Problems in Quantum and Classical Statistical Mechanics

-PI：Tom Kennedy，亚利桑那大学DMS 0201566摘要：该项目的第一部分涉及二维自回避行走及其与Schramm随机Loewner演化过程的关系。Schramm过程被认为描述了各种二维模型的尺度极限，包括自回避行走，枢轴算法提供了一种快速模拟自回避行走的方法。最近，首席研究员发现了这个算法的一个新的实现，在二维中快了80倍，这将被用来测试自避免行走的约束共形不变性及其与随机Loewner演化的等价性。一个版本的弱自避免行走，其中惩罚自交的自相交所产生的循环的长度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。主要研究者最近的工作发展了一种研究一维准粒子态和界面态色散关系的方法。这种方法是基于对态的波函数作一定的近似，然后用压缩映射论证证明确实存在这种形式的本征态，这种方法将用于研究二维界面态及其上的激发态。自回避游动是一种随机游动，它不允许访问同一个地方超过一次。它们提供了线性聚合物在稀溶液中的模型。在这个模型中的兴趣是，但是，更广泛的，因为它是最简单的模型之一，表现出临界现象和二维共形不变性。最近有一个爆炸性的理论联系的自我回避行走和其他二维模型，以一个新的二维随机过程，称为随机Loewner evolution，介绍了Schramm。该项目的一部分将研究许多这些理论的自我回避行走的Monte Carlo模拟和微扰方法。模拟将使用最近由主要计算器实现的枢轴算法来完成，该算法比以前的实现快得多。这种快速算法也将用于研究物理化学中重要的自避免行走的版本。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。These are models模型of the behaviorof the electron电子spins自旋in crystals晶体.在低温下，电子自旋倾向于排列。然而，可以形成几个区域，在这些区域内自旋对齐，但是在这些区域之间自旋指向相反的方向。这些领域之间的接口将研究的方法，已被证明是非常成功的研究一个准粒子状态。特别是，将调查的性质的激发上面的界面状态。

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