Irrational Versus Rational Charge and Statistics in Two-Dimensional Quantum Systems

We show that quasiparticle excitations with irrational charge and irrational exchange statistics exist in tight-binding 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 rerationalizes to the value $1/2$ and the exchange statistics to that of “quartons” (half-semions).

Figure 1

The 2 sphere spanned by the masses $\mathit{n}\equiv (n_1,n_2,n_3)=m^{-1}({\Delta }_1,{\Delta }_2,{\mu }_s)$. A vortex of charge 1 in the dimerizations ${\Delta }_1$ and ${\Delta }_2$ corresponds to a parallel on the sphere whose height is fixed by the asymptotic value of the staggered chemical potential ${\mu }_s(\infty )$. The spherical area red (or yellow) enclosed by this parallel gives the fractional charge induced by the fermionic zero mode localized at the core of the vortex.

Figure 2

(a) The columnar dimerization pattern, as indicated by the coloring (shading) of the bonds, for the nearest-neighbor hopping on a square lattice in the background of the $\pi$-flux phase that opens up the gap $\Delta (\infty )$. A staggered chemical potential, as indicated by the distinction between sites of sublattice $A$ and $B$ opens up the gap ${\mu }_s(\infty )$. (b) The staggered dimerization pattern for the nearest-neighbor hopping on a square lattice that generates an axial gauge field. (c) A charge-1 vortex in the columnar dimerization (a). (d) A charge-$1/2$ vortex in the staggered dimerization (b) with core radius $c=0.3$. (e) The fermion density profile of (c) for a square lattice with open boundary conditions and $144\times 144$ sites. (f) The fermion number as a function of the scaling variable ${\mu }_s(\infty )/\Delta (\infty )$ in the presence of the single charge 1 vortex (c) or with the addition of the axial charge $1/2$ vortex (d) with core radius $c=0.01$. The staggered chemical potential ${\mu }_s$ takes the values $0.01t$ (black, points 1–3), $0.03t$ (red, points 4–7), $0.06t$ (green, points 8–10), and $0.1t$ (blue, points 11–15). The thick and thin lines are the prediction (5) without the axial vortex and $Q=1/2$, respectively.