2025: Geometry and Fourier transform: Kekule patterns and intervalley orders (with Dmytro Oriekhov)

Project motivation: A number of recent experimental results in bilayer / trilayer graphene suggest that the so-called "intervalley" orders help to achieve superconductivity. Intervalley is a term that describes ground state property in momentum space. But how does it look like in real space? Imaging such patterns, and expesially understand how the new patterns look in real space is a challenging task. Solution could help experimentalist recognize what is the origin of superconductivity in particular material.

Motivating image - Ref.[1] experiment, Kekule pattern in graphene:

Project goal: Understand how simple and complicated functions in momentum space - f(k) create patterns on a real space lattice. And how making Fourier transform of image could help to identify the structure of physical state.

Short example: Let's solve example problem - we have a function of f(k)*g(k+D). How does the Fourier transform of this function looks like? And if we take discrete Fourier?

Requirements: Fourier transform, calculus. This project is good for 1,2,3-course students. Programming skills might be helpful if we decide to play with images.

[1] Imaging inter-valley coherent order in magic-angle twisted trilayer graphene, Hyunjin Kim, et.al.  Nature 623, 942 (2023) https://doi.org/10.1038/s41586-023-06663-8