Dimensional crossover in the quasi-two-dimensional Ising-O(3) model

We present results of our Monte Carlo simulation of the Ising-O(3) model on the two-dimensional (2D) and quasi-2D lattices. This model is an effective classical model for the stacked square-lattice $J_1$-$J_2$ Heisenberg model where the nearest neighbor ($J_1$) and next-nearest neighbor ($J_2$) couplings are frustrated and we assume that $J_2$ is dominant. We find an Ising ordered phase where the O(3) spins remain disordered in a moderate quasi-2D region. There is a single first order transition for a sufficiently large 3D coupling in agreement with a renormalization group treatment. The subtle region where the single transition splits into two transitions is also discussed and compared against recent measurements of two very close transitions in BaFe$_2$As$_2$. Our results can provide a qualitative explanation on the experiments on ferropnictides, namely observed sequence and orders of the structural and magnetic transitions, in terms of the ratio between the inter-layer and intra-layer coupling.