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

TL;DR

This paper introduces a variational tensor network method for accurately representing the Gibbs operator in 2D quantum systems at finite temperature, enabling precise analysis of phase transitions.

Contribution

The authors develop a variational tensor network renormalization algorithm that optimizes isometries with full environment consideration to construct finite bond dimension PEPOs for 2D quantum systems.

Findings

Estimated critical temperature for the quantum compass model as 0.0606(4)J.

Demonstrated the method's effectiveness near a symmetry-breaking phase transition.

Achieved accurate thermodynamic property calculations using tensor network coarse-graining.

Abstract

Progress in describing thermodynamic phase transitions in quantum systems is obtained by noticing that the Gibbs operator $e^{-\beta H}$ 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) $\beta$. Coarse-graining the network along $\beta$ 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 ${\cal T}_c=0.0606(4)J$, where $J$ is the isotropic coupling constant between $S=1/2$ pseudospins.