The Klein bottle ratio of two-dimensional ferromagnetic Potts models

TL;DR

This study uses advanced numerical methods to analyze the phase transition nature of the 2D 5-state Potts model, revealing signatures of weakly first-order behavior through the Klein bottle ratio and conformal field theory insights.

Contribution

It introduces a novel approach using Klein bottle ratio and tensor-network methods to distinguish weakly first-order from continuous phase transitions in the Potts model.

Findings

Finite-size scaling of g locates critical points for q=4,5,6.

Central charge for q=5 close to complex CFT prediction.

Divergence of g confirms first-order transition at q=5.

Abstract

The weakly first-order nature of the two-dimensional 5-state ferromagnetic Potts model poses challenges for numerical study. Using density-matrix and tensor-network renormalization group methods, we investigate these transitions of the Potts-$q$ model via the Klein bottle ratio $g$ on original and dual lattices. Finite-size scaling of $g$ as a function of transverse system size $L_y$ accurately locates the critical points for $q = 4, 5, 6$. We further examine the transfer-matrix spectra and entanglement entropy, extracting central charges through toroidal and Klein bottle boundary conditions. For $q = 5$, the extracted central charge ($c \approx 1.14811$) is close to the real part of the theoretical value $c_{5\text{-Potts}} = 1.1375 \pm 0.0211 i$ predicted by complex conformal field theories. The observed drift in the scaling exponent $b$ effectively distinguishes the continuous transition from the weakly first-order regime. Furthermore, the extrapolated divergence of $g$ confirms the first-order nature of the $q=5$ Potts model.