Electron fractionalization for two-dimensional Dirac fermions

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.