Tianyu Kong

Tianyu Kong

William H. Kruskal Instructor

University of ChicagoJones 316

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Bistritzer-MacDonald dynamics in twisted bilayer graphene

Journal of Mathematical Physics, 2023

The Bistritzer-MacDonald (BM) model attempts to capture the electronic properties of twisted bilayer graphene (TBG), even at incommensurate twist angles, by an effective periodic model over the bilayer moiré pattern. Starting from a tight-binding model, we identify a regime where the BM model emerges as the effective dynamics for electrons modeled as wave-packets spectrally concentrated at the monolayer Dirac points, up to error that can be rigorously estimated. Using measured values of relevant physical constants, we argue that this regime is realized in TBG at the first “magic” angle.

Modeling of electronic dynamics in twisted bilayer graphene

SIAM Journal on Applied Mathematics, 2024

We consider the problem of numerically computing the quantum dynamics of an electron in twisted bilayer graphene. The challenge is that atomic-scale models of the dynamics are aperiodic for generic twist angles because of the incommensurability of the layers. The Bistritzer-MacDonald PDE model, which is periodic with respect to the bilayer’s moiré pattern, has recently been shown to rigorously describe these dynamics in a parameter regime. In this work, we first prove that the dynamics of the tight-binding model of incommensurate twisted bilayer graphene can be approximated by computations on finite domains. The main ingredient of this proof is a speed of propagation estimate proved using Combes-Thomas estimates. We then provide extensive numerical computations which clarify the range of validity of the Bistritzer-MacDonald model.

Interacting twisted bilayer graphene with systematic modeling of structural relaxation

Electronic Structure, 2025

We introduce and compute solutions of a many-body model of the electronic properties of twisted bilayer graphene which systematically accounts for the effects of structural relaxation. We model mechanical relaxation by coupling linear elasticity to a stacking energy that penalizes disregistry. Minimizers of the resulting functional are then input into a tight-binding model of twisted bilayer graphene, from which a single-particle continuum moiré-scale (Bistritzer-MacDonald-like) model is systematically derived. We then project this model together with a Coulomb electron-electron interaction term into the single-particle model’s flat moiré bands. We numerically compute Hartree-Fock ground states of this model, comparing the relative energies of competing many-body ground states.

Higher-order continuum models for twisted bilayer graphene

Journal of Mathematical Physics, 2025

The first-order continuum PDE model proposed by Bistritzer and MacDonald accurately describes the single-particle electronic properties of twisted bilayer graphene (TBG) at small twist angles. In this paper, we obtain higher-order corrections to the Bistritzer-MacDonald model via a systematic multiple-scales expansion. We prove that the solution of the resulting higher-order PDE model accurately approximates the corresponding tight-binding wave function under a natural choice of parameters and given initial conditions that are spectrally localized to the monolayer Dirac points. Numerical simulations of tight-binding and continuum dynamics demonstrate the validity of the higher-order continuum model. Symmetries of the higher-order models are also discussed. This work extends the analysis of Watson et al., which rigorously established the validity of the (first-order) BM model.

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