Mean-field Modelling of Moiré Materials: A User's Guide with Selected Applications to Twisted Bilayer Graphene

Abstract

We review the theoretical modelling of moiré materials, focusing on various aspects of magic-angle twisted bilayer graphene (MA-TBG) viewed through the lens of Hartree-Fock mean-field theory. We first provide an elementary introduction to the continuum modelling of moiré bandstructures, and explain how interactions are incorporated to study correlated states. We then discuss how to implement mean-field simulations of ground state structure and collective excitations in this setting. With this background established, we rationalize the power of mean-field approximations in MA-TBG, by discussing the idealized "chiral-flat" strong-coupling limit, in which ground states at electron densities commensurate with the moiré superlattice are exactly captured by mean-field ansätze. We then illustrate the phenomenological shortcomings of this limit, leading us naturally into a discussion of the intermediate-coupling incommensurate Kekulé spiral (IKS) order and its origins in ever-present heterostrain. IKS and its placement within an expanded Hartree-Fock manifold form our first "case study". Our second case study involves time-dependence, and focuses on the collective modes of various broken-symmetry insulators in MA-TBG. As a third and final case study, we return to the strong-coupling picture, which can be stabilized by aligning MA-TBG to an hBN substrate. In this limit, we show how mean field theory can be adapted to the translationally non-invariant setting in order to quantitatively study the energetics of domain walls in orbital Chern insulating states. We close with a discussion of extensions and further applications. Used either as a standalone reference or alongside the accompanying open-source code, this review should enable readers with a basic knowledge of band theory and many-body physics to systematically build and analyze detailed models of generic moiré systems.

Figures (9)

• Figure 1: a) Real-space schematic of TBG with $\theta=5^\circ$ and zero interlayer shift $\bm{d}=\bm{0}$. The symmetry elements $\hat{C}_{6z}$ and $\hat{C}_{2x}$ are indicated. b) Extended zone scheme of the mBZ in each valley superposed on the rotated monolayer BZ's (red and blue). Right circle zooms into the mBZ in valley $K$, showing the high-symmetry momenta and the moiré reciprocal lattice vectors $\bm{b}_{1,2}$.
• Figure 2: Rigidly twisted bilayer graphene forms regions with local AA, AB, and BA stacking configurations, named for the placement of sites in one layer relative to the other layer. Orange (blue) dots denote carbon atoms on layer 1 (2). Lattice relaxation effects cause the AB and BA regions to expand at the expense of AA regions.
• Figure 3: Band structure of the BM model along a cut in the mBZ for different values of the chiral ratio $\kappa=w_{\text{AA}}/w_{\text{AB}}$. The twist angle is held constant at the magic angle appropriate for the chiral limit $\kappa=0$ ($\theta\simeq 1.06^\circ$). Black (red) lines denote valley $K$ ($K'$).
• Figure 4: The Hamiltonian $\hat{H}_{U_\text{S}}$ satisfies a $U(4)_{C=1}\times U(4)_{C=-1}$ symmetry corresponding to arbitrary rotations between bands in the same Chern sector.
• Figure 5: The real-space structure of an IKS with $\bm{q}_{\text{IVC}} = -\bm{G}_1/3$. The main plot shows charge density in color (dark spots correspond to AA regions) and complex IVC order parameter $\sim\braket{\tau_x\sigma_x} + i\braket{\tau_y\sigma_x}$ in arrows. Bloch spheres schematically illustrate the different IVC angles across three AA regions. Insets show the graphene scale charge patterns at three different AA regions. Blue (red) dots correspond to positive (negative) values of $\braket{\hat{c}^\dagger_A\hat{c}^{\space}_B} + \braket{\hat{c}^\dagger_B\hat{c}^{\space}_A}$, with the cross marking the center of each AA region. Adapted with permission from Kwan et al., Phys. Rev. X. 11, 041063 (2021) Kwan2021IKS. Copyright (2021) American Physical Society.