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Stable Optimization of Tensor Product Variational Functions

Andrej Gendiar, Nobuya Maeshima, Tomotoshi Nishino

Published Mar 19, 2003• 0 citations
OpenAlex

Abstract

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

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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
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
A variational problem for three-dimensional (3D) classical lattice models is considered with trial state given by two-dimensional (2D) uniform product of local variational weights. This approach, the tensor product variational approach (TPVA), has been applied to 3D classical models (the Ising and the Potts models). We consider a way of estimating the magnetic critical exponent βββ for the simple 3D Ising model assuming a functional form of the spontaneous magnetization in the off critical region, where the TPVA provides reliable data.
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.