Superfluid-insulator transition of the Josephson junction array model with commensurate frustration

We have studied the rationally frustrated Josephson-junction array model in the square lattice through Monte Carlo simulations of $(2+1)$D XY-model. For frustration $f=1/4$, the model at zero temperature shows a continuous superfluid-insulator transition. From the measurement of the correlation function and the superfluid stiffness, we obtain the dynamical critical exponent $z=1.0$ and the correlation length critical exponent $\nu=0.4 \pm 0.05$. While the dynamical critical exponent is the same as that for cases $f=0$, 1/2, and 1/3, the correlation length critical exponent is surprisingly quite different. When $f=1/5$, we have the nature of a first-order transition.