A comprehensive study of the two-dimensional (2D) compass model on the square
lattice is performed for classical and quantum spin degrees of freedom using
Monte Carlo and quantum Monte Carlo methods. We employ state-of-the-art
implementations using Metropolis, stochastic series expansion and parallel
tempering techniques to obtain the critical ordering temperatures and critical
exponents. In a pre-investigation we reconsider the classical compass model
where we study and contrast the finite-size scaling behavior of ordinary
periodic boundary conditions against annealed boundary conditions. It is shown
that periodic boundary conditions suffer from extreme finite-size effects which
might be caused by closed loop excitations on the torus. These excitations also
appear to have severe effects on the Binder parameter. On this footing we
report on a systematic Monte Carlo study of the quantum compass model. Our
numerical results are at odds with recent literature on the subject which we
trace back to neglecting the strong finite-size effects on periodic lattices.
The critical temperatures are obtained as Tc​=0.1464(2)J and
Tc​=0.055(1)J for the classical and quantum version, respectively,
and our data support a transition in the 2D Ising universality class for both
cases.Comment: 8 pages, 7 figures, differs slightly from published versio