The order of a phase transition is usually determined by the nature of the symmetry breaking at the phase transition point and the dimension of the model under consideration. For instance, q-state Potts models in two dimensions display a second order, continuous transition for q = 2,3,4 and first order for higher q.
Tamura et al recently introduced Potts models with "invisible" states which contribute to the entropy but not the internal energy and noted that adding such invisible states could transmute continuous transitions into first order transitions. This was observed both in a Bragg-Williams type mean-field calculation and 2D Monte-Carlo simulations. It was suggested that the invisible state mechanism for transmuting the order of a transition might play a role where transition orders inconsistent with the usual scheme had been observed.
In this paper we note that an alternative mean-field approach employing 3-regular random ("thin") graphs also displays this change in the order of the transition as the number of invisible states is varied, although the number of states required to effect the transmutation, 17 invisible states when there are 2 visible states, is much higher than in the Bragg-Williams case. The calculation proceeds by using the equivalence of the Potts model with 2 visible and r invisible states to the Blume-Emery-Griffiths (BEG) model, so a by-product is the solution of the BEG model on thin random graphs.
Potts Models with Invisible States on General Bethe Lattices
为了使Potts模型的相变从连续相变变为一阶相变，需要在Potts模型中加入多少个所谓的不可见态，这一问题引起了人们的关注。在q=2的情况下，布拉格-威廉姆斯平均场方法需要4个这样的不可见态，而3-正则随机图形式主义需要17个。在这两种情况下，从二阶到一阶的行为引起的不可见状态的转换是通过一个等效的布卢姆-金刚砂-格里菲斯模型的三临界点确定的。在这里，我们调查广义Potts模型上的Bethe晶格z邻居。我们证明了在q=2的情况下，rc（z）=[4 z / 3（z-1）] [（z-1）/（z-2）]^z不可见态是等价的Blume-Emery-Griffiths三临界性的必要条件.当z=3时，恢复了3-正则随机图的结果，而无限z极限提供了布拉格-威廉姆斯平均场的结果。
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。
