The Pfaffian quantum Hall state made simple--multiple vacua and domain walls on a thin torus

We analyze the Moore-Read Pfaffian state on a thin torus. The known six-fold degeneracy is realized by two inequivalent crystalline states with a four- and two-fold degeneracy respectively. The fundamental quasihole and quasiparticle excitations are domain walls between these vacua, and simple counting arguments give a Hilbert space of dimension $2^{n-1}$ for $2n-k$ holes and $k$ particles at fixed positions and assign each a charge $\pm e/4$. This generalizes the known properties of the hole excitations in the Pfaffian state as deduced using conformal field theory techniques. Numerical calculations using a model hamiltonian and a small number of particles supports the presence of a stable phase with degenerate vacua and quarter charged domain walls also away from the thin torus limit. A spin chain hamiltonian encodes the degenerate vacua and the various domain walls.Comment: 4 pages, 1 figure. Published, minor change

Pedestrian index theorem a la Aharonov-Casher for bulk threshold modes in corrugated multilayer graphene

Zero-modes, their topological degeneracy and relation to index theorems have attracted attention in the study of single- and bilayer graphene. For negligible scalar potentials, index theorems explain why the degeneracy of the zero-energy Landau level of a Dirac hamiltonian is not lifted by gauge field disorder, for example due to ripples, whereas other Landau levels become broadened by the inhomogenous effective magnetic field. That also the bilayer hamiltonian supports such protected bulk zero-modes was proved formally by Katsnelson and Prokhorova to hold on a compact manifold by using the Atiyah-Singer index theorem. Here we complement and generalize this result in a pedestrian way by pointing out that the simple argument by Aharonov and Casher for degenerate zero-modes of a Dirac hamiltonian in the infinite plane extends naturally to the multilayer case. The degeneracy remains, though at nonzero energy, also in the presence of a gap. These threshold modes make the spectrum asymmetric. The rest of the spectrum, however, remains symmetric even in arbitrary gauge fields, a fact related to supersymmetry. Possible benefits of this connection are discussed.Comment: 6 pages, 2 figures. The second version states now also in words that the conjugation symmetry that in the massive case gets replaced by supersymmetry is the chiral symmetry. Changes in figure

Quantum corrections in the Boltzmann conductivity of graphene and their sensitivity to the choice of formalism

Semiclassical spin-coherent kinetic equations can be derived from quantum theory with many different approaches (Liouville equation based approaches, nonequilibrium Green's functions techniques, etc.). The collision integrals turn out to be formally different, but coincide in textbook examples as well as for systems where the spin-orbit coupling is only a small part of the kinetic energy like in related studies on the spin Hall effect. In Dirac cone physics (graphene, surface states of topological insulators like Bi_{1-x}Sb_x, Bi_2Te_3 etc.), where this coupling constitutes the entire kinetic energy, the difference manifests itself in the precise value of the electron-hole coherence originated quantum correction to the Drude conductivity $\sim e^2/h * \ell k_F$. The leading correction is derived analytically for single and multilayer graphene with general scalar impurities. The often neglected principal value terms in the collision integral are important. Neglecting them yields a leading correction of order $(\ell k_F)^{-1}$, whereas including them can give a correction of order $(\ell k_F)^0$. The latter opens up a counterintuitive scenario with finite electron-hole coherent effects at Fermi energies arbitrarily far above the neutrality point regime, for example in the form of a shift $\sim e^2/h$ that only depends on the dielectric constant. This residual conductivity, possibly related to the one observed in recent experiments, depends crucially on the approach and could offer a setting for experimentally singling out one of the candidates. Concerning the different formalisms we notice that the discrepancy between a density matrix approach and a Green's function approach is removed if the Generalized Kadanoff-Baym Ansatz in the latter is replaced by an anti-ordered version.Comment: 31 pages, 1 figure. An important sign error has been rectified in the principal value terms in equation (52) in the vN & NSO expression. It has no implications for the results on the leading quantum correction studied in this paper. However, for the higher quantum corrections studied in arXiv:1304.3929 (see comment in the latter) the implications are crucia

Interacting non-Abelian anyons as Majorana fermions in the honeycomb lattice model

We study the collective states of interacting non-Abelian anyons that emerge in Kitaev's honeycomb lattice model. Vortex-vortex interactions are shown to lead to the lifting of the topological degeneracy and the energy is discovered to exhibit oscillations that are consistent with Majorana fermions being localized at vortex cores. We show how to construct states corresponding to the fusion channel degrees of freedom and obtain the energy gaps characterizing the stability of the topological low energy spectrum. To study the collective behavior of many vortices, we introduce an effective lattice model of Majorana fermions. We find necessary conditions for it to approximate the spectrum of the honeycomb lattice model and show that bi-partite interactions are responsible for the degeneracy lifting also in many vortex systems.Comment: 22 pages, 12 figures, published versio

