The Kosterlitz-Thouless transition
The Kosterlitz-Thouless transition
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科斯特利茨-托利斯转变
DOI：
10.1007/3-540-11192-1_5
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发表时间：
1982
期刊：
影响因子：
--
通讯作者：
T. Spencer
中科院分区：
文献类型：
--
作者：
J. Fröhlich;T. Spencer
本次演讲的目的是描述一种证明相变存在的新方法。特别是，我们将讨论二维平面旋转器的 Kosterlitz-Thouless 转变以及具有 i/r 2 相互作用的一维 Ising 模型的自发磁化的存在。基本上有三种方法可以确定相变的存在：（a）精确解。该技术适用于非常有限的一类模型，例如二维伊辛模型和冰模型，但给出了转变本质的详细描述。参见 [i]。(b) Peierls 论证 (1936) 适用于最多具有离散对称群的各种模型。这些模型包括三维伊辛模型和各向异性海森堡模型。(c)红外界限[2]适用于具有连续内部对称性的三维或更多维自旋系统，例如各向同性最近邻海森堡模型。该方法给出了相当好的临界温度下限，但仅限于反射正自旋系统。在（b）和（c）中，需要一个阶次参数（通常是长程阶次）来表征转变。我们将在本次讲座中讨论的方法将对偶变换与块自旋方法结合起来。对偶变换基本上是傅立叶级数展开，并用离散对称性代替了旋转器的 0 (2) 对称性。我们的基本估计本质上是佩尔斯论证中出现的能量熵不等式的更精确版本。该方法适用于多种模型，包括实体叠加实体模型、四维 U (1) 规范模型 [另请参阅 Guth [3] 了解早期结果] 和 2 维时钟模型，n
The purpose of this talk is to describe a new method for proving the existence of phase transitions. In particular we shall discuss the Kosterlitz-Thouless transition for the two dimensional plane rotator and the existence of a spontaneous magnetization for the one dimensional Ising model with i/r 2 interaction. There are basically three ways to establish the existence of phase transitions:(a) Exact solution. This technique applies to a very limited class of models such as the two dimensional Ising model and the ice model, but gives a detailed description of the nature of the transition. See [i].(b) The Peierls argument (1936) applies to a wide variety of models which have at most a discrete symmetry group. These models include the 3 dimensional Ising model and the anisotropic Heisenberg model.(c) The infrared bounds [2] are applicable to three or more dimensional spin systems with continuous internal symmetry, such as the isotropic nearest neighbor Heisenberg model. The method gives reasonably good lower bounds on the critical temperature but is restricted to reflection positive spin systems. In both (b) and (c) an order parameter, typically long range order, is needed to characterize the transi£ ion. The method we shall discuss in this lecture combines a duality transformation with block spin methods. The duality transformation is basically a Fourier series expansion and replaces the 0 (2) symmetry of the rotator, by a discrete symmetry~. Our basic estimates are essentially a more refined version of the energy entropy inequalities which appear in the proof of the Peierls argument. This method is applicable to a wide variety of models including the solid on solid model, the four dimensional U (1) gauge model [see also Guth [3] for earlier results], and the 2 dimensional~ clock models, n