量子霍尔跃迁：基于共形限制的精确理论
Quantum Hall transitions: An exact theory based on conformal restriction
摘要：
我们重新讨论了整数量子霍尔效应中的平台跃迁问题。在这里，我们开发了一个分析方法，这种转变的基础上，共形限制的理论。这是一个数学理论，最近开发的背景下的施拉姆-Loewner演化，描述了随机几何的分形曲线和其他随机几何分形对象在2D空间。阐明与平台过渡的联系的观测值包括在样品边界上的点之间的所谓的点接触电导（PCC），在Chalker-Coddington网络模型的语言中描述。我们发现，无序平均PCC的特点是经典概率的某些几何对象在平面（图片），发生积极的统计权重，满足至关重要的限制属性的变化与吸收边界的样品的形状。在结合这种限制属性与预期的共形不变性在过渡点，我们采用共形限制措施的数学理论相关联的无序平均PCC的共形场论（中心荷$c=0$）的初级运营商的关联函数。我们展示了如何可以用来计算这些功能在一些几何形状与各种边界条件。由于我们的结果只采用共形约束性质，它们同样适用于其他一些临界无序电子系统的二维。对于这些系统中的大多数，我们还预测了与各种无序平均PCC的空间行为相关的临界指数的精确值。
英文摘要：
We revisit the problem of the plateau transition in the integer quantum Hall effect. Here we develop an analytical approach for this transition, based on the theory of conformal restriction. This is a mathematical theory that was recently developed within the context of the Schramm-Loewner evolution which describes the stochastic geometry of fractal curves and other stochastic geometrical fractal objects in 2D space. Observables elucidating the connection with the plateau transition include the so-called point-contact conductances (PCCs) between points on the boundary of the sample, described within the language of the Chalker-Coddington network model. We show that the disorder-averaged PCCs are characterized by classical probabilities for certain geometric objects in the plane (pictures), occurring with positive statistical weights, that satisfy the crucial restriction property with respect to changes in the shape of the sample with absorbing boundaries. Upon combining this restriction property with the expected conformal invariance at the transition point, we employ the mathematical theory of conformal restriction measures to relate the disorder-averaged PCCs to correlation functions of primary operators in a conformal field theory (of central charge $c=0$). We show how this can be used to calculate these functions in a number of geometries with various boundary conditions. Since our results employ only the conformal restriction property, they are equally applicable to a number of other critical disordered electronic systems in 2D. For most of these systems, we also predict exact values of critical exponents related to the spatial behavior of various disorder-averaged PCCs.
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