Projected and Solvable Topological Heavy Fermion Model of Twisted Bilayer Graphene

TL;DR

This work develops the projected-limit topological heavy-fermion (THF) model for magic-angle twisted bilayer graphene, connecting real-space localized $f$-orbitals to itinerant Dirac $c$-electrons and showing how a topological flat band emerges when $U_1<γ$. It derives an analytic topological-flat-band wavefunction with real-space charge concentration and Berry-curvature localization, and analyzes correlation effects using Hubbard-I, demonstrating stable local moments for $U_1 \gg Δ(ω)$ and providing a controlled self-energy for the flat bands. The study also derives an effective ferromagnetic coupling between flavor moments and demonstrates consistency with non-local-moment descriptions, supported by quantitative parameterization that aligns with experimental observations such as cascade peaks and band gaps. Overall, the THF framework offers a coherent, real-space approach to topology and strong correlations in MATBG, bridging momentum-space projections and local-moment physics with experimentally accessible scales.

Abstract

We investigate the topological heavy-fermion (THF) model of magic-angle twisted bilayer graphene (MATBG) in the projected limit, where only the flat bands are present in the low-energy spectrum. Such limit has been previously analyzed in momentum-space Bistritzer-MacDonald-type continuum models, but not in a real-space formalism. In this regime, the Hubbard interaction ($U_1$) of the $f$-electrons is larger than the bandwidth ($2M$) of the flat bands but smaller than the gap ($γ$) between the flat and remote bands. In the THF model, concentrated charge (in real space) and concentrated Berry curvature (in momentum space) are respectively realized by exponentially localized $f$-orbitals and itinerant Dirac $c$-electrons. Local moments naturally arise from $f$-orbitals. Hybridizing the $f$-electrons with $c$-electrons produces power-law tails of the flat-band Wannier functions, raising the question of relevance of the local moment picture in the projected $U_1\ll γ$ limit. Nonetheless, we find that the local moments remain stable as long as $U_1 \gg Δ(ω)$ for $|ω|\lesssim U_1$, where $Δ(ω)\sim γ^2 N(ω)$ is the hybridization function seen by each $f$-site, and $N(ω)$ is the density of states of the Dirac $c$-bands. Notably, the comparison between $U_1$ and $γ$ is irrelevant to the local moment formation if $N(ω)$ is unknown. Within the framework of THF, we also derive the correlated self-energy of the flat bands using the Hubbard-I approximation and estimate the coupling strength between the local moments. Finally, we comment that, in the regime of extremely concentrated Berry curvature, the single-particle gap between flat bands and remote bands vanishes and the interaction is always larger than the gap.

Figures (2)

• Figure 1: Spectrum obtained from Hubbard-I approximation at integer fillings $\nu=0,-1,-2,-3$ (from left to right). To realize the limit of $|v_\star \Lambda_c|\gg \gamma \gg U_1$, we have taken the parameter values given in Ref. song_magic-angle_2022, and then rescaled all the interaction strength by a factor of $1/3$.
• Figure 2: Hartree-Fock bands of the THF model. (a) and (c) are Hartree-Fock bands at CNP with parameters $w_0/w_1=0.8, 0.7$, respectively. (b) and (d) are Hartree-Fock bands at $\nu=-4$ with parameters $w_0/w_1=0.8$, 0.7, respectively. The $U_1=$58.0, 51.7meV for $w_0/w_1=0.8$, 0.7, respectively.