Non-Abelian gauge fields in the gradient expansion: generalized Boltzmann and Eilenberger equations

We present a microscopic derivation of the generalized Boltzmann and Eilenberger equations in the presence of non-Abelian gauges, for the case of a non-relativistic disordered Fermi gas. A unified and symmetric treatment of the charge $[U(1)]$ and spin $[SU(2)]$ degrees of freedom is achieved. Within this framework, just as the $U(1)$ Lorentz force generates the Hall effect, so does its $SU(2)$ counterpart give rise to the spin Hall effect. Considering elastic and spin-independent disorder we obtain diffusion equations for charge and spin densities and show how the interplay between an in-plane magnetic field and a time dependent Rashba term generates in-plane charge currents.Comment: 11 pages, 1 figure; some corrections and updated/extended reference

Bose-Einstein condensates in strong electric fields -- effective gauge potentials and rotating states

Magnetically-trapped atoms in Bose-Einstein condensates are spin polarized. Since the magnetic field is inhomogeneous, the atoms aquire Berry phases of the Aharonov-Bohm type during adiabatic motion. In the presence of an eletric field there is an additional Aharonov-Casher effect. Taking into account the limitations on the strength of the electric fields due to the polarizability of the atoms, we investigate the extent to which these effects can be used to induce rotation in a Bose-Einstein condensate.Comment: 5 pages, 2 ps figures, RevTe

Charge and Current in the Quantum Hall Matrix Model

We extend the quantum Hall matrix model to include couplings to external electric and magnetic fields. The associated current suffers from matrix ordering ambiguities even at the classical level. We calculate the linear response at low momenta -- this is unambigously defined. In particular, we obtain the correct fractional quantum Hall conductivity, and the expected density modulations in response to a weak and slowly varying magnetic field. These results show that the classical quantum Hall matrix models describe important aspects of the dynamics of electrons in the lowest Landau level. In the quantum theory the ordering ambiguities are more severe; we discuss possible strategies, but we have not been able to construct a good density operator, satisfying the pertinent lowest Landau level commutator algebra.Comment: 12 pages, no figures; a logical error below the proposed density operator (46) in version 1 is corrected, and the claim that this density operator satisfy the magnetic algebra (2) is withdrawn. Some formulations have been changed and a few misprints correcte

Solitons and Quasielectrons in the Quantum Hall Matrix Model

We show how to incorporate fractionally charged quasielectrons in the finite quantum Hall matrix model.The quasielectrons emerge as combinations of BPS solitons and quasiholes in a finite matrix version of the noncommutative $\phi^4$ theory coupled to a noncommutative Chern-Simons gauge field. We also discuss how to properly define the charge density in the classical matrix model, and calculate density profiles for droplets, quasiholes and quasielectrons.Comment: 15 pages, 9 figure

Kaleidoscope of topological phases with multiple Majorana species

Exactly solvable lattice models for spins and non-interacting fermions provide fascinating examples of topological phases, some of them exhibiting the localized Majorana fermions that feature in proposals for topological quantum computing. The Chern invariant $\nu$ is one important characterization of such phases. Here we look at the square-octagon variant of Kitaev's honeycomb model. It maps to spinful paired fermions and enjoys a rich phase diagram featuring distinct abelian and nonabelian phases with $\nu= 0,\pm1,\pm2,\pm3$ and $\pm4$. The $\nu=\pm1$ and $\nu=\pm3$ phases all support localized Majorana modes and are examples of Ising and $SU(2)_2$ anyon theories respectively.Comment: 6 pages, 5 figures. The second version has a new title, reflecting a change of focus of the presentation in this version. The third version contains minor changes and is essentially the one published in New Journal of Physic

Matrix Model Description of Laughlin Hall States

We analyze Susskind's proposal of applying the non-commutative Chern-Simons theory to the quantum Hall effect. We study the corresponding regularized matrix Chern-Simons theory introduced by Polychronakos. We use holomorphic quantization and perform a change of matrix variables that solves the Gauss law constraint. The remaining physical degrees of freedom are the complex eigenvalues that can be interpreted as the coordinates of electrons in the lowest Landau level with Laughlin's wave function. At the same time, a statistical interaction is generated among the electrons that is necessary to stabilize the ground state. The stability conditions can be expressed as the highest-weight conditions for the representations of the W-infinity algebra in the matrix theory. This symmetry provides a coordinate-independent characterization of the incompressible quantum Hall states.Comment: 31 pages, large additions on the path integral and overlaps, and on the W-infinity symmetr