  In this work, we show that the quantum compass model on an square lattice can
be mapped to a fermionic model with local density interaction. We introduce a
mean-field approximation where the most important fluctuations, those
perpendicular to the ordering direction, are taken into account exactly. It is
found that the quantum phase transition point at $J_x=J_z$ marks a first order
phase transition. We also show that the mean field result is robust against the
remaining fluctuation corrections up to the second order.
