Fractional vortices in the $\mathrm{XY}$ model with $\pi$ bonds

We define a new set of excitations in the $\mathrm{XY}$ model which we call “fractional vortices.” In the frustrated $\mathrm{XY}$ model containing $\pi$ bonds, we make the ansatz that the ground state configurations can be characterized by pairs of oppositely charged fractional vortices. For a chain of $\pi$ bonds, the ground state energy and the phase configurations calculated on the basis of this ansatz agree well with the results from direct numerical simulations. Finally, we discuss the possible connection of these results to some recent experiments by Kirtley et al. [Phys. Rev. B 51, R12 057 (1995)] on high-$T_c$ superconductors where fractional flux trapping was observed along certain grain boundaries.