $w$. To sort out the non-trivial
competition between candidate ground states we analyze the following 4
criteria: its overlap with trial wave functions; the magnitude of energy gaps;
the sign of the expectation value of an order parameter for particle-hole
symmetry breaking; and the entanglement spectrum. We conclude that the ground
state is in the universality class of the Moore-Read Pfaffian state, rather
than the anti-Pfaffian, for $\kappa < {\kappa_c}(w)$, where ${\kappa_c}(w)$ is
a $w$-dependent critical value $0.6 \lesssim{\kappa_c}(w)\lesssim 1$. We
observe that both Landau level mixing and non-zero width suppress the
excitation gap, but Landau level mixing has a larger effect in this regard. Our
findings have important implications for the identification of non-Abelian
fractional quantum Hall states.
