Disorder and Topology: Anderson Transitions

根据传统的理论，在Anderson转变(即无序电子系统的定域和离域能量本征态之间的量子相变)的临界行为是由单参数标度假设来捕捉的。这一假设最早是在Abrahams等人的一篇著名论文中提出的，它得到了Wegner和Efetov的非线性sigma模型的弱耦合场论计算(在d=2的较低临界维度附近进行)的精彩证实。现在，虽然Anderson相变在两维以上的弱耦合区域得到了很好的控制，但有越来越多的分析和数值证据表明，传统的图景需要在高维情况下进行修改，更广泛地说，在强耦合情况下(即对于电导量子量级的临界电导)，Anderson相变需要修改。新出现的修正相当惊人：超越了单参数标度的范式，Altshuler、Kravtsov和合作者(独立地，本文作者)认为应该存在与两个已知相(即金属和安德森绝缘体)不同的第三个稳定相。此外，在我们最近的工作中，我们已经指出，第三相被三个相互关联的特征所分开：(I)分形本征态，(Ii)(无限体积中无序电子哈密顿量的)奇异连续能谱，(Iii)无序电子系统场论表述中的部分对称性破缺(在相和相变的朗道理论意义下)的新场景。本项目的首要目标是开发一种可行的场论方法，能够以受控的方式处理强耦合安德森跃迁(和相关问题)。在短期内，我们的具体目标是(I)推导高维Anderson跃迁的双参数场论(取代非线性Sigma模型)，(Ii)发展整数量子霍尔跃迁的完全标度理论(包括相关和无关微扰)，(Iii)建立描述自旋量子霍尔跃迁标度极限的共形场论。C类)，以及(Iv)解释表面态的全光谱量子临界性，这已被预测用于对称类AIII的拓扑绝缘体。我们的远大目标是将分析理论发展到如此成熟的阶段，以至于现有的谜题和争议都可以通过进一步的数值工作来解决(例如：Anderson相变是否违反共形对称原理？)，“在树状图的情况下，使用双参数缩放的正确方法是什么？”

According to traditional theory, the critical behavior at the Anderson transition (i.e. the quantum phase transition between localized and delocalized energy eigenstates of a disordered electron system) is captured by a one-parameter scaling hypothesis. First put forward in a celebrated paper by Abrahams et al., this hypothesis received brilliant confirmation from weak-coupling field-theory computations (carried out near the lower critical dimension of d=2) for the nonlinear sigma model of Wegner and Efetov. Now, while the Anderson transition is under good control in the weak-coupling regime just above two dimensions, there exists mounting analytical and numerical evidence that the traditional picture needs modification in high dimension and, more generally, in situations where the Anderson transition occurs at strong coupling (i.e. for a critical conductance of the order of the conductance quantum). The emerging modification is rather striking: going beyond the paradigm of one-parameter scaling, Altshuler, Kravtsov and collaborators (and independently, the present author) have argued that there should exist a third stable phase distinct from the two known phases (namely, the metal and the Anderson insulator). Moreover, in our recent work we have pointed out that the third phase is set apart by three characteristic features which are inter-related: (i) fractal energy eigenstates, (ii) singular continuous energy spectrum (of the disordered electron Hamiltonian in infinite volume), and (iii) a novel scenario of partial symmetry breaking (in the sense of the Landau theory of phases and phase transitions) in the field-theory formulation of the disordered electron system. The overarching objective of the present project is to develop a viable field-theory approach that is able to handle strong-coupling Anderson transitions (and related problems) in a controlled way. In the short term, our concrete goals are (i) to derive a two-parameter field theory (superseding the nonlinear sigma model) for Anderson transitions in high dimension, (ii) to develop the complete scaling theory (including relevant and irrelevant perturbations) of the integer quantum Hall transition, (iii) to formulate the conformal field theory describing the scaling limit of the spin quantum Hall transition (a.k.a. class C), and (iv) to explain the spectrum-wide quantum criticality of surface states, which has been predicted for topological insulators of symmetry class AIII. Our far goal is to develop the analytical theory to such a mature stage that the existing puzzles and controversies ("Do Anderson transitions violate the principle of conformal symmetry?", "What is the correct ansatz to use with two-parameter scaling in the case of tree-like graphs?", to name a few) can all be decided by further numerical work.