Many-Body Theory of Nonlinear Responses in Topological Quantum Materials

量子材料表现出独特的性质，源于量子力学、拓扑学和强多体相互作用的相互作用。例子包括拓扑绝缘体、狄拉克和韦尔半金属、超导体和相关绝缘体。二维量子材料，如石墨烯、过渡金属二卤化物(如MoS_2和WTe_2)以及它们的异质结构，由于其原子薄的结构和高度的可调性和灵活性，在纳米技术应用方面显示出巨大的潜力。尽管如此，非线性响应函数在揭示材料属性中隐藏的对称性和拓扑特征方面提供了更大的通用性。非线性光谱学，包括光电流效应、高次谐波产生和拉曼光谱，研究材料如何响应强烈的外场，如激光、温度梯度和压力。在过去的十年里，在揭示非线性光学效应的拓扑相关性方面已经取得了相当大的进展，例如拓扑光电流的产生，但是粒子间相互作用对二维量子材料中非线性光学响应函数的影响在很大程度上还没有被探索。这一博士项目旨在填补这一空白。该博士项目的主要目标是发展拓扑量子材料中非线性响应的多体理论，并研究这些系统的非线性光学和光诱导操纵。我们计划建立引人注目的多体框架和数值模拟来计算非线性光学效应，如高次谐波产生和光电流。特别是，我们将研究以振动量子声子为特征的非线性光学和晶格动力学的相互作用。我们将为非线性光学和声子学领域中的几个关键和仍然悬而未决的问题寻找系统的答案，这些问题包括：像声子和等离子体这样的集体模如何在2D材料中调节非线性光学？这些材料中的光致非线性电流与电子拓扑结构之间是什么关系？晶格扭曲和应变如何影响2D材料中的非线性光学和输运？为了解决这些问题，我们将使用有效的晶格和连续介质模型来表征系统中电子和声子的动态行为。我们的方法将利用半经典输运理论、量子场论和动力学理论来计算非线性响应函数。前两年的暂定计划时间表如下：在第一年，我们深入研究拓扑2D材料中的非线性光电流和力，然后探索电子-声子相互作用对二阶光学的影响。第二年将我们的研究扩展到声子介导的三阶光学，以及双层系统中光感生拉曼力的有趣效应。这是一个暂定的时间表，根据领域的演变，我们可能会重新安排不同任务的优先级。出于培训的目的，我们可能会在第一个月从理想化系统中的基本非线性响应问题开始，然后继续到项目的实际目标。

Quantum materials exhibit distinct properties arising from the interplay of quantum mechanics, topology, and strong many-body interactions. Examples include topological insulators, Dirac and Weyl semimetals, superconductors, and correlated insulators. Two-dimensional quantum materials, such as graphene, transition metal dichalcogenides (e.g., MoS2 and WTe2), and their heterostructures, demonstrate significant potential for nanotechnology applications owing to their atomically thin structures and high degree of tunability and flexibility.Linear response measurements, like electrical conductance and dielectric function, have significantly advanced our understanding of materials under weak perturbations. Nonetheless, nonlinear response functions provide an even greater versatility in revealing hidden symmetries and topological features within material properties. Nonlinear spectroscopy, including photogalvanic effects, high harmonic generation, and Raman spectroscopy, studies how materials respond to intense external fields like lasers, temperature gradient, and pressure. It helps uncover emergent orders and collective excitations in the system.During the last decade, there have been considerable progress in revealing the topological relevance of nonlinear optical effects, such as topological photocurrent generation, however the impact of inter-particle interaction on nonlinear optical response functions in 2D quantum materials is largely unexplored. This PhD project aims to fill this gap.The primary objective of this Ph.D. project is to develop many-body theories for nonlinear responses in topological quantum materials and study nonlinear optics and light-induced manipulation of these systems. We plan to build compelling many-body frameworks and numerical simulations to compute nonlinear optical effects such as high harmonic generations and photocurrents. In particular, we will investigate the interplay of nonlinear optics and the crystalline lattice dynamics, characterised by the vibration quanta phonon. We will search for systematic answers to several crucial and still open questions in the status of nonlinear optics and phononics fields that include: How do collective modes like phonons and plasmons mediate nonlinear optics in 2D materials? What is the relationship between light-induced nonlinear currents and the electronic topology in these materials? How does lattice distortion and strain affect nonlinear optics and transport in 2D materials?To address these questions, we will employ effective lattice and continuum models to characterise the dynamic behaviour of electrons and phonons within the system. Our approach will leverage semiclassical transport theory, quantum field theory, and kinetic theory to compute nonlinear response functions.The tentative project timeline for the first two years follows: In Year 1, we delve into nonlinear photocurrent and force in topological 2D materials, and then explore the impact of electron-phonon interactions on second-order optics. Year 2 extends our investigation to phonon-mediated third-order optics, along with the intriguing effects of light-induced Raman forces in bilayer systems. This is a tentative timeline, and depending on the evolution of the field, we may rearrange the priority of different tasks. For training purposes, we may begin with basic nonlinear response problems in idealized systems during the first month, and then progress to the actual objectives of the project.