Using the corner-transfer matrix renormalization group approach, we revisit the three-state chiral Potts model on the square lattice, a model proposed in the eighties to describe commensurate-incommensurate transitions at surfaces, and with direct relevance to recent experiments on chains of Rydberg atoms. This model was suggested by Huse and Fisher to have a chiral transition in the vicinity of the Potts point, a possibility that turned out to be very difficult to definitely establish or refute numerically. Our results confirm that the transition changes character at a Lifshitz point that separates a line of Pokrosky-Talapov transition far enough from the Potts point from a line of direct continuous order-disorder transition close to it. Thanks to the accuracy of the numerical results, we have been able to base the analysis entirely on effective exponents to deal with the crossovers that have hampered previous numerical investigations. The emerging picture is that of a new universality class with exponents that do not change between the Potts point and the Lifshitz point, and that are consistent with those of a self-dual version of the model, namely correlation lengths exponents $ν_x=2/3$ in the direction of the asymmetry and $ν_y=1$ perpendicular to it, an incommensurability exponent $\bar β=2/3$, a specific heat exponent that keeps the value $α=1/3$ of the three-state Potts model, and a dynamical exponent $z=3/2$. These results are in excellent agreement with experimental results obtained on reconstructed surfaces in the nineties, and shed light on recent Kibble-Zurek experiments on the period-3 phase of chains of Rydberg atoms.
Biskup等人[Phys. Rev. Lett. 84（2000）4794]最近提出，可以有利地将配分函数零点的轨迹视为复温度或场平面中的相边界。我们得到的Fisher零点的Ising和Potts模型的非平面（“薄”）定期随机图使用这种方法，并注意到，费歇尔零点的轨迹上的Bethe格是相同的相应的随机图。由于状态数在Potts解中作为一个参数出现，色零点的极限轨迹也是可以接近的。
