Electron fractionalization is intimately related to topology. In one-dimensional systems, fractionally charged states exist at domain walls between degenerate vacua. In two-dimensional systems, fractionalization exists in quantum Hall fluids, where time-reversal symmetry is broken by a large external magnetic field. Recently, there has been a tremendous effort in the search for examples of fractionalization in two-dimensional systems with time-reversal symmetry. In this letter, we show that fractionally charged topological excitations exist on graphenelike structures, where quasiparticles are described by two flavors of Dirac fermions and time-reversal symmetry is respected. The topological zero-modes are mathematically similar to fractional vortices in p-wave superconductors. They correspond to a twist in the phase in the mass of the Dirac fermions, akin to cosmic strings in particle physics. In low dimensional systems, the excitation spectrum sometimes contains quasiparticles with fractionalized quantum numbers. A famous example of fractionalization was obtained in one-dimension (1D) by Jackiw and Rebbi [1] and by Su, Schrieffer and Hegger [2]. They showed the existence of charge e/2 states, with polyacetelene as a physical realization of such phenomena. In these systems, a charge density wave develops and the ground state is two-fold degenerate. The fractionalized states correspond to mid-gap or zero-mode solutions that are sustained at the domain wall (a soliton) interpolating between the two-degenerate vacua.The fractional quantum Hall effect provides an example of fractionalization in two-dimensions (2D). Not only do the Laughlin quasiparticles have fractional charge [3], but they also have fractional (anyon) statistics [4,5]. Time-reversal symmetry (TRS) is broken due to the strong magnetic field, leaving as an outstanding problem the search for systems where fractionalization is realized without the breaking of TRS. The motivation for such a quest stems from speculations that fractionalization may play a role in the mechanism for high-temperature superconductivity [6,7,8]. Progress has been made on finding model systems, such as dimer models, in which monomers defects act as fractionalized (and deconfined, in the case of the triangular lattice) excitations.In this letter, we present a mechanism to fractionalize the electron in graphenelike systems that leaves TRS unbroken. The excitation spectrum of honeycomb lattices, which have been known theoretically for a few decades to be described by Dirac fermions [9,10], is now the subject of many recent studies since single and few atomic-layer graphite samples have been realized experimentally [11]. Quantum number fractionalization is intimately related to topology and here we find that a twist or a vortex in an order parameter for a mass gap gives rise to a single mid-gap state at zero energy. Such twist in the mass of the Dirac fermions in graphenelike structure is the analogous in 2+1 space-time dimensions of a cosmic string in 3+1-dimensions [12].The zero-...

Proposals to measure non-Abelian anyons in a superconductor by quantum interference of vortices suffer from the predominantly classical dynamics of the normal core of an Abrikosov vortex. We show how to avoid this obstruction using coreless Josephson vortices, for which the quantum dynamics has been demonstrated experimentally. The interferometer is a flux qubit in a Josephson junction circuit, which can nondestructively read out a topological qubit stored in a pair of anyons -even though the Josephson vortices themselves are not anyons. The flux qubit does not couple to intra-vortex excitations, thereby removing the dominant restriction on the operating temperature of anyonic interferometry in superconductors.

We propose and analyze interedge tunneling in a quantum spin Hall corner junction as a means to probe the helical nature of the edge states. We show that electron-electron interactions in the one-dimensional helical edge states result in Luttinger parameters for spin and charge that are intertwined, and thus rather different from those for a quantum wire with spin rotation invariance. Consequently, we find that the four-terminal conductance in a corner junction has a distinctive form that could be used as evidence for the helical nature of the edge states.

Fermion-number fractionalization without breaking of time-reversal symmetry was recently demonstrated for a field theory in $(2+1)$-dimensional space and time that describes the couplings between massive Dirac fermions, a complex-valued Higgs field carrying an axial gauge charge of 2, and a U(1) axial gauge field. Charge fractionalization occurs whenever the Higgs field either supports vortices by itself, or when these vortices are accompanied by half-vortices in the axial gauge field. The fractional charge is computed by three different techniques. A formula for the fractional charge is given as a function of a parameter in the Dirac Hamiltonian that breaks the spectral energy-reflection symmetry. In the presence of a charge $\pm1$ vortex in the Higgs field only, the fractional charge varies continuously and thus can take irrational values. The simultaneous presence of a half-vortex in the axial gauge field and a charge $\pm1$ vortex in the Higgs field re-rationalizes the fractional charge to the value 1/2.Comment: 18 pages, 2 figure

We show that quasiparticle excitations with irrational charge and irrational exchange statistics exist in tightbiding systems described, in the continuum approximation, by the Dirac equation in (2+1)-dimensional space and time. These excitations can be deconfined at zero temperature, but when they are, the charge re-rationalizes to the value 1/2 and the exchange statistics to that of "quartons" (half-semions).Introduction -It was shown by Jackiw and Rebbi [1] and by Su, Schrieffer, and Heeger [2] that excitations with fermion number 1/2 (or charge 1/2) exist at domain walls in the dimerization pattern of electrons hopping along a chain, as is believed to occur in polyacetylene. For electrons hopping on the honeycomb lattice, the lattice relevant to graphene, a topological defect in a dimerization pattern that is realized by a vortex was shown to lead to a topological zero-mode and bind the fermion number 1/2 to the vortex [3]. Fermionnumber fractionalization in both polyacetylene and graphene can be understood in terms of the spectral properties of onedimensional (1D) and two-dimensional (2D) massive Dirac Hamiltonians, respectively, that describe the low energy limit of the electronic tight-binding Hamiltonians. These fractionally charged topological excitations are generically deconfined in 1D. Their deconfinement in 2D relies on a mechanism for the screening of the 2D Coulomb potential by thermal [3] or quantum fluctuations involving an axial gauge field [4].Applying different potentials to odd and even sites of the linear chain results in a continuously varying fractional fermion number [5,6,7]. At the level of the Dirac Hamiltonian, this perturbation is represented by a second (gapopening) mass term, that adds in quadrature to the mass due to the hopping dimerization of the chain. A complex order parameter is constructed from these two masses as its real and imaginary pieces, and the fractional fermion number is related to the phase twist of this order parameter as it sweeps through a domain wall, a result that has a natural interpretation within a bosonization scheme [5]. Although exchange statistics is ill-defined in 1D, this varying phase twist also implies a continuously varying exclusion statistics [8].Can the charge and the exchange statistics of fractionalized quasiparticles in 2D be continuously varied as well? Here we show that they can. The fermion numbers of quasiparticles bound to a vortex can thus be irrational. Remarkably, if an axial gauge field supporting a half vortex is added to (precisely) screen the interaction potential between quasiparticles, their fractional fermion number re-rationalizes to the value Q = 1/2 and their statistical angle (for when timereversal symmetry is broken) to the value θ/π = 1/4. These results are first derived at the level of an effective field theory in (2 + 1)-dimensional space and time. We then discuss the relevance of this analysis for planar tight-binding models.