The interplay between topology, geometry and correlations in novel materials

在这个项目中，我们将考虑各种强关联的二维材料中的流体动力学电子输运。我们的目标是推导出描述这些系统中电子运动的方程，并针对某些简单的几何形状进行求解。一种耐人寻味的可能性是，将有点“半经典”的流体动力学描述与粒子的量子性质结合起来，粒子的量子性质实际上具有离散的自旋、伪自旋、山谷和层状自由度。一般来说，这些都与粒子动量相耦合，并赋予电子非平凡的贝利相。反过来，当电子动量变化时(由于外加电势或碰撞的结果)，粒子在垂直方向上“扭曲”，产生可测量的效应。扭曲二维材料中粒子的强关联、流体动力学输运和Berry相之间的相互作用是一个新的和未被探索的领域。在这个项目中，我们将：-推导电子在扭曲的双层石墨烯中的运动方程，在平带和上面带，其中迁移率非常高，考虑了强相互作用、带重整化以及伪自旋/层/谷指数与动量的耦合的作用；-用简单但实验上相关的几何(通道、收缩等)求解上述方程，解决从费米液体到电子-空穴-等离子体区域的广泛的密度和温度范围；-了解Berry相如何影响碰撞积分，从而影响出现在电子流体动力学方程中的粘性。因此，我们将研究“两体侧面跳跃”现象的出现，并计算输运系数和弛豫时间；-将这些结果推广到其他平带材料，如过渡金属二卤化物，其中电子具有自旋-轨道耦合，并描述自旋电子学的新机制。在这个项目中，学生将获得2D材料、现代输运理论和流体力学的知识，这些都是目前的热门研究课题。此外，他/她将学习如何使用现代场论方法(格林函数、微扰理论、凯尔德什)来解决凝聚态物理中的多体输运问题。最后，他/她还将发展和/或增强计算技能，使用数学/Python来数值解决一些问题。

In this project we will consider hydrodynamic electron transport in various strongly-correlated 2Dmaterials. The goal is to derive equations describing the motion of electrons in these systems andsolve them for certain simple geometries. An intriguing possibility is to combine the somewhat"semiclassical" hydrodynamic description with the quantum nature of particles, which are in factcharacterized by discrete spin, pseudospin, valley, layer degrees of freedom. These are, quitegenerally, coupled to the particle kinetic momentum and endow electrons with non-trivial Berryphases. In turn, when the electron momentum varies (as a consequence of external potentials orcollisions), particles "skew" in the orthogonal direction, producing measurable effects. The interplaybetween strong correlations, hydrodynamic transport and the Berry phases of particles in twisted 2Dmaterials is a new and unexplored field.In this project we will:- derive equations of motions for electron in twisted bilayer graphene, both in flat and upper bands,where mobility is very high, accounting for the role of strong interactions, band renormalization,and the coupling between pseudospin/layer/valley indices with kinetic momentum;- solve the above-derived equations in simple but experimentally relevant geometries (channels,constrictions, etc.), addressing a wide range of densities and temperatures - from Fermi liquid toelectron-hole-plasma regime;- understand how Berry phases impact on the collision integral and therefore on viscositiesappearing in electrons' hydrodynamic equations. We will therefore study the emergence of "twobodyside jump" phenomena and calculate transport coefficients and relaxation times;- extend these results to other flat-band materials, such as transition-metal dichalcogenides, whereelectrons feature spin-orbit coupling and describe a new regime for spintronics.During this project the student will gain a knowledge of 2D materials, modern theory of transport andhydrodynamics, all of them hot research topics at the moment. Moreover, he/she will learn how toemploy modern field theoretical methods (Green functions, perturbation theory, Keldysh) to tacklemany-body transport problems in condensed matter physics. Finally, he/she will also develop and/orenhance computational skills by using Mathematica/Python to solve some of the problem numerically.