Variational Corner Transfer Matrix Renormalization Group Method for Classical Statistical Models

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