Tensor Network Based Finite-Size Scaling for Two-Dimensional Classical Models

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 $T_c$ 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 cut-off in bond dimension and elucidate its physical meaning in this
context.