Many-Body Theory of Nonlinear Responses in Topological Quantum Materials

量子材料表现出独特的性质，这些性质来自量子力学、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、狄拉克和外尔半金属、超导体和相关绝缘体。二维量子材料，例如石墨烯、过渡金属二硫属化物（例如，MoS 2和WTe 2）及其异质结构由于其原子薄的结构和高度的可调谐性和灵活性而在纳米技术应用中表现出巨大的潜力。线性响应测量，如电导和介电函数，显著地推进了我们对材料在弱扰动下的理解。尽管如此，非线性响应函数在揭示材料特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，包括光电效应，高次谐波产生和拉曼光谱学，研究材料如何响应强外部场，如激光，温度梯度和压力。在过去的十年里，人们在揭示非线性光学效应的拓扑相关性方面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还很大程度上没有被探索。本博士项目旨在填补这一空白。项目是发展拓扑量子材料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，我们将研究非线性光学和晶格动力学的相互作用，其特征在于振动量子声子。我们将寻找系统的答案，在非线性光学和声子领域的地位，其中包括几个关键的和仍然悬而未决的问题：集体模式，如声子和等离子体激元如何调解非线性光学在2D材料？这些材料中光致非线性电流与电子拓扑结构之间的关系是什么？晶格畸变和应变如何影响二维材料中的非线性光学和输运？为了解决这些问题，我们将采用有效的晶格和连续介质模型来模拟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。前两年的暂定项目时间轴如下：在第一年，我们深入研究拓扑二维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，沿着双层系统中光诱导拉曼力的有趣效应。这是一个暂定的时间轴，根据领域的发展，我们可能会重新安排不同任务的优先级。为了训练的目的，我们可以在第一个月从理想系统的基本非线性响应问题开始开始，然后逐步达到项目的实际目标。

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.