Problems in Quantum and Classical Statistical Mechanics

Schramm的过程被认为描述了包括自我回避行走在内的各种二维模型的缩放极限。Pivot算法提供了一种快速模拟自我回避行走的方法。最近，主要研究人员发现了该算法的一个新的实现，它在二维上的速度高达80倍，这将被用来测试猜想的自回避行走的共形不变性以及它与随机Loewner演化的等价性。我们将用模拟和微扰的方法来研究一种形式的弱自回避行走，其中自相交的惩罚随自相交产生的环的长度而衰减。该项目的第二部分涉及量子自旋系统中的激发态。最近主要研究者的工作发展了一种研究一维准粒子态和界面态的色散关系的方法。这种方法的基础是假设这些态的波函数具有一定的平稳性，然后通过压缩映射证明确实存在这种形式的本征态。这种方法将被用来研究二维界面态以及这些界面态之上的激发态。自我回避漫步是指不允许多次访问同一个地方的随机漫步。它们为稀溶液中的线性聚合物提供了一个模型。然而，人们对这个模型的兴趣要广泛得多，因为它是最简单的模型之一，它展示了临界现象和二维共形不变性。最近，将自回避行走和其他二维模型与Schramm提出的一种新的二维随机过程--随机Loewner演化--联系起来的猜想激增。该项目的一部分将用蒙特卡罗模拟和微扰方法研究其中许多关于自回避行走的猜想。该项目的首席研究员最近实现了枢轴算法，该算法比以前的实现要快得多。该快速算法还将被用于研究物理化学中重要的自回避行走的版本。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，电子自旋往往排列成一条直线。然而，可能会形成几个区域，自旋在这些区域内对齐，但它们之间的自旋指向相反的方向。我们将用一种已被证明非常成功的方法来研究这些域之间的界面，这种方法已被证明是非常成功地研究了一个准粒子态。特别是，我们将研究界面态上方激发的性质。

---------------------------------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.