General Many-Body Perturbation Framework for Moiré Systems

TL;DR

The study tackles the limitations of Hartree-Fock in moiré systems by introducing a general many-body perturbation framework that adds dynamical screening via $E_c^{\text{RPA}}$ and $\Sigma^{GW}$ corrections to all-band HF in a continuum moiré model. Applying it to $R5G$-$hBN$ and magic-angle TBG, the approach reveals a phase diagram and single-particle spectra that align quantitatively with experiments, attributing improvements to high-energy remote bands and inhomogeneous screening. GW corrections yield reduced gaps and bandwidths and quasiparticle weights near unity, validating the use of (nearly) mean-field starting points for integer fillings while capturing beyond-HF physics. The framework, which relies on a single static dielectric constant as the main fitting parameter, provides a scalable, largely ab initio method for systematic beyond-mean-field studies of generic moiré superlattices.

Abstract

Moiré superlattices host a rich variety of correlated topological states, including interaction-driven integer and fractional Chern insulators. A common approach to study interacting ground states at integer fillings is the Hartree-Fock mean-field method. However, this method neglects dynamical correlations, which often leads to an overestimation of spontaneous symmetry breaking and fails to provide quantitative descriptions of single-particle excitations. This work introduces a general many-body perturbation framework for moiré systems, combining all-band Hartree-Fock calculations with random phase approximation (RPA) correlation energies and $GW$ quasiparticle corrections. We apply this framework to hexagonal boron nitride aligned rhombohedral pentalayer graphene and magic-angle twisted bilayer graphene. We show that incorporating RPA correlation energy and $GW$ self-energy corrections yields phase diagrams and single-particle spectra that quantitatively align with experimental measurements. Our versatile framework provides a systematic beyond-mean-field approach applicable to generic moiré systems.

Figures (11)

• Figure 1: All-band HF phase diagrams for $\epsilon_r=8,10,12$ at $D$ fields from 0.6 to 1.1 V/nm, including the Chern number of the first conduction band (left top), its bandwidth (left bottom), the indirect gaps (right top) and the Berry curvature standard deviation (right bottom). Metallic and insulating phases are represented by squares with and without shades, respectively. The standard deviation in the Brillouin zone of the normalized Berry curvature is only shown for $C=1$ gapped phases.
• Figure 2: Comparison between (a) the evolution of the Chern number and the overall gap as a function of the $D$ field from the HF+RPA calculations, (b) the evolution of transverse and longitudinal resistance observed experimentally for R5G-hBN fqah-julong-nature24data-lu-fqah-nature-2024 and (c) those obtained from HF+$GW$+RPA calculations, where we include the lowest 20 valence and 20 conduction $GW$ bands. Different phases are separated by vertical dashed lines. Inset: RPA total energy around $D=0.7$ V/nm. The dielectric constant is $\epsilon_r=10$.
• Figure 3: HF (red dashed) and $GW$ (blue solid) band structures for TBG of twist angle $\theta=1.08^\circ$ at $\nu=0$. (a) Bands plotted along a high-symmetry path, indicated on the right side. (b) and (c) are the energy profile of HF and $GW$ bands in the first Brillouin zone, respectively. Here, we set $\epsilon_r=8$.
• Figure S1: Non-interacting band structures for TBG at $\theta=1.08^{\circ}$ for two plane-wave cutoffs $n_D=13$ (left) and $n_D=5$ (right). The bandwidth of the flat bands are also given on the panels.
• Figure S2: Comparison between the numerically calculated inverse dielectric function (dashed lines) and the MPA-fitted inverse dielectric function (solid lines). The real part is shown in red, and the imaginary part is shown in blue. Left upper panel: small $\mathbf{q}$ with $\mathbf{G} = 0$ near the $\Gamma$ point; right upper panel: non-zero $\mathbf{G}$ away from the $\Gamma$ point; two lower panels: off-diagonal elements near (left) and away from (right) the $\Gamma$ point.