Fractionalization in a square-lattice model with time-reversal symmetry

We propose a two-dimensional time-reversal invariant system of essentially noninteracting electrons on a square lattice that exhibits configurations with fractional charges $\pm e∕2$. These are vortexlike topological defects in the dimerization order parameter 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.

Figure 1

(Color online) The model: (a) square lattice with $\frac{1}{2}{\Phi }_0$ magnetic flux per plaquette and dimerized hopping amplitudes. The ± on the left for each row show the choice of gauge for the Peierls phase factors. The solid (dashed) bonds indicate an increased (decreased) hopping amplitude in the $\widehat{x}$ (blue/gray) and $\widehat{y}$ (green/light gray) directions as explained in the text. The four sites of the unit cell are marked. (b) The ${\mathrm{Z}}_4$ vortex. The dashed lines indicate the domain walls sharing the center of the vortex. The phase of the local dimer order parameter $f_i$ (see text) around (c) the U(1) and (d) the ${\mathrm{Z}}_4$ vortices. The × shows the center of each vortex where $f=0$.

Figure 2

(Color online) [(a) and (b)] The charge density at half-filling with zero mode filled in the U(1) vortex with $m=0.15$. [(c) and (d)] The same plots for the $Z_4$ vortex and $m=0.30$. The lattice size is $51\times 51$ and the gridline spacing is 3. The top-view density plots on the left are included to show the symmetry of the solution for each vortex and the support on a single sublattice.

Figure 3

[(a) and (c)] The charge density at half-filling with both midgap states filled and [(b) and (d)] the low-energy (including midgap) eigenvalues of a system [(a) and (b)] with two U(1) vortices and [(c) and (d)] with a vortex and an antivortex. The gap $m=0.2$, the lattice size is $52\times 51$, and the gridline spacing is 3.