Quantum phase transition in the quantum compass model

In this work, we show that the quantum compass model on a 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