A Numerical Perspective on Moiré Superlattices: From Single-Particle Properties to Many-Body Physics

TL;DR

This work presents a practical numerical workflow for moiré materials that starts from lattice-relaxation–informed continuum models and proceeds through Hartree-Fock, MBPT, and exact diagonalization to study correlated and topological states. It explicitly addresses subtleties including remote-band renormalization, inhomogeneous and dynamical screening, double counting, and multiband effects to ensure quantitative reliability. HF captures symmetry-breaking ground states; GW and RPA improve quasiparticle energies and total energies; ED probes fractional Chern insulators, with all-band HF required for convergence. Applied to representative moiré systems (e.g., R5G-hBN, twisted TMDs), the workflow yields results in quantitative agreement with experiments and offers a bridge between theory and experiment for predicting correlated phenomena in moiré materials.

Abstract

Moiré superlattices in two-dimensional materials provide a versatile platform to explore strongly correlated and topological phases. This work presents a practical theoretical workflow for studying the correlated and topological states in moiré systems, combining continuum modeling, Hartree-Fock mean-field approximations, many-body perturbation theory, and exact diagonalizations. We focus on the numerical implementation of these methods, emphasizing subtleties such as remote band effects, inhomogeneous and dynamical screening, double counting problem, etc., which are often swept under the rug in theoretical works. The workflow enables a systematic investigation of symmetry-breaking ground state properties, quasiparticle excitation properties and fractional Chern insulator phases emerging from moiré superlattices, providing insights that are directly relevant to experimental observations. By bridging technical details and physical interpretations, this work aims to guide both theorists and experimentalists in understanding and predicting correlated phenomena in moiré materials.

Figures (7)

• Figure 1: Workflow to study the non-interacting and interacting effects in moiré systems. The large shaded rectangular frames with bold title nearby denote different computational levels, which corresponds one-to-one the sections in the present work. Small rectangular boxes in each frame denote the input and output data for methods, which are represented by elongated hexagonal boxes. Arrows indicate data dependencies.
• Figure 2: (a) The lattice structure and reciprocal lattice of a moiré superlattice. Moiré superlattice system consists of several layered materials. The lattice mismatch gives rise to an enlarged moiré supercell, preserving an additional translational symmetry, and the corresponding mini Brillouin zone. (b) The real space distribution of the lattice relaxation pattern of twisted bilayer graphene with twist angle $\theta\!=\!1.05^{\circ}$. $\textbf{u}^{-}$ represents the relative in-plane distortion. The colorbar encodes the amplitudes of the distortion fields and the vectors show the directions of the distortion fields. $h^{-}$ represents the out-of-plane relative distortions (the interlayer distance), with the colorbar encoding the amplitudes. (c) The band structure of twisted bilayer graphene with twist angle $\theta\!=\!1.05^{\circ}$. The upper sub-figure shows the energy band structures calculated by the continuum model. The red dashed lines represent the band structures based on rigid lattice, while the blue lines represent energy band structure considering the lattice relaxation effects. The lower sub-figure shows the energy band structure based on relaxed lattice structure, the red lines show the band structure calculated by tight binding model, while the blue lines are calculated by the continuum model considering both the lattice relaxation effects and the non-local moiré potential terms. (d) Schematic to construct a generic continuum model. We distinguish the twist angle independent intrinsic properties and the twist angle dependent external inputs. The intrinsic properties can be fitted via small-size DFT calculation and the external inputs are obtained by Deep Potential Molecule Dynamics (DPMD). Based on one set of universal parameters, the electronic properties of the given moiré superlattice can be accurately obtained. Subfigure (a-c) are extracted from Bo Xie and Jianpeng Liu, Phys. Rev. B, Vol.108, 094115, 2023 xie_phonon_prb23; licensed under a Creative Commons Attribution (CC BY) license.
• Figure 3: (a) Schematic illustration of the moiré superlattice consisted of nearly aligned hBN and pentalayer graphene with rhombohedral stacking. $C_A$, $C_B$ denote carbon atoms in $A$, $B$ sublattices, and $B$, $N$ denote boron and nitrogen atoms. The moiré Brillouin zone is plotted. (b)-(d) show continuum-model band structures of hBN-pentalayer graphene moiré superlattice with twist angle $0.77\,^{\circ}$ with $D=0.97\,$V/nm. (b) The bare non-interacting band structures, where the green dashed lines mark the low-energy window $E_c^*$ (see text), and the Green area denotes the remote energy bands. The solid and dashed blue lines represent the bands from $K$ and $K'$ valleys, respectively. (c) Low-energy band structures within $E_c^*$ with renormalized continuum model parameters given by Eqs. \ref{['eq:RG_eq1']}-\ref{['eq:RG_eq4']}. The solid and dashed blue lines represent the bands from $K$ and $K'$ valleys, respectively. (d) Hartree-Fock band structures at filling $\nu=1$, where the gray dashed line marks the chemical potential, and the occupied and unoccupied subspaces within $E_c^*$ are marked. The orange line represents the isolated flat band right below the chemical potential of filling 1. Figure adapted from Guo et al., Phys. Rev. B, Vol.110, 075109, 2024 guo-HFfci-prb-2024; licensed under a Creative Commons Attribution (CC BY) license.
• Figure 4: The wavevector dependence of the cRPA dielectric constant calculated for TBG. Figure extracted from Zhang et al., Phys. Rev. Lett. Vol.128, 247402, 2022 zhang-liu-tbg-prl22; licensed under a Creative Commons Attribution (CC BY) license.
• Figure 5: Feynman digrams that are considered in HF total energy, RPA correlation energy $E_{c}^{\text{RPA}}$, RPA screened Coulomb potential $W$ and $GW$ self-energy $\Sigma$. Single wiggling lines represents bare Coulomb interaction, double wiggling line is the RPA screened Coulomb interactions and solid lines with an arrow are the bare Green's function.