Singh Vijay, Brunson C T, Boettcher Stefan
Department of Physics, Emory University, Atlanta, Georgia 30322, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Nov;90(5-1):052119. doi: 10.1103/PhysRevE.90.052119. Epub 2014 Nov 13.
We analyze the phase transitions that emerge from the recursive design of certain hyperbolic networks that includes, for instance, a discontinuous ("explosive") transition in ordinary percolation. To this end, we solve the q-state Potts model in the analytic continuation for noninteger q with the real-space renormalization group. We find exact expressions for this one-parameter family of models that describe the dramatic transformation of the transition. In particular, this variation in q shows that the discontinuous transition is generic in the regime q<2 that includes percolation. A continuous ferromagnetic transition is recovered in a singular manner only for the Ising model, q=2. For q>2 the transition immediately transforms into an infinitely smooth order parameter of the Berezinskii-Kosterlitz-Thouless type.
我们分析了某些双曲网络递归设计中出现的相变，例如，普通渗流中会出现不连续（“爆发性”）转变。为此，我们使用实空间重整化群，在非整数q的解析延拓中求解q态Potts模型。我们找到了描述该转变剧烈变化的这一参数模型族的精确表达式。特别地，q的这种变化表明，不连续转变在包括渗流的q<2区域中是普遍存在的。只有对于伊辛模型（q = 2），连续铁磁转变才以奇异的方式恢复。对于q>2，该转变立即转变为Berezinskii-Kosterlitz-Thouless型的无限平滑序参量。
