Flat-band projected versus fully atomistic twisted bilayer graphene

TL;DR

This work tests the validity of the flat-band projection method for MATBG by benchmarking it against a full atomistic tight-binding Hartree-Fock calculation across several symmetry-breaking phases at charge neutrality. The authors find good quantitative agreement in band structures and energies, with small energy advantages for the atomistic model and a slightly larger K-point gap, justifying the projection when remote bands are effectively frozen. They introduce a novel set of local real-space order parameters derived from wave-function overlaps to visualize and quantify symmetry breaking, enabling clear identification of phases such as KIVC, TIVC, QAH, OP, NSM, and VP directly in real space. Collectively, the results establish a computationally efficient framework that preserves essential physics of MATBG while capturing the influence of remote bands, with potential applicability to related honeycomb systems and future extensions to include interlayer overlaps and electron-phonon effects.

Abstract

We benchmark the recently proposed projection method [Phys. Rev. B 111, 205133 (2025)] for magic-angle twisted bilayer graphene (MATBG) across various symmetry-breaking phases at charge neutrality. The flat-band projected solutions agree well with the full tight-binding, with band structures and total energies differing by only a few meV. The projection to the flat bands is justified, owing to the increased gap to the remote bands in the normal state. Moreover, we employ a novel set of order parameters that allow us to visualize the wave functions locally in real space and quantify the breaking of various symmetries in the correlated phases. These order parameters are suitable for characterizing MATBG and generic honeycomb systems.

Figures (5)

• Figure 1: Band structures of the normal state (SYM), the nematic semimetal (NSM), quantum anomalous Hall (QAH), Kramers intervalley coherent (KIVC), orbital polarized (OP), time-reversal intervalley coherent (TIVC) and valley polarized (VP) states. Bands from the tight-binding and flat-band models are shown in gray and blue, respectively. We also plot the non-interacting bands in the SYM panel in light gray, for comparison. In the NSM, the protected Dirac nodes are located close to the $K \Gamma K'$ line
• Figure 2: Microscopic loops used to compute the wave function overlaps ($\omega = e^{2\pi i/3}$).
• Figure 3: Local order parameters. (a) Intrasublattice, intervalley coherence of TIVC (real part) and KIVC (imaginary part). (b) Density of the OP state on the two sublattices; the density on the $B$ sublattice is multiplied by $10$ for better visualization. (c) Valley polarization on sublattice $B$ for QAH and sublattice $A$ for VP. (d) NSM state. Left: density on sublattice $A$. Right: intersublattice, intravalley order parameter. the transparency and color indicate the magnitude and the phase of the complex number, respectively. All quantities are shown for the top layer.
• Figure A1: Lattice of monolayer graphene. We label the $A$ and $B$ sublattices, the unit vectors ($\boldsymbol{a}_1,\boldsymbol{a}_2$), the nearest neighbors vectors ($\boldsymbol{\delta}_i$) and the triangular and hexagonal loops used to compute the wave function overlaps.
• Figure C2: Local order parameters as a function of the number of bands included. The first two rows show the valley $K$ density on the $A$ sublattice $A$ and top layer of the SYM and VP states. The color scales are identical, and the maximal (minimal) values are labeled for each state. The bottom row depicts the valley polarization on the $B$ sublattice and top layer of QAH. All three plots are qualitatively similar.