Order in Quantum Compass and Orbital eg Models

We investigate thermodynamic phase transitions in the compass model and in e(g) orbital model on an infinite square lattice by variational tensor network renormalization (VTNR) in imaginary time. The onset of nematic order in the quantum compass model is estimated at T-c/J = 0.0606(4). For the e(g) orbital model one finds: (i) a very accurate estimate of T-c/J = 0.3566 +/- 0.0001 and (ii) the critical exponents in the Ising universality class. Remarkably large difference in frustration results in so distinct values of T-c, while entanglement influences the quality of T-c estimation.