Order in Quantum Compass and Orbital $e_g$ 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 ${\mathcal{T}}_c\mathrm{/}J=0.0606(4)$. For~the $e_g$ orbital model one finds: ($i$) a very accurate estimate of ${\cal T}_c/J=0.3566\pm 0.0001$and($ii$)~the~critical exponents in the Ising universality class. Remarkably large difference in frustration results in so distinct values of ${\mathcal{T}}_c$, while entanglement influences the quality of ${\mathcal{T}}_c$ estimation.