Tensor network based finite-size scaling for two-dimensional Ising model

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 = 70, the absolute errors of the critical temperature Tc and the critical exponent nu, beta are at the order of 10-7, 10-5, 10-4, 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.

Automated summary

We propose a scheme to perform finite-size scaling analysis for two-dimensional classical models using tensor network. The higher-order tensor renomalization group (HOTRG) method is used to coarse grain the weight tensor in the tensor network representation of the partition function. 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, magnetization, and energy density, which are used to determine the critical temperature and critical exponents in finite-size scaling analysis. The results show accurate determination of the critical temperature and critical exponents can be achieved.