Three New Phases in the Two-Dimensional Potts Model at Complex Temperature
Abstract
The complex-temperature phase structure of the two-dimensional Potts model has remained largely unresolved despite decades of study. Here we determine the complex-temperature phase diagram of the 3-state Potts model, and uncover three gapless phases beyond the familiar ferromagnetic and paramagnetic phases. This is done by combining tensor-network free energies with the Poisson equation to compute the Fisher zero density in the thermodynamic limit. The resulting zero distribution unifies and revises long-standing conjectures on the zero loci, and outlines each phase. One is a non-conformal gapless phase, with an exponentially growing number of low-lying transfer-matrix levels distributed across all momentum sectors. The other two are Luttinger-liquid phases with complex-valued Luttinger parameters related by the Kramers-Wannier duality. At the antiferromagnetic critical point, the Luttinger parameter equals 3, while its limit from the adjacent Luttinger-liquid phase is 6. These findings reveal rich phase structures and unconventional critical behavior in a classical textbook model.
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BibTeX
@misc{wang2026three,
title = {{Three New Phases in the Two-Dimensional Potts Model at Complex Temperature}},
author = {Fang-Cheng Wang and Qi-Jun Ye and Xin-Zheng Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01771},
eprint = {2611.01771},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01771},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Three New Phases in the Two-Dimensional Potts Model at Complex Temperature
AU - Fang-Cheng Wang
AU - Qi-Jun Ye
AU - Xin-Zheng Li
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01771
UR - https://arxiv.org/abs/2611.01771
AB - The complex-temperature phase structure of the two-dimensional Potts model has remained largely unresolved despite decades of study. Here we determine the complex-temperature phase diagram of the 3-state Potts model, and uncover three gapless phases beyond the familiar ferromagnetic and paramagnetic phases. This is done by combining tensor-network free energies with the Poisson equation to compute the Fisher zero density in the thermodynamic limit. The resulting zero distribution unifies and revises long-standing conjectures on the zero loci, and outlines each phase. One is a non-conformal gapless phase, with an exponentially growing number of low-lying transfer-matrix levels distributed across all momentum sectors. The other two are Luttinger-liquid phases with complex-valued Luttinger parameters related by the Kramers-Wannier duality. At the antiferromagnetic critical point, the Luttinger parameter equals 3, while its limit from the adjacent Luttinger-liquid phase is 6. These findings reveal rich phase structures and unconventional critical behavior in a classical textbook model.
ER -