진학프로석·박사 채용 접수 플랫폼

Mathematics of Moiré materials - Density Functional Theory and beyond

Type of contract : 12-month research contract (full time, no teaching), renewable once

Monthly gross salary: from 2700 euros, depending on experience

Expected starting date: June 1st (negociable)

Application process: please send a CV and a cover letter, contact information of at least two academics who can provide reference letters upon request, and examples of past projects to eric.cances@enpc.fr and david.gontier@enpc.fr with "MMM postdoc application'' as a topic by March 31st, 2026. Candidate selection and interviews might be conducted as soon as the application is received, and might last until April 30th, 2026.

Research topic

The study of multilayer 2D materials is one of the hottest topics in contemporary condensed matter physics. Shortly after the experimental exfoliation of monolayer graphene (2010 Nobel Prize in Physics), and other monolayer 2D materials, a natural idea arose: stack a few monolayers of same or different 2D materials in order to obtain new materials with hopefully new properties.

Twisting produces moiré patterns with diameter scaling as $\theta^{-1}$ for small twist angle $\theta$ and gives rise to new mesoscale phenomena. This was shown first theoretically, then experimentally on the case of twisted bilayer graphene (TBG) at the so-called ``magic'' angle ~1.1° (2020 Wolff Prize in Physics).

Density-Functional Theory - DFT - (1998 Nobel Prize in Chemistry) is currently the most popular model to compute the electronic structure of molecules and materials. For a molecular systems with $N$ electrons, the DFT equations read
$$
\left( -\frac 12\Delta + V_{\rm nuc} + V_{Hxc}[\rho] \right)\varphi_i=\epsilon_i\varphi_i, \quad \int_{\mathbb R^3} \varphi_i\varphi_j=\delta_{ij}, \quad \rho(x)=2\sum_{i=1}^N |\varphi_i(x)|^2,
$$
where $V_{\rm nuc}$ is the Coulomb potential generated by the nuclei and $V_{\rm Hxc}[\rho]$ a universal functional of the electronic density $\rho$.

However, applying DFT to moiré materials raises a number of theoretical and practical challenges:

• • Moiré materials are typically modeled as infinite systems which, in most cases (for instance, for almost all twist angles in TBG), are not periodic. While DFT models are mathematically well defined for periodic infinite systems, they are far from being fully understood in the aperiodic setting
• • DFT performs very well for weakly correlated systems but fails in the strongly correlated regime. Some moiré materials are weakly correlated, whereas others - including TBG at the magic angle - are strongly correlated. In such cases, one must rely on models that are based on DFT but go beyond it, such as the quantum embedding framework DFT+DMFT. The mathematical theory of quantum embedding methods is still in its infancy, and this area is a rich source of challenging mathematical and numerical questions.
• • From a practical standpoint, aperiodic moiré structures are approximated by periodic ones. Yet, in the most relevant parameter regimes (e.g. small twist angles in TBG), the resulting supercell has a very large period and contains too many atoms for a direct application of DFT, let alone DFT+DMFT. In this situation, effective models at the moiré scale must be employed. Formal derivations of non-interacting moiré-scale models from DFT have been proposed, and a rigorous mathematical analysis of this approach is currently under way. By contrast, the construction of interacting effective models, which is essential to capture subtle phenomena such as unconventional superconductivity, is a delicate model-reduction problem that has not yet been investigated from a mathematical perspective.

The goal of the CERMICS research group is to advance the development of models for computing the electronic properties of moiré materials, along with their mathematical analysis and the design of numerical methods for their simulation. Research directions include:

• • Constructing DFT models for aperiodic structures, analyzing them mathematically, and developing numerical methods for their simulation;
• • Developing mathematically rigorous methods to derive interacting effective models at the moiré scale from DFT, computing their parameters numerically from DFT calculations, and simulating the resulting models, using e.g. quantum embedding methods.

The precise focus and balance between theory and numerics will be tailored to the background and interests of the selected candidate.

Scientific environment: the research program will be carried out in the framework of the Moiré Materials Magic project funded by the Simons Foundation. The postdoctoral researcher will be based at CERMICS, the applied mathematics department of Ecole des Ponts - Institut Polytechnique de Paris. He/she will interact on a weekly basis with senior and junior mathematicians, as well as physicists working on moiré materials.

Keywords: linear and nonlinear partial differential equations, self-adjoint operators, variational methods, numerical analysis, numerical methods, quantum mechanics, condensed-matter physics, Julia programming language