Theory of Correlated Hofstadter Spectrum in Magic-Angle Graphene

TL;DR

This work addresses the origin of correlated Chern insulator states in the Hofstadter spectrum of magic-angle twisted bilayer graphene by proposing a unified framework where correlation-enhanced spin and valley Zeeman terms shape a fourfold, flavor-polarized Hofstadter spectrum. The key contributions are the identification of two gap-closing mechanisms—Zeeman-splitting–driven flavor crossings and rational-flux interruptions at $\Phi/\Phi_0=1/q$—and the demonstration that a small substrate-induced mass term $\Delta$ together with twist-angle–dependent bandwidth reconstructs the spectrum to produce experimentally observed CCI states. The model, with $g_s\approx3.4$, $g_v\approx7.7$, and $\Delta\approx1$ meV, explains a wide range of experiments and makes testable predictions, such as tuning spectrum intersections with an in-plane field $B_{\parallel}$. These results highlight a Zeeman-driven, flavor-resolved Hofstadter landscape as the central mechanism behind correlated Hofstadter physics in MATBG.

Abstract

The magnetic-field-induced correlated Chern insulator (CCI) states in magic-angle twisted bilayer graphene (MATBG) have been intensively studied in experiments, but a simple and clear understanding of their origin is still lacking. Here, we propose a unified theoretical framework for the CCI states in MATBG that successfully explains most experimental observations. The key insight of our theory is that, due to the very narrow bandwidth of MATBG, correlation-enhanced valley and spin Zeeman terms are critical for shaping the intricate Hofstadter spectrum, resulting in an interwoven, flavor-resolved (spin and valley) Hofstadter spectrum that can well describe the observed CCI states. Crucially, due to the Zeeman effect, the crossings between these flavor-polarized Hofstadter spectra are magnetic-field-dependent, causing certain CCI states to emerge only above a critical field. This is the main mechanism underlying the critical field phenomenon of the CCI states observed in experiments. Our theory provides a clear and unified physical picture for the correlated Hofstadter spectrum in MATBG.

Figures (10)

• Figure 1: Hofstadter spectrum of MATBG with twist angle $\theta=1.06^\circ$. (a) Wannier diagram. (b) Flavor-polarized Hofstadter spectra. Each gap (CCI state) in Hofstadter spectrum corresponds to a linear trajectory in Wannier diagram denoted by two integers $(t,s)$, where $t$ is the Chern number and $s$ represents the band filling. In Wannier diagram, the observed CCI states in experiment are colored according to their value of $s$, with the same color scheme as that in Ref. yu2022correlated. The integers represent the value of $t$ for these CCI states. The minimum energy resolution for gaps is $0.01$ meV.
• Figure 2: Hofstadter spectrum of MATBG with twist angle $\theta=1.03^\circ$. (a) Wannier diagram. The observed CCI states in experiment are highlighted with the same color scheme as that in Ref. he2025strongly. (b) Flavor-polarized Hofstadter spectra. The minimum energy resolution for gaps is $0.02$ meV. All the other parameters are the same as Fig. \ref{['fig1']}.
• Figure 3: (a) is the evolution of the gaps of the $(-1,-3)$ states in both Fig. \ref{['fig1']} and Fig. \ref{['fig2']} with magnetic field. (b) and (c) are the magnified versions of the Hofstadter spectrum with $\theta=1.06^\circ$ and $\theta=1.03^\circ$, respectively. The blue shaded region marks the gap of the $(-1,-3)$ state. (d) is the magnified version of the Hofstadter spectrum with $\theta=1.06^\circ$ and an in-plane magnetic field $B_\parallel=8$ T. (e) is the evolution of gaps of the $(-1,-3)$ states with $\theta=1.06^\circ$, under different in-plane fields.
• Figure S1: Single-particle band structures of MATBG at twist angles (a) $\theta = 1.02^\circ$, (b) $\theta = 1.05^\circ$, (c) $\theta = 1.08^\circ$, (d) $\theta = 1.12^\circ$, and (e) $\theta = 1.16^\circ$, calculated within the BM model including a small substrate-induced mass term $\Delta = 1~\text{meV}$.
• Figure S2: Hofstadter spectra of MATBG at $\theta = 1.06^\circ$ with a substrate-induced mass term $\Delta = 1~\text{meV}$ while neglecting the valley and spin Zeeman terms. Without this mass term, the spectra of the two valleys would be degenerate under the magnetic field.