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We use high-temperature series expansions to obtain thermodynamic propertiesof the quantum compass model, and to investigate the phase transition on thesquare and simple cubic lattices. On the square lattice we obtain evidence fora phase transition, consistent with recent Monte Carlo results. On the simplecubic lattice the same procedure provides no sign of a transition, and weconjecture that there is no finite temperature transition in this case.

The Quantum Compass Model on the Square and Simple Cubic Lattices