In recent years, tensor networks have become popular as a numerical means to investigate quantum/classical systems. Especially in one-dimensional quantum systems, they have contributed widely to classifying critical phenomena and topological phases due to their compatibility with conformal field theory. [1-2] Similarly, in the two-dimensional classical critical model, it has been shown that the scaling dimension and central charge can be calculated using the fixed point tensor obtained by tensor network renormalization, and has attracted attention as a "new way to solve old problems." [3]

In this presentation, we will show that the renormalization group flow (RG flow), the phase diagram around a critical point, can also be computed quantitatively using tensor network renormalization.[4] As a practical example, we will calculate the coupling constants in field theory at each scale from the classical XY model and show that they are consistent with the Kosterlitz RG flow. If time permits, we will introduce applications to the RP^2 model, a more complex classical two-dimensional system. [5]

[1] P. Calabrese and J. Cardy, J. Phys. A: Math. Theor. 42, 504005 (2009).
[2] F. Pollmann et al, Phys. Rev. B 81, 064439 (2010).
[3] ZC. Gu and XG. Wen, Phys. Rev. B 80, 155131 (2009).
[4] A. Ueda and M. Oshikawa, Phys. Rev. B 104, 165132 (2021).
[5] A. Ueda and M. Oshikawa, arXiv:2202.07042 (2022).