The $\ensuremath{\nu}=5/2$ fractional quantum Hall effect is a system of intense experimental and theoretical interest as its ground state may host non-Abelian excitations, but the exact nature of the ground state is still undetermined. We present the results of an exact diagonalization study of an electron system in the disk configuration, including the effects of Landau level (LL) mixing and the finite thickness of the quantum well confining the electrons. The degeneracy between the two leading candidates for the ground state, the Pfaffian and anti-Pfaffian, is broken by interactions with a neutralizing background, in addition to the inclusion of two- and three-body interactions via LL mixing. As a result of the neutralizing background in the disk configuration, there is a phase transition from the anti-Pfaffian to the Pfaffian as LL mixing is turned on, in stark contrast to what is observed in spherical geometry. This behavior is in agreement with existing experiments, showing the appearance of the Pfaffian state at strong LL mixing before the system enters a compressible phase. The inclusion of LL mixing leads to an increased charge
e/4 quasihole size. LL mixing interactions are also shown to overcome the effects of edge reconstruction. Due to finite thickness effects, these properties are enhanced dramatically. We also find that only the Pfaffian and anti-Pfaffian states continue to possess energy gaps at finite width, while gaps for compressible stripe states close, which is in agreement with available experimental data.
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 [special characters omitted] fractional quantum Hall effect (FQHE) is a unique and interesting experimental and theoretical state. A great deal of experimental, theoretical and numerical work suggests that this state may support quasihole excitations with non-Abelian statistics, where the order of particle exchange influences the final state of the system. Thus, the [special characters omitted] FQHE offers a system in which the properties of the particles may be explored experimentally and theoretically. Additionally, by controlling the exchange of such particles, it is possible to create a topologically-protected quantum computer. In order to make this possible, however, we must first understand the nature of the ground state. The two leading candidates, the Moore-Read Pfaffian and the anti-Pfaffian, both support non-Abelian excitations, but there has not been a clear answer for which state is realized in experiment. In the present work, we present results of exact diagaonlization calculations which strive to answer this question using a disk geometry. What we find is that the ground state of the system is dependent upon device specific quantities and thus we may be able to engineer samples which will have specific ground state properties.
We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau-level filling factor 1/2 bilayer problem and make predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong-tunneling regime. In the lowest Landau level, we find that although, in principle, increasing (decreasing) tunneling strength (layer separation) could lead to a transition from the Abelian two-component Halperin 331 to non-Abelian one-component Moore-Read Pfaffian state, the FQHE excitation gap is relatively small in the lowest Landau level Pfaffian regime, and we establish that all so far observed FQHE states in half-filled lowest Landau level bilayers are most likely described by the Abelian Halperin 331 state. In the second Landau level we make the prediction that bilayer structures would manifest two distinct branches of incompressible FQHE corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Moore-Read Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the possible quantum phase transition between them will be compelling evidence supporting the existence of the non-Abelian Moore-Read Pfaffian state in the second Landau level. We discuss our results in the context of existing experiments and theoretical works.
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.