We describe and analyze in detail our recent theoretical proposal for the realization and manipulation of anyons in a weakly interacting system consisting of a two-dimensional electron gas in the integer quantum Hall regime adjacent to a type-II supe
rconducting film with an artificial array of pinning sites. The anyon is realized in response to a defect in the pinned vortex lattice and carries a charge pm e/2 and a statistical angle pi/4. We establish this result, both analytically and numerically, in three complementary approaches: (i) a continuum model of two-dimensional electrons in the vortex lattice of the superconducting film; (ii) a minimal tight-binding lattice model that captures the essential features of the system; and (iii) an effective theory of the superconducting vortex lattice superposed on the integer quantum Hall state. We propose a novel method to measure the fractional charge directly in a bulk transport experiment and an all-electric setup for an ``anyon shuttle implementing the braiding operations. We briefly discuss conditions for fabricating the system in the lab and its potential applications in quantum information processing with non-Abelian anyons.
A real-space formulation is given for the recently discussed exciton condensate in a symmetrically biased graphene bilayer. We show that in the continuum limit an oddly-quantized vortex in this condensate binds exactly one zero mode per valley index
of the bilayer. In the full lattice model the zero modes are split slightly due to intervalley mixing. We support these results by an exact numerical diagonalization of the lattice Hamiltonian. We also discuss the effect of the zero modes on the charge content of these vortices and deduce some of their interesting properties.
I study the midgap spectrum of the fermion-vortex system in two spatial dimensions. The existence of bound states, in addition to the zero modes found by Jackiw and Rossi, is established. For a singly quantized vortex, I present complete analytical s
olutions in terms of generalized Laguerre polynomials in the opposite limits of vanishing and large vortex core size. There is an infinite number of such bound states, with a spectrum that is, when squared, given by, respectively, the Coulomb potential and the isotropic harmonic oscillator. Possible experimental signatures of this spectrum in condensed-matter realizations of the system are pointed out.
We propose a two-dimensional time-reversal invariant system of essentially non-interacting electrons on a square lattice that exhibits configurations with fractional charges e/2. These are vortex-like topological defects in the dimerization order par
ameter describing spatial modulation in the electron hopping amplitudes. Charge fractionalization is established by a simple counting argument, analytical calculation within the effective low-energy theory, and by an exact numerical diagonalization of the lattice Hamiltonian. We comment on the exchange statistics of fractional charges and possible realizations of the system.
We show that fractional charges bound to topological defects in the recently proposed time-reversal-invariant models on honeycomb and square lattices obey fractional statistics. The effective low-energy description is given in terms of a `doubled lev
el-2 Chern-Simons field theory, which is parity and time-reversal invariant and implies two species of semions (particles with statistical angle pi/2) labeled by a new emergent quantum number that we identify as the fermion axial charge.
