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1Two-Dimensional Tensor Product Variational Formulation

Tomotoshi Nishino, Yasuhiro Hieida, Kouichi Okunishi, Yasuhiro Akutsu, Andrej Gendiar

Published Jan 1, 2000• 59 citations
OpenAlex

Abstract

We propose a numerical self-consistent method for 3D classical lattice models, which optimizes the variational state written as two-dimensional product of tensors. The variational partition function is calculated by the corner transfer matrix renormalization group (CTMRG), which is a variant of the density matrix renormalization group (DMRG). Numerical efficiency of the method is observed via its application to the 3D Ising model. Variational states represented as products of local tensols have been used widely for calculating the ground-state energy in one-dimensional (1D) quantum systems and the free energy in two-dimensional (2D) classical statistical systems. In 1941 Kramers and Wannier calculated a lower bound of the partition function of the 2D

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We propose a numerical self-consistent method for 3D classical lattice models, which optimizes the variational state written as two-dimensional product of tensors. The variational partition function is calculated by the corner transfer matrix renormalization group (CTMRG), which is a variant of the density matrix renormalization group (DMRG). Numerical efficiency of the method is observed via its application to the 3D Ising model. Variational states represented as products of local tensols have been used widely for calculating the ground-state energy in one-dimensional (1D) quantum systems and the free energy in two-dimensional (2D) classical statistical systems. In 1941 Kramers and Wannier calculated a lower bound of the partition function of the 2D
We investigate the Kramers-Wannier approximation for the three-dimensional (3D) Ising model. The variational state is represented by an effective 2D Ising model, which contains two variational parameters. We numerically calculate the variational partition function using the corner transfer matrix renormalization group (CTMRG) method, and find its maximum with respect to the variational parameters. The value of the calculated transition point, Kc = 0.2184, is only 1.5% less than the true Kc. This result is better than that obtained using the corner transfer tensor renormalization group (CTTRG) approach. The calculated phase transition is mean-field like.
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.
We consider a variational problem for three-dimensional (3D) classical lattice models, where the trial state is given by a uniform 2D product of local factors. Maximization of the variational partition function draws a self-consistent equation for the local factor. We propose a stable algorithm to solve the equation numerically when the variational function contains many degrees of freedom. Numerical stability and efficiency of the new algorithm is examined through its application to the 3D Ising model. 1
Three-dimensional (3D) q-state Potts models (q=3, 4, and 5) are studied by the tensor product variational approach, which is a recently developed variational method for 3D classical lattice models. The variational state is given by a 2D product of local factors, and is improved by way of self-consistent calculations assisted by the corner transfer matrix renormalization group. It should be noted that no a priori condition is imposed for the local factor. Transition temperatures and latent heats are calculated from the observations of thermodynamic functions in both ordered and disordered phases.