Machine-learning approach to real-space renormalization of the 2D Potts model

The ordering transition of the q-state Potts model on a square lattice changes from continuous to first-order at q=4. This changeover has been analyzed in real-space renormalization group (RSRG) studies[1] but a clear physical picture is lacking. We have previously conjectured that the weak first order transition at q>4 is related to the four-color theorem in graph theory[2]. Here we implement a recently proposed deep-learning scheme that maps configurations at successive scales to each other[3]. Parameters of the mapping are learned by maximizing mutual information. We take into account symmetry properties of the model and analyze the resulting RG flow to extract critical exponents and other properties. The emergence of "mosaic domains" which represent dilution of the original Potts model under the RG transformation will be discussed[4].
[1] B. Nienhuis, A. N. Berker, E. K. Riedel and M. Schick, Phys. Rev. Lett. 43, 737-740 (1979).
[2] L.-H. Tang, unpublished.
[3] M. Koch-Janusz and Z. Ringel, Nat. Phys. 14, 578-582 (2018).
[4] C.M. Chan, L. Tian and L.-H. Tang, in preparation.