二维系统中基于四色定理的一级相变物理机制的统计力学研究

项目名称： 二维系统中基于四色定理的一级相变物理机制的统计力学研究

项目编号： No.11505095

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 田亮

作者单位： 南京航空航天大学

项目金额： 18万元

中文摘要： 该项目着眼于一类低温相破坏不连续对称性的一级有序-无序相变的物理机制。典型的模型是二维q态铁磁Potts模型。该模型在q≤4时表现出连续相变，但在q>4时转变为一级相变。出乎意料的是，虽然对Potts模型的研究非常广泛，但针对在q=4处发生的转变至今还没有清晰的物理图像。4这个数字使我们联想起图论中的四色定理：我们只需要四种颜色就可以将一个二维随机平面图染色使得相邻的区域具有不同的颜色。这意味着多余的颜色会带来额外的染色方法，也就是“颜色熵”。基于此我们认为二维Potts模型在q>4时的一级相变是由大规模的颜色熵所驱动的，并伴随具有随机马赛克结构的相的出现。通过细致的数值模拟和理论分析，我们将深入的研究马赛克相的热力学稳定性，并建立起由颜色熵驱动的一级相变的物理机制。颜色熵的物理图像将对于我们理解凝聚态系统中熵驱动的一级相变的物理机制提供新的视角。

中文关键词： 相变；Potts模型；临界现象；蒙特卡洛模拟；重整化群

英文摘要： This proposal focuses on the physical mechanism of a class of first-order order-disorder phase transition, whose low temperature phase breaks a discrete symmetry. A typical model is the two dimensional q-state ferromagnetic Potts model. This model at q≤4 shows a continuous phase transition, while for q>4 it exhibits a first-order phase transition. Surprisingly, although the study on the Potts model is very extensive, until now there is still no clear physical picture of the changeover occurring at q=4. The figure “4” reminds of the four-color theorem in graph theory: we can paint a two-dimensional random graph with only four colors so that any adjacent regions have different colors. This means that the excess color will bring additional solutions, which is the “coloring entropy”. Based on this consideration, we suggest that the first-order phase transition in Potts model at q>4 is driven by a large-scale coloring entropy, accompanied by the emergence of a phase with random mosaic structure. We in details study the thermodynamic stability of the mosaic phase, and establish the physical mechanism of coloring entropy driven first-order phase transition. The physical picture of coloring entropy will help us understand the entropy-driven first-order phase transition in condensed matter systems.

英文关键词： phase transition;Potts model;critical phenomena;Monte Carlo simulation;renormalization group