arXiv subjects
Publications and source records attributed to Michael Stone.
We show that a version of the covariant gauge anomaly for a 3+1 dimensional chiral fermion interacting with a non-Abelian gauge field can be obtained from the classical Hamiltonian flow of its probability distribution in phase space. The only quantum input needed is the Berry phase that arises from the direction of the spin being slaved to the particle's momentum.
We numerically solve the coupled Landau-Ginzburg-Maxwell equations for a model of a spin triplet px + ipy superconductor in which whole or half-quanta of flux thread through a hole. We recover the pattern of stable and unstable regions for the half-flux quanta observed in a recent experiment.
We use the identification of the edge mode of the filling fraction $ν=1$ quantum Hall phase with a 1+1 dimensional chiral Dirac fermion to construct an analogue model for a chiral fermion in a space-time geometry possessing an event horizon. By solving the model in the lowest Landau level, we show that the event horizon emits particles and holes with a thermal spectrum. Each emitted quasiparticle is correlated with an opposite-energy partner on the other side of the event horizon. Once we trace out these "unobservable" partners, we are left with a thermal density matrix.
It has been suggested that a temperature gradient will induce a Leduc-Righi, or thermal Hall, current in the Majorana quasiparticles localized on the surface of class DIII topological insulators, and that the magnitude of this current can be related {\it via} an Einstein argument to a Hall-like energy flux induced by gravity. We critically examine this idea, and argue that the gravitational Hall effect is more complicated than its familiar analogue. A conventional Hall current is generated by a {\it uniform} electric field, but computing the flux from the gravitational Chern-Simons functional shows that gravitational field {\it gradients} - i.e. tidal forces - are needed to induce a energy-momentum flow. We relate the surface energy-momentum flux to a domain-wall gravitational anomaly {\it via} the Callan-Harvey inflow mechanism. We stress that the gauge invariance of the combined bulk-plus-boundary theory ensures that the current in the domain wall always experiences a "covariant" rather than "consistent" anomaly. We use this observation to confirm that the tidally induced energy-momentum current exactly accounts for the covariant gravitational anomaly in $(1+1)$ dimensional domain-wall fermions. The same anomaly arises whether we write the Chern-Simons functional in terms of the Christofflel symbol or in terms of the the spin connection.
We consider the adiabatic evolution of Kramers degenerate pairs of spin states in a half-integer spin molecular magnet as the molecule is slowly rotated. To reveal the full details the of the quantum evolution, we use Majorana's parametrisation of a general state in the $2j+1$ dimensional Hilbert space in terms of $2j$ Majorana points. We show that the intricate motion of the Majorana points may be described by a classical hamiltonian which is of the same form, but of quite different origin, as that which appears in the spin-coherent-state path integral. As an illustration we consider molecular magnets of the $j=9/2$ Mn4 family and compute the frequency with which the magnetization varies. This frequency is generally different from the frequency of the rotation.
We consider a simple model for an SNS Josephson junction in which the "normal metal" is a section of a filling-factor $ν=2$ integer quantum-Hall edge. We provide analytic expressions for the current/phase relations to all orders in the coupling between the superconductor and the quantum Hall edge modes, and for all temperatures. Our conclusions are consistent with the earlier perturbative study by Ma and Zyuzin [Europhysics Letters {\bf 21} 941-945 (1993)]: The Josephson current is independent of the distance between the superconducting leads, and the upper bound on the maximum Josephson current is inversely proportional to the perimeter of the Hall device.
We provide an account of some of the mathematics of Bott periodicity and the Atiyah, Bott, Shapiro construction. We apply these ideas to understanding the twisted bundles of electron bands that underly the properties of topological insulators, spin Hall systems, and other topologically interesting materials.
We consider the low-energy electronic properties of graphene cones in the presence of a global Fries-Kekulé Peierls distortion. Such cones occur in fullerenes as the geometric response to the disclination associated with pentagon rings. It is well known that the long-range effect of the disclination deficit-angle can be modelled in the continuum Dirac-equation approximation by a spin connection and a non-abelian gauge field. We show here that to understand the bound states localized in the vicinity of a pair of pentagons one must, in addition to the long-range topological effects of the curvature and gauge flux, consider the effect the short-range lattice disruption near the defect. In particular, the radial Dirac equation for the lowest angular-momentum channel sees the defect as a singular endpoint at the origin, and the resulting operator possesses deficiency indices $(2,2)$. The radial equation therefore admits a four-parameter set of self-adjoint boundary conditions. The values of the four parameters depend on how the pentagons are distributed and determine whether or not there are zero modes or other bound states.
