Variational tensor network renormalization in imaginary time: Two-dimensional quantum compass model at finite temperature

Progress in describing thermodynamic phase transitions in quantum systems is obtained by noticing that the Gibbs operator e^-βH for a two-dimensional (2D) lattice system with a Hamiltonian H can be represented by a three-dimensional tensor network, the third dimension being the imaginary time (inverse temperature) β. Coarse-graining the network along β results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension D. The coarse-graining is performed by a tree tensor network of isometries. The isometries are optimized variationally --- taking into account full tensor environment --- to maximize the accuracy of the PEPO. The algorithm is applied to the isotropic quantum compass model on an infinite square lattice near a symmetry-breaking phase transition at finite temperature. From the linear susceptibility in the symmetric phase and the order parameter in the symmetry-broken phase the critical temperature is estimated at {cal T}_c=0.0606(4)J, where J is the isotropic coupling constant between S=1/2 pseudospins.