Mathematical Sciences: Statistical Mechanics of Classical and Quantum Lattice Systems

摘要肯尼迪9623509 三种类型的晶格系统将被研究-量子自旋系统，弱自避免行走和经典自旋系统的真实的空间重整化群变换。我们将要研究的主要量子自旋系统是阶梯上的海森堡反铁磁体（2个站点宽，无限多个站点长）。这个特殊的模型包含几个有趣的量子自旋模型作为极限情况，应该有助于我们理解这些模型中发生的各种奇异相位之间的关系。弱自回避游动是格上的最近邻游动，其概率根据其包含的自相交的数量而减少。这里的目标是开发真实的空间重整化群变换的通常的模型和一个模型，其中的能量惩罚为一个自相交的相交所形成的环的长度上升到一个权力衰减。后一种模型即使在一维中也有有趣的行为。对于像伊辛模型这样的经典晶格自旋系统，我们 将继续我们的研究真实的空间重整化群变换，如多数规则。当这个变换被应用到具有幂律相互作用的一维伊辛模型时，当幂接近于1时，有一个很好的简化。这种简化应该是严格定义转换的开始。我们还将致力于在各种二维模型中建立变换的存在性。 该项目致力于物理学中起源的几个系统的数学研究-量子自旋系统，自避免行走和经典自旋系统。量子和经典自旋系统是晶体中电子自旋相互作用的模型。这些系统包含大量的自旋，但通常每个自旋只与附近的自旋相互作用。尽管如此，这些微观的局部相互作用可以 产生宏观效应，例如，磁性和超导性。自回避行走是非常长的聚合物的模型。同样，有大量的基本单元，但相互作用是局部的-聚合物的不同部分不能占据相同的位置。这种局部相互作用影响聚合物的尺寸如何随着基本聚合物的数量而增长。 构成聚合物的单元增加。研究所有这些模型的一个共同动机是理解在具有非常大的自由度的数学模型中，简单的局部相互作用如何产生全局效应。物理学中最引人注目的发现之一是，这些全局效应的某些性质并不依赖于微观相互作用的细节。有几个可量化的属性， 宏观行为被称为临界指数，其被认为对于各种微观相互作用是完全相同的。这个项目的目标之一是进一步加深我们对这种“普遍性”的数学理解。"

Abstract Kennedy 9623509 Three types of lattice systems will be studied - quantum spin systems, weakly self-avoiding walks and real space renormalization group transformations for classical spin systems. The primary quantum spin system that we will study is the Heisenberg antiferromagnet on a ladder (2 sites wide, infinitely many sites long). This particular model contains several interesting quantum spin models as limiting cases and should help us understand the relationship between the variety of exotic phases that occur in these models. Weakly self-avoiding walks are nearest neighbor walks on the lattice with the probability of a walk reduced according to the number of self intersections it contains. The goal here is to develop real space renormalization group transformations for the usual model and for a model in which the energy penalty for a self intersection decays as the length of the loop formed by the intersection raised to a power. This latter model has interesting behavior even in one dimension. For classical lattice spin systems like the Ising model, we will continue our study of real space renormalization group transformations like majority rule. When this transformation is applied to the one dimensional Ising model with a power law interaction, there is a nice simplication when the power is close to one. This simplification should be the start of a rigorous definition of the transformation. We will also work on establishing the existence of the transformation in various two dimensional models. This project is devoted to the mathematical study of several systems that originate in physics - quantum spin systems, self avoiding walks and classical spin systems. Quantum and classical spin systems are models for the interactions of the spins of electrons in crystals. These systems contain a huge number of spins, but typically each spin only interacts with nearby spins. Nonetheless these microscopic local interactions can produce macroscopic effects, e.g., magnetism and super conductivity. Self avoiding walks are models for very long polymers. Again, there are a large number of basic units, but the interaction is local - different sections of the polymer cannot occupy the same location. This local interaction affects how the size of the polymer grows as the number of basic units making up the polymer increases. A common motivation for studying all of these models is to understand how simple local interactions in a mathematical model with a very large number of degrees of freedom produce global effects. One of the most remarkable discoveries in physics is that certain properties of these global effects do not depend on the details of the microscopic interactions. There are several quantifiable properties of the macroscopic behavior know as critical exponents which are believed to be exactly the same for a wide variety of microscopic interactions. One of the goals of this project is to further our mathematical understanding of this "universality."