The Klein bottle ratio of two-dimensional ferromagnetic Potts models

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.

Figures (12)

• Figure 1: Tensor-network representation of the partition function for the $q$-state Potts model on a square lattice:(a) Original lattice; (b) Dual lattice. The dual spin variable $\sigma_{\alpha}$ is defined on each bond and takes a value depending on whether the two Potts spins at the ends of the bond are equal. For example, $\sigma_{i} = \text{mod}(s_i-s_{i-1}+q,q)$, where $s_i$ denotes a Potts spin and $s_0=s_4$.
• Figure 2: The calculation of the Klein bottle ratio $g$ is implemented by the ratio of two tensor network contractions. Along $L_y$ direction, the crosscap contraction is adopted. $\mathcal{T}$ means the column transfer matrix.
• Figure 3: The Klein bottle ratio $g$ at critical temperature $T_c$ for the Potts-$q$ models ($q=2,3,4,5,6$). $L_y$ ranges from $16$ to $70$ in steps of $4$. The truncation dimension $D=200$. The dash lines from CFT description are drawn for reference.
• Figure 4: four-state Potts model.Temperature dependence of the Klein bottle ratio $g$ with the length $L_y$ ranging from $16$ to $60$. (a) original lattice; (b) dual lattice; (c) data collapse on the original lattice: $T_c = 0.91035$, $b = 1.39581$; (d) data collapse on the dual lattice: $T_c = 0.91011$, $b = 1.36798$.
• Figure 5: five-state Potts model. Temperature dependence of the Klein bottle ratio $g$ with the length $L_y$ ranging from $16$ to $60$. (a) original lattice; (b) dual lattice; (c) data collapse on the original lattice, giving $T_c = 0.85167$, $b = 1.59605$; (d) data collapse on the dual lattice, giving $T_c = 0.85140$, $b = 1.57144$.