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A comprehensive study of the two-dimensional (2D) compass model on the squarelattice is performed for classical and quantum spin degrees of freedom usingMonte Carlo and quantum Monte Carlo methods. We employ state-of-the-artimplementations using Metropolis, stochastic series expansion and paralleltempering techniques to obtain the critical ordering temperatures and criticalexponents. In a pre-investigation we reconsider the classical compass modelwhere we study and contrast the finite-size scaling behavior of ordinaryperiodic boundary conditions against annealed boundary conditions. It is shownthat periodic boundary conditions suffer from extreme finite-size effects whichmight be caused by closed loop excitations on the torus. These excitations alsoappear to have severe effects on the Binder parameter. On this footing wereport on a systematic Monte Carlo study of the quantum compass model. Ournumerical results are at odds with recent literature on the subject which wetrace back to neglecting the strong finite-size effects on periodic lattices.The critical temperatures are obtained as $T_\mathrm{c}=0.1464(2)J$ and$T_\mathrm{c}=0.055(1)J$ for the classical and quantum version, respectively,and our data support a transition in the 2D Ising universality class for bothcases.

Monte Carlo simulations of the directional-ordering transition in the two-dimensional classical and quantum compass model