Variational corner transfer matrix renormalization group method for classical statistical models

TL;DR

This paper introduces a variational reformulation of the corner transfer matrix renormalization group method within tensor networks, enabling efficient and precise evaluation of classical statistical models and their phase transitions.

Contribution

It presents the first variational bilevel optimization approach to CTMRG, improving accuracy and extending applicability to higher-dimensional and quantum models.

Findings

Accurate residual entropy calculation for the dimer model.

Critical points and exponents match known results.

Method is straightforward to extend to 3D and quantum models.

Abstract

In the context of tensor network states, we for the first time reformulate the corner transfer matrix renormalization group (CTMRG) method into a variational bilevel optimization algorithm. The solution of the optimization problem corresponds to the fixed-point environment pursued in the conventional CTMRG method, from which the partition function of a classical statistical model, represented by an infinite tensor network, can be efficiently evaluated. The validity of this variational idea is demonstrated by the high-precision calculation of the residual entropy of the dimer model, and is further verified by investigating several typical phase transitions in classical spin models, where the obtained critical points and critical exponents all agree with the best known results in literature. Its extension to three-dimensional tensor networks or quantum lattice models is straightforward, as also discussed briefly.