Low-energy excitations and phase transitions in the frustrated two-dimensional $\mathrm{XY}$ model

We study the critical properties of the two-dimensional (2D) $\mathrm{XY}$ model in a transverse magnetic field with filling factors $f=1/3\mathrm{}$ and $2/5.$ To obtain a comparison with recent experiments, we investigate the effect of weak quenched bond disorder for $f=2/5.\mathrm{}$ A finite-size scaling analysis of extensive Monte Carlo simulations strongly suggests that the critical exponents of the phase transition for $f=1/3\mathrm{}$ and for $f=2/5\mathrm{}$ with disorder are those of the pure 2D Ising model. The relevant low-energy excitations are domain walls, and we show that their properties determine the nature of the phase transition.