Interacting Twisted Bilayer Graphene with Systematic Modeling of Structural Relaxation

TL;DR

This work tackles the challenge of incorporating structural relaxation in twisted bilayer graphene beyond simple tuning of interlayer hopping ratios. It develops a systematic continuum relaxation model by coupling linear elasticity to a stacking-energy penalty and embeds the resulting relaxation into a moiré-scale relaxed BM single-particle Hamiltonian, which is then projected to the flat bands to form an interacting many-body problem. HF and CCSD calculations at charge neutrality reveal quantitative differences in ground-state competition and symmetry-breaking behavior between the systematic relaxation model and the conventional relaxed BM/AA-AB-tuning approach, including earlier symmetry-breaking transitions tied to the relaxed structure. The findings underscore the importance of full relaxation physics for accurately predicting TBG’s many-body phase diagram and point to future extensions incorporating strain and spin. The methodology provides a robust framework for systematically assessing how lattice relaxation shapes correlated electronic states in moiré materials.

Abstract

Twisted bilayer graphene (TBG) has drawn significant interest due to recent experiments which show that TBG can exhibit strongly correlated behavior such as the superconducting and correlated insulator phases. Much of the theoretical work on TBG has been based on analysis of the Bistritzer-MacDonald model which includes a phenomenological parameter to account for lattice relaxation. In this work, we use a newly developed continuum model which systematically accounts for the effects of structural relaxation. In particular, we model structural relaxation by coupling linear elasticity to a stacking energy that penalizes disregistry. We compare the impact of the two relaxation models on the corresponding many-body model by defining an interacting model projected to the flat bands. We perform tests at charge neutrality at both the Hartree-Fock and Coupled Cluster Singles and Doubles (CCSD) level of theory and find the systematic relaxation model gives quantitative differences from the simplified relaxation model.

Figures (7)

• Figure 1: \ref{['fig:graphene']} Hexagonal graphene lattice, monolayer lattice vectors $\mathbf{a}_1, \mathbf{a}_2$, a unit cell (gray), and the relative shift between A and B sublattices $\bm \tau^B$. \ref{['fig:mBZ']} Brillouin zones of individual layers (red and blue), the Dirac points $\mathbf{K}_\ell, \mathbf{K}_\ell'$ and the two moiré Brillouin zones around $\mathbf{K}$ and $\mathbf{K}'$ valleys (green) with a set of moiré reciprocal lattice vectors $\mathbf{b}_1, \mathbf{b}_2$. We expand our wave function around $\tilde{\mathbf{K}}$. \ref{['fig:mom_hops']} The momentum hops around $\mathbf{K}_1$ and $\mathbf{K}_2$. The original BM model has only three interlayer momentum hops $\mathbf{s}_1$, $\mathbf{s}_2 = \mathbf{s}_1 + \mathbf{b}_1$ and $\mathbf{s}_3 = \mathbf{s}_1 - \mathbf{b}_2$. The relaxed BM model allows further interlayer hopping with momentum shifts $\mathbf{s}_1 + \mathcal{R}_\mathrm{m}^*$ (red), and also intralayer hopping with momentum shifts $\mathcal{R}_\mathrm{m}^*$ (blue). We keep the first three shells of interlayer hopping, and first two shells of intralayer hopping (12 hoppings each).
• Figure 2: \ref{['fig:disregistry']} Illustration of local stacking configuration of layer 2 (red) with respect to layer 1 (blue). The disregistry is a vector in the unit cell $\Gamma_1$ of layer 1. We also account for the sublattice shift when computing the disregistry, atoms in A sublattice are compared to A sublattice of the other layer. \ref{['fig:GSFE']} GSFE as a function of disregistry. It is minimized at AB configuration, and maximized at AA configuration.
• Figure 3: Original and relaxed atom configuration for TBG at $\theta = 1.5^\circ$. To generate the relaxed figure, the GSFE was scaled to be 50 times stronger to emphasize the triangular pattern after relaxation. Regions of AA stacking shrink in size, while AB and BA stacking regions grow.
• Figure 4: Single particle band structures for $\mathbf{K}$ valley of \ref{['fig:BM_bands']} BM model, \ref{['fig:chiral_bands']} Chiral Model, \ref{['fig:relaxed_BM_bands']} Relaxed BM model at $1.05^\circ$. The projected bands are highlighted in red. Note that the gap between the projected bands and the "remote bands" is small in the BM model \ref{['fig:BM_bands']}. To avoid these bands touching we consider $\kappa \leq .95$ in this work. This is justified since relaxation tends to decrease this ratio significantly; in our relaxed Bistritzer-MacDonald model we calculute this parameter as $\approx .7$, see Appendix \ref{['sec:relax-bm-inter']}.
• Figure 5: Energy of converged state per moiré site, energy difference to ground state (minimum of the five candidate states), and HOMO-LUMO gap with respect to interpolation parameter $\alpha$ for the BM model (\ref{['fig:bm_e']}, \ref{['fig:bm_e_diff']}, \ref{['fig:bm_gap']}) and for the relaxed BM model (\ref{['fig:relax_e']}, \ref{['fig:relax_e_diff']}, \ref{['fig:relax_gap']}). The gray line ($\kappa=0.7$) represents the physical ratio of relaxation in the BM model.

Theorems & Definitions (1)

• Remark 3.1