Progress in describing thermodynamic phase transitions in quantum systems is obtained by noticing that the Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ 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) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ 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
Tc=0.0606(4)J, where
J is the isotropic coupling constant between
S=21 pseudospins.
In the present paper, we investigate quantum phase transition in the one-dimensional compass model introduced by Brzezicki et al. [Phys. Rev. B 75, 134415 (2007)]. Unlike the previous approach based on the Jordan-Wigner transformation, we use directly the standard pseudospin representation in our investigation. Therefore, our method can be potentially applied to study rigorously the properties of the same model in higher dimensions. By applying the reflection positivity technique, we are able to determine degeneracy of the global ground state and the quantum phase-transition point of this model. We find also that the transition is of the first order. These results are consistent with the previous conclusions. However, some differences are also uncovered.