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

TL;DR

This paper introduces a tensor network-based finite-size scaling method for 2D classical models, accurately determining critical points and exponents using HOTRG, with results improving as bond dimension increases.

Contribution

The paper presents a novel tensor network approach combining HOTRG and transfer matrix techniques for finite-size scaling analysis of 2D classical models.

Findings

Accurate determination of critical temperature and exponents for 2D Ising model.

Errors in critical parameters decrease with higher HOTRG bond dimension.

Systematic improvement of results by increasing bond dimension.

Abstract

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.