Overcoming the sign problem at finite temperature: Quantum tensor network for the orbital eg model on an infinite square lattice

The variational tensor network renormalization approach to two-dimensional (2D) quantum systems at finite temperature is applied for the first time to a model suffering the notorious quantum Monte Carlo sign problem --- the orbital $e_g$ model with spatially highly anisotropic orbital interactions. Coarse-graining of the tensor network along the inverse temperature $\beta$ yields a numerically tractable 2D tensor network representing the Gibbs state. Its bond dimension $D$ --- limiting the amount of entanglement --- is a natural refinement parameter. Increasing $D$ we obtain a converged order parameter and its linear susceptibility close to the critical point. They confirm the existence of finite order parameter below the critical temperature $T_c$, provide a numerically exact estimate of~$T_c$, and give the critical exponents within $1\%$ of the 2D Ising universality class.