Tensor Network Finite-Size Scaling for Two-Dimensional 3-state Clock Model

Abstract

We benchmark recently proposed tensor network based finite-size scaling analysis in Phys. Rev. B {\bf 107}, 205123 (2023) against two-dimensional classical 3-state clock model. Due to the higher complexity of the model, more complicated crossover behavior is observed. We advocate that the crossover behavior can be understood from the perspective of finite bond dimension inducing relevant perturbation. This leads to a general strategy to best estimate the critical properties for a given set of control parameters. For the critical temperature $T_c$, the relative error at the order of $10^{-7}$ can be reached with bond dimension $D=70$. On the other hand, with bond dimension $D=60$, the relative errors of the critical exponents $\nu, \beta, \alpha$ are at the order of $10^{-2}$. Increasing the bond dimension to $D=90$, these relative errors can be reduced at least to the order of $10^{-3}$. In all cases our results indicate that the errors can be systematically reduced by increasing the bond dimension and the stacking number.