Exotic Phases and Loop Models in Condensed Matter

在理论和实验凝聚态物理中，当前一个非常有趣的话题是寻找“奇异”相。这种相存在于二维电子气体中，并且有大量证据表明它们也存在于某些种类的磁铁中。许多工作都集中在两个空间维度的系统上。人们对一维系统有很多理论上的了解，但在实验室中观察它们的行为是非常困难的。有效的二维系统比一维系统具有更丰富的理论可能性，也更容易实现。一个著名的例子是量子霍尔效应，当二维电子气体被置于强磁场中时，就会产生这种效应。一个引人注目的实验证实的预测是分数电荷的出现。这意味着即使系统完全由电子组成，也存在准粒子，其电荷是电子的一小部分。例如，在最简单的情况下，从系统中移除一个电子会产生三个相同的局域“准粒子”，它们的电荷必须是电子的三分之一。研究奇异相的一个动机是找到一个能够充当拓扑量子计算机的系统。量子计算的总体思路是利用量子力学的特殊性质来构建一种新型计算机，能够解决传统计算机无法解决的大问题。例如，量子计算机可以分解比传统计算机大得多的数字，从而使常用的加密方案得以破解。在实际建造量子计算机的道路上存在许多障碍，最重要的是，大多数方法都需要设计一个能够重复产生精确到10万分之一的结果的系统。虽然这样的精度不是不可想象的，但这样做的困难导致了对拓扑量子计算的极大兴趣，从而避免了这一困难。拓扑量子计算机的“量子位”（或量子位，在计算的基本步骤中处理）是由带有分数电荷的粒子组成的。当然，研究新问题往往需要开发新的理论工具。幸运的是，其中一些是近在咫尺的，使用的是最近几年才发展起来的数学。这给我们提供了一种新的方法来描述这些系统的微观行为，从它们的几何特性来看——从某些角度来看，它们看起来像一团纠结的意大利面，或者随机曲线。这些随机曲线的精确特征将使我们能够了解我们希望了解的奇异相的大尺度特性。还有其他几种强有力的理论方法可以运用，在这些方法中，研究人员和访问研究人员之间有着丰富的经验。

A topic of great current interest in both theoretical and experimental condensed matter physics is the search for 'exotic' phases. Such phases exist in two-dimensional electron gases, and there is substantial evidence that they also occur in certain kinds of magnets. Many efforts are focused on systems in two spatial dimensions. Much is understood theoretically about one-dimensional systems, but seeing their behaviour in the lab is very difficult. Systems effectively in two dimensions have even more rich theoretical possibilities than in one, and are also easier to realise. A famous example is the quantum Hall effect, which arises when a two-dimensional electron gas is placed in a strong magnetic field. One spectacular experimentally verified prediction was the appearance of fractional charge. This means that even though the system is made up entirely of electrons, there are quasiparticles whose charge is a fraction of an electron. For example, in the simplest case, removing one electron from the system creates three identical localized 'quasiparticles', which necessarily have charge one third of the electron's.One motivation for studying exotic phases is to find a system capable of acting as a topological quantum computer. The general idea of quantum computation is to exploit peculiar properties of quantum mechanics to build a new sort of computer, able to solveproblems too large for a conventional computer. A quantum computer, for example, could factor much larger numbers than conventional ones, enabling commonly used encryption schemes to be cracked. Many obstacles stand in the way of actually building a quantum computer, the most substantial being that most approacheswould require devising a system capable of repeatedly producing results accurate on the order of one part in 100,000. While such precision is not inconceivable, the difficulty in doing so has resulted in a great deal of interest in topological quantum computation, which avoids this difficulty. The 'qubits' (or quantum bits, which are processed in the elementary steps of a computation) of a topological quantum computer are formed from particles with fractional charge.Of course, studying new problems often involves developing new theoretical tools. Fortunately some of these are at hand, using mathematics developed only in the last few years. This gives us new ways of describing the microscopic behaviour of many of these systems in term of their geometrical properties - from certain points of view they look like a tangled mass of spaghetti, or random curves. The precise characterisation of these random curves will give us access to the large scale properties of the exotic phases we wish to understand. There are several other powerful theoretical methods which can by brought to bear, in which the investigators and the visiting researcher, between them, have a great deal of experience.