Problems in Quantum and Classical Statistical Mechanics

摘要：该项目的第一部分研究了二维自回避行走及其与Schramm随机Loewner进化过程的关系。Schramm的过程被认为描述了各种二维模型的缩放极限，包括自我回避行走。该算法提供了一种快速模拟自避行走的方法。最近，首席研究员发现了这种算法的一种新实现，它在二维上的速度要快80倍。这将用于测试自回避行走的保形不变性及其与随机洛厄纳进化的等价性。一个弱自避行走的版本，其中自交的惩罚随着自交产生的环路的长度而衰减，将通过模拟和微扰手段进行研究。该项目的第二部分涉及量子自旋系统中的激发态。最近，首席研究员开发了一种研究一维准粒子态和界面态色散关系的方法。该方法是基于对状态的波函数假设一个确定的ansatz，然后通过收缩映射论证证明确实存在这种形式的特征态。该方法将用于研究二维界面态和这些界面态之上的激发态。自我回避散步是随机散步，不允许多次访问同一个地方。它们为稀溶液中的线性聚合物提供了一个模型。然而，对这个模型的兴趣要广泛得多，因为它是展示临界现象和二维保形不变性的最简单模型之一。最近，关于自我回避行走和其他二维模型与Schramm引入的一种新的二维随机过程（称为随机Loewner进化）之间的联系的猜想激增。该项目的一部分将通过蒙特卡罗模拟和摄动方法研究许多这些关于自我避免行走的猜想。模拟将通过首席研究员最近实现的pivot算法来完成，该算法比以前的实现快得多。这种快速算法也将用于研究在物理化学中很重要的自我回避行走。该项目的另一部分致力于研究各种量子自旋系统中的低能量激发。这些是晶体中电子自旋行为的模型。在低温下，电子自旋趋向于排列。然而，可以形成几个区域，其中自旋是对齐的，但在这些区域之间，自旋指向相反的方向。这些领域之间的界面将通过一种方法来研究，这种方法已被证明是非常成功的研究一个准粒子态。特别是，将研究界面态以上的激发的性质。

---------------------------------PI: Tom Kennedy, University of ArizonaDMS 0201566Abstract:The first part of the project concerns self-avoiding walks in two dimensions and their relation to Schramm's stochastic Loewner evolution process. Schramm's process is believed to describe the scaling limit ofa variety of two-dimensional models, including the self-avoiding walk.The pivot algorithm provides a fast method for simulating self-avoiding walks. Recently, the principal investigator has found a new implementation of this algorithm that is as much as eighty times faster in two dimensions.This will be used to test both the conjectured conformal invariance of the self-avoiding walk and its equivalence with stochastic Loewner evolution. A version of the weakly self-avoiding walk in which the penalty for self-intersections decays with the length of the loop produced by the self-intersection will be investigated by simulations and perturbative means. The second part of the project concerns excited states in quantum spin systems. Recent work by the principal investigator has developed a method for studying the dispersionrelation of one quasi-particle states and interface states in one dimension. The method is based on assuming a certain ansatz for the wave function of the states and then proving there is indeed an eigenstate of thisform by a contraction mapping argument.This method will be used to study interface states in two dimensions and the excited states above these interface states. Self-avoiding walks are random walks which are not allowed to visitthe same place more than once. They provide a model for linear polymers in a dilute solution. The interest in this model is, however, much broader since it is one of the simplest models that exhibits critical phenomena and in two dimensions conformal invariance. Recently there has been an explosion of conjectures relating the self-avoiding walk and other two-dimensional models to a new two dimensional stochasticprocess, called stochastic Loewner evolution, introduced by Schramm.Part of the project will study many of these conjectures for the self-avoiding walk by Monte Carlo simulations and by perturbative methods. The simulations will be done with a recent implementation of the pivot algorithm by the principal investigatorthat is much faster than previous implementations.This fast algorithm will also be used to study versions of the self-avoiding walk that are important in physical chemistry.Another part of the project is devoted to studying the low energy excitationsin a variety of quantum spin systems. These are models of the behaviorof the electron spins in crystals. At low temperatures the electronspins tend to align. However, several domains may form within which the spins are aligned, but between which the spins point in opposite directions. The interfaces between these domains will be studied by an approach which has proved very successful for studying one quasi-particle states. In particular, the nature of the excitations just above the interface states will be investigated.