First-order transition in the stacked-$J_1$-$J_2$ Ising model on a cubic lattice

We investigate critical properties of the stacked-$J_1$-$J_2$ Ising model on a cubic lattice. Using Monte Carlo simulations and renormalization group, we find a single phase transition of the first order for $J_2/J_1>1/2$. The renormgroup approach predicts that a transition can be of the second order from the universality class of the $O(2)$ model, but the Monte Carlo results show another set of critical exponents: exponents continuously vary form the values typical for a first-order transition in the finite-size scaling theory at $1/2<J_2/J_1<1$ to the Ising values in the limit $J_2/J_1\to\infty$. We also exclude the pseudo-first-order behavior observed in the $J_1$-$J_2$ Ising model on a square lattice for $0.67\lesssim J_2/J_1\lesssim0.9$.