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Computation of correlation functions by tensor renormalization group method

Norihiro Nakamoto, Takeda, Shinji, 真滋 武田

Published Jan 1, 2016• 6 citations
Publisher

Abstract

We demonstrate how to compute the correlation function by using the tensor renormalization group for 2D Ising model. From the correlation function around the critical temperature, we extract the correlation length and the spontaneous magnetization and then determine some critical exponents. Attractive feature of our method to extract the critical exponents is that numerical derivative, that causes a loss of significant digits, and additional simulation parameters, that demand more computational cost and complicated analysis, are not required. On the other hand, our approach requires an additional treatment of impurity tensors, but the additional cost is still tolerable.

DOI
10.24517/00011105

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We propose a scheme to perform tensor network based finite-size scaling analysis for two-dimensional classical models. In the tensor network representation of the partition function, we use higher-order tensor renormalization group (HOTRG) method to coarse grain the weight tensor. The renormalized tensor is then used to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain the correlation length, the magnetization, and the energy density, which are used in finite-size scaling analysis to determine the critical temperature and the critical exponents. As a benchmark we study the two-dimensional classical Ising model. We show that the critical temperature and the critical exponents can be accurately determined. With HOTRG bond dimension D=70D=70D=70, the absolute errors of the critical temperature Tc{T}_{c}Tc​ and the critical exponent $\ensuremath{\nu}$, $\ensuremath{\beta}$ are at the order of ${10}^{\ensuremath{-}7},{10}^{\ensuremath{-}5}$, ${10}^{\ensuremath{-}4}$, respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cutoff in bond dimension and elucidate its physical meaning in this context.
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