Magnetic Ordering in Moiré Graphene Multilayers from a Continuum Hartree+U Approach

TL;DR

This work develops a self-consistent continuum Hartree+U framework that embeds atomistic short-range Hubbard interactions into a moiré graphene continuum model, coupled with long-range Coulomb screening, to study magnetic ordering in twisted bilayer and trilayer graphene near the magic angle. By seeding the continuum model with atomistic instability inputs and solving for self-consistent order parameters, the authors map the magnetic phase diagram as a function of doping $\nu$ and twist angle $\theta$, recovering qualitative agreement with prior atomistic Hartree+U calculations. They identify leading instabilities—FM, MAFM, and NAFM in tBLG and FM/MMAFM in tTLG—and show how Hartree interactions enhance certain orders near $\nu=\pm1$ while generally modifying band structures, including opening gaps at $K/K'$ for AFM states and spin-splitting for FM. The results demonstrate a computationally efficient route to incorporate short-range exchange into continuum moiré models, enabling exploration of magnetism across moiré graphene multilayers and guiding future extensions to include exchange effects and more complex orders.

Abstract

Recently, symmetry-broken ground states, such as correlated insulating states, magnetic order and superconductivity, have been discovered in twisted bilayer graphene (tBLG) and twisted trilayer graphene (tTLG) near the so-called magic-angle. Understanding the magnetic order in these systems is challenging, however, as atomistic methods become extremely expensive near the magic angle and continuum approaches fail to capture important atomistic details. In this work, we develop an approach to incorporate short-ranged Hubbard interactions self-consistently in a continuum model. In addition, we include long-ranged Coulomb interactions, which are known to be important when doping the flat bands of tBLG and tTLG. Therefore, for the first time, magnetic order in moiré graphene multilayers is self-consistently explored in a continuum model with atomistic detail. With this approach, we perform a systematic analysis of the magnetic phase diagram of tBLG as a function of doping level and twist angle, near the magic angle. Our results are consistent with previous perturbative atomistic Hartree+U calculations. Furthermore, we investigated magnetic order of tTLG, which were found to be similar to those in tBLG. In the future, the developed continuum model can be utilized to investigate magnetic ordering tendencies from short-range exchange interactions in other moiré graphene multilayers as a function of doping, twist angle, screening environment, among other variables.

Figures (5)

• Figure 1: Magnetic instabilities obtained from atomistic RPA calculations. The ferromagnetic order (FM), moiré modulated antiferromagnetic (MAFM), and nodal antiferromagnetic (NAFM) orders are shown in a), b) and c), respectively. Note these graphs are obtained from the eigenvectors of these calculations, corresponding to the most negative eignevalues, where the largest positive value was chosen to be 1.
• Figure 2: Magnitudes of Hubbard potential order parameters $|\delta_0| + |\delta_1|$, for the considered magnetic ground states, to establish where these phases can exist at twist angle ($\theta$) close to the magic angle and for integer doping levels within the flat bands ($\nu$). With Hartree interactions, we show results for MAFM in (a), NAFM in (b), and FM in (c). Results without Hartree interactions are shown in (d) for MAFM, (e) for NAFM, and (d) for FM. $U =2$ meV throughout. We note that $\delta_0=0$ for the NAFM order.
• Figure 3: The self-consistent band structures at 1.08° for different fillings $\nu$ for (a): Hartree corrections only, (b): MAFM, (c): NAFM, and (d): FM. We note that a large Hubbard parameter ($U=4$ meV) was chosen to demonstrate the effects of gap opening on Hartree-corrected bands. Color represents different filling factors: light pink for $\nu=0$; light blue for $\nu=1$; blue for $\nu=2$ and dark green for $\nu=3$.
• Figure 4: Magnetic instabilities obtained from atomistic RPA calculations. Here we only consider the ferromagnetic order (FM) and MAFM/MAFM (MMAFM) order. Since the outside and insider layers are inequivalent, we show them separately, and since the outside layers are equivalent we only show one of them. The outside and inside layers of FM are shown in a) and c), respectively, while the outside and inside layers of MMAFM are respectively in b) and d). . Note these plots are from the eigenvectors of the RPA calculations, where the largest positive value was chosen to be $\pm$1. These values are proportional to the spin-polarised electron density.
• Figure 5: Self-consistent band structures obtained of tTLG at charge neutrality at $\theta=1.54^{\circ}$ for (a) FM and (b) MMAFM. The band structure at charge neutrality without any Hubbard interactions is included in orange.