The interplay between topology, geometry and correlations in novel materials

在这个项目中，我们将考虑各种强相关二维材料中的流体动力学电子传递。目标是推导出描述这些系统中电子运动的方程，并求解某些简单几何形状的方程。一种有趣的可能性是将有点“半经典”的流体力学描述与粒子的量子特性结合起来，粒子实际上具有离散自旋、伪自旋、谷和层自由度。它们通常与粒子的动能相耦合，并赋予电子非平凡的贝里相。反过来，当电子动量变化时（作为外部电位或碰撞的结果），粒子在正交方向上“倾斜”，产生可测量的效应。在扭曲二维材料中，强相关、流体动力输运和粒子的贝里相之间的相互作用是一个新的未知领域。在这个项目中，我们将：-推导电子在扭曲双层石墨烯中的运动方程，包括平面和上带，其中迁移率非常高，考虑到强相互作用，带重正化以及赝自旋/层/谷指数与动能之间的耦合；-在简单但实验相关的几何（通道，收缩等）中解决上述导出的方程，解决广泛的密度和温度范围-从费米液体到电子-空穴-等离子体状态；-理解贝里相如何影响碰撞积分，从而影响电子流体动力学方程中出现的粘度。因此，我们将研究“两体侧跳”现象的出现，并计算输运系数和弛豫时间；-将这些结果扩展到其他平带材料，如过渡金属二硫族化合物，其中电子具有自旋轨道耦合并描述了自旋电子学的新制度。在这个项目中，学生将获得二维材料，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/她将学习如何运用现代场理论方法（格林函数，微扰理论，Keldysh）来解决凝聚态物理中的多体输运问题。最后，他/她还将通过使用Mathematica/Python以数字方式解决一些问题，从而发展和提高计算技能。

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.