Using exact diagonalization we show that the spin-polarized Coulomb ground state at nu=5/2 is adiabatically connected with the Moore-Read wave function for systems with up to 18 electrons on the surface of a sphere. The ground state is protected by a large gap for all system sizes studied. Furthermore, varying the Haldane pseudopotentials v{1} and v{3}, keeping all others at their value for the Coulomb interaction, energy gap and overlap between ground- and Moore-Read state form hills whose positions and extent in the (v{1},v{3}) plane coincide. We conclude that the physics of the Coulomb ground state at nu=5/2 is captured by the Moore-Read state. Such an adiabatic connection is not found at nu=1/2, unless the width of the interface wave function or Landau level mixing effects are large enough. Yet, a Moore-Read-phase at nu=1/2 appears unlikely in the thermodynamic limit.
We numerically study the spin polarization of the fractional quantum Hall state at filling factor $\ensuremath{\nu}=5/2$. By using both exact diagonalization and the density matrix renormalization group methods on the sphere, we are able to analyze more values of partial spin polarization (in addition to fully polarized and unpolarized) than any previous studies. We find that for the Coulomb interaction the exact finite-system ground state is fully polarized, for shifts corresponding to both the Moore-Read Pfaffian state and its particle-hole conjugate (anti-Pfaffian). This result is found to be robust against small variations in the interaction and change of shift. The low-energy excitation spectrum is consistent with spin-wave excitations of a fully magnetized ferromagnet.
Using exact diagonalization in the spherical geometry, we investigate systems of localized quasiholes at $\ensuremath{\nu}=5/2$ for interactions interpolating between the pure Coulomb and the three-body interaction for which the Moore-Read state is the exact ground state. We show that the charge
e/4 quasihole can be easily localized by means of a $\ensuremath{\delta}$-function pinning potential. Using a tuned smooth pinning potential, the quasihole radius can be limited to approximately three magnetic length units. For systems of two quasiholes, adiabatic continuity between the Moore-Read and the Coulomb limit holds for the ground state, while for four quasiholes, the lowest two energy states exhibit adiabatic continuity. This implies the existence of a Majorana fermion for pure Coulomb interaction. We also present preliminary results in the Coulomb limit for braiding in systems containing four quasiholes, with up to 14 electrons, diagonalizing in the full spin-polarized sector of the second Landau-level Hilbert space.
We compare ground-state energies by variational Monte Carlo of the spin unpolarized Halperin 331 and the spin polarized Moore-Read (MR) Pfaffian fractional quantum Hall states at half filling of the lowest Landau level (LLL) and the second Landau level (SLL) as a function of small deviations around the Coulomb point via the finite-thickness effect and direct alterations to the the first two Haldane pseudopotentials. In the comparison we find that in the LLL, either the 331 state or the MR Pfaffian may be lower in energy depending on the deviations. In the SLL, however, the MR Pfaffian is consistently lower in energy except for large deviations. These results suggest that even under moderate deviations in the interaction potential (through various physical processes such as finite thickness, Landau-level mixing, etc.), the MR Pfaffian description is more energetically favorable than the Halperin 331 state in the half-filled SLL (i.e., $\ensuremath{\nu}=5/2$), consistent with recent experimental investigations.
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<κc(w), where
κc(w) is a
w-dependent critical value
0.6≲κc(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.
The fractional quantum Hall effect (FQHE) observed at half filling of the second Landau level is believed to be caused by a pairing of composite fermions captured by the Moore-Read Pfaffian wave function. The generating Hamiltonian for the Moore-Read Pfaffian is a purely three-body model that breaks particle-hole symmetry and lacks other properties, such as dominate two-body repulsive interactions, expected from a physical model of the FQHE. We use exact diagonalization to study the low-energy states of a more physical two-body generator model derived from the three-body model. We find that the two-body model exhibits the essential features expected from the Moore-Read Pfaffian: pairing, non-Abelian anyon excitations, and a neutral fermion mode. The model also satisfies constraints expected for a physical model of the FQHE at half-filling because it is short range, spatially decaying, particle-hole symmetric, and supports a roton mode with a robust spectral gap in the thermodynamic limit. Hence, this two-body model offers a bridge between artificial three-body generator models for paired states and the physical Coulomb interaction and can be used to further explore properties of non-Abelian physics in the FQHE.