Correlations and impurities in topological insulators

拓扑绝缘体与普通绝缘材料的根本不同之处在于，它们的特点是本体上有绝缘间隙，但表现出坚固的金属表面状态。后者可以归因于体波函数的拓扑性质，并且可以在实验上观察到。大多数描述拓扑绝缘体的理论方法忽略了电子之间的库仑相互作用。将这些项添加到具有自旋-轨道耦合的石墨烯的紧束缚模型中-这是该领域的一种范例，因为它描述了拓扑顺序的最简单实现-产生了两带Hubbard模型。然而，众所周知，Hubbard模型显示了从有序磁性类型到Mott绝缘行为的各种强关联相。蜂窝几何已经在一些工作中得到了解决，但关于不同(常规)类型的有序如何从拓扑阶段(在没有库仑相互作用的情况下持续存在)演化出来的完整图景还没有。作为这个提议的第一部分，我想用一个框架来研究这个问题，这个框架为Hubbard模型的各种不稳定性--泛函重整化群--提供了一种公正的方法。正常金属的低能物理被极大地改变了，即使是添加了一些磁性杂质。由于近藤效应，单个自旋自由度可以完全被传导电子屏蔽。拓扑绝缘体的表面呈现螺旋状金属，但近藤效应是否持续还不明显。我想使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--来研究拓扑绝缘体顶部的杂质问题。此外，我还想研究量子点结构中两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.