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arXiv:cond-mat/0412192

Estimation of the magnetic critical exponent by Tensor Product Variational Approach

Andrej Gendiar, Tomotoshi Nishino, René Derian

Published Dec 8, 2004• 2 citations
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Abstract

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 $β$ 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.

DOI
10.48550/arxiv.cond-mat/0412192

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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.
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
Values of dynamic critical exponents are numerically estimated for various models with the nonequilibrium relaxation method to test the dynamic universality hypothesis. The dynamics used here are single-spin update with Metropolis-type transition probabities. The estimated values of nonequilibrium relaxation exponent of magnetization λ m (=β/ z ν) of Ising models on bcc and fcc lattices are estimated to be 0.251(3) and 0.252(3), respectively, which are consistent with the value of the model on simple-cubic lattice, 0.250(2). The dynamic critical exponents of three-states Potts models on square, honeycomb and triangular lattices are also estimated to be 2.193(5), 2.198(4), and 2.199(3), respectively. They are consistent within the error bars. It is also confirmed that Ising models with regularly modulated coupling constants on square lattice have the same dynamic critical exponents with the uniformly ferromagnetic Ising model.