We explore the classical version of the mapping, due to Abanov and Wiegmann, of Calogero-Sutherland hydrodynamics onto the Benjamin-Ono equation ``on the double.'' We illustrate the mapping by constructing the soliton solutions to the hydrodynamic equations, and show how certain subtleties arise from the need to include corrections to the naive replacement of singular sums by principal-part integrals.
We study various geometrical aspects of the propagation of particles obeying fractional statistics in the physical setting of the quantum Hall system. We find a discrete set of zeros for the two-particle kernel in the lowest Landau level; these arise from a combination of a two-particle Aharonov-Bohm effect and the exchange phase related to fractional statistics. The kernel also shows short distance exclusion statistics, for instance, in a power law behavior as a function of initial and final positions of the particles. We employ the one-particle kernel to compute impurity-mediated tunneling amplitudes between different edges of a finite-sized quantum Hall system and and find that they vanishes for certain strengths and locations of the impurity scattering potentials. We show that even in the absence of scattering, the correlation functions between different edges exhibits unusual features for a narrow enough Hall bar.
At half filling, the electronic structure of graphene can be modelled by a pair of free two-dimensional Dirac fermions. We explicitly demonstrate that in the presence of a geometrically induced gauge field, an everywhere-real Kekule modulation of the hopping matrix elements can correspond to a non-real Higgs field with non-trivial vorticity. This provides a natural setting for fractionally charged vortices with localized zero modes. For fullerene-like molecules we employ the index theorem to demonstrate the existence of six low-lying states that do not depend strongly on the Kekule-induced mass gap.
We consider a graphene sheet folded in an arbitrary geometry, compact or with nanotube-like open boundaries. In the continuous limit, the Hamiltonian takes the form of the Dirac operator, which provides a good description of the low energy spectrum of the lattice system. We derive an index theorem that relates the zero energy modes of the graphene sheet with the topology of the lattice. The result coincides with analytical and numerical studies for the known cases of fullerene molecules and carbon nanotubes and it extend to more complicated molecules. Potential applications to topological quantum computation are discussed.
We construct the wave functions for the Moore-Read $ν= 5/2$ quantum Hall state on a torus in the presence of two quasiholes. These explicit wave functions allow us to compute the monodromy matrix that describes the effect of quasihole motion on the space of degenerate ground states. The result agrees with the recent discussion by Oshikawa {\it et al.} Our calculation provides a conformal field theory explanation of why certain transitions between ground states are forbidden. It is because taking a quasihole around a generator of the torus can change the fusion channel of the two quasiholes, and this requires a change of parity of the electron number in some of the ground states.
We show that the form of the chiral condition found by Abanov et al., in the quantum hydrodyamics of the Sutherland model arises because there are two distinct inner products with respect to which the chiral Hamitonian hermitian, but only one with respect to which the full, non-chiral, Hamiltonian is hermitian.
We derive single-particle and two-particle correlators of anyons in the presence of a magnetic field in the lowest Landau level. We show that the two-particle correlator exhibits signatures of fractional statistics which can distinguish anyons from their fermionic and bosonic counterparts. These signatures include the zeroes of the two-particle correlator and its exclusion behavior. We find that the single-particle correlator in finite geometries carries valuable information relevant to experiments in which quasiparticles on the edge of a quantum Hall system tunnel through its bulk.
We present a simple two-dimensional model of a $p_x+ip_y$ superfluid in which the mass flow that gives rise to the intrinsic angular momentum is easily calculated by numerical diagonalization of the Bogoliubov-de Gennes operator. We find that, at zero temperature and for constant director $\bf l$, the mass flow closely follows the Ishikawa-Mermin-Muzikar formula ${\bf j}_{\rm mass}= \frac 12 {\rm curl} (ρ\hbar {\bf l}/2)$.
In this article we consider a graphene sheet that is folded in various compact geometries with arbitrary topology described by a certain genus, $g$. While the Hamiltonian of these systems is defined on a lattice one can take the continuous limit. The obtained Dirac-like Hamiltonian describes well the low energy modes of the initial system. Starting from first principles we derive an index theorem that corresponds to this Hamiltonian. This theorem relates the zero energy modes of the graphene sheet with the topology of the compact lattice. For $g=0$ and $g=1$ these results coincide with the analytical and numerical studies performed for fullerene molecules and carbon nanotubes while for higher values of $g$ they give predictions for more complicated molecules.
To detect non-abelian statistics in the $ν= 12/5$ quantum Hall state through interferometry, we apply an analysis similar to the ones proposed for the non-abelian $ν= 5/2$ quantum Hall state. The result is that the amplitude of the Aharonov-Bohm oscillation of this interference is dependent on the internal states of quasiholes, but, in contrast to the $ν= 5/2$ quantum Hall state, independent of the number of quasiholes. However, if the quasiholes are in a superposition state, it is necessary for the interferometer to have certain additional features to obtain the coefficients.