Disorder and Topology: Anderson Transitions

根据传统理论，安德森跃迁的临界行为（即无序电子系统的局域和非局域能量本征态之间的量子相变）是用单参数标度假设捕获的。这一假设首先由Abrahams等人在一篇著名的论文中提出，并在Wegner和Efetov的非线性sigma模型的弱耦合场论计算（在d=2的下临界维附近进行）中得到了出色的证实。现在，虽然安德森跃迁在二维以上的弱耦合状态下得到了很好的控制，但有越来越多的分析和数值证据表明，在高维，更一般地说，在安德森跃迁发生在强耦合的情况下（即对于电导量子阶的临界电导），传统的图像需要修改。新出现的修改是相当惊人的：超越了单参数缩放的范式，Altshuler， Kravtsov和合作者（以及独立的，本作者）认为应该存在第三个稳定相，不同于两个已知相（即，金属和安德森绝缘体）。此外，在我们最近的工作中，我们已经指出，第三阶段是由三个相互关联的特征特征分开的：(i)分形能量特征态，（ii）奇异连续能谱（无限体积中的无序电子哈密顿量），以及（iii）在无序电子系统的场理论公式中部分对称性破缺的新场景（在朗道相和相变理论的意义上）。本项目的总体目标是开发一种可行的场理论方法，能够以可控的方式处理强耦合安德森转换（及相关问题）。在短期内，我们的具体目标是：(i)推导出高维Anderson跃迁的双参数场理论（取代非线性sigma模型），（ii）发展整数量子霍尔跃迁的完整缩放理论（包括相关和不相关的扰动），（iii）制定描述自旋量子霍尔跃迁（又称C类）缩放极限的共形场理论，（iv）解释表面态的全谱量子临界性，这是对对称类AIII的拓扑绝缘体的预测。我们的长远目标是将解析理论发展到如此成熟的阶段，以至于现有的难题和争议（“安德森转换是否违反保形对称原则？”，“在树状图的情况下，使用双参数缩放的正确ansatz是什么？”，仅举几例）都可以通过进一步的数值工作来决定。

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.