Correlations and impurities in topological insulators

拓扑绝缘体从根本上不同于普通的绝缘材料，因为它们的特点是在体中有绝缘间隙，但表现出坚固的金属表面状态。后者可归因于体波函数的拓扑性质，并可在实验中观察到。大多数描述拓扑绝缘体的理论方法都忽略了电子之间的库仑相互作用。将这些术语添加到具有自旋轨道耦合的石墨烯紧密结合模型中——这是该领域的一个范例，因为它描述了拓扑秩序的最简单实现——产生了一个双波段哈伯德模型。然而，众所周知，哈伯德模型表现出各种强相关的相位，从有序的磁性类型到莫特绝缘行为。蜂巢的几何结构在一些作品中得到了解决，但是关于不同（传统）种类的秩序是如何从拓扑阶段（在没有库仑相互作用的情况下持续存在）进化出来的完整图景仍然缺失。作为本提案的第一部分，我想使用一个框架来研究这个问题，该框架为哈伯德模型的各种不稳定性提供了一种无偏的方法——功能重整化群。普通金属的低能物理特性会因加入少量磁性杂质而发生剧烈变化。由于近藤效应，单个自旋自由度可以被传导电子完全屏蔽。拓扑绝缘体表面呈现螺旋形金属，但近藤效应是否持续存在并不明显。我想用两个互补的重整化群框架——泛函重整化群和实时重整化群来研究拓扑绝缘子上的杂质问题。此外，我想研究量子点设置中双端输运几何的实时和稳态动力学。

Topological insulators are fundamentally different from ordinary insulating materials in that they feature an insulating gap in the bulk but exhibit robust metallic surface states. The latter can be attributed to topological properties of the bulk wave functions and can be observed experimentally. Most theoretical approaches to describe topological insulators ignore the Coulomb interactions between electrons. Adding such terms to the tight-binding model of graphene with spin-orbit couplings -- which is a paradigm in the field as it describes the most simple realization of topological order -- yields a two-band Hubbard model. Hubbard models, however, are well-know to exhibit a variety of strongly-correlated phases ranging from magnetic types of order to Mott insulating behavior. The honeycomb geometry was addressed in a few works, but a complete picture of how the different (conventional) kinds of order evolve out of the topological phase (persisting in absence of Coulomb interactions) is yet missing. As the first part of this proposal, I want to study this problem using a framework which provides an unbiased approach to the various instabilities of the Hubbard model -- the functional renormalization group.The low-energy physics of a normal metal is drastically modified by the addition of even a few magnetic impurities. A single spin degree of freedom can be screened completely by the conduction electrons by virtue of the Kondo effect. The surface of topological insulators represents a helical metal, but it is not obvious whether the Kondo effect persists. I want to study this problem of impurities on top of topological insulators using two complementary renormalization group frameworks - the functional and real-time renormalization group. Moreover, I want to investigate the real-time and steady state dynamics of a two-terminal transport geometry in a quantum dot setup.