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

Abstract

Interesting non-Abelian states, e.g., the Moore-Read Pfaffian and the anti-Pfaffian, offer candidate descriptions of the ν=5/2 fractional quantum Hall state. But the significant controversy surrounding the nature of the ν=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 ν=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 κ , 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 κ 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 κ<κ c (w) , where κ c (w) is a w -dependent critical value 0.6≲κ c (w)≲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.