Hohenberg-Mermin-Wagner type theorems for equilibrium models of flocking
Hal Tasaki
备注：
8 pages. Refs. [27,28,29] were added and the discussion was improved in version 2. Supplemental material, which should be useful, and Ref [18] are added in version 3. There is a 18.5 minutes video on YouTube in which I discuss the background, motivation, and main results of the present paper: https://youtu.be/CD0WEkNl9XA
We study a class of two-dimensional models of classical hard-core particles with Vicsek-type "exchange interaction" that aligns the directions of motion of nearby particles. By extending the Hohenberg-Mermin-Wagner theorem for the absence of spontaneous magnetization and the McBryan-Spencer bound for correlation functions, we prove that the models do not spontaneously break the rotational symmetry in their equilibrium states at any nonzero temperature. We thus conclude that the mobility of particles alone does not account for the spontaneous symmetry breaking in Vicsek type models. The origin of the symmetry breaking must be sought in the absence of detailed balance condition, or, equivalently, in the nonequilibrium nature.
Two-Dimensional Phase Transitions in Classical Systems: 60 Years after the Hohenberg-Mermin-Wagner Theorem
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；State Key Laboratory of Surface Physics and Department of Physics, Fudan University（复旦大学表面物理国家重点实验室和物理系）
The inverse Mermin-Wagner theorem for classical spin models on graphs
在这封信中，我们提出的Mermin-Wagner定理在图上的反转，通过证明在有限温度下的自发磁化的存在性的经典自旋模型的瞬态平均（TOA）图，即图中的随机步行者返回到其起点的平均概率$\bar F < 1$。这个结果，这是证明了模型的O（n）对称性，包括作为一个特殊的情况$n=1$，提供了一个非常普遍的条件，自发对称性破缺的非均匀结构，即使是伊辛模型。
On the Hohenberg-Mermin-Wagner theorem and its limitations
就在50多年前，皮埃尔·霍恩伯格（Pierre Hohenberg）对有限温度下二维超流体或超导体中不存在长程序进行了严格的证明。这个证明立即被N. D. Mermin和H.瓦格纳到海森堡铁磁体和反铁磁体，此后不久，梅尔明证明了二维晶体中不存在平移长程序，无论是在量子力学还是经典力学中。本文推广了Hohenberg-Mermin-Wagner定理，严格证明了在没有自旋轨道耦合和磁偶极相互作用的巡游电子系统中不可能存在长程铁磁有序.我们还评论了一些情况下，有令人信服的论点，长程序是不可能的，但没有严格的证明已经给出，以及情况下，如一个磁铁与长程相互作用，或在一个二维晶体的取向顺序，其中长程序可以发生，打破了连续的对称性。
