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arXiv:cond-mat/0011103 · Progress of Theoretical Physics

Two-Dimensional Tensor Product Variational Formulation

T. Nishino, Y. Hieida, K. Okunishi, Nobuya Maeshima, Y. Akutsu, A. Gendiar

Published Mar 1, 2001• 152 citations
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Abstract

We propose a numerical self-consistent method for 3D classical lattice models, which optimizes the variational state written as a two-dimensional product of tensors. The variational partition function is calculated using the corner transfer matrix renormalization group (CTMRG), which is a variant of the density matrix renormalization group (DMRG). The numerical efficiency of the method is exemplified in its application to the 3D Ising model.

DOI
10.1143/ptp.105.409

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We propose a numerical self-consistent method for 3D classical lattice models, which optimizes the variational state written as a two-dimensional product of tensors. The variational partition function is calculated using the corner transfer matrix renormalization group (CTMRG), which is a variant of the density matrix renormalization group (DMRG). The numerical efficiency of the method is exemplified in its application to the 3D Ising model.
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