Functional renormalization group for fermions in three dimensions

这个理论项目的目标是设计和应用一个用于三维晶格中电子相互作用的泛函重整化群(FRG)方案。在过去的二十年里，FRG被用来探索许多零维、一维和二维模型系统中的相互作用效应，但由于数值限制，3D系统基本上超出了范围。FRG的独特优势是它对低尺度下的主要相互作用趋势是不带偏见的，而且流动方程的功能性质允许人们获得有效低能物理的相当详细的图景，例如对于有效相互作用。这些强度当然也有助于分析3D相互作用的电子系统。FRG的最新进展，最重要的是一种称为顶点分解的技术，使有效相互作用的描述变得更加有效，无论是在数值方面还是在物理上对其结构的理解都是如此。因此，可以寻址新的扩展。在这里，我们计划开发一般三维相互作用电子系统的顶点分解技术，并将它们应用于重要的模型案例。作为3D FRG方法的首次应用，我们设想研究线性和二次能带交叉点模型，这些模型最近被认为包含非费米液体行为、非传统电荷屏蔽和相互作用诱导的拓扑态。

The goal of this theory project is to devise and apply a functional renormalization group (fRG) scheme for interacting electrons in three-dimensional (3D) lattices. Over the last two decades, fRG has been used to explore interaction effects in a multitude of zero-, one- and two-dimensional model systems, but 3D systems were essentially out of range because of numerical limitations. Particular advantages of the fRG are its unbiasedness as to what the dominant interaction tendencies at low scales are, and that the functional nature of the flow equations allows one to obtain a rather detailed picture of the effective low-energy physics, e.g. for the effective interactions. These strengths would of course also be helpful in the analysis of 3D interacting electron systems. Recent advances in the fRG, most importantly a technique called vertex decomposition, have made the description of the effective interaction much more efficient, both regarding numerical aspects as well as for the physical understanding of its structure. Hence, new extensions can be addressed. Here, we plan to develop vertex decomposition techniques for general three-dimensional interacting electron systems and to apply them to important model cases. As a first application of the 3D fRG approach, we envisage the study of linear and quadratic band crossing point models that have recently been considered as hosting non-Fermi liquid behavior, unconventional charge screening and interaction-induced topological states.