anti-Pfaffian States Research Articles

Odd-even crossover in a non-Abelianν=5/2interferometer

We compute the backscattered current in a double point-contact geometry of a Quantum Hall system at filling fraction $\nu=5/2$ as a function of bias voltage in the weak backscattering regime. We assume that the system is in the universality class of either the Pfaffian or anti-Pfaffian state. When the number of charge $e/4$ quasiparticles in the interferometer is odd, there is no interference pattern. However, the coupling between a charge $e/4$ quasiparticle and the edge causes it to be absorbed by the edge at low energies. Consequently, an interference pattern appears at low bias voltages and temperatures, as if there were an even number of quasiparticles in the interferometer. We relate this problem to that of a semi-infinite Ising model with a boundary magnetic field. Using the methods of perturbed boundary conformal field theory, we give an exact expression for this crossover of the interferometer as a function of bias voltage. Finally, we comment on the possible relevance of our results to recent interference experiments.

Long tunneling contact as a probe of fractional quantum Hall neutral edge modes

We study the tunneling current between edge states of quantum Hall liquids across a single long-contact region and predict a resonance at a bias voltage set by the scale of the edge velocity. For typical devices and edge velocities associated with charged modes, this resonance occurs outside the physically accessible bias domain. However, for edge states that are expected to support neutral modes, such as the $\ensuremath{\nu}=\frac{2}{3}$ and $\ensuremath{\nu}=\frac{5}{2}$ Pfaffian and anti-Pfaffian states, the neutral velocity can be orders of magnitude smaller than the charged mode and if so the resonance would be accessible. Therefore, such long tunneling contacts can resolve the presence of neutral edge modes in certain quantum Hall liquids.

Multichannel Kondo Models in Non-Abelian Quantum Hall Droplets

We study the coupling between a quantum dot and the edge of a non-Abelian fractional quantum Hall state which is spatially separated from it by an integer quantum Hall state. Near a resonance, the physics at energy scales below the level spacing of the edge states of the dot is governed by a k-channel Kondo model when the quantum Hall state is a Read-Rezayi state at filling fraction nu=2+k/(k+2) or its particle-hole conjugate at nu=2+2/(k+2). The k-channel Kondo model is channel isotropic even without fine-tuning in the former state; in the latter, it is generically channel anisotropic. In the special case of k=2, our results provide a new venue, realized in a mesoscopic context, to distinguish between the Pfaffian and anti-Pfaffian states at filling fraction nu=5/2.

Charge-statistics separation and probing non-Abelian states

Several states were proposed as candidates for the $\ensuremath{\nu}=5/2$ quantum Hall plateau. We suggest an experiment which can determine the physical state. The proposal involves transport measurements in the geometry with three quantum Hall edges connected by two quantum point contacts. In contrast to interference experiments, this approach can distinguish the Pfaffian and anti-Pfaffian states as well as different states with identical Pfaffian or anti-Pfaffian statistics. The transport is not sensitive to the fluctuations of the number of the quasiparticles trapped in the system.

Spontaneous Particle-Hole Symmetry Breaking in theν=5/2Fractional Quantum Hall Effect

The essence of the nu=5/2 fractional quantum Hall effect is believed to be captured by the Moore-Read Pfaffian (or anti-Pfaffian) description. However, a mystery regarding the formation of the Pfaffian state is the role of the three-body interaction Hamiltonian H3 that produces it as an exact ground state and the concomitant particle-hole symmetry breaking. We show that a two-body interaction Hamiltonian H2 constructed via particle-hole symmetrization of H3 produces a ground state nearly exactly approximating the Pfaffian and anti-Pfaffian states, respectively, in the spherical geometry. Importantly, the ground state energy of H2 exhibits a "Mexican-hat" structure as a function of particle number in the vicinity of half filling for a given flux indicating spontaneous particle-hole symmetry breaking. This signature is absent for the second Landau level Coulomb interaction at 5/2.

Fractional quantum Hall effect atν=5∕2: Ground states, non-Abelian quasiholes, and edge modes in a microscopic model

We present a comprehensive numerical study of a microscopic model of the fractional quantum Hall system at filling fraction $\ensuremath{\nu}=5∕2$, based on the disk geometry. Our model includes Coulomb interaction and a semirealistic confining potential. We also mix in a three-body interaction in some cases to help elucidate the physics. We obtain a phase diagram, discuss the conditions under which the ground state can be described by the Moore-Read state, and study its competition with neighboring stripe phases. We also study quasihole excitations and edge excitations in the Moore-Read-like state. From the evolution of the edge spectrum, we obtain the velocities of the charge and neutral edge modes, which turn out to be very different. This separation of velocities is a source of decoherence for a non-Abelian quasihole and/or quasiparticle (with charge $\ifmmode\pm\else\textpm\fi{}e∕4$) when propagating at the edge; using numbers obtained from a specific set of parameters, we estimate the decoherence length to be around $4\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$. This sets an upper bound for the separation of the two point contacts in a double point-contact interferometer, designed to detect the non-Abelian nature of such quasiparticles. We also find a state that is a potential candidate for the recently proposed anti-Pfaffian state. We find the speculated anti-Pfaffian state is favored in weak confinement (smooth edge), while the Moore-Read Pfaffian state is favored in strong confinement (sharp edge).

Gauge symmetry in Kitaev-type spin models and index theorems on odd manifolds

We construct an exactly soluble spin- 1 2 model on a honeycomb lattice, which is a generalization of Kitaev model. The topological phases of the system are analyzed by study of the ground state sector of this model, the vortex-free states. Basically, there are two phases, A phase and B phase. The behaviors of both A and B phases may be studied by mapping the ground state sector into a general p-wave paired states of spinless fermions with tunable pairing parameters on a square lattice. In this p-wave paired state theory, the A phase is shown to be the strong paired phase, an insulating phase. The B phase may be either gapped or gapless determined by the generalized inversion symmetry is broken or not. The gapped B is the weak pairing phase described by either the Moore–Read Pfaffian state of the spinless fermions or anti-Pfaffian state of holes depending on the sign of the next nearest neighbor hopping amplitude. A phase transition between Pfaffian and anti-Pfaffian states are found in the gapped B phase. Furthermore, we show that there is a hidden SU(2) gauge symmetry in our model. In the gapped B phase, the ground state has a non-trivial topological number, the spectral first Chern number or the chiral central charge, which reflects the chiral anomaly of the edge state. We proved that the topological number is identified to the reduced eta-invariant and this anomaly may be cancelled by a bulk Wess–Zumino term of SO(3) group through an index theorem in 2 + 1 dimensions.