Many-Body Theory of Nonlinear Responses in Topological Quantum Materials

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